Intermediate Macroeconomics (26276) Dr Keshab Bhattarai University of Hull Business School, 2016
Abstract This workbook contains a concise presentation of the classical, Keynesian, real business cycle, new Keynesian and DSGE models for closed and open economies that are often applied in analysis of …scal, monetary, …nancial, and labour market policies. It contains general equilibrium models for comparative static and dynamic analyses policies and other structural features of the economy. Theories and models of economic growth, …nance, …scal and monetary policies, input-output and general equilibrium analysis, trade and exchange rates are presented along with strategic models of cooperative and non-cooperative games, bargaining, signalling and mechanism design under asymmetric information as required for evaluation of economic decisions made by households, …rms and government in modern economies. It takes problem solving approach to learning macro economics. Contributions of eminent UK economists such as Ricardo, Keynes, Hicks, Stone, Meade, Buchanan, Mirrlees and Pissarides are discussed along with those of Solow, Lucas, Prescott, Sargent, Romer, Jones, Taylor, Blanchard, Kotliko¤, Mankiw, Stiglitz, Miller, Fama, Hansen, Shiller and Weale, and strategic tax and trade models of Nash, Spence, Akerlo¤, Vickery, Hansen, Tirole, Whalley, Rutherford and many other economists while developing analytical frameworks aimed to solve problems on stabilisation, growth, redistribution, coordination and cooperation in economic policy making. This workbook is disigned for one semester and contains details on derivations to make it accessible to students and interested readers. JEL Classi…cation: D, E Keywords: macroeconomics, growth, general equilibrium, game theory, trade. Dr. Keshab Bhattarai: Business School, University of Hull, HU6 7RX, Hull, UK. email:
[email protected] ; phone: 01482 463207; fax: 01482463484.
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Contents 1 L1: Neoclassical Growth Model 1.1 Solow growth model . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Stylized facts of economic growth . . . . . . . . . . . . 1.1.2 Steady state . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Summary of Solow Model . . . . . . . . . . . . . . . . 1.1.4 Golden rule of saving and capital accumulation . . . . 1.1.5 Literature on economic growth . . . . . . . . . . . . . . 1.1.6 Computer Lab1: Growth and Macro Models (optional) 1.1.7 Problem 1:Neoclassical Growth Model . . . . . . . . . 2 L2: Endogenous Growth Model: Knowledge is powerful 2.1 AK endogenous growth model . . . . . . . . . . . . . . . . . . 2.1.1 Human capital and economic growth . . . . . . . . . . 2.2 Human Capital and Di¤erences in Income Per Capita . . . . . 2.2.1 Do countries with lower per capita income grow faster? 2.2.2 Clean air maximising growth rate . . . . . . . . . . . . 2.2.3 Human capital augmented Solow model . . . . . . . . . 2.2.4 Technology and inventions . . . . . . . . . . . . . . . . 2.2.5 Problem 2: Endogenous Growth Model . . . . . . .
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3 L3: Ramsey Growth Model 3.1 Optimisation under in…nite horizon . . . . . . . . . . . . . . . . . 3.1.1 Solution of an in…nite horizon problem . . . . . . . . . . . 3.1.2 Steady state in a Ramsey model . . . . . . . . . . . . . . . 3.1.3 Long run scenarios from a Ramsey model . . . . . . . . . . 3.1.4 Empirical evidence: a panel regression on economic growth 3.2 Theories on share of labour in total income . . . . . . . . . . . . . 3.2.1 Marxian theory the surplus value (S) . . . . . . . . . . . . 3.2.2 Kaldorian theory of functional distribution . . . . . . . . . 3.2.3 Hahn’s Dynamic Theory of Wage Share . . . . . . . . . . .
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4 L4: Classical macroeconomic model; Pure exchange and two period general equilibrium models 4.1 Two Good Pure Exchange General Equilibrium Model . . . . . . . . 4.1.1 Main Features of an Applied General Equilibrium Model . . . 4.1.2 Pure exchange model of general equilibrium . . . . . . . . . . 2
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A general equilibrium model with labour leisure choice and production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Two period general equilibrium model . . . . . . . . . . . . . . . . . 4.2.1 Ricardian equivalence . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Ricardian equivalence and macro-micro impacts of …scal policy
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5 L5: Keynes-Hicks-Samuelson-Mundell-Fleming Models of Macroeconomic Fluctuations 88 5.1 Classical macroeconomic model . . . . . . . . . . . . . . . . . . . . . 96 5.1.1 Numerical example of a classical macroeconomic model . . . . 97 5.1.2 Classical macro model: A numerical example . . . . . . . . . . 98 5.2 Keynes (1936)-Hicks (1937) macroeconomic model . . . . . . . . . . . 98 5.2.1 Comparative static analysis in macroeconomic models . . . . . 100 5.2.2 An example of a Keynesian macroeconomic model: goods market104 5.2.3 Estimation Keynesian model using the PcGive . . . . . . . . . 106 5.2.4 Lessons from a simulated macroeconmetric model . . . . . . . 109 5.2.5 Samuelsonian Multiplier-Accelerator Model . . . . . . . . . . . 111 5.3 ISLM equilibrium: a new approach . . . . . . . . . . . . . . . . . . . 113 5.3.1 Open economy macroeconomic model . . . . . . . . . . . . . . 115 5.3.2 An Example of Mundell-Fleming Small Open Economy Macroeconomic Model . . . . . . . . . . . . . . . . . . . . . . 115 5.3.3 An Example of Mundell-Fleming Two Country Interdependent Global Economy Macroeconomic Model . . . . . . . . . . 117 5.3.4 Solution of the two country global economy model . . . . . . . 118 5.3.5 Numerical example of a small open economy model . . . . . . 118 5.3.6 Excercise on new approach to the ISLM model . . . . . . . . . 120 5.3.7 Literature in macroeconomic modelling . . . . . . . . . . . . . 123 5.3.8 Problem 3: Analysis of Macroeconomic Fluctuations . . 128 6 L6: Rational Expectation Model 6.1 Rational expectation model of in‡ation and output . . . . . . . . . . 6.1.1 Three ways of forming expectations about unknown variables: 6.1.2 An example of rational expectation model: . . . . . . . . . . . 6.1.3 Solution procedure . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Rational expecation ( example 2): solution procedure . . . . . 6.1.5 Problem 5: Rational Expectation . . . . . . . . . . . . . . . .
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136 136 137 137 140 141 146
7 L7: New Keynesian DSGE Model: Nominal and Real Rigidities and Economy 149 7.1 New Keynesian macroeconomic model with real and nominal rigidities 149 7.1.1 A Basic DSGE model . . . . . . . . . . . . . . . . . . . . . . . 150 7.2 New Keynesian Model: a prototype example . . . . . . . . . . . . . . 155 7.3 Basics of monopolistic competition in a new Keynesian model . . . . 158 8 L8: Real business cycle model: TFP shocks 8.1 What is a business cycle? . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 What are the facts about business cycles . . . . . . . . . . . . 8.1.2 Real Business Cycle (RBC) models on macroeconomic ‡uctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 RBC model for a two period economy . . . . . . . . . . . . . . . . . . 8.2.1 How are wages and interest rates determined? . . . . . . . . . 8.2.2 Linear RBC Model: Macroeocnomic ‡uctuations only due to shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Proof for the Linear RBC Model . . . . . . . . . . . . . . . . 8.2.4 Prudent policies implied by RBC models: intertemporal balance in budgets of households, government and …rms . . . . .
160 163 165
9 L9: Tax, Spending and Fiscal Policy 9.0.5 Key tax rates and allowances in the UK . . . . . . . . . . . . 9.0.6 Who bears the burden of tax? Microeconomic Approach . . . 9.0.7 Golden rule of …ne tuning: an example . . . . . . . . . . . . . 9.0.8 Revenue maximising tax rate: La¤er Curve . . . . . . . . . . . 9.1 Net e¤ects of …scal policy . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Progressive tax system for fairness: Mirrleeian idea on progressive taxation to achieve equity . . . . . . . . . . . . . . . . . . 9.1.2 Optimal size of the state . . . . . . . . . . . . . . . . . . . . . 9.1.3 Objectives of the …scal policy . . . . . . . . . . . . . . . . . . 9.1.4 Maintaining macroeconomic stability . . . . . . . . . . . . . . 9.1.5 Major …scal issues of the current government . . . . . . . . . . 9.1.6 Trends of …scal policy in UK . . . . . . . . . . . . . . . . . . . 9.1.7 Balanced budget multiplier: Lump-sum tax case . . . . . . . . 9.1.8 Automatic stabiliser: proportional tax case . . . . . . . . . . . 9.1.9 Samuelson’s Theory on optimal public spending . . . . . . . . 9.1.10 Ricardian Equivalence Theorem: Questions . . . . . . . . . . . 9.1.11 Debt dynamics . . . . . . . . . . . . . . . . . . . . . . . . . .
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188 190 199 200 200 202 203 204 205 206 207
9.1.12 Debt Dynamics with In‡ation Tax (Seigniorage) . . . . . . . . 209 9.1.13 Keynesian model of debt . . . . . . . . . . . . . . . . . . . . . 210 10 L10: Role of Financial Markets in the Economy 10.0.14 Principles of …nance . . . . . . . . . . . . . . . . . . . . 10.1 Growth and e¢ ciency of the …nancial sector . . . . . . . . . . . 10.1.1 Integration of …nance and real economy . . . . . . . . . . 10.2 Determinants of asset prices . . . . . . . . . . . . . . . . . . . . 10.2.1 Price of console: in…nite horizon . . . . . . . . . . . . . . 10.2.2 Students and tuition: life cycle optimisation problem . . 10.2.3 Investors, marginal productivity of capital and tax credit 10.2.4 Investment problem . . . . . . . . . . . . . . . . . . . . . 10.3 Essentials of capital asset price model . . . . . . . . . . . . . . . 10.3.1 Relevant web pages: . . . . . . . . . . . . . . . . . . . . 10.4 Problem 7: Financial markets . . . . . . . . . . . . . . . . . .
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215 218 218 219 220 222 223 224 225 227 232 233
11 L11: Monetary Policy 236 11.0.1 Transmission process of the monetary policy . . . . . . . . . . 238 12 Weale (2014) model of Interest rate rule 12.1 Interest rate determination rule: Taylor rule . . . . . 12.1.1 Transmission Mechanism of Monetary Policy . 12.1.2 Long run natural rate of interest: steady state 12.1.3 Three cases in interest rate rule model . . . . 12.2 Policy Rule versus Optimal Discretion . . . . . . . . 12.3 Policy game models . . . . . . . . . . . . . . . . . . . 12.3.1 Blake-Weale (1994) model of debt . . . . . . . 12.4 Problem 8: Interest rate rule . . . . . . . . . . . . 13 Class Test: A Samples 13.1 Class Test : Sample 1 . 13.2 Class Test : Sample 2 . 13.3 Class Test : Sample 3 13.4 Class Test : Sample 4 .
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14 L12: A General Equilibrium Tax Model with Labour-Leisure Choice279 14.0.1 Supply Side of the General Equilibrium Model . . . . . . . . . 280 14.1 Impact of taxes in consumer demand: income and substitution e¤ects 283 5
14.1.1 Impacts of tax reforms: Hicksian compensating and equivalent variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.2 Consumer optimisation model: a numerical example . . . . . . 14.1.3 Cobb-Douglas preferences . . . . . . . . . . . . . . . . . . . . 14.1.4 Problem 4: Impacts of Fiscal Policies . . . . . . . . . . . . . .
283 284 287 290
15 L13: Phillips Curve: A Link Between the Supply and Demand Sides in the Keynesian Macroeconomic Model 293 15.1 Literature on unemployment and in‡ation . . . . . . . . . . . . . . . 296 15.1.1 Wage price spiral . . . . . . . . . . . . . . . . . . . . . . . . . 297 15.1.2 Derivation of expectation augmented Phillips curve . . . . . . 298 15.2 Theories of unemployment . . . . . . . . . . . . . . . . . . . . . . . . 299 15.2.1 Classical minimum wage theory of unemployment . . . . . . . 300 15.2.2 Wage setting and price setting or union-…rm bargaining theory of unemployment . . . . . . . . . . . . . . . . . . . . . . . . . 300 15.2.3 Union-…rm bargaining theory of unemployment . . . . . . . . 303 15.2.4 Search and matching theory: job creation and destruction . . 304 16 L14: Macroeconomic Stabilisation 16.1 Why stabilisation? . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Needs for stabilisation: costs of unemployment . . . . . . . 16.1.2 Mean, variance and covariance . . . . . . . . . . . . . . . . 16.2 Measurement of business cycles . . . . . . . . . . . . . . . . . . . 16.3 Dyanmic Aggregate Demand-Aggregate Supply (AS-AD) Model Business Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 Stabilisation model . . . . . . . . . . . . . . . . . . . . . . 16.4 Problem 6: Price Wage Dynamics and Stabilisation . . . . .
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17 L15: Overlapping Generation Model 331 17.0.1 Speci…cation of an overlapping generation model . . . . . . . . 332 18 L16: General Equilibrium in Ricardian Trade Model 336 18.1 A question on Ricardian trade model: gains from the trade . . . . . . 336 18.1.1 Two country Ricardian trade model: Explaining patterns of China-US trade . . . . . . . . . . . . . . . . . . . . . . . . . . 344 19 L17 Exchange Rates 350 19.1 Nominal, real and e¤ective exchange rates . . . . . . . . . . . . . . . 352 6
19.1.1 Demand and supply model for foreign assets and the Exchange Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Marshall-Lerner condition . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 Purchasing Power Parity Theory . . . . . . . . . . . . . . . . 19.2.2 Interest Parity Theory and the Exchange rate: . . . . . . . . . 19.2.3 Rational Expectation model of Determination of Exchange Rate in the Short Run . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.4 Impact of changes of exchange rate in an economy . . . . . . . 19.2.5 Impact of expansionary …scal policy on exchange rate . . . . . 19.2.6 An Empirical model of impact of changes in the exchange rates in economic growth . . . . . . . . . . . . . . . . . . . . . . . . 19.2.7 Real exchange rates . . . . . . . . . . . . . . . . . . . . . . . . 19.2.8 Exchange Rate Policies . . . . . . . . . . . . . . . . . . . . . . 19.2.9 Wrong Fundamentals and Debt Accumulation . . . . . . . . . 19.2.10 Exchange rate arrangements . . . . . . . . . . . . . . . . . . . 19.2.11 Conclusion about the exchange rate . . . . . . . . . . . . . .
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20 L18: National Macroeconomic Policy Co-ordination: Central Bank Independence and In‡ation 370 20.0.12 Optimisation Approach to Setting of the target in‡ation rate . 376 20.0.13 Model for Fiscal and Monetary Policy Co-ordination . . . . . 379 20.0.14 Is this model applicable in the UK? . . . . . . . . . . . . . . . 385 20.0.15 Blake-Weale (1998) Model of Fiscal and Monetary Policy Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 20.0.16 Tinberginian instrument-target analysis of policy choices . . . 387 21 L19: International Macroeconomic Policy Co-ordination 389 21.1 Mundell-Fleming Two Country Model of the Global Economy . . . . 389 21.1.1 Game Theoretic Model of Policy Coordination . . . . . . . . . 394 21.1.2 Canzoneri M. B. and J A Gray (1985) . . . . . . . . . . . . . 396 21.1.3 Choice of optimal policy . . . . . . . . . . . . . . . . . . . . . 398 21.1.4 Under Nash equilibrium . . . . . . . . . . . . . . . . . . . . . 398 21.1.5 Stackleberg solution . . . . . . . . . . . . . . . . . . . . . . . 398 21.1.6 Recent Micro-Founded Models of International Policy Co-ordination400 22 L20: Game Theory 403 22.1 Problem 9: Strategic Models and Optimal Tax . . . . . . . . . . 414 22.2 Problem 10: General Equilibrium Model: Pure Exchange . . . . . . . 417 22.3 Problem 12: Bargaining and Cooperative Game . . . . . . . . . 420 7
23 L21: Bargaining Game 23.1 Nash Product in Bargaining Game . . . . . . . 23.1.1 Coalition possibilities . . . . . . . . . . . 23.1.2 Superadditivity of a coalition . . . . . . 23.1.3 Shapley value . . . . . . . . . . . . . . . 23.1.4 Equivalence of core in games and general 24 L22: Signalling and Principal Agent Model: plete) Information 24.1 Signalling for managing a company . . . . . . 24.1.1 Signalling and incentives . . . . . . . . 24.1.2 Spence model of education . . . . . . . 24.2 Problem 13: Signalling and mechanism . . 24.3 Principal agent games . . . . . . . . . . . . .
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422 422 426 429 430 431
Asymmetric (incom432 . . . . . . . . . . . . . 435 . . . . . . . . . . . . . 436 . . . . . . . . . . . . . 438 . . . . . . . . . . . . . 441 . . . . . . . . . . . . . 443
25 L23: Repeated Games and Auctions 449 25.1 Auction Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 26 L24: Mechanism Design Game 455 26.1 Mechanism to ensure high e¤orts by a CEO . . . . . . . . . . . . . . 457 26.1.1 Mechanism design in renting lands . . . . . . . . . . . . . . . 458 26.1.2 Hire contract . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 27 L25: E¢ ciency conditions of the market system 27.1 E¢ ciency in consumption . . . . . . . . . . . . . 27.1.1 E¢ ciency in production . . . . . . . . . . 27.1.2 E¢ ciency of trade (Exchange) . . . . . . . 27.1.3 E¢ ciency in public goods . . . . . . . . . 27.1.4 Theory of second best . . . . . . . . . . .
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28 L26: Externality 464 28.1 Negative externality . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 28.1.1 Positive externality . . . . . . . . . . . . . . . . . . . . . . . . 465 29 L27: Uncertainty and Expected Utility 467 29.1 Problem 11: Uncertainty and insurance . . . . . . . . . . . . . . . 474 30 L28: Uncertainty and Insurance 31 L29: Input-Output Model
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32 L30: Linear Programming
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33 Assignment 2016 493 33.1 Topic 1: Why do the economic growth rates di¤er across countries or regions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 33.2 Topic 2: The mechanism of coordinating monetary and …scal policies 495 33.3 Topic 3: Does direct or indirect tax promote growth, e¢ ciency and equality? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 33.4 Topic 4: Are institutional reforms key to solve the unemployment problems? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 33.5 Topic 5: The development of the …nancial sector and economic growth500 33.6 Topic 6: Globalisation and the distribution of income . . . . . . . . . 501 34 Popular Databases 34.1 Econometric and Statistical Software . . . . . . . . . . . . . . . . . . 34.2 Mathematical software . . . . . . . . . . . . . . . . . . . . . . . . . . 34.3 MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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35 Final Exam Samples 507 35.0.1 Section A: Short Problems [28 marks] . . . . . . . . . . . . . . 508 35.0.2 Section B: Problems [12 marks for each question] . . . . . . . 509 35.0.3 Section C: Essay [36 marks] . . . . . . . . . . . . . . . . . . . 515 36 Tutorials 36.1 Tutorial 36.2 Tutorial 36.3 Tutorial 36.4 Tutorial 36.5 Tutorial 36.6 Tutorial 36.7 Tutorial 36.8 Tutorial 36.9 Tutorial 36.10Tutorial 36.11Tutorial 36.12Tutorial 36.13Tutorial
1: Neoclassical Growth Model . . . . . . . . . . . . . 2 : Endogenous Growth Model . . . . . . . . . . . . . 3: Intertemporal optimisation and Ramsey Model . . 4: Macro Fluctuations . . . . . . . . . . . . . . . . . 5: New Keynesian (DSGE) and Rational Expecation 6: RBC model . . . . . . . . . . . . . . . . . . . . . 7: Fiscal Policy and General Equilibrium Analysis . . 8: Financial markets . . . . . . . . . . . . . . . . . . 9: Price Wage Dynamics . . . . . . . . . . . . . . . . 10: Trade and welfare . . . . . . . . . . . . . . . . . . 11: Bargaining and Cooperative Game . . . . . . . . 12: Signalling and mechanism . . . . . . . . . . . . . 13: Uncertainty and insurance . . . . . . . . . . . . .
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515 . . . . 515 . . . . 517 . . . . 520 . . . . 522 Models 525 . . . . 528 . . . . 529 . . . . 531 . . . . 533 . . . . 536 . . . . 538 . . . . 541 . . . . 543
37 Recommended texts: 37.1 Best twenty articles in 100 years in the American Economic Review 37.2 Classics in Economics and the Economic Journal . . . . . . . . . . . 37.3 Ten Best articles in the Journal of European Economic Association 37.4 Best 40 articles in the Journal of Economic Perspectives . . . . . . 37.5 Some articles by economists of Hull University . . . . . . . . . . . . 37.6 Model Codes and Computations . . . . . . . . . . . . . . . . . . . .
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L1: Neoclassical Growth Model
Economic growth is a recent phenomenon in human civilisation. The annual average growth rate of GDP in the UK was 0.2 percent on average (0.08 percent in per capita) between AD 1 and 1830 and 2.05 percent (1.5 percent in per capita) between the years 1830 and 2008 based on data provided by Maddison at http://www.ggdc.net/maddison/. Per capita income in the Western Europe, which was at about the same or even at slightly lower level than that in China and India up to 15th century, had become more than 30 times higher by 2000 (about ten times in terms of purchacing power parity (PPP)). This extraordinary growth in living standards and longevity in the West was made possible by the use of scienti…c technology (machines and ideas) in production and use of more sophisticated means of transportation and communications in trade after the industrial revolution that started in late the 16th century (Maddison (1991)). Growth in Asian economies including Taiwan, Singapore, Hong Kong and South Korea and more recently in China and India have been very exemplary and transforming the structure of the global economy. In a seminal study Kaldor (1961) has summarised this growth process for the US and the majority of Western countries succinctly in terms of four stylised facts of economic growth (see https://pwt.sas.upenn.edu/). First, labour input has grown more slowly than capital and output. Therefore capital per capita (K/L) and output per capita (Y/L) have increased secularly. Second, the capital output ratio (K/Y) has remained fairly constant, has had no discernible trend and has more or less converged across industrialised economies. Third, the rate of return on capital (pro…t) and the real interest rate have no trend whereas real wage rates have followed a rising trend as the secular rises in productivity (Y/L) and per capita capital (K/L) have translated into higher real wages. Fourth, the share of income devoted to capital (rK/Y) and to labour (wL/Y) show no trend and remain fairly constant as the capital stock per person tends to grow along with output per person. Romer (1986) and Jones (2002) also illustrate these issues with various numerical examples.
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Figure 1: Fig 2
Fig. 1 The wide variation in the living standards seen today across countries was caused by di¤erences in the rates of economic growth across countries over time and it has created imbalance between distributions of population and that of income around the globe. In 2002 seventy eight percent of global population living in developing economies enjoyed only about 21 percent of the global income. Even a small di¤erence in annual growth rates across countries in the past has made a big di¤erence in the 12
level of income over time. If an economy grows at 5 percent per year, it takes 14 years to double its per capita income, while it takes 70 years if it grows by 1 percent per year. Income level can rise 10 times the current level if an economy grows continuously at the rate of 4 percent per year in 58 years, or takes only 23 years if it grows by the rate of 10 percent per year. This means if the poorest country had only 5 percent of the level of income of the richest country, it could still catch up that country in 50 years time if it could maintain the rate of growth of its income at the rate higher than the six percent of that rich country. Such growth rates are not impossible to sustain over a signi…cant time period as is shown by the experiences of the newly emerging economies of Japan and the East Asia. Relative positions in terms of distribution of per capita incomes in the world have changed signi…cantly in the last century for many newly industrialised countries, including China and India in recent years. The “poor”economies have become “rich”in 50 years time. Many growth studies have identi…ed accumulation of physical and human capital, as well as improvement in the production technology, and sound economic policies as the most important factors for economic growth (Ramsey (1928), Harrod (1939), Domar (1947), Solow (1956), Cass (1965), Koopman (1965), Lucas (1988), Romer (1989), Barro (1998) and Barro and Sala-i-Martin (1995), Quah (1997), Rodrik (1999), Uhling (2002)). International trade has played a big role in the growth process as more open economies have grown faster as small economies have gained proportionately more from the specialisation. Di¤erences in the prospects of economic growth rates between countries have also caused a signi…cant reallocation of both skilled and unskilled workers as they have tended to migrate from poor countries or regions with low wage rates to rich countries or regions with higher wage 13
rates. Growth theories analyse these broad experiences of economic growth across the world.
Classical economists from Adam Smith to Marshall attributed higher rates of economic growth mainly to the capital accumulation (Roll (1938), Hahn and Matthew (1964), Madison (1991)). New and better equipment raises productivity of workers by improving the technology of production. Higher rates of saving and investment allow more accumulation of capital and generate higher rates of economic growth. They believed in the general equilibrium mechanisms of economic growth and thought that an increase in productivity automatically translates into an increase in the wage rate. They argued that the economy works better if left to itself as supply creates its own demand in the long run (Say’s law). They favoured absolute economic freedom and minimum role of the government in economic activities. These liberal economic ideas were behind the great success of industrial revolution in Great Britain and many other Western economies. The classical ideas have got more accurate analytical framework in works of economists in the subsequent generation. Ramsey’s (1928) mathematical theory of saving showed the optimal saving rate for an individual (or a benevolent social planner) who likes to maximise utility over its life time. The dynamic programming approach by Bellman (1957), Solows’ neoclassical model of economics growth, the optimal growth framework of Cass (1965) Koopman (1966) added more complete dynamics in the macroeconomics that was focusing more on short run business cycle issues such as ‡uctuations in employment and output and determination of interest rate and money supply under Keynes (1936) rather than on the long run issue of economic growth. Harrod (1939) and Domar (1946, 1947) 14
following Keynesian model assumed a constant rate of saving and capital output ratio in deriving a simple formula for economic growth. In their model the rate of growth of output related to the rate of saving and capital output ratio as g = vs , where g is the growth rate of output, s is the saving rate and v is the capital output ratio. Under Harrod-Domar models a country with v equal to 4 but wanting to grow at the rate of 5 percent per year, has to save and invest 20 percent of its national income. Such simple calculations were used to formulate national economic plans in many countries after World War II. The lack of substitutability between capital and labour, depending on relative prices of capital and labour, made the Harrod-Domar model less applicable for market economies as it meant a permanent unemployment or growth restrained by the capacity constraint if the warranted rate of growth were di¤erent from the actual growth rates. Solow (1956) avoided this shortcoming by incorporating explicit substitutability between capital and labour, illustrating this using either a Cobb-Douglas or a CES (constant ealsticity of substitution) production functions. He adopted a constant saving rate (as in the standard Keynesian model) rather than a variable optimal saving rate as in Ramsey (1928). Cass (1965), Koopman (1965) and Uzawa (1968) derived optimal economic growth models in which saving and investment rates also became endogenous to the lifetime utility maximisation behavior of households and …rms (Hahn (1964)). Growth models have become more prominent in recent years as the most of the economies have realised limitations of demand management policies and focused on reform of goods and factor markets to reform the supply side of their economies. Theoretical models are supported by more in depth empirical growth studies (Maddison (1991), Mankiw et. al. (1992), Barrow and Sala-i-Martin (1995), Temple (1999)). Institutional and policy regimes are considered important determinants of growth in recent years (Barro (1995), Rodrik (1999), Quah (1997), Levin et.al. (1997), Temple (1999)). The basic version of the neo-classical growth model has been extended in many directions for analysis of international trade (Bhagwati (1969), Krugman (1990), Grossman and Helpman (1991)), education and human capital (Dennison (1962), Jorgenson and Fraumeni (1992), technological advancement and assimilation (Parente and Prescott (1991)), fertility and population growth (Tamura (1991), Miles (1999)), economic policies and institutional factors (Turnovskey (1993), Cooley (1995)). Some studies have integrated analysis of long run growth and the analysis of short-run ‡uctuations (Blackburn (1999), Blake and Weale (2003)).
1.1
Solow growth model
Solow’s neoclassical growth model includes maximisation of pro…t by producers, who pay remuneration to factors equal to their marginal products; maximisation of utility 15
by consumers, who save …xed fraction of their income for future use; model closure by equality of amount of savings to investment; substitution between capital and labour depending on the wage rental ratios. Economic policies can make one input cheaper or expensive relative to another input by distorting their relative prices. Solow’s model assumes a smooth and continuous and twice di¤erential production function. Productivity of capital is high with a low level of capital stock and productivity of labour is higher with lower level of labour inputs (Inada conditions). Productivity of capital is higher in labour abundant developing economies than in capital abundant developed economies. Public policy that increases the rate of saving can only in‡uence the level of output but not the rate of growth of output, which can increase only from the exogenous rate of technological progress. This section presents a version of the neoclassical growth model in which output for year t (Yt ) is produced using capital (Kt ), labour (Lt ) and technology (At ) with productivity parameters and and a constant returns to scale production function + = 1 as: Yt = A t Kt L t
Saving (St ) is a …xed fraction (0 < s < 1) of output and given by 16
(1)
(2)
St = sYt
The required level of investment for year t (It ) ;depends on the population growth rate (n), depreciation ( ) and the capital stock as: (3)
It = (n + ) Kt Capital accumulation relates to investment and depreciation rate as: Kt = (1
) Kt
1
(4)
+ It
The market clears in the sense that output (Yt ) is either consumed (Ct ) or saved (St ). The model is closed by balance between the saving and investment as: Yt = Ct + St = Ct + It ;
(5)
It = St
Growth Accounting Sources of growth can be derived by taking log and differentiating with respect to time; taking log both sides of the production function, Yt = At Kt Lt : ln (Kt ) + ln (Lt )
(6)
dA=dt dK=dt dL=dt dY =dt = + + Y A K L
(7)
ln (Yt ) = ln (At ) +
growth in discrete notations: gy = In contineus terms: gy =
Y Y
; gA =
Yt
Yt Yt
A ; A
1
1
gk =
K K
and gL = LL ; or
Y Y
gy = gA + gK + gL Growth rate of output = Growth rate of TFP +capital share growth rate of capital + labour share growth rate of labour. TFP (A) indicates Hicks neutral total factor productivity.
17
=
A A
+
K K
+
L L
(8)
1.1.1
Stylized facts of economic growth labour input has grown more slowly than capital and output. Therefore capital per capita (K/L) and output per capita (Y/L) have increased secularly. t Kt 0 (1+gk ) 0 => K > K as gk > gL . Lt L0 L (1+g )t 0
L
capital output ratio (K/Y) has remained fairly constant, has had no discernible t trend and has more or less converged across industrialised economies. K => Yt K0 (1+gk )t Y0 (1+gY )t
'
K0 Y0
as gk = gY .
rate of return on capital (pro…t) and the real interest rate have no trend whereas real wage rates have followed a rising trend. Secular rises in productivity (Y =L) and per capita capital (K=L) have translated into higher real wages. [proof: y = Ak ; r = Ak 1 ; w = (1 )Ak Thus w rises as k rises. share of income devoted to capital (r:K=Y ) and to labour (w:L=Y ) show no trend and remain fairly constant as the capital stock per person tends to grow along with output per person. Proof: Given output Y = AK L ; capital share: r:K = AK 1 L :K; labour share: wL = AK L 1 L then Y = rK+wL = AK L + AK L = ( + ) AK L = Y * ( + ) = 1 by constant return to scale.
18
1.1.2
Steady state
Percapita output per e¤ective worker (yet ) : where
K L
yet =
Yt At Kt Lt K = = 1t At Lt At Lt Lt
is per capita capital, k =
K : L
Kt Lt
=
L = L = sy k
with
(9)
= kt
n and K = (sY
K)
L =K = (sY K K) LL = sY ( + n) ( + n) = sAk 1 ( + n) K L K Fundamental equation of economic growth in the neoclassical growth model: k k
k = syt
(n + ) kt
(10)
where dk=dt = k There is no change in per capita capital in the steady state: k dk=dt = =0 kt kt dk=dt kt
> 0 before the steady state and
dk=dt 19 kt
< 0 after the steady state.
(11)
Solution of the model for the steady state k = syt
(n + ) kt = sAe yt
k = sAkt kt
1
(n + ) kt
(12)
(13)
(n + ) = 0
(14)
sAk = (n + ) k Per capita capital in the steady state: k=
sA (n + )
1 1
(15)
Per capita output in the steady state: ye =
y=A
sA (n + ) sA (n + )
1
(16) 1
(17)
Percapita consumption in the steady state: c = (1
s) y = (1
s) A
sA (n + )
1
(18)
With technical progress the fundamental equation becomes sAk = (n + a + ) k: Now the investment also should make up for growth in the technology. 1.1.3
Summary of Solow Model
1. Countries with higher (lower) saving rate have higher (lower) steady state level of output. 2. Countries with higher (lower) level of technology have higher (lower) level of output in the steady state. 3. Countries with higher rate of population growth rate have lower level of output in the steady state. 4. Countries with higher capital share have higher output in the steady state. 20
5. Countries which di¤er in the initial capital stock eventually reach to the same output level in the steady state. Global economy converges in percapita income if all countries have the same technology. 6. Growth of per capita income is determined by the rate of technical progress in the steady state; (saving rate only determines the level of output not its growth rate). 1.1.4
Golden rule of saving and capital accumulation
First de…ne consumption as the di¤erence between output and investment. Then under goldern rule means choosing the amount of capital stock that maximises consumption as: max c = y k
(n + )k = k
(n + )k
(19)
with y = k First order condition for the golden rule: @c = k @k
1
1 1
kG =
(n + )
;
(20)
(n + ) = 0
kss =
s (n + )
1 1
(21)
Setting the saving rate (s) according to the marginal productivity of capital ( ) is optimal, = s:
21
c = y (n + ) k = 0:5 k (n+ )k = 0:5 250:5 (0:02 + 0:03) 25 = 2:5 1:25 = 1:25: Thus 1:5 = 0:5: As shown in the …gure. s = YS = Y Y C = 2:52:51:5 = 2:5 1.1.5
Literature on economic growth
One or two sector growth models Ramsey (1928), Harrod (1939), Domar (1947), Solow (1956), Cass (1965), Koopman (1965), Uzawa (1962), Lucas (1988), Rebelo (1991) Romer (1989), Barro (1998) and Barro and Sala-i-Martin (1995), Quah (1997), Parente and Prescott (1993) Rankin (1992) Dixon H (1987), Jones (1995) Greenwood and Jovanovic (1990). Empirical growth study Maddison (1991). Nicolas Kaldor (1961) Rodrik (1999) Mankiw et al. (1992), Hendry (1997), Temple (1999), McMahon and Squire (2003), Holly and Weale (2000), Perroti (1996), Grossmand and Helpman (1991) HM Treasury (2002).
22
References [1] Acemoglu D. (2010) Introduction to Modern Economic Growth, Princeton University Press. [2] Aghion P. and P. Howitt (1998) Endogenous Growth Theory, MIT Press, Cambridge MA. [3] Alesina A.and D. Rodrik (1994) “Distributive Politics and Economic Growth”, Quarterly Journal of Economics, 109:2:465-90. [4] Arrow K. J. (1962) Economic Implications of Learning by Doing, Review of Economic Studies, 29:155-73. [5] Barro R. J.(1991) Economic Growth in Cross Section of Countries, Quarterly Journal of Economics, May, 407-433. [6] Barro R. and X. Sala-i-Martin (1992) Public Finance in Models of Economic Growth, Review of Economic Studies, 56:4:645-61. [7] Barro R. J. and Sala-I-Martin (1995) Economic Growth, McGraw Hill. [8] Basu P. and K. Bhattarai (2012) Government Bias in Education, Schooling Attainment and Long-run Growth, Southern Economic Journal, 79(1), 127-143 [9] Basu P. and K. Bhattarai (2012) Cognitive Skills, Openness and Growth, Economic Record, 88: 280: March, 18-38 [10] Becker G.S. (1975) Human Capital, Columbia University, New York. [11] Blanchard O. and D R Johnson (2013) Macro Economics, 6th ed, Pearson. [12] Bruce Neil and Stephen J. Turnovsky (1999) Budget Balance, Welfare, and the Growth Rate: "Dynamic Scoring" of the Long-Run, Journal of Money, Credit and Banking, 31, 2, 162-186. [13] Bruno M. and W. Easterly (1998) “In‡ation Crises and Long-run Growth”Journal of Monetary Economics, 4: 3-26. [14] Cass, D. (1965): Optimum Growth in Aggregative Model of Capital Accumulation, Review of Economic Studies, 32:233-240. [15] Cobb C. W. and P. H. Douglas (1928) A Theory of Production, American Economic Review. 18:1:139-165. [16] Denison E F (1962) The sources of economic growth in the United States, Committee for Economic Development, New York. [17] Deverajan S., W R Easterly and H. Pack (2003) Low Investment is not the Constraint on African Development, Economic Development and Cultural Change, 52:3:547-571, April. [18] Dixon Huw (1988) Controversy: the macroeconomics of unemployment in the OECD, Economic Journal, 108:779-781, May. [19] Dollar, D. (1992) “Outward-Oriented Developing Economies Really Do Grow More Rapidly: Evidence from 95 LDCs, 1976-1985” Economic Development 23
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and Cultural Change, 40:3:523-544. Domar E. (1947) Capital Expansion, Rate of Growth and Employment”Econometrica, 14:2:137-147. Domar E. (1947) Expansion and Employment , American Economic Review, 37:1:35-55. Doornik J A and Hendry D.F. (2001) Econometric Modelling Using PcGive vol. I-III. Timberlake Consultants, London. Dri¢ ll J and M. Miller (2013) Liquidity when it matters: QE and Tobin’s q, Oxf. Econ. Papers, 65 (S1), 115-145. Easterly W and R.Levine (2001) “It’s not factor accumulation: Stylized facts and Growth Models”World Bank Economic Review, 2001. Fisher S (1993) “The Role of Macroeconomic Factors in Growth” Journal of Monetary Economics, 32:485-511. Fullerton, D., J. Shoven and J. Whalley (1983) “Dynamic General Equilibrium Impacts of Replacing the US Income tax with a Progressive Consumption Tax,” Journal of Public Economics 38: 265-96. Greenwood J., B. Jovanovic (1990) Financial Development, Growth, and the Distribution of Income, Journal of Political Economy, 98, 5, pt.1:1076-1107 Grossman G and H Helpman (1991) Quality Ladders in the Theory of Growth, Review of Economic Studies,18:43-61. Grossman G. and E. Helpman (1991) Innovation and Growth in the Global Economy, Cambridge Mass. MIT Press. Hall R. and C Jones (1999) “Why do some countries produce so much more output than others?”Quarterly Journal of Economics, February, 104:1:83-116. Hahn F.H. and R.C.O. Matthews (1964) The Theory of Economic Growth: A Survey, Economic Journal, 296:74:779-902. Harrod R (1939) An Essay in Dynamic Theory, Economic Journal, 49:193:14-33. Hendry D.F. (1997) The Econometrics of Macroeconomic Forecasting , Economic Journal, 107, 444, 1330-1357 Holly S and M Weale Eds.(2000) Econometric Modelling: Techniques and Applications, Cambridge University Press. Jones, Charles (2015) Macroeconomics, Third edition, W. W. Norton (ISE). Jones, Charles (2002) Introduction to economic growth, 2002, 2nd Edition, Norton. Jones C. I. (1995) R & D-Based Models of Economic Growth, Journal of Political Economy, 103:4:759-784 Jogenson D W and B. Fraumeni (1992) Investment in Education and US Economic Growth, Scandenevian Journal of Economics 94:51-70, September. 24
[39] Kaldor N. (1961) Capital Accumulation and Economic Growth in F.A. Lutz and D.C. Hague ed. The Theory of Capital, New York, St. Martin. [40] Keynes J.M. (1936) The General Theory of Employment, Interest and Money, MacMillan and Cambridge University Press. [41] Koopmans, T. C. (1965) On the concept of Optimal Growth, The Econometric Approach to Development Planning, Chicago, Rand McNally. [42] Krugman P.R and A.J. Venables (1995) “Globalisation and Inequality of Nations”Quarterly Journal of Economics, November 857-80. [43] Krugman P. and R Wells (2009) Macroeconomics, 2nd edition, Worth. [44] Levine, R and S. Zervos (1998) “Stock markets, banks and economic Growth” American Economic Review, 88:537-58. [45] Lin Justin Yifu (2003) Development Strategy, Viability and Economic Convergence, Economic development and Cultural Change, 51:2:277-308, January. [46] Lucas, R E. (2002) Lectures on Economic Growth, Harvard University Press. [47] Lucas R.E. (1988) “On the Mechanics of Economic Development”, Journal of Monetary Economics, 22:1: 3-42. [48] McMahon G and L. Squire(2003) Explaining Growth: A Global Research Project, Palgrave Macmillan Great Britain. [49] Maddison A. (1991) Dynamic of Capital Accumulation and Economic Growth, Oxford. [50] Mankiw N.G. (2016) Macroeconomics, Ninth Edition, Worth Publishers, macmillan reprint. [51] Mankiw N.G., D. Romer and D. N. Weil (1992) “Contribution to the Empirics of Economic Growth”Quarterly Journal of Economics, 107:407-437, May. [52] Mankiw N. G. and M. P. Taylor (2008) Macroeconomics: European Edition, Worth Publishers. [53] Miles D. and A. Scott (2002) Understanding the Wealth of Nations, John Wiley and Sons. [54] NIESR (various issues) UK and World Economies Forecast, National Institute of Economic Review, http://www.niesr.ac.uk [55] Nelson R R and H. Pack (1999) The Asian Miracle and Modern Growth Theory, Economic Journal, 109:416-436, July. [56] Parente S.L.(1994) Technology Adoption, Learning-by-Doing, and Economic Growth, Journal of Economic Theory, 63:346-369. [57] Parente S.L. and Prescott E. C. (1993) Changes in the Wealth of Nations, Federal Reserve Bank of Minneapolis, Quarterly Review, 17: 3-16, Spring. [58] Perroni, C. (1995), “Assessing the Dynamic E¢ ciency Gains of Tax Reform When Human Capital is Endogenous, International Economic Review 36:90725
[59] [60] [61] [62] [63] [64] [65] [66] [67]
[68] [69] [70] [71] [72] [73]
[74] [75]
925. Perotti Roberto.1993. "Political Equilibrium, Income Distribution, and Growth." Review of Economic Studies, 60(4): 755-776 Quah D. (1997) “Increasingly Weightless Economies”, Bank of England Quarterly Bulletin, 37:1:49-56. Ramsey, F.P. (1928) A Mathematical Theory of Saving, Economic Journal 38:543-559. Rebelo, S. (1991) “Long-run Policy Analysis and Long-run Growth,”Journal of Political Economy 99, 500-521. Rodrik D. (1999) “Where did all the growth go? External Shocks, Social con‡ict, and Growth Collapse”Journal of Economic Growth, 4: 4:385-412. Roll E (1938) History of Economic Thought, Oxford. Romer, Paul (1989) Endogenous Technological Change, Journal of Political Economy,98:5: Pt. 2: S71-S102. Restuccia D. (2004) Barriers to Capital Accumulation and Aggregate Total Factor Productivity, International Economic Review, 45:1:225-38. Rutherford, T.F. (1995) “Extension of GAMS for Complementary Problems Arising in Applied Economic Analysis,” Journal of Economic Dynamics and Control 19:1299-1324. Schultz T W (1961) Investment in Human Capital, American Economic Review, 51:1-17. Solow, R. M.(1956) A Contribution to the Theory of Economic Growth, Quarterly Journal of Economics, 70:1:65-95. Tamura R (1991) Income convergence in an endogenous growth model, Journal of Political Economy, 99:522-540, June. Temple J. (1999) The New Growth Evidence, Journal of Economic Literature, 37:112-156, March. Turnovsky S.J. (1993) Macroeconomic Policies, Growth, and Welfare in a Stochastic Economy, International Economic Review, 34:4: 953-981. Uhling H. (1999) A toolkit for analysing non-linear dynamic stochastic models easily, in Marimon and A, Scott ed. Computational Methods for the Study of Dynamic Economics pp. 30-61, Oxford University Press. Uzawa, H. (1962) On a Two-Sector Model of Economic Growth, Review of Economic Studies 29, 40-47. Weil D.N. (2013) Economic Growth, 3rd edition, Pearson.
26
1.1.6
Computer Lab1: Growth and Macro Models (optional)
This is to help understanding of models in further economic analysis. This lab exercise has four components. It may require sometime to get used to numerical and programming skills required in doing this exercise. 1. Exercises in Excel calculations. Be familiar on how to compute models in excel. First download the Solow1.xls …le from the macroeconomic model folder in ebridge to your own directory. Prepare a list of parameters ( ; ; n; and A ) that de…ne the functional forms of the model. Then drag down the formula to compute the model till you …nd the steady state, where variables such as output, capital stock, investment stop changing. Table 1: Parameters of a growth model s n k0 Values of parameters
Table 2: Steady state solution of the growth model K Y K I S L Scenario 1 .. .. Scenario n Do similar exercise for the endogenous growth and the Ramsey models. Then look at the keynesian model.xls and try to construct similar macro models. After that have a look into microeconomic models folders and do excel based computations of various models. 2. Be familiar to programming software used to compute economic models such as GAMS or MATLAB. Open GAMS in start/network applicationsneconomicsngams. Demo version of GAMS can be downloaded free from the www.gams.com. First try to understand the logic of programming. As in any other programming languate GAMS has …ve elements in the programme: Declaring parameters and variables. declaring and assigning equations. 27
declaring models. Solving models and reporting results. Interpreting results and analysisng if they con…rm to the logics in economic theories. For exercise you need to …rst open a project. Give a newname, anyname such as "aa" or "zz" is …ne. Type the code in a microsoft notepad or in text editors like PFE32 or Emacs. Use …lename.gms extension. Save the programme and compile. Model should generate feasible and optimal solutions. Once you understand one programme, others are a lot easier. Consult GAMS and GAMS/MPSGE User Manuals, GAMS Development Corporation, 1217 Potomac Street, Washington D.C or www.gams.com or www.mpsge.org for GAMS/MPSGE. MATLAB is widely used for solving models. It has script and function …les used in computations. Both have *.m extensions. Its syntax are case sensitive. Solving a system of linear equations and handling matrices Example 1 Write a programme …le matrix.m like the following and try to run. % now solve a linear equation % 5x1 + 2x2 =20 % 3x2 + 4x2 =15 k =[5 2;3 4]; n = [20 15]; kk = inv(k) x = kk*n’ "Dynare" programme (www.dynare.com) interacts with the MATLAB and is good for business cycle analysis and simulations. 3. Book Exercises (do regularly as much as you can: links better from the weblinks at the Ebrige) a) http://wps.prenhall.com/bp_blanchard_macro_4/ b) http://wps.aw.com/aw_perlo¤_microcalc_1/76/19536/5001253.cw/index.html c) http://www.worthpublishers.com/krugmanwells/ http://www.economicsnetwork.ac.uk/ http://www.aeaweb.org/rfe/ 28
Follow links http://www.hull.ac.uk/php/ecskrb/Confer/research.html; 4. Constructing Data for Analysis: Step by Step Guidelines Following can be useful for providing empirical evidence for the essay or for general idea on a topic at the early stage. Connect to http://www.esds.ac.uk/international/ Choose direct links to macro data Important Steps for extracting data I. World Bank Data (World Bank data Indicator 1. Click on Direct Links to Macro Data 2. Choose World Bank Data 3. Select University of Hull /put user name and pass word (if necessary); 4. Complete the registration process required by data 5. Select World Bank Development Indicators 6. Select Year 1960 -2012 (all can be selected by a tick mark) 7. Select a country (e.g. UK, US, China, India, Japan, South Africa, Brazil) 8. Select a series (e. g. Population between 15-64; and population growth rate) ; can search for population 9. Click on show Table 10. Adjust row and column dimensions of the table by moving around icons 11. Download data in *.CSV (MS-DOS) format 12. Open the data …le just created 13. Make some time series graph 14. Next time; add few more variables like DGP per capita constant 2000 dollars; Gross …xed capital formation % of GDP; Final consumption Expenditure as a % GDP; Current account balance as a % of GDP; General Government …nal consumption % of GDP; GDP constant 2000 $ II.
IMF World Economic Outlook (WEO) data 29
1. Steps 1 -5 as above 2. Select IMF WEO data 3. Select World Economic Outlook 4. Select Euro Aria 5. Select crude oil price, output gap, unemployment rate, in‡ation average consumer price. 6. Select all years 1991-2010 7. Show Table; Download the data; Open and Excel. 8. Do macro analysis. III.
Eurostat New Cronos (Database of EU countries)
1. Steps 1 -5 as above 2. Select Eurostat New Cronos 3. Select Economy and Finance 4. Eurostat 5. Exchange rates 6. Nominal e¤ective exchange rates 7. Real e¤ective exchange rates IV. growth data in EU economies: http://www.euklems.net/ V. Datastream/ Bloomberg
30
1.1.7
Problem 1:Neoclassical Growth Model
Q1. Consider a version of the neoclassical growth model in which output for year t (Yt ) is produced using capital (Kt ), labour (Lt ) and technology (At ) with productivity parameters and and a constant returns to scale production function + = 1 as:
(22)
Yt = At Kt Lt Saving (St ) is a …xed fraction (s) of output and given by
(23)
St = sYt
The required level of investment for year t (It ) ;depends on the population growth rate (n), depreciation ( ) and the capital stock(Kt ) as: It = (n + ) Kt
(24)
Capital accumulation relates to investment and the rate of depreciation as: Kt = (1
) Kt
1
+ It
(25)
The market clears in the sense that output (Yt ) is either consumed (Ct ) or saved (St ). The model is closed by balancing saving and investment as: Yt = Ct + St = Ct + It ; =) It = St
(26)
1. Write the production function in intensive form, y = Ak and represent is in a diagram. 2. Find the steady state value of per capita capital and per capita output in terms of share of capital in output , the saving rate , the depreciation rate and population growth rate. What are the values of the capital stock and output in the steady state in terms of parameters ; ; n; ; s and A? 3. What will be value of per capita capital and per-capita output if the savings rate was 32%, the depreciation rate was 8%, the population was growing at 2% per annum, = 0:3 and A = 1? 4. How would an increase in the foreign direct investment a¤ect the steady state in this model? 31
5. Now assume that the computer hackers and terrorists destroy computing system and other infrastructure. The e¤ective capital stock is virtually reduced to half of its previous stock. How does this a¤ect the steady state obtained in (3) above? 6. Global warming causes ‡oods that contaminates all food stocks and brings wide spread diseases. Many people die. As a result the labour force reduces by one quarter. How does this a¤ect the income and capital stock in the steady state? 7. Compute the transitional dynamics in the Solow model and show how it is sensitive to ; parameters. 8. What will happen to the growth process if
+
> 1?
Q2. Decompose the sources of economic growth in the human capital augmented Solow model. Yt = At Kt Lt H
(27)
where + + = 1 and At is an index of Hicks neutral technical knowledge A certain country with parameter values = 0:3; = 0:5 and = 0:2 had output (Yt ) growing by 6 percent, capital (Kt ) by 3 percent, labour (Lt ) by 2 percent and human capital (Ht ) by 2 percent. What was the growth rate of technical progress (gA ) in this country? Q3. What is the golden rule of savings and consumption in the Solow model? How do output and capital di¤er in than in the steady state? What prevents a higher saving rate from leading to a higher growth rate in the steady state in the Solow model? [hints: y = Ak ; c = y (n + )k; S = sY ] Q4. How fast should a poor country that has only 5 percent of the per capita income of the richest country should grow if its wants to catch up the level of income in the rich country which is growing at 2 percent in coming 50 years? What do they need to do for this? [hint: Y0p = 0:05 Y0R ; Ytp = Y0P (1 + gp )t = YtR = Y0R (1 + gR )t ; 0:05 Y0R (1 + gp )t = Y0R (1 + gR )t ]. Q5. Compare the level of GDP (in PPP US dollars) for a number of countries around the world in 1980 and 2005 and average annual growth rates in the following table.Why do countries di¤er so much in terms of economic growth rates? 32
Table 3: Growth Rates and Di¤erence Between GDP at constant price and GDP in PPP G D P in P P P 1 9 8 0
C h in a
N epal
In d ia
S o u th A fric a
UK
G e rm a ny
USA
B ra z il
Japan
A u stra lia
774
825
1165
10051
17025
17216
22568
6767
17026
17427
G D P in P P P 2 0 0 5
6012
1379
3072
9884
29571
26210
37267
7475
27817
28286
G row th ra te
0 .0 8 2
0 .0 2 1
0 .0 3 9
-0 .0 0 1
0 .0 2 2
0 .0 1 7
0 .0 2
0 .0 4
0 .0 2 0
0 .0 1 9
C o n sta nt G D P 2 0 1 2 ($ 2 0 0 5 )
3348
400
1106
6003
37850
37479
43063
5721
36938
37288
G D P in P P P in 2 0 1 2
7958
1279
3340
9860
32723
34766
43063
10264
31425
35669
P P P / c o n sta nt G D P
2 .3 7
3 .2 0
3 .0 2
1 .6 4
0 .8 6
0 .9 3
1 .0
1 .7 9
0 .8 5
0 .9 6
G row th ra te P P P 2 0 0 5 to 2 0 1 2
0 .0 4
-0 .0 1 2
0 .0 1 1
-0 .0 0 0 3
0 .0 1 4
0 .0 4 0 3
0 .0 2 1
0 .0 4 5
0 .0 1 7
0 .0 3 3
G D P a t c o n sta nt $ a n d G D P P P P
D a ta S o u rc e : M IM A S / E S D S / Wo rld B a n k D e ve lo p m e nt In d ic a to rs.
What should the annual growth rate in Nepal (one of the lowest income country) be in order to catch up to India by 2055 and to the UK by 2105 if growth rates of those countries are likely to remain unchanged from the ones given above throughout the next century? Case studies Summers and Heston data set on growth http://pwt.econ.upenn.edu/; See links to other sources of data in: Department of Business Innovations and Skills: http://www.bis.gov.uk/; HM Treasury http://www.hm-treasury.gov.uk/ Yorkshire and Humber http://www.humberlep.org/ http://www.lse.ac.uk/researchAndExpertise/units/growthCommission/home.aspx Links at http://www.hull.ac.uk/php/ecskrb/Confer/research.html
33
2
L2: Endogenous Growth Model: Knowledge is powerful
Many economists were not happy with the exogenous nature of technological advancement in the Solow model which is the major source of growth in the steady state. Taking on the learning by doing literature (Schultz (1961), Arrrow (1962) and Becker (1975)), Lucas (1988) and Romer (1989), have made technology endogenous by introducing knowledge or human capital in the production function along with the physical capital. Lucas models human capital accumulation to be a direct outcome of time spent on studying and learning rather than at work. If people spend more time in studying, they learn more and become more skilled. This raises per capita human capital available in the economy, which complements with physical capital and raises the skill and the productivity of workers at work. Such a rise in productivity is the major source of economic growth.
34
Human capital (H) refers to the average level of skills of individual workers who use physical capital such as buildings, machines and computers to produce goods and services. More skilled workers generally have more years of schooling and on the job experience (learning by doing), are more productive. Human capital combined with physical capital may generate a constant marginal product of physical capital as physical and human capital become complementary to each other. When the marginal product of capital is constant and not subject to the law of diminishing returns, then each marginal unit of capital adds more to the output. Economies with a larger stock of human and physical capital can grow faster than economies with smaller stocks of both of these capitals.
2.1
AK endogenous growth model
Romer (1989) introduces a role for accumulated knowledge in the production process which results from work of researchers in universities or research laboratories. The stock of knowledge that exists in the form of designs, formulas or models is a non-rival good with positive externality as it can be borrowed from the library. He assumes separate production functions for research, intermediate and the …nal goods sector while illustrating the endogenous process of technical progress and its impact in economic growth. Workers in the research sector produce new ideas that they sell to an intermediate sector, which apply them in production of …nal goods. Productivity of workers in the …nal goods sectors rises when they get better tools to work with. Economic growth is ultimately the result of human resources employed in the research sector such as universities and research laboratories. 35
Production Technology (28)
Yt = At Kt Capital Accumulation Kt = (1
) Kt
1
+ It
(29)
Market Clearing Yt = Ct + St = Ct + It
(30)
No diminishing returns to capital, marginal productivity of capital at time t is: @Yt = At @Kt
(31)
ln (Yt ) = ln (At ) + ln (Kt )
(32)
Endogenous Growth
Growth rate of output (by log di¤erentiation w.r.t. time) gy = gA + gK Growth rate of output = Growth rate of TFP+growth rate of capital
36
(33)
Higher rate of saving implies more investment. (34)
It = St
@K=@t sY = (35) K K That means larger capital stock [Kt = (1 ) Kt 1 + It ] and higher level of output [Yt = At Kt ]. E¢ ciency of the …nancial system is important for capital formation. What Determines the Stock of Technical Knowledge (A)? Two sector Model: 1) …nal goods sector and 2) knowledge sector. Final goods are produced by capital, labour and technology gK =
Yt = Kt (At Lt )
(36)
Workers work either in the …nal goods sectors (e.g. agriculture, manufacturing) knowledge sector (universities, research labs) L = LY + LA
(37)
Technology is endogenous here determined inside the model. @A=@t = gA = A LA A
(38)
More workers in the knowledge sector means greater stock of technical knowledge (A), > 1. Knowledge generates more knowledge knowledge.
> 1. There is increasing returns to
Increasing returns to knowledge means economic growth continues forever. Thus education raises productivity and income of workers. Knowledge is the core sector - it involves processing information, derivations, inductive-deductive reasoning. =)Thinking=) Turning ideas into actions. Ideas empower workers, makes them more productive. They bring new products. This raises the growth rate of the economy. Thus the growth rate of output equals to the growth rate of knowledge and that of capital as: 37
sY K Growth rate is driven both by higher TFP and higher rate of saving. gy = gA + gK = A LA +
2.1.1
(39)
Human capital and economic growth
Countries that focus on educating younger people can grow faster, they can employ more workers in the research sector and get more ideas and put more workers to translate those ideas into …nal products whereas countries that neglect education sector do poorly in economic growth. Education, education and education!!! Cognitive skills and growth are linked; time spent on studies (lHt ) or at work (lGt ) matters for growth; Basu and Bhattarai (2012) link human capital and cognitive skill to growth as: Human capital (ht ) ht+1 = (1
h )ht
+ Qt ht
(40)
Cognitive skill (Qt ): Qt = AHt :lHt
(41)
Final good sector (yt ): yt = AGt kt (lGt ht )1
(42)
See Southeast Asia, Western Europe, North America. Note that larger population size of a country is not necessarily a burden to its economic growth if young can be educated properly in the endogenous growth model. This is in sharp contrast to the Solow model that proved that countries with higher growth rate of population had lower per capita income. Investment in human capital (skills) is the best way to raise the growth rate and to tackle the poverty. Both public and private sectors should invest for better schooling. Educate people to bring them out of poverty.
2.2
Human Capital and Di¤erences in Income Per Capita
Consider the steady state condistions in a growth model y = k h with human and physical capital as sK k h = (n + )k 38
sh k h = (n + )h Log-linearise this function and show the values of k, h and y in the steady state in terms of ; ; n; ; sK and sh : ln sK +
ln k + ln h = ln(n + ) + ln k
(43)
ln sh +
ln k + ln h = ln(n + ) + ln h
(44)
ln k
ln k
ln h = ln sK
ln(n + )
(45)
ln h = ln sh
ln(n + )
(46)
ln k + ln h In matrix notation 1
ln k ln h
1 ln k ln h
=
ln sK ln sh
= 1
1 1
ln(n + ) ln(n + )
ln sK ln sh
ln(n + ) ln(n + )
(47)
(48)
Easy to solve this by Cramer’s rule
ln k =
ln(n + ) ln(n + ) 1
ln sK ln sh
(49)
1 1 1
ln h =
ln(n + ) ln(n + )
ln sK ln sh
(50)
1 1
Evaluate the determinants ln k = ln h =
(1
) (ln sK
ln(n + )) + (ln sh (1 )
ln(n + ))
(1
) (ln sh
ln(n + )) + (ln sK (139 )
ln(n + ))
(51) (52)
ln k =
(1
)
(1
ln h =
) (1
ln sK +
)
(1
)
(1
ln sh +
)
(1
1
ln sh
)
(1
)
ln(n + )
1
ln sk
(1
)
ln(n + )
(53)
(54)
Log of production function: ln y =
ln k + ln h
Substituting above values of capital and labour in y function ln y = ln k + ln h = (1 ) ln sK + (1 ) (1 (1 ) + ln sh + (1 ) (1 ln y =
(1
In Solow model
)
ln sk +
)
)
(1
)
1
ln sh
(1
ln sk
) 1
(1
) +
ln sh
ln(n + )
(1
)
ln(n + )
ln(n + )
= 0 it degerates to ln y =
(1
)
ln sk
(1
)
ln(n + )
While the model with human capital can explain big di¤erences in income among countries but Solow model cannot. Assume = 0:35; = 0:4 and two countries. Let sK2 and sh2 be twice as big than in country 1 and (n + ) be 20 percent less country 2; then how big will be di¤erence in their income levels:
(ln y2
ln y1 ) =
(1
)
(ln sk2
+ (1
)
ln sk1 ) +
[ln(n2 + )
40
(1
ln(n1 + )]
)
(ln sh2
ln sh1 )
(ln y2
ln y1 ) =
(ln y2
0:35 ln (2) + (1 0:35 0:4) (1 0:35 + 0:4 [ln(0:8)] (1 0:35 0:4)
ln y1 ) = 1:4 ln (2) + 1:6 ln (2)
0:4 0:35
0:4)
ln (2)
3 [ln(0:8)] = 2:25
taking antilog: y2 = e2:25 = 15:6 y1 Twice higher saving rates and human capital formation rate causes 15.6 times more steady state output in country 2 than that in country 1. Solow model with = 0 cannot show such big di¤erence in the steady state income. (ln y2
ln y1 ) =
(ln sk2 ln sk1 ) [ln(n2 + ) ln(n1 + )] ) (1 ) 0:35 0:35 ln (2) ln(0:8) = 0:54 [ln (2) ln(0:8)] = (1 0:35) (1 0:35) = 0:54 0:92 = 0:49 (1
y2 = e0:49 = 1:63 y1 Thus introducing human capital in the model is essential to explain the big differences in the levels of income across countries. Romer (1995) Advanced Macroeconomics, McGraw Hill. 2.2.1
Do countries with lower per capita income grow faster?
Prove that countries lower in their per capita income should grow faster in the neoclassical model with: y = k1
k = sk 1 Steady state:
(n + x + ) k =) 41
k sk 1 = k k
(n + x + )
k sk 1 = k k
(n + x + ) = 0 =) k
=
s n+x+
1
s n+x+
=) k =
Growth rate of output is driven by growth rate of capital y = (1 y Now substituting the
y = (1 y
y = (1 y
)
k = (1 k
)
k k
)
k k
from above: sk 1 k
(n + x + )
) (n + x + ) k k
1 = (1
= (1
s k (n + x + )
) (n + x + )
) (n + x + )
"
y y
1
#
1
Check y 1 = k (1 ) 1 = k and y 1 = k (1 ) 1 = k : If y > y then yy > 0. Farther away an economy is from the steady state it should grow faster; this is called convergence towards the steady state. Skills in UK: http://sandbox.bis.gov.uk/ Growth studies in UK: http://www.hm-treasury.gov.uk/ Growth in Yorkshire and Humber: http://www.yorkshire-forward.com/ In countries with large size of population, like China and India, can a¤ord more people could be employed in the research sector. They are growing fast now and have potentials for very high rate of economic growth. Dynamic General Equilibrium Models and stochastic DGEM Debreau (1954), King and Fullerton (1984), Miller and Spencer (1977), Aeurbach and Kotliko¤ (1987), Ballard, Fullerton, Shoven and Whalley(1985), Hutton and Kenc (1994), Perroni (1995), Rutherford (1995), Cooly (1995), Harberger (1962), Basu and Bhattarai (2012). 42
1
References [1] Arrow K.J. (1962) The Economic Implications of Learning by Doing, Review of Economic Studies, 29, 3, 155-173 [2] Auerbach, A.J. and L.J. Kotliko¤ (1987) Dynamic Fiscal Policy. Cambridge University Press. [3] Ballard, C.L., D. Fullerton , J.B. Shoven and J. Whalley (1985) A General Equilibrium Model for Tax Policy Evaluation. University of Chicago Press, Chicago. [4] Blankenau, W & G. Camera (2009) "Public Spending on Education and the Incentives for Student Achievement," Economica, vol. 76 (303), pages 505-527, 07 [5] Cooly Thomas F (1995) Frontiers of Business Cycle Research, Princeton. [6] Debreu, G. (1954) The Theory of Value. Yale University Press, New Haven. [7] Glomm, Gerhard and B. Ravikumar. (1992). "Public versus private investment in human capital: endogenous growth and income inequality." Journal of Political Economy, 100, 818-834. [8] Harberger A.C. (1962) “The Incidence of the Corporation Income Tax,”Journal of Political Economy 70, 215-40. [9] Hutton, J. and T. Kenc (1994) “Representative UK Income Tax with a Progressive Consumption Tax,” University of Cambridge Department of Applied Economics Working Paper 9413, June. [10] Lewis, W. A. (1954) ‘Economic Development with Unlimited Supplies of Labour’,The Manchester School, 22, 2,139–191. [11] Lucas R.E., Jr. (1988), "On the Mechanics of Economic Development", Journal of Monetary Economics, 22:1: 3-42. [12] Perroni, C. (1995), “Assessing the Dynamic E¢ ciency Gains of Tax Reform When Human Capital is Endogenous," International Economic Review 36, 907925. [13] King M. A. and D. Fullerton (1984)The taxation of Income from Capital, The University of Chicago Press Ltd. [14] Rutherford, T.F. (1995) “Extension of GAMS for Complementary Problems Arising in Applied Economic Analysis,”Journal of Economic Dynamics and Control 19, 1299-1324. [15] Tournemaine F. and C. Tsoukis (2009) Status Jobs, Human Capital, and Growth: The E¤ects of Heterogeneity, Oxford Economic Papers, 61, 3, 467-493 2.2.2
Clean air maximising growth rate
There are growing concerns about the environmental impacts of economic growth. A higher growth rate requires burning more fossil fuel, this puts more hazardous gases like carbon dioxide in the atmosphere and is harmful for human welfare. There is 43
an optimal rate of growth that maximises the clean air and minimises the pollution. Let L be a clean air function of growth rate of output and parameters c; a and b: L = c + agy
bgy2
(55)
Then the growth rate that maximises the amount of clean air (…rst order condition): @L =a @gy
2bgy = 0 =) gy =
a 2b
(56)
if a = 0:2 and b = 2 then gy = 5%: This growth rate maximises the clean air: implied by the negative second order condition as: @ 2L = @gy2
2b < 0
(57)
Kyoto agreement to reduce the level of emissions and maximise the clean air. O¢ ce of Climate Change https://www.gov.uk/government/organisations/departmentof-energy-climate-change Emission Minimising Growth Rate Let E be amount of emission. E = 2gy2
0:2gy + 5
(58)
Optimal growth rate that minimises the emission by the …rst order condition: @E = 4gy @gy
0:2 = 0 =) gy =
0:2 = 0:05 = 5% 4
(59)
This minimises the emission (second order condition is positive) @ 2E =4>0 @gy2
(60)
Stern Review on the Economics of Climate Change http://www.hm-treasury.gov.uk/sternreview_index.htm Copenhagen summit 2009 and Durban (South Africa) Summit 2011.
44
Table 4: Sources of Green-House Gas (GHG) Emission in UK Percentage Electricity Generation 35 Household Consumption 14 Industry and Business 17 Transport 20 Waste 3 Agriculture 8 Public Sector 3 Total 100 Source: DECC Carbon Plan, London, (2010).
Figure 4
45
Figure 1: Emission Greenhouse Gases by Industry in UK, 2001
[Unit: thousand tones of CO2 Equivalent; Source: DEFRA] Figure 2: Emission Carbon Dioxide by Industry in UK, 2001
[Unit: thousand tones of CO2; Source: DEFRA] See: Aldy J.E, A. J. Krupnick, R. G. Newell, I. W. H. Parry and W. A. Pizer (2010) Designing Climate Mitigation Policy, Journal of Economic Literature, 48:4, 903–934. 2.2.3
Human capital augmented Solow model
Output is function of technology, capital, labour and human capital as: Yt = At Kt Lt Ht
(61)
where constant return to scale implies share parameters sum to on1: + + = 1: 46
Now by log-di¤erenitation we can decompose the growth rate of output in its components as: (62)
gy = gA + gK + gL + gH
Growth rate of output = Growth rate of TFP+capital share growth rate of capital + labour share growth rate of labour.+ human capital share growth rate of human capital. Now to …nd the per capital growth rate use the parameter condition + + = 1 to replace as:
gy = gA + gK + (1
(63)
) gL + g H
By rearrangement:
(gy
gL )
(gH
gL ) = gA +
(gK
(64)
gL )
Human capital adjusted percapita growth of output is still determined by the per capita physical capital. Marginal Productivity of Human and Physical Capital Here is an another way to relate the output and the human and physical capital: (65)
Yt = At Kt Lt Ht Assuming
+
= 1 and H = K =
Y H
(66)
L H =
Y K
(67)
@Y = AK L H @H @Y = AK @K
1
Yt = AK L H = AK L ( K) = @Y = @K
AL 47
1
AK
1
+
L =
AKL
(68)
(69)
The marginal product of capital does not diminish but may lead to increasing return to scale. With human capital more saving, means more capital that means more output. Human Capital and Growth in the Lucas Model Division of time between learning and earning (working) is very important for economic growth. When people learn more they create more human capital h and higher h means more output: Y = K ( hL)1
(70)
h = human capital per worker. = fraction of time spent on working. 1 = fraction of time spent on studies. L = labour supply –(assume this as given). Example: If K=100, L=100, h=3, =0.8, =0.3 Y = K ( hL)1
= 1000:3 (0:8
Y = K (L)1
100)1
3
= 1000:3 (100)1
0:3
0:3
= 185
= 100
(71) (72)
World Economic Forum http://www.weforum.org/en/knowledge/Themes/index.htm. Stock of human capital starting with the initial stock of human capital h0 : ht = h0 e
(1
)t
(73)
Growth rate of human capital (gh ) depends on fraction of time spent on studies (1 ) and the rate of human capital created by per unit of time spent on studying : gh = if h0 =1, =0.4, (1
(1
(74)
)
)=0.2, time (t)=20
ht = h0 e
(1
)t
=1
e0:4
0:2 20
= e1:6 = 4:95
(75)
How are cognitive skilled measured? Programme for International Student Assessment (PISA) from the OECD measures ability to solve problems in math and science and ability in English. 48
Indices Literacy numeracy Health Income HDI 2.2.4
Application planning programming organising implementing monitoring
Information Gathering Summarising Analysing links Modelling Testing
Research new ideas new product innovation Research Generalising
Technology and inventions
Table 5: Innovations: Product of Human Capital Algebra - Arabs, India (0) Photography , Wireless (1896) Printing Press -Gutenberg 1440 Telephone Bells (1876) Calculus - Newton (1684) Cinematography - Daguerre (1839) Steam Engine - James Watt (1765) Electro Magnetic Telegraph (1833) Electricity - Edison (1879) Powered ‡ight (1903) Computer- Babbage (1820, 1984) X-ray, Jet Engine Radium,Radio, TV, 3D printing PC, Internet, Ipod (2006) ipad(2010) Source: Economist, December 31, 1999
Why research need to be subsidized? Take an example based on Jones (2002) Introduction to Economic Growth. Consider an economy with production function Y = 10(L
F)
where F is …xed labour, L is labour, w the wage rate. Then the cost of production is C = wL and the cost function by substituting L from the production function: C=w
Y +F 10
Under the marginal cost pricing rule: @C w = =P @Y 10 49
. Average cost declines with production: C = Y
w wF + 10 Y
but the producers experience negative pro…t: =R
C = P:Y
C =)
w Y 10
w
Y +F 10
=
wF < 0
Firms undertaking research will not implement projects on their own. Government needs to subsidise to produce optimal amount of research. Whether new technology creates or destroys jobs is a very old question. During the Industrial Revolution in England (1750-1850) a group of workers led by Ned Ludd deliberately smashed new factories being afraid that new machines substitute capital for labour and take away existing jobs and keep them out of work. This also happened in recent past with the invention of computers. However, over years new technology by increasing productivity has helped to expand economic activities and to bring new jobs. Technology and Employment: Luddites??
References [1] Aldy J.E, A. J. Krupnick, R. G. Newell, I. W. H. Parry and W. A. Pizer (2010) Designing Climate Mitigation Policy, Journal of Economic Literature, 48:4, 903– 934 . [2] Barker P.,R. Blundell and J. Micklewright(1989) Modelling household energy expenditure using micro data, Economic Journal 99:397:720-738. [3] BIS (2009) National Skills Strategy, Department of Business, Innovation and Skills, November. 50
[4] Becker G.S. (1975) Human Capital, Columbia University, New York. [5] Grubb, Michael (2004): Kyoto and the future of International Climate Change Responses: From here to Where? International Review for Environmental Strategies, Summer 2004, 5:1:15-38. [6] Jones C. I. (1995) R & D-Based Models of Economic Growth, Journal of Political Economy, 103:4:759-784 [7] Lucas R.E. (1988) On the Mechanics of Economic Development, Journal of Monetary Economics, 22:1: 3-42. [8] Nordhaus, W.D. and Yang, Z( 1996) A Regional Dynamic General-Equilibrium Model of Alternative Climate-Change Strategies, American Economic Review, September, 86:4:741-65 [9] Perroni, C. and T. F. Rutherford (1993) International Trade in Carbon Emission Rights and Basic Materials: General Equilibrium Calculations for 2020, Scandinavian Journal of Economics, 95:3:257-78 [10] Poterba J. M. (1993) Global Warming Policy: A Public Finance Perspective, Journal of Economic Perspectives, 7:4:47-63 [11] Romer, Paul (1989) Endogenous Technological Change, Journal of Political Economy, 98:5: Pt. 2: S71-S102. [12] Stern N. (2008) The Economics of Climate Change, American Economic Review, 98:2:1-37. [13] Schultz T W (1961) Investment in Human Capital, American Economic Review, 51:1-17. [14] Uzawa, H. (1962) On a Two-Sector Model of Economic Growth, Review of Economic Studies 29, 40-47. Web pages (for new economic ideas): http://www.aeaweb.org/webcasts/index.php http://www.res.org.uk/view/conference.html http://www.eeassoc.org/ http://www.unesco.org/new/en/unesco/; http://www.hefce.ac.uk/ http://www.erantis.com/events/denmark/copenhagen/climate-conference-2009/index.htm http://www.app.collinsindicate.com/worldbankatlas-global/en http://web.worldbank.org/
51
http://www.weforum.org/
52
2.2.5
Problem 2: Endogenous Growth Model
Q1. Show possibility of an unbounded growth with higher rate of savings when an economy has “AK”production technology: Y = AK. How can A be generated here? Use diagrams to illutrate. Q2. Consider a two sector model of growth in which output for year t of the …nal goods sector is given by:
Yt = Kt (At Lt )
(76)
Here output (Yt ) is produced using capital (Kt ) ; labour (Lt ) ; and technology (At ). Now Yt is output of the …nal goods sectors (e.g. agriculture, manufacturing) whereas; (At ) is generated in the knowledge sector (universities, research labs). The growth rate of knowledge is gA and is a function of existing knowledge (A) and the number of people employed in the knowledge sector (LA ) gA = A LA
(77)
All three parameters are positive; > 0; 0 < < 1 and 0 < < 1 There is increasing returns to knowledge; knowledge generates more knowledge. Total amount of labour (L) time available in the economy is divided between the …nal goods (LY ) and knowledge sector (LY ) as: L = LY + LA (78) Using this model, brie‡y explain the role played by universities and research laboratories in enhancing economic growth. 1. How is the technical knowledge generated or endogenous in this model? 2. How do research in universities and laboratories raise rate of growth of output? 3. Illustrate graphically how the technical advancement has an impact on the marginal productivity of capital? 4. Graph an “AK”type production function, expressing output in terms of capital and technology underlying an endogenous growth model. Explain how the elasticity of output to capital input, in the production function, di¤ers in a neoclassical growth model as presented in problem 1 and in the endogenous model like this. 53
5. Explain the complementarity between physical and human capital using their marginal productivity curves. 6. Prove that a higher rate of saving generates higher rate of growth in the long run in the endogenous growth model where as there is no e¤ect of higher rate of saving in the growth rate in the steady state in a neoclassical growth model. 7. Using Summers and Heston data set (PennWorld Tables) compare average rates of growth from 1960 to 2010 of the UK (or economy of your choice) to that of newly emerging economies of China, India, Japan, South Korea, Brazil, Russia, Australia and New Zeeland, South Asia, Sub-Saharan Africa, Germany, France, Spain, Portugal, Canada and USA. Put time series graphs of growth rates together. Analyse factors that have contributed to faster, slower or lack of growth of per capita income in these economies. Q3. Substitute all constraints in the objective function and write the resulting reduced form equation appropriate for intertemporal optimisation in the Ramsey model given below.
max Uo = Ct
Subject to production technology ( 0
0 or say when t = 0.40. What will the solution of the model?
4.2
Two period general equilibrium model
Pure exchange model was illustrated with an Edgeworth box diagram followed by demands derived from the utility maximisation problems of households in the above section. This pure exchange model can be extended to a two period general equilibrium constrained optimisation problem as illustrated in this section. This allows to do some inter-temporal analysis of consumption and saving, lending and borrowing and the determination of the real interest rate depending on the preferences of households for the current and future consumption and the endowments in each period. General lesson is that people who have more today are willing to lend to people who have less today or people who have less today borrow against their future income. 79
The terms of this borrowing and lending is determined by the rate of interest at equilibrium. Let the model be speci…ed as: Households, A and B. Each lives today and tomorrow. Each is endowed with goods in both periods; one has more and another has less in the …rst period Objective of each is to maximise life time utility subject to budget constraints in period 1 and 2. Financial market allows lending and borrowing. Equilibrium interest rate is price that determines intertemporal allocations. Problems of representative households with the subjective discount factor consumption in period 1 and 2 M ax U (C1i ; C2i ) = ln C1i + ln C2i
i = A; B
for
(167)
Subject to First period budget constraint: C1i + bi = ! i1
(168)
Second period budget constraint: C2i = bi (1 + r) + ! i2
(169)
here C1i and C2i are consumption in period 1 and 2 by householdi = A; B; bi is borrowing or lending by household i. ! i1 and ! i2 are endowments in period 1 and 2 of household i = A; B; r is the interest rate and is the discount factor. Intertemporal budget constraint is by solving the period 2 budget constraint for i b and using the resulting value in the period 1 budget constraint and by slight rearrangement as: From (169) bi =
C2i 1+r 80
! i2 1+r
(170)
substituting (170) in (168) gives the intertemporal budget constraint C2i ! i2 = ! i1 + 1+r 1+r Lagrangian for the constrained optimisation is C1i +
L = ln C1i + ln C2i +
! i1 +
! i2 1+r
(171)
C1i
C2i 1+r
(172)
First order conditions for optimisation 1 @L = i i @C1 C1 @L = i @C2i C2
(173)
=0
1+r
(174)
=0
@L ! i2 C2i = ! i1 + C1i =0 (175) @ 1+r 1+r Now dividing (779) by (780) gives the marginal rate of substitution between current and future consumption 1 C1i
C2i
=
1+r
Demand and market clearing conditions From (1286) 81
=) C2i =
(1 + r) C1i
(176)
!i
Ci
2 2 = ! i1 + 1+r Putting (176) in it gives C1i + C1i + 1+r The demand for consumption in period 1 is
C1i =
1 (1 + )
! i1 +
(1+r)C1i 1+r
!i
2 = (1 + ) C1i = ! i1 + 1+r
! i2 1+r
(177)
Similarly the demand for consumption in period 2 is obtained by putting (177) in (176) C2i =
(1 + r) C1i =
(1 + r) (1 + )
! i1 +
! i2 1+r
(178)
Market clears each period B C1A + C1B = ! A 1 + !1
(179)
B C2A + C2B = ! A 2 + !2
(180)
Market clearing interest rate: Put (177) and (178) in (179)
C1A
+
C1B
1 !A A = !1 + 2 (1 + ) 1+r A B = !1 + !1
1 + (1 + )
!B 1
!B + 2 1+r
(182)
!B 1 !A 2 2 B B + !A + ! + = !A 1 1 + !1 (1 + ) 1 1+r 1+r 1 A B !A + !B 2 = (1 + ) ! 1 + ! 1 1+r 2 1+r =
B 1 !A 2 + !2 ; B !A 1 + !1
(181)
B !A 1 + !1 =
r=
B !A 1 + !1
B 1 !A 2 + !2 B !A 1 + !1
1
(183)
(184)
(185)
is the value of income in terms of utils. Proof of Walras’Law =
1 = C1i
1 1 (1+ )
!A 1 +
!A 2 1+r 82
=
1 1 (1+ )
!B 1 +
!B 2 1+r
(186)
By Walras law when one market clears other market automatically clears. Check this by putting (959) and (178) in (180) (1 + r) (1 + ) B = !A 2 + !2
C2A + C2B =
!A 1 +
!A 2 1+r
+
(1 + r) (1 + )
!B 1 +
!B 2 1+r
(188)
(1 + r) A !A !B 2 2 B !1 + !B + + = !A 1 2 + !2 (1 + ) 1+r 1+r Proof of Walras’Law (1 + r) A A B !1 + !B 1 = !2 + !2 (1 + ) (1 + r) A !1 + !B 1 = 1 (1 + ) (1 + r) =
(187)
(1 + )
B !A 2 + !2
(190)
1 B !A 2 + !2 (1 + )
(191)
(1 + )
B !A 2 + !2 =
(189)
B B 1 !A 1 !A 2 + !2 2 + !2 ; r = B B !A !A 1 + !1 1 + !1
1
(192)
QED Summary of results Summary of results: a numerical example Homework:do this exercise when Above two example’s on Walrasian general equilibrium models are based on perfectly competitive market clearing assumptions. These have been widely extended in dynamics and with stochastic shocks in recent macroeconomic models. Stickiness of prices and wages and preference or technology parameters characterise many macro models to study the business cycle. Stochastic shock to technology (TFP) is the major element under the real business cycle (RBC) literature. This issue is taken up in lectures 5 to 7 next. 4.2.1
Ricardian equivalence Are households better o¤ if the government …nances budget de…cit now by borrowing or by raising taxes? David Ricardo, a British economist, wrote around 1820 that borrowing now or taxing more are equivalent in terms of welfare loss to households as equal amount of money is taken away from them in both regime. 83
Table 10: Summary of two period general equilibrium model: Analytical solutions Individual A Individual B A A B Endowments !1 ; !2 !B 1 ; !2 h A Bi ! +! 1 Equilibrium interest rate r = 1 !2A +!2B 1
Life time income
Consumption in period 1 Consumption in period 2 Saving/borrowing period 1 Saving/borrowing period 2 Life time utility Shadow price
!A 1
+
!A 2 1+r !A (1+r) 2 !A 1 + 1+r (1+ ) S1A = ! A C1A 1 C2A S2A = ! A 2 U (C1A ; C2A ) = ln C1A + ln C2A 1 = C1i = !A 1 2 1 !A 1 + 1+r (1+ ) 1 (1+ )
1
!A 2 1+r
!A 1 +
!B 1 +
1 (1+ ) (1+r) (1+ )
!B 2 1+r
!B 2 1+r !B 2 !B 1 + 1+r
!B 1 +
S1 = ! 1 C1 S2 = ! 2 C2 B U (C1 ; C2B ) = ln C1B + ln C2B 1 = !B 1 (1+ )
2 !B 1 + 1+r
Table 11: Parameters Individual A Individual B A A B Endowments ! 1 ; ! 2 = f50; 100g !B = f150; 200g 1 ; !2 Discount rate = 0:9 = 0:9 Households care their lifetime consumption and welfare. Impacts of more taxes now or borrowing to pay for de…cit by the government have no impact on intertemporal choice of households. Ricardian Equivalence Theorem: borrowing more from private sector now or taxing them more now have equivalent outcomes. Intertemporal budge constraint of households remains the same and people anticipate a rise in tax tomorrow and save to pay for it now if government borrows now to …nance the de…cit. 1. Consider a two period economy with the preferences of households given by U (C1 ; C2 ) = lnC1 + ln C2
(193)
endowments for period 1 and 2 are given by fw1 ; w2 g and the public policy is fG1 ; G2 ; T1 ; T2 ; Bg. 84
Table 12: Summary of two period general equilibrium model: Numerical example Individual A h Individual B i A +! B ! 1 = 0:667 Equilibrium interest rate r = 1 !A2 +!B2 1
1
!A 2 = 110 1+r !A 1 2 !A 1 + 1+r = 57:895 (1+ ) !A (1+r) 2 !A 1 + 1+r = 86:842 (1+ ) S1A = ! A C1A = 7:895 1 C2A = 13:158 S2A = ! A 2 A A A U (C A 1 ; C 2 ) = ln C1 + ln C2 = 8:076 1 = C1i = = 0:0017 A 1 A + !2 1 ! 1 (1+ ) 1+r
!A 1 +
Life time income
Consumption in period 1 Consumption in period 2 Saving/borrowing period 1 Saving/borrowing period 2 Life time utility Shadow price
!B 2 1+r !B 2 !B 1 + 1+r !B 2 !B 1 + 1+r
!B 1 +
1 (1+ ) (1+r) (1+ )
1 (1+ )
2 !B 1 + 1+r
(a) What is the equilibrium consumption in period 1 and 2,fC1 ; C2 g when r = 0:01; = 0:95; fw1 ; w2 g = f100; 150g ; fG1 ; G2 g = f20; 30g ; fT1 ; T2 g = f20; ?g : The budget need to be balanced intertemporally. C1 +
C2 1+r
(w1
T1 )
w2 T2 1+r
w2 T2 1 (w1 T1 ) + ; 1+ 1+r w2 T2 (1 + r) C2 = (w1 T1 ) + 1+ 1+r
(194)
C1 =
Here intertemporal endowment is w1 + G2 30 G1 + 1+r = 20 + 1:01 = 49:7.
85
w2 1+r
= 100 +
(195) 150 1:01
= 142:11 = 213:16
S1 = ! 1 C1 = 7:895 S2 = ! 2 C2 = 13:158 U = ln C1B + ln C2B = 9:782 1 = = 0:007 !B
Table 13: Parameters Individual A Individual B A B ; ! = f100; 50g = f200; 150g Endowments !A !B 2 1 1 ; !2 Discount rate = 0:9 = 0:9
L = lnC1 + ln C2 +
= 270
= 248:52 and
(b) Assume budget balance in each period 1 150 30 1 120 (100 20) + = 80 + = 101:95; 1 + 0:95 1 + 0:01 1 + 0:95 1:01 0:95 (1 + 0:01) 150 30 (100 20) + = 0:95 (1:01) 101:95 = 97:82 (196) C2 = 1 + 0:95 1 + 0:01
C1 =
if T1 = 0 T2 should be very high to meet the government expenses Borrowing will be 20 in the …rst period and zero in the second period. T2 = 20 (1:01) + 30 = 50:2. (c) Prove the Ricardian Equivalence. Comment what this implies to the expansionary …scal policy observed around the world in the current recession. Is this plausible? G1 is rising T1 is falling ; this means G2 has to fall and T2 has to rise. T2 G2 = T1 + 1+r This means G1 + 1+r 4.2.2
Ricardian equivalence and macro-micro impacts of …scal policy
Q1. Consider a two period economy in which preferences of households, endowments and government policy are given by the following set of equations. Preference: U (C1; C2 ) = ln C1 + ln C2
(197)
Endowments: fw1; w2 g Government policy: fG1; G2 ; T1; T2 ; Bg where C1 and C2 are consumption in period 1 and period 2; G1 and G2 are government expenditure in period 1 and period 2; w1 and w2 are endowments in period 1 and period 2; T1 and T2 are tax revenues in period 1 and period 2 and B is the borrowing by government in period 1. 1. Provide a graphical explanation of inter-temporal optimisation process for a representative household in this economy. Indicate choices between current and future consumption for this representative consumer who faces inter-temporal budget constraint using correctly labelled diagrams. 2. What is the role of parameter in the utility function and how would the point of inter-temporal choice that you showed in (a) change with a rise in the interest rate? 86
3. Construct inter-temporal budget constraints for a representative household and the government in real terms for this economy. Explain in simple words what do two sides of this budget constraint represent for the household and for the government? 4. Given government policy fG1; G2 ; T1; T2 ; Bg and endowments fw1; w2 g and preferenceU (C1; C2 ) = ln C1 + ln C2 , what are the conditions for optimal choices for the representative household? Solve for optimal choices available to this representative consumer in period 1 and period 2 in terms of endowment, interest rate and model parameters. 5. Find the equilibrium rate of interest and consumption choices and using the market clearing condition if fw1; w2 g = f100; 150g, fT1; T2 g=f20; 30g ,fG1; G2 g = f20; 30g, is 0.95 and there is no borrowing by the government, B =0. 6. Prove the Ricardian equivalence that states that the consumption choices of households are not a¤ected whether the government pursues a (i) tax only policy as in (e) or (ii) borrowing only policy as in (f) above. 7. How would your result in (d) change if the representative household faces a borrowing constraint such that she will not be able to borrow more than 10 in period 1? Hints: L = lnC1 + ln C2 +
C1 +
C2 1+r
(w1
T1 )
1 w2 T2 (w1 T1 ) + ; 1+ 1+r (1 + r) w2 T2 C2 = (w1 T1 ) + 1+ 1+r
w2 T2 1+r
(198)
C1 =
(199)
G1 is rising T1 is falling ; this means G2 has to fall and T2 has to rise. This means T2 G2 = T1 + 1+r G1 + 1+r
87
5
L5: Keynes-Hicks-Samuelson-Mundell-Fleming Models of Macroeconomic Fluctuations
Macroeconomic models have been in use for formulation of economic policies almost in every country in the world and for coordination of macroeconomic policies at regional and global levels. These models not only provide analytical frameworks to link the demand and supply sides and the resource allocation processes in an economy but also help in reducing ‡uctuations and in enhancing the economic growth, two major aspects of any economy. Classical, Keynesian, new classical (or real business cycle (RBC)) and new Keynesian approaches have evolved over time to analyse ‡uctuations of output, employment and price level over years (see Keynes (1936), Hicks (1937), Samuelson (1939), Phillips (1958), Friedman (1968), Phelps (1968), Tobin (1969), Barro and Gordon (1983), Sargent (1986), Goodhart (1989), Nickell (1990), IMF (1992)). Empirical validity of these models are tested using either macroeconometric simulation models, applied multisectoral general equilibrium models or by using stochastic dynamic general equilibrium models (Wallis (1989), MPC (1999), Pagan and Wickens (1989), Kydland and Prescott (1977)). There is a considerable controversy about the causes, consequences and remedies for the macroeconomic ‡uctuations in the short run in the literature (Mankiw (1988). New classical and new Keynesian models use rational expectations and market imperfections and frictions in the labour market or technological shocks in explaining these ‡uctuations. There is less controversy in the literature about the economic events in the long run despite plenty of work that has taken place in the area of endogenous and exogenous growth models (Solow (1956), Lucas (1988), Romer (1990), Mankiw, Romer and Weil (1992), Parente and Prescott (1993), Temple (1999)). Price system played a crucial role in the classical macroeconomic models. Real wages, that equate demand for labour to its supply, determines the level of employment. That then determines the level of output. Income is either spent on the current consumption or saved for the future consumption. The real sector equilibrium is guaranteed by equality between the saving and investment. The price level is proportional to supply of money. The monetary neutrality is maintained by perfectly ‡exible real prices. Unemployment or glut cannot happen in the classical system because of the perfect ‡exibility in prices. Aggregate demand always equals aggregate supply. The major objective of government, in the classical world, is to ensure law and order so that business enterprises could thrive. As such less intervention is considered better. Capital accumulation and saving, that mainly occurs through the private sector, drives the dynamics of economy in the classical system. More saving means more investment and larger amount of capital stock and higher output. 88
Rapid rate of economic growth during the industrial revolution through out the 19th century, except few interruptions, provided a strong support for the classical free market system. Such theories in‡uenced thoughts of policy makers from the time of Adam Smith (1776) up to Marshall or Pigou in 1920s ( and in Thatcher-Reagan area of liberal market in recent past). This was the period of industrial revolution and structural transformation and unprecedented improvement in production technology as well as in the living standards of the majority of people in the Western economies. The assumption of equality between aggregate demand and aggregate supply did not hold true in late 1920s. Factories were producing a lot more than they could sell. Firms laid o¤ employees. Individuals became pessimistic about their future prospects and started spending less. This further reduced the aggregate demand. Production capacity became under utilised. More than a quarter of working population became unemployed causing both social and economic problems. Ine¢ ciency of aggregate demand had far reaching consequences on lowering the level of employment and output. This is the starting point of the Keynesian macroeconomic analysis. Keynes showed how de…ciency in aggregate demand may continue for a long period if the government does not step in to solve the problem. In the national income identity, sum of consumption, investment, government spending and net exports, the aggregate demand, should equal aggregate supply. If the aggregate demand remains less than aggregate supply the productive capacity is under utilised. Keynes spent a signi…cant amount of time in explaining consumption and investment behavior of the economy. Values of multiplier and accelerator coe¢ cients were determined based on the key structural parameters such as the marginal propensity to consume, productivity of capital and the sensitivity of imports to the national income. While the ratios of consumption, investment, government expenditure and trade balance to the GDP provide broad indicators of resource constraints, the behavioral assumption behind each of these demand components give a good framework for analysing how …scal, monetary and exchange rate policies interact in determining the level of aggregate output, employment and savings and investment activities in an economy. Though the supply shocks, such as the rise in the oil prices in 1970s, gave rise to new classical and new Keynesian approaches with more focus in the supply side of the economy, the basic structure of Keynesian model are still very useful in policy analysis. These models are popular because they are simple and easy to understand. They can be used to compute the impact multipliers of various policy scenarios such as tax cuts, increase in spending, increase in money supply or increase in external demand or changes in the behavior of consumers and producers in an economy. Macro-econometric models aim to test macroeconomic theories with time series or cross section data on major macro-economic variables and forecast the future 89
course of the economy (Wallis (1989), MPC (1999), Pagan and Wickens (1989), Hendry (1995), Holly Weale (2000)). Once these models can mimic or replicate an actual economy then they are used for policy analysis. Structural parameters such as the marginal propensity of consume and imports, elasticities of investment to the interest rate or change in aggregate demand, elasticity of production to capital and labour inputs. When real parameters are known, then a model can be used for policy simulation. Policy makers have control over policy instruments such as the tax rate, or government spending or exchange rate or the interest rate. They like to know how the real output, employment and trade balances change when a certain policy measure or a set of policy measures are implemented such as the debt reduction programme or bene…t reform scenario or opening up the economy for international trade. Macro models provide a systematic framework to analyse these questions. These exogenous policy variables include model predictions as well as judgement and decisions of policy makers regarding taxes, money supply, exchange rate, division of resources across various sectors or between consumption and saving. Some of them are also linked to the manifestos of ruling political party. Further more they may include perceptions of people regarding in‡ation and expected wage rates and their labour supply behaviors. Traditional econometric models were based on Keynesian structural models as found in the Macro Modelling Bureaus. Partial or adaptive expectation mechanism or the non-micro foundations were heavily criticised in the form of the Lucas Critique (Sargent and Wallace (1975), Lucas (1976), King and Plosser (1984)). More e¤orts have taken place since then in modelling the supply side, dynamic optimisation, and incorporating …scal or monetary or exchange rate policy rules based on fundamentals of an economy. The applied general equilibrium models of an economy have more elaborate speci…cation about the price mechanism, consumption, production and the trade in an economy. These models use input-output tables that provide micro consistent data set on income, expenditure and demand side of the more decentralised economy (Shoven and Whalley (1984), Auerbach and Kotliko¤ (1987), Bhattarai (1999), Rutherford (1995), Perroni (1995), Bhattarai and Whalley (1999 and 2003), Kehoe, Srinivasan and Whalley (2005)). A calibrated applied general equilibrium model can reproduce the su¢ ciently decentralised benchmark economy as its solution and can act as a laboratory of economic policy analyses in which one can estimate impacts of various policy alternatives available to the policy makers. These models can be single country, multiple country or the global economy models which can be used to analyse the impacts of not only of domestic policies but also of the multiplier impacts of external events in the domestic economy. The general equilibrium impacts of tax, trade, labour market, …nancial sector policies, monetary and …scal policy measures 90
can be quite deep and penetrating when all sorts of chain reactions of policy actions are taken into account. These general equilibrium assessments aim to capture these impacts. Stochastic dynamic general equilibrium models are outcome of the research programme of new classical economists who oppose the interventionist idea of Keynes to contain economic ‡uctuations (Prescott 1986). These models claim that economies are always in equilibrium and ‡uctuations are outcome of technological shocks and optimising behavior of economic agents. Economic policies are ine¤ective in generating real impacts in an economy. The shocks to the production technology or government spending are outcome of random processes. Workers supply more hours when wage rates are high due to technological breakthrough and less hours when wage rates are low. The degree of response depends upon the inter-temporal substitution of labour supply. The model generated solutions are often used to analyse the underlying factors behind macroeconomic series by comparing their variance or covariances to actual time series. In…nite period economy is approximated by steady state characterisation of the …rst order conditions linking two consecutive periods. Macro economic models analyse factors responsible for long run growth and ‡uctuations around them over time. The growth in the factors of production and technology bring long run growth in output, shocks in technology, economic policies or con…dence of consumers and producers or in the trade and exchange rate system are often responsible for short run ‡uctuations.
91
Burning macroeconomic questions of 2015 to 2020 What are the causes of slow recovery in UK and in European countries? Are there challenges from emerging economies? how long will it take before OECD economies to gain a ful recovery from aftermath of shocks due to the …nancial crisis in 2008? How long will the zero lower bound in the basic interest rates, at 0.5% by the Bank of England and 0-0.25% by the Fed and to 1% by the ECB last? How e¤ective were the quantitative easing (QE) policies in …ghting recessions or slow growth rates? How e¤ective was the …scal stimulus in …ghting recession? How much in‡ation did it generate or when is in‡ation likely to rise? How do these expansionary monetary and …scal policies interact? What will happen to growth and the price level? 92
Are there any sensible exit strategies for these expansionary policies? What is the theoretical framework to analyse such business cycles? How have savings and investments been a¤ected by the credit crises? How relevant is the Keynesian demand side economics today? What alternatives are on the supply side? Summary of macro models Modern macroeconomics has four distinguishing features a) dynamic models b) competitive equilibrium c) micro-foundation d) rational expectation. It is important to have a clear historical perspective on this from the very beginning in order appreciate developments of policies and the economy. Classical view Ideas of Adam Smith (1776) in "An Inquiry into the Nature and Causes of the Wealth of Nations" Economy adjusts automatically towards its long run equilibrium if the price system is perfectly ‡exible and government policy is liberal. Division of labour creates productivity. Higher rates of saving and investment are key for capital accumulation and rising living standards and output. The invisible hand, price system, plays a crucial role in allocating resources. Ideas of Ricardo (1817), Say (1817), Malthus (1790) Mill (1844), Marx (1859), Marshall (1922) and Pigou (1918) Invisible hand sets prices to equate demand and supply in each market. No excess demand or no excess supply can persist. No glut or shortages in goods market. No unemployment or labour pressure in the labour market. Money is neutral (quantity theory of money). Prices proportional to money supply. It is a long run view. Balanced budget recommended. Free and open market economy is competitive in the global economy. 93
Laisser faire: minimum government is the best government. Downward sloping aggregate demand and vertical supply curve Keynesian Revolution (Short run analysis) Gaps between supply and demand may persist for a log time. Markets (prices) may not work automatically itself because of de…ciency in demand: massive unemployment of labour and under utilisation of capital is possible. Costs of waiting to return to the natural level; irresponsible to do so. Balancing budget is a stupid and dangerous policy. Active role by government can mitigate de…ciency in private demand (consumption and investment). Positive role of …scal policy and monetary policy. Multiplier e¤ect of demand on output Aggregate supply is horizontal in the short run. Animal spirits –importance of expectations. Consensus on the IS-LM models (Keynes-Hicks-Hansen) (1940-1970s) Exact speci…cation of relations among economic variables discussed in Keynes; Klein’s macro economic model, Macro Modelling Bureau, dynamic models of UK economy IS curve thought to be too steep: active …scal policy recommended Theory of consumption: Modigliani-(life cycle), Friedman –permanent income ; importance of expectations Theory of investment (Tobin’s q, user cost of capital) Phillips curve; Okun’s law, Trade-o¤ between unemployment and in‡ation; more empirical models adaptive expectations Theory of unemployment (worker mis-perception, ine¢ cient markets, employer mis-perception, 94
E¤ectiveness of monetary policy Mundel-Flemming open economy model Criticism of IS-LM and new macroeconomics Monetarist approach: Natural rate of unemployment of Milton Friedman (1968) vertical Phillips and aggregate supply curve Lucas critique (1976): rational expectations No money illusion possible, people are clever and process current information to know what the government is doing; government cannot fool people, when prices change –nominal wages change accordingly; no trade-o¤ between in‡ation and unemployment Stag‡ation –no growth and in‡ation simultaneously Revenue may decrease if tax rate increases (La¤er curve) Supply side policies recommended (subsidies, manpower, investment tax credit, technological development, e-commerce, low tari¤) IS-LM are ad-hoc models; need more Micro-foundation Real business cycle view: fully ‡exible prices Business cycle are optimal response not due to ‡uctuations in demand New Keynesian view: classical model may be right in the long run but the Keynesian hypothesis still valid because of contracts and staggering wages, menu costs: it takes time for prices to adjust More powerful mathematical tools applied to analyse these di¤erent claims Growth Theory: Supply Side Harrod-Domar model Solow-Swan Neo-Classical Model (assumes saving to be …xed) Sources of growth (labour, capital and technology) Balanced growth path of the economy (steady state) 95
Dynamically e¢ cient savings rate: golden rule Role of human capital in enhancing technology Optimal growth theory (Cass-Koopman) Endogenous growth models (Lucas-Romer) Forecasting macro time series (leading lagging and coincident indicators) Above theories need to be tested by the data. Macro data has coincident indicators (move together with the real GDP): GDP and GDP components ; leading indicators (moves before the real GDP): inventories, capacity utilisation, stock prices, real money balances; and lagging indicators: unemployment rate, in‡ation.
5.1
Classical macroeconomic model
Classical View: Free Market and Minimum Government (Ideas of Adam Smith (1776), Ricardo (1817), Say (1821), Malthus (1798), Mill (1873), Marshall (1925) Market is always in equilibrium: Demand = Supply both in goods and factor markets; No excess demand or no excess supply can persist. Perfectly ‡exible prices (invisible hands) make this happen. No glut or shortages in goods market. No unemployment or labour pressure in the labour market. It is long run view (growth model). Prices proportional to money supply. Money is neutral (quantity theory of money). Balanced budget recommended. Laisser faire: minimum government is the best government. Downward sloping aggregate demand and vertical supply curve. See http://socserv2.socsci.mcmaster.ca/~econ/ugcm/3ll3/ 96
5.1.1
Numerical example of a classical macroeconomic model
Assumption: perfectly competitive market with full employment (Hicks (1937)) Output (Y ) is function of labour (N ) : (200)
Y = F (N ) ): Labour demand is downward sloping function of the real wage rate ( W P W ) P Labour Supply is an opward sloping fundtion of the ( W ): P N = N(
W ) P Classical model assumes full employment:
(201)
(202)
L = N(
(203)
L=N
Money is super neutral; price level is directly proportional to money supply: (204)
M = mP Y Perfectly competitive capital market set interest rate (i) where: Saving relates positively to the interest rate (i)
(205)
S = S(i) More investment occurs with lower interest rate, (i) :
(206)
I = I(i) Capital market equilibrium implies
(207)
S=I Capital (Kt ) accumulation process for a given rate of depreciation ( ): Kt = (1
) Kt
1
+ It
97
0
0
"
(425)
Elasticity of labour demand to the real wage is thus: @Li Wi =P = @ (Wi =P ) Li
Y nA
"
(1
)A
"
(
mp Y nA
"
(1
Wi P
) )A
mp
"
Wi P
" 1
Wi =P
"
=
"
(426)
Thus higher mark up by …rms results in lower demand for labour. Hence lower employment and output. The expansionary monetary policy can still have any positive impacts when prices or nominal wages are sticky. See:
Blanchard O. and J. Galí (2013) Labor Markets and Monetary Policy: A New Keynesian Model with Unemployment American Economic Journal: Macroeconomics 2 (April 2010): 1-30 Smet F. and R. Wouters (2003) An estimated dynamic stochastic general equilibrium model of the Euro Area, Journal of European Economic Association, Sept, 1(5):1123-1175. See: website of Dynare programs a number of applications of DSGE models http://www.douglaslaxton.org/d
8
L8: Real business cycle model: TFP shocks
Two major features distinguish the real business cycle (RBC) model from the Keynesian models. First it assumes market clearing prices and general equilibrium in the economy in each period. Secondly the cyclical volatilities in macro variables are caused by technological (TFP) shocks in production. In addition most RBC models assume perfect foresight among economic agents. Technological shocks a¤ect productivity and income which trigger intertemporal and intratemporal substitutions (Prescott and Kydland(1982), Prescott(1986)) by households and …rms. Prices are ‡exible to clear markets in Walrassian way. Demand and supply in goods, factor and …nancial markets re‡ect optimising behaviour of households and …rms. Stochastic process of technology or public policy such as the government spending causes ‡uctuations of output employment and prices around the trend. More recent versions of RBC models includes non-Walrassian features, such as imperfect competition, externalities, asymmetric information, departure from rationality and failure of market to clear- while explaining economic ‡uctuations (Black (1995), Cooley (1995), Romer 160
(1996, p.151)). Technical innovation (Schumpeterian) leads to a productivity shock, investments become pro…table. Demand for investment goods rises along with output and savings. Economy slows down with negative shocks in productivity.
Real business cycle models aim to capture mean, variance and covariance of the actual GDP series by model generated series. These models calibrate the …rst and second moments of time series to analyse movements in procyclical, anticyclical and acyclical macro-variables. GDP and its components move together; increase in consumption or investment or exports or government expenditure raise GDP. This makes them procyclical. Anticyclical variables move in the opposite direction of the GDP such as in‡ation and unemployment rate; lower unemployment means a pressure for the upward movement in prices. Acyclical variables do not have any relation with the GDP. In order to identify which variables are pro, anti or a-cyclical, RBC analysts use correlation coe¢ cient between the GDP and concerned variable, X as:
y;x
T P
cov (Yt ; Xt ) p = rPt=0 =p var (Yt ) var (Xt ) Yt
Yt
Y Y
2
Xt rP
X Xt
X
2
(427)
a variable is pro-cyclical if y;x > 0; counter-cyclical if y;x < 0 and acyclical if y;x = 0; lagging procyclical if yt ;xt 1 > 0 leading procyclical if yt 1 ;xt > 0. See UK_demand_2015.xlsx Measurement of business cycles Standard deviation of growth rate of output, consumption, investment and hours worked C o rre la tio n s to …n d o u t c y c lic a l, p ro c y c lic a l a n d a ntic y c lic a l m a c ro va ria b le s
161
Table 19: Contemporaneous 2021:1 C C 1.000 G 0.985 I 0.994 X 0.959 M 0.952 Y 1.000
Correlations among Macro Variables in th UK 2009:1G
I
X
M
Y
1.000 0.982 0.952 0.946 0.987
1.000 0.937 1.000 0.928 0.994 1.000 0.995 0.958 0.950 1.000
Table 20: Percentage standard deviation of macro variables GDP Consumption Investment Hours worked % standard deviation 0.0097 0.0109 0.0279 0.0060 Relative % standard deviation 1 1.1251 2.8786 0.6142 See excel …les RBC-UK-Calculations.xls calculation of standard devition and leadlag correlations. Growth rate with a straight line trend imposes a constant slope (for any log trend ln yt ) yt yt 1 (428) ln yt ln yt 1 = g ' yt 1 Hodrick-Prescott …lter generates trend instead by minimises following function T X t=1
ln yt
ln yt
2 1
+
T X
ln yt
ln yt
2 1
(429)
t=1
= 1600 standard for quarterly series. Get trends and HP tred for a GDP variable and see the di¤erence. Table 21: Lag, contemporaneous and lead correlations among macro variables GDPt ; xt 1 GDP 0.9995 Consumption 0.9988 Investment 0.9955 Hours worked 0.6628 162
GDPt ; xt 1 0.9988 0.9860 0.6728
GDPt ; xt+1 0.9995 0.9986 0.9856 0.6832
Burns, A and W. Michell, (1946), “Measuring Business Cycles” NBER, New York.
8.1
What is a business cycle?
Economies tend to grow over time but in an uneven fashion. The aggregate output, employment, investment, consumption and net exports all grow in good times and economy is close to full utilisation of resources. Consumers, producers and government are optimistic about future course of the economy. Such con…dence propels economy forward. However, such optimism suddenly dies out when pessimistic outlook predominates in bad times. Downward swing continues until some con…dence is restored among investors and consumers. Business planners, economic forecasters and policy planners study the economic scene intensely to detect signals of future macroeconomic developments. They are especially concerned with turning points, those times when the economic cycle reaches a peak or a trough in the cycle of economic activity. To that end, they watch a number of variables that tend to move systematically with, or even anticipate the business cycle. Business cycles are usually studied using quarterly data, in order to reveal enough details within the time period. In theory these macro time series tend to ‡uctuate around their long-term trends. Popular discussions in the UK emphasise the short run ‡uctuations associated with business cycles, to the point that cycles can have signi…cant economic and political repercussions including bringing the governments down at home and a set of chains of upward or downward movement in the global economy. The classical methods of business cycle analysis pioneered by Burns and Michell (1946) were successful at uncovering a broad set of statistical regularities observed among hundreds of economic time series. This massive body of empirical work provides the dominant background to most academic and policy discussion of macroeconomic developments. The qualitative features of business cycle behaviour they described have become the empirical foundation for modern business theory and set the standards for measuring and de…ning the business cycle characteristics of economic data. Insights provided by their methods of business cycle analysis serve as the foundation for theoretical models of business cycle behaviour and provide a standard to evaluate those models. For example, early Keynesian models were judged to be quantitatively successful in part because of the Adelman (1959) …nding that these models produced arti…cial time series whose cyclical properties could not be distinguished from historical business cycle behaviour on the basis of results generated using Burns and Mitchell methods. In order to investigate and measure the characteristics of di¤erent series over the business cycle, it is necessary to de…ne the business cycle and to determine when 163
it occurs. Burns and Mitchell adopted the working de…nition proposed earlier by Mitchell: Business cycles are a type of ‡uctuations found in the aggregate economic activity of nations that organise their work mainly in business enterprises: A cycle consists of expansions occurring at about the same time in many economic activities, followed by similar general recessions, contractions, and revivals which merge into the expansion phase of the next cycle; this sequence of changes is recurrent but not periodic; in duration business cycles vary from more than one year to ten or twelve years; they are not divisible into shorter cycles of similar character with amplitudes approximating their own. Burns and Mitchell investigated the business cycle characteristics of economic time series by constructing reference cycle patterns of cyclical behaviour, which compactly summarise business cycle properties of the data. These reference cycle pattern are the major tool used to describe the cyclical behaviour of economic time series within their framework. The …rst step in constructing these reference cycle patterns is to delineate periods of economic expansion and contraction and determine the speci…c dates of business cycle peaks and troughs. This is complicated by the fact that business cycles are characterised by the co-movements of many economic time series which are not perfectly in phase. However, peaks and troughs in a majority of individual series tend to cluster together with some regularity. Reference business cycle peaks and troughs are selected at the dates of these clusters. As McCallum (1989) points out, the real business cycle approach to modelling business cycle behaviour has gained attention for both theoretical and quantitative reasons. Growing dissatisfaction with the theoretical monetary misperception models of Lucas (1972) and Barro (1976) led to greater emphasis being placed on alternative equilibrium type models which could account for business cycle ‡uctuations. Further, the real business cycle approach was indirectly supported by the empirical results of Sims (1980) and Nelson and Plosser (1982) which implied that nominal (monetary) shocks are incapable of generating typical business cycle behaviour in economic time series. However the most in‡uential and quantitative support for real business cycle models was presented by Kydland and Prescott (1982). Their work was the …rst to show explicitly that a real business cycle model, driven by exogenous shocks to technology, was capable of generating time series with statistical properties characteristic of post war U.S. business cycles. Variability in the model’s data as well as co-variability between output and other variables in the model appeared to “matchup” well with the corresponding summary measure for similar U.S. economic time series. Kydland and Prescott (1982) showed that one could account for two-thirds of 164
the U.S. economic ‡uctuations with a dynamic stochastic general equilibrium model from which nominal variables were totally absent, (Kydland and Zarazaga (1997)). They obtained this result using a variation of the same basic theoretical model economists had been using time and time again to study economic growth issues. What many economists found attractive about the Real Business Cycle (RBC) theory proposed by Kydland and Prescott was that for the …rst time, a business-cycle theory pointed to the possibility that the same analytical tools used to address economic growth issues could be used to address business-cycle questions as well. Beginning with Kydland and Prescott (1982), RBC theory initially concentrated on the closed economy and tried to construct a dynamic general equilibrium model, which mimics the business cycle features of the real economy. These models generate arti…cial data, such as output, consumption, investment and employment, from a model economy whose ‡uctuations are driven solely by technological shocks. Researchers then compare the co-movements and volatility of these simulated series with those present in actual data. Many proponents of RBC models tend to judge the success or failure of a model on how close these two measures calculated from arti…cial data are to their real counterparts. The real business cycle model is an extraordinarily bold conjecture in that it describes each stage of the business cycle – the troughs as well as the peaks as an equilibrium, Hartley et al. (1997). The real business cycle model does not present a descriptively realistic account of the economic process, but a highly stylised or idealised account. This is a common feature of many economic models, but real business cycle practitioners are bold in their conjecture that such models nevertheless provide useful quanti…cation of the actual economy. In particular, the real business cycle programme is part of a larger new classical economics, which is argued to provide satisfactory microfoundations for macroeconomics in a way that Keynesian models conspicuously failed to do (e.g. Lucas and Sargent. (1979)). The claims that new classical models in general, and real business cycle models in particular provide microfoundations that is largely based on their use of a representative agent who solve a single dynamic optimisation problem on behalf of all the consumers, workers, and …rms in the economy. Economic agents maximize their objective function (utility or pro…ts) subject to the resource and technology constrains and supply and demand in all markets are equal. 8.1.1
What are the facts about business cycles
Before the real business cycle models can be tested, we must know precisely what they are meant to explain. Following Prescott (1986), advocates of real business cycle models have de…ned the explanandum of business cycles. Business cycle theory 165
has traditionally tried to explain what causes output to fall and then rise again. To be sure when output declines one expects employment, income, and trade to decline as well. Nevertheless the central fact to be explained was believed to be the decline and the subsequent recovery and not the co-movements of the aggregate time series. Two important measures of business cycles are the co-movement of economic variables and their volatility. While volatility issues, such as the smoothness of consumption and the magnitude of investment ‡uctuations, have been continuously subject to examination within business cycle research, co-movement issues have not always been emphasised. For example, time series studies often concentrate on single variable, such as output, to characterise the business cycle –e.g Hamilton (1989) and Hess and Iwata (1997). An early de…nition by Burns and Mitchell (1946) highlighted that expansions occurred simultaneously across many economic activities, followed by similar general recessions, contractions and revivals, which then merge into the expansion phase of the next cycle. Fortunately, the early tradition of emphasising the co-movement of economic variables at business cycle frequencies has been revived in real business cycle (RBC) theory. Interest in the long run e¤ect of business cycles was revived recently as part of the interest in endogenous growth. While the long run growth was traditionally thought of as entirely driven by some unexplained trend of technical progress, endogenous growth theory explicitly takes into account the fact that technical progress itself has economic determinants and depends on the incentives to innovate, to acquire education and on the acquisition of knowledge as a by-product of economic activity. Since the basic claim is that the business cycle is potentially as much a product of the permanent as the transitory component, it will be apparent that the “policy” one needs to discuss is not just monetary and …scal settings but also actions designed to in‡uence the longer-term performance of the economy. From the perspective of this paper, there seem to be three things that policy might aim at in in‡uencing the cycle – the long-term growth rate, the degree of persistence of shocks, and the standard deviation of growth rates. The long-term growth rate is generally regarded as a phenomenon associated with growth in output and technical change. While growth in capital and labour is predictable, arrival of technology is stochastic. While increased R&D may be important for it, creating an environment that is conducive to the adoption of innovations, such as various micro-economic reform measures, is possibly just as important. A study of the factors that lead to an improvement in long-run growth potential is therefore a study of how to mitigate the classical cycle, although it would do nothing for the growth cycle. A second dimension along which policy may modify the cycle is through the 166
degree of persistence of shocks. Whether policy can in‡uence this depends a great deal upon the source of the persistence and it is here that one needs to be able to ‡esh out the parametric statistical model with some economic structure. If the RBC is correct, the persistence derives from the fact that a TFP shock remains for a long time. One way of categorising the post war literature on models of business cycles is into deterministic approach and stochastic approach, Muellbauer (1997). In the former category come simple multiplier/accelerator models of aggregate demand, typically ignoring the supply side, which generate second-order di¤erencing equations. Less convincing are the aggregate demand models with ‡oor and ceilings (Barnett et al., 1989 and Grandmont 1991, 1993). Real business cycle (RBC) models can be seen as the attempt to account for observed ‡uctuations without appeal to phenomena such as sticky wages or lagged responses, which characterise approaches emphasising the demand side. The process of verifying, sharpening or refuting the real-shock account of business cycles has generated a large body of theoretical and empirical research. This framework assumes that the macroeconomy usually obeys simple behavioural relationships but is occasionally disrupted by large “shocks,”which force it temporarily away from these relationships and into a recession. The behavioural relationships then guide the orderly recovery of the economy back to full employment, where the economy remains until another signi…cant shock upsets it. There are controversies about the methods used in empirical research on business cycles Quah (1997), Gregory and Smith (1997). There are criticisms of business cycle research made by econometricians and criticisms of econometricians made by business cycle researchers who use calibration and the heavy use of simulations to derive operating characteristics of models. These researchers argue that such practice allow one to focus on the important economic issues and makes transparent the economic signi…cance of particular failures of their models. An Empirical Model Though the research on business cycle, either in the traditional sense or in the multiplier accelerator settings or in the real business cycle tradition has gone much further, a very simple model based on IS-LM or AS-AD analysis is still more popular among the business cycle practitioners. A very simple business cycle model should be able to show the e¤ect of …scal policy (budget de…cit), movements in the real exchange rates and real interest rates on output. Beside these demand side e¤ects it should also explain formation of prices and its e¤ect on outputs in the supply side. Furthermore, some macroeconomists have advanced the idea that shocks to the supply side or “real” factors cause many, if not most of the ups and downs in the economy. This idea could be captured by shocks either on the demand 167
or the supply sides in this simple model. Thus this simple models takes account of the traditional macroeconomic notion that changes in the aggregate demand causes most of the ‡uctuations on the one hand and e¤ect of real disturbances through shocks on the other. While some shocks are amenable to policy corrections others are immediate and have no clear policy options before such shocks hit the economy. Our simple business cycle model consists of four equations that explain aggregate supply, aggregate demand, prices, and unemployment rate. We also incorporate four policy variables: budget de…cit, real exchange rate, real interest rate and core in‡ation. Thus this model is a useful framework to study the impacts of …scal, monetary, exchange rate policies and e¤ect of trade unions and labour market conditions in price setting process. A formal presentation of the model underlying the AD-AS framework with underlying IS and LM analysis is as following. We assume an autoregressive process for aggregate prices Pt = Pt 1 (1 +
t)
This implies that the real exchange rate to be t
=
E t Pt ; Pt
t
=
(1 + t )(1 + 1+ t
t)
where is the (expected and actual) rate of change of the exchange rate and is the foreign in‡ation assumed to be constant. Thus equilibrium value of the real exchange rate , is in‡uenced by the nominal as well as the domestic and foreign in‡ation. Given these relations between the past and current prices as well as the relation between domestic and foreign in‡ation in determining the competitiveness of the economy the aggregate demand consistent with goods market equilibrium given by the regular IS curve, under the Mundell Flemming assumptions is: Yt
Y = F Pt + (
t
t)
'( i
t)
+ s1t
(430)
where a bar above a variable denotes its long run value, ie Y . These long run trends are given by Yt =
0
+
1 tt
+ ut
, linearised around the long-run equilibrium. Here, F Pt is a …scal policy variable (budget de…cit), t t represents the exchange rate policy, and i t represents monetary policy. Finally the last term in s1t represents shocks in aggregate demand. 168
Thus …scal, monetary and exchange rate policies can in‡uence aggregate demand. Parameters , , and ' explain the relation of …scal, exchange rate and monetary policy on aggregate demand their expected signs are > 0 , < 0, and ' < 0. Thus this model summarises the e¤ects of internal and external balances on demand side of the economy in the short run. Analysis of supply side of the economy is di¤erent in the short and long run. Output is determined by the amount of capital and labour inputs as well as the level of technology in the long run. However, business cycle models emphasise short run factors in determining the supply side of the economy. The short run supply side factors are often represented by the deviation of general prices from the core underlying price structure. In this tradition a simple aggregate supply is given by the combination of in‡ation and output as explained by the following equation t
=
t
+ b1 (Yt
(431)
Y ) + s2t
where t = core in‡ation in t, Y = trend (equilibrium) level of output s2;t = one-o¤ supply shock (e.g. oil price increases, indirect tax increases). It is assumed that b1 > 0 and that s2;t represents a random shock with expected value equal to zero. Core in‡ation depends on wage bargaining along with increase in productivity and mark-up practices among the monopolistic …rms. In simple form it captures both past experience and future expectations. A simple price setting process that incorporates both backward and forward looking expectations can be summarised by the following equation: t
=
t+1
+ (1
)
t 1
where t = core in‡ation in t. The constant is restricted between zero and one so that 0 < < 1. Thus the core in‡ation is higher when the economy is overheating as given by (2) and lagged term t 1 in equation (3) and the expectation of future by the economic agents (particularly the employers and employees) in the economy is captured by t+1 . Our last equation relates to the relation of unemployment rate with the output gap on the one hand and di¤erence between foreign and domestic in‡ation on the other. It can be presented as following. :
Ut = Ut _
1 (Yt
Y t)
2( t
t)
+ s3t
(432)
Here UUtt is the change in the unemployment rate which depends upon the output gap Yt Y (in line with the Okun’s (1962)) and di¤erences between domestic and 169
foreign prices (Phillips (1958)), t t and shocks in the labour market, s3;t . We expect unemployment to be lower when the current output, Yt , is higher than the trend output, Y and vice-versa. Similarly, higher domestic prices relative to foreign prices due to an expansion in aggregate demand implies lower unemployment rate. Labour market shocks can either be positive or negative. Aggregate Demand The results for the aggregate demand is Y Yt = 0:01 0:06DEFt 0:09RRt + 0:03RIRt + 1:00Y Yt 1 t-ratio -0.31 -1.22 2.01** 39.65*** R2 = 0.94; F (4,113) = 463.1***; DW-stat. = 1.81 Aggregate Supply: The results for the aggregate supply is RP It = 0:05 + 1:00CIN Ft + 0:07Y Yt t-ratio 82.43*** 2.97*** 2 R = 0.98; F (2,116) = 3398.0***; DW-stat. = 1.06 unemployment rate The results for unemployment is U RAT Et = 7:98 0:87CY Yt 0:15DIN Ft t-ratio -11.4*** -2.57*** R2 = 0.53; F (2,116) = 65.24***; DW-stat. = 1.05 The results obtained using a simple model of aggregate supply/demand equation is plausible. For the aggregate demand YYt 1 term is included to account for serial correlation. The deviation of output is inversely related to the budget de…cit (DEF) and the deviation of real exchange rate from equilibrium (RR) and positively related to the real interest rate. The aggregate supply equation shows that in‡ation is positively related to core in‡ation and deviation of output from its equilibrium level. In the unemployment equation, U.K. unemployment is negatively related to the deviation of output from its equilibrium level and negatively related to the in‡ation di¤erential between the domestic and foreign in‡ation (DINF). High in‡ation will cause a reduction in output as in the stag‡ation period of the 1980s. In the 1990s the U.K. economy has been growing steadily with the lowest levels of unemployment recorded. VAR model of unemployment and in‡ation 170
Supply Shock: uit =
11 Luit
+
12 L it
+ e1;it
=
21 Luit
+
22 L it
+ e2;it
Demand Shock: it
These VAR equations can be estimated by any econometric software (see Bhattarai and Jones (2001)). 8.1.2
Real Business Cycle (RBC) models on macroeconomic ‡uctuations
Simple speci…cation of a real business cycle (RBC) model is similar to the perfect foresight model analysed in the Ramsey model but it includes stochastic technology to explain macro ‡uctuations of output and employment that is observed in real economy. These technological shocks a¤ect productivity and income and result in intertemporal and intratemporally substitutions (Prescott and Kydland (1982), Prescott (1986)) by households and …rms. Prices are ‡exible clear markets in Walrassian way. Demand and supply in goods, factor and …nancial markets re‡ect optimising behavior of households and …rms. Stochastic process of technology or public policy such as the government spending causes ‡uctuations of output employment and prices around the trend. More recent versions of RBC models includes nonWalrassian features, such as imperfect competition, externalities, asymmetric information, departure from rationality and failure of market to clear- while explaining economic ‡uctuations (Black (1995), Cooley (1995), Romer (1996, p.151)). Technical innovation (Shumpetarian) leads to a productivity shock, investments become pro…table. Demand for investment goods rises along with output and interest rates and savings. Economy slows down with slow down in productivity.
8.2
RBC model for a two period economy
A simple RBC model can be illustrated for a two period economy in which households choose consumption and leisure in period 1 and 2 to maximize utility: M ax U (C1 ; l1 ; C2 ; l2 ) = ln C1 + ln (1
L1 ) + e
[ln C2 + ln (1
>0 Subject to intertemporal budget constraint C1 +
w2 L2 C2 = w1 L1 + 1+r 1+r 171
L2 )]
(433)
The constrained optimisation function of household problem is:
L = ln C1 + ln (1
L1 )+e
[ln C2 + ln (1
L2 )]+
C1 +
C2 1+r
w1 L1
w2 L2 1+r
Demand for goods and factors are determined by constrained optimisation by households and …rms. 1 @L = + =0 @C1 C1 1 = C1 @L e 1 = = @C2 C2 1+r e (1 + r) = C2 @L = @l1 (1 =
L1 )
w1 (1
= w1
L1 )
@L e w2 = = @l2 (1 L2 ) 1+r =
@L C2 = C1 + @ 1+r Eliminating
e (1 + r) w2 (1 L2 )
w1 L1
w2 L2 l2 =0 1+r
term in the FOC with respect to C1 and C2 leads to C2 = e
and l2 (= 1
l1
(1 + r) C1
Similarly the demand for leisure from the FOC w.r.t, l1 (= 1 L1 ) L2 ) leads to 172
w1 e l1 (1 + r) w2
l2 = Using the value of = consumption relate as:
1 C1
and
=
w1 l1
the optimal demand for leisure and
C1 w1 It is now possible to solve explicitly for C1 by putting values of C2 L1 and L2 in intertemporal budget as: l1 =
C1 +
e
C1 +
C2 1+r
(1 + r) C1 1+r
w1
w2 l2 w2 L2 = w1 L1 + 1+r 1+r
w1 l1
w2 w1 e (1 + r) C1 w2 L2 = w1 L1 + 1+r w2 w1 1+r
C1 w1
C1 =
w1 L1 +
w2 L2 1+r
C2 =
w1 L1 +
w2 L2 1+r
l1 =
l2 =
w1
e w1
w1 L1 +
(1 + e
1 + +e
)
e (1 + e
(1 + r) + +e
)
w2 L2 1+r
w1 L1 +
w2 L2 1+r
e (1 + e e (1 + e
(1 + r) + +e (1 + r) + +e
)
)
There is no di¤erence between the perfect foresight and RBC models on the consumption side of the economy. Consumption and leisure demands in period 1 2 L2 and 2 depend upon income in the life period, w1 L1 + w1+r .
173
8.2.1
How are wages and interest rates determined?
Firms solve the pro…t maximization problem every period. t
= Yt
rt K t
wt Lt
Subject to technological constraints and stochastic process of production technology Yt = Kt (At Lt ) with 0
0g if the amount for debt servicing (rBt 1 ) is above current saving (Yt Ct ). 177
Bt
Bt
1
= Yt
Ct + rBt
(467)
1
Yt + Ct C t Yt Bt = + 1+r 1+r 1+r By successive iteration forward Bt
Bt =
1
Bt
=
Bt+1
Bt+1 =
Yt+1 + Ct+1 Ct+1 Yt+1 Bt+1 = + 1+r 1+r 1+r
Bt+2
Bt =
(468)
(469)
Yt+2 + Ct+2 Ct+2 Yt+2 Bt+2 = + 1+r 1+r 1+r
(470)
Bt+2 Ct+1 Yt+1 Ct+2 Yt+2 + + 2 1+r (1 + r) (1 + r)2
(471)
Ct+1 Yt+1 Ct+2 Yt+2 Bt+n + + :::: + 2 1+r (1 + r)n (1 + r) For the budget balance over the life time
(472)
Bt =
1 1 X X Ct+2 Yt+2 Bt+n Lim =0 i = B0 (1 + r) + i; ) t ! 1 (1 + r)n t=1 (1 + r) t=1 (1 + r)
(473)
The life time budget should balance, though the household is free to borrow and lend in the …nancial markets. The present value of expenditure should equal the present value of income (endowments). When productive …rms are included this life 1 P t+1 which time budget set is slightly modi…ed and includes value of the …rm V = (1+r)i t=1
is derived from the Tobin’s q.
1 1 X X Ct+2 Yt+2 Bt+n Lim =0 i = B0 (1 + r) + i +V;t ! 1 (1 + r)n t=1 (1 + r) t=1 (1 + r)
(474)
Similar logic applies for the governments debt dynamics: Dt Dt
1
=
Dt
Dt
1
= Tt
Gt + rDt
1
Tt + Gt Gt Tt Dt = + 1 + r 178 1+r 1+r
(475) (476)
Dt =
Gt+1 Tt+1 Dt+1 + ; 1+r 1+r
Gt+2 Tt+2 Dt+2 + 1+r 1+r
(477)
Gt+1 Tt+1 Gt+2 Tt+2 Dt+2 + + 2 1+r (1 + r) (1 + r)2
(478)
Dt+n Gt+1 Tt+1 Gt+2 Tt+2 + + :: + 2 1+r (1 + r)n (1 + r)
(479)
Dt = Dt =
Dt+1 =
1 1 X X Gt+2 Tt+2 Dt+n Lim = D (1 + r) + 0 n = 0 i i; t ! 1 (1 + r) (1 + r) (1 + r) t=1 t=1
(480)
Tobin’s q and investment For …rms investment decisions are guided by inter temporal optimisation. It depends on the ratio of market value to the cost of capital assets (Tobin’s q): 1
I1 = K1 No investment occurs if q
Lt (It ; Kt ; qt ) =
1 X
(q
1) ;
I1 = 0 if q
(481)
1
1. Here qt represents excess return on investment: t t
(1 + r) t=0
1 X
qt t [Kt+1 (1 + r) t=0
(1
) Kt
It ]
(482)
In multisectoral and multi-household model the inter-temporal balance conditions usually are explained as following: Households: prevent value of expenditure = present value of income T X N X
h Pi;t 1 + thci Ci;t =
t=0 i=1
T X
rt (1
tk ) Kth + Rth + wh Lh
(483)
t=0
Firms: Present value of revenue = present value of cost # " H T T X X X Pi;t Yi;t = rt (1 tk ) Ki;t + wth Lhi;t t=0
t=0
(484)
h=i
Government: present value of public spending = present value of revenue ! T T H X X X G= Gt = RVt + (485) Rth t=1 t=1 h=1 179
where RVt is the total tax revenue from direct and indirect taxes: T X N h X X h h RVt = Pi;t tci Ci;t + rt tk Kth + wh Lh t=0 i=1
(486)
h=1
Economy: present value of exports = present value of imports T X N T X N X X P Ei;t Ei;t = P Mi;t Mi;t t=0 i=1
(487)
t=0 i=1
For any period: current account de…cit (surplus) = capital in‡ow (out‡ow) N X i=1
P Ei;t Ei;t
N X P Mi;t Mi;t =
F
(488)
i=1
Exchange rate appreciation (depreciations) should follow trade surplus (de…cit) but not all countries follow this rule.
9
L9: Tax, Spending and Fiscal Policy
Common beliefs on e¢ cient tax and spending policies in UK " The core of our proposal is for a progressive, neutral tax system; that minimises economic distortions and is a right tool for achieving distributional objectives" "There are taxes that are fairer, less damaging, and simpler than those that we have now. To implement them will take a government, ....., willing to put long term strategy ahead of short term tactics". " .. the cost of not doing so are very large. .. Economic welfare could be improved by many billions of pounds if the taxation of income, expenditure, pro…ts, environmental externalities and saving were reformed..." (1) Mirrlees J., and S. Adam, T. Besley, R. Blundell, S. Bond, R. Chote, M. Gammie, P. Johnson, G. Myles, J. Poterba.(2010) Dimensions of Tax Design: the Mirrlees review, Oxford: Oxford University Press.; (2) Mirrlees J., and S. Adam, T. Besley, R. Blundell, S. Bond, R. Chote, M. Gammie, P. Johnson, G. Myles, J. Poterba. (2011) Tax by Design: the Mirrlees review, Oxford: Oxford University Press. "The Government’s strategy is designed to protect the economy through this period of global uncertainty, to maintain market con…dence in the UK and to lay the foundations for a stronger, more balanced economy in the future."- Budget, 2013; page 16. Economic and social justice: Rawls and Bentham?
Fiscal Policy Questions 180
1. What is the optimal Revenue/GDP ratio? 2. What is the optimal Spending/GDP ratio? 3. What is the optimal De…cit/GDP ratio? 4. What is the optimal Debt/GDP ratio? 5. What are the deadweight losses of taxes and which kind of tax is the most e¢ cient? 6. Which components of spending could be cut and which cannot? 7. What is the golden rule of …scal policy? 8. Whis is hurt most from the Austerity? 9. Why is the structure of public …nance so di¤erent across countries?
See http://www.ukpublicspending.co.uk/index.php; http://www.res.org.uk/view/lectureEduTraining.htm
9.0.5
Key tax rates and allowances in the UK No tax is paid in the …rst £ 11000 of individual income from April 2016 (£ 10,600 from 2017 and to be £ 12500 by 2020). Then basic tax rate of 20% is paid on additional income up to £ 32,000 ( total income of £ 42700; to be £ 43,000 by 2016). 181
40% rate from income £ 42,700 up to £ 150,000 (from 43,700 by 2017 and 50,000 by 2020). Top rate income tax of 45% is paid in all income above £ 150,000. National insurance contribution rate is 12% for every employee. Council tax rate vary by the value of property in A to H bands, A paying a two third and H paying the twice of band D which liable for amount £ 1332 of council tax for each year. VAT is 20 % on goods and services and corporation tax rate is 20% (19 % in 2017 (becoming 18% from 2020). Excise and business tax-subsidy rates vary by product, going up to 95 percent. 9.0.6
Who bears the burden of tax? Microeconomic Approach
In partial equilibrium analysis, it depends on elasticities of demand or supply. Consider linear demand and supply model D = 150
3P
S = 30 + 2P 182
(489) (490)
Equilibrium D =S implies P=24 and Q = 78. Now let there be a sales tax in this commodity so that consumers pay more P D and suppliers get P S it is less than P D because of the tax wedge. PD = PS + t
(491)
where t is tax imposed per unit. Let t = 2. D = 150
3P D = 150
3 PS + 2
(492)
Burden of Taxes in Partial Equilibrium Analysis (it depends on elasticities) D = 150
3 P S + 2 = S = 30 + 2P S
(493)
P S = 22:8; P D = P S + t = 24:8 Q = 75.6 Table 22: Impact of sales tax on equilibrium No tax case Tax case P 24 24.8 Q 78 75.6 Deadweight loss of taxes =loss of consumer surplus+loss of producer surplus = = 0.5(0.8 2:4) + 0:5 (1:2 2:4) = 0:96 + 1:44 = 2:4 183
P P Elasticity of demand : @Q = 3 24 = 0:92 Elasticity of supply: @Q = @P Q 78 @P Q 24 2 78 = 0:61. Thus more burden is taken by producers, as supply is less elastic. (see TAXBEN model of IFS; www.ifs.org.uk).
Impacts of Tax Reforms Hicksian Compensating and Equivalent Variations (changes in money metric utility): Table 23: Summary of Equivalent and Compensating Variation Fall in Price Rise in Price EV + CV + Need to consider income and substitution e¤ects in individual markets and take account of multiple rounds of knock on e¤ects to measure the impacts of tax changes more accurately. EV is measured in after tax prices and CV measured in before tax prices. A general equilibrium (GE) model is needed to evaluate tax reforms to capture all knock on impacts; see UK GE model in excel. G E im p a c ts o f su ch ta x e s c a n b e h ig h e r o r low e r th a n p a rtia l e q u ilib riu m im p a c ts.
C o m p u te a la rg e sc a le G E ta x m o d e l c a n b e
b u ilt w ith m ic ro -c o n siste nt d a ta fro m th e in p u t-o u tp u t ta b le s u sin g fu ll ve rsio n o f G A M S o r M P S G E o r M AT L A B ; (d e m o o f G A M S is d ow lo a d a b le fro m http :/ / w w w .g a m s.c o m c a n so lve sm a ll m o d e ls).
Impacts of Tax Reforms: Hicksian Compensating Variations 1
1
Base utility u = x12 x22 , with budget m = p1 x1 + p2 x2 if m = 100; (p1 ; p2 ) = (1; 1) demand x1 = 2pm1 ; x2 = 2pm2 = (50; 50) 1
1
1
1
u = x12 x22 = 50 2 50 2 = 50: Now there is a tax on good 1 and new prices are (p1 ; p2 ) = (2; 1) income does not change. new demand (x1 ; x2 ) = (25; 50). How much income need to be compensated to this consumer to maintain at the old level of utility? 1 2
1 2
u0 = x1 x2 =
m0 2p1
1 2
m0 2p2
1 2
p = 50: Here m0 = 2 2
50 =141.4
CV =141.4-100=41.4. Compensating variation is positive for a price rise is positive. Impacts of Tax Reforms: Hicksian Equivalent Variations 184
How much money should be taken away from the consumer in the original prices to make him/her achieve the utility level after the price change. 1
1
1
1
0
1
0
1
u0 = x12 x22 = (25) 2 (50) 2 = 35:35; m2 2 m2 2 = 35:35; =) m0 =70.7 EV = 70.7-100 = -29.3. Equivalent variation in negative for a rise in price level is negative. This consumer would have got 35.35 utility by paying 70.7 if prices were (1,1) as before. Macroeconomic Stabilisation role of …scal policy 9.0.7
Golden rule of …ne tuning: an example Output at full employment is 16; Tax rate is 0.4.
Government budget is balanced at full employment. (T=0.4(16)= 6.4 = G)) 1) Currently it is recession and economy is producing 12. How much should be the public budget de…cit? (T=0.4(12); G=6.4); budget de…cit required 6.4 - 4.8 = 1.6. 2) Economy is overheating and producing at 18. How much should be the budget surplus? (T=0.4(18)= 7.2; G =6.4;) Surplus = 7.2 - 6.4 = 0.8; 185
9.0.8
Revenue maximising tax rate: La¤er Curve
Let the revenue function be: R = 50t
2t2
(494)
Where R is revenue in billion of pounds, t is the tax rate. The tax rate that maximises the revenue is given by @R = 50 @t
4t = 0
(495)
t = 12:5 There are two tax rates that can raise the same revenue. Let 200 = 50t 2t2 t2 25t + 100 = 0 p 25 252 4 100 t1 ; t2 = = 5; 20 (496) 2 Reasons: tax evasion, tax avoidance. (higher rate of taxes creates hidden economy: tax heaven; smuggling,corruptions).
Reasons why a higher tax rate (tH ) raises lower revenue (R-low): tax evasion, tax avoidance (higher rate of taxes create a hidden economy: tax heavens; smuggling, corruptions; e.g search for tax heavens by multinational …rms, e.g.Straw Bucks, Banks). Mirrlees’(1971) Theory of Optimal Taxation Society has distribution of highly skilled people and non-skilled people. 186
There is an incentive compatible allocation in which highly productive people earn more and pay higher taxes. Highly productive individuals have material incentive to work hard even though their net of tax income may not be proportional to their labour. Incentive compatible consumption maximises the social welfare and tax system can be designed to obtain this. Impact of taxes in output and distribution of income Millions of working men and women in every country pay local and national taxes on their labour, capital or other incomes and in consumption. They receive public goods and services including health, education, unemployment insurance, pension and social security or income subsidy from national and local public institutions. Ratios of revenue and public spending to GDP vary enormously across these countries as are the generousness of the social security system or the rates of economic growth. Consider few relevant facts. The republic of Ireland grew impressively at 7.9 percent during 1994-2004 maintaining revenue and spending ratios just around 35 percent of GDP; South Korea had about 5 percent annual growth rate during that period with even smaller public sector of around 31 percent of its GDP. In contrast Japan grew only by 1.2 percent despite a large public sector de…cit which separated its revenue and spending ratios by a whopping 7 percent of it GDP (30.3 and 38.2 percents respectively). Sweden had about the same rate of growth of 2.8 percent as of UK, despite having about 17 percent higher revenue GDP ratio than of UK. In contrast, growth rate of Denmark was just 2.1 percent with the relative size of the public sector even larger than that of Sweden. Sources of revenue and sectors of public spending vary in their nature and magnitudes among then. About 59 percent of public spending was classi…ed as social spending for Germany but only 18 percent in Korea. Why are the sizes of public sector and growth rates so di¤erent among these countries? How far do the variations in the sizes of public sector explain variation in their growth rates? These are never ending questions and will remain as the main issues of public debate as long as economies exist with thier public and private sectors of the economy.
9.1
Net e¤ects of …scal policy Taxes distort prices: reduce labour supply, consumption, income, saving investment and growth. Public spending can generate positive externality by provision of public goods and public investment in infrastructure, redistribute income from rich to poor 187
and across generations. Net bene…t of …scal policy depends whether bene…ts from public spending are larger than costs from taxation. How to …nd this net e¤ect: – Compare social welfare after tax-transfer and spending and before them. What kind of social welfare function (W ) should be applied? It is function of utilities of household 1 to N as: W = W fU1 ; U2 ; :::UN g
(497)
Is there any optimal size of public debt? What are the consequences of in‡ationary taxes? 9.1.1
Progressive tax system for fairness: Mirrleeian idea on progressive taxation to achieve equity
Let R be the amount of revenue to be raised from taxes on income of the poor and rich people: R = t1 B (y1 ) + t2 B (y2 ) t1 is tax rate in for low income, B (y1 ) bene…t to poor from their low income ; t2 is tax rate for high income and B (y2 ) the bene…t for the rich from their income. Now consider revenue neutral tax reforms by setting tax rates appropriately. Take the total di¤erention of the above equation as: dR = t1 M B1 dy1 + t2 M B2 dy2 = 0 Let increase in income of low income person be equal to fall in income of high income (dy1 = dy2 ) and the marginal value of income for poor be higher than for the rich, M B1 > M B2 : t1 M B1 dy1 = t2 M B2 dy2 = 0; =)
M B2 t1 = < 1 =) t1 < t2 t2 M B1
This implies tax rate on income of poor (t1 ) should be less than than the tax rate in the income of rich (t2 ) : Tax system then could be progressive to achieve vertical and horizontal equity. 188
b u d g e t: http :/ / w w w .h m -tre a su ry.g ov .u k / ;
G re e n B u d g e t: http :/ / w w w .ifs.o rg .u k / ; http :/ / w w w .u k p u b lic sp e n d in g .c o .u k / in d e x .p h p
Consider a table for the numerical example how the tax transfers system can redistribution income and eliminate poverty (Bhattarai (2008)): Let Y be the original income. Lowest income person has 10 and richest person has 400. First consider the income gap being (y - y). Average income is 100. If this y is the target income then income gap (y - y) of the lowest income group is -90 and for the highest group is 300. This requires taxing second richest by 100 and the highest income group by 300 and redistributing it to cover income gap of each income group. This is very drastic income redistribution scheme. Here tax revenue (R) equals 400 and equals transfers to eliminate the gap, every one has equal income 100 after redistribution. Is this fair for the people who work hard to earn more? Less drastic measure is to use half of the average income 1 y as the basic target income for everyone. Then the income gap would be 2 (y - 12 y) as in the last column in the table. Only 100 need to be redistributed. Tax 100 to the richest person and transfer it to households in the bottom four deciles. No one remains in the poverty after this redistribution. Most countries in the world have this second type of the redistribution scheme in operation.
Table 24: Redistribution of income among households Deciles Y Pop (y - y) Redistribute for y Redistribute for 12 y H1 10 1 -90 100 (90) 50 (40) H2 20 1 -80 100(80) 50(30) H3 30 1 -70 100(70) 50(20) H4 40 1 -60 100(60) 50(10) H5 50 1 -50 100(50) 50 H6 60 1 -40 100(40) 60 H7 90 1 -10 100(10) 90 H8 100 1 0 100(0) 100 H9 200 1 100 100(-100) 200 H10 400 1 300 100(-300) 300 Gini 0.528 0 0.368
In real world such easy schemes are not practical; incentives are distorted. Countries di¤er in redistribution programmes. 189
Gini is the measure of inequality (area between the Lorez curve and equlity line): [0 < Gini < 1]. See Gini_Redistribution_2014.xls A general equilibrium model is required to evaluate such e¤ects for an economy (see uk16.gms; ptax.gms for UK)
9.1.2
Optimal size of the state
What should be the optimal size of public sector? How does it impact on the economy, particularly the economic growth?The …rst one is a political economic question that relates basically to the freedom of choice of individual citizens in these countries between private and public goods. From very ancient times states have been raising public funds to provide public goods. Tax rate was six percent of income even in ancient India as in Europe (Kautilya (300 BC)). Sizes of governments have increased as the responsibilities of states have risen out of proportions. Enough debates have taken place regarding the optimal size of the government (Pigou (1947), Samuelson (1954), Buchanan (1965), Atksinson and Stern (1974), Feldstein (1974), Whalley (1975), Boadway (1979), Summer (1980), Blomquest (1985), Bovenberg (1989), Benabou (2002) and Taveres (2004), Fullerton and Heutel (2007), Chen (2007). In more modern times classical or new classical economists favoured a smaller size of government that only focuses on providing pure public goods such as national defence and internal law and order. The Keynesian or new Keynesians implicitly have argued for larger economic roles for public sectors to stabilize economy from vagaries of market ‡uctuations. Optimal size of the public sector:
190
Private good Y
B A
C
Publ ic good, G Ut ili ty t o t he median vo ter
A’
C’
B’
G1
Go
G2 Public good, G
Utility, Disutility and Net Utility of Public Sector
Utility
Utility from spending
Go O Tax/spending
Net utility from public sector Disutility
Disutility from tax
There is extensive literature on Pareto optimality, Benthamian utilitarian analyses on social welfare, Arrows’impossibility theorem of equity and e¢ ciency by means of voting mechanism or the Rawalsonian principle of social justice judged from the 191
welfare of the lowest income person to Little-Mirrlees principles of social cost bene…t analyses. These entrust public authorities as guarantor of e¢ ciency in resource allocation and bringing reasonable amount of equity of income among citizens by means of tax and transfer mechanism. They recommend proper use of public funds in providing in kind bene…ts and other public goods. In its extreme version, in Marxists or communists thinking, state is at the forefront of economic management in which governments of proletariats takes control over almost every economic decision. State owns most of the assets and reaps their pro…ts, uses them in creating monolith infrastructure irrespective of demand of the consumers. In contrast, consumers are sovereign in the capitalist system where almost all productive activities are guided by invisible hands of market prices that provide enough signals to producers who supply various commodities that enter into consumption baskets of individuals.
Fig. 3 source: http://www.ukpublicspending.co.uk/index.php
Fig. 4 192
Only pure public goods are provided by the state. Despite this theoretical dichotomy both private and public sectors remain active in reality for providing commodities and services in almost all countries. Therefore a clear view on principles of optimal size of public sector, optimal taxation and public spending and factors are not only relevant for a major political parties contesting for power or running a government but also for economic and political thinkers who are active in theorising on optimal size of the government with su¢ cient degree of individual freedom. Table 25: Fiscal policy, growth and inequality 1950s 1960s 1970s Revenue/GDP 41.1 40.0 41.1 Spending/GDP 39.0 40.2 44.4 De…cit/GDP 2.0 -0.2 -3.3 Debt GDP ratio 145.0 89.6 49.9 Growth rate 2.5 3.1 2.4 Gini of original income 41.3 32.1 43.3 Gini of post tax income 35.4 25.1 28.6 In‡ation 4.2 3.6 13.6
in the UK: Recent 1980s 1990s 42.5 37.1 44.7 40.4 -2.2 -3.3 40.6 34.6 2.5 2.2 48.8 52.4 33.8 38.6 7.6 3.6
Trends 2000s 37.3 40.7 -3.4 34.2 1.7 51.7 38.3 2.5
Data source: ONS, OBR, IFS, and http://www.ukpublicspending.co.uk/index.php Gini for 1950 and 1960 rely on Stark (1972), Barna (1945), Nicholson (1964).
193
Table 26: Ratios of Revenue, Speding and De…cit to GDP (OBR) 2009
2010
2011
2013
2014
2015
2016
2017
2018
2019
2020
R e ve nu e / G D P
3 6 .5
3 6 .4
3 6 .5
3 5 .9
3 5 .6
3 5 .4
3 5 .8
3 6 .9
3 6 .9
3 6 .9
3 7 .1
S p e n d in g / G D P
4 5 .7
4 4 .9
4 3 .4
4 3 .0
4 0 .3
3 9 .7
3 9 .1
3 8 .1
3 7 .2
3 6 .5
3 6 .4
D e …c t/ G D P
1 0 .2
8 .6
7 .0
7 .1
5 .7
4 .9
3 .9
2 .5
1 .2
0 .2
-0 .5
D e b t/ G D P
6 2 .3
6 8 .8
7 2 .1
7 5 .8
7 8 .0
8 0 .0
8 2 .5
8 1 .7
7 9 .9
7 7 .3
7 4 .3
G row th ra te (re a l)
-4 .2
1 .5
2 .0
1 .2
2 .2
2 .9
2 .4
2 .4
2 .5
2 .4
2 .3
N o m in a l G D P
1504
1575
1629
1679
1755
1829
1903
1980
2065
2157
2251
D G D d e ‡a to r
9 3 .4
9 4 .8
9 6 .5
9 8 .6
100
1 0 1 .4
103
1 0 4 .9
107
1 0 9 .2
1 1 1 .6
so u rc e : E F O , O ¢ c e o f B u d g e t R e sp o n sib ility, N ov 2 0 1 5 .
The next question relates to the impact of public sector on economic growth. All kinds of taxes are distortionary on one or other side but they also create public goods and economic infrastructure required for the smooth functioning of the economy. Which one of these two e¤ects is stronger is not clear at the outset. Is the larger size of public sector necessarily harmful for economic growth? Do the bene…ts generated by public goods compensate enough for those distortions? What levels of public services generate enough infrastructures and maintain good incentives required for a healthy economy? How can one make collections of taxes and allocations of spending more e¤ectively? What are the criteria for e¢ cient amounts of surplus, de…cit or debt? Ideas of Harberger (1962), Uzawa (1962), Cass (1965), Atkinson (1971), Goulder and Summers (1989), King and Rebelo (1993), Perroni (1995), Cummins, Hasset and Hubbard (1996), Rust and Phelan (1997), Dhillon, Perroni and Scharf(1999), Wagsta¤ (1999), Caucutt, Imrohoroglu and Kumar (2006), Krueckner (2006), Di Tella and MacCullock (2006) and Mirrlees et al. (2011) have further illuminated on this debate.
194
Table 27: Net E¤ects of Tax and Transfer to an Average 2009 Bene…ts Taxes Cash In Kind Total Direct Indirect Bottom 6883 7555 14,438 -1195 -2965 2nd 8280 7252 15,535 -2200 -3466 3rd 6139 7088 13,227 -4850 -4459 4th 3949 6162 10,111 -8403 -5386 Top 1992 5123 7,115 -19500 -7441 Average 5448 6636 12,084 -7230 -4743
Household by Quintile in Net Total Gain or Loss -4,160 10,278 -5,666 9,866 -9,309 3,918 -13,789 -3,678 -26,941 -19,826 -11,973 111
Data source: O¢ ce of the National Statistics; in £ .
Summary of the e¤ects of taxes and bene…ts on ALL households, 2009/10 (Average per household, £ per year)
We can say that it is optimal to have a large public sector if there are more preferences for public good among citizens of a country. Very high presence of public goods seen in Scandinavian countries and Germany is indicative of preferences of households. Similarly countries with lower public sector, such as Mexico or Korea rely more on private sectors rather than in public sector for providing semi public goods. Equity, e¢ ciency, sanction and economy, were four major cannons of taxations that Adam Smith had taught more than two centuries ago. The ability to pay 195
Table 28: Share of origina and post tax income by quintile in UK, 2009 Original income share Post-tax income share Impacts of tax and transfers, % Bottom 2.46 6.75 4.29 2nd 6.92 11.33 4.41 3rd 15.04 15.92 0.88 4th 24.92 22.25 -2.67 Top 50.71 43.75 -6.96 Data source: O¢ ce of the National Statistics
principle and bene…ts to tax payers are modern major principles of public spending that Mirrlees (1971), Mirrlees et.al (2010) and Meade (1978) have propounded. These theories underpin the annual …scal programmes of governments around the world. To be more precise these age old principles are succinctly summarised in terms of major objectives of maintaining macroeconomic stability, reforming the …nancial services, supporting the business and growth, achieving fairness and providing opportunities for all, protecting the public services, supporting the low carbon growth. Listen to the Royal Economic Society Public Lecture 2011 by Robert Chote, the Director of the OBE at http://www.res.org.uk/view/res2011AnnualPublicLecture.html.
196
Table 29: Source of Revenue in UK (GBP Billion) 2009 S o u rc e s o f R e ve nu e
2010
2011
2012
2013
2015
R e ve nu e
P e rc e nt
R e ve nu e
P e rc e nt
R e ve nu e
P e rc e nt
R e ve nu e
P e rc e nt
R e ve nu e
P e rc e nt
R e ve nu e
P e rc e nt
146
0 .2 7
150
0 .2 7
158
0 .2 7
150
0 .2 7
155
0 .2 7
170
0 .2 5
N a tio n a l in su ra n c e
97
0 .1 8
99
0 .1 8
101
0 .1 7
99
0 .1 8
107
0 .1 7
115
0 .1 7
C o rp o ra tio n ta x
42
0 .0 8
43
0 .0 8
48
0 .0 8
43
0 .0 8
39
0 .0 6
47
0 .0 7
E x c ise ta x
46
0 .0 9
46
0 .0 8
46
0 .0 8
46
0 .0 8
47
0 .0 8
42
0 .0 6
VAT
78
0 .1 4
81
0 .1 5
100
0 .1 7
81
0 .1 5
103
0 .1 7
133
0 .2 0
B u sin e ss ta x
25
0 .0 5
25
0 .0 5
25
0 .0 4
25
0 .0 5
27
0 .0 4
28
0 .0 4
C o u n c il ta x
26
0 .0 5
25
0 .0 5
26
0 .0 4
25
0 .0 5
27
0 .0 4
28
0 .0 4
O th e r
81
0 .1 5
79
0 .1 4
85
0 .1 4
79
0 .1 4
107
0 .1 7
65
0 .1 0
541
1 .0 0
548
1 .0 0
589
1 .0 0
548
1 .0 0
612
1 .0 0
673
1 .0 0
In c o m e ta x
To ta l
S o u rc e : B u d g e t R e p o rt (M a rch 2 0 1 1 ) H M Tre a su ry, http :/ / w w w .h m -tre a su ry.g ov .
197
Table 30: Elements of Public Expenditure in UK (GBP Billion) 2009 E x p e n d itu re Ite m s S o c ia l p ro te c tio n P e rso n a l so c ia l se rv ic e s
2010
2011
2012
2013
2015
S p e n d in g
P e rc e nt
S p e n d in g
P e rc e nt
S p e n d in g
P e rc e nt
S p e n d in g
P e rc e nt
S p e n d in g
P e rc e nt
S p e n d in g
190
0 .2 8
194
0 .2 8
200
0 .2 8
194
0 .2 8
220
0 .3 1
231
29
0 .0 4
32
0 .0 4
32
0 .0 5
32
0 .0 4
31
0 .0 4
30
119
0 .1 8
122
0 .1 8
126
0 .1 8
122
0 .1 8
137
0 .1 9
141
E d u c a tio n
88
0 .1 3
89
0 .1 3
89
0 .1 3
89
0 .1 3
97
0 .1 3
99
Tra n sp o rt
23
0 .0 3
22
0 .0 3
23
0 .0 3
22
0 .0 3
21
0 .0 3
28
D e fe n c e
38
0 .0 5
40
0 .0 6
40
0 .0 6
40
0 .0 6
40
0 .0 6
45
In d u stry, A g r, E m p loy m e nt
21
0 .0 3
20
0 .0 3
20
0 .0 3
20
0 .0 3
19
0 .0 3
24
H o u sin g a n d E nv iro n m e nt
30
0 .0 4
27
0 .0 4
24
0 .0 3
27
0 .0 4
23
0 .0 2
28
P u b lic o rd e r a n d sa fe ty
36
0 .0 5
35
0 .0 4
33
0 .0 5
33
0 .0 5
33
0 .0 3
34
D e b t a n d inte re st
43
0 .0 6
44
0 .1 1
50
0 .0 7
50
0 .0 7
50
0 .0 7
36
O th e rs
74
0 .1 1
73
0 .1 0
74
0 .1 0
74
0 .1 0
74
0 .1 0
48
704
1 .0 0
696
1 .0 0
711
1 .0 0
696
1 .0 0
720
1 .0 0
742
H e a lth
To ta l
S o u rc e : B u d g e t R e p o rt (J u ly 2 0 1 5 ) H M Tre a su ry, http :/ / w w w .h m -tre a su ry.g ov .
There is a long literature in assessing who really pays taxes and who actually bene…ts from the public spending. As can be seen from the …gures the table below about 60 percent of total revenue comes from direct taxes (income tax, national insurance, corporate tax and council tax) and 40 percent from indirect taxes. Incidence and impacts of these taxes vary signi…cantly. Income and corporation taxes are more progressive, national insurance contribution, council taxes are proportional and VAT, business and other taxes are generally regressive. Around 60 percent of public spending is transfer of resources and remaining 40 percent provides for basic public goods that very essential for a viable economy. Thus it is important to consider both the revenue and spending sides simultaneously while considering the impact of …scal policy on welfare of households and in the e¢ cacy of allocations resources in the economy. It is well known that the burden of indirect taxes shifts from producers to consumers or to households. In theory the incidence and impact of the direct taxes should be in the same place, however, who bears the burden of taxes depends on more complicated economy-wide income and substitution e¤ects in the presence of these taxes. Initial guess of these burdens are measured in terms of, the total deadweight loss, loss of consumer and producer surpluses because of taxes. Obviously these measures depend on elasticities of demand and supply (Blundell 2010). 198
9.1.3
Objectives of the …scal policy Macroeconomic stabilisation
1. (a) Higher growth rate of output (b) Full employment (c) Stable prices: (d) low rate of in‡ation (e) stable interest and exchange rates Equity: horizontal and vertical (tax/transfer) E¢ ciency in resource allocation Provision of public goods Externality: max positive externality, minimum negative externality Controlling Market failure Public private partnership (education, health, R&D) Instruments of Fiscal Policy Taxes Direct: income, pro…t, wealth Indirect: VAT, tari¤, excise, business Subsidies: goods/services and for use of inputs Spending Public goods: defence, law / order, national parks Semi-public goods: education, health, Rn&D Debt Borrowing from the private sector : Crowding out From the central banks : in‡ation Theory of Ricardian equivalence
199
9.1.4
Maintaining macroeconomic stability
Traditional macroeconomic theory suggests expansionary policy during recession and contractionary policy in booming years. Con…dence of consumers and producers is very low during recession as they su¤er from dimmed expectation about future. As predicted in the Keynesian model elaborated by Hicks and Samuelson in the form of IS-LM and multiplier accelerator model low volumes of consumption and investment cause reduction in demand and excess capacity in the economy. Similarly MundellFlemming open economy model predicts inter-link among economies that are linked to each other through net exports and exchange system. In the 2008-09 recession the household consumption in UK declined by 3 percent, business investment reduced by 14.25 percent and net exports declined by 10.75 percent. When each of these three components of aggregate demand were low the government expanded its own consumption by 2 percent. When private sector spends less, produces and earns less the public revenue also shrinks. It is not wise to raise extra resources from taxes in the wake of very low expectation. Government has no alternative than borrowing to …nance extra spending. Such de…cit spending cannot continue forever as it would raise the debt GDP ratio and can make government weaker and ine¤ective in future. Therefore the current government adopted an debt reduction plan (DRP) that aims to reduce de…cit from the 11 percent of GDP to a surplus of 0.6 percent of by 2020-21. Success of DRP depends on how realisation of revenue and spending target that in turn depend on the projected growth rate of the economy that is mainly determined by millions of consumers and producers in the economy. It is important that the con…dence among them is restored as quickly as possible and the economy moves quickly on steady expansionary phase. 9.1.5
Major …scal issues of the current government
The current conservative government is continuing the debt reduction plan (DRP) that started with the its coalition with the Liberal Democrats in June 2010. It has put economic and national security as the topi priority. In a spending review of November 2015 it has forecasted its revenue and matched its spending plans that is consistent to the projection of economic growth rates as provided by the O¢ ce of Budget Responsibility. UK is projected to grow more than any other advanced economies between 2015 and 2020 as it has successfully mitigated the impacts the …nancial crisis and European debt crises. Debt/GDP ratio are projected to peak at 82.5 percent by 2016 and to decrease to 74.3 percent by 2020. The growth, fairness and reforms are the major objectives of …scal policy of the current government. Growth
200
1. broad based economy supporting private sector jobs 2. maintenance and investment in the transport system 3. world leadership in science and research 4. low carbon economy 5. sustainable footing for higher education 6. enhanced adult partnership Fairness 1. (a) Enhancing social mobility (b) Fairness premium (c) Promoting work and responsibility through universal credit system Reform 1. Decentralisation of power and devolution revolution with autonomy of mayors to use revenues from the business taxes 2. Transparency, e¢ ciency and accountability 3. Defence and security For the latest on UK budget see Autumn Statement 2015 from http://www.hmtreasury.gov.uk/ Targets and achievment of public services in recent years Protecting the Public Services – A &E treatment within 4 hours; hospital treatment within 18 months, refer to a cancer specialist within 2 weeks – Number of schools with less than 30 percent pupils achieving less than …ve good grades in GCSEs including English and mathematics has reduced from 1600 to 270 in the last 12 years. – 16,000 police o¢ cers in the street Supporting the Low Carbon Growth 201
Stern Review (2006) UK is leading in setting up global agenda for climate change and low carbon growth Obligations, initiation of carbon capture and storage (CCS) demonstration projects UK is committed to 34 percent reduction in emission by 2020 compared to 1990 levels and has pushed for global agreements through UN Copenhagen conference; Insure development aid to remain at least 0.7 % of GDP. 9.1.6
Trends of …scal policy in UK
S u m m a ry o f th e e ¤e c ts o f ta x e s a n d b e n e …ts o n A L L h o u se h o ld s, 2 0 0 9 / 1 0
All policy makers agree that de…cit/GDP and debt/GDP ratios should be kept at sustainable rate but there is some sensitive debate on whether austerity measures including the spending reductions across departments and cut back the tax credits are appropriate for the economy particularly to the vulnerable low income households.
Fig. 10 Source: George Osborne (Nove 2015) -http://www.hm-treasury.gov.uk/junebudget_diagrams.htm 202
and Alistair Darling (2009), Pre-Budget Report, December, HM Treasury, p. 12. http://www.hm-treasury.gov.uk/.; http://budgetresponsibility.independent.gov.uk/economicand-…scal-outlook-november-2011/
Fig. 9 Bhattarai (2011) Fiscal policy, growth and income distribution in UK available at http://www.aeaweb.org/aea/2012conference/program/preliminary.php. 9.1.7
Balanced budget multiplier: Lump-sum tax case
A change of 100 in both G and T also raises income by 100. Balanced change in G and T is not macro economically neutral. This can be shown as follows: Consumption function: C=
0
+
1
(Y
1
T0 )
(498)
T0 ) + I + G0
(499)
(Y
Macro balance: Y =
0
+
Solving for the national output: Y = Tax and spending multipliers:
0
1 T0
1 203
+ I + G0 1
(500)
@Y = @T0 1
1
@Y 1 = @G0 1
; 1
(501) 1
Net multiplier: @Y @Y + = @T0 @G0 1
1
+ 1
1 1
(502)
=1 1
Thus unit change in tax and public spending results in unit change in output. 9.1.8
Automatic stabiliser: proportional tax case
Consumption function: C=
0
+
1
(Y
T);
0
R Cost And Earning 18,000
Break Even Earning (R)
C 0 the central bank would choose lower rate of in‡ation than the government. Question 3: Now assume that both central bank and the government co-operate to each other as given expressed in equation (1024). t
yt
Mt = LCB + (1
) LG
(1024)
where is 1 if the central bank is fully independent and 0 if it is fully under the spell of the government and between 0 and 1 in the intermediate case. What will be the in‡ation target if the monetary policy decision is taken with due account of government and the central bank as in (1024), when 0 < < 1?
Mt =
1+ 2
Mt =
2 t
+
b (( 2
e t)
t
(1 + ) 1 + 2 2 Mt =
2 t
1+ 2
@Mt = (1 + @ t t
=
yt )2 +(1
+ ut
+ 2 t
)
t
b 1+
+
)
b b (1 + 2 2 b (( 2
+ [b (( yt + b375
t
)
e t)
t
e t)
1 2
2 t
((
t
b 1+
+b
b (( 2
t
e t)
+ ut
yt )2
+ ut
+ ut
+
yt )] = 0 ut
e t)
+ ut
yt )2
yt )2
Thus the in‡ation targets are low when both the central bank and the governments are committed to reduce the in‡ation, more so when the central bank is completely independent from the spell of the government (Eij¢ nger and Haan (2000)). Is this model relevant? Delegating the monetary authority to a conservative central bank produces lower in‡ation rates as shown above. This is the reason why the MPC of the Bank of England (BOE) has been made independent in determining the interest rate. The BOE predicts the expected in‡ation rate using a macroeconomic model for the economy (as shown by its popular “fan chart” ) and lowers the interest rate when demand slows down and raises it when prices are expected to rise. Same applies to the European Central Bank or the Federal Reserve Bank. Take a supply function given by t
=
e
+ (yt
y) + st
Where e is the expected in‡ation and s is the supply shock y is actual output and y is the trend output. Suppose that the supply shock s is 1, 0, -1 respectively. Question: How do responses of an independent or less independent central bank di¤er? A “Hard-nosed”central banks permits no in‡ation, “wet nosed”would accept no decrease in in‡ation at the cost of output. Importance of the credibility of the central bank and ‡exibility of wage rates. Question: A central bank is trusted to reduce in‡ation and it wants to reduce the current in‡ation rate to its half. How does the credibility of the central bank and the ‡exibility of wage rate a¤ect this outcome? How can price stability be achieved? Bind the central bank with a zero in‡ation rate target Appoint the most conservative central banker Make the central bank as independent as possible from the government Peg the exchange rate to the currency of a country with one or more of the above characteristics. 20.0.12
Optimisation Approach to Setting of the target in‡ation rate
Alternatively in‡ation can be set to minimize a social cost function that includes ‡uctuations in unemployment and in‡ation rate as in Barro and Gordon (1983, JPE 91:4:589-610) as following. 376
Phillips curve: Ut = Utn
(
t
e t) ;
(1025)
00
k
(1027)
1
Overall objective function given information set "1 # X Zt Zo = E I0 (1 + r)t t=0
(1028)
Private sectors expectation of in‡ation e t
= he (It 1 )
(1029)
here Ut is the actual unemployment rate, Utn is the natural rate of unemployment rate, U is the long run natural rate of unemployment, t is the actual in‡ation rate, e t is the expected in‡ation rate Zt is the social cost function, Z0 is the current value of the social cost function, I0 is the current information set, r is the interest rate, E is expectation operator, t is a kind of supply shock that a¤ects the unemployment rate. Parameters , a; b; , k have signs as restricted above. By substituting (2) and (5) in (1) )U + t ( t he (It 1 )) Ut = Utn 1 + (1 Now substitute (6) into the objective function (3) Zt = a
Utn 1 + (1
)U +
t
(
t
he (It 1 ))
kUtn
2
+ b ( t )2
(1030)
Again substituting term Utn and by rearranging this alters to 2 Zt = a (1 k) Utn 1 + (1 )U + t ( t he (It 1 )) + b ( t )2 Now policy makers like to minimise the social welfare function in (7) by choosing the optimal in‡ation rate @Zt = @ t
2a (1
k) Utn 1 + (1
)U + t 377
(
t
he (It 1 )) + 2b
t
= 0 (1031)
^t =
a b
(^ t
he (It 1 )) + (1
k)
Utn 1 + (1
)U +
t
(1032)
Under the rational expectation people expect in‡ation rate to be the same as determined by the policy makers ^ t = he (It 1 ) Therefore ^t =
a b
(1
k) ^t =
Utn 1 + (1 a (1 b
)U +
t
(1033)
k) Utn
However notice that policy makers have an incentive to cheat and may set in‡ation rate higher than people’s expectations once these expectations are made ^t
he (It 1 ) 6= 0
Barro R.J. and D. B. Gordon (1983) A Positive Theory of Monetary Policy in a Natural Rate Model, Journal of Political Economy, 91 4, 589-610. Alberto Alesina, Lawrence H. Summers (1993) Central Bank Independence and Macroeconomic Performance: Some Comparative Evidence Journal of Money, Credit and Banking, 25, 2. (May, 1993), 151-162. Alex Cukierman (1994) Policy Forum: The Banking System and Monetary Control Central Bank Independence and Monetary Control The Economic Journal, 104, 427. November, 1437-1448 Bank of England (www.bankofengland.co.uk) The Transmission Mechanism of Monetary Policy. Barro R.J. and D. B. Gordon (1983) Rules, Discretion and Reputation in a Model of Monetary Policy, Journal of Monetary Economics, 12: 101-121, North-Holland. Eij¢ nger SCW and J.D. Haan (2000) European Monetary and Fiscal Policy,Oxford University Press. 378
Gartner Manfred, Macroeconomics, Prentice Hall, 2003, chapter 13. Goodhard Charles AE (1994) Game Theory for Central Bankers: A Report to the Governor of the Bank of England, Journal of Economic Literature, March 101-115. Goodhart C.E.A. (1994) What should central banks do? What should be their macroeconomic objective and operations?, Economic Journal, 104, November, 1424-1436. Gregorio Jose De (1995) Policy Accommodation and Gradual Stabilisations, Journal of Money, Credit and Banking, 27, 3. August, 727-741. HM Treasury (2002) “UK Model of Central Bank Independence: An Assessment”in Reforming Britain’s Economic and Financial Policy, Palgrave 85-109. http://www.fsa.gov.uk/ Kydland F.E and E.C. Prescott (1977) Rules rather than discretion: the Inconsistency of Optimal Plans, Journal of Political Economy, 85:3: 473-491. Marcus Miller, Mark Salmon (1985) Dynamic Games and the Time Inconsistency of Optimal Policy in Open Economies The Economic Journal, 95, 124-137. Taylor J (1993) Discretion versus policy rules in practice, Carnegie Rochester Conference Series on Public Policy 29 Amsterdam. John B. Taylor (1995) Symposia: The Monetary Transmission Mechanism : An Empirical Framework The Journal of Economic Perspectives, 9, 4. Autumn, 11-26. 20.0.13
Model for Fiscal and Monetary Policy Co-ordination
The major objective of both the …scal and monetary authorities is to achieve economic stability with low rates of in‡ation and unemployment. Fiscal authority uses counter cyclical tax and spending instruments and cares more for lower unemployment rather than lower rate of in‡ation. It declares …scal de…cit when revenues are not enough to meet spending and surpluses arise when spending are lower than revenue. Normally de…cits raise aggregate demand and are expansionary and lead to higher interest rate as well as higher level of prices. Budget surpluses are contractionary and lower the aggregate demand, the interest rate and prices. Fiscal authority 379
alters budget surplus or de…cit in order to achieve employment and in‡ation objectives. Though the monetary authority also prefers lower rate of unemployment and in‡ation it more prefers lower in‡ation rate rather than low unemployment. Instruments of monetary authority include the interest rate, money supply or the exchange rates. It lowers the interest rate in recession and raises in case of in‡ationary pressure on prices. Would it be better to the economy if the …scal or monetary authority decides its own policies independently without any regards policy choice of another authority or would it be better if they cooperate and consult each other while making policy decisions? What would be the value of in‡ation and unemployment rates, output and interest rate when they do not cooperate to each other or when they cooperate and consult each other? Research works of Mundell (1962), Niehans (1968) Krugman (1979), Barro and Gordon (1983), Miller and Salmon (1990), Nordhaus (1994), Corder and Weale (2012), Weale (2014) have proposed models that can be used to investigate these issues. Here we base discussion using the Nordhaus (1994) model for …scal and monetary policy games. This model can be illustrated by means of the above diagram. The FF curve is an iso-aggregate demand curve in space of …scal and monetary policy instruments, the budget de…cit in the vertical axis represents the …scal policy instrument and interest rate on the horizontal axis represents monetary policy instrument. North-eastern side re‡ects lower aggregate demand than other points in this diagram. The point FB is the optimal choice of budget surplus and the interest rate, the bliss point, for the …scal authority. The circle around this point represents a 380
policy indi¤erence curve of the …scal authority that shows the combination of budget surplus and the interest rate consistent with a given targets of surplus and interest rate that aim to moderate the aggregate demand. Fiscal authority wants higher level of aggregate demand, and used expansionary …scal policy, therefore higher level of de…cit. Given …scal authorities choice monetary authority takes a decision regarding the interest rate. At optimal point, given everything else, …scal authority prefers slight budget de…cit at the bliss point. Higher budget de…cit raises aggregate demand and raises the interest rate crowding out some private investment. FF is downward sloping to show higher interest rates with rising level of de…cit. The MM curve shows iso aggregate demand curve consistent to budget surplus and the interest rate from the point of the monetary authority. MB represents the optimal choice of the interest rate and the …scal de…cit, the bliss point, from the point of the monetary authority. The circle around the MB point shows indi¤erence curve of the monetary authority. The monetary authority likes to achieve lower in‡ation rate with a surplus budget and an interest rate higher than that chosen by the …scal authority. It is more concerned with lower in‡ation rate and so adopts a higher interest rate even if it may raise the unemployment rate. If the …scal authority chooses higher the budget de…cit it responds with a higher the rate of interest to neutralise the in‡ationary impacts of expansionary …scal policy. Given this set up of the game there are three possible ways of the monetary and …scal policy game. 1. When …scal and monetary authorities operate independently disregarding each other. In this case they tend to choose their own bliss points. 2. When they play a non-co-operative game with each other. The Nash equilibrium is achieved at point N in this case. 3. When they co-operate each other the actual choice of the budget surplus and the interest rate lies in the contract curve between FB and MB can be chosen. This is Pareto dominating strategy than both of above cases and generates the economic stability with low in‡ation and low in‡ation rate. Consider a Nordhaus (1994) model on …scal and monetary policy co-ordination in which the …scal and monetary authorities have di¤erent policy preferences. U F = V F (u; p; g; S) where U F and U M are social welfare (utility) functions of the …scal and monetary authorities respectively, u is the actual unemployment rate, p is the in‡ation rate 381
and S is the actual government budget de…cit or surplus. U M = V M (u; p; g) (2) More speci…cally consider a quadratic social welfare function of the following form. UF =
u
uF
2
F
p
pF
M
p
pM
2
F
S
SF
2
(3) UM =
u
uM
2
2
M
SM
S
2
(4) SF and SM represent the budget de…cit targets for the …scal and monetary authorities, uF and uM represent the target unemployment rate of the …scal and monetary authorities. Similarly pF and pM are the target in‡ation rates of these two authorities. The terms F and M and F and M represent weights that these authorities attach to the in‡ation and the budget de…cit. Actual unemployment rate and price levels are functions of …scal and monetary policies. Unemployment response function: u=
sS
+
rr
(5) In‡ation response function: p=
u+k =
sS
rr
+k
(6) where S and r are multipliers with respect to the budget de…cit and the interest rates. The term is the coe¢ cient of the Phillips curve, k is a constant. Fiscal authority prefers the budget surplus or de…cit as a policy tool and the monetary authority prefers it to be the interest rate. Fiscal authority chooses the surplus or de…cit and the monetary authority sets the interest rate. Maximising (3) by the …scal authority wrt S yields @U F = 2 u uF @S Dividing both sides by 2 S
S
+2
F
p 382
pF
S
2
F
S
SF = 0
F F
uF
u
SF
S
pF
p
=0
S
(7) Similarly maximising (4) by the monetary authority wrt r yields @U M = @r
uM
2 u
M
+2
r
pM
p
r
=0
(unknown char) M
uM
u
pM
p
=0
(8) From (6) it is obvious that p
pF =
u + k + uF
0
k =
uF
u
Using this information in (7) and solving for the actual unemployment rate " # F F S S 0 u = uF + F uF 2 + = 1+ F 2 S
sS
+
rr
"
= uF +
F
uF
F
0
2
+
SF
S S
#
= 1+
(9) For the monetary authority sS +
For simplicity assume that
h r = uM + r s
M
uM
F
2
M
2
1+ 1
For monetary authority
2
= 1 and
1
S=
0
1+ h F r + uF + r
F
= 1+
=
F
=
M
uF
383
i
0
2
F
S
M
2
SF
i
F
2
S=
rr +
M
h uM +
M
uM
0
2
i
The policy reaction function for the …scal authority is given by S=
1+
rr F
F F
+
F
1+
F
h
uF +
F
uF
0
(10) The reaction function for the monetary authority is h 0 M uM + M uM S= r (S) + r
2
2
(11) The slopes of these reaction functions are: @S F = @r
1+
r F
+
F
SF
i
i
F
and @S = @rM
r
(12) Here S S F when all parameters are the same for both the …scal and monetary authority except the fact that uM > uF , these reaction functions simplify to h i F F F0 2 F SF = r (S) + u + u r (12)
S=
r r (S)
+
M
uM +
M
uM 02
(13) The budget surplus along the monetary reaction function in equation (13) evaluated at the same level of r is higher by the amount M
uM
uF
,which implies that the monetary bliss point is above the …scal bliss point as long as the optimal …scal surplus desired by the monetary authority is higher than the optimal surplus desired by the …scal authority. Monetary authority has lower desired 384
interest rate than the …scal authority. The equations (12) and (13) also can be used to show that the …scal and monetary authorities have higher target in‡ation rates than the monetary authority. Central banks’ …scal preferences do not a¤ect the central banks’reaction functions. 20.0.14
Is this model applicable in the UK?
Nordhaus model is useful in explaining the cooperation between the Treasury and the Bank of England in the UK. High in‡ation and higher unemployment in 1970s and 1980s were direct result of non-cooperation between monetary and …scal authorities. Tight monetary policies were used with rising …scal de…cits and prices. This resulted in a non-cooperative solution as shown by the diagram. After the independence in the bank of England, there is more co-operation between …scal and monetary authorities with a pleasant result of low in‡ation, higher employment and lower interest rates. Nordhause model also was applicable in analysing the impacts of de…cit reduction programme under the Clinton adiministration in the US and post-uni…cation …scal monetary policy mix in Germany. 20.0.15
Blake-Weale (1998) Model of Fiscal and Monetary Policy Coordination
More analysis of policy uncertainty is found in Corden and Weale (2012) and Weale (2014).
385
386
20.0.16
Tinberginian instrument-target analysis of policy choices
Internal balance refers to equality between aggregate supply and demand. Internal balance line is downward sloping because in‡ationary reduction in government surplus need to be accompanied by a tight interest rate policy to achieve stability (see Shaw, McCrostie and Greenaway (2001)). The external balance is downward sloping because a reduction in the budget de…cit releases some resources for imports, BOP deteriorates. More capital in‡ow can be induced only with the higher interest rate. Points Internal balance External balance Budget Surplus Change in i 1 In‡ation De…cit + + 2 Unemployment De…cit + 3 Unemployment Surplus 4 Unemployment Surplus 5 Unemployment Surplus 6 In‡ation Surplus + 7 In‡ation De…cit + + 8 In‡ation De…cit + +
References [1] Barro R.J. and D. B. Gordon (1983) A Positive Theory of Monetary Policy in a Natural Rate Model, Journal of Political Economy, 91 4, 589-610. [2] Blake A. P., M. Weale (1998) Costs of Separating Budgetary Policy from Control 387
of In‡ation: A Neglected Aspect of Central Bank Independence, Oxford Economic Papers, 50, 3, 449-467 [3] Corder M and M. Weale (2012) Uncertain Uncertainty, British Actuarial Journal, 17, part 3, 542–561. [4] Chrystal K. A. and Simon Price (1994) Controversies in Macroeconomics, Harvester Wheatsheaf, chapter 6. [5] Krugman Paul (1979) A Model of Balance of Payment Crisis, Journal of Money Credit and Banking, 11,Aug. [6] HM Treasury (2002) Reforming Britain’s Economic and Financial Policy, Palgrave. [7] Miller, Marcus; Salmon, Mark When Does Coordination Pay? Journal of Economic Dynamics and Control, July-Oct. 1990, v. 14, iss. 3-4, 553-69 [8] Mundell R. A (1962) Capital mobility and stabilisation policy under …xed and ‡exible exchange rates, Canadian Journal of Economic and Political Science, 29, 475-85. [9] John Nash (1953), Two-Person Cooperative Games Econometrica, 21, 1.Jan., 128-140. [10] Jurg Niehans (1968) Monetary and Fiscal Policies in Open Economies under Fixed Exchange Rates: An Optimizing Approach The Journal of Political Economy, 76, 4, Part2, 1967. (Jul. - Aug., 1968), 893-920. [11] Nordhaus W.D. (1995) Policy Games: Co-ordination and placeCityIndependence in Monetary and Fiscal Policeis, Brookings Papers on Economic Activity 2:1994: 139-216. 388
[12] Shaw G.K., M. J. McCrostie and D. Greenaway (2001) Macroeconomics: Theory and Policy in the UK, Blackwell. [13] Petit Maria Luisa (1989) Fiscal and Monetary Policy Co-Ordination: A Di¤erential Game Approach Journal of Applied Econometrics, 4, 2. (Apr. - Jun., 1989), 161-179. [14] Rogo¤ K. (1985) Can International Monetary Policy Cooperation Be Counterproductive? Journal of International Economics, 18 199-217, placeNorth Holland. [15] Weale M (2014) What to do when we don’t know: policy-making when spare capacity is uncertain,JSG Wilson Lecture, Hull, 15 October 2014 http://www.bankofengland.co.uk/pra/Pages/default.aspx
21
L19: International Macroeconomic Policy Co-ordination
The global economy consists of many interdependent individual and regional economies. Expansionary or contractionary monetary and …scal policies in one country most often a¤ects economic events signi…cantly in another economies. For instance expansionary …scal or monetary policies or higher growth rate in the UK economy does only a¤ect the prices and level of investment and output only in the UK but also in economies of France and Germany and other European economies which are her major trading partners. Similarly policies in the Europe and the US as well many other economies in Asia, Middle East andLatin America can have signi…cant impacts on the UK economy. Stability and prosperity as well as economic shocks and crises, any bigger economies, have wide-ranging impacts across all other economies around the globe. It is important to consider both domestic as well as international aspects of opportunities and constraints while setting an economic policy. Here we analyse the gains from co-operation and losses from non-co-operation using four di¤erent tools for international macroeconomic policy making (1) MundellFleming two country global economy model (2) Policy co-ordination diagram (3) Canzoneri and Gray (1985) (4) More recent micro-founded computable general equilibrium model of interenational monetary policy co-ordination.
21.1
Mundell-Fleming Two Country Model of the Global Economy
Assuming a …xed supply of money as in the gold standards or the Bretton Woods or the monetary union such as in the European Monetary union the policy of one country can spill over to another country as shown in the following diagrams in (y, i) space. 389
Here start with a global equilibrium in which the interest rates are same across two econoies, i1=i1* and output in country 1 is y0 and that in the country 2 is y20. Then suppose that the country 1 adopts an expansionary …scal policy shifting its IS curve to the right giving rise to the interest rate and output at c. It has two di¤erent impacts in country 2. First it can export more to country 1 and secondly increase in interest rate in country 1 induces out‡ow of capital from country 2 to country 1. Thus the IS curve of country 2 shifts to the right and LM curve shifts to the left because of decrease in the money supply due to a loss of reserves. New global interest will be set when all impacts of initial expansionary …scal policy has been worked through and the country is attains equilibrium at d and country 2 at e. These impacts can be summarised brie‡y by derivatives of domestic and foreign > 0; output and reserves with respect of change in the government spending as @Y @G @Y @R @R > 0; > 0; < 0. @G @G @G The impact of country one policy in country 2 may not be always positive as shown above. In fact Japan was in recession through out 1990s and blamed the expansionary …scal policy in the US, which caused an appreciation of Yen to be the major reason for its economic problems. Similar arguments can be heard between the EU and theUS. Many times an interest cut by the Federal Reserves is followed by similar cut by the ECB or the Bank of England or vice versa, or steel tari¤. Let us apply a multi-country multiplier and accelerator model opening up the Samuelson (1939) model for international trade shocks. Macro balance of country i is given by: 390
(1034)
Yi;t = Ci;t + Ii;t + N Xi;t + Gi;t Consumption function for this country is Ci;t =
i Yi;t 1 ;
0
0
Stackleberg solution
Use (7) into (4) and substitute the resulting equation in the utility function (3) and take a derivative wrt g, domestic monetary policy and solve for it. x=
x=
x=
1
( +
1
g+
2
)
1
u=
1
g+ (
2
g
(
2
2
2
g +
g+ +
g+ +
( x)2 ( 398
3 2 1 0
3 2 1
3
q
q )
q )+ + g)2
+ 3
3
q
q( +
1
2
)
u=
( +
@u = @g ( 0 + g) = 0
1
2
)
1
( +
g=
g =
1 3
q [( +
g
1
2
(
)
2
g+ +
g
1
(
3 2 1
2
q )+
g+ +
q ( + 21 ( + 21 ) +
1 3
+ 21 ( +
2 1) (
3 2 1
3
q )+
1 2) ( 2 1( +
1 2) ( 2 2 1) + 1 (
q( +
2
2
)
(
q( +
3
2 1
+
+ +
1
2 2)
2 1
2 2)
2 1 2 1
2 2) 2 2)
1
2
)
0
+ g)2
[( +
1
2
)
1
+ +
>0 2 1 2
(
1
2 )]
>0
Under the …xed exchange rate rule monetary policy is the same at home and abroad g= g Substituting this condition the output function in x=
1
x=(
g+
2
g +
+
2)
g+
1
q
3
3
q
. Substitute this in the utility function and maximise wrt g u= @u = @g
(( [(
1
1
+
+
2)
2)
@g =
g+ g+ (
3
3
q)2 q]
+ 2) +( 1+ 1
( 2 (
0
0
+ g)2
+ g) = 0
q 2 2)
3
The major contribution of Canzonery and Gray (1985) model lies in showing how monetary policy in one country can have a detrimental, or favourable or neutral e¤ects in another economies depending how the policy games is played. 399
3 1
q( + 2
1
2
)]
Money growth rate is smaller in the …xed rate than the Stackleberg equilibrium which is small than the Nash equilibrium. The welfare gains from the co-operation in the monetary policy can be obtained by taking the di¤erences of utilities under these two di¤erent regimes. 21.1.6
Recent Micro-Founded Models of International Policy Co-ordination
Corsetti and Pesenti (2001) assess the welfare impacts of macroeconomic interdependence among countries. The use a model in which utility function has three arguments; composite good made of domestic and foreign goods, real money balances and leisure. Output in each country is produced using domestic labour. Goods market is monopolistic and there are imperfections in the labour market so that wage rate is not completely ‡exible. Agents are free to hold domestic money or international bonds of combination to two. Terms of trade e¤ect and impacts of monopolistic structures are considered with an expansionary monetary policy. A competitive equilibrium is a set of prices and quantities that maximise utility for households in both domestic and foreign countries. Using closed form solutions they conclude that the terms of trade e¤ects may welfare dominate aggregate demand externalities. These models correct lack of micro-foundation in the traditional MundellFleming-Dornbusch models for analysis of economic policy of open and interdependent economies including issues such as “beggar thyself”or beggar-thy-neighbour or prosper-thy-neighbour e¤ects. In a follow up study Benigno (2002) uses this new framework to establish new results. He proves that Nash equilibrium is not the Pareto e¢ cient allocation and the co-operative solution is not credible. He shows rationale for delegating monetary policy to a central institution. Interdependence among these economies and interactions could be studied using bargaining, signalling and mechanism designing concepts. Cooperative and noncooperative games with complete and incomplete information among nations, households and …rms could be used to conceptualize the issues and solutions to the problems of growth and development in these economies. There are three generations of literature in the policy coordination. First generation models include studies such as Kydland and Prescott (1977), Dri¢ l (1988), Currie and Levine (1986) and Obstfeld and Rogo¤ (2000). These had found gains from coordination to be small. Cooper (1969) and Hamada (1976) and Kydland (1975) showed inferiority of the non-cooperative Nash equilibrium compared to a cooperative solution. Lucas (1976), and Kydland and Prescott (1977) used rational expectations and argued for the advantage of rule-based policies to create rational expectations equilibrium solution. Petit (1989) used di¤erential games as did the studies of Obstfeld (1994), Sutherland (1996), Senay (1998), Martin and Rey (2000). Obstfeld (2001) and Rogo¤ (2002) 400
provide an excellent review of some of the models used for policy coordination with Mundell-Fleming-Dornbush type models with little gains from coordination. Second generation models of policy coordination in Pappa (2004), Canzoneri, Cumby and Diba (2005), Clerc, Dellas and Loisel (2011), Juillard and Villemot (2011) and Goyal (2007) …nd pay o¤ from monetary and …scal policy coordination to be bigger. Supply and strategic modelling has much improved in recent literature on the policy coordination showing more gains from coordination as stated by Conzoneri et. al.(2005), Evans and Hnatkovska (2007), Douglas and Laxton in dynare. Aarle et.al. (2002) examine the coalition formation in EMU. Recent models such as Kempf and von Thadden (2013), Dedola et al. (2013) add asymmetric information and commitment where the welfare gains can be bigger as the number of countries increase in such deals.
References [1] Beetsma Roel M.W.J., Henrik Jensen (2005) Monetary and …scal policy interactions in a micro-founded model of a monetary union Journal of International Economics, 67, Issue 2, December 2005, 320-352 [2] Bratsiotis, G (2008) In‡uential Price and Wage Setters, Monetary Policy and Real E¤ects, European Journal of Political Economy, 24, 2:503-517 [3] George J. B (2007) Monetary policy responses and strategic price setting, Economics Letters, 95, 3, 327–333 [4] Bullard J, Aarti Singh (2008) Worldwide macroeconomic stability and monetary policy rules Journal of Monetary Economics,55, Supplement, October 2008, S34S47 [5] Chang, Roberto. (1997) Financial integration with and without international policy coordination ,International Economic Review. Aug97, Vol. 38 Issue 3, p547. 18p. [6] Canzoneri Matthew B. , Robert E. Cumby, Behzad T. Diba (2005) The need for international policy coordination: what’s old, what’s new, what’s yet to come? Journal of International Economics,66, 2, 363-384 [7] Clerc L, Harris Dellas, Olivier Loisel (2011) To be or not to be in monetary union: A synthesis Journal of International Economics,83, Issue 2, 154-167 [8] Clarida R, Jordi Galí, Mark Gertler (2002) A simple framework for international monetary policy analysis, Journal of Monetary Economics,49, 5, 87–904 [9] Conconi P, Carlo Perroni (2009) Do credible domestic institutions promote credible international agreements? Journal of International Economics,79, 1, 160-170 [10] Cooper, Russell; DeJong, Douglas V.; Forsythe, Robert; Ross, Thomas W. (1992) Communication in coordination games, Quarterly Journal of Economics. 401
107, 2, 33p. [11] Currie D adn P Levine (1986) Time inconsistency and optimal policies in deterministic and stochastic worlds Journal of Economic Dynamics and Control, 10, 1–2,191-199 [12] Dedola Luca , Peter Karadi, Giovanni Lombardo (2013) Global implications of national unconventional policies Journal of Monetary Economics, 60, 1, 66-85 [13] Evans, M. D., and V. V. Hnatkovska (2007): “Financial Integration, Macroeconomic Volatility, and Welfare,”Journal of the European Economic Association, 5(2-3), 500–508. [14] Fender, J. and Rankin, N. (2003), A Small Open Economy with Staggered Wage Setting and Intertemporal Optimization: The Basic Analytics. The Manchester School, 71: 396–416. [15] Fender, J. And Yip, C. K. (1995), Fiscal Policy In An Imperfectly Competitive Macroeconomy With Nominally Rigid Unemployment Bene…t. The Manchester School, 63: 257–273. [16] Fratzscher M (2009) How successful is the G7 in managing exchange rates? Journal of International Economics,79, Issue 1, September 2009, 78-88 [17] Goodfriend, Marvin; King, Robert G. (1997) The New Neoclassical Synthesis and the Role of Monetary Policy NBER/Macroeconomics Annual (MIT Press). 12 1, p231-283. [18] Juillard M, Sébastien Villemot (2011)Multi-country real business cycle models: Accuracy tests and test bench Journal of Economic Dynamics and Control, 35, 2„178-185 [19] Levine P, Andrew Brociner (1994) Fiscal policy coordination and EMU: A dynamic game approach Journal of Economic Dynamics and Control,18, Issues 3–4, May–July 1994, 699-729 [20] Hansen Lars Peter , Thomas J. Sargent (2003) Robust control of forward-looking models Journal of Monetary Economics,50, Issue 3, 581-604 [21] Kempf Hubert , Leopold von Thadden (2013) When do cooperation and commitment matter in a monetary union? Journal of International Economics,91, Issue 2, November 2013, 252-262 [22] Lubik, T. A. and F. Schorfheide (2005): A Bayesian Look at New Open Economy Macroeconomics, NBER Macroeconomics Annual, Volume 20. [23] Liu Z, Evi Pappa (2008) Gains from international monetary policy coordination: Does it pay to be di¤erent? Journal of Economic Dynamics and Control,32, Issue 7, July 2008, 2085-2117 [24] Marquez J (1988) International policy coordination and the reduction of the US trade de…cit, Journal of Economic Dynamics and Control,12, Issue 1, March 1988, 402
19-25 [25] Martin, P., and H. Rey (2000): “Financial integration and asset returns,” European Economic Review, 44(7), 1327–1350. [26] Michelle R. Gar…nkel (1989) Global macroeconomics: Policy con‡ict and cooperation: A review essay Journal of Monetary Economics,23, Issue 2, March 1989, 345-352 [27] Obstfeld, M. (1994): “Risk-Taking, Global Diversi…cation, and Growth,” American Economic Review, 84(5), 1310–29.
[28] Pappa Evi (2004) Do the ECB and the fed really need to cooperate? Optimal monetary policy in a two-country world Journal of Monetary Economics,51, Issue 4, 753-779 [29] Kose M. Ayhan , Christopher Otrok, Charles H. Whiteman (2008) Understanding the evolution of world business cycles Journal of International Economics,75, Issue 1, May 2008, 110-130 [30] Sheen Je¤rey (1992) International monetary and …scal policy cooperation in the presence of wage in‡exibilities: Are both counterproductive? Journal of Economic Dynamics and Control,16, Issue 2, April 1992, 359-387 [31] Senay, O. (1998): “The E¤ects of Goods and Financial Market Integration on Macroeconomic Volatility,” Manchester School of Economic & Social Studies, 66(0), 39–61. [32] Sutherland, A. (1996): “Financial Market Integration and Macroeconomic Volatility,”Scandinavian Journal of Economics, 98(4), 521–39.
22
L20: Game Theory Economic activities of consumers, producers, governments and nations or regions are interdependent. Game theory provides tools to study the strategic interactions among such economic agents where decisions taken by one individual depend on actions taken by others. Each game has a number of players who choose a set of strategies and rules. .Optimal choices available to one depend on choices made by others. Pay-o¤s are clearly de…ned for each player strategy pairs. Strategic modelling like this started with classics such as Cournot (1838), Bertrand (1883), Edgeworth (1925) von Neumann and Morgenstern (1944), Nash (1950). It is developing very fast in recent years following works of Kuhn 403
(1953), Shapley (1953),Selten ( 1965) Aumann (1966) Scarf (1967), Shapley and Shubic (1969), Harsanyi (1967), Spence (1974), Kreps (1990), Fundenberg and Tirole (1991) and Binmore (1992); see Handbook of Game Theory, from Elseviers. Elements of a Game Rational Players Strategies Payo¤ matrix Table 69: Structure of a Game Player A Strategy 1 Strategy 2 Player B R C R C Strategy 1 1;1 ; 1;1; 1;2 ; 1;2 C C R R Strategy 2 2;1 ; 2;1; 2;2 ; 2;2 R 1;1
is pay-o¤ to row player if he plays strategy 1 and the column player plays strategy 1. Players like to maximise their own pay-o¤ given opponent’s strategy; B will choose strategy 1 or 2 that maximises his/her payo¤ looking at the choice of player A. Most games have equilibrium from which players do not have any incentive to move away. Types of Games Table 70: Zero Sum Game Player A Strategy 1 Strategy 2 Player B Strategy 1 (10; 10) ( 10; 10) Strategy 2 ( 10; 10) (10; 10) zero sum game: one’s gain = loss of another ; sports ; market shares two or many players; Chess, football Cooperative Games: Global climate change; bargaining game 404
Non-cooperative Games: two or many players Competition and Collusion oligopoly/competition between opposing political parties, countries Single period of multiple period: static and dynamic Full information or incomplete information :Firms and consumers; government and public;Among individuals, clubs, parties; nations Solution of Games by the Dominant Strategy Dominant strategy
Table 71: Advertisement Game Player A Advert Dont Advert Player B Advert (10; 5) (15; 0) Dont Advert (6; 8) (10; 2) Dominant strategy is to advertise for both A and B. With a slight change Table 72: Advertisement Game Player A Advert Dont Advert Player B Advert (10; 5) (15; 0) Dont Advert (6; 8) (20; 2) Dominant strategy is to advertise for A but B has no dominant strategy. Solution of Games by Nash Equilibrium (Prisoner’s Dilemma) Punishment structure for a crime F in d in g N a sh so lu tio n (u n d e rsc o re th e b e st stra te g y to a p laye r i g ive n th e ch o ic e o f th e o p p o n e nt.
Nash Equilibrium: Prisoner’s Dilemma Fact: both players did a crime together. Police suspects and arrest both of them. Playing non cooperatively each convicts another. Game results in Nash solution (confess, Confess) = ( 5; 5); Each ends up with 5 years in prison. 405
Table 73: Prisoners’Dilemma Game Player A Confess Dont Confess Player B Confess ( 5; 5) ( 1; 10) Dont Confess ( 10; 1) ( 2; 2) Table 74: Prisoners’Dilemma Game Player A Confess Dont Confess Player B Confess ( 5; 5) ( 1; 10) Dont Confess ( 10; 1) ( 2; 2) By confessing, each gives evidence to the police to determine the highest possible punishment. If they had cooperated remaining silent, police would not have enough evidence. Each would have been given only two years of prison ( 2; 2) : This is Pareto optimal outcome, "where no one could be made better o¤ without making someone worse-o¤". Cooperation is better but each think that other player will cheat and therefore doesn’t cooperate. Therefore stay longer in jail. There are many example of prisoner’s dilemma game in real world -pricing and output in a cartel, pollution, tax-revenue. Solution by the mixed strategy Table 75: Game of matching penny: mixed strategy Player A Head Tail Player B Head (1; 1) ( 1; 1) Tail ( 1; 1) (1; 1) This game does not have equilibrium in pure strategy. Player B will play H if A plays H but A will play T if B plays H. If A plays T it is optimal to play T for B, then it is optimal for B to play H. Game goes in round and in a circle again. 406
It can be solved my the mixed strategy. Le p be probability of playing H and (1-p) be the probability of playing T by player B. Then the expected payo¤ when A plays H should equal the expected payo¤ when A plays T resulting in the value of p to 0.5 as: 1 2 If each played H or T half of the times optimal payo¤ is zero to both players. Probability of playing H or T is 0.5. Another way is to ‡ip the coin to randomise the chosen strategies. Solution by mixed strategy another example: p
(1
p) =
p + (1
p) =) p =
Table 76: Competitive Game Player A Left Right Player B Top (50; 50) (80; 80) Bottom (90; 90) (20; 20) B plays Top p times and Bottom (1 p) times if A plays Left. B plays Top p times and Bottom (1 p) times if A plays Right. B likes to be equally well o¤ no matter what A plays. Expected pay-o¤ for B if A plays Left E(
B;L )
= 50p + 90(1
p)
(1058)
p)
(1059)
Expected pay-o¤ for B if A plays Right E(
B;R )
= 80p + 20(1
Making these two payo¤s equal 50p + 90(1
p) = 80p + 20(1 p = 0:7
p) =) 100p = 70
(1060) (1061)
B plays Top 70 % of times and Bottom 30% of times. Subsidy Game Between the Airbus and Boeing If both Boeing and Airbus produce a new aircraft each will lose -10. If Airbus does not produce and only Boeing produces Boeing will make 100 pro…t. If Airbus 407
Table 77: Subsidy Game Airbus Produce Don’t produce Boeing Produce ( 10; 10) (100; 0) Don’t produce (0; 100) (0; 0) does not produce Airbus can make 100 but then Boeing will decide to produce even at a loss of 10 so that Airbus does not enter in that market. EU countries want Airbus to produce, they change this by subsidising 20 to Airbus. Table 78: Subsidy Game Airbus Produce Don’t produce Boeing Produce ( 10; 10) (100; 0) Don’t produce (0; 120) (0; 0) Producing new aircraft is dominant strategy for Airbus now, no matter whether Boding produces or not. Entry Deterrence Game Table 79: Subsidy Game Entrant Enter Dont Enter Incumbent Enter ( 10; 10) (100; 0) Dont Enter (0; 100) (0; 0) In‡ation and unemployment game between public and private sectors Higher payo¤ is good. First element represents payo¤ to the row-player (Government). Second element represents payo¤ to the column-player (private sector). Nash solution is (H; H) = (4; 4) Cooperative solution would have been better with (L; L) = (5; 5). Cost of Cheating Cooperative solution would have been better with (L; L) = (5; 5) but distrusting each other results in (H; H) = (4; 4) . 408
Table 80: Subsidy Game Entrant Enter Dont Enter Incumbent Enter ( 10; 10) (100; 0) Dont Enter (0; 120) (0; 0) Table 81: In‡ation and unemployment game Private Sector H L Government H (4; 4) (6; 3) L (3; 6) (5; 5) If the game is plaid repeatedly what will be value of the game? It is given by the discounted present value of the game for any discount rate 0 < < 1: 2
n
=
5
(1062) 1 However, there is an incentive to cheat to get 6 instead of 5. when one player deviates from the cooperative strategy in this way another player will …nd out being cheated next period. Then he/she will punish the cheater by playing noncooperatively next period. So the value of game: P V (cooperate) = 5 + 5 + 5
+ :::: + 5
P V (cheat) = 6 + 4 + 4
2
+ :::: + 4
n
(1063)
P V (cheat) = 6 + 4 + 4
2
+ :::: + 4
n
(1064)
Cost of Cheating
Taking the sum P V (cheat) = 6 +
4+4 +4
2
+ :::: + 4
n
(1065) (1066)
P V (cheat) = 6 + 4
(1 ) Whether a person cheats or not depends on discount factor 5 1
=6+4
(1
)
or 5 = 6 (1
)+4 409
=)
1=
2 ;
=
1 2
(1067)
A player with > 12 will cheat and a player with 6 21 will not cheat. Extensive form of the game Solution by Backward Induction (Is there any …rst movers advantage?)
410
In‡ation and unemployment game in a diagram In‡ation and unemployment game in a diagram
Economic policy game between the …scal and monetary authority 411
412
References [1] Binmore K. (1990) Fun and Games: A text on Game Theory, Lexington, Heath. [2] Cripps, M.W.(1997) Bargaining and the Timing of Investment, International Economic Review, 38:3 :Aug.:527-546 [3] Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. [4] Gardener R (2003) Games of Business and Economics, Wiley, Second Edition. [5] Holt Charles (2007) Markets, Games and Strategic Behaviour, Pearson, . [6] Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. [7] Kreps D. M. (1990) A Course in Microeconomic Theory, Princeton. [8] Ok Efe A. (2007) Real Analysis with Economic Applications, Princeton. [9] Osborne M.J. and A. Robinstein (1994) A course in game theory, MIT Press. [10] Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with Calculus, Pearson. [11] Rasmusen E(2007) Games and Information, Blackwell,. [12] Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th ed.
413
22.1
Problem 9: Strategic Models and Optimal Tax
Q1. Only two …rms supply products in a certain market in which the market demand for the product is:
P = 150
(1068)
(q1 + q2 )
Cost of production for each of the two …rms is . Ci = 11qi f or
(1069)
i = 1; 2
a) What is the total pro…t when these two …rms collude? b) What is the output in Cournot equilibrium? What kind of game is this? Q2. Consider a …rm in monopolistically competitive industry Q=A
(1070)
B P
Prove that its marginal revenue is given by MR = P
Q B
(1071)
(a) If the cost function is C = F + cQ then prove that the average cost declines because of the economy of scale. (b) Further assume that the output sold by a …rm, number of …rms, its own price and average prices of …rms are given by Q=S
1 N
b P
P
(1072)
show that the average cost rises to number of …rms in the industry when all …rms charge same price. AC = n:F +c s (c) Prove that price charged by a particular …rm declines with the number of …rms P = c + b1n 414
(d) Determine the number of …rms and price in equilibrium. Explain entry exit behavior and prices when number of …rms are below or above this equilibrium point. (e) Collusive and strategic behaviors may limit above conclusions. Discuss. (f) Apply above model to explain international trade and its impact on prices and number of …rms in a particular industry. (g) Use this model to explain interindustry and intra-industry trade. (h) Use monopolistic competition model to analyse consequences of dumping practices in international trade. Q3. Nature left 1000 pounds on the table to be split between two players. What is the optimal solution from a symmetric bargaining game if the threat point is given by d(0,0)? Q4. Use of Game Theory for Analysis of Macroeconomic Policy Consider an in‡ation-unemployment strategic policy game between the government and the private sector. Both sectors can play high (H) or low (L) in‡ation rate strategy. The H and L for private sector denotes their expectation of high or low in‡ation depending on their perception about the actual policy to be taken by the government. Government sector’s H or L strategy refer to the choice of actual rate of in‡ation by the government. The …rst element in the pay-o¤ matrix is the gain for the government (row player) and the second element refers to the gain for the private sector (column player). Pay-O¤ Matrix for In‡ation-unemployment Policy Game Private Sector H L Public Sector H 4; 4 6; 3 L 3; 6 5; 5
(1073)
a. Represent this policy game using a set of expectation augmented Phillips curves. An equation for such a curve can be written to show how the actual and the expected rates of in‡ation are in relation to the actual and the natural rate of unemployment. b. What is the pay-o¤ for the private and the government sectors in the Nash equilibrium? Show that the non-cooperative Nash equilibrium is Pareto inferior to the cooperative solution in this game. Argue why policy of cooperation is not credible. 415
c. Write this game in an extensive form assuming that the government sector makes its decision …rst on whether to choose high or low rate of actual in‡ation and the private sector then selects its strategic move looking at the actions of the government. Solve this game using the backward induction technique. d. What would be the solution of this game if it is played in…nite number of times? Show how the deviation from the co-operative solution either by the private or the government sector player would be punished by another sector player. What would be the discount rate consistent with equilibrium of the game? Q5. Consider a two person zero sum (TPZS) framework. Such game can be given by a matrix such as Strong Hand H L Strong Leg H 10; 10 10; 10 L 10; 10 10; 10 a) b) c) d)
(1074)
Explain this TPZS game. Solve it by using minmax = maxmin method. Find a mixed strategy for players strong-leg and strong-hand. What is the value of the game? Why this game is not realistic in modern world?
Q6. Prove that optimal tax rate is independent of market structure, optimal tax rate is the same whether market is under monopoly or oligopoly. Pro…t function of a monoplist with taxes = PQ
TC
P =a
T
bQ
(1075) (1076)
total cost with marginal cost c and …xed cost f T C = cQ + f
(1077)
T = tQ
(1078)
Tax revenue
416
22.2
Problem 10: General Equilibrium Model: Pure Exchange
Q1. Consider a pure exchange general equilibrium model for an economy with individuals A and B with the set of parameters in the table given below. Lagrangian for constrained optimisation for Household A : LA = X1A
A
X2A
1
A
+
A P1 ! A 1 + P2 ! 2
P1 X1A
P2 X2A
(1079)
P2 X2B
(1080)
Lagrangian for constrained optimisation for Household B : LB = X1B
B
X2B
1
B
+
B P1 ! B 1 + P2 ! 2
P1 X1B
Table 82: Parameters in Pure Exchange Model Household A Household B A A B = f0; 200g Endowments ! 1 ; ! 2 = f100; 0g !1 ; !B 2 Preference for X1 ( ) 0:6 0:4 Preference for X2 (1 ) 0.4 0.6 You may assume Walrasian numeraire: P1 = 1 with this speci…cation, and implied incomes for A and B are: I A = !A 1
I B = P2 ! B 2
(1081)
Derive the demand functions for both A and B individuals and …nd the relative price that clears the markets for both X1 and X2 : Q2. Consider a pure exchange economy in which the utility of households A and 1 1 A B B are given by U A = X1A A X2A and U B = X1B B X2B . Here U A and U B are levels of utilities of household A and B respectively, and A and B denote preferences of these households for the consumption of good 1. Similarly X1A and X2A , and X1B and X2B are consumptions of good 1 and good 2 by household A and B respectively. Only household A has an endowment of good 1 and it is ! A 1 = 100; and only household B has an endowment of good 2 and it is ! B = 200; is A 0.4 and B is 0.6. 2 417
a. Represent the initial endowment position of goods A and B of these two households using the Edgeworth box diagram with a number of indi¤erence curves for each. b. Formulate the Lagrangian function for constrained optimisation for A and B. c. Provide the …rst order conditions necessary for optimisations by both households. d. Derive demand functions for both products by both households. e. State the market clearing conditions for both goods. f. Use good 1 as a numeraire. Find the relative price of good 2 that clears both markets and is consistent with maximization of utility (satisfaction) by both households given their budget constraints. g. Determine the income of each household. h. Evaluate optimal demands X1A and X2A , and X1B and X2B for those endowments and preferences. i. Check whether your solutions satisfy the market-clearing conditions required for a general equilibrium. j. What are the levels of utility for A and B at equilibrium? k. Represent the general equilibrium (optimal quantities, relative prices) in another Edgeworth box diagram. Q3. Consider a general equilibrium model with taxes in which a representative household maximises utility subject to its budget constraint and the …rm maximises pro…t subject to a technology constraint as given below.
max U = C L
(1082)
p (1 + t) C + wL = wL
(1083)
Subject to
The …rm’s pro…t maximisation problem is: max
= p:Y
w:LS
(1084)
Subject to Y = LS 0:5 You may select p = 1 as a numeraire. 418
(1085)
Find expressions for the wage rate, consumption, output, labour supply and demand for labour consistent with the general equilibrium.
References [1] Auerbach Alan J., Laurence J. Kotliko¤ and Jonathan Skinner. 1983. "The E¢ ciency Gains from Dynamic Tax Reform." International Economic Review, 24(1): 81-100. [2] Balasko Y. (2011) General Equilibrium Theory of Value [3] Bhattarai Keshab 2007. "Welfare Impacts of Equal-Yield Tax Experiment in the UK Economy." Applied Economics, 39(10-12): 1545-1563. [4] Diamond P. A. and J. A. Mirrlees .1971. "Optimal taxation and public producton II: Tax Rules.", American Economic Review, 61(3-1):261-278. [5] Hansen Gary D.and Edward C. Prescott.2002. "Malthus to Solow." American Economic Review, 92(4): 1205-1217 [6] Hicks J. R. 1939. Value and Capital: An inquiry into some fundamental principles of economic theory, Oxford: Oxford University Press. [7] Kuznets Simon.1955. Economic Growth and Income Inequality." American Economic Review, 45(1): 1-28. [8] Meade James E. and Richard Stone (1941) The Construction of Tables of National Income, Expenditure, Savings and Investment, Economic Journal, 51(202/203):216-233. [9] Meade James E., Ironside, Jones, Bell, Flemming, Kay, King, Macdonald, Sandford and Whittington, Willis.1978. The Structure and Reform of Direct Taxation. IFS, London: George Allen and Unwin. [10] Mirlees, J.A. 1971. "An exploration in the theory of optimum income taxation." Review of Economic Studies,38:175-208. [11] Mirrlees James, S. Adam, T. Besley, R. Blundell, S. Bond, R. Chote, M. Gammie, P. Johnson, G. Myles and J. Poterba.2010. Dimensions of tax design: the Mirrlees review, Oxford: Oxford University Press. [12] Rutherford, Thomas F. 1995 “Extension of GAMS for Complementary Problems Arising in applied Economic Analysis.” Journal of Economic Dynamics and Control 19 1299-1324. [13] Sen Amartya. 1974. "Informational bases of alternative welfare approaches: Aggregation and income distribution.", Journal of Public Economics, 3(4): 387-403. [14] Stone Richard. 1961. Input-output and National Accounts, Paris:OECD. [15] Whalley John. 1975. "A General Equilibrium Assessment of the 1973 United Kingdom tax reform." Economica, 42(166): 139-161. 419
22.3
Problem 12: Bargaining and Cooperative Game
Q1. Find the Nash equilibrium in the prisoner’s dilemma game given below.
Table 83: Prisonar’s Dilemma Game Player A Confess Don’t Confess Player B Confess ( 7; 7) ( 1; 10) Don’t Confess ( 10; 1) ( 2; 2) [Negative sign indicates bad payo¤; -10 is worse than -7]. What would have been cooperative and the Pareto optimal solution? Q2. One common example for a bargaining game is splitting a pie between two individuals, i and j. The total amount to be divided is 1. Their shares in this pie are given by i and j respectively and they should not claim more than what is on the table, i.e. i + j 1: This implies a meaningful solution of the game requires i 0 and j 0 . If the sum of claims is more than what is on the table each gets zero i.e. when i + j > 1 then i = 0 and j = 0 . Thus the Nash bargaining problem is given by max U = (
0) (
i
j
0)
(1086)
subject to i
+
j
=1
(1087)
Formulate the constrained optimisation of this problem. Find the optimal values of i and j that satisfy the Nash equilibrium. Q3. In Spence’s model of signalling, type 1 workers are less productive than type 2 workers. Workers signal their productivity type by choosing years of education to maximise their utility. As given below the utility of a worker is positively related to the wage rate (w) and negatively to the e¤ort for education (e) but it is less costly for more productive workers to get education. p ut (wt ; e) = 42 wt
kt e1:5 with k1 = 3; k2 = 1 w1 = e; w2 = 2e
(1088)
Given the values of kt and the above utility function …nd the optimal choice of e for each type of worker. 420
Q4. Consider a game in which player B has top and bottom strategies and player A has left and right strategies as following.
Table 84: Game of Mixed Strategy Player A Left Right Player B Top (30; 30) (70; 70) Bottom (80; 80) (10; 10) Probability of playing Top by B is p and playing Bottom is (1 p) if A plays Left . Similarly probability of B playing Top is p and playing Bottom (1 p) if A plays Right. B likes to be equally well o¤ no matter what A plays. Find the optimal probability p of playing Top by player B solving this game by the mixed strategy.
421
23
L21: Bargaining Game
In many circumstances optimal decisions of an economic agent depends on decisions taken by others. Dominants …rms competing for a market shares, political parties contesting for power and research and scienti…c discoveries aimed for path-breaking innovations are in‡uenced by decision taken by others. In all these circumstances there are situations where collective e¤orts rather than individual ones generate the best outcome for the group as a whole and for each individual members of the group. Concepts of bargaining, coalition and repeated games developed over years by economists such as Cournot (1838), Bertrand (1883), Edgeworth (1925) von Neumann and Morgenstern (1944) and Nash (1950, 1953) is developing very fast in recent years following works of Kuhn (1953), Shapley (1953),Shelten ( 1965) Aumman (1966) Scarf (1967), Shapley and Shubik (1969), Harsanyi(1967), Spence (1974), Hurwicz (1973), Myerson (1986), Maskin and Tirole (1989), Kreps (1990), Fundenberg and Tirole (1991) and Binmore (1992), Rubinstein (1982) Sutton (1986) Cho and Kreps (1987) Sobel (1985) Machina (1987) Riley (1979) McCormick (1990), Ghosal and Morelli (2004) and others. These have generated models that can be applied to analyse the relative gains from coalitions rather than without these coalitions. The very common example for bargaining game is splitting a pie between two individuals. The sum of the shares of the pie claimed by both cannot exceed more than 1, otherwise each will get zero. If we denote these shares by i and j then i + meaningful solution of the game where each get i When i + j > 1 then and i = 0 and j = 0 .
j
1 is required for a 0 and j 0 payo¤.
Standard technique to solve this problem is to use the concept of Nash Product
23.1
Nash Product in Bargaining Game max U = (
0) (
i
j
0)
(1089)
subject to i
or by non-satiation property
i
+
+
j
=1 422 j
1
(1090)
Using a Lagrangian function L ( i;
j;
)=(
0) (
i
j
0) + [1
i
j]
(1091)
First Order Conditions First order conditions of this maximization problem are L ( i; @ L ( i; @
j;
)
=
j
=0
(1092)
=
i
=0
(1093)
i j;
)
j
L ( i; j ; ) =1 (1094) i j = 0 @ From the …rst two …rst order conditions j = i implies j = i and putting this into the third …rst order condition j = i = 12 . This is called focal point. Thus Nash solution of this problem is to divide the pie symmetrically into two equal parts. Any other solution of this not stable. Roy Gardner (2003) and Charles Holt (2007) have a number of interesting examples on bargaining game. Application of Bargaining Game Money to be divided between two players M = u1 + u2
(1095)
The origin of this bargaining game is the disagreement point d(0; 0), the threat point. Here the utility of player one (u1 ) is plotted against the utility of player two u2 and the line u1 u2 is the utility possibility frontier (UPF). Starting of bargaining can be (0; M ) or (M; 0) where one player claims all but other nothing. But this is not stable. O¤ers and counter o¤ers will be made until the game is settled at u 1 M; 21 M where each player gets equal share. 2 423
=
Numerical Example of Bargaining Game Suppose there is 1000 in the table to be split between two players. What is the optimal solution from a symmetric bargaining game if the threat point is given by d(0,0)? Using a Lagrangian function for constrained optimisation L (u1; u2 ; ) = u1 u2 + [1000
u1
(1096)
u2 ]
First order conditions of this maximization problem are L (u1; u2 ; ) = u2 @u1
=0
(1097)
L (u1; u2 ; ) = u1 @u2
=0
(1098)
L (u1; u2 ; ) = 1000 u1 u2 = 0 (1099) @ From the …rst two …rst order conditions u2 = u1 implies u2 = u1 and 1000 putting this into the third …rst order condition u2 = u1 = 2 = 500. This is called focal point. The Nash bargaining solution is the values of u1 and u2 that maximise the value of the Nash product u1 u2 subject to the resource allocation constraint,u1 +u2 = 1000. This bargaining solution ful…ls four di¤erent properties: 1) symmetry 2) e¢ ciency 3) linear invariance 4) independence of irrelevant alternatives (IIA). Symmetry implies that equal division between two players and e¢ ciency implies no wastage of resources u1 + u2 = M or maximisation of the Nash product, u1 u2 . Linear invariance refers to the location of threat point as can be shown in a bankruptcy game say dividing 50000. If u is a solution to the bargaining game then u + d is a solution to the bargaining problem with disagreement point d. L (u1; u2 ; ) = (u1
d1 ) (u2
d2 ) + [50000
u1
u2 ]
(1100)
Suppose the player 1 has side payment d1 = 15000 L (u1; u2 ; ) = (u1
15000) (u2
d2 ) + [50000
u1
u2 ]
(1101)
First order conditions of this maximization problem are L (u1; u2 ; ) = u2 @u1 424
=0
(1102)
L (u1; u2 ; ) = u1 @u2
L (u1; u2 ; ) = 50000 @
u1
From the …rst two …rst order conditions u2 implies u2 = u1
=0
(1103)
u2 = 0
(1104)
15000
= u1
15000
15000 and
putting this into the third …rst order condition u2 + 15000 = u1 ; u2 =
50000 15000 2
= 17500; u1 = 15000 + u2 = 32500.
Then u1 + u2 = 17500 + 32500 = 50000. Risk and Bargaining A risk averse person looses in bargaining but the risk neutral person gains. Suppose the utility functions of risk averse person is given byu2 = (m2 )0:5 but the risk neutral person has a linear utility u1 = m1 . m1 + m2 = M .u1 + u22 = 100 Using a Lagrangian function for constrained optimisation L (u1; u2 ; ) = u1 u2 + 425
100
u1
u22
(1105)
First order conditions of this maximization problem are L (u1; u2 ; ) = u2 @u1 L (u1; u2 ; ) = u1 @u2
=0 2 u2 = 0
(1106) (1107)
L (u1; u2 ; ) = 100 u1 u22 = 0 (1108) @ From the …rst two …rst order conditions uu1;2 = 2 u2 implies u1 2u22 and putting this into the third …rst order condition .3u22 = 100 ; u22 = 100 = 33:3 ; u2 = 5:77 3 2 2 u1 = 2u2 = 2 (5:77) = 66:6 u1 + u22 = 66:6 + 33:3 = 100 Thus the risk nuetral player’s utility is 66.7 and risk averse player’s utility is only 5.7. Morale: do not reveal anyone if you are risk averse, otherwise you will lose in the bargaining. 23.1.1
Coalition possibilities
N
2 -1 rule for possible coalition Consider Four Players A,B,C,D A, B, C, D AB, AC, AD BC, BD, CD ABC, ABD,ACD, BCD, ABCD 16 -1=15 What is core of a coalition? It is a coalition that cannot be blocked by any coalition. Detected by the shapley value. Shapley value is the amount a particular player can bring in the team - di¤erence a particular player makes in the game. Weakest link game. What is empty core? When non-cooperatively playing coalition partners all end up in a zero payo¤. Examples of empty core. Global warming game; Why no constitution in Nepal? I. The 123 Game with a rotating pivotal player Rule: a player located in the middle is pivotal. II. The 123 Game with only one Pivotal Player 426
Table 85: No Pivotal Player in a Bargaining Game orderings M(1,S) M(2,S) M(3,S) 1 123 0 1 0 2 132 0 0 1 3 213 1 0 0 4 231 0 0 1 5 312 1 0 0 6 321 0 1 0 Table 86: Pivotal Player in a Bargaining Game orderings M(1,S) M(2,S) M(3,S) 1 123 1 0 0 2 132 1 0 0 3 213 1 0 0 4 231 1 0 0 5 312 1 0 0 6 321 1 0 0 Shapley value of player 1 is 1 no matter which position it is in the coalition and it is 0 for players 2 and 3. Bargaining and coalition in a Venn Diagram
By 2N 1 rule for possible number of coalitions in N person games, there are seven possible coalitions in 123 gaem: f1g,f2g ,f3g ,f1; 2g , f1; 3g,f2; 3g ,f1; 2; 3g . Core of a three player game Rationality of a coalition 427
428
A coalition of players should ful…ll individual rationality group rationality and coalition rationality. 1
=
v (N ) =
2
;
X
i
=
N X
(1109)
i
(1110)
i
i2N
i
n
; ::::;
v (fig)
i2N
(1111)
In the dynamic context players like to maximise the present value, V, of the gain from in…nite period, with a given discount rate r over years: V =
t=1 Z
v (i) e
rt
dt
(1112)
t=0
23.1.2
Superadditivity of a coalition
Superadditivity of a coalition Superadditivity condition implies that the value of the game in a coalition is greater than the sum of the value of the game of playing alone by those individual members. v (1 [ 2 [ 3) v1 (1) + v (2) + v (3) (1113) Coalitions (parties) playing together generate more value for each of its member than by playing alone. Team spirit generates extra bene…ts. When normalised to 0 and 1 the value of the gains from a coalition are: v (1) = 0; v (N ) = 1 for n = 1; :::; N Power of individual i in the coalitions is measured by the di¤erence that person makes in the value of the game v (S [ fig
v (S)) = 1
429
(1114)
23.1.3
Shapley value
Payo¤ for coalition of empty set: v ( ) = 0 Payo¤ from players acting alone: v (1) = 0; v (2) = 0; v (3) = 0 Payo¤ from alternative coalitions: v (1; 2) = 0:1; v (1; 3) = 0:2; v (2; 3) = 0:2; Payo¤ from the grand coalition: v (1; 2; 3) = 1 For each coalition with s members: i
=
X
n
S2N
(S) v (S [ fig
v (S)) ;
n
(S) =
s! (n
s n!
1)
(1115)
v (1) v ( ) = 0 v (1; 2) v (1) = 0:1 v (1; 3) v (1) = 0:2 0 = 0:2 v (1; 2; 3) v (2; 3) = 1 0:2 = 0:8 s! (n
s n!
1)
s! (n
s n!
1)
0! (3
0 3!
1)!
1! (3
1 3!
1)!
2! (3 2 s 1) = n! 3! Shapley value for player 1 is thus
1)!
0
(S) =
1
(S) =
2
1
=
(S) =
X
n
S2N
= =
s! (n
(S) v (S [ fig
v (S)) =
=
2! 2 = 3! 6
(1116)
=
1! 1 = 3! 6
(1117)
=
2! 2 = 3! 6
(1118)
2 1 1 (0) + (0:1) + (0:2) 6 6 6
19 2 + (0:8) = 6 60
(1119)
For player 2 2
=
X
S2N
2 19 + (0:8) = 6 60
n
(S) v (S [ fig
v (S)) =
430
2 1 1 (0) + (0:1) + (0:2) 6 6 6 (1120)
Note as before v (2) v ( ) = 0; v (1; 2) v (1) = 0:1 v (2; 3) v (1) = 0:2 0 = 0:2; v (1; 2; 3) For player 3
3
=
X
S2N
n
(S) v (S [ fig
v (2; 3) = 1
0:2 = 0:8
1 1 2 (0) + (0:2) + (0:2) 6 6 6
v (S)) =
2 22 + (0:9) = 6 60
(1121)
v (3) v ( ) = 0 v (1; 3) v (1) = 0:2 0 = 0:2 v (2; 3) v (2) = 0:2 0 = 0:2 v (1; 2; 3) v (1; 2) = 1 0:1 = 0:9 23.1.4
Equivalence of core in games and general equilibrium
Equivalence of Core in Games and Core in a General Equilibrium Model X
xi =
i
n X X
i T =fSg S fig
=
X
S2T
S
X
i2S
S i xi
X
=
S
T =fSg
X
i2S
n X X !i = !i i
431
xSi
T =fSg S fig
S
=
X !i i
(1122)
24
L22: Signalling and Principal Agent Model: Asymmetric (incomplete) Information
When intention cannot be directly revealed or stated players can signal indirectly to other players. These signals can take many forms. Signalling plays important roles in strategic choices of individuals, parties, communities, regions, national and the global community as a whole. Formation of payo¤ discussed above depends on signalling - players do not know the moves of their opponents but based on their interpretation of signal they can however, put some numerical values to the payo¤. Impacts of Asymmetric (incomplete) Information on Markets Equilibrium is ine¢ cient relative to full information case Signalling can improve the e¢ ciency: warranty and guarantee Screening: revealing the risk type of agent Credit history from credit card companies Government can improve the market by setting high standards of business contracts or bailing out troubled ones (Northern Rock, Bear Stearns, Lehman Brothers) Right regulations –Financial Services Authority, Fair trade commissions; O¢ ce of standards; Bank of England Moral hazard (hidden action) Probability of bad event is raised by the action of the person People who have theft insurance are likely to haven low quality locks that are easy to break (in cars, houses, bicycle (car)) most likely to claim insurances Remedy: deductible amount; to ensure that some customers take care in security. Adverse Selection (hidden information) Problem 432
Uncertainty about the quality of good or services – honest borrowers less likely to borrow at higher interest rates. – low quality items crowd out high quality items – risky borrowers drive out gentle borrowers in the …nancial market. Theft insurance; health insurance; – people from safe area are less likely to buy theft insurance; only – unsafe customers end up buying theft insurance – healthy people are less likely to buy health insurance Adverse Selection (hidden information) Problem Uncertainty about the quality of good or services – honest borrowers less likely to borrow at higher interest rates. – low quality items crowd out high quality items – risky borrowers drive out gentle borrowers in the …nancial market. Theft insurance; health insurance; – people from safe area are less likely to buy theft insurance; only – unsafe customers end up buying theft insurance – healthy people are less likely to buy health insurance Asymmetric information in Used Car Market -Akerlof’s Model of Asymmetric Information Sellers know exactly quality of cars but buyers do not. Equilibrium is a¤ected when sellers have more information than buyers. Market has plums: good cars and lemons: bad cars Seller knows his quality of cars but buyers do not Market for good cars disappear because of existence of bad cars in the market. 433
Demand for high quality car falls and demand for low quality cars rise. Ultimately only low quality cars remain in the market. Asymmetric information in Used Car Market -Signalling solves the Problem signals: warranty and Guarantee Providing warranty less costly for high quality cars as they last long. Warranty is costly for low quality cars as they frequently break down. Buyers can decide whether a car is good or bad looking at the warranty and pay appropriately. Right signalling can remove ine¢ ciency due to incomplete information. Markets for both types of car can operate e¢ ciently by right signals of warranty and Guarantee Pooling, Separating and Mixed Equilibrium Complete market failure pooling equilibrium (same price for good and bad cars; good cars disappear from the market) Complete market success Separating equilibrium where players act as they should according to the signal (prices according to quality) Partial market success (both good and bad cars are bought, some feel cheated) Near Market failure (mixed strategies) Bayesian updating mechanism at work Education Level- A Signal of Productive Worker An employer does not know is more productive and who is less productive It pays the same wage rate to both productive and unproductive workers market is ine¢ cient, it drives out more productive workers. 434
Workers can signal their quality by the level of educational attainment, then market may work well. Less costlier for high quality worker to get education. costlier for low quality worker to get the speci…ed education. so the low quality worker gets no education, but the higher quality worker gets education. Employers pay according to the level of education. Education works as a signalling device and makes the market e¢ cient. Education separates the equilibrium.
References [1] Ross, Stephen A. (1973) “The Economic Theory of Agency: The Principal’s Problem.” American Economic Review, 63(2): 134–39.
24.1
Signalling for managing a company
Owners of a company are concerned about a project that would earn them 600,000 if successful. Probability of success with normal e¤ort from the manager is 60 percent and this can increase up to 80 percent if the manager puts extra e¤orts. The basic salary of the manager is 100,000. He would put extra e¤orts only if he is paid additional amount of at least 50,000. Owners cannot monitor whether the manager is putting high or low e¤orts. a) Is it pro…table to pay extra for the manager? Pro…t without paying extra 0.6 * 600,000 - 100,000 = 260, 000 Pro…t with extra incentive payment 0.8 * 600,000 - 150,000 = 330, 000 Extra payment can make up to 70,000 with probability of 0.8. Once extra payment is made how can owners make sure that he puts extra e¤orts? This requires evaluation of incentive compatibility and participation constraints. a) Incentive compatibility constraint (s + 0:8b)
(s + 0:6b)
50000 =) 0:2b 435
50000
(1123)
b = 250,000 b) Participation constraint: s 150000 =) s = 150000 0:8b =) s = 150000 0:8 (250; 000) = 50; 000
(1124)
It is not possible to hire manager with negative salary. At most managers can be conditioned to bonus payment but with zero salary. 0 + 0:8 (250; 000)
150000 =) 200; 000
150000
(1125)
Pay 200,000 and the manager will put maximum e¤ort. c) Is it pro…table to pay extra 200,000 as an incentive payment? Pro…t with incentive payment 0.8 * 600,000 - 200,000 = 280, 000 Pro…t without incentive payment 0.6 * 600,000 - 100,000 = 260, 000 Reference: Dixit A and S Skeath (1998) Games of Strategy, New York Norton. Education Level- A Signal of Productive Worker Consider a level of education e c1 e c2 e =) c1 c2 (1126) Cost of education of unproductive worker is much higher c2 e < (a2
a1 ) < c1 e
(1127)
Cost of education relative to productivity of low and high quality workers for education e (a2 24.1.1
a1 )
c1 Signalling and incentives
c
(1133)
This is possible if the cost of education is 5000; then wage net of education cost for high quality is 35000 which is above the pooling wage rate. This makes sense to signal by choosing higher education. Signalling is optimal in this case; fraction of workers will signal by going to education. 437
Aggregate labour cost will be the same but wages will be paid according to the productivity of workers as re‡ected by the level of education of workers. Figure: Separating and Pooling Equilibrium
Excel calculations. While making a hiring decision employers take level of education as a signal of quality of workers. Government Policy and Signalling It is important to have optimal amount of signalling – too little or too much signalling generates ine¢ cient result. Empirical …nding on signalling is mixed. Public policy could be designed to generate right amount of signalling as following: 1. It can create separating equilibrium by subsidizing education of more able workers. It can ban on wasteful signalling by banning schools that do not produce good workers. 2. High education provides signals, employers pay according to this signal, this will a¤ect the distribution of wages. 24.1.2
Spence model of education
Players consisting of {workers, …rms and nature}. There are two types of workers [t = f1; 2g]. Type 1 is less productive and type 2 more productive. 438
Employer does not know which one is low or high quality worker but sees level of education Nature decides whether a worker is high or low productivity type. Level of education signals the quality of worker Spence model of education: Preferences over wage and level of education Workers choose level of education according to their beliefs about its impact on wage o¤er: wt (e). Utility from wage and education is given by ut (w; e). Utility is rising in wage received Utility falls in work e¤orts
@ut (w;e): @w
@ut (w;e): @e
0 t = 1; 2
(1134)
kt . indicates the cost of education for the worker type t. It is more expensive for less productive worker to produce education signal k1 > k2 More Speci…cally p ut (w; e) = 42 w
kt e1:5
k1 = 2; k2 = 1 w1 = e; w2 = 2e
(1135)
Level of education chosen by less productive worker In perfect information equilibrium, …rms pay according to the marginal productivity Wage of less productive worker: w1 = e; 439
The type 1 worker’s optimisation problem p p Max ut (w; e) = 42 w kt e1:5 = 42 e e
@ut (w; e) : 1 = 42 p @e 2 e
2e1:5
1
(1136)
(1137)
3e 2 = 0
1 1 42 42 p = 3e 2 =) e1 = =7 6 2 e
(1138)
It is optimal for the less productive worker to takes only seven years of education Level of education chosen by more productive worker Wage of less productive worker: w2 = 2e; The type 1 worker’s optimisation problem p p Max ut (w; e) = 42 w kt e1:5 = 42 2e e
1 @ut (w; e) : = 42 p @e 2 2e
2
e1:5
1
1:5e 2 = 0
1 1 1 42 42 p = 1:5e 2 ; 42 p = e =) e2 = = 19:8 2:121 2e 1:5 2
(1139)
(1140) (1141)
It is optimal for the more productive worker to takes 19.8 years of education. Government Policy and Signalling It is important to have optimal amount of signalling – too little or too much signalling generates ine¢ cient result. Empirical …nding on signalling is mixed. Public policy could be designed to generate right amount of signalling as following It can create separating equilibrium by subsidizing education of more able workers. It can ban on wasteful signalling by banning schools that do not produce good workers. High education provides signals, employers pay according to this signal, this will a¤ect the distribution of wages. 440
24.2
Problem 13: Signalling and mechanism
Q1. Consider a principal-agent game in a certain job market. It includes self selection and participation constraints for good and bad workers. The self selection constraint for a good worker in terms of utility (UG ), wage rate (WG ) ; work e¤orts (eG ) and output per worker (qG ) is given by:
U G = wG
qG 3
e2G = wG
2
U G = wB
qB 3
e2B = wB
2
(1142)
The self selection constraint for a bad worker in terms of utility (Ub ), wage rate (Wb ) ; work e¤orts (eb ) and output per worker (qb ) is: UB = wB
e2B = wB
(qB )2
UB = wG
e2G
(1143)
The participation constraint for a good worker is given by: qG 2 0 3 The participation constraint for a bad worker is given by: UG = wG
(qB )2
UB = wB
0
(1144)
(1145)
What are the binding constraints in this game and why? Q2. Let there be two bidders bidding b1 and b2 but with true values v1 and v2 . The highest bidder wins the auction at the price of the second-highest bid. The expected value for bidder 1 is then given by prob (b1 > b2 ) (v1 b2 ). Prove that honesty is the best policy in this game. Q3. A22 is a taxi company in a certain city. There are two options for owners of the company. Option one is to lend all taxis to taxi drivers on a …xed fee (F ) basis. Option two is to collide with the taxi drivers for maximisation of joint pro…t which could be divided between taxi drivers and owners according to their mutual agreement. The market demand and cost functions for this company are given as:
P = 24
0:5q; 441
C = 12q
(1146)
Prove the solutions of output, price, revenue, cost and pro…t are the same whether this taxi company operates under the …xed fee (F ) contract or under the joint pro…t maximisation agreement. [Hints: revenue: R = P:q pro…ts: (q) = P:q C vs. (q) = P:q F C] : Q4. Productivity of a worker with the level of education e is a2 and it is a1 without education e ; i.e. productivity di¤erence is a2 a1 between educated and non-educated workers. c1 e
c2 e =) c1
c2
(1147)
Show how the cost of education relative to the productivity di¤erences a2 a1 is lower for the high quality worker than for the low quality worker. Q5. Level of education signals quality of a worker. Spence (1973) model was among the …rst to illustrate how to analyse principal agent and role of signalling in the job market.Consider a situation where there are N individuals applying to work. In absence of education as the criteria of quality employers cannot see who is a high quality worker and who is a low quality worker. Employers know that proportion of workers is of high quality and (1- ) proportion is of bad quality. Therefore they pay each worker an average wage rate as: w = wh + (1
) wl
(1148)
more productive worker is worth 70000 and less productive worker is worth 30000 and =0.5 then the average wage rate will be 50000. Prove separating equilibrium is more e¢ cient than the pooling equilibrium and that it is worth for high quality workers to signal their quality by the standard of their education. Q6. Owners of a company are concerned about a project that would earn them £ 600,000 if successful. Probability of success is 60 percent if the manager puts in normal e¤ort. This probability can rise to 80 percent if the manager puts in extra e¤ort. The manager will put in extra e¤ort only if an additional payment of £ 50,000 is made above the basic salary of £ 100,000. However, it is di¢ cult for owners to monitor whether the manager is putting in extra e¤ort even if they pay an additional amount of £ 50,000. 442
a) Is it pro…table for owners to pay an extra £ 50,000 for the manager? Why does such an extra payment not automatically guarantee higher probability of pro…t? b) Design incentive compatibility and participation constraints in terms of the basic salary and a bonus so that the manager puts in extra e¤ort in return for extra payments. c) Based on above information what is the minimum payment required by the manager to put in extra e¤ort? Do owners …nd it pro…table to pay such extra payments as an incentive device? d) Consider now the case where managers can be of low or high productivity type. How can the level of education of a prospective manager signal to the employers whether he or she is of high productivity type? e) How can owners signal to the manager that they cannot be fooled by a manager who pretends to the owners of putting in extra e¤ort while actually putting in only normal e¤orts? [hint: survey of customers]
24.3
Principal agent games
Players often do not have enough information about other players in the game. They have to guess intention of other players looking at their choices. People are principals of a political game, they want better standard of living, peace and prosperity in a country but they do not have enough information about the true intention of the members of political parties act as their agents and should in principle be responsible for their principals - the common people who elect political parties frequently in the parliament. Once elected party with majority forms the government and tries to ful…l its collective interest. Political contracts are as similar as wage contracts in a labour market that are designed to match e¤orts put by a worker to their productivities in the labour maker. Political parties know their type but the people do not. The principals know the distribution of quality of various political parties , where s denotes either good or bad signal. For simplicity one can assign probability of 0.5 for observing good and of 0.5 for bad signal. Popular Principal Agent Games Principal Agent Model in Job Market: Incomplete Information and Adverse Selection Principal wants to produce output employing workers with a scheme of wage contract that matches e¤orts put by a worker to produce. Worker knows his type but the principal does not. Principal knows the distribution of quality of workers F(s), where s denotes 443
Table 87: Principal Agent Games Principal Agent Action Shareholders CEO Pro…t maximisation Landlord Tenants work e¤ort People Government Political power Manager Workers Work e¤ort Central Banks Banks Quality of credit Patient Doctor Intervention Owner Renter Maintenance Insurance company Policy holder Careful behavior either good or bad state such as probability of observing good is 0.5 and of bad 0.5. Principal o¤ers the agent a wage contract W(q). Worker accepts or rejects this contract based on self-selection and participation constraints. Objective of Principal and Agents Basically worker evaluates the utility from the wage and disutility from work and decides the amount of work to put in. Output from good workers is q (e; good) = 3e and from bad state is q (e; bad) = e If agent rejects the contract there is no work both worker and principal get zero payo¤. If worker accepts the contract Agent’s utility: e2
UA (e; w; s) = w
(1149)
Principal’s pro…t: . . Vp (q; w) = q
w
Optimal level of e¤orts by good and bad workers 444
(1150)
Good worker maximises M ax UG = wG eG
e2G = 3eG
e2G
(1151)
The …rst part is wage income and the second part of disutility of work. The optimal level of e¤orts by good agent is: 3
(1152)
2eG = 0 =) eG = 1:5
Bad worker’s Objective and Optimal E¤orts M ax UB = wB eB
1
e2B = eB
e2B
(1153) (1154)
2eG = 0 =) eG = 0:5
The principal does not know what levels of e¤orts are appropriate for good and bad workers. Principal’s Objective Principal maximises expected pro…t M ax
qG ;qB ;wG ;wB
UP = [0:5 (qG
wG ) + 0:5 (qB
(1155)
wB )]
by designing separate contracts for good (qG ; wG ) and bad workers (qB ; wB ) and . Wage for good worker: wG = q (e; good) = 3e or e = q3G Wage for bad worker: wB = q (e; bad) = e or e = qB Incentive Compatibility Constraints for Agents Self selection constraint for good worker
U G = wG
e2G = wG
qG 3
2
U G = wB
qB 3
e2B = wB
2
(1156)
Self selection constraint for bad worker UB = wB
e2B = wB
(qB )2
Participation constraints for good worker 445
UB = wG
e2G
(1157)
qG 3
U G = wG
2
0
(1158)
0
(1159)
Participation constraint for bad worker (qB )2
UB = wB Binding Constraints Participation constraint of bad worker
wB = qB2
(1160)
Self selection constraint for good worker qB qG 2 + wB 3 3 Principal’s Optimal Solution
2
=) wG =
wG =
qG 3
2
+ qB2
qB 3
2
(1161)
Principal includes agents’ optimal choices into his utility function
M ax
qG ;qB ;wG ;wB
UP = [0:5 (qG
wG ) + 0:5 (qB
wB )]
Including binding constraints of agents: h qG 2 M ax UP = 0:5 qG + qB2 3 qG ;qB ;wG ;wB
qB 2 3
+ 0:5 (qB
qB2 )
i
Now principal decides how much to produce from each type of worker First order conditions with respect to qG and qB 2qG 9
@UP = 0:5 1 @qG @UP = 0:5 @qB
2qB
2qB 9
+ 0:5 (1
= 0 =) qG = 4:5
(1162)
2qB ) = 0 =) 34qB = 9 =) qB = 0:265 (1163)
Incentive Compatible First Best Choices of Good and Bad Worker
446
Now wages can be found from the constraints wB = qB2 = (0:265)2 = 0:07
wG =
qG 3
2
+ qB2
qB 3
2
=
4:5 3
2
+ (0:265)2
(1164)
0:265 3
2
= 2:32 (1165)
Thus in the presence of information asymmetry , the e¤orts by the good worker is at the …rst best level as the bad e¤ort by him is not as attractive as the good e¤ort. It is not pro…table for good worker to pretend as a bad worker. Good worker is not attracted by the contract for bad worker. It is very costly for the bad worker to accept the contract of good worker. Bad worker’s …rst best to put low e¤ort. Incentive compatible game on renting a piece of agricultural land If a worker puts x amount of e¤ort, the land produces y = f (x) Then the land owner pays worker s(y). The land owner wants to maximise pro…t = f (x) s(y) = f (x) s(f (x)) Worker has cost of putting e¤ort c(x) and has a reservation utility, u The participation constraint is given by . s(f (x)) c(x) u Including this constraint maximisation problem becomes max = f (x) s(f (x)) subject to sf (x) c(x) u Solution: marginal productivity equals marginal e¤orts f 0 (x)) c0 (x) Incentive compatible game on renting a piece of agricultural land (a) renting the land where the workers pays a …xed rent R to the owner and takes the residual amount of output, at equilibrium f (x )
c(x )
R=u
(1166)
(b) Take it or leave it contract where the owner gives some amount such as B
c(x ) = u 447
(1167)
(c) hourly contract s(f (x)) = wx + K
(1168)
(d) sharecropping, in which both worker and owner divide the output in a certain way. In (a)-(c) burden of risks due to ‡uctuations in the output falls on the worker but it is shared by both owner and worker in (d). Which of these incentives work best depends on the situation.
448
25
L23: Repeated Games and Auctions
Cooperative Solution Market demand for a product is P = 130
(1169)
(q1 + q2 )
Cost of production for each of two …rms is . (1170)
Ci = 10qi
If played in…nite number of time two …rms form a cartel and monopolise the market. Each will supply only 30, set market price to monopoly level at £ 70 and divide total pro…t £ 3600 equally; each getting £ 1800. This is shown by (1800,1800) point in the diagram. It pays to cooperate in the long run; it is sub-game perfect equilibrium. = (130
Q) Q
10Q = 130Q
Q2
10Q
120 @ = 130 2Q 10 = 0 =) Q = = 60 @Q 2 P = (130 Q) = 130 60 = 70; C = 10Q = 10 60 = 600; = P Q C = 60 70 600 = 3600 Non-Cooperative Nash Equilibrium
449
(1171) (1172)
If any one …rm cheats and tries to supply more in order to get more pro…t; it will be found out by another …rm. It will react to this. Game will be non-cooperative with resulting in a Cournot Nash equilibrium. with each …rm producing 40 units, market price of 50 and each getting £ 1600 pro…ts. 1
= (130
(q1 + q2 )) q1
10q1 and
2
= (130
(q1 + q2 )) q2
10q2
with reaction functions 2q1 + q2 = 120 and q1 + 2q2 = 120 Total supply is 80, each supplying 40 and making pro…t of 1600 and market price 50. Trigger Strategy and Perpetual Punishment If …rm 1 plays Cournot game but …rm 2 still plays cartel and supply just 30. Then from the …rm 1’reaction function . 2q1 + q2 = 120 q1 = 60
1 q2 = 60 2
1 (30) = 45 2
(1173)
If …rm 1 supplies 45, market price will be . P = 130
(q1 + q2 ) = 130
45
(1174)
30 = 55
This makes pro…t margin of …rm 1 to be 45 and its pro…t . 45q1 = 45 45 = 2025
1
= (55q1
10q1 ) =
Firm 2 will …nd out that …rm 1 has cheated. It will also produce according to its reaction curve. Thus the Nash equilibrium will result with each …rm producing 40 and earning 1600 pro…t for the rest of the periods and the market price will be 50.
450
For whom is it pro…table to Cheat? Does …rm 1 gain or lose by deviation from the agreement. For this evaluate the in…nite series of pro…ts in deviation and in compliance with agreement. Present value of pro…t in case of cheating 2 + :::: + ::: 1 = 2025 + 1600 + 1600 2025 1600 + 1600 + 1600 + 1600 2 + :::: + ::: 1 = (Note just with –and + 1600) Using operator to maintain a constant payo¤ from the game i h 1600 ) + 1600] = 425 425 + 1600 = (1 ) 1 = (1 ) 425 + (1 ) = [425 (1 2025 425 By comparing pro…ts with and without cheating 9 2025 425 < 1800 or ; 425 > 2025 1800; > 225 =) > 17 425 Whether the …rm 1 will stick to agreement or not depends on whether its discount 9 9 factor if greater than > 17 . For discount factor < 17 it is bene…cial to stick to the agreement, which is very high, about 53 percent.
Home Work 451
Show above results in a diagram Illustrate repeated game for multiple periods using branch nodes Workout Bertrand type competition for above game and illustrate "cut-throat" price competition in a diagram.
25.1
Auction Game
Types of Auction First price, sealed-bid: person who bids the highest amount gets the good. Second-price, Sealed-bid: Each submit a bid. Higher bidder wins and pays second-highest bid for the good. Dutch Auction: Seller begins from very high price and reduces it until someone raises a hand. English Auction: Begins with very low price, bigger drops out by raising a hand. Which one of these four mechanism is good for the seller?? Auction: Vickrey-Clerk-Grove (VCG) mechanism Honesty is the best policy in Vickery auction; truth telling is the winning strategy. Proof Let there be two bidders bidding b1 and b2 but with true values v1 and v2 . Highest bidder wins the auction at the price of the second-highest bid. English auctions and second-highest sealed-bid auctions are equivalent. Expected value for bidder 1 is then given by prob (b1 > b2 ) (v1
b2 )
(1175)
If (v1 > b1 ) it is in the best interest of bidder 1 to raise the probability of winning prob (b1 > b2 ) , this can happen when (v1 = b1 ) 452
Similarly If (v1 < b2 ) then it is in the interest of bidder 1 to make prob (b1 < b2 ) as small as possible. It happens when .(v1 = b1 ) Thus the truth telling is the best interest in such auction. Auction: Financing Mechanism for Public Goods Let x be a public good such as streetlight or road; x = 1 if it is provided x = 0 if not. If state knew that how much each person is willing to pay for this it could bill e¢ ciently. Each would pay according to the value they put in such public good. Unfortunately it is impossible to know preferences of individuals. Individuals do not tell true value when asked that how much they are ready to pay for this. Let N individuals be indexed by i. Then the utility from the public good to an individual i is given by Ui (x). There is free rider problem with public goods. Individuals may underreport their utility thinking that others will pay higher for it if they act like this but they will have opportunity of full bene…t. Under Veckrey-Clark-Grove mechanism it is in the best interest of individuals to tell the truth. Under Grove mechanism each individual is asked to reported his her utility; which is ri (x). . Then the state chooses x* that maximises the sum of reported utilities N N P P R = ri (x): Each individual receives a side-payment Ri = ri (x):. i=1
j6=1
With side payment the total utility of an individual is Ui (x) +
N X
ri (x)
(1176)
i=1
State chooses x to maximise ri (x) +
N X ri (x)
(1177)
i=1
Therefore it is in the best interest of individual to tell the truth Ui (x) = ri (x). All agents tell truth like this and this mechanism generates e¢ cient outcome (see Varian HR (2010):36). 453
References [1] Binmore K. (1990) Fun and Games: A text on Game Theory, Lexington, Heath. [2] Cripps, M.W.(1997) Bargaining and the Timing of Investment, International Economic Review, 38:3 :Aug.:527-546 [3] Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. [4] Gardener R (2003) Games of Business and Economics, Wiley, Second Edition. [5] Hey J. D. (2003) Intermediate microeconomics, McGraw Hill. [6] Holt Charles (2007) Markets, Games and Strategic Behavior, Pearson, . [7] Lambartini Luca (2011) Game Theory in the Social Sciences: A Reader-friendly Guide, Routledge. [8] Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. [9] Kreps D. M. (1990) A Course in Microeconomic Theory, Princeton. [10] Osborne M.J. and A. Robinstein (1994) A course in game theory, MIT Press. [11] Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with Calculus, Pearson. [12] Rasmusen E(2007) Games and Information, Blackwell,. [13] Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th ed.
454
26
L24: Mechanism Design Game
Why Mechanism Design for Price Discrimination: Low Cost Airlines Example Economy and Business Class Ticket Problem for Airlines (Based on Dixit et. al. (2009)) Two types of travellers: economy and business Assume 100 travellers and 70 of them economy type tourists and 30 business type …rst class. Cost of Reservation Price Airline’s Pro…t the Airlines Tourists Business Tourists Business Economy 100 140 225 40 125 First Class 150 175 300 25 150 Economy class tickets cost less than the business class. Business traveller is ready to pay higher price than economy class for both economy and …rst class but the airlines cannot separate them out. Business traveller may well buy economy class ticket rather then business class. Airlines likes to build a mechanism so that business class customers buy business class tickets and economy class customers buy economy class ticket. What is the pro…t to the airlines if it knows reservation prices of tourists and business group of travellers? How would this pro…t change in business type buy the economy class ticket? What is the incentive compatible price that the airlines can o¤er to the business group? What would happen if the split between the business and economy class is 50/50? What will be the optimal reaction of the airlines? Incentive Compatible Mechanism Pro…t in an ideal scenario ( perfect price discrimination; if the airlines knew each customer type) 455
30(300
150) + (140
100) (70) = 30 150 + 40 70 = 4500 + 2800 = 7300
(1178)
Business travellers have consumer surplus of 225 -140 = 85 in economy class ticket. For this all 30 of may decide to buy economy class ticket. Then the pro…t of the airlines when the airlines fails to screen customers will be (140
100) (100) = 4000
(1179)
Airlines should give consumer surplus of 85 to business traveller and charge them (300-85) = 215. This will alter their pro…t 30(215
150) + (140
100) (70) = 30 65 + 40 70 = 1950 + 2800 = 4750
(1180)
Incentive Compatible and Participation Constraints Airline initially does not have enough information on types of customers It should design incentive compatible pricing scheme so that business class travellers do not defect to economy class. This requirement is contained in the incentive compatible constraint. If it charges 240 for the business class then the their consumer surplus will be equal (300-240) = 60 from business class travel and (225-165)=60 However 140 is the maximum the tourist class traveller is ready to pay. If the airline raises price to 165 they will lose all tourist travellers. Mechanism requires ful…llment of the participation constraint. Airlines should operate taking account of the participation constraint of tourists and incentive compatible constraint of the business travellers. X < 140 is the participation constraint; incentive compatible constraint is 225 -X < 300-Y or Y < X+75 Mechanism when the composition of travellers change Charging 215 for the business class and 140 for the economy class is the solution to the mechanism design problem. 30(215-150)+(140-100) (70) =30 65+40 70 =1950+2800=4750 456
Suppose the composition of travellers changes to 50% of each. Pro…t with the above price mechanism
50(215
150) + (140
100) (50) = 50 65 + 40 50 = 3250 + 2000 = 5250
(1181)
It is more pro…table to scrap the tourist class tickets instead and charge the business class its full reservation price 50(300
150) = 50
150 = 7500
(1182)
There are relatively few customers but all are willing to pay higher price. There is no problem of screening as the airlines now does not serve to the tourist class at all.
26.1
Mechanism to ensure high e¤orts by a CEO Owners of a company are concerned about a project that would earn them 600,000 if successful. Probability of success with normal e¤ort from the manager is 60 percent and this can increase up to 80 percent if the manager puts extra e¤orts. The basic salary of the manager is 100,000. He would put extra e¤orts only if he is paid additional amount of at least 50,000. Owners cannot monitor whether the manager is putting high or low e¤orts.
a) Is it pro…table to pay extra for the manager? Pro…t without paying extra: 0.6 * 600,000 - 100,000 = 260, 000 Pro…t with extra incentive payment: 0.8 * 600,000 - 150,000 = 330, 000 Extra payment can make up to 70,000 with probability of 0.8. Once extra payment is made how can owners make sure that he puts extra e¤orts? This requires evaluation of incentive compatibility and participation constraints. Mechanism to ensure high e¤orts by a CEO a) Incentive compatibility constraint (s + 0:8b)
(s + 0:8b) > 50; 000 457
(1183)
0:2b > 50; 000
(1184)
(s + 0:8b) > 150; 000
(1185)
b = 250,000 b) Participation constraint:
s = 150; 000
0:8b;
s = 150; 000
0:8 (250; 000) =
50; 000
(1186)
It is not possible to hire manager with negative salary. At most managers can be conditioned to bonus payment but with zero salary. Mechanism to ensure high e¤orts by a CEO (0 + 0:8b)
(250; 000) > 150; 000
200; 000 > 150; 000
(1187) (1188)
Pay 200,000 and the manager will put maximum e¤ort. c) Is it pro…table to pay extra 200,000 as an incentive payment? Pro…t with incentive payment 0.8 * 600,000 - 200,000 = 280, 000 Pro…t without incentive payment 0.6 * 600,000 - 100,000 = 260, 000 Thus pro…t increases by 20,000 with the incentive payments. 26.1.1
Mechanism design in renting lands
Proposition 1: Results of …xed fee contract and joint pro…t maximisation are equivalent Proposition 2: Hire contract is incentive incompatible and leads to production ine¢ ciency Proposition 3: Moral hazard problem and production ine¢ ciency exists in revenue sharing contingent contract Proposition 4: Pro…t sharing contract is e¢ cient and free of moral hazard problem Price and cost P = 24 Revenue
0:5q 458
C = 12q
(1189)
(1190)
R = P:q Mechanism design in renting lands Under the joint pro…t maximisation agreement (q) = P:q
C = (24
0:5q) q
12q = 24q
0:5q 2
(1191)
12q
Under the …xed fee contract tenant maximises (q) = P:q
C
F = (24
0:5q) q
12q
F = 24q
0:5q 2
12q
F
(1192)
Under both these arrangements 0
(q) = 24
q
(1193)
12 = 0
q = 12; p = 18; R = 216; C = 144;
(1194)
(q) = 72
This is the total pro…t. It is divided between the tenant and the landlord by their mutually agreed arrangement. Under the …xed fee contract landlord may …x the amount that he needs at 48. Then the residual 24 pro…t goes to the tenant. This arrangement achieves production e¢ ciency, is incentive compatible, ful…ls the participation constraint and motivates to put the optimal e¤ort and solves the moral hazard problem. 26.1.2
Hire contract
Landowner can hire workers in …xed fee basis, say 12 per unit of output a. This does not motivate tenant to work because his cost per a is also 12 and so does not make any pro…t. Landlord has to raise payment to tenant to say 14 to motivate him to work. Then the pro…t maximisation problem of the landlord will be (q) = P:q
C = (24
0:5q) q 459
14q = 24q
0:5q 2
14q
(1195)
0
(q) = 24
q
(1196)
14 = 0
q = 10; p = 19; R = 190; C = 120;
LL
(q) = 50;
T
(q) = 20
(1197)
The tenant has incentive to overproduce whenever is paid more than 12. Revenue sharing contract Let the landlord enter into a revenue sharing contract whereby she gets 41 th of the revenue and leavening 34 of revenue to the tenant who also bears all production cost. The pro…t function of the tenant is now modi…ed as 3 (q) = P:q 4 0
(q) = 6
C=
3 (24 4
3 4 q=0)q= 4 3
q = 8; p = 20; R = 160; C = 96; = 120;
LL
(q) =
0:5q) q
12q
(1199)
6=8
T
(q) =
(1198)
3 (160) 4
1 (160) = 40 4
(1200)
Pro…t of tenant = 120 - 96 =24 This level of production is not incentive compatible for the land-lord who would be interested in maximising revenue by producing 24 Pro…t sharing contract Now let us assume the landlords and tenants enter into a pro…t sharing deal, say 1/3rd of pro…t goes to the tenant and 2/3rd to the landlord. 1 3
(q) =
0
1 (P:q 3
(q) = 4
C) =
1 24q 3
1 q=0 )q=3 3 460
0:5q 2
4 = 12
12q
(1201)
(1202)
q = 12; p = 18; R = 216; C = 144; LL (q) = 48; T (q) = 24
(q) = 72; (1203)
There are many other situations, including optimal tax designs, optimal price discrimination, fund management, management of theme-park, renting of buildings, collection of taxes or tari¤s, union-management contracts, where these types of models have been applied. Problem 1. Given the market demand and cost functions P = 24
0:5q
C = 12q
(1204)
Prove following four propositions regarding e¢ cient contract. Proposition 1: Results of …xed fee contract and joint pro…t maximisation are equivalent Proposition 2: Hire contract is incentive incompatible and leads to production ine¢ ciency Proposition 3: Moral hazard problem and production ine¢ ciency exists in revenue sharing contingent contract Proposition 4: Pro…t sharing contract is e¢ cient and free of moral hazard problem
461
27
L25: E¢ ciency conditions of the market system
27.1
E¢ ciency in consumption
Marginal rate of substitution between two products should equal price ratios for a certain consumer Ux Uy Un = = ::::: = Px Py Pn
(1205)
Allocation is Pareto e¢ cient if it is not possible to make one person better o¤ without making another worse o¤. L = U (X; Y ) + [T (X; Y )]
(1206)
@L @U @T = + =0 @X @X @X
(1207)
@L @U @T = + =0 @Y @Y @Y
(1208)
@L = @
(1209)
[T (X; Y )] = 0 @U @X @U @Y
=
@T @X @T @Y
(1210)
@Y = RP TX;Y (1211) @X This is the optimal point in the production possibility frontier. See the trade model. M RSX;Y =
27.1.1
E¢ ciency in production
If it is not possible to more of one good without reducing the production of another good. X = f1 (K1 ; L1 ) + f2 (K2 ; L2 )
(1212)
K1 + K2 = K
(1213)
L1 + L2 = L 462
(1214)
X = f1 (K1 ; L1 ) + f2 (K
L1 )
(1215)
@f2 =0 @K2
(1216)
K1 ; L
@X @f1 @f2 @f1 = + = @K1 @K1 @K2 @K1
@X @f1 @f2 @f1 @f2 = + = =0 (1217) @L1 @L1 @L2 @L1 @L2 Marginal productivity of capital and labour inputs are same across both sectors:
27.1.2
@f2 @f1 = @K1 @K2
(1218)
@f1 @f2 = @L1 @L2
(1219)
E¢ ciency of trade (Exchange)
If it is possible to increase welfare of one country without harming another country. M RSx;y = 27.1.3
Px Ux = Uy Py
= ::::::: =
M RSx;y =
1
Ux Px = Uy Py
(1220) N
E¢ ciency in public goods
When i = 1; :::N individuals live in a society then, the social marginal utility of public goods is the sum of the utilities from public good for individuals SM UP = SM UP1 + SM UP2 + :::: + SM UPN SM RSP;G =
SM UP1 SM UP2 SM UPN SM UP = + + :::: + M UGi M UGi M UGi M UGi SM RSP;G = RP TP;G
(1221) (1222) (1223)
Rate of transformation of private to public goods should equal the social rate of substitution of private to public goods. 27.1.4
Theory of second best
When the optimal point is not achievable, other points in the e¢ ciency frontier are not necessarily optimal. (draw a diagram to prove). 463
28
L26: Externality What would happen to public parks if city councils do not maintain? Personal and social bene…ts of a beautiful garden? Why private market does not produce e¢ cient amount of education and health? Why will market produce excessive amount of water, air or noise pollution? Why many cities in England are introducing congestion charges?
28.1
Negative externality
Negative externality production of electricity and pollution and food production Electricity production using coal generates electricity as well as pollution. This pollution raises production cost in the food industry. Cost of electricity production when the environment is not taken into account Ce = e2 (x 3) and its pro…t function is: e = Pe e e2 (x 3)2 the cost of food production Cf = f 2 + 2x and its pro…t f = Pf f f 2 2x. Pollution adds extra cost in food production. Private market solution e2 (x 3)2 e = Pe e @ e = Pe 2e = 0 =) Pe = 2e and hence supply of electricity @e e= = Pf f @ f = Pf @f f
Pe 2
(1224)
f 2 2x 2f = 0 =) Pf = 2f and hence supply of electricity f=
Pf 2
(1225)
Here pollution is produced more than optimal. @ e = 2 (x @x Socially optimal solution : =
e
+
e
= Pe e
(1226)
3) = 0 =) x = 3:
e2
(x 464
3)2 + Pf f
f2
2x
(1227)
@ e @e
= Pe
2e = 0 =) Pe = 2e and hence supply of electricity e=
@ f @f
= Pf
Pe 2
(1228)
2f = 0 =) Pf = 2f and hence supply of electricity f=
Pf 2
(1229)
@ = 2 (x 3) + 2 = 0 =) x = 2: @x Social solution generates less pollution than the market solution. 28.1.1
(1230)
Positive externality
A classic example of positive externality: bees pollinate apple trees and they get materials for honey from apples. For instance cost of producing apples is Ca = a2 and cost of producing honey Ch = h2 a, Private market solution Firms maximise own pro…t independently: a
by marginal cost pricing rule and supply of apples
a2
= Pa a @ a @a
a=
(1231) 2a = 0 =) Pa = 2a and hence
= Pa Pa 2
(1232)
Similarly h
= Ph h
h2 + a
(1233)
Supply of honey by the private market @ a = Ph 2h = 0 =) Ph = 2h .and @a Ph : (1234) 2 Private market does not consider positive externality. Now consider a social planner that produces both to maximise joint pro…t: h=
= Pa a Then optimal apple supply is :
a2 + P h h 465
h2 + a
(1235)
@ a @a
= Pa
1 .and
2a + 1 = 0 =) Pa = 2a a=
Pa 1 + 2 2
(1236)
Ph 2
(1237)
Optimal honey supply is @ h = Ph 2h = 0 =) Ph = 2h .and @h h=
It is optimal to produce more apples taking account of its positive externality.
466
29
L27: Uncertainty and Expected Utility
Future is uncertain. Some times it turn outs to be better than normally expected and other times it can be a lot worse. People assign probabilities to good and bad events while making decisions and protect themselves by buying insurances against any unwanted contingencies. Individuals calculate expected wealth and utility from expected wealth in order to …nd out the optimal level of insurance. This section brie‡y explains the degree of risk aversion and the amount of insurance premium that individuals like to pay to ensure against uncertainty. Future is uncertain; two states - high wealth and low wealth. Contingent wealth in high state is WH and in low state is WL : Probability of high wealth
H
and low wealth
L
:
Utilities from high wealth u (WH ) and low wealth u (WL ) : Expected wealth EW = Expected utility EU =
L WL
+
L u (WH )
H WH :
+
H u (WL ) :
Faced with uncertainty people maximise expected utility (von-Neumann-Morgentern preferences). People are ready to pay some amount to insure themselves against the possible risk. Preferences of risk averse consumer Utility functions of risk averse individual (1238)
U (W ) = ln(W ) U (W ) =
p
1
(1239)
W =W2
Expected utility theorem: utilities under uncertainty are additively separable (von-Neumann-Morgenstern Utility) M ax EU =
H
:u (WH ) +
Utility from expected wealth 467
L:
u (WL )
(1240)
U (EW ) = ln (
H
WH +
L
WL ) = ln (EW )
(1241)
Certainty equivalent wealth CEW = exp (ln (EU ))
(1242)
Maximum insurance against risk and a measure of risk aversion Maximum insurance that person is ready to pay to cover risk: Insurance = EW
CEW
Table 88: Uncertainty of Income High Probability 0.75 Income 5000 Expected Income 3750
(1243)
and Wealth Low 0.25 1000 250
Expected wealth EW = L WL + H WH = 0:75 (5000) + 0:25 (1000) = 4000 Do people maximize expected wealth? No. They maximize expected utility. Maximum insurance against risk and a measure of risk aversion EU = H :u (WH ) + L : u (WL ) = H : ln (WH ) + L : ln (WL ) = 0:75 ln (5000) + 0:25 ln (1000) = 6:388 + 1:727 = 8:115
(1244) (1245)
Certainty equivalent wealth CEW = exp (ln (EU )) = exp(8:115) = 3344:26
(1246)
Maximum insurance that person is ready to pay to cover risk: Insurance = EW
CEW = 4000
3344:26 = 655:74
(1247)
After paying 655.74 for the insurance company, this person can be sure that no matter high or low state 3344:26 is guaranteed. Can sleep well! Risk pooling is possible. If 100 people ensure like this revenue of insurance company is 65575; only 468
25 percent people claim (2444.26 25 = 61106:5). Pro…t to the insurer is 65575 61106.5 = 4468.5. Measure of risk aversion Arrow-Pratt (1964) measure of risk aversion r(W ) =
U 00 (W ) >0 U 0 (W )
(1248)
r(W ) =
U 00 (W ) 0 W
(1252)
with Cobb-Douglus type preferences 1
(1253)
U (W ) = W 2
r(W ) =
U 00 (W ) = U 0 (W )
1 2
1 W 2 1 W 2
1 2
1 2
1
=
1 1 >0 2W
Maximum insurance against risk and a measure of risk aversion Risk lovers U (W ) = exp(aW ) r(W ) =
U 00 (W ) = U 0 (W )
a2 W 2 exp(aW ) = aW exp(aW ) 469
aW < 0
(1254)
(1255) (1256)
Risk neutral (1257)
U (W ) = aW r(W ) =
0 U 00 (W ) = =0 0 U (W ) a
(1258)
St Petersberg Paradox (Bernoulli Game) People are ready to play a small amount for a lottery but do not want to risk a huge amount in it. People care about utility. How much should one pay to play a game that promises to pay 2n if the head turns up in the nth trial? Answer 1.39. How? Expected payo¤ is in…nite E( )=
1
2+
22 +
2
3
23 + ::: +
n
2n = 1 + 1 + 1 + ::: + 1 = 1
1 1 1 > 2 = 2 > 3 = 3 > :::::: > 2 2 2 but the Expected Utility is …nite here 1
E (u) =
E (u) =
=
n
=
1 2n
(1259)
(1260)
ln (2n ) < 1
(1261)
1 1 1 1 ln (2) + 2 ln 22 + 3 ln 23 + ::: + n ln (2n ) < 1 2 2 2 2
(1262)
1
ln (2) +
E (u) =
2
1 X 1 i=1
2i
ln 22 +
3
ln 23 + ::: +
n
1 X i i ln (2) = ln (2) = ln (2) 2 = 1:39 2i i=1
(1263)
People buy lotteries for small amount but not for more (Allais Paradox) St Petersberg Paradox (Bernoulli Game)
E (u) =
1 1 1 1 ln (2) + 2 ln 22 + 3 ln 23 + ::: + n ln (2n ) < 1 2 2 2 2 E (u) =
1 X 1 i=1
2i
1 X i i ln (2) = ln (2) = ln (2) 2 = 1:39 2i i=1 470
(1264)
(1265)
People are ready to pay small amount to buy lotteries but do not want to risk large sums (Allais Paradox) Asset Markets Utility is derived from return and risk (return is measured by mean and risk by standard deviation) as: U (W ) = U ( w ; W ) (1266) Average return from portfolio of risky and risk free assets ( for 0 < x < 1 S P
s
= 1)
s=1
rx =
S X
(xms + (1
x) rf )
s
=
s=1
S X
xms
s
+ (1
x) rf
S X
(1267)
s
s=1
s=1
rx = xrm + (1
(1268)
x) rf
Variance of the portfolio is
2 x
=
S X
[(xms + (1
x) rf )
rx ]
2 s
=
S X
(xms
xrm )2
s
= x2
2 m
(1269)
s=1
s=1
Price of risk x
=x
(1270)
m
price of risk from return-risk diagram p=
rm
rf
(1271)
m
Marginal rate of substitution between return and risk should equal this price ratio M RS
i
=
@U (W )=@ @U (W )=@
=
rm
risk of asset i risk of stock market m ;cost of risk i mp i
amount of risk Price of risk
;
=
471
rf
(1272)
m
(1273)
Risk adjustment =
i mp
rm
=
i m
rf
=
m
i
(rm
rf )
(1274)
Risk adjusted returns should be equal in all assets ri
i
(rm
rf ) = rj
j
(rm
rf )
(1275)
f
(rm
rf )
(1276)
If one of the assets is the risk free asset ri f
i
(rm
rf ) = rf
=0 ri = rf +
i
(rm
rf )
(1277)
This the theory behind the capital asset price model (CAPM). Homework Crops are worth 50000. Probability of storm or ‡ood is 0.01 percent. Crop is completely destroyed if a storm or ‡ood occurs. How much insurance is ideal in this business? An individual has a car worth £ 10000. Probability of accident is 0.01 percent and the car will be useless if it meets any accident. How much insurance should this person pay? John has a house worth 200,000. Probability of …re is 0.05 percent. House is worthless after the …re. How much insurance should John pay for the …re insurance? Jane has a business worth 1 million. Probability of bankruptcy is 0.02 percent. How much insurance should Jane pay to protect against such bankruptcy?
References [1] Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. [2] Harsanyi J.C. (1967) Games with incomplete information played by Baysian Players, Management Science, 14:3:159-182. [3] Machina M. (1987) Choice under uncertainty: problems solved and unsolved, Journal of Economic Perspective, 1:1:121-154. [4] Moore J. (1988) Contracting between two parties with private information, Review of Economic Studies, 55: 49-70. 472
[5] Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th edition.
473
29.1
Problem 11: Uncertainty and insurance
Q1. What are the measures of risk aversion for consumers with following utility functions? Which of these consumers is risk-averse, which one is risk neutral and which one is a risk lover? (a) Logarithmic utility in wealth:
U (W ) = ln(W ) 1
(b) Cobb-Douglus type utility:
U (W ) = W 2
(c) Linear utility:
U (W ) = aW
(d) Exponential utility:
U (W ) = exp(aW )
Q2. The amount of wealth in the good state is W . If a bad event occurs there will be a loss (L) and the probability of a loss is p: The owner of the property can insure for amount (q) paying premium rate (m) : The expected utility maximisation problem of the individual is implicitly written as: maxEU = p:u (W q
L
mq + q) + (1
p) u (W
mq)
(1278)
The pro…t maximising condition of the insurance company with perfect competition in the insurance market is: p (1
m) q
(1
(1279)
p) mq = 0
A risk averse consumer likes to get the same marginal utility whether in the good or bad state u0 (W
L
mq + q) = u0 (W
mq)
(1280)
Prove that the optimal premium rate equals the probability of loss (p) ; and that it is optimal for an individual facing uncertainty in this way to purchase full insurance. Q3. Utility function of an individual is given by U (W ) = ln(W ) , where U is the utility and W is the level of wealth. Is this a risk loving, risk averse or risk neutral individual? a. Draw this utility function in a diagram in space and explain the economic meaning underlying the curvature of the utility function. 474
b. Find expected utility of this individual if probability of high wealth is and that of low wealth is . Show what is the certainty equivalent income and the amount of insurance that this person is ready to pay against income uncertainty. c. Probability of getting high wealth of 5000 is 0.4 against 0.6 probability of getting low wealth of 2500. What is the expected wealth of this person? d. What is the utility of expected wealth? e. What is the value of expected utility form high and low values of wealth? f. Find the certainty equivalent income and the maximum amount that this individual will be ready to pay for the insurance? Q4. Utility from wealth for a person living in Fair…eld village is given by , where U is the utility and W is the level of wealth. This person has a prospect of good income of 4000 with probability 0.4 and prospect of low income of 1000 with probability of 06. How much would this person pay to insure against such income uncertainty?
475
30
L28: Uncertainty and Insurance
Uncertainty of Good Times and Bad Times Future is uncertain; can be good or bad; two states. Contingent consumption in good times Cg and in bad times Cb Probability of good times
g
and of bad times
b
Prices of good times pg and of bad times pb Utilities from contingent consumption in good times u (Cg ) and in bad times u (Cb ) Budget constraint .I = Pg Cg + Pb Cb Consumer problem under uncertainty Expected utility theorem: utilities under uncertainty are additively separable (von-Neumann-Morgenstern Utility) M ax EU =
g u (Cg )
+
(1281)
b u (Cb )
Subject to (1282)
I = P g C g + Pb C b Lagrangian for constrained optimisation L=
g u (Cg )
+
b u (Cb )
+ [I
P g Cg
P b Cb ]
(1283)
First order conditions for optimisation @L = @Cg
0 gu
(Cg )
Pg = 0
(1284)
@L = @Cb
0 bu
(Cb )
Pb = 0
(1285)
@L = I P g Cg P b Cb = 0 (1286) @ Dividing (1284) by (1285) gives the marginal rate of substitution between good and bad times 476
gu b
0
u0
Pg (Cg ) = ; (Cb ) Pb
Pg = Pb
g
(1287)
b
Fair market for contingent goods implies ratio of prices in good and bad states equals ratio of respective probabilities. Utility and allocation in good and bad times u0 (Cg ) =1 u0 (Cb )
(1288)
u0 (Cg ) = u0 (Cb )
(1289)
Since preference are symmetric over the states (1290)
Cg = Cb
consumer likely to fully insure against any risk; like to have same consumption in both good and bad states. Represent above result in a diagram with certainty line. budget line and indi¤erence curve u (Cg ; Cb ) : It is possible that individuals like to consume a bit more in good times and a bit less in bad times. Optimal Demand for Insurance There is certain wealth (W ), if an event occurs there will be a loss (L). probability of loss is (p) : Owner of the property can insure for amount (q) paying premium (m) Expected utility maximisation problem is maxEU = p:u (W q
L
mq + q) + (1
mq)
(1291)
mq) m = 0
(1292)
p) u (W
Choose q to maximise EU using the …rst order condition as: @EU = p:u0 (W @q
L
mq + q) (1
m)
(1
p) u0 (W
Optimal condition u0 (W L u0 (W
mq + q) (1 p) m = mq) p (1 m)
Pro…t function of the insurance company 477
(1293)
= (1
p) mq
p (1
(1294)
m) q
Assume perfect competition in the insurance business, pro…t is zero p (1
m) q
(1
(1295)
p) mq = 0
The premium rate equals the probability of loss in equilibrium (1296)
p=m This is actuarially fair insurance. Insert (??) into (1292) p:u0 (W
L
mq + q) (1
p)
u0 (W
(1 L
p) u0 (W
mq) p = 0
mq + q) = u0 (W
mq)
(1297) (1298)
Optimal demand for insurance For risk averse consumer u00 (W ) < 0 W
L
mq + q = W
mq
q=L
(1299) (1300)
Consumer completely insures (q) against the loss (L). Risk spreading and risk diversi…cation Risk can be spread among individuals. Imagine a society with 1000 individuals each endowed with £ 35000. Each faces a risk of losing £ 10000 with probability of 1 percent. Only 10 person in aggregate face this risk. It is a big loss for each individual as it can happen to each of them. Now they create an insurance market. Each contributes 100 to mitigate this uncertainty. This creates 100,000 insurance fund. This is enough to ensure each for any eventual loss. 478
Every one will be certain (ensured) to have 34,900.: endowment minus insurance contribution. This is an example of risk spreading. Risk is spread (divided) among all. Each pays 100 to ensure against loss of 10000. Risk spreading and risk diversi…cation Risk can be diversi…ed by choosing an appropriate portfolio. Consider an excellent example from Varian (2010) on sunglasses and raincoat. You have 100 to invest. Probability of rain or shine is equally likely. You can invest only in sunglasses or raincoats or split 50/50 in each. Value of sunglass investment will double if it is sunny or down by half if it is rainy. Similarly value of investment will be double if rainy and down by half if sunny. If invested all in one then at the end of the day the expected value is 0.5(50)+0.5(200)=125. There is considerable risk. If case of splitting 50/50 the expected value of investment is [0:5(25) + 0:5(100)]+[0:5(100) + 0:5(25)]=125. Thus 125 is guaranteed no matter rainy or shiny. Diversi…cation has ensured 125. Do not put all your eggs in one basket. Homework A person has wealth worth £ 35000. There is 1 percent probability of loss. If this event occurs there is a loss of 10,000. This individual is risk neutral. 1) What is expected wealth without insurance? 2) This person can buy insurance equal to amount K to cover insurance by paying K insurance premium, where is the premium rate. Write individuals budget in case of accident and in case of no accident. 3) Write the expected utility function of this person. Assume that person receives utility from the wealth that he has. 4) What is expected pro…t of the insurance company? 5) Prove that premium rate equals the probability of the event. 6) Prove that consumption is same in both states with insurance. L) Prove that it is optimal to fully insurance against the loss and that is actuarially fair insurance. 479
References [1] Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. [2] Hirshleifer J and J G Riley (1992) The Analytics of Uncertainty and Information, Cambridge. [3] Perlo¤ J. M. (2008) Microeconomics: Theory and Applications with Calculus, Pearson. [4] Nicholson W. (1985) Microeconomic Theory: Basic Principles and Extensions, Norton. (Holt-Saunders). [5] Rasmusen E(2007) Games and Information, Blackwell, ISBN 1-140513666-9. [6] Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th edition [7] http://cepa.newschool.edu/het/home.htm.
480
31
L29: Input-Output Model
The input output tables can provide a comprehensive framework to think about the much complicated economic relations among producers, consumers, investors and the public sector and the Rest of the World sectors and also for the distribution of resources between workers and the owners of capital among production sectors in an economy. These can also be applied to estimate negative or positive impacts of public policy measures such as taxes and transfers that a¤ect on output, employment and investment. The short run impacts of changes in consumption, government spending, investment or the net exports in output, employment, capital stock and distribution of resources in the economy are captured by a multiplier matrix. IO model can also be applied to project manpower requirements across sectors. More important use of the IO table remains as a micro consistent data set to calibrate the general equilibrium model of an economy allowing income and substitution impacts to occur across sectors of any changes in relative prices and to illustrate how various components of the economy are coordinated by the market mechanism for allocation of scarce economic resources. This section provides a simple introduction on how an IO model could be applied for policy analysis. An example of input-Output Table Structure of an input-output table (snap-shot of the economy for a given time) IO F V A T rasf ers
(1301)
Table 89: Leontief Coe¢ cients Intermediate demand
X1 X1 10 X2 30 Labour input 40 Capital input 20 Total 100
Final Demand
Total
F 70 150
Y 100 200 90 130
X2 20 20 50 110 200
220
Leontief coe¢ cients Input-Output Model: Structural Equations X1 = X11 + X12 + F1 481
(1302)
Table 90: Leontief Technology and Primary Input Coe¢ cients Intermediate demand
X1 X1 0:1 X2 0:3 Labour input 0:4 Capital input 0:2 Total 1:0
X2 0:1 0:1 0:25 0:55 1:0
(1303)
X2 = X21 + X22 + F2 a11 =
X12 X21 X22 X11 ; a12 = ; a21 = ; a22 = ; X1 X2 X1 X2
(1304)
X1 = a11 X1 + a12 X2 + F1
(1305)
X2 = a21 X1 + a22 X2 + F2
(1306)
X1
(1307)
Input-Output Model a11 X1
a12 X2 = F1
a21 X1 + X2 (1
X1 X2
a11 ) a12 a21 (1 a22 ) =
(1
X1 X2
A)
1
F1 F2
=
a11 ) a12 a21 (1 a22 )
X = (I
(1308)
a22 X2 = F2
F
1
F1 F2
(1309)
(1310) (1311)
Employment (labour income) L = l1 Capital stock (capital income)
X1 + l2 482
X2
(1312)
K = k1
X1 + k2
(1313)
X2
Solution of the input - output model by Cramer’s Rule jAj =
(1
a11 ) a12 = 1 a21 (1 a22 )
F1 a12 F2 (1 a22 ) X1 = = (1 a11 ) a12 1 a21 (1 a22 ) a11 ) F1 a21 F2 X2 = = (1 a11 ) a12 1 a21 (1 a22 )
a1;1
1
a21 a12
(1314)
F1 (1 a1;1
a22 ) + a12 F2 1 a2;2 a21 a12
(1315)
F2 (1 a1;1
a11 ) + a21 F1 1 a2;2 a21 a12
(1316)
a2;2
(1
Numerical Example of Input Output Model X1 X2
X1 =
=
(1
0:1) 0:1 0:3 (1 0:1)
1
70 150
(1317)
70 0:1 150 0:9 63 + 15 78 = = = 100 0:81 0:03 0:78 0:9 0:1 0:3 0:9
(1318)
Numerical Example of Input Output Model
X2 =
0:9 70 0:3 150 135 + 21 156 = = = 200 0:81 0:03 0:78 0:9 0:1 0:3 0:9
(1319)
Solutions reproduce the benchmark data. Model is calibrated. Solving the Input-Output Model by Matrix Inverse X = (I
A)
1
(1320)
F
1
(I
A)
1
=
1 0:9 0:1 = adj (I 0:3 0:9 jI Aj 483
A)
(1321)
A) = C 0
adj (I
(1322)
For C cofactor matrix. For this cross the row and column corresponding to an element and multiply by ( 1)i+j C=
j1
a22 j ja21 j 0:9 0:3 = ja12 j j1 a11 j 0:1 0:9 0
0:9 0:3 C = 0:1 0:9 0
=
(1323)
0:9 0:1 0:3 0:9
(1324)
Inverse of A Inverse of the Leontief technology matrix is the major element of the InputOutput model A) 1 = 1 a22 a12 = a21 1 a11 (I
1 1
a1;1
1
a2;2
a21 a12
1 0:81
X = (I
A)
1
F =
0:9 0:1 = 0:03 0:3 0:9 1 0:9 0:1 = 0:78 0:3 0:9
1 0:9 0:1 0:78 0:3 0:9
70 150
0:9 0:78 0:3 0:78
0:1 0:78 0:9 0:78
(1325)
Inverse of A
X = (I =
1 0:78
1 0:9 0:1 70 0:3 0:9 150 0:78 1 63 + 15 78 = = 21 + 135 156 0:78
A)
1
F =
Model is calibrated Impact analysis 484
100 200
(1326)
X1 X2
(1
=
a11 ) a12 a21 (1 a22 )
X = (I
A)
1
1
F1 F2
(1327) (1328)
F
Impact Analysis If the …nal demand of sector X1 changes by 15 percent X1 X2
X1 X2
1 70 0:15 0:9 0:1 150 0 0:78 0:3 0:9 1 0:9 0:1 10:5 = 0:3 0:9 0 0:78 1 9:45 12:11 = = 3:15 4:03 0:78
=
(1329)
Employment (labour income) L = l1
X1 + l2
X2 = 0:4
(1330)
100 + 0:25
200 = 40 + 50 = 90
100 + 0:55
200 = 20 + 110 = 130 (1331)
Capital stock (capital income) K = k1
X1 + k2
X2 = 0:2
A 15 pecent change in the …nal demand of sector will change gross output of both sector. change in capital and labour demand could be found out by using the capital and labour coe¢ cients. Backward and forward linkages cause this to happen. Real world input- output model can be easily computed using Matrix routines in Excel.
485
References [1] Bhattarai K. (2007) Economic Models of Hull and Humber Region, Atlantic Economic Journal, 35:473-490 December. [2] Leontief, W. (1949) Structural Matrices of National Economy, Econometrica, 17: 273-282, Suppl. [3] Piggott, J. and J.Whalley (1985) UK Tax Policy and Applied General Equilibrium Analysis, Cambridge University Press. [4] O¢ ce for National Statistics (ONS (1995)) Input Output: Tables for the United Kingdom, HMSO, London. [5] Stone Richard.1942-43. National Income in the United Kingdom and the United States of America, American Economic Review, 10(1): 1-27. [6] Stone Richard. 1961. Input-output and National Accounts, Paris:OECD. [7] Thijs ten Raa. (2005)The economics of input-output analysis Cambridge : Cambridge University Press [8] OECD IO database: http://stats.oecd.org/index.aspx
486
32
L30: Linear Programming
Linear Programming: Maximisation Problem 1. Solve the following linear programming problem using a simplex method. What are the optimal value of R; X1 and X2 ? (1332)
max R = 10X1 + 5X2 Subject to
where X1
0 and X2
25X1 + 10X2
1000
(1333)
20X1 + 50X2
1500
(1334)
0;
2. Write the dual of the above problem. Show that optimal solution of dual is equivalent to optimal solution of the primal problem. 3. Show that LP problem given above is a special case of non-linear problem. Linear Programming: Simplex Algorithm for Maximisation
R Row0 1 Row1 0 Row2 0
Table 91: X1 X 2 -10 -5 25 10 20 50
Simplex Table 1 S1 S2 Constant Ratios 0 0 0 1 0 1000 40 0 1 1500 75
Basic feasible solution R X1 X2 S1 S2 = 0 0 0 1000 1500 Linear Programming: Simplex Algorithm for Maximisation
R Row0 1 Row1 0 Row2 0
Table 92: X1 X2 0 -1 1 2/5 0 42
Simplex Table 2 S1 S2 Constant Ratios 2/5 0 400 1/25 0 40 100 -4/5 1 700 16.7
Basic feasible solution R X1 X2 S1 S2 = 400 40 0 Linear Programming: Simplex Algorithm for Maximisation 487
0 16:7
R Row0 1 Row1 0 Row2 0
Table 93: X1 X2 0 0 1 0 0 1
Basic feasible solution R X1 Linear Programming: Duality
Simplex S1 8/21 1/21 -2/105
X2
S1
Table 3 S2 1/42 -1/105 1/42
Constant 17500/42 700/21 700/42
S2 = 17500/42 700/21 700/42 0
Every maximisation problem has corresponding minimisation problem. The revenue maximisation problem above has equivalent to the cost minimisation problem. Primal max R = 10X1 + 5X2 (1335) Subject to 25 10 20 50
X1 X2
1000 ; X1 1500
0; X2
(1336)
0
This is equivalent to minimising the cost (1337)
M in C = 1000Y1 + 1500Y2 subject to: 25 20 10 50
Y1 Y2
10 ; Y1 5
0; Y2
(1338)
0
Linear Programming: Fundamental Theorems of Duality Two fundamental theorems of duality: (1) Optimal values of the primal and the dual objective functions are always identical, provided that optimal feasible solution does exist. (2) If a certain choice variable in a linear programme is optimally nonzero then the corresponding dummy variable should be equal to zero. Similarly if a certain choice variable in a linear programme is optimally zero then the corresponding dummy variable in the linear programme should be non-zero. Lagrangian for the constrained optimisation (linear program (LP) as a special case of non-linear program (NLP)) L = 10X1 + 5X2 +
1
[1000
25X1
10X2 ] + 488
2
[1500
20X1
50X2 ]
(1339)
0
Solution of the nonlinear program will be equivalent to the solution of the nonlinear program: @L = 10 @X1
25
@L =5 @X2
10
20
1
50
1
2
2
(1340)
=0
=0
(1341)
@L = 1000 @ 1
25X1
10X2 = 0
(1342)
@L = 1500 @ 2
20X1
50X2 = 0
(1343)
1000 = 25X1 + 10X2 =) 200 = 5X1 + 2X2 1500 = 20X1 + 50X2 =) 150 = 2X1 + 5X2 From these two 1000 = 25X1 + 10X2 300 = 4X1 + 10X2 Then 700 = 21X1 100 700 = 3 21 + 5X2 =) 5X2 = 150
X1 = 150 = 2X1 + 5X2 =) 150 = 2
100 3
X2 =
R = 10X1 +5X2 = 10
700 +5 21
700 42
50 700 = 3 42
=
(1344) 200 3
=
250 ; 3
(1345)
14000 + 3500 17500 = = 416:67 (1346) 42 42
Find the shadow prices 25
1
+ 20
2
= 10
(1347)
=5
(1348)
10
1
+ 50
5
1
+4 2 =2 489
2
(1349)
2 10 10
1
+ 10
2
=1
(1350)
+8
2
=4
(1351)
=5
(1352)
1
1
+ 50
2
2
1
+ 10
2
= 1 =)
1
2
=
1 42
=
1 2
(1353) 1
10 42
=
16 8 = 42 21
(1354)
Linear Programming: Minimisation Problem 1. One family wants to …nd the minimum expenditure with optimal amounts of vegetarian (X1 ), meat (X2 ) and fat (X3 ) contents in its food mix for a month. Per unit market price of these items is £ 5, £ 3 and £ 2 respectively. Suppose that nutritionists recommend 1000 units of carbohydrate, 1000 units of protein and 200 units of fat. One unit of vegetable item gives 5 units of carbo, 3 units of protein and 0.3 units of fat; one unit of meat item gives 3 units of carbo, 6 units of protein and 1 unit of fat; one unit of dairy product gives 2 units of carbo, 2 units of protein and 5 units of fat. Using a simplex method …nd the optimal amounts of vegetarian (X1 ), meat (X2 ) and fat (X3 ) items that ful…ls the nutrition constraints that minimises food expenditure for this family. Hint: formulate the problem as follows: min E = 5X1 + 3X2 + 2X3
(1355)
Subject to 5X1 + 3X2 + 2X3
1000
(1356)
3X1 + 6X2 + 2X3
1000
(1357)
200
(1358)
0:3X1 + 1X2 + 5X3 1. where X1
0 , X2
0 and X3
0
What are the optimal values of X1 X2 ;and X3 ? What is the optimal expenditure? 490
Basic feasible solution E
X1
X2
X3
= 1000 127.06 87.43 51.2
2. Solve the following minimisation problem using a simplex method. min C = 0:6X1 + X2
(1359)
Subject to 10X1 + 4X2 5X1 + 5X2 2X1 + 6X2 where X1
0 and X2
20
(1360)
1500
(1361)
12
(1362)
0
What are the optimal values of X1 and X2 ?
References [1] Chiang A.C. (1984) Fundamental Methods of Mathematical Economics, 3rd edition, McGraw Hill. [2] Hoy M., J. Livernois, C McKenna, R. Rees and T Stengos (2001) Mathematics for Economics, MIT Press.
491
Write an essay in 1500 words in any one of the following topics. Support your statements with some derivations based on economic theories and evidenced from the real world. T1. Apply theories of economic growth to explain why the economic growth rates di¤er across countries or regions within a country and over time? Illustrate how the accumulation of physical and human capital have enhanced growth but tended to increase inequality in the distribution of income after 1980s? (Hints: Piketty (2014)). T2. The Bank of England is likely to raise the base rate at some time in 2016. How will impact on growth, employment and in‡ation in the UK economy? T3. Should governments raise indirect taxes to …nance public expenditure by reducing direct taxes for growth, e¢ ciency and equality? Evaluate this issue using arguments from the classical, Keynesian, new Keynesian and real business cycle and Walrasian general equilibrium models. T4. Institutional reforms are key to solve the unemployment problems in advanced,emerging and developing economies. Apply the Beveridge curve as in Pisserides (2011 and 2014) to analyse how these institutions could impact on ‡ows in and out of labour force in reducing the equilibrium rate of unemployment in these economies. T5. Explain mechanisms on how the development of the …nancial sector is essential for raising growth rates in an economy. How the …nancial policy committee and monetary policy committee could enhance …nancial deepening optimal for a steady growth of an economy. T6. Cooperative strategies generate gains for all but the non-cooperative strategies create imbalances and tensions in the global economy. Process of globalisation in recent years has made rich individuals even richer and poor persons even poorer. Illustrate with evidence. This essay accounts for 20 percent of the module marks. Write in your own words referring to existing economic theories and evidences available to you. Be critical, analytical and precise. Submit the electronic copy of essay through Turnitin and a hard copy through the undergraduate o¢ ce. Class ID and password and the Turnitin procedure are given in the module handbook ready to be downloaded from the resources folder in eBridge site for this module. 492
This is expected to be a professional piece of work and must contain a model and analysis. The elements of marks will broadly be based on the originality of the motivation to the question (15%), statement of the relevant model (15%), derivations (15%), analysis based on derivations (15%), application of the model (40%). Students are allowed to ask any question on the chosen topic in any teaching sessions. You are allowed and encouraged to use any notes from lectures or tutorials and these are placed in the resources folder in the eBridge of this module. Students are expected to digest relevant models and derivations presented in Bhattarai (2014). Tentative list of articles from the literature for each topic is listed below. However, students should try to …nd most recent articles as the study progresses using electronic databases such as JSTOR and Econlit in the library. Policy documents from the central bank and the treasury or …nance ministries or planning agencies could be used for analysis.
33 33.1
Assignment 2016 Topic 1: Why do the economic growth rates di¤er across countries or regions?
Apply theories of economic growth to explain why the economic growth rates di¤er across countries or regions within a country and over time? Illustrate how the accumulation of physical and human capital have enhanced growth but tended to increase inequality in the distribution of income after 1980s? (Hints: Piketty (2014)). May read papers such as Allen (2011) Barro (1991 Basu and Bhattarai. (2012), Becker (1975), Blankenau and Camera (2009), Glomm and Ravikumar (1992) Guryan, Hurst, and Kearney (2008), Hanushek, and Woessmann (2008), Lee and Barro (2001), Pritchett (2001), Sylwester (2000), Temple (2001), Grossman, and Stiglitz (1980) for this topic.
References [1] Allen R. C. (2011) Global Economic History: A Very Short Introduction, Oxford:Oxford University Press. [2] Barro, R. J.(1991), ‘Economic Growth in Cross Section of Countries’, Quarterly Journal of Economics, 106, May, 407-433. [3] Basu, P and K. Bhattarai. 2012. Cognitive Skills, Openness and Growth. Economic Record, 88: 280: March, 18-38. [4] Becker, G.S. (1975), Human Capital. University of Chicago Press, Chicago. [5] Blankenau, W & G. Camera (2009), ‘Public Spending on Education and the Incentives for Student Achievement’, Economica, 76,303,505-527, 07. 493
[6] Glomm, G. and B. Ravikumar.1992. Public versus private investment in human capital: endogenous growth and income inequality. Journal of Political Economy 100:4:818-834. [7] Grossman, S. J., and J. E. Stiglitz. 1980. “On the Impossibility of Informationally E¢ cient Markets.”American Economic Review, 70(3): 393–408. [8] Guryan J., E. Hurst, and M. Kearney (2008), ‘Parental Education and Parental Time with Children’, Journal of Economic Perspectives, 22, 3,Summer, 23–46. [9] Hanushek, E. A. and L. Woessmann (2008), "The Role of Cognitive Skills in Economic Development." Journal of Economic Literature, 46:3:607-668, September. [10] IFS (2014) Green Budget, London. [11] Jones, Charles (2002) Introduction to economic growth, 2nd edition, Norton. [12] Jones C. I. (1995) R & D-Based Models of Economic Growth, Journal of Political Economy, 103:4:759-784 [13] Lee J. W. and R. J. Barro (2001) "Schooling Quality in a Cross-Section of Countries", Economica, New Series, 68 : 272: Nov.: 465-488 [14] Maddison A. (1991) Dynamic of Capital Accumulation and Economic Growth, Oxford. [15] Mankiw N. G. and M. P. Taylor (2008) Macroeconomics: European Edition, Worth Publishers. [16] Perroni C. (1995), Assessing the Dynamic E¢ ciency Gains of Tax Reform When Human Capital is Endogenous, International Economic Review . 36, 4, 907-925. [17] Pritchett, L. 2001. Where has all the education gone?. The World Bank Economic Review. 15:3:367-391. [18] Solow, R. M.(1956) A Contribution to the Theory of Economic Growth, Quarterly Journal of Economics, 70:1:65-95. [19] Sylwester, K. 2000. Income inequality, education expenditures and growth. Journal of Development Economics. 63:2:379-398. [20] Temple, J. R.W. 2001. Generalizations that aren’t? Evidence on education and growth. European Economic Review 45:4-6:May:905-918. [21] Weil D.N. (2013) Economic Growth, 3rd edition, Pearson.
Web pages: http://cep.lse.ac.uk/;http://www.lse.ac.uk/researchAndExpertise/units/growthCommission/home http://www.bis.gov.uk/; http://pwt.econ.upenn.edu/; http://www.app.collinsindicate.com/worldbankatlas-global/en
494
33.2
Topic 2: The mechanism of coordinating monetary and …scal policies
The European Central Bank has now brought a big programme of quantitative easing for the Euro area. How will this bring macroeconomic stability and growth in Euro area? What will be its impact in the UK, US and emerging economies? What will be the mechanism of coordinating monetary and …scal policies among Euro members? What will be implications of sovereign debts in the long run? Feel free to focus on a particular economy or compare across economies in your analysis. May read papers such as Bean (2009, 1998), Benati (2008), Bernanke, Boivin and Eliasz (2005), Dri¢ ll and Schultz (1992),Ellison and Pearlman (2011), Goodhart (1989), Lucas(1973), Martin and Milas (2004), McCallum and Nelson(1999), Miller and Weller (1991), Pain,Weale and Young (1997), Rudebusch (2006), Sawyer (2003), Woodford (2011) for this.
References [1] Barro R.J. and D. B. Gordon (1983) Rules, Discretion and Reputation in a Model of Monetary Policy, Journal of Monetary Economics, 12: 101-121, North-Holland. [2] Bean C. (2009) The meaning of internal balance’ Thirty years on, Economic Journal, 119 (November), F442–F460. [3] Bean C. (1998) The New UK Monetary Arrangements: A View from the Literature, Economic Journal, 108, 451, 1795-1809 [4] Benati, Luca. (2008) “The ‘Great Moderation’in the United Kingdom.”Journal of Money,Credit and Banking, 40, 121–47. [5] Bernanke Ben S., J. Boivin and P. Eliasz (2005) Measuring the E¤ects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach, Quarterly Journal of Economics, 120,1,387-422. [6] Bernheim B. D. (1989) A Neoclassical Perspective on Budget De…cits, Journal of Economic Perspectives, 3: 2 Spring: 55-72 [7] Blake A. P. and M. Weale (1998) Costs of Separating Budgetary Policy from Control of Inzation: A Neglected Aspect of Central Bank Independence, Oxford Economic Papers, 50, 3, 449-467. [8] Chadha J. S. & C. Nolan (2002) In‡ation and price level targeting in a new Keynesian model, Manchester School, 70(4), pp. 570-595. [9] Corden W.M. (1985) Macroeconomic Policy Interaction under Flexible Exchange Rates: A Two-Country Model, Economica, New Series, 52:205:Feb. 9-23 [10] Dri¢ ll J., C. Schultz (1992) Wage Setting and Stabilization Policy in a Game with Renegotiation, Oxford Economic Papers, 44, 3, 440-459. 495
[11] Ellison M. and J. Pearlman (2011) Saddlepath learning, Journal of Economic theory 146, 1500-1519. [12] Frankel J. A., K. E. Rockett (1988) International Macroeconomic Policy Coordination When Policymakers Do Not Agree on the True Model, American Economic Review, 78, 3,Jun.,: 318-340. [13] Goodhart C.(1989) The Conduct of Monetary Policy, Economic Journal, 99, 396, 293-346. [14] Holly S and M Weale Eds.(2000) Econometric Modelling:Techniques and Applications,Cambridge University Press. [15] King Mervyn (2004) The Institutions of Monetary Policy, American Economic Review - Proceedings, 94, 2,1-13 [16] Kirsanova T., C. Leith and S. Wren-Lewis ( 2009) Monetary and …scal policy interaction: The current consensus assignment in the light Of recent developments, Economic Journal,119,Nov,F482–F496. [17] Krugman P. (1979) A Model of Balance of Payment Crisis, Journal of Money Credit and Banking, 11, 3, 311-325. [18] Kydland F.E and E.C. Prescott (1977) Rules rather than discretions: the Inconsistency of Optimal Plans, Journal of Political Economy, 85:3: 473-491. [19] Lucas R. E. (1973) Some International Evidence on Output In‡ation Trade-O¤s, American Economic Review, 63:3:326-334. [20] Mankiw N.G. (1989) Real Business cycle: A New Keynesian Perspective, Journal of Economic Perspectives, 3,3 :79-90. [21] Martin, C. and Milas, C. (2004) Modelling monetary policy: in‡ation targets in practice, Economica, 71, 209-21. [22] McCallum, B. and Nelson, E. (1999) An optimizing IS-LM Speci…cation for monetary policy and business cycle analysis, Journal of Money, Credit, and Banking, 31, 296-316. [23] Miller M. and P. Weller (1991) Exchange Rate Bands with Price Inertia, Economic Journal, 101:409:1380-1399. [24] Monetary Policy Committee Bank of England (1999) The Transmission Mechanism of Monetary Policy, www.bankofengland.co.uk. [25] Mundell R. A (1962) Capital mobility and stabilisation policy under …xed and ‡exible exchange rates, Canadian Journal of Economic and Political Science, 29, 475-85. [26] Monetary Policy Committee Bank of England (1999) The Transmission Mechanism of Monetary Policy, www.bankofengland.co.uk. [27] Nordhaus W.D. (1994) Policy Games: Cooperation and Independence in Monetary and Fiscal Policy, Brookings Papers on Economic Activity, 2: pp. 139-216. 496
[28] Obstfeld M. and K. Rogo¤ (1996) Exchange rate dynamics redux, Journal of Political Economy, 103:3:June: 624-660. [29] Pain Nigel; Martin Weale; Garry Young (1997) Britain’s Fiscal Problems, Economic Journal, 107, 443, 1142-1156. [30] Rudebusch, G. (2006) Monetary policy inertia: fact or …ction? International Journal of Central Banking, 2, 85-135. [31] Sawyer M (2003) Employer of Last Resort: Could It Deliver Full Employment and Price Stability? Journal of Economic Issues, 37, 4, 881-907 [32] Taylor M. P. (1995) The Economics of Exchange Rates, Journal of Economic Literature, March, 33:1:13-4 [33] Woodford, M. (2011) Simple Analytics of the Government Expenditure Multiplier, American Economic Journal: Macroeconomics, 3(1): 1–35. [34] Wren-Lewis S., J. Darby, J. Ireland, O. Ricchi (1996) The Macroeconomic Effects of Fiscal Policy: Linking an Econometric Model with Theory, Economic Journal, 106, 436, 543-559
33.3
Topic 3: Does direct or indirect tax promote growth, e¢ ciency and equality?
Should governments raise indirect taxes to …nance public expenditure by reducing direct taxes for growth, e¢ ciency and equality? Evaluate this issue using arguments from the classical, Keynesian, new Keynesian and real business cycle and Walrasian general equilibrium models. May base your essay reading papers such as Beveridge (1942), Bhattarai and Whalley (2009) Blanchard abd Tirole (2008), Blundell , Fry and Walker (1988) Blundell (2001), Blundell, Mike, Haan, Shephard (2009), Brewer, Francesconi, Gregg and Grogger (2009), Card, Chetty, and Weber. 2007; DWP (2010),Easton (1979), Hohman (1934) , Layard and Nickell (1986), Meade, and Ironside, Jones, Bell, Flemming, Kay, King, Macdonald, Sandford and Whittington, Willis.(1978) Mirrlees, and Adam, Besley, Blundell, Bond, Chote, Gammie, Johnson, Myles, Poterba.(2010) Mo¢ tt and Nicholson (1982), Mortensen and Pissarides (1994) Pallage, Scruggs and Zimmermann (2009), Pissarides(1985), Pissarides (2000), Smith (2011).
References [1] Beveridge William (1942) Social Insurance and Allied Services (Beveridge Report), Nov., HMSO, London. 497
[2] Bhattarai K. (2012) Fiscal Policy, Growth and Income Distribution in UK, http://www.aeaweb.org/aea/2012conference/program/preliminary.php [3] Bhattarai K and J. Whalley (2009) Redistribution E¤ects of Transfers, Economica 76:3:413-431 July. [4] Blundell R. V. Fry and I. Walker (1988) Modelling the Take-Up of Means-Tested Bene…ts: the Case of Housing Bene…ts in the United Kingdom, Economic Journal, Conference Vol. pp. 58-74. [5] Blundell, Richard (2001). Welfare reform for low income workers, Oxford Economic Papers, 53(2):189–214. [6] Blundell Richard, Mike Brewer, Peter Haan, Andrew Shephard (2009) Optimal Income Taxation of Lone Mothers: An Empirical Comparison of the UK and Germany ,Economic Journal, 119,(535), Feb., F101-F121. [7] Brewer Mike, Marco Francesconi, Paul Gregg and Je¤rey Grogger (2009) Feature: in-work bene…t reform in a Cross-national perspective introduction, Economic Journal, 119, Feb, F1–F14. [8] DWP (2010) 21st Century Welfare, Department of Work and Pension, www.dwp.uk21st-century-welfare. [9] Meade J. E., and Ironside, Jones, Bell, Flemming, Kay, King, Macdonald, Sandford and Whittington, Willis.(1978). The Structure and Reform of Direct Taxation. IFS, London: George Allen and Unwin. [10] Mirrlees J., and S. Adam, T. Besley, R. Blundell, S. Bond, R. Chote, M. Gammie, P. Johnson, G. Myles, J. Poterba.2010. Dimensions of tax design: the Mirrlees review, Oxford: Oxford University Press. [11] Pissarides C. A. (1985):Taxes, Subsidies and Equilibrium Unemployment,Review of Economic Studies, 52,1,Jan., 121-133
33.4
Topic 4: Are institutional reforms key to solve the unemployment problems?
Institutional reforms are key to solve the unemployment problems in advanced,emerging and developing economies. Apply the Beveridge curve as in Pissarides (2011 and 2014) to analyse how these institutions could impact on ‡ows in and out of labour force in reducing the equilibrium rate of unemployment in these economies.
References [1] Bhattarai K and H. Dixon (2014) Equilibrium Unemployment in a General Equilibrium Model with Taxes, The Manchester School, 82, S1, 90-128 [2] Blanchard,Olivier J. Jean Tirole (2008)The Joint Design of Unemployment In498
surance and Employment Protection: A First Pass, Journal of the European Economic Association, 6,1 (Mar.,), 45-77. [3] Card, David, Raj Chetty, and Andrea Weber. 2007. "The Spike at Bene…t Exhaustion: Leaving the Unemployment System or Starting a New Job?" American Economic Review, 97(2): 113–118. [4] Easton Stephen T. (1979) Aggregate Aspects of the Poor Law, Unemployment Insurance and Unemployment in Britain, 1855-1940, The Journal of Economic History, 39, 1, Tasks of Economic History (Mar.). 326-329. [5] Hohman Helen Fisher (1934) The Status of Unemployment Insurance in Great Britain, Journal of Political Economy, 42, 6,Dec., 721-752. [6] Layard R and S. Nickell (1986) Unemployment in Britain, Economica, 53: S12169. [7] Mortensen D T and C. A. Pissarides (1994) Job Creation and Job Destruction in the Theory of Unemployment, Review of Economic Studies, 61:3:397-415. [8] Mo¢ tt Robert , Walter Nicholson (1982) The E¤ect of Unemployment Insurance on Unemployment: The Case of Federal Supplemental Bene…ts, Review of Economics and Statistics, 64, 1 (Feb.,), 1-11 [9] Pallage Stéphane , Lyle Scruggs, Christian Zimmermann (2009) Unemployment Insurance Generosity: A Transatlantic Comparison, Annals of Economics and Statistics No. 95/96, Labor Market Outcomes: A Transatlantic Perspective pp. 15-23 [10] Phelps, Edmund S. (1968), Money-Wage Dynamics and Labor-market equilibrium, Journal of Political Economy, vol. 76, pp. 678-710. [11] Phillips, A. W., (1958) The Relation Between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom, 1861-1957, Economica, pp.283-299. [12] Pissarides, C. A. (2013) Unemployment in the Great Recession, Economica, 80: 385–403 [13] Pissarides, C. A. (2011) Equilibrium in the Labor Market with Search Frictions, American Economic Review, 101(4): 1092-1105. [14] Pissarides C A (2000) Equilibrium Unemployment Theory, MIT Press. [15] Smith Jennifer C. (2011) The ins and outs of UK unemployment, Economic Journal, 121, 402–444. [16] Rankin Neil (1992) Imperfect competition, expectations and the multiple e¤ects of monetary growth, Economic Journal 102: 743-753.
499
33.5
Topic 5: The development of the …nancial sector and economic growth
Explain mechanisms on how the development of the …nancial sector is essential for raising growth rates in an economy. How the …nancial policy committee and monetary policy committee could enhance …nancial deepening optimal for a steady growth of an economy.
References [1] Banks J (1977) A note on life-cycle income and consumption pro…les: recent results from UK household data, Institute of Fiscal Studies.London. [2] Banks J, R. Blundell, A. Brugiavini (2001) Risk Pooling, Precautionary Saving and Consumption Growth The Review of Economic Studies, 68:4 :Oct.:757-779 [3] Bhattarai K (2015) Financial Deepening and Economic Growth, Applied Economics, Vol. 47, No. 11, 1133-1150 [4] Davies, J (1997) Uncertain lifetime consumption, and retirement, Journal of Political Economy, 89:3:561-577. [5] Fama E.F. (2014) Two Pillars of Asset Pricing, American Economic Review, 104(6): 1467–1485 [6] Gruber J and D Wise (1998) Social Security and Retirement: An International Comparison, American Economic Review, 88, 2,158-163 [7] Hansen L P (2014) Nobel Lecture: Uncertainty Outside and Inside Economic Models, Journal of Political Economy, 122, 5,945-987 [8] Hey J. D. and J. A. Knoll (2011) Strategies in dynamic decision making: An experimental investigation of the rationality of decision behaviour, Journal of Economic Psychology 32 399–409 [9] King, R. G. and R. Levine (1993) Finance and Growth: Schumpeter Might Be Right, Quarterly Journal of Economics, Aug. pp.717-737. [10] Pagano Marco (1993) Financial Markets and Growth: An Overview, European Economic Review, 37(2-3, April) 613-622. [11] Poterba J., S. Venti, D. Wise (2011) The Composition and Drawdown of Wealth in Retirement, Journal of Economic Perspectives, 25, 4, 95-117 [12] Roth A. E. (2008) What have we learned from market design?, Economic Journal, 118, 285–310. [13] Shiller R. J. (2014) Speculative Asset Prices, American Economic Review, 104(6): 1486–1517 [14] Shiller, Robert J. (1981) “Do Stock Prices Move Too Much to Be Justi…ed by Subsequent Changes in Dividends?”American Economic Review, 71(3): 421–36. 500
[15] Spencer P. D. (1984) Precautionary and Speculative Aspects of the Behaviour of Banks in the United Kingdom Under Competition and Credit Control 1972-1980, Economic Journal, 94, 375 :Sep.,554-568 [16] von Weizsäcker R.K (1996) Distributive implications of an aging society, European Economic Review, 40:729-746. [17] Wise D (1999) Economics of aging, NBER.
33.6
Topic 6: Globalisation and the distribution of income
Cooperative strategies generate gains for all but the non-cooperative strategies create imbalances and tensions in the global economy. Process of globalisation in recent years has made rich individuals even richer and poor persons even poorer. Illustrate with evidence. May read papers such as Barro and Gordon (1983) , Corden (1985) Frankel , Rockett (1988), Nordhaus (1994), Holly and Weale .(2000),King (2004) Kirsanova , Leith and Wren-Lewis ( 2009), Krugman (1979) Kydland and Prescott (1977), Mundell (1962) Monetary Policy Committee Bank of England (1999) Obstfeld and Rogo¤ (1996) for this.
References [1] Abrego L and J. Whalley (2000) The choice of structural model in trade-wages decompositions, Review of International Economics, 8(3):462-477. [2] Beaulieu E, M. Benarroch and J. Gaisford (2004) Trade barriers and wage inequality in a North-South model with technology-driven intra-industry, trade, Journal of development Economics, 75:113-136 [3] Bhattarai K and J Whalley (2006), Division and Size of Gains from Liberalization of Trade in Services, Review of International Economics, 14:3:348-361, August. [4] Baldwin R.E., J.F. Francois, R. Portes, D. Rodrik, I.P. Székely (1997) The Costs and Bene…ts of Eastern Enlargement: The Impact on the EU and Central Europe, Economic Policy, 12 : 24,Apr. ; 125-176 [5] Bernard Andrew B., J. Bradford Jensen, Stephen J. Redding, Peter K. Schott (2007) Firms in International Trade, The Journal of Economic Perspectives, 21, 3, 105-130 [6] Cripps, M.W.(1997) Bargaining and the Timing of Investment, International Economic Review, 38:3 :Aug.:527-546 [7] Dixit A., S. Skeath and D. F. Reiley (2009) Games of Strategy, Norton. [8] Dixit A K and J E. Stiglitz (1977) Monopolistic Competition and Optimum Product Diversity, American Economic Review, 67:3:297-308. 501
[9] Farrell J. (1987) Cheap talk, coordination and entry, Rand Journal of Economics, 18:1:34-39. [10] Gardener R (2003) Games of Business and Economics, Wiley, Second Edition. [11] Greenaway D. W. Morgan and P. Wright (2002) Trade Liberalisation and Growth in Developing Countries, Journal of Development Economics, vol. 67 229244. [12] Haskel J, and M.J. Slaughter (2001). Trade, technology and UK wage inequality, Economic Journal, pages 163-187. [13] Helpman E (1976) Macroeconomic Policy in a Model of International Trade with a Wage Restriction, International Economic Review, 17:2:262-277. [14] Hine R.C. and P.W. Wright (1998) Trade with Low Wage Economies, Employment and Productivity in UK Manufacturing, Economic Journal, 108:450:15001510. [15] Holt Charles (2007) Markets, Games and Strategic Behaviour, Pearson. [16] Jenkins, S.P. (1996) Recent trends in UK income distribution, Oxford Review of Economic Policy, 12:1:29-46. [17] Krugman P. (1980) Scale Economies, Product Di¤erentiation and the Pattern of Trade, American Economic Review, 70:5:950-959. [18] Mailath G. J. and L. Samuelson (2006) Repeated Games and Reputations: long run relationship, Oxford. [19] Melitz M. T. (2003) The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity, Econometrica, 71:6:1695-1725. [20] Mirrlees James A. (1997) Information and Incentives: The Economics of Carrots and Sticks, Economic Journal ,107, 444 Sep., 1311-1329 [21] Moore J. (1988) Contracting between two parties with private information, Review of Economic Studies, 55: 49-70. [22] Perlo¤ J. M. (2013) Microeconomics: Theory and Applications with Calculus, Pearson, 3rd Edition. [23] Rasmusen E(2007) Games and Information, Blackwell,. [24] Roe T and H. Mohladi (2001) International Trade and Growth: An Overview Using the New Growth Theory, Review of Agricultural Economics, 23:2:423-440 [25] Roth Alvin E. (2008) What have we learned from market design?, Economic Journal, 118, 285–310. [26] Shapley L (1953) A Value for n Person Games, Contributions to the Theory of Games II, 307-317, Princeton. [27] Sutton J. (1986) Non-Cooperative Bargaining Theory: An Introduction The Review of Economic Studies, Vol. 53, No. 5., pp. 709-724 [28] Taylor M. P. (1995) The Economics of Exchange Rates, Journal of Economic 502
Literature, March, vol 33, No. 1, pp. 13-47. [29] Varian HR (2010) Intermediate Microeconomics: A Modern Approach, Norton,8th ed. [30] Watt R (2011) The microeconomics of risk and information, Palgrave , macmillan. [31] Winchester N, D. Greenaway, G.V. Reed (2006). Skill Classi…cation and the E¤ects of Trade on Wage Inequality, Review of World Economics, 142(2): 287306. Conferences: http://www.hull.ac.uk/php/ecskrb/Confer/research.html http://www.eea-esem.com/eea-esem/2014/ https://www.aeaweb.org/ http://www.res.org.uk/view/index.html http://www.hull.ac.uk/php/ecskrb/Confer/Confer.html http://www.wto.org Bhattarai K. (2016) Intermediate Macroeconomics, University of Hull Business School.
34
Popular Databases
Constructing Data for Analysis: Step by Step Guidelines Connect to http://www.esds.ac.uk/international/ Choose direct links to macro data Important Steps for extracting data I. World Bank Data (World Bank data Indicator 1. Click on Direct Links to Macro Data 2. Choose World Bank Data 3. Select University of Hull 4. Put Athense user name and pass word; 5. Complete the registration process required by data 6. Select World Bank Development Indicators 7. Select Year 1960 -2008 (all can be selected by a tick mark) 8. Select a country (e.g. South Africa) 9. Select a series (e. g. Population between 15-64; and population growth rate) ; can search for population 10. Click on show Table 11. Adjust row and column dimensions of the table by moving around icons 12. Download data in *.CSV (MS-DOS) format 503
13. Open the data …le just created 14. Make some time series graph 15. Next time; add few more variables like DGP per capita constant 2000 dollars; Gross …xed capital formation % of GDP; Final consumption Expenditure as a % GDP; Current account balance as a % of GDP; General Government …nal consumption % of GDP; GDP constant 2000 $ II. IMF World Economic Outlook (WEO) data 1. Steps 1 -5 as above 2. Select IMF WEO data 3. Select World Economic Outlook 4. Select Euro Aria 5. Select crude oil price, output gap, unemployment rate, in‡ation average consumer price) 6. Select all years 1991-2010 7. Show Table; Download the data; Open and Excel. 8. Do macro analysis. III. Eurostat New Cornos (Data for EU countries) 1. Steps 1 -5 as above 2. Select Eurostat New Cornos 3. Select Economy and Finance 4. Eurostat 5. Exchange rates 6. Nominal e¤ective exchange rates 7. Real e¤ective exchange rates IV.
34.1
Datastream
Econometric and Statistical Software Excel OX-GiveWin/PcGive/STAMP Eviews Shazam micro…t RATS 504
LIMDEP GAUSS STATA/SPSS http://www.feweb.vu.nl/econometriclinks/; https://www.aeaweb.org/rfe/ 1. Excel Spreadsheets are very user friendly and could be used for algebraic calculations and statistical analyses for many kinds of economic models. First prepare an analytical solution by hand then use Excel formula to compute. Excel has constrained optimiser routine at tool/goal seek and solver commend. It also contains matrix routines to get determinants of matrices and to multiply and invert them using multiple cell options. Koop (2007) is a brilliant text for analysis of economic data using excel. Koop G (2007) Analysis of Economic Data, Wiley, UK. 2. OX-GiveWin/PcGive/STAMP (www.oxmetrics.net) is a very good econometric software for analysing time series and cross section data. This software is available in all labs in the network of the university by sequence of clicks Start/applications/economics/givewin. Following steps are required to access this software. a. save the data in a standard excel …le. Better to save in *.csv format . b. start give win at start/applications/economics/givewin and pcgive (click them separately) c. open the data …le using …le/open data…le command. d. choose PcGive module for econometric analysis. e. select the package such as descriptive statistics, econometric modelling or panel data models. d. choose dependent and independent variables as asked by the menu. Choose options for output. e. do the estimation and analyse the results, generate graphs of actual and predicted series. A Batch …le can be written in OX for more complicated calculations using a text editor such as pfe32.exe. Such …le contains instructions for computer to compute several tasks in a given sequence.
References [1] Doornik J A and D.F. Hendry ((2003) PC-Give Volume I-III, GiveWin Timberlake Consultants Limited, London 505
34.2
Mathematical software
4. GAMS is good particularly in solving linear and non-linear problems. It has widely been used to solve general equilibrium models with many linear or non-linear equations on continuous or discrete variables. It comes with a number of solvers that are useful for numerical analysis. For economic modelling it can solve very large scale models using detailed structure of consumption, production and trade arrangements on unilateral, bilateral or multilateral basis in the global economy where the optimal choices of consumers and producers are constrained by resources and production technology or arrangements for trade. It is a user friendly software. Any GAMS programme involves declaration of set, parameters, variables, equations, initialisation of variables and setting their lower or upper bounds and solving the model using Newton or other methods for linear or non-linear optimisation and reporting the results in tables or graphs (e.g. ISLM.gms ). Full version of GAMS/MPSGE program is good for large scale standard general equilibrium models. GAMS programme can be downloaded from demo version of GAMS free from www.gams.com/download). The check whether the results are consistent with the economic theory underlying the model such as ISLM-ASAD analysis for evaluating the impacts of expansionary …scal and monetary policies. Use knowledge of growth theory to explain results of the Solow growth model from Solow.gms. Consult GAMS and GAMS/MPSGE User Manuals, GAMS Development Corporation, 1217 Potomac Street, Washington D.C or www.gams.com or www.mpsge.org for GAMS/MPSGE. For other relevant software visit: http://www.feweb.vu.nl/econometriclinks/ or https://www.aeaweb.org/rfe/
34.3
MATLAB
MATLAB is widely used for solving models. It has script and function …les used in computations.
506
Both have *.m extensions. Its syntax are case sensivite. Solving a system of linear equations and handling matrices Example 1 Write a programme …le matrix.m like the following and try run. % now solve a linear equation % 5x1 + 2x2 =20 % 3x2 + 4x2 =15 k =[5 2;3 4]; n = [20 15]; kk = inv(k) x = kk*n’ One more example of system of equation and factorisation of matrices A=[1 2 3; 3 3 4; 2 3 3] b=[1; 2; 3] %solve AX=b X = inv(A)*b %eigen value and eigenvectors of A [V,D]=eig(A) %LU decomposition of A [L,U]=lu(A) %orthogonal matrix of A [Q,R]=qr(A) %Cholesky decomposition (matrix must be positive de…nite) %R = chol(A) %Singular value decomposition [U,D,V]=svd(A) Contents.m for list of …les in MATLAB demo. MATLAB demo available in http://www.youtube.com/.
35
Final Exam Samples INSTRUCTIONS
Answer all questions in sections A. Then answer either all six questions from section B OR three questions from section B and one essay from section C. Two questions in section A are worth 28 marks and each question in section B is worth 12 marks, and one essay from section C is worth 36 marks. Marks for each 507
question are indicated in [ ] after each question. Use diagrams to illustrate your answers. 35.0.1
Section A: Short Problems [28 marks]
Q1. Consider a version of the neoclassical growth model in which output for year t (Yt ) is produced using capital (Kt ), labour (Lt ) and technology (At ) with productivity parameters and and a constant returns to scale production function + = 1: Let:
(1.1)
Yt = At Kt Lt Saving (St ) is a …xed fraction (s) of output and is given by
(1.2)
St = sYt
The required level of investment for year t (It ) ; depends on the population growth rate (n), rate of depreciation ( ) and the capital stock. So: It = (n + ) Kt
(1.3)
Capital accumulation relates to investment and the depreciation rate as follows: Kt = (1
) Kt
1
+ It
(1.4)
The market clears in the sense that output (Yt ) is either consumed (Ct ) or saved (St ). The model is closed by balancing the savings to investment. Thus: Yt = Ct + St = Ct + It ; =) It = St
(1.5)
1. What are the values of output and consumption in the steady state in terms of parameters ; ; n; and A? [9 marks] 2. Modify this model so that it can explain large di¤erences in the level of GDP per capita observed across countries in the world today. [9 marks] Q2. Consider a simple endogenous growth model with production technology that shows how the level of output (Yt ) depends on the amount of technical knowledge (At ) and the stock of physical capital (Kt ) as: 508
(2.1)
Yt = At Kt
The capital accumulation equation relates investment to the capital stock and the rate of depreciation ( ) as: It = Kt+1
(1
) Kt
(2.2)
Output (Yt ) is either consumed or saved as shown by the market clearing equation: (2.3)
Yt = Ct + St
How much of total savings (St ) is turned into investment (It ) is indicated by the e¢ ciency parameter ( ) of the …nancial system as: It = St
0
0
(4.2)
a1 > 0
(4.3)
Here random shocks to the demand t N (0; 2 ) and those to the supply t N (0; 2 ) are distributed normally with zero mean and constant variance. Aggregate demand equals aggregate supply in equilibrium: 510
ytd = yts = yt
(4.4)
1. Prove that the level of price and output deviate from the steady state either due to monetary policy errors (mt Et mt+1 ) or due to shocks to aggregate demand and aggregate supply ( t ; vt ) such that:
yt = yn + pt =
a0
a1 b1 (mt Et mt+1 ) b1 t + a1 vt + a1 + b 1 a1 + b 1 yn
a1
+
(4.5)
a1 mt + b1Et mt+1 vt t + a1 + b 1 a1 + b 1
(4.6) [6 marks]
2. Di¤erencing the price function, prove that the variance of in‡ation (its volatility) depends on the variance of monetary policy errors and the variances of demand and supply shocks as:
var ( t ) = var ( ) + where
= (mt
Et
var (
t
t 1)
+ var (vt (a1 + b1 )2
vt 1 )
(4.7)
mt+1 ) measures the monetary policy error. [6 marks]
Q5. Consider a new approach to solving the IS-LM model for analysing the macroeconomic ‡uctuations from the steady state (as in Jones (2011) or in the section 5.3 of the workbook for this module). As in the earlier versions of the IS-LM models the starting point is the macroeconomic balance equation where the aggregate demand (Yt ) consists of consumption (Ct ), investment (It ), government spending (Gt ), exports (Xt ) minus imports (Mt ) as: Yt = Ct + It + Gt + Xt
(5.1)
Mt
Then this model focuses on the "share form" of these demand components, obtained by dividing both sides of this equation by a steady state output Y as: Yt Ct It Gt Xt = + + + Y Y Y Y Y
Mt = ac + ai Y 511
b (R
r) + ag + ax
am
(5.2)
Here ax de…nes the share of xth demand component; as a , ac ; ai ; ag ; ax ; am are values of Yt Ct ; It ; Gt ; Xt ; Mt relative to Yt . In (R r) ; R is the real interest rate and r is the marginal productivity of capital in equilibrium, b is the slope of investment function on the excess real return, (R r). Let Ybt = YYt 1 term measure the deviation of output from the steady state. Then new IS curve is: Yt 1 = Ybt = a b (Rt r) Y where a = ac + ai + ag + ax am and a = a 1:
(5.3)
1. How does the real interest rate relate to the marginal productivity of capital in the steady state in this set up? [3 marks] 2. When a central bank changes the interest rate (R) ; how will this impact on aggregate demand (Yt ). Illustrate in a diagram with Yb = 0 when the economy is in the steady state. [3 marks]
In the supply side, in‡ation is positively linked to the output gap Ybt subject to a supply shock ("t ) as: t
= Ybt + "t
and is
(5.4)
Assume a backward looking in‡ation expectation t = et + Ybt + "t or t = t 1 + Ybt + "t . Monetary policy rule (or Taylor rule) as in Weale (2014) is to alter the real interest rate to control in‡ation towards its target ( ) by linking the interest rate to ): the in‡ation gap as (Rt r) = m ( t 3. Relate this in‡ation targeting interest rate rule to Ybt (the deviation in demand away from the steady state) for this economy. [3 marks]
4. What sort of nominal interest rate rule can a monetary policy authority adopt based on the Fisher equation and the analysis above to achieve the targeted in‡ation in this economy? [3 marks] Q6. Consider a two period model of consumption for an economy with two individuals ( i = A; B) both living for two periods. Given a common that measures the subjective discount factor for future consumption relative to current consumption, each individual aims to maximise its utility subject to its own budget constraint as: (6.1) M ax U (C1i ; C2i ) = ln C1i + ln C2i 512
Subject to: the …rst period budget constraint: C1i = bi + ! i1
(6.2)
and the second period budget constraint: C2i + bi (1 + r) = ! i2
(6.3)
where C1i and C2i are consumption in period 1 and 2 by household i = A; B; ! i1 and ! i2 are endowments in period 1 and 2 of household i = A; B; r is the interest rate and is the subjective discount factor for future consumption. The …nancial market is assumed to be friction-less, it is also assumed that none of these consumers are borrowing constrained. 1. Represent this model in a diagram in (C1i , C2i ) space for an individual i. [4 marks] 2. Formulate the Lagrangian function for intertemporal optimisation in a two-period model for consumers A and B. [4 marks] 3. Apply Walras’law to characterise the interest rate (the price in the model) that determines the equilibrium allocations in this economy in terms of model parameters and the endowments. [4 marks] Q7. Consider a general equilibrium model of an economy with a representative household that maximises utility (U ) from consumption (C) and leisure (L). This household is endowed with L of time and pays VAT rate (t) on consumption (C).
max U = C L
(7.1)
p (1 + t) C + wL = wL
(7.2)
Subject to
The representative …rm maximises pro…t ( ) subject to the linear technology constraint as follows: 513
max
= p:Y
w:LS
(7.3)
Subject to Y = LS
(7.4)
where Y is output and LS is labour demanded by the …rm (labour supplied by the household in equilibrium). Both the household and the …rm take the market price (p) and wage (w) as given. For simplicity assume p = 1. Assume that the tax revenue is spent on public consumption (G). Assume L = 100 and the commodity tax rate of 25 percent, t = 0:25. 1. What is the market clearing wage rate consistent with the general equilibrium in this economy? [6 marks] 2. What are the optimal amounts of leisure, labour supply, output, consumption, government revenue and household utility? [6 marks] Q8. Consider a Ricardian trade model with two goods Y1 and Y2 traded between two 2 = countries. The production possibility frontier (PPF) of country 1 is Y1;1 +Y1;2 2 1000 and that of country 2 is Y2;1 + Y2;2 = 2000: Here the second subscript is for a good and the …rst subscript is for a country; i.e. Y1;2 is output of good 2 in country 1. These PPFs imply that country 1 has comparative advantage in producing good 1 and country 2 has it in producing good 2. The demand for good j in each country i is denoted by Xi;j and is derived by constrained optimisation. The preferences of representative consumers in each 1 1 country are given by Cobb-Douglas utility functions as U1 = X1;11 X1;2 and U2 = 1 2 2 X2;1 X2;2 : The representative household in each country maximises its own utility, subject to its own budget constraint I1 = P1 X1;1 + P2 X1;2 or I2 = P1 X2;1 + P2 X2;2 , given prices P1 and P2 of those two goods in the global market. With specialisation in production, the income is I1 = P1 Y1 for country 1 and I2 = P2 Y2 for country 2. The global market clearing condition for good 1 is X1;1 +X2;1 = Y1 and that for good 2 is X1;2 +X2;2 = Y2 where Y1 = Y1;1 +Y2;1 and Y2 = Y1;2 +Y2;2 . Each country produces only one good because of specialisation and thus Y2;1 = Y1;2 = 0: The preference for good 1 in country 1 is 1 = 0:6 and that for good 1 in country 2 is 2 = 0:4. Good 2 is considered as a numeraire in the global market, P2 = 1: 514
1. What is the relative price of good 1 that clears the global market? [3 marks] 2. What are the levels of income (I1 ; I2 )? [3 marks] 3. What are levels of demands (X1;1 ; X2;1 ; X1;2 and X2;2 ) for goods 1 and 2 in countries 1 and 2? [3 marks] 4. Find which one of these two countries is better o¤ from such trade, evaluating their utility levels, U1 and U2 . [3 marks] 35.0.3
Section C: Essay [36 marks]
Instructions. Write one essay on any one of the following, applying economic models with real world examples. E1. Critically discuss why unemployment rates are so di¤erent across countries. Refer to frictional, search and matching models of unemployment, and e¢ ciency wage theory, in your answer. [36 marks] E2. Countries which are committed to high quality research through universities and scienti…c laboratories cause the amount of human capital available to increase the growth rate of output. Critically discuss this with reference to the theory of endogenous growth. [36 marks] E3. ‘Despite developments in the purchasing power parity (PPP) and uncovered interest parity (UIP) theories of nominal exchange rates, real exchange rates are determined by the fundamentals of demand and supply and macroeconomic policies in an economy. However, governments that borrow to spend more have expansionary impacts in the economy only if the sum of the elasticities of imports and exports are greater than one.’ Discuss this statement with reference to the Marshall-Lerner condition on the e¤ectiveness of depreciation in an economy. [36 marks] E4. Coordination between …scal and monetary policies is essential for macroeconomic stability. Critically discuss this. [36 marks]
36 36.1
Tutorials Tutorial 1: Neoclassical Growth Model
Q1. Consider a version of the neoclassical growth model in which output for year t (Yt ) is produced using capital (Kt ), labour (Lt ) and technology (At ) with 515
productivity parameters function + = 1 as:
and
and a constant returns to scale production
(1366)
Yt = At Kt Lt Saving (St ) is a …xed fraction (s) of output and given by
(1367)
St = sYt
The required level of investment for year t (It ) ;depends on the population growth rate (n), depreciation ( ) and the capital stock as: It = (n + ) Kt
(1368)
Capital accumulation relates to investment and the rate of depreciation as: Kt = (1
) Kt
1
+ It
(1369)
The market clears in the sense that output (Yt ) is either consumed (Ct ) or saved (St ). The model is closed by balance between saving and investment as: Yt = Ct + St = Ct + It ; =) It = St
(1370)
What are the values of the capital stock and output in the steady state in terms of parameters ; ; n; and A? Q2. Decompose the sources of economic growth in the human capital augmented Solow model.
Yt = At Kt Lt H
(1371)
where + + = 1 and At is an index of Hicks neutral technical knowledge A certain country with parameter values = 0:3; = 0:5 and = 0:2 had output (Yt ) growing by 6 percent, capital (Kt ) by 3 percent, labour (Lt ) by 2 percent and human capital (Ht ) by 2 percent. What was the growth rate of technical progress (gA ) in this country?
516
36.2
Tutorial 2 : Endogenous Growth Model
Q1. Consider a two sector model of growth in which output for year t of the …nal goods sector is given by (1372)
Yt = Kt (At Lt )
Here output (Yt ) is produced using capital (Kt ) labour (Lt ) and technology (At ). Now Yt is output of the …nal goods sectors (e.g. agriculture, manufacturing) whereas; (At ) is generated in the knowledge sector (universities, research labs). The growth rate of knowledge is gA and is a function of existing knowledge (A) and the number of people employed in the knowledge sector (LA ) (1373)
gA = A LA
All three parameters are positive; > 0; 0 < < 1 and 0 < < 1 There is increasing returns to knowledge; knowledge generates more knowledge. Labour is divided between the …nal goods and knowledge sectors as: (1374)
L = LY + LA
Using this model, brie‡y explain the role played by universities and research laboratories in enhancing economic growth. Q2. Substitute all constraints in the objective function and write the resulting reduced form equation appropriate for intertemporal optimisation in the Ramsey model given below.
max Uo = Ct
Subject to production technology ( 0
