MACROECONOMIC THEORY (MA)
Bart Hobijn
COURSE NOTES
COURSE NOTES
Macroeconomic Theory
© Bart Hobijn
[email protected]
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Table of Contents CHAPTER 1: INTRODUCTION
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1.1 Historical U.S. macroeconomic evidence
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1.2 Structure of this class
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1.3 An introduction to the main toolkit
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1.4 Here we go…
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CHAPTER 2: THE HOUSEHOLD
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2.1 The household’s objective: Utility
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2.2 The labor supply decision
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2.3 The savings decision
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2.4 Measuring the aggregate household
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2.5 Application: CCAPM
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CHAPTER 3: THE FIRM
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3.1 The firm’s objective: Profits
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3.2 The firm rents its capital stock
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3.3 The firm owns its capital stock
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3.4 Measuring the aggregate firm
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3.5 Application: Productivity measurement
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CHAPTER 4: EQUILIBRIUM
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4.1 Structure of economy
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4.2 Equilibrium definition
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4.3 Equilibrium conditions
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4.4 Equilibrium dynamics
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4.5 Steady State
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4.6 Transitional dynamics
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4.7 Measuring equilibrium
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CHAPTER 5: ECONOMIC GROWTH
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T A B L E
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5.1 Neoclassical Growth Model
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5.2 Measuring economic growth
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5.3 Application: Convergence Hypothesis
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CHAPTER 6: REAL BUSINESS CYCLES
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6.1 Measuring business cycles
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6.2 RBC Hypothesis
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6.3 Application: Bringing the model to the data
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6.4 Merits and shortcomings of RBC theory
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CHAPTER 7: MONEY DEMAND AND SUPPLY
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7.1 Fiat money
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7.2 Money demand: Cash-in-advance model
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7.3 Money supply: The central bank
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7.4 Measurement: Money and prices
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7.5 Application: The Friedman Rule
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CHAPTER 8: THE NEW KEYNESIAN MODEL
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8.1 A new perspective on ‘Keynesian’ theory
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8.2 The New Keynesian Model
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8.3 Measurement: ‘New Keynesian Facts’
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8.4 Application: Monetary policy in practice
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CHAPTER 9: FISCAL POLICY
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9.1 Government budget constraint
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9.2 Taxes and distortions
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9.3 Fiscal policy in equilibrium
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9.4 Interaction of fiscal and monetary policy
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9.5 Measurement: Government in the NIPAs
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9.6 Application: The U.S. federal budget
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CHAPTER 10: UNEMPLOYMENT
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10.1 Unemployment and job flows
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10.2 Labor supply with search frictions
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10.3 Labor demand with real rigidities
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10.4 Frictional unemployment in equilibrium
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10.5 Measurement: Unemployment and job flows
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10.6 Application: Eurosclerosis
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C O N T E N T S
APPENDIX A: SOLUTIONS
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APPENDIX B: MATHEMATICAL TECHNIQUES
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APPENDIX C: EXCEL
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E X E R C I S E S
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A N S W E R S
Table of Exercises and Answers Exercise 1.1: Macroeconomic aggregates quiz
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Exercise 1.2: Identify the stylized facts
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Exercise 1.3: Is there a core of macroeconomics?
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Exercise 2.1: Change in θ and its effect on preferences
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Exercise 2.2: Real versus nominal interest rates
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Exercise 2.3: The shape of the budget constraint
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Exercise 2.4: Non-binding upperbound on Lt
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Exercise 2.5: Perfectly inelastic labor supply
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Exercise 2.6: Graphical representation of labor supply choice
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Exercise 2.7: Labor supply choice partials
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Exercise 2.8: Logarithmic utility
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Exercise 2.9: PDV of consumption and income
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Exercise 2.10: Savings decision with θ = 1
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Exercise 2.11: Consumption smoothing
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Exercise 2.12: Logarithmic preferences and saving
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Exercise 2.13: Logikwiz
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Exercise 2.14: Personal Income and Outlays data lookup
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Exercise 2.15: Employment Situation data lookup
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Exercise 2.16: Wealth inequality and aggregate consumption
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Exercise 2.17: Equity premium puzzle
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Exercise 2.18: Excess volatility puzzle
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Exercise 2.19: Price earnings ratio
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Exercise 3.1: Output elasticity of capital
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Exercise 3.2: Total factor productivity
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Exercise 3.3: Zero profits
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Exercise 3.4: Constant returns to scale
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Exercise 3.5: Estimating the output elasticity of capital
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Exercise 3.6: Capital stock and past investment levels
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Exercise 3.7: Optimal capital stock, Kt
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Exercise 3.8: Dropping time to build
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Exercise 3.9: Tobin’s Q
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Exercise 3.10: GDP and its sectoral components
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Exercise 3.11: Establishment survey data
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Exercise 3.12: Investment and GDP
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Exercise 3.13: GDP and its sectoral components
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Exercise 3.14: Consumer durables
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Exercise 3.15: DIY growth accounting
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Exercise 3.16: Determining capital service flows
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Exercise 3.17: Resurgence of Growth in the Late 1990s
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Exercise 3.18: Will the ‘new economy’ last?
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Exercise 4.1: Rental versus ownership of capital in equilibrium
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Exercise 4.2: Invisible hand
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Exercise 4.3: Comparative statics in steady state
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Exercise 5.1: Real GDP as a measure of the standard of living
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Exercise 5.2: Kaldor’s growth facts and the BGP
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Exercise 5.3: Kaldor in practice
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Exercise 5.4: Transitional dynamics and convergence
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Exercise 5.5: Convergence for a bigger set of countries
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Exercise 5.6: Why doesn’t capital flow from rich to poor?
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Exercise 6.1: Find the stylized facts
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Exercise 6.2: Persistence of productivity shock
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Exercise 6.3: Impulse responses and stylized facts
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Exercise 7.1: Returns on money and assetholdings
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Exercise 7.2: Money growth and inflation in the long run
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Exercise 7.3: Is U.S. velocity constant?
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Exercise 7.4: CES in a bit more detail
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Exercise 7.5: What’s in your wallet?
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Exercise 7.6: Calculation Example of Price Indices
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Exercise 7.7: CPI and PCE inflation
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Exercise 7.8: Optimal money supply: the 2-good case
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Exercise 8.1: Evidence on sticky prices
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Exercise 8.2: Monetary policy in the NKM: Graphically
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Exercise 8.3: Marginal utility and logarithmic preferences
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Exercise 8.4: Demand curve under monopolistic competition
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Exercise 8.5: The U.S. monetary policy rule
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Exercise 8.6: Stabilizing monetary policy
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Exercise 8.7: On the stickiness of ketchup (prices)
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Exercise 8.8: Short-run effects of monetary policy
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Exercise 8.9: The U.S. Phillips Curve
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Exercise 8.10: Credibility and monetary policy
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Exercise 8.11: Monetary policy and the great inflation
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Exercise 9.1: Solving the government budget constraint
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Exercise 9.2: Utility maximization under taxation
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Exercise 9.3: Optimal policy and distortionary taxes
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Exercise 9.4: Laffer curve
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Exercise 9.5: Ricardian equivalence
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Exercise 9.6: Are government bonds net wealth?
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Exercise 9.7: German hyperinflation in 1923
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Exercise 9.8: Government facts from the NIPAs
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Exercise 9.9: International comparison of government sectors
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Exercise 9.10: The current U.S. budget
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Exercise 10.1: Optimal labor supply decision with search
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Exercise 10.2: Equilibrium unemployment
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Exercise 10.3: Random marginal productivity of labor
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Exercise 10.4: s,S-labor demand decision rule
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Answer to Exercise 2.1: Change in θ and its effect on preferences
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Answer to Exercise 2.2: Real versus nominal interest rates
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Answer to Exercise 2.3: The shape of the budget constraint
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Answer to Exercise 2.4: Non-binding upperbound on Lt
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Answer to Exercise 2.5: Perfectly inelastic labor supply
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Answer to Exercise 2.6: Graphical representation of labor supply choice
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Answer to Exercise 2.7: Labor supply choice partials
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Answer to Exercise 2.8: Logarithmic utility
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Answer to Exercise 2.9: PDV of consumption and income
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Answer to Exercise 2.10: Savings decision with θ = 1
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Answer to Exercise 2.11: Consumption smoothing
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Answer to Exercise 2.12: Logarithmic preferences and saving
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Answer to Exercise 2.13: Logikwiz
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Answer to Exercise 2.16: Wealth inequality and aggregate consumption
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Answer to Exercise 2.17: Equity premium puzzle
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Answer to Exercise 2.18: Excess volatility puzzle
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Answer to Exercise 2.19: Price earnings ratio
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Answer to Exercise 3.1: Output elasticity of capital
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Answer to Exercise 3.2: Total Factor Productivity
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Answer to Exercise 3.3: Zero profits
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Answer to Exercise 3.4: Constant returns to scale
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Answer to Exercise 3.5: Estimating the output elasticity of capital
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Answer to Exercise 3.6: Capital stock and past investment levels
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Answer to Exercise 3.7: Optimal capital stock, Kt
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Answer to Exercise 3.8: Dropping time to build
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Answer to Exercise 3.9: Tobin’s Q
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Answer to Exercise 3.14: Consumer durables
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Answer to Exercise 3.15: DIY growth accounting
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Answer to Exercise 3.16: Determining capital service flows
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Answer to Exercise 4.1: Rental versus ownership of capital in equilibrium
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Answer to Exercise 4.2: Invisible Hand
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Answer to Exercise 4.3: Comparative statics in steady state
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Answer to Exercise 5.1: Real GDP as a measure of the standard of living
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Answer to Exercise 5.2: Kaldor’s growth facts and the BGP
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Answer to Exercise 5.3: Kaldor in practice
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Answer to Exercise 5.4: Transitional dynamics and convergence
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Answer to Exercise 5.5: Convergence for a bigger set of countries
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Answer to Exercise 5.6: Why doesn’t capital flow from rich to poor?
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Answer to Exercise 6.1: Find the stylized facts
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Answer to Exercise 6.2: Persistence of productivity shock
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Answer to Exercise 6.3: Impulse responses and stylized facts
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Answer to Exercise : Returns on money and assetholdings
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Answer to Exercise 7.2: Money growth and inflation in the long run
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Answer to Exercise 7.3: Is US velocity constant?
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Answer to Exercise 7.4: CES in a bit more detail
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Answer to Exercise 7.7: CPI and PCE inflation
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Answer to Exercise 7.8: Optimal money supply: the 2-good case
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Answer to Exercise 8.1: Evidence on sticky prices
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Answer to Exercise 8.2: Monetary policy in the NKM: Graphically
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Answer to Exercise 8.3: Marginal utility and logarithmic preferences
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Answer to Exercise 8.4: Demand curve under monopolistic competition
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Answer to Exercise 8.5: The U.S. monetary policy rule
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Answer to Exercise 8.6: Stabilizing Monetary Policy
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Table of Figures Figure 1.1: Marginal analysis illustrated
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Figure 2.1: Indifference curve
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Figure 2.2: Shape of the budget constraint
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Figure 2.3: Graphical representation of the optimal consumption and leisure choice
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Figure 2.4: Marginal utility schedules
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Figure 2.5: Personal income release summary – July 2003
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Figure 2.6: Consumption smoothing in practice
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Figure 2.7: Employment situation press release, household survey indicators – July 2006
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Figure 2.8: Historical real S&P index
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Figure 2.9: Historical S&P 500 price earnings ratio
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Figure 3.1: Historical U.S. labor share
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Figure 3.2: Postwar behavior of Tobin’s Q (Source: Robert E. Hall, “The Stock Market and Capital Accumulation,” American Economic Review, December 2001)
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Figure 3.3: Table 1.3.5 from the NIPA
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Figure 3.4: Index of U.S. labor productivity for the postwar period.
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Figure 3.5: Index of U.S. MFP during the postwar period.
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Figure 3.6: Historical price declines in computer prices
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Figure 4.1: Flow diagram of the model economy
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Figure 4.2: Production account for the model economy
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Figure 4.3: Appropriation account for the model economy
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Figure 4.4: Investment and savings account for the model economy
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Figure 5.1: Level and logarithm of U.S. real GDP per capita (1990 dollars)
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Figure 6.1: Real GDP in deviation from its trend
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Figure 6.2: Real GDP, real personal consumption expenditures, and fixed private investment in deviation from trend
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Figure 6.3: Detrended hours and output
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Figure 7.1: Long run relationship (1961-2005) between average money growth and inflation
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Figure 7.2: Personal moneyholdings diagram (for exercise 7.5)
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Figure 8.1: IS-LM curves and the determination of aggregate demand
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Figure 8.2: Equilibrium output and price level in the IS-LM model
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Figure 8.3: Graphical representation of the New Keynesian Model.
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Figure 8.4: Monetary policy in the New Keynesian Model.
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Figure 8.5: Demand under perfect and monopolistic competition
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Figure 9.1: Banknotes and the price of meat from the German hyperinflation.
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Figure 9.2: Government sector in the flow diagram of our model economy.
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Figure 10.1: Joint existence of unemployment and vacancies in the United States.
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Figure 10.2: s,S-hiring and firing rule.
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Figure A.1: Effect of change in preferences on indifference curves
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Figure A.2: Effect of change in wages and interest rate on the budget constraint
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Figure A.3: Effect of change in real wage and interest rate on the labor supply choice
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Figure A.4: Consumption smoothing and marginal utility schedules
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Figure A.5: Monetary policy in the New Keynesian model.
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Figure B.1: One-variable optimization
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Figure B.2: Graphical illustration of Taylor approximation of ln(1+x).
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Figure C.1: Choose the security item from the tools>macro menu.
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Figure C.2: Set the macro security level to medium (recommended) or low.
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Figure C.3: Choose enable macros in the security warning window to activate the macros.
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Chapter
Introduction This chapter contains an overview of the approach taken in these course notes. It explains the structure and focus of this course and how these notes are intended to help you to understand the formal theory, evidence, and tools that are introduced.
Macroeconomics is the study of the behavior of economic aggregates. Economic aggregates are measures of overall economic activity that summarize the size, state, and changes of the overall economy. The aim of this course is to introduce you to the main theories, evidence, and tools that modern macroeconomics focuses on and uses to explain the behavior of economies, and of the U.S. economy in particular. The theories that we will focus on in this course are known as microfounded macroeconomic theories. These theories aim to build an explanation of the overall economy from the ground up. They start of by explaining how individual decision units, also known as agents, in the economy make their own decisions. For example, how do households decide on how much of their income they spend and how much they save? How do firms decide on how many new computers to buy today versus tomorrow? After having developed a theory of the decisions that individual agents make, microfounded macroeconomics then combines these theories and describes what will happen if these agents interact in a (stylized) market environment. Stylized fact
We will look at macroeconomic evidence and data in this course in three ways. First of all, we will consider what is known as ‘stylized’ facts. Stylized facts are general patterns in macroeconomic data that most macroeconomists agree on. Because of their generality, a good theory of the macroeconomy should be able to explain them. Secondly, we will consider the sources of U.S. macroeconomic data by studying some of the main data releases on the U.S. economy produced by U.S. government and statistical agencies. The releases that we will look at are, among others, the GDP Release, the Labor Market Report, and the Productivity Release. Finally, you, as a student, will be expected to work with some of the data that makes up this evidence and use basic statistical tools to analyze them and draw your own conclusions. There are two main sets of tools that are covered in this class. The first set of tools consists of the mathematical methods that we use to formalize modern
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macroeconomic theories and to analyze the predictions of our macroeconomic models. The second set is a set of basic statistical methods that are often used to summarize the main stylized facts that a large part of the theory focuses on. This introductory chapter is intended to give you a brief flavor of the material that will be covered in this course as well as explain the structure and the approach taken. We will start of by considering the historical patterns of some of the most important macroeconomic aggregates for the U.S. economy. Doing so allows us to consider a broad set of evidence from the data that macroeconomic theory sets out to explain. In the second part of this chapter we will consider the basic structure of the theory that we will develop. Finally, in the third part of this chapter we will briefly preview some of the tools that we will be using. In this part we will particularly focus on the concept of marginal analysis, which is the approach we take in some of the mathematically more challenging parts of this course. I C O N
K E Y
Data information Exercise Computer exercise Reading material
The only way to become comfortable with the theory, evidence, and tools in this course is through practice. In order to provide you with practice on the theory, these notes contain a large number of theoretical exercises. To give you some experience with studying the data, there is also a set of data related exercises. Some of the exercises require the use of a computer with Microsoft Excel (version 97 or higher). The icon key in the left margin details which icons pertain to exercises and to computer exercises. Besides exercises, these notes also contain information on macroeconomic data. The parts of the text that describe macroeconomic data releases for the U.S. economy are also indicated by an icon in the left margin. Finally, you are expected to read a set of reading materials that consists of academic papers. These materials are meant to supplement some of the theory and evidence as well as to put some of the results and theory in this course in a critical perspective. References to reading materials are also accompanied by an icon in the left margin.
1.1 Historical U.S. macroeconomic evidence Macroeconomics is an empirical social science. It aims to explain the observed outcomes that are the result of behavior and interactions of the individuals that make up households, firms, the stock market, government institutions, central banks, unions, and other economic entities. Macroeconomic aggregates
What puts the macro in macroeconomics is that it focuses on the overall outcome for the whole economy rather than the outcomes that pertain to specific households or firms. Economic variables that are studied by macroeconomists are often referred to as macroeconomic aggregates.
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Gross Domestic Product (GDP)
‘To aggregate’ literally means ‘to sum up’. Many macroeconomic aggregates are indeed the sum of the underlying amounts for households and firms. The most important example of this is Gross Domestic Product (GDP), which measures the total value added to goods and services by households, firms, and other economic entities in a country. Other macroeconomic aggregates, however, do not represent the sum of measures of economic activity. Some reflect the average outcome for firms, households, and other economic entities. For example, the inflation rate measures the percentage increase in the price of a basket of goods that reflects the consumption pattern of the ‘average’ household. Others, like the unemployment rate, measure ratios. That is, the unemployment rate is the ratio of the number of persons looking for a job divided by the total number of people that want to and/or work. In this section, we will have a look at the historical patterns for a set of main macroeconomic indicators for the U.S. economy. We will do so in the form of a quiz in which you are asked to match up the economic indicators with a set of increasingly informative figures. Exercise 1.1: Macroeconomic aggregates quiz
This exercise uses the Excel file Chapter1.xls. This file contains a quiz, which makes you identify eight macroeconomic aggregates. The detailed description of the quiz is in the file1. Once you have finished the quiz above, have a detailed look at the time series and do the following exercise. Exercise 1.2: Identify the stylized facts
For each of the eight macroeconomic aggregates included in the quiz in exercise 1.1 what are their three most obvious properties. Do the aggregates increase over time? Do they fluctuate? Do they move together? etc. The above two exercises did not only expose you to some of the important macroeconomic aggregates, they also made you extract some of the stylized facts. There are many different degrees to which one can call facts ‘stylized’. However, for
Note that this file uses macros. Please refer to Appendix C for more information on the Excel security settings in order to run these macros and do the quiz.
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the purpose of the class here we will mainly focus on facts that are glaringly obvious. Where this is not the case, the statistical tools necessary to identify these facts will be introduced as we go along.
1.2 Structure of this class There tends to be a big difference between the way macroeconomics is taught at the undergraduate and at the graduate level. At the undergraduate level, the emphasis tends to be more on familiarizing students with the main facts, issues, and arguments. At the graduate level, however, the emphasis is much more on a formal underpinning of the arguments. This is done by making a set of assumptions and then deriving their theoretical implications for observed macroeconomic phenomena. These assumptions are best formalized in the form of mathematical models. A graduate student in macroeconomics is thus expected to make a big leap from informal arguments to the formal mathematical theoretical analysis of economic models. In this section we argue why making such a big leap is useful, how we will make this big leap, and which topics we will address while doing so. The big leap that we will make consists of going from an informal theory of macroeconomic relationships, as presented in most advanced undergraduate texts, to a formal theory of macroeconomic relationships. For example, most undergraduate texts just assume that savings are decreasing in the interest rate and then use this assumption to see what it implies in the macroeconomy. In this course, however, we will not make any assumptions about important relationships, like that between the interest rates and savings. Instead we will develop a detailed theory of why this is the case. Microfoundations
The type of theory that we will develop is known as microfounded macroeconomic theory. Microfoundations of macroeconomics are assumptions and theories that aim to explain the behavior of individual agents, like households and firms, in the economy. The resulting microfounded macroeconomic theory combines these theories of individuals to consider the resulting behavior of macroeconomic aggregates when these individuals interact in markets. The advantage of using microfoundations to develop a macroeconomic theory is that they allows us to consider the behavior of the macroeconomy as a function of fundamentals, namely the preferences of households and the production technologies used by firms. This allows us to do two important things. First of all, being very explicit about the assumptions that we make for our theory, allows us to identify the assumptions that are the most important for the explanations of the facts. Secondly, the mathematical treatment of these assumptions yields a parameterization of them which we can use to quantify their relative importance in the data. The structure of these course notes is as follows:
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In the first part, consisting of chapters 2 and 3, we develop the microfoundations for the two most important types of agents in the economy, the household and the firm. In chapter 2, we look at the theory of the household. This theory aims to explain how households decide how much they would like to work, how much they would like to consume, and how much they would like to save in each period. In chapter 3, we consider the firm. We look at what determines how many workers firms decide to hire, how many machines, buildings, and other equipment firms decide to use in production, and how much output firms produce. General equilibrium
In the second part, consisting of chapters 4, 5, and 6, we consider the behavior of macroeconomic aggregates in our theoretical economy that come about when the firms and households that we consider interact in markets. In Chapter 4 we describe how we derive what happens in these markets and define the concept of general equilibrium in the macroeconomy. In Chapter 5 we consider the trend properties of output, consumption, and investment in this economy when the efficiency with which buildings, equipment, and labor are used to produce output, known as total factor productivity, grows at a constant rate over time. In Chapter 6 we study the cyclical behavior of the economy, i.e. the fluctuations around this trend, when productivity does not grow smoothly but instead fluctuates itself around its trend. The first two parts of these notes complete our derivation of the basic microfounded model. This model is also known as the neoclassical model of the macroeconomy. In the subsequent parts we will basically do two things. First of all, we will drop some of the assumptions underlying the neoclassical model. The neoclassical model basically assumes that markets work perfectly in the sense that everyone has full information, prices and inputs that firms set and use adjust immediately to changes in the economic environment, etcetera. We will drop this assumption in many directions. Mechanisms, like adjustment costs to prices and the time it takes to search for a job, are known as rigidities and frictions. Hence, we will spend a large part of our time considering the effect of such rigidities and frictions on the economy. Secondly, we extend the neoclassical model by adding additional types of economic agents and institutions beyond the households and firms. For example, we add a central bank that conducts monetary policy, a government that conducts fiscal policy and provides unemployment benefits, an education sector in which skills are accumulated, an R&D sector that generates new ideas and technologies, as well as consider what happens when we open the economy to trade with other countries. The first extension of our model that we will consider in chapters 7 and 8 is the addition of money and a central bank. We consider the two mainstreams in the theory of the effect of monetary policy on the economy. The first theory explicitly considers what determines why people hold money and how it affects the overall allocation of resources in the macroeconomy. The second theory actually ignores the demand for
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money and instead considers what happens when prices can only adjust slowly in response to changes in the interest rate. The latter theory is known as the New Keynesian model and is the theoretical framework that has become the basis for many of the monetary policy decisions that are currently taken around the World. In Chapter 9 we add a government that performs fiscal policy to our model. We consider the relative costs, in terms of the efficiency of the allocation of resources, of different types of taxes. Furthermore, we also look at the interaction between monetary policy and fiscal policy. Governments do not only decide on taxes but also regulate other markets. In Chapter 10 we consider how labor market regulations and the welfare state affect equilibrium rates of unemployment. We do so in the context of a model of the labor market with frictions due to costs of searching for a job as well as hiring costs. We use this model to get a handle on the persistent differences in unemployment rates among many of the leading industrialized economies in the World. In Chapter 11 we revisit the issue of trend productivity growth and consider mechanisms that would endogenously generate continuous growth of productivity. We do so by adding two sectors to our model economy. The first is a sector that produces human capital; an education sector. This sector accommodates the consistent accumulation of skills that spur long-run economic growth. The second sector is an R&D sector that generates new and improved ideas and technologies. In Chapter 12, we open up our model economy to interact with the rest of the world. That is, we introduce a foreign sector in our model. Finally, we conclude in Chapter 13 by looking at the main lessons we learned, the important issues we addressed, the answers we came up with, extensions of the theory that were not covered, and the many issues that remain unanswered by modern macroeconomics. Each of the following chapters in these notes has a similar structure. The first part of the chapter introduces the issues addressed and the theoretical framework applied to address them. The second part of the chapter contains the formal derivation of the main results. The third part of the chapter deals with the data on macroeconomic aggregates that are relevant to the topic at hand. The final part of each chapter contains an application of the theory. These applications serve two purposes. The first is to show the practical importance of the theory introduced. The second is to use a repetition of the main results to increase your understanding of them. In short, in each chapter the theory is introduced first. Then the data and the stylized facts are addressed. Finally, the two of them are combined in an empirical application, which allows you to both get re-exposed to the theoretical model as well as apply simple statistical tools to the data.
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Throughout this course it is important to always focus on the issues being addressed. It turns out that, over the past 25 years, academic macroeconomists have basically come to agree on the proper methodological foundations of their theory. However, they still disagree a lot about which assumptions underlying the different versions of the theory and their resulting predictions are most important in the data. A great example of this disagreement is the set of papers that you are asked to read in the following exercise. Exercise 1.3: Is there a core of macroeconomics?
During the first weekend of every calendar year, many, if not most, academic macroeconomists meet at a large conference organized by the American Economic Association and the Econometric Society, as well as other social science organizations. Many of the papers presented at this meeting are subsequently published in leading journals. During this conference in 1997, five of the World’s leading macroeconomists were asked to each give their answer to the question: ‘Is there a core of practical macroeconomics that we should all believe?’ The economists on the panel were: (i) Robert Solow, who won the Nobel prize for his groundbreaking work on long-run economic growth, (ii) John Taylor, who has been a leading scholar on monetary policy, (iii) Martin Eichenbaum, who has done important research on sources of business cycle fluctuations, (iv) Alan Blinder, who is an advocate of the Keynesian school of economics that emphasizes to role of market imperfections as a contributor to economic fluctuations, (v) Olivier Blanchard, who has also done research on the importance of market imperfections for macroeconomic outcomes, especially in the context of labor markets. Each of these economists summarized their opinion in a short paper. These five papers were published in the American Economic Review, Papers and Proceedings in May 1997, 230-246. (i) Read each of the five short papers that these economists wrote and try to fill in Table 1.1. The first line of the table is filled in as an example. It is the point that long-run economic growth is mainly determined by the supply side of the economy. Try to fill in the subsequent four lines with the other four issues that are most often mentioned as a part of core macroeconomics by these economists. For each topic, classify each of the economists according to: not mentioned, which is when the topic is not mentioned in their paper,
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agree, is when they adhere to the majority view, disagree, is when they don’t. (ii) What would be your anwer to the question ‘Is there a core of practical macroeconomics that we should all believe?’, based on the opinions of these five economists? Do they all believe in the same core of practical macroeconomics? Table 1.1: Core topics and who agrees or disagrees with them.
Topic
Solow
Taylor
Eichenbaum
Blinder
Blanchard
1. Long-run growth is determined by the supply side of the economy
Agree
Agree
Not mentioned
Not mentioned
Agree
2. 3. 4. 5.
The exercise above gives you a quick overview of many of the issues that we will address in the subsequent chapters. You are strongly advised to return to this exercise at the end of the course and see whether you have a better grasp of the issues and are better able to assess the opinions of these economists.
1.3 An introduction to the main toolkit In the previous section we noted that graduate students in macroeconomics are expected to make a big leap from their undergraduate work to the formal theory presented in their graduate class. The big difference between undergraduate material in macroeconomics and the material presented in this class is the formal mathematical underpinnings of the theory presented here. As we argued in the previous section, the formal mathematical framework behind our theory will allow us to isolate the assumptions that drive the main results in it, as well as parameterize and quantify the important factors in our theory. However, learning such a formal mathematical framework is a big investment and can often look like a daunting task.
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Marginal analysis
As it turns out, below the complicated notation, there is a basic set of principles that return over and over again. This set of principles was first formalized by Alfred Marshall who laid out the ‘principles of modern economics’ in 1890. It is known as the concept of marginal analysis and is central to many of the formal mathematical economic theories developed since. Figure 1.1 provides a graphical illustration of marginal analysis. At the heart of our microfounded theory of the macroeconomy is that the economic decisions in the economy are made to maximize particular objectives. The top panel of figure 1.1 depicts the choice of x to maximize the objective plotted in the panel. The choice, x*, that maximizes the objective is that for which the objective function peaks, which is the choice of x for which the slope of the objective function is zero. The objectives of economic agents generally are the difference between benefits and costs of the decision, x. The bottom panel plots the benefits and cost functions associated with the objective function in the top panel.
Marginal benefit Marginal cost
Marginal analysis relates the slopes of the benefits and costs curves to the optimal choice, x*. These slopes are called marginal benefit and marginal cost of x and reflect the change in the benefits and costs that result from a small change in the choice variable, in this case x. To see what is so special about the optimal choice, x*, consider two choices, x1 and x2, that are suboptimal. Let us consider x1 first. At x1 marginal benefit exceeds marginal cost. That is, if the household would increase x a little bit, to x1+∆x, then the additional benefits from the increase ∆x are higher than the increase in the costs. This means that the difference between benefits and costs, i.e. the objective, must be higher in x1+∆x than in x1. Thus, x1 can not be the choice that maximizes the objective. The choice x2 is suboptimal for a very similar reason. In x2 marginal costs exceed marginal benefits. This implies that if the one would decrease x a little bit, to x2-∆x, then the additional benefits from the decrease ∆x are higher than the increase in the costs. This means that the objective must be higher in x2-∆x than in x2. Thus x2 can not be the choice that maximizes the objective. What is so special about the optimal choice x* is that in x* the marginal benefits equal the marginal costs. Hence, for both a small increase as well as a decrease in x from x* the changes in the costs offset the changes in the benefits. Thus a deviation from x* does not increase the objective either way.
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Figure 1.1: Marginal analysis illustrated
You might not realize it, but we have basically taken four steps to find the solution, x*, to the optimization problem depicted above. Step 1: Identify the costs and benefits We first identified the costs and the benefits that underly the objective that we tried to maximize. Step 2: Calculate the marginal costs and marginal benefits
We then determined
the slopes of both the costs and benefits parts of the objective. We found the condition that the optimal choice has to satisfy by equating marginal costs to marginal benefits. This is known as the optimality condition associated with the choice of x.
Step 3: Equate the marginal costs and marginal benefits
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Optimality condition
We found the choice x* for which the marginal costs and marginal benefits are equal and the optimality condition is thus satisfied. We argued why it intuitively makes sense to be the optimizing choice.
Step 4: Solve the optimality condition and interpret it
These four steps are not only applicable to the illustrative example above, they turn out to be generally applicable to most of the economic theory that tries to explain the decisions that economic agents make. For this reason, we will take these four steps in many different directions in what follows. Whenever we do so, the math can often look intimidating. However, bear in mind that the underlying principles are as simple as the example we considered above.
1.4 Here we go… Before we go… A bit of advice on how to tackle the material in these notes. First and foremost, stick with it! The biggest challenge of the theory presented here is, in many respects, the translation of the math into the underlying economic concepts. To make you comfortable with the math, most of the equations in Chapters 2, 3, and 4 are derived in great detail. This might look daunting the first time you read the text. However, you are strongly advised to reread the material and try to derive the main results. When doing so, ask yourself for every equation: ‘What does each term in here mean?’. That is, make sure that you are able to ‘tell the story’ behind the equations. Secondly, do the exercises! Translating the math into the economic concepts that we discuss takes a lot of practice. This is what the exercises are there for. The exercises do not only allow you to practice some of the math, they often are an integral part of the arguments made and theory covered. They also teach you about looking up data from common statistical releases that pertain to the U.S. and World economy. Hence, the exercises are a must-do. Without doing them, you will not understand the material. Thirdly, use the resources provided! These notes contain a series of hyperlinks to webpages and are accompanied by a set of readings from academic journals. These links and readings are an integral part of the material and you are strongly encouraged to use them to your advantage. Finally, the ultimate goal is not for you to do well in this class but to become a better economist! For this purpose, it is important to learn how the knowledge in this class is applied on a daily basis. Many of the major economic newssources, like the Economist Magazine, Wall Street Journal, Bloomberg Television, etc., discuss both the data as well as apply the theory that is covered in this class. Tying the economic news back to the material in this class will help you not only in this class but, most importantly, to grow as an economist in general.
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Additional Readings and Resources
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2
Chapter
The household In this chapter we consider the decisions that households make in our stylized version of the overall economy that is our macroeconomic model. We do so in three parts. In the first part we consider the household’s problem in the absence of time and look at the trade off between consumption and working. This trade off is what determines the labor supply decisions of households. In the second part, we add time to the environment in which the household makes its decision. Time adds another dimension to the household’s choices. It allows a household to decide whether it wants to use its income and wealth for consumption now or in the future. This is known as the savings decision. Finally, we end this chapter with an empirical application of the theory in this chapter and consider what it predicts for the behavior of share prices.
2.1 The household’s objective: Utility In principle, the macroeconomy is made up of a broad spectrum of very different households. Some are single women or men, families where both parents and children living at home, single-parent families, grandparents that take care of their grandchildren, college roommates, etc. The aim of the macroeconomic theory of household behavior is to capture the basic determinants and mechanisms that affect the common choices that the households in the economy are facing. They might not always make the same decisions. However, we will assume that for the study of the aggregate economy the underlying differences between the households do not matter much and that their overall (average) behavior represents that of a ‘representative’ household. Representative household
In order to make this representative household assumption work, we assume, throughout this, that all the households are identical. As it turns out, this simplifying assumption does not affect our ability to explain many of the macroeconomic facts that we are interested in. Where it does so, it will be briefly pointed out further on in these notes.
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Utility function
In order to develop a theory of the decisions that the representative household makes, we need to define what the household cares about when it makes its decisions. For our representative household we assume that it values both consumption and leisure. It chooses its level of consumption and leisure in order to maximize its ‘happiness’, called utility in the context of economic theory. We assume that the utility that a household derives in period t from its consumption and leisure can be represented by a function, known as the utility function. Let Ct denote the household’s consumption level at time t and let 0≤Lt≤1 be the fraction of time that the household works. We assume that there is only one representative good that the household consumes. Given this notation, (1-Lt) is the fraction of time that the household/worker has left for leisure. Utility derived from consuming Ct and having leisure time (1-Lt) equals U(Ct,(1-Lt)).
Constant Relative Risk Aversion preferences
Throughout these notes we assume that the preferences of the representative household, which determine its utility level, are given by the following Constant Relative Risk Aversion utility function U (Ct , Lt ) =
[C σ −1 σ
1−θ σ ] t (1 − Lt )
θ
σ −1
where 0 < θ ≤ 1 and σ≠1
(2.1)
The parameter θ determines the relative importance of consumption and leisure in the representative household's preferences. The parameter σ, which is called the intertemporal elasticity of substitution affects the path of consumption and leisure over time when we consider the household's dynamic optimization decision. Indifference curve
Before we consider the optimization decision that the household faces, we first consider the form of this utility function in more detail. An indifference curve for this household is a set of combinations of consumption, Ct, and leisure, (1-Lt), for which the household obtains a particular level of utility U*. Because it is the utility level that determines the household’s choices, the household would be indifferent between all points on the indifferent curve since they give the same utility. Hence the name. In order to consider the effect of θ on the household’s preferences, consider the following exercise. Exercise 2.1: Change in θ and its effect on preferences
Figure 2.1 shows the indifference curve corresponding to θ=1/2 and the constant utility level U*=σ/(σ-1). In the figure, draw what would happen to the indifference curve when (i) θ 1/2. Explain.
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Figure 2.1: Indifference curve
Intratemporal decision
With the household’s objective pinned down, we can consider how the household takes it into account when it makes its decisions. We will start of by considering what is generally called the household’s intratemporal decision. It is the decision that involves the trade off between utility the household derives from current consumption and from current leisure. Since both the benefit and cost of this decision are obtained and incurred within the same period, this is called the intratemporal decision problem of the household.
2.2 The labor supply decision If the household would be unconstrained, it would decide not to work and consume an infinite amount. Unfortunately, it is limited by its budget constraint, which implies that it can only spend what it earns. We will assume that the household has two sources of income. The first is labor income, i.e. the money it earns as a compensation for the time it works. The second is capital income, i.e. the money it earns on the assets it has invested in. Let Pt>0 be the price of a consumption good and Wt>0 denote the wage paid for work such that labor income equals WtLt. Let At-1 be the asset holdings of the household at the end of the previous period (and thus at the beginning of the current period, i.e. period t ) and rt be the interest rate paid on these holdings. We will look at what
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determines the path of At-1 over time later in this chapter when we consider the household’s savings decision. For the moment, we will take At-1 as given. The capital income of the household consists of the interest paid on its assets2, which equals rtAt-1. Besides the capital income the consumer can also decide to consume the initial asset holdings At-1 itself. Consequently, the consumer faces the budget constraint that implies that consumption expenditures cannot exceed the sum of labor income, capital income and the initial asset holdings. Mathematically, this boils down to Pt Ct ≤ Wt Lt + (1 + rt )At −1
(2.2)
Throughout this section, we assume that the household only considers current consumption and that there is no future that it takes into account and derives utility from. In that case, the inequality above turns out always to be binding. This is because utility is strictly increasing in consumption, such that if the household would choose Pt Ct < Wt Lt + (1 + rt )At −1
(2.3)
then it would not be maximizing utility. In that case, it could increase utility by simply increasing its consumption to the level where the restriction is binding. Hence, the budget constraint can effectively be written as an equality such that Pt Ct = Wt Lt + (1 + rt )At −1
(2.4)
The left and right sides of this equation are measured in dollar-terms. However, we can rewrite the equation such that both sides are measured in terms of units of the consumption goods. That is, ⎛W ⎞ P A Ct = ⎜⎜ t ⎟⎟ Lt + (1 + rt ) t −1 t −1 Pt Pt −1 ⎝ Pt ⎠ Real budget constraint
(2.5)
This form of the budget constraint is known as the real budget constraint because it expresses the budget constraint in terms of quantities of the consumption good rather than in terms of money. Let us focus for a moment on the term
(1 + rt ) Pt −1
At −1 Pt Pt −1
(2.6)
It represents the value of the asset holdings that equalled At-1/Pt-1 units of the consumption good at the end of period t-1 in terms of units of the consumption good at the end of period t. It considers two factors that affect the value of At-1/Pt-1. The first In principle, assetholdings can be negative, which implies the household has borrowed money. Our assumption about interest income/payments implies that the interest rate paid to savings is the same as the interest paid on debt.
2
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is the potential change in the price level of the consumption good, i.e. Pt-1/Pt. That is, if prices change between periods t and t-1, then one can buy a different amount of the consumption good for At-1 in each period. The second takes into account the interest payments, (1+rt). The combination of these two terms,
(1 + rt ) Pt −1 Pt
(2.7)
represents the gross return on the assets between the end of period t-1 and period t measured in terms of units of the consumption goods. Inflation rate
We can relate the price change, Pt-1/Pt back to the inflation rate. The inflation rate in period t, which we will denote by πt, is the percentage change in the price level between periods t-1 and t, such that
Pt − Pt −1 Pt −1
(2.8)
Pt −1 1 = Pt 1+πt
(2.9)
πt = We thus obtain that
such that
(1 + rt ) Pt −1 Pt
Real interest rate
=
1 + rt 1+πt
(2.10)
Denote the net return on the assets measured in terms of the consumption good, called the real interest rate, as ~ rt . The above equation implies that
(1 + ~rt ) = 1 + rt
1+πt
(2.11)
Thus, the real interest rate is the nominal interest rate corrected for the change in the real value of the assets due to inflation. Exercise 2.2: Real versus nominal interest rates
In most undergraduate textbooks and in many economic applications, the real interest rate is approximated as the difference between the nominal interest rate and the inflation rate, i.e.
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~ rt = rt − π t Fisher identity
(2.12)
This is known as the Fisher identity. (i) Under what condition on the product of ~ rt and πt is this a good approximation? (ii) What about countries with hyperinflation where πt is very large. Does the above approximation over- or underestimate the real interest rate in that case? Given this definition of the real interest rate, we can now simplify the real budget constraint in (2.5) by writing ⎛W ⎞ A Ct = ⎜⎜ t ⎟⎟ Lt + (1 + ~ rt ) t −1 Pt −1 ⎝ Pt ⎠
(2.13)
Throughout this course, we will follow the convention that we denote real variables, that are measured in units of the consumption good, with a ~. That is, the real wage, which reflects the number of consumption goods one can buy for, say, an hour of, work, will be written as ~ Wt = Wt / Pt
(2.14)
Similarly, the real asset holdings at the end of period t-1 are defined as ~ At −1 = At −1 / Pt −1
(2.15)
Hence, in real variables the real budget constraint can be written as ~ ~ Ct = Wt Lt + (1 + ~ rt )At −1
(2.16)
In order to get some feeling for the shape of this budget constraint, it is worthwhile to do the following exercise Exercise 2.3: The shape of the budget constraint
In the leisure-consumption space, the budget constraint can be represented as a straight line as depicted in Figure 2.2. Draw in this diagram what happens to the budget constraint (i)
when the wage increases and
(ii)
when the nominal interest rate, rt, decreases
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Figure 2.2: Shape of the budget constraint
Given this budget constraint we can now consider the household's utility maximization problem. The household chooses consumption, Ct, and its labor supply, 0≤Lt≤ 1, to maximize its utility
U (Ct , Lt ) =
[C σ −1 σ
1−θ σ ] t (1 − Lt )
θ
σ −1
(2.17)
subject to the real budget constraint ~ ~ Ct = Wt Lt + (1 + ~ rt )At −1
(2.18)
Before we jump into the algebra of things, there are a few things we can prove without doing the math. This is what you are asked to do in the next two exercises. Exercise 2.4: Non-binding upperbound on Lt
Show that if 00 are of a similar form. If you are not convinced, it is a good exercise to derive them yourself! In order to derive the costs and benefits of Xt it turns out to be convenient to write out explicitly the first two terms of the summation (2.95). That is, we use the budget constraint, (2.96), as ~ ~ ~ ~ ~ Ct + s = Wt + s + Dt + s X t + s + (1 + ~ rt + s )At + s −1 − At + s − ( X t + s − X t + s −1 )S t + s
(2.97)
and substitute this into the summation for Ct and Ct+s, such that (2.96) can be written as
(
)
~ ~ ~ ~ ~ U Wt + Dt X t + (1 + ~ rt )At −1 − At − ( X t − X t −1 )S t ~ ~ ~ ~ ~ + βU Wt +1 + Dt +1 X t +1 + (1 + ~ rt +1 )At − At +1 − ( X t +1 − X t )S t +1
(
)
(2.98)
∞
+ ∑ β sU (Ct + s ) s=2
This representation of the objective allows us to identify benefits and costs of investments in the stock market in a similar way as we did with the benefits and costs of saving. Just like for regular savings, the cost of investing in the stock market is that the amount invested cannot be used towards current consumption. That is, the costs consist of the current consumption sacrificed to invest in the stock market.
⎛ ⎛ ⎞ ⎞ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ~ ~ ⎜ ~ ~ ~ ~ ⎜ U Wt + Dt X t + (1 + rt )At −1 − At − X t − X t −1 ⎟ S t ⎟ { ⎜ ⎜ cost ⎟ ⎟ : ⎜ ⎜ sacrificed ⎟ ⎟ current ⎜ consumptio ⎟ ⎟ ⎜ n ⎝ ⎠ ⎠ ⎝ ~ ~ ~ ~ ~ + βU Wt +1 + Dt +1 X t +1 + (1 + ~ rt +1 )At − At +1 − ( X t +1 − X t )S t +1
(
+
σ
)
(2.99)
∞
∑ β U (C ) s
σ − 1 s =2
t+s
The benefits consist of two parts. First of all, current share holdings yield dividends, which can be used towards current consumption. Secondly, current share holdings can be sold in the next period and used for next period’s consumption expenditures. Mathematically these two types of benefits show up as
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⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ~ ~ ~⎟ ~ ~ ~ ⎜ + (1 + rt )At −1 − At − ( X t − X t −1 )S t U Wt + Dt X {t ⎜ ⎟ benefits: ⎜ ⎟ current consumption increase due to dividend ⎜ ⎟ income ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜~ ⎜ ⎟ ~ ~ ~ ⎟ ~ + βU ⎜Wt +1 + Dt +1 X t +1 + (1 + ~ rt +1 )At − At +1 − ⎜ X t +1 − Xt S t +1 ⎟ (2.100) ⎟ { ⎜ ⎟ ⎜ ⎟ benefits: increase in next ⎜ ⎟ ⎜ ⎟ period's consumption due to income ⎟ ⎜ ⎟ ⎜ from sale of shares ⎠ ⎝ ⎝ ⎠ +
σ
∞
∑ β U (C ) s
σ − 1 s =2
t +s
Now that we have identified the costs and benefits of the stock market investment decision we can unleash our partial derivatives on this problem again to derive the marginal benefits and marginal costs. For the marginal costs of the stock market investment we find that they are composed of the amount of current consumption lost due to the purchase of an additional stock times the marginal utility value of this amount of consumption. That is, in terms of partial derivatives Step 2: Calculate the marginal costs and marginal benefits
−
∂C ∂ U (Ct ) t ∂Ct ∂X t
where we only vary the X t in consumption that is part of the costs and not the X t that shows up in the dividend income
⎡ ∂ ⎤~ U (Ct )⎥ S t =⎢ ⎣ ∂Ct ⎦
(2.101)
where we again inserted the minus sign at the beginning to express marginal costs as a positive, rather than a negative, number. The marginal benefits of the stock market investment are made up of two parts, each of which corresponds to one of the channels through which the household benefits from its stock market investment. The marginal benefits of the dividends paid on the shares equals the amount of dividends received on an additional share that could be invested in, which is the amount of current consumption obtained per share from dividend income, times the marginal utility of current consumption. Mathematically, this can be expressed as
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β
∂C ∂ U (Ct ) t ∂Ct ∂X t
where we only vary the X t that is part of the dividend income
⎡ ∂ ⎤~ U (Ct )⎥ Dt =⎢ ⎣ ∂Ct ⎦
(2.102)
The second part of the marginal benefits consists of the marginal benefits of the sale of an additional share tomorrow. This equals the additional consumption that can be obtained from the sale of an extra share times the marginal utility of consumption tomorrow.
β
⎡ ∂ ⎤~ ∂C ∂ U (Ct +1 ) t +1 = β ⎢ U (Ct +1 )⎥ S t +1 ∂Ct +1 ∂X t ⎣ ∂Ct +1 ⎦
(2.103)
Combining these three terms, we can derive the optimality condition. Step 3: Equate marginal costs and marginal benefits Equating the marginal benefits
and marginal costs, we find that the optimality condition for the stock portfolio decision is ⎡ ∂ ⎤~ ⎡ ∂ ⎤~ ⎡ ∂ ⎤~ U (Ct )⎥ S t = ⎢ U (Ct )⎥ Dt + β ⎢ U (Ct +1 )⎥ S t +1 ⎢ ⎣ ∂Ct ⎦ ⎣ ∂Ct ⎦ ⎣ ∂Ct +1 ⎦
(2.104)
This is known as an arbitrage condition. If the left hand side would be higher than the right hand side of this equation the costs of investing in stocks exceed the benefits and the household would reduce its exposure to stocks in its portfolio, if it would be lower households would invest more in stocks. There is nothing that prevents you from convincing yourself that you also get this condition when you take to total (partial) derivative of (2.95) with respect to Xt. This works in the same way we did for savings. Anyway, the above is the optimality condition for the investment in shares. The question that remains is what do we learn from it? Fundamental value of a share
The optimality condition (2.104) actually teaches us a lot. It allows us to derive what is known as the fundamental value underlying the share price. The fundamental value of a share is derived by solving (2.104) by forward recursion. This works as follows. Equation (2.104) implies Step 4: Solve the optimality condition
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⎡ ∂U (Ct +1 ) ∂Ct +1 ⎤ ~ ~ ~ S t = Dt + β ⎢ ⎥ S t +1 ( ) ∂ ∂ U C C t t ⎦ ⎣ ⎡ ∂U (Ct +1 ) ∂Ct +1 ⎤ ⎧ ~ ⎡ ∂U (Ct + 2 ) ~ = Dt + β ⎢ ⎥ ⎨ Dt +1 + β ⎢ ⎣ ∂U (Ct ) ∂Ct ⎦ ⎩ ⎣ ∂U (Ct +1 ) ⎡ ∂U (Ct +1 ) ∂Ct +1 ⎤ ~ ~ 2 ⎡ ∂U (Ct + 2 ) = Dt + β ⎢ ⎥ Dt +1 + β ⎢ ⎣ ∂U (Ct ) ∂Ct ⎦ ⎣ ∂U (Ct )
∂Ct + 2 ⎤ ~ ⎫ ⎥ St +2 ⎬ ∂Ct +1 ⎦ ⎭
(2.105)
∂Ct + 2 ⎤ ~ ⎥ St +2 ∂Ct ⎦
The above illustrates one step of the forward recursive solution. It is obtained by ~ substituting the solution to S t +1 into (2.104). Now, we can proceed taking similar steps ~ by substituting the solution for S t + 2 in the above equation, etc.. When we do so, we obtain the following solution for the real stock price.
~ k −1 ⎡ ∂U (Ct + s ) ∂Ct + s ⎤ ~ k ⎡ ∂U (Ct + k ) ∂Ct + k ⎤ ~ St = ∑ β s ⎢ ⎥ Dt + s + β ⎢ ⎥ St +k s =0 ⎣ ∂U (Ct ) ∂Ct ⎦ ⎣ ∂U (Ct ) ∂Ct ⎦
(2.106)
We will assume that the stock price will not increase too fast, such that ⎡ ∂U (Ct + k ) ∂Ct + k ⎤ ~ lim β k ⎢ ⎥ St +k = 0 k →∞ ( ) ∂ ∂ U C C t t ⎣ ⎦
(2.107)
If this condition holds5, then we can write the share price as ∞ ⎡ ∂U (Ct + s ) ∂Ct + s ⎤ ~ ~ St = ∑ β s ⎢ ⎥ Dt + s s =0 ⎣ ∂U (Ct ) ∂Ct ⎦
(2.108)
The interpretation of this result goes as follows. A share in a company is nothing more than the entitlement to a stream of dividends paid to the holder of that share. Hence, the price someone is willing to pay for that share should be equal to the present discounted marginal utility value of that stream of dividends. This is exactly what this equation implies. That is, the share price on the left hand side equals the present discounted (marginal utility) value of the dividends on the right hand side. The right hand side is often referred to as the fundamental value of the share. S&P 500
Let's have a quick look at the data and see what some of the major issues are in our theory of stock prices. For the empirical part of this section we will focus our attention on the Standard and Poor's 500 (S&P 500) index.
5 There is a technical reason why this condition has to hold. That is that if it doesn’t then the households objective is unbounded and the dynamic maximization problem is ill-defined.
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The S&P 500 index can be interpreted as reflecting the value of a portfolio in which one holds shares in the 500 most valuable companies in the U.S. stock market. This portfolio is such that the portfolio-share held in each company is proportional to the company’s market capitalization. Furthermore, this portfolio is adjusted every period without incurring any transaction costs. Figure 2.8 depicts the value of the S&P 500 index, corrected for inflation and normalized to be 100 in January 2006. The value of this index was 10.89 in 1947. This can be interpreted as follows. If one would have invested $10.89, measured in inflation adjusted 2006 dollars, in the biggest firms in the market in 1947 and one would have consumed the dividends paid on these shares over time, then the increase in their share prices would have given you $100 in January 2006. 140
real S&P 500 index (Jan 2006=100)
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Figure 2.8: Historical real S&P index
Equity premium puzzle Excess volatility puzzle
The S&P 500 index has two puzzling properties. First of all, if one would calculate the average return, including the dividends paid, on the S&P 500 then this is much higher than those obtained when one would have invested in bonds. So much higher that the difference in returns seems to be more than just the fact that stocks generally yield higher returns because they are a more risky investment than bonds. Secondly, the stream of dividends that, according to our theory, determines the value of stocks does not seem to be able to account for the fluctuations, known as volatility, in the returns on stocks. The first puzzle is known as the equity premium puzzle, while the second is known as the excess volatility puzzle.
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These two ‘puzzles’ are the topics of the following two exercises Exercise 2.17: Equity premium puzzle
The household that we consider has perfect foresight and exactly knows the path of future interest rates, inflation, as well as dividends and stock prices. Consequently, in order to hold both stocks and bonds their return must be identical. This exercise is meant to prove that. We will consider the household investing in stocks and bonds as we did in this section, with preferences
U (Ct ) = (i)
σ σ −1
σ −1 σ
Ct
(2.109)
Show that the optimal savings decision implies that 1
⎡ ⎤σ (1 + ~rt +1 ) = 1 ⎢ Ct +1 ⎥ β ⎣ Ct ⎦ (ii)
Using (2.104) and the form of the preferences, show that 1 ~ S t +1 1 ⎡ Ct +1 ⎤ σ ~ ~ = ⎢ ⎥ S t − Dt β ⎣ Ct ⎦
(iii)
(2.110)
(2.111)
So, parts (i) and (ii) imply that ~ S t +1 ~ ~ ~ = (1 + rt +1 ) S t − Dt
(2.112)
Interpret this result. Limited participation puzzle
In reality, the returns on stocks are higher than the interest paid on bonds, which is known as the equity premium puzzle. Part of this difference can be explained by the fact that people tend to be risk averse. Since stock market investments are more risky than those in bonds, one would expect equity investments to pay a risk premium. However, even when one corrects for an empirically plausible value for the risk premium, average stock returns do still exceed those on bonds. This actually suggests another puzzle. Why, if stock market investments are so profitable on average, does
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not everyone invest in the stock market? This is sometimes referred to as the limited participation puzzle.
We already referred to stock market investments being much more risky than bond market investments. Figure 2.8 clearly shows major fluctuations in the S&P 500 index. There is one question that arises from these fluctuations combined with our theory. It is whether the discounted dividend stream, that underlies the stock price according to our theory, is just as volatile as the stock price. Our theory would suggest that the answer to this question should be yes because it is this dividend stream that represents the fundamental value of the share. In practice, however, this is not the case. For plausible values of the intertemporal elasticity of substitution, σ, the present discounted value of dividends is not as volatile as the stock price. This is known as the excess volatility puzzle and is the topic of the next exercise. The analysis in the exercise is based on Grossman and Shiller (1981). Exercise 2.18: Excess volatility puzzle
This exercise uses the Excel file Chapter2.xls for its calculations. Upload this file to your computer and open it in Excel. This exercise uses the worksheet called ‘Grossman-Shiller’. It contains a description of this exercise as well as the calculated results. For this exercise we use data running through 1997. According to our theory, the S&P 500 has to satisfy the equation
~ St =
1997 −t −1
∑ s =0
⎛ C β s ⎜⎜ t ⎝ Ct + s
1
1
⎞σ ~ ⎛ C ⎞σ ~ ⎟⎟ Dt + s + β 1997−t ⎜⎜ t ⎟⎟ S1997 ⎠ ⎝ C1997 ⎠
(2.113)
~ ~ We have data for the S&P 500 index, S t , the dividends paid on it, Dt , as well as consumption Ct. The sample of the data runs through 1997. This is why the summation in the equation above is developed up till 1997.
The left hand side of the above equation can be directly measured from the data. The right hand side of the equation cannot. It depends on both the intertemporal elasticity of substitution σ as well as the discount factor β. What we would like to do is to calculate the right hand side and compare it with the left hand side. What we will do is the following. (1) We will choose an intertemporal elasticity of substitution, σ. (2) Conditional on the value of σ, we estimate the implied value of the discount factor, β. We will do so by choosing β such that on average in our data the optimality condition
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~ ~ S −D β = t~ t S t +1
1
⎛ Ct +1 ⎞ σ ⎜⎜ ⎟⎟ C ⎝ t ⎠
(2.114)
holds. (3) We then plot the implied path of the left hand side of (2.113), i.e. the ‘real S&P 500 index’, and the right hand side of (2.113), i.e. the ‘present discounted value of the dividends’. Choose different values for σ in the `Grossman-Shiller' worksheet. Try values for σ between 0.5 and 10. Based on the results try to answer the following questions: Experiment:
(i)
For any value of σ that you choose, does the right hand side of (2.113) show as much volatility (fluctuations) as the left hand side?
(ii)
When you increase σ does the right hand side become less or more volatile? Why?.
The possible answers to the equity premium and excess volatility puzzles are beyond the scope of this class. However, they make for high protein food for thought, spurring a very wide ranging research field in financial economics. Price earnings ratio
One of the things that is often considered when analyzing stock prices is the price earnings ratio. Earnings per share and dividends are generally not the same because often firms decide to retain part of their earnings to finance some of their subsequent investments. However, if the firm uses earnings to finance investments this will also add to stockholders equity by raising the assets of the firm without raising its liabilities. In our simple example here we assume that dividends and earnings coincide. Given this assumption we use the following exercise to consider what determines price earnings ratios. Before we do so, consider the S&P 500's price-earnings ratio series in Figure 2.9.
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50 45
S&P 500 price earnings ratio
40 35 30 25 20 15 10 5
Figure 2.9: Historical S&P 500 price earnings ratio
Exercise 2.19: Price earnings ratio
We have derived that the stock price is the present discounted value of the stream of future dividends. That is, ∞ ⎛ C ~ S t = ∑ β s ⎜⎜ t s =0 ⎝ Ct + s
1
⎞σ ~ ⎟⎟ Dt + s ⎠
(2.115)
Furthermore, the optimality condition for consumption implies that 1
⎛ C ⎞σ 1 β ⎜⎜ t + s ⎟⎟ = (1 + ~rt + s+1 ) ⎝ Ct + s +1 ⎠ (i)
(2.116)
Show that for s≥1,
⎛ Ct ⎜⎜ ⎝ Ct + s
1
s ⎛ ⎞σ s 1 ⎟⎟ β = ∏ ⎜ ~ ⎜ j =1 ⎝ 1 + rt + j ⎠
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⎞ ⎟ ⎟ ⎠
(2.117)
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(ii)
Use the result of part (i) to show that ∞ ⎡ s ⎛ ~ ~ 1 S t = Dt + ∑ ⎢∏ ⎜ ~ ⎜ s =1 ⎣ ⎢ j =1 ⎝ 1 + rt + j
⎞⎤ ~ ⎟⎥ Dt + s ⎟⎥ ⎠⎦
(2.118)
In order to keep things tractable, let's assume that real dividends grow at a constant rate g>0 such that ~ s ~ Dt + s = (1 + g ) Dt
(2.119)
and furthermore, let's assume that the real interest rate is constant and equals ~ r and that ~ r >g. (iii)
Show that in that case s ∞ ~ r ⎞~ ⎛1+ g ⎞ ~ ⎛ 1 + ~ ⎟⎟ Dt St = ∑ ⎜ ⎟ Dt = ⎜⎜ ~ ~ s =0 ⎝ 1 + r ⎠ ⎝r −g⎠
(2.120)
Where the second step uses that for 00 and therefore none of the costs and benefits of the current labor choice of the firm affect future profits. The costs and benefits of the capital inputs, when they are rented, show up in a similar manner to those of the labor inputs. That is, the cost firm’s capital input decision is nothing more than the reduction in flow profits due to the rents on capital paid. The benefit is the contribution of capital to output through the production function. In terms of the firm’s objective and the choice of Kt, this boils down to
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⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ∞ ⎡ s ⎛⎜ 1 1−α α P Z K L W L R K − − t t t t t t ⎟ + ∑ ⎢∏ ⎜ ⎜ t t { { s =1 ⎢ benefits: costs: ⎟ ⎣ j =1 ⎝ 1 + rt + j ⎜ contributi on of rental ⎜ ⎟ capital to output costs ⎠ ⎝
⎞⎤ ⎟⎥ Π t + s ⎟ ⎠⎥⎦
(3.11)
Again, note that Πt+s does not depend on Kt for any s>0 and, therefore, none of the costs and benefits of the current capital input choice of the firm affects future profits. Because neither any of the costs nor any of the benefits of the input choice decisions in this case show up in future profits, the optimality conditions associated with the input choices in this case are intratemporal. In order to derive these optimality conditions, we derive the marginal costs and marginal benefits in the next step. For the marginal costs of the labor demand decision, we find that they are the increase in costs induced by hiring an additional unit of labor. This turns out to be equal to the wage. That is, Step 2: Calculate the marginal costs and marginal benefits
∂ (Wt Lt ) = Wt ∂Lt Marginal revenue of labor Marginal product of labor
(3.12)
For the marginal benefits of the labor demand choice, we obtain that they consist of the additional revenue generated by an additional unit of labor. This is also known as the marginal revenue of labor. The marginal revenue of labor is the output price times the additional units of output produced by an additional unit of labor6. The latter is known as the marginal product of labor. Mathematically, this is ⎛ ∂ ⎞ Y Pt ⎜⎜ Yt ⎟⎟ = Pt (1 − α ) t ∂Lt ⎠ Lt ⎝1 42 4 3
(3.13)
marginal product of labor
Here, the right hand side of this equation is derived by solving for the partial derivative ⎛ ∂ ⎞ Y ⎜⎜ Yt ⎟⎟ = (1 − α )Z t K tα L−t α = (1 − α ) t ∂Lt ⎠ Lt ⎝1 42 4 3
(3.14)
marginal product of labor
In this case, when capital is rented, its marginal costs and benefits are very similar to those of labor. That is, the marginal costs of the capital input decision consist of the
It matters here that the firm is a price taker, such that the price does not change in response to a change in output produced with different inputs. If this is not the case and the price changes, then the marginal revenue of labor also includes the effect of the labor input on the price.
6
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increase in the rental costs due to the rental of an additional unit of capital. This turns out to be equal to the rental rate, as you can see from
∂ (Rt K t ) = Rt ∂K t Marginal revenue of capital Marginal product of capital
(3.15)
The marginal benefits consist of the additional revenue generated by an additional unit of capital inputs. This is also known as the marginal revenue of capital. The marginal revenue is the output price times the additional units of output produced by an additional unit of capital. The latter is the marginal product of capital. Mathematically, this is ⎛ ∂ ⎞ Y Pt ⎜⎜ Yt ⎟⎟ = Ptα t ∂K t ⎠ Kt ⎝1 42 4 3
(3.16)
marginal product of capital
This allows us to solve for the optimality conditions for the capital and labor input choices. Equating the marginal costs and marginal benefits for the labor demand and capital input decisions yields the following optimality condition for labor Step 3: Equate marginal costs and marginal benefits
⎛ ∂ ⎞ Pt ⎜⎜ Yt ⎟⎟ = Wt ⎝ ∂Lt ⎠
(3.17)
which implies that is optimal to hire workers until ⎛ ∂ ⎞ W ~ ⎜⎜ Yt ⎟⎟ = t = Wt ⎝ ∂Lt ⎠ Pt
(3.18)
Hence, the firm will hire workers as long as the increase in output due to an additional worker, i.e. the marginal product of labor, exceeds the amount of output paid to this worker, i.e. the real wage. The optimality condition that results from equating the marginal costs and marginal benefits of capital rental is very similar. It reads ⎛ ∂ ⎞ Pt ⎜⎜ Yt ⎟⎟ = Rt ⎝ ∂K t ⎠
which implies that it is optimal to rent more capital until
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⎛ ∂ ⎞ R ~ ⎜⎜ Yt ⎟⎟ = t = Rt ⎝ ∂K t ⎠ Pt
(3.20)
Hence, the firm will rent more capital as long as the increase in output due to an additional unit of capital, i.e. the marginal product of capital, exceeds the amount of output paid to rent this capital unit, i.e. the real rental price of capital. Step 4: Solve the optimality condition There are several important results that can be
derived from solving the optimality conditions. We will do so mostly in the form of exercises. Right off the bat, let’s determine what the amount of flow profits, i.e. Πt, is that results from the optimal choices of labor and capital of the firm. Exercise 3.3: Zero profits
Show that the above results imply that the firm will choose its capital and labor inputs such that… (i)
…its total labor costs, WtLt, make up a share (1-α) of its revenue, PtYt.
(ii)
…its total rental costs of capital, RtKt, make up a share α of its revenue, PtYt.
(iii)
Use the result of parts (i) and (ii) to derive the profits of the firm at its profit maximizing inputs level.
Exercise 3.4: Constant returns to scale
Constant returns to scale (CRS)
The Cobb-Douglas production function that we used throughout these notes has a property called constant returns to scale. In order to illustrate this property, consider the output level Yt produced with inputs Kt and Lt, such that
Yt = Z t K tα L1t−α (i)
(3.21)
Suppose the firm would instead choose input levels λKt and λLt where λ>0. What would be the output level in that case? How does it relate to Yt?
The reason that we assume that the production function exhibits this property is because we think it is reasonable to assume that when we have twice as much off all inputs we could just do what we did before
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and replicate it such that we get twice as much output. Now consider what constant returns to scale imply for the profit function. Consider the level of profits Πt corresponding to the inputs Kt and Lt, such that
Π t = Pt Yt − Wt Lt − Rt K t
(3.22)
(ii)
Suppose the firm would instead choose input levels λKt and λLt where λ>0. What would be the profit level in that case? How does it relate toΠt?
(iii)
Show that the level of profits per worker, i.e. Πt/Lt, equals
⎛K Πt = Pt Z t ⎜⎜ t Lt ⎝ Lt
α
⎞ ⎛K ⎟⎟ − Rt ⎜⎜ t ⎠ ⎝ Lt
⎞ ⎟⎟ − Wt ⎠
(3.23)
(iv)
Suppose that the wage Wt is such that the firm could choose its capital labor ratio, Kt/Lt, such that it would make strictly positive profits per worker? How many workers would firms decide to hire? What would happen to the wage rate in that case?
(v)
Suppose that the wage Wt is such that the firm could only choose its capital labor ratio, Kt/Lt, such that it would make negative profits per worker? Would any firm decide to produce? What would happen to the wage rate in that case?
Hence, the only equilibrium outcome in this case can be the one in which the wage rate is such that each firm makes zero profits. In that case, no firms will enter the market in pursuit of profits and no firms will exit due to losses they make. We will consider the equilibrium outcome in great detail in the next chapter. What we have derived so far is that if the production function of firms in the economy exhibits constant returns to scale and there is free entry and exit for firms into the market, then in equilibrium these firms need to make zero profits. It is time to consider some empirical evidence. In the first section of this chapter the Cobb-Douglas production function was introduced with the promise that it actually fits an important macroeconomic fact. Now that we have solved the firm’s input demand decision, we can finally consider which fact the Cobb-Douglas production function fits well. We do so in the next exercise.
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Exercise 3.5: Estimating the output elasticity of capital
We assume that the firm uses a technology that can be represented by a Cobb-Douglas production function. In this problem we will argue that this assumption is consistent with a stylized fact in the data.
(i)
Show that if a firm that uses a Cobb-Douglas technology chooses its labor demand to maximize its profits, then its wage bill, WtLt, as a fraction of its total revenue, i.e. PtYt, does not depend on Zt, Pt, or Wt. What is the constant value of WtLt/PtYt? How does it depend on the output elasticity of capital, α.
(ii)
Figure 3.1 depicts total labor compensation as a fraction of GDP, also known as the labor share. Make an estimate of the average labor share during the postwar period? Explain why the data in this graph suggest that the Cobb-Douglas production function is not an unreasonable approximation.
(iii)
Combine your answers to parts (i) and (ii) to obtain an estimate of the output elasticity of capital based on the data presented in Figure 3.1.
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90.0%
80.0%
70.0%
60.0%
50.0%
40.0%
30.0%
20.0%
10.0%
2002
1997
1992
1987
1982
1977
1972
1967
1962
1957
1952
0.0% 1947
employee compensation and proprietor's income as a fraction of GDP
100.0%
Quarter
Figure 3.1: Historical U.S. labor share
This completes our analysis of the firm’s profit maximization problem when the firm rents its capital inputs. It is now time to consider the same problem when the firm owns its capital inputs, also referred to as its capital stock.
3.3 The firm owns its capital stock In the previous section we saw that when the firm rents its capital inputs and can freely adjust them, then its profit maximization problem is essentially static in the sense that it leads to an optimality condition for the capital input choice that is intratemporal rather than intertemporal. Usercost equation of capital
This changes in an important way when the firm actually owns its capital stock. In this section we set up the firm’s profit maximization problem when the firm owns its capital stock. We derive the optimality condition for the capital input decision. This is an intertemporal condition that is often referred to as the usercost equation of capital. The reason that the capital input decision yields an intertemporal optimality condition in case the firm owns its capital stock is because capital inputs are durable. Durability means that even though one buys a factory, office, or machine at one point in time, its productive capacity will last for many periods. In fact, durability is what defines an input as a capital input. Nondurable inputs are called intermediate inputs.
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Hence, a theory of capital inputs will be one that compares the current cost of an investment with its expected stream of future benefits. This is why time is crucial in studying investment and the optimal investment decision is intertemporal because it compares costs and benefits at different points in time. In order to get an idea how the future benefits of investment relate to the current investment level, we have to make an assumption about how the current capital stock depends on past levels of investment. Take a truck, for example. While one can only use the gasoline consumed by the truck once, the truck itself can be used for many trips. However, the longer it is used the more it deteriorates and the less productive it becomes. Depreciation Capital consumption Perpetual inventory method Depreciation rate of capital
Economists call this deterioration of capital goods over time depreciation. Some people like to refer to it as capital consumption. The latter interprets the deterioration of capital due to its usage as consumption. Throughout this class we assume that the capital stock can be calculated using a rule called the perpetual inventory method. The perpetual inventory method assumes that in each period a constant fraction, 00, then the corresponding output level is
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Z t F (λLt , λK1,t ,K, λK m,t ) = λZ t F (Lt , K1,t ,K, K m ,t ) = λYt Homogenous of degree one Euler’s theorem
(3.61)
Mathematically, this implies that the production function F(.) is homogenous of degree one. What we will use for our generalized growth accounting framework is a result for functions that are homogenous of degree one, called Euler’s theorem. Euler’s theorem is explained in general in the mathematical appendix. Here, we will just use the fact that the theorem implies that because F(.) is homogenous of degree one, it satisfies m ⎡ ⎤ ⎡ ∂ ⎤ ∂ F (Lt , K1,t ,K, K m ,t ) = ⎢ F (Lt , K 1,t ,K, K m ,t )⎥ Lt + ∑ ⎢ F (Lt , K 1,t ,K, K m ,t )⎥ K s ,t ∂ ∂ L K s =1 ⎣ ⎢ s ,t ⎣ t ⎦ ⎦⎥
(3.62)
Hence, for output we find that m ⎡ ⎧⎪⎡ ∂ ⎫⎪ ⎤ ⎤ ∂ Yt = Z t ⎨⎢ F (Lt , K1,t ,K, K m ,t )⎥ Lt + ∑ ⎢ F (Lt , K1,t ,K, K m ,t )⎥ K s ,t ⎬ ⎪⎩⎣ ∂Lt ⎪⎭ s =1 ⎣ ∂K s ,t ⎦ ⎦
The generalized growth accounting method uses the following approximation to account for changes in output.
first order Taylor
∆Yt +1 ≈ F (Lt , K1,t ,K, K m ,t )∆Z t +1 m ⎡ ⎤ ⎡ ∂ ⎤ ∂ +⎢ Z t F (Lt , K1,t ,K, K m,t )⎥ ∆Lt +1 + ∑ ⎢ Z t F (Lt , K1,t ,K, K m,t )⎥ ∆K s ,t +1 s =1 ⎣ ∂K s ,t ⎣ ∂Lt ⎦ ⎦
First difference operator
(3.63)
(3.64)
The operator ∆ is known as the first difference operator, such that ∆Xt= Xt- Xt-1. The above equation can be rewritten as ∆Yt +1 ≈ [Z t F (Lt , K1,t , K , K m ,t )]
∆Z t +1 Zt
(3.65)
⎤ ∆K s ,t +1 ⎡ ⎡ ⎤ ∆L ∂ ∂ F (Lt , K1,t , K , K m ,t )Lt ⎥ t +1 + ∑ ⎢ Z t F (Lt , K1,t , K , K m ,t )K s ,t ⎥ + ⎢Zt L L K ∂ ∂ = 1 s ⎥⎦ K s ,t ⎢ t s ,t ⎣ ⎦ t ⎣ m
Dividing both sides of this equation by Yt yields an approximation of the growth rate of output between t+1 and t, which is ∆Yt +1 ⎡ Z t F (Lt , K1,t , K , K m ,t )⎤ ∆Z t +1 ≈⎢ ⎥ Yt Yt ⎣ ⎦ Zt
(3.66)
∂ ⎤ ⎡ ∂ ⎤ ⎡ Z F (Lt , K1, t , K , K m ,t )K s ,t ⎥ m ⎢ t ⎢ Z t ∂L F (Lt , K1,t , K , K m , t )Lt ⎥ ∆L ∆K s , t +1 ∂K s , t t +1 t ⎥ ⎥ +∑⎢ +⎢ Yt Yt ⎥ K s ,t ⎥ Lt ⎢ s =1 ⎢ ⎥ ⎢ ⎦⎥ ⎣⎢ ⎦ ⎣ ∂ ⎤ ⎡ ∂ ⎤ ⎡ Zt F (Lt , K1, t , K , K m , t )K s ,t ⎥ Zt F (Lt , K1,t , K , K m ,t )Lt ⎥ ∆K s , t +1 ∆Lt +1 m ⎢⎢ ∂K s ,t ∆Z t +1 ⎢ ∂Lt ⎥ ⎥ +∑ = +⎢ Zt Yt Y K s ,t ⎥ ⎢ ⎥ Lt ⎢ s =1 t ⎥ ⎢ ⎥⎦ ⎢⎣ ⎦ ⎣
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This equation shows how the growth rate of output, i.e. ∆Yt+1/Yt, can be decomposed in the parts that are due to TFP growth, i.e. ∆Zt+1/Zt, growth of the labor inputs, i.e. ∆Lt+1/Lt, and growth of the various capital stocks, i.e. ∆Ks,t+1/Ks,t. In order to make this decomposition practically implementable in the sense that we can actually assign amounts to the parts of output growth attributed to these inputs and TFP, we need to be able to measure ∂ ⎡ ⎢ Z t ∂L F (Lt , K 1,t , K , K m ,t )Lt t ⎢ Yt ⎢ ⎢ ⎣
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
and
∂ ⎡ ⎢ Z t ∂K F (Lt , K 1,t , K , K m ,t )K s ,t s ,t ⎢ ⎢ Yt ⎢ ⎣
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
for s=1,…,m (3.67)
This is where our theory of the firm that we have developed in this chapter comes in. The theory of the firm implies that the firm hires workers until the marginal product of labor equals the real wage rate. That is,
W ∂Yt ∂ F (Lt , K1,t ,K, K m,t ) = t = Zt ∂Lt ∂Lt Pt
(3.68)
such that we can write the first term of (3.67) as the labor share. This is because it implies
Zt
∂ F (Lt , K1,t ,K, K m ,t )Lt WL ∂Lt = t t = s L ,t Yt Pt Yt
(3.69)
The second term of (3.67) can be derived using the usercost equation of capital. The usercost equation of capital that allows us to calculate the marginal product of capital as MPK s ,t = Z t
∂ F (Lt , K1,t ,K, K m ,t ) ∂K s ,t
q ⎡ (1 + rt ) ⎤ = s ,t ⎢ − (1 − δ s )⎥ qs Pt ⎣ 1 + π t ⎦
(
(3.70)
)
where qs,t is the price of capital good s, δs is that capital good’s depreciation rate, and π tq s is the capital goods price inflation. Hence, if our theory of the firm is correct, the second term of (3.67) can be written as
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⎡ ⎤ ⎡ (1 + rt ) ⎤ ∂ F (Lt , K1,t ,K, K m ,t )⎥ K s ,t ⎢ − (1 − δ s )⎥ q s ,t K s ,t ⎢Z t qs 1+ π t ⎣⎢ ∂K s ,t ⎦⎥ ⎦ =⎣ = s K s ,t Pt Yt Yt
(
Capital service flow
(3.71)
The numerator of the right hand side of this equation, i.e. ⎡ (1 + rt ) ⎤ − (1 − δ s )⎥ q s ,t K s ,t ⎢ qs ⎣ 1+ π t ⎦
(
Replacement value of capital Capital share
)
(3.72)
)
represents the revenue that can be attributed to the capital stock of type s. This amount is known as the capital service flow of Ks,t. The amount qs,tKs,t is the replacement value of Ks,t. The fraction s K ,t is the share of revenue that can be attribute to the s capital service flows of Ks,t. It is therefore often referred to as the capital share of Ks,t. In order to see how these concepts are applied in practice, try the following exercise Exercise 3.16: Determining capital service flows
In this exercise we will plug numbers into the usercost equation of capital to obtain estimates of the capital service flows of various types of capital. As a basis for this exercise, we will use that the capital service flows of Ks,t are given by ⎡ (1 + rt ) ⎤ − (1 − δ s )⎥ q s ,t K s ,t ⎢ qs ⎣ 1+ π t ⎦
(
)
(3.73)
This is often approximated by
[r − π t
qs t
]
+ δ s q s ,t K s ,t
(3.74)
We will use this equation to compare what part of output is produced by equipment versus the part produced by structures. We will do so for 1997. Consider the data in table 3.1 and use them to answer the following questions (i)
Calculate the capital service flows of structures and equipment in 1997.
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F I R M
(ii)
Calculate the ratio of the capital service flow to the replacement value of capital for both structures and equipment. What accounts for the difference of this ratio between structures and equipment, is it investment price inflation or depreciation, or both?
Table 3.1: User cost variables for structures and equipment
1997: rt=6% replacement value in billions of dollars
π tq δs
s
structures 6027 2.47%
equipment 3959 -2.62%
3%
13%
At the end of this derivation, what we are left with is the following equation that allows us to do some generalized growth accounting exercises.
∆K s ,t +1 ∆Yt +1 ∆Z t +1 ∆Lt +1 m ≈ + sL , t + ∑ sK s , t Yt Zt Lt K s ,t s =1
(3.75)
In practice, it is common to rewrite this as
∆K s ,t +1 ∆Z t +1 ∆Yt +1 ∆Lt +1 m ≈ − s L ,t − ∑ s K s ,t Zt Yt Lt K s ,t s =1
(3.76)
because it is TFP growth, i.e. ∆Zt+1/Zt, that is the unobserved variable in this equation. Note, furthermore, that Euler’s theorem implies that m
1 = s L ,t + ∑ s K s ,t
(3.77)
s =1
Now that we have figured out how to do growth accounting for a more general production function than the Cobb-Douglas production function that we used in the previous part of this chapter, it is time to see how the generalized growth accounting method derived in this section has been applied to the ‘new economy’ hypothesis. In order to do so, read the article “The Resurgence of Growth in the Late 1990s: Is Information Technology the Story?” from the Journal of Economic Perspectives by Oliner and Sichel (2000) and try to do the following exercise
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T H E
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Exercise 3.17: Resurgence of Growth in the Late 1990s
Answer the question for the nonfarm business sector (i)
What was the average value of ∆Yt/ Yt-1 for 1974-1990?
(ii)
What was the average value of this variable for the late 1990’s?
(iii)
What was the average value of s K s ,t −1
∆K s ,t K s ,t −1
for the period 1974-
1990 where s=hardware? (iv)
What was the average value of this variable in the late 1990’s?
(v)
Which sectors were most responsible for the growth of aggregate multi factor productivity (MFP) in the nonfarm business sector in the late 1990’s?
Hence, computers, software, and related equipment contributed in two ways to the productivity acceleration in the late 1990’s. First of all, the capital deepening in terms of the IT capital stock in the nonfarm business sector added to output and labor productivity growth. Secondly, the sectors that produce the IT capital goods saw increased productivity themselves, which also contributed to aggregate productivity. The results in Oliner and Sichel (2000) fed another discussion, namely on how long the ‘new economy’ will last. In order to get some insight in this discussion, which tends to be rather speculative, consider the following exercise. Exercise 3.18: Will the ‘new economy’ last?
For this exercise, read the articles by Gordon (2000) and Brynjolfsson and Hitt (2000). Try and answer the following questions (i)
Which is the article that is most critical of the ‘new economy’ hypothesis?
(ii)
What makes the author(s) of this article so sceptic?
(iii)
Which article is most positive about the staying power of the ‘new economy’? Which arguments in this article support this optimistic view.
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F I R M
(iv)
Can you find a study that updates the results in Oliner and Sichel (2000) to include evidence from after 2000? What does this study find for the role of IT capital in U.S. productivity growth for the post-2000 period?
(v)
Where do you stand? Why?
This completes our analysis of the theory of the firm. The next step is to combine our theories of the household and the firm and to see what will happen when these two types of entities interact with each other in our model economy.
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4
Chapter
Equilibrium This class is on macroeconomics. Macroeconomics is the study of economic aggregates. So far, we have not really developed a theory of aggregates. We have simply developed a theory of how individual households and firms make their consumption, investment, labor supply, and labor demand decisions. The macroeconomic aggregates that we observe in reality are the result of the interaction between firms and households. Hence, an essential part of macroeconomic theory should be the theory of what the outcome will be when the micro-decision units in the economy interact with each other. The aim of this chapter is to introduce you to the main concepts and techniques used in the analysis of the macroeconomic equilibrium outcome of this interaction. Dynamic general equilibrium (DGE)
The equilibrium outcome of our model economy is something called a Dynamic general equilibrium (DGE). First of all, it is called an equilibrium because we will assume that prices adjust to the level where markets clear. Secondly, it is called a general equilibrium because we look at the joint conditions that have to hold for all markets to clear at the same time. That is, we take into account the interactions between equilibria in different markets, like the labor- and capital markets. Finally, it is called dynamic because our general equilibrium concept is not about the outcome for the economy at one particular point in time, but instead involves the path of all macroeconomic equilibrium variables over time. Before we consider the DGE of our model economy, we first have to define the structure of the model economy in which we assume that the firms and consumers, we studied in the previous two chapters, interact. We then consider what constitutes an equilibrium outcome in that context.
Steady state Transitional dynamics
Subsequently, we derive the equilibrium outcome. We consider two important concepts often used in the analysis of macroeconomic models. The first is that of the steady state. The steady state is the outcome in which the dynamic equilibrium turns out to be constant over time. The second is that of transitional dynamics. It is the
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path of the economy towards the steady state. These two concepts are the topic of the last two sections of this chapter.
4.1 Structure of economy Our model economy is made up of three elements. The first two are the ones that we have already spent considerable time on; the consumers and firms. The final one is the market environment in which the households and firms interact. Before we consider these three elements, we first have to start with some simplifying assumptions.. Because of the introductory nature of this class, we stick to the most basic structure of the model economy used in macroeconomic theory. One-sector economy
This structure is called a one-sector economy. In the real world, there is a tremendous number of different goods in the economy. However, for simplicity, we assume that in our model economy there is only one type of good. This single good is used for consumption purposes, for asset holdings, as well as a capital input. The result of this assumption is that investment and capital goods are identical to consumption goods and thus have the same price, i.e. Pt. We briefly review the households and firms here and then proceed by describing the market structure in which we assume that these households and firms interact. Households
Households in our model economy do four things. First of all, they are the consumers of the economy who buy consumption goods and derive utility from this consumption. Secondly, households save and rent out their assets to firms. This yields them capital income. Thirdly, they are the shareholders of the companies and get paid back the profits the firms make. Finally, consumers supply labor to firms which earns them labor income, i.e. wages. Hence, the income of the households is made up of the following components: (i) wages and salaries, i.e. their labor income; (ii) capital income, obtained from renting out their asset holdings as capital inputs to firms; (iii) proprietor's income, i.e. the profits that they obtain from the businesses that they own. The expenditures of the households consist of two parts. The goods that they buy are either consumed or they are saved and added to their asset holdings. Hence, the expenditures of the households consist of consumption expenditures and savings. Firms
Because of its simplicity, we will assume throughout that firms rent the capital that they use in production. In that case, firms choose their labor and capital demands to maximize their flow profits. The income from the firms' side consists of their revenue. While their expenditures are the factor costs made up of wages and rental costs. As we
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have derived in Exercises 3.3 and 3.4, in equilibrium these firms will make zero profits, i.e. Πt=0. Markets
The households and the firms in this economy interact in three markets. The first is the market for the goods that the firms produce and that the households buy to consume and save. This is the goodsmarket of this economy. The quantity supplied on this market is the output produced by the firms, Yt, while the demand is determined by the combined consumption and savings levels, i.e. Ct+It, of the households. The price in this market is the goods price, which we will denote by Pt. The second market is the labor market, in which firms hire workers, which are members of the households. The price that equalizes labor demand to labor supply is the wage rate, Wt. Finally, there is the capital market in which firms hire the assets saved by the households and use them as capital inputs in their production process. Figure 4.1 contains a diagram that clarifies the structure of the economy. It also illustrates something else. There are three ways of measuring the size of this economy.
Business sector (firms)
Labor Market
labor costs, WtLt labor demand, Lt
labor income, WtLt labor supply, Lt
Goods Market
nominal output, PtYt output, Yt
personal consumption expenditures, PtCt
consumption, Ct
capital expenditures,PtIt
investment, It
(nominal investment)
rental cost of capital, RtKt
Capital Market
capital consumption, δKt
capital demand: Kt ~ capital supply: At-1
Flow of goods/labor Income approach Expenditure approach Product approach
corporate profits, Πt
Figure 4.1: Flow diagram of the model economy
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savings, At-At-1 capital income, rtAt-1
Household sector (households)
Markets
E Q U I L I B R I U M
Statistical discrepancy
These three ways follow from the identities
Pt Yt = Pt C t + Pt I t = Rt K t + Wt Lt + Π t
(4.1)
If we would like to know the size of the economy, we could consider looking at how much output the firms produce. The nominal amount of this would be the total revenue of the firms and is equal to PtYt. This is known as the product approach to national accounting. We could also measure how much households and firms actually spend on consumption and investment. This would be considered the expenditure approach to national accounting and it involves measuring consumption and investment expenditures. It is represented by the second part of the equation above. Finally, we could measure how much households earn from their different sources of income. This would be the income approach to national accounting. This is the third part of the equation. Because of this identity, theory suggests that these three different ways of measuring output and the size of the economy would have to yield the same outcome. In practice, however, this turns out not to be the case. The difference is often referred to as the statistical discrepancy in the National Income and Product Accounts. The Bureau of Economic Analysis describes the statistical discrepancy as follows “Gross domestic product (GDP) measures output as the sum of final expenditures, consumer spending, private investment, net exports, and government consumption and investment. Gross domestic income (GDI) measures output as the sum of the costs incurred and the incomes earned in the production of GDP. In theory, GDP should equal GDI; in practice, they differ because their components are estimated using largely independent and less-than-perfect source data. In the national income and product accounts (NIPAs), the difference between GDP and GDI is called the ‘statistical discrepancy’; it is recorded in the NIPAs as an ‘income’ component that reconciles GDI with GDP (see NIPA table 1.7.5). Recently, there has been considerable public debate about the growth rate of the U.S. economy because since the early 1990's, growth measured by real GDI has increased faster than growth measured by real GDP. Some analysts maintain that the higher rate of growth of real GDI is more consistent with declines in the unemployment rate and with anecdotal information about increases in productivity in servicesproducing industries. This debate has important implications for market participants and policymakers. BEA views GDP as a more reliable measure of output than GDI, because it considers the source data underlying the estimates of GDP to be more accurate. For example, most of the annual source data used for
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estimating GDP are based on complete enumerations, such as Federal Government budget data, or are regularly adjusted to complete enumerations, such as the quinquennial economic censuses and census of governments. In addition, all the expenditure components of GDP are revised every 5 years to reflect BEA's benchmark input-output accounts, which are prepared within an internally consistent framework that tracks the input and output flows in the economy. For GDI, only the annual tabulations of employment tax returns and Federal Government budget data are complete enumerations, and only farm proprietors' income and State and local government budget data are regularly adjusted to complete enumerations. For most of the remaining components of GDI, the annual source data are tabulations of samples of income tax returns.'' Table 4.1 summarizes the basic structure of the model economy. There are 7 quantity variables that are determined in equilibrium, namely Lt, Yt, Ct, It, At, Kt, and Πt. The four prices in the markets are Wt, Pt, rt, Rt. There are many more quantities than there are markets. This is because many of the quantities are related through identities. Furthermore, the capital market equilibrium seems to be determined by two prices. This must imply that rt and Rt are closely related. Table 4.1: Markets and prices in the model economy
Market Labor market Goods market Capital market (Profits)
Quantities Lt Yt, Ct, and It At, Kt, and It
Prices Wt Pt rt, and Rt
Πt
Before we solve for it, we first need to consider what exactly we mean by a dynamic general equilibrium.
4.2 Equilibrium definition Definition of DGE
Given an initial assetholdings, At-1 and the path of technology, {Z t + s }s =0 , a dynamic general equilibrium in this economy is defined as a path of quantities {Lt + s , Yt + s , Ct + s , I t + s , At + s , K t + s +1 , Π t + s }∞s =0 and prices {Wt + s , Pt + s , Rt + s , rt + s }∞s =0 , that satisfies the following three conditions: ∞
1.
Utility maximization
Given the path of the prices {Wt + s , Pt + s , rt + s }s =0 , the profits, {Π t + s }s =0 , and its initial asset holdings, At-1, the representative household that we studied in Chapter 2 chooses ∞
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E Q U I L I B R I U M
its labor supply, consumption and savings, i.e. {Lt + s , C t + s , At + s , Π t + s }s =0 ,such that it maximizes its present discounted value of utility. ∞
2.
Profit maximization
Given the path of the prices, {Wt + s , Pt + s , Rt + s , rt + s }s =0 , and the state of the technology, ∞
{Z t + s }∞s =0 the representative firm chooses its path of output, labor demand, and capital
inputs to maximize the present discounted value of the stream of its profits, as derived in the previous chapter. 3.
Market clearing
At any point in time t+s for s=0,…,∞ , all markets clear. That is, labor supply equals labor demand, output equals consumption and investment, and firms' capital inputs equal the households' asset holdings. There are 7 quantities and 4 prices that are jointly determined in equilibrium. This implies that our equilibrium definition must result in 11 equations that pin down these variables. Below, we derive these 11 equations. We do so in three parts, each corresponding to a condition that is part of our equilibrium definition. All these equations are solved for the variables at time t and relate these variables at time t to those at time t+1.
4.3 Equilibrium conditions Utility maximization
Utility maximization by the household implies three equations that have to hold in equilibrium. The first is determined by the optimal labor choice that the household makes. This equation is the optimality condition for the labor supply. That is, the marginal utility of consumption equals the marginal disutility of working times the real wage rate. Given the CRRA preferences introduced in Chapter 2, this can be rewritten as
(1 − θ ) θ
Ct ~ = Wt 1 − Lt
(4.2)
The second is the intertemporal optimality condition associated with the household's optimal savings decision. It is the consumption Euler equation
∂ ∂ U (C t , Lt ) = β (1 + ~ rt +1 ) U (C t +1 , Lt +1 ) ∂C t +1 ∂C t
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(4.3)
E Q U I L I B R I U M
Finally, the third equation determined by the household's utility maximization decision is the asset accumulation equation, i.e. the real flow budget constraint, subject to which the household maximizes utility. ~ ~ ~ ~ At = (1 + ~ rt )At −1 + Wt Lt + Π t − C t
(4.4)
What is different from the dynamic budget constraint that we considered in the ~ previous chapter is that this one contains the real profit income, Π t = Π t Pt , that households can earn from their ownership of firms. Profit maximization
The firm’s profit maximization problem yields four equilibrium conditions. The first equation defines the firm’s real profit level ~ ~ ~ Π t = Yt − Wt Lt − Rt K t
(4.5)
The second equation is the production technology that the firm uses. That is, in equilibrium, output must be produced using the assumed Cobb-Douglas production function
Yt = Z t K tα L1t−α
(4.6)
The third is the one that determines the firm's optimal labor demand. This condition implies that the firm equates the real wage rate to the marginal product of labor. For the specific Cobb-Douglas technology that we consider, this implies Y ~ ∂Y Wt = t = (1 − α ) t ∂Lt Lt
(4.7)
The fourth equilibrium condition implied by the firm’s profit maximizing behavior is determined by its capital demand decision. Since we assumed that firms in this economy rent their capital inputs, firms equate the marginal product of capital to its real rental costs. Mathematically, this boils down to ∂Y Y ~ R Rt = t = t = α t Pt ∂K t Kt
(4.8)
given the Cobb-Douglas production function. Market clearing
What remains are the market clearing conditions as well as the condition relating the interest rate to the rental rate of capital. Equilibrium in the goods market requires that output equals consumption plus savings. We simply denote savings by It because we later impose that savings equal investment anyway. Mathematically, equilibrium in the goods market requires
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Yt = Ct + I t
(4.9)
The equilibrium condition for the capital market implies that the firms rent all the capital that the households hold as assets. The asset holdings that the households rent out in period t are those they held at the end of period t-1. This implies that equilibrium in the capital market boils down to ~ K t = At −1
(4.10)
There are, in principle, two prices in the capital market, namely rt as well as Rt. It must thus be that rt and Rt are somehow directly related. The reason that the interest rate and the rental rate of capital are related is because they represent the returns to two options that households have for dealing with their assets. Because there are many households, households can, in principle, lend their assets to other households. If they do so, for every unit of assets that they lend to another household they obtain a return of ~ rt . If, instead, they rent out their assets to firms on the other hand, then for every unit they ~ rent out, they get Rt units of return in terms of the rental price. However, if the firm uses the assets as capital input, they also depreciate at the rate δ. Consequently, the net ~ return of renting out a unit of capital are Rt − δ . In equilibrium, the net returns to lending capital as well as renting out capital should be the same. That is, ~ ~ rt = Rt − δ
(4.11)
Suppose this was not the case. If the right hand side would be bigger than the left hand side, then all households would get more by lending money to other households than by renting out their assets to firms. Because we have assumed that all households are identical, this would mean that all households would like to lend money. Thus their would be a large supply of assets but no demand. This would imply that ~ rt would go down and can thus not be an equilibrium outcome. On the other hand, if the right hand side of this equation is bigger than the left hand side, then all households would like to borrow an infinite amount of money and then rent it out to firms. This is known as an arbitrage opportunity. This can obviously not persist in an equilibrium situation. The final equation is the one that pins down the price level. None of the previous ten equations, (4.2) through (4.11), contains the price level, Pt. In fact, all variables in these equations are real variables that are measured in units of the good traded in this economy. Consequently, the price level in this economy is undetermined. We will return to this result in much more detail when we will consider the possible channels and effects of monetary policy. For the moment, we will just normalize the price level to equal 1. That is, the final equation in equilibrium is
Pt = 1
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(4.12)
E Q U I L I B R I U M
Classical dichotomy
The reason that the price level does not matter in this economy is that the allocation of resources in this economy only depend on the relative price of labor services, i.e. the real wage, and the real interest rate. An economy in which the price level does not affect the allocation of resources satisfies the classical dichotomy between the real and nominal sides of the economy. We return to what determines the price level in the economy in much more detail in chapter 7 when we introduce money and the price level is determined in the market for liquidity. For our solution of the equilibrium equations we have, so far, assumed that the firm rents it capital inputs. The following exercise considers the case in which the firms own their capital stocks. Exercise 4.1: Rental versus ownership of capital in equilibrium
In the derivation of the equilibrium in our model economy we have assumed that the firm rents its capital stock. In this problem we will investigate whether the equilibrium dynamics of this economy differ from the ones derived when we assume instead that the firm owns its capital inputs. (xvii) Explain why, when the firm rents its capital inputs, its capital demand is determined by the optimality condition ⎛ ∂Yt ⎜⎜ ⎝ ∂K t
⎞ Rt ~ ⎟⎟ = = Rt ⎠ Pt
(4.13)
(xviii) Explain why in equilibrium it must be the case that the real ~ rental price of capital, i.e. Rt , equals the real interest rate, i.e. ~ rt , plus the depreciation rate of capital, i.e. δ. Suppose instead that firms own their capital stocks. In that case investment is determined by the user cost equation of capital, which reads ⎛ 1 ⎞ ⎛ ∂Yt +1 ⎞ ⎛ 1 − δ ⎞ ⎟⎟ Pt +1 ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟qt +1 qt = ⎜⎜ ⎝ 1 + rt +1 ⎠ ⎝ ∂K t +1 ⎠ ⎝ 1 + rt +1 ⎠
(4.14)
In the one-sector model that we considered, we assumed that capital and consumption goods are 1 for 1 transferable, such that qt=Pt=1 for all t.
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(xix) Show that this assumption implies that the nominal interest rate, i.e. rt, equals the real interest rate ~ rt . (xx)Apply the assumption that qt=Pt=1 for all t to the user cost equation of capital above and show that, given this assumption, the user cost equation yields that the marginal product of capital in a period equals the real interest rate plus the depreciation rate. (xxi) Given the above results, do you think its matters of the equilibrium equations of our model whether capital is owned or rented by firms? Why, or why not?
4.4 Equilibrium dynamics Hence, our equilibrium definition yields a set of eleven equations, (4.2) through (4.12). Luckily, it turns out that we can actually simplify these equations and reduce the description of how the dynamic equilibrium of this economy behaves to a set of five equations. This is what we do in this section. After that, we consider one of the main properties of the equilibrium that we consider. Namely, that it can be interpreted as an example of Adam Smith's ‘invisible hand’ hypothesis. That is, the market mechanism underlying the equilibrium outcome leads to an outcome that would have been chosen by a social planner for this economy. First things first, let us first simplify our equilibrium equations. When we combine (4.8), (4.11), and (4.3) we obtain that ⎛ ⎞ ∂ Y ∂ U (C t +1 , Lt +1 ) U (C t , Lt ) = β ⎜⎜1 + α t +1 − δ ⎟⎟ ∂C t K t +1 ⎝ ⎠ ∂C t +1
(4.15)
Equation (4.9) does not change, such that
Yt = C t + I t
(4.16)
Equations (4.2) and (4.7) imply that
(1 − θ ) θ
Ct Y = (1 − α ) t 1 − Lt Lt
(4.17)
The production function does also not simplify, it remains
Yt = Z t K tα L1t−α
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(4.18)
E Q U I L I B R I U M
The equation that turns out to simplify the most is the flow budget constraint. ~ ~ ~ ~ At = (1 + ~ rt )At −1 + Wt Lt + Π t − C t
(4.19)
The first thing that cancels out of this situation is the profits, which are zero in equilibrium. This reduces the above equation to ~ ~ ~ At = (1 + ~ rt )At −1 + Wt Lt − Ct
(4.20)
~ Imposing capital market equilibrium, K t = At −1 , as well as the equibrium real interest ~ rate, ~ r = R − δ , then we find that t
t
(
)
~ ~ K t +1 = 1 + Rt − δ K t + Wt Lt − Ct ~ ~ = (1 − δ )K t + Rt K t + Wt Lt − Ct = (1 − δ )K t + Yt − Ct
(4.21)
where, in the last line, we have used the fact that firms make zero profits. Finally, we use the goods market equilibrium, which implies that investment equals output minus consumption, i.e. It=Yt-Ct. Doing so, we obtain the equilibrium law of motion of capital
K t +1 = (1 − δ )K t + I t
(4.22)
We thus obtain that the equilibrium dynamics of our economy can be described by the equations (4.15), (4.16), (4.17), (4.18), and (4.22). Note that this simplification does not only reduce the number of equations, it also reduces the number of variables to five, namely {Yt , K t , Ct , Lt , I t }. Decentralized equilibrium
Invisible hand
We derived the equilibrium dynamics in the form of what is known as a decentralized That is, we assumed that all micro-decision units in this economy decide on their own what is their utility or profit maximizing choice, taking prices as given. The prices then equalize the marginal benefits for the agents on the supply side of the market to the marginal costs for the agents on the demand side of the market.
equilibrium.
One of the most powerful claims that is that the root of economic science is Adam Smith’s (1776) ‘invisible hand’ hypothesis. Smith, in his ‘Wealth of Nations’, argued that the market mechanism and its decentralized equilibrium act as an ‘invisible hand’ that leads to an outcome that is close to the optimal allocation of resources that a social planner would choose that aims to maximize the utility of the agents in the economy.
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First Welfare Theorem
It took economists another 175 years to prove Adam Smith’s claim in a stylized theoretical economic model. Arrow and Debreu (1954) proved a version of Smith’s claim. Their result is known as the First Welfare Theorem of economics. The First Welfare Theorem is generally not covered in a course on macroeconomics. However, it is important that it holds in a particular form in the DGE setup that we consider. In the exercise below, you are asked to prove this particular case, derived originally and more generally by Prescott and Mehra (1980). Exercise 4.2: Invisible hand
In our model economy, households and firms each decide on their own what is optimal for them. The market mechanism then coordinates their activities through the adjustment of prices paid in the markets. In 1776 with ‘The Wealth of Nations’ Adam Smith started the science of economics. One of Smith's major claims was that markets coordinate decentralized decisions such that they lead to a desirable outcome. He revered to this coordination mechanism as the ‘invisible hand’. This exercise is meant to consider how Smith's claim is applicable to the model economy we study here. In order to do so, we consider the outcome that a social planner would choose, whose objective is to maximize the welfare of the representative household subject to the technological constraints imposed by the production function, the capital accumulation equation, and the resource constraint. We consider a planner that, given Kt and a path of productivity, {Z t + s }∞s=0 , chooses the path of
{Yt + s , K t + s , Ct + s , Lt + s , I t + s }∞s =0
(4.23)
to maximize the present discounted value of utility ∞
∑ β U (C s
s =0
t+s
, Lt + s )
(4.24)
subject to the production function
Yt = Z t K tα L1t−α the law of motion of capital
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(4.25)
E Q U I L I B R I U M
K t +1 = (1 − δ )K t + I t
(4.26)
and the resource constraint that output can either be consumed or invested, i.e.
Yt = C t + I t
(4.27)
We first eliminate a lot of variables from this optimization problem. (i)
Combine (4.25) through (4.27) to show that the planner is bound by the restriction that
Ct = Z t K tα L1t−α − [K t +1 − (1 − δ )K t ]
(4.28)
Given the result to part (i), the planner's problem simplifies to choosing {K t + s Lt + s }∞s=0 to maximize (4.24) subject to (4.28). We focus on the planner’s choices of Lt and Kt+1. The optimal choices of these variables are again derived using the four-step method that we have applied for all the optimization problems before. Before taking these four steps, let us first write out the first two terms of the objective function. That is, given the restriction (4.28), we can write the objective (4.24) as
(
)
U Z t K tα L1t−α − [K t +1 − (1 − δ )K t ], Lt +
βU (Z t +1 K L α
1−α t +1 t +1
∞
∑ β U (C s
s =2
t+s
)
− [K t + 2 − (1 − δ )K t +1 ], Lt +1 +
(4.29)
, Lt + s )
(ii)
Step 1: Identify the costs and benefits Use
the above equation to identify the costs and benefits of working, i.e. Lt, and saving, i.e. Kt, for the social planner.
(iii)
Step 2, part I: Calculate the marginal costs and benefits
Show that the marginal costs of the choice of Lt are
−
∂ U (Ct , Lt ) ∂Lt
and that the marginal benefits are
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(4.30)
E Q U I L I B R I U M
⎤ ⎡ ∂Y ⎤ ⎡ ∂ U (Ct , Lt )⎥ ⎢ t ⎥ ⎢ ⎦ ⎣ ∂Lt ⎦ ⎣ ∂Ct
(iv)
(4.31)
Step 2, part II: Calculate the marginal costs and benefits
Show that the marginal costs of the choice of Kt+1 are
∂ U (Ct , Lt ) ∂Ct
(4.32)
and that the marginal benefits are ⎤⎛ ∂Y ⎞ ⎡ ∂ U (Ct +1 , Lt +1 )⎥⎜⎜ t +1 + 1 − δ ⎟⎟ ⎦⎝ ∂K t +1 ⎠ ⎣ ∂Ct +1
β⎢ (v)
Step 3, part I: Solve for the optimal labor choice
(vi)
Step 3, part II: Solve for the optimal savings decision
(4.33)
Equate the marginal costs and benefits of Lt and show that the resulting optimality condition is identical to (4.17). Equate the marginal costs and benefits of Kt+1 and show the the resulting optimality condition is identical to (4.15).
Hence, the social planner chooses the paths of ∞ {Yt + s , K t + s , Ct + s , Lt + s , I t + s }s =0 to satisfy the constraints (4.25), (4.26), and (4.27) as well as the optimality conditions (4.15) and (4.17).
Steady state Transitional dynamics
(vii)
Step 4: Interpret
How do these five equations compare to the five equations (4.15), (4.16), (4.17), (4.18), and (4.22) that determine the paths of these variables in the decentralized equilibrium?
(viii)
What do the above results imply for the choice that the social planner would make in this economy compared to the decentralized equilibrium outcome?
Now that we have derived the equilibrium dynamics of the economy and shown that they coincide with those chosen by a social planner, it is time to introduce two concepts used in the study of equilibrium dynamics. The first, which is generally interpreted as a long-run (average) outcome, is known as the steady state of the economy. The second, which describes the short run behavior of the economy on its
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way to the steady state is called the economy's transitional dynamics. Each of these concepts is the topic of its own section below.
4.5 Steady State In which state would our equilibrium dynamics imply that the economy is steady? According to Webster's Dictionary (1) Stable, not liable to shake or totter: a steady ladder. (2) Unfaltering, constant: a steady light. Steady
A steady state is a stable state of the economy. The equilibrium dynamics in the previous section imply a relationship between the equilibrium variables now, i.e. {Yt , K t , Ct , Lt , I t }, and those same variables one period from now, , i.e. {Yt +1 , K t +1 , Ct +1 , Lt +1 , I t +1 } . Define the variable Ψt = {Yt , K t , Ct , Lt , I t }, then the equilibrium dynamics derived in the previous section imply a functional relationship
0 = F (Ψt , Ψt +1 )
(4.34)
that has to hold at every t on the equilibrium path. Here F( .) consists of the five equations (4.15), (4.16), (4.17), (4.18), and (4.22). In particular, they imply that
⎛ ⎞ ∂ Y ∂ U (Ct , Lt ) − β ⎜⎜1 + α t +1 − δ ⎟⎟ U (Ct +1 , Lt +1 ) K t +1 ∂Ct ⎝ ⎠ ∂Ct +1 0 = Yt − Ct − I t 0=
0=
(1 − θ ) θ
Ct Y − (1 − α ) t 1 − Lt Lt
(4.35)
0 = Yt − Z t K tα L1t−α
0 = K t +1 − (1 − δ )K t − I t The steady state of this economy is that equilibrium path for which all variables are constant over time. It can be interpreted as the long run equilibrium, or average equilibrium outcome of the economy.Formally, the steady state is Ψ = {Y , K , C , L , I } such that
0 = F (Ψ , Ψ )
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In terms of the equilibrium equations that we derived for the dynamics the variables {Y , K , C , L , I } solve
⎛ ⎞ ∂ ∂ Y U (C , L ) − β ⎜⎜1 + α − δ ⎟⎟ U (C , L ) ∂Ct K ⎝ ⎠ ∂Ct +1 0 =Y −C − I 0=
(1 − θ )
C Y − (1 − α ) L θ 1− L α 1−α 0=Y −K L 0=
(4.37)
0 = K − (1 − δ )K − I Here Zt dropped out because we assume that Zt→1 far in the future and thus we assume that Zt=1 is the steady state level of total factor productivity. We return to the ∞ path of {Z t + s }s =0 in the next two chapters when we consider economic growth and business cycles. For now, we will assume that TFP will converge to a constant value and, without loss of generality, will normalize that value to equal one. Let us solve for the steady state and try to interpret the solution. The first steady state condition implies that if the capital stock has to be constant in steady state then net investment must be zero, such that gross investment equals capital consumption. That is
I = δK
(4.38)
Because it is evaluated in the same arguments in each period, the marginal utility of consumption is constant over time, such that
∂ ∂ U (C , L ) = U (C , L ) ∂Ct +1 ∂Ct
(4.39)
and the Consumption Euler equation in steady state reduces to K α = Y 1 β − (1 − δ )
(4.40)
The left hand side of this equation basically measures the capital intensity of the production process. Let's see how the right hand side relates this capital intensity to the underlying parameters. First of all, α is the capital elasticity of output which essentially measures the importance of capital in the production process. K Y is increasing in α because the more important capital is in production, the more capital intensive firms will choose their production process to be. β is the discount rate. The higher β, the less
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future utility is discounted relative to current utility and the more willing consumers are to save. Since savings equal investment in equilibrium, the more consumers are willing to save, the higher the steady state capital stock. That is, the right hand side is increasing in β. Finally, the higher the depreciation rate, δ, the lower the capital intensity of output. Basically, an increase in capital depreciation implies an increase in the cost of investment/capital and thus will lead to a reduction in the capital intensity of production. The resource constraint implies that
C =Y −I =
⎤ 1 ⎡1 − 1 + (1 − α )δ ⎥ K ⎢ α ⎣β ⎦
(4.41)
which is a straigthforward implementation of the two results derived above and does not add that much intution. However, we can substitute this equation into the labor market equilibrium condition. When we do so, a bit of algebra yields that 1 ⎤ 1 ⎛ 1 − θ ⎞⎡ αδ 1+ ⎜ ⎟ ⎢1 − 1 − α ⎝ θ ⎠ ⎣ 1 β − (1 − δ ) ⎥⎦ 1 β − (1 − δ ) = 1 ⎛ 1 −θ ⎞ 1 β − (1 − δ ) + ⎜ ⎟[1 β − 1 + (1 − α )δ ] 1−α ⎝ θ ⎠
L=
(4.42)
A few things stand out from this result. The more important consumption is in the utility function, i.e. the higher θ, the higher the steady state employment level. That is, in the long run people who like to consume rather than like to have time off will work more. The higher α, the lower the steady state employment level. That is, if capital is more important in production, then the economy will be more capital intense and workers will choose to have more leisure. The same intuition applies to δ. The lower δ, the lower the cost of capital and the higher the capital labor ratio and the lower the steady state employment level. Finally, the lower β, the lower the steady state employment level. The main intuition behind this is that part of the motive to work is that current labor provides productive resources for the creation of savings. However, the lower the discount rate the less value people attribute to delayed consumption and thus to savings. Hence, the lower the incentive to work to generate more savings. When we substitute the above results into the production function, we obtain that
Y = K α L1−α
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which, when combined with the other results, yields the steady state capital labor ratio
⎤ K ⎡ α =⎢ L ⎣1 β − (1 − δ ) ⎥⎦
1 1−α
(4.44)
To which much of the same intuition applies that we have considered above. Exercise 4.3: Comparative statics in steady state
This question compares two model economies, economy A and B. Their only difference is that they have different underlying parameters. Answer the following questions without using any math and solely by economic reasoning. (i)
Suppose economies A and B are the same, except A has a perfectly inelastic labor supply and B's labor supply is elastic. Which economy has a higher steady state output level?
(ii)
Suppose economies A and B are the same, except A has a higher depreciation rate than B. Which economy has a higher steady state capital labor ratio?
(iii)
Suppose economies A and B are the same, except A has a higher discount factor, β, than B. Which economy has a higher steady state employment level?
(iv)
Suppose economies A and B are the same, except A has a higher depreciation rate than B. Which economy has a higher investment to capital ratio in steady state?
The steady state, considered in this section, gives us a nice idea about long run (average) outcomes of the simple macroeconomy that we consider. However, it does not help us understand the short run behavior of the economy towards it. The path of the economy towards its steady state is known as the economy’s transitional dynamics and is what we will consider in the following section.
4.6 Transitional dynamics Transitional dynamics
We have considered the long run outcome of the economy through the concept of the steady state. The question that remains is ‘how does the economy actually get to this
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steady state?’. The path towards the steady state is often referred to as the transitional dynamics of the economy. In order to answer the question on how the economy gets to the steady state, we first need to define where the economy starts off from. That is, we need to pin down what describes that current state of the economy. Our equilibrium definition stated that the equilibrium is a path of consumption, output, investment, employment, and the capital stock that, given the current capital stock and the path of the technology, is consistent with utility and profit maximization as well as markets clearing. This suggests that the current state of the economy is the current capital stock and the exogenously given path of future technological progress. As it turns out it is easiest to measure variables in percentage deviations from their steady state values. The main reason for this is that percentages are independent of units of measurement. If we define things in percentage deviations from the steady state, then the current state of the economy is given by 1. The percentage deviation of the current capital stock from its steady state value. 2. The path of the future percentage deviations of TFP, i.e. Zt, from its long run value, i.e. Zt=1. Given this current state, we are interested in how the economy moves towards the steady state. Just like for the variables that determine the state of the economy, it is most convenient to describe the economy's transitional dynamics in terms of percentage deviations from the steady state. The exact transitional dynamics of the model economy that we consider here turn out to be hard to solve exactly because equations (4.15), (4.16), (4.17), (4.18), and (4.22) are non-linear. Log-linearization
For this reason, macroeconomists often rely on an approximation technique called loglinearization. The technique of log-linearization is too technical for this class. Therefore, we rely on the computer to perform it for us. Whenever necessary, we use Microsoft Excel to perform the calculations for us. Log-linearization is essential when we consider the dynamic properties of our model economy in the next chapters and we therefore rely on a whole set of Excel files to learn about the model’s dynamic properties.
4.7 Measuring equilibrium In principle, the title of this section is rather tautological. That is, if you believe that what we observe is the equilibrium outcome of the U.S. economy, then what we measure is by definition the equilibrium outcome. However, this section is called
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‘measuring equilibrium’ because it focuses on how the National Income and Product Accounts measure the various sides of the macroeconomic equilibrium outcome for the U.S. Figure 4.1 showed the flow diagram of our stylized economy and related the flows back to their conceptual measurement in the NIPAs. In this section we will look in more detail at the flow diagram of the U.S. economy and consider which flows are measured where in the NIPAs. The material in this section is essentially a summary of the BEA’s publication “An Introduction to National Economic Accounting”. The aim of this section is to introduce you to the basic setup of the National Income and Product Accounts and to relate this setup back to the equilibrium flow diagram of Figure 4.1. The NIPAs distinguish four sectors in the U.S. macroeconomy, namely the household sector, the business sector (firms), the government, and the foreign sector. The latter measures the magnitude of inflows and outflows of goods, capital, and money in the U.S. economy. For each of these four sectors, the NIPAs report three accounts. Instead of reviewing these accounts for each sector separately, we will consider the structure of these accounts for the overall economy. How these accounts are disaggregated to represent the activities of each sector in the economy is very well explained in the BEA publication referred to above. Just like a balance sheet, each of the NIPA accounts can be represented as a ‘Taccount’. The difference is that the NIPA accounts list the sources, rather than liabilities, of a particular type of income on the right hand side and the uses of the income, rather than the assets, on the left-hand side. Production account
The, arguably, most important of the three types of accounts in the NIPAs is known as the production account. The right-hand side of the production accounts in the NIPA represents the different types of production that are attributable to the sector to which the account is applicable. The left hand side lists the payments to the various factors of production, i.e. capital and labor, as well as the profits. In terms of the flow diagram of figure 4.1, the production account for our overall model economy looks like WtLt RtKt
Πt
Uses Wages and salaries Rental costs of capital Profits
PtYt
Sources Sales
Figure 4.2: Production account for the model economy
The actual production accounts in the NIPAs are more complicated because they treat capital as being owned by firms and thus consider the cost of capital the value of
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capital consumption. Furthermore, the left-hand side of the production account would also include interest paid on debt. Finally, the production accounts in the NIPAs also track tax flows from the various sectors to the government. Since our model economy does not yet include a government sector, these are abstracted from in the figure above. Appropriation account
The appropriation account measures the sources and the uses of the income appropriated by the various sectors in the economy. For the business sector, income equals its profits which are used to pay shareholders, pay corporate income tax, as well as retain earnings. For the household sector income and its uses are a bit more easily defined. Households have three sources of income, namely wages and salaries, interest received, and dividends received. The households use this income for consumption expenditures and savings, as well as tax payments. In our simple model economy, the appropriation account looks as follows PtCt At-At-1
Uses Consumption expenditures Savings
WtLt RtKt
Πt
Sources Wages and salaries Rental costs of capital Profits
Figure 4.3: Appropriation account for the model economy
Here we do not have tax payments because we do not have a government sector in our model economy. When we worked out the model we already took into account that equilibrium in the capital market implied that investment equals savings and we never formally used St to denote savings. Here, it turns out to be more convenient to explicitly do so. Savings and invesmtent account
The final account is known as the Savings and Investment Account. This account has on its sources side the net change in the financial assets of the sector, i.e. the increment of the sectoral balance sheet. On the uses side, it lists the investments, i.e. capital expenditures, that these financial assets are used for. For our simple model, this account looks like PtIt
Uses Capital expenditures
Figure 4.4: Investment and savings account for the model economy
At-At-1
Sources Savings
In principle, the NIPAs could just give the three types of accounts for each of the four sectors in the U.S. economy. In practice, however, this is not done so. This would amount to twelve accounts, many of which would contain the same numbers. Therefore, the NIPAs contain a consolidated version of these accounts for the U.S. economy. This consolidated version contains five, rather than twelve, tables.
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The following table lists the five NIPA accounts and how they relate to the twelve sectoral accounts described above. Table 4.2: Structure of the National Income and Product Accounts
NIPA account
Contains…
National Income and Product Account
All sectors’ production accounts and the business sector appropriation account Personal Income and Outlay Account Household sector appropriation account Government Receipts and Expenditures Government sector appropriation Account account Foreign Transactions Account Foreign appropriations and savings and investment accounts Gross Savings and Investment Account Business, household, and government sectors’ savings and investment accounts The National Income and Product Accounts are released at a quarterly frequency. They do not only contain versions of the above accounts in current dollars, but also versions in constant dollars that reflect real variables rather than dollar amounts. Flow of Funds Accounts
It is important to realize that because the NIPAs present consolidated versions of the twelve sectoral accounts, some economic variables of interest that affect sectoral balance sheets might not be reported in the NIPAs. If you need more detailed information about sectoral balance sheets, like the amount of debt outstanding by firms or households, you would probably like to refer to the Flow of Funds Accounts. The Flow of Funds Accounts, published quarterly by the Federal Reserve Board of Governors, contains detailed information on the levels and changes in the holdings of financial assets by the various sectors of economic activity of the U.S. economy. This exposition of National Economic Accounting only scratched the surface of what data is out there and how it is collected. The U.S. National Income and Product Accounts are truly a remarkable achievement. The website of the Bureau of Economic Analysis contains much more information on how this achievement is realized and what all the data mean. In the previous two chapters, we followed up on the measurement section with a section that contained an applied example of the theory introduced. This chapter does not contain such a section. The reason is that the two following chapters are applications of the equilibrium concept and definition introduced here. In the next chapter we consider how our model behaves in terms of long run economic growth when Zt grows at a constant rate. In the subsequent chapter we will consider a theory of the cyclical behavior of the economy due to cyclical fluctuations of Zt around a constant mean.
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5
Chapter
Economic growth Now that we have developed a way to study the theoretical equilibrium outcome of our model economy, it is time to bring it to the data and consider what it teaches us about real life economic facts. In this chapter we will focus on the theory and facts related the long run trend behavior of economies. We will try and reconcile the long run trend behavior observed in the macroeconomic data for many countries with a version of our model in which technological change continuously drives the level of total factor productivity upwards. This version of our model is often referred to as the Neoclassical Growth Model. Economists generally distinguish between trends and cycles. That is, macroeconomics basically consists of two major fields. The first studies why the economy grows, compares the long run growth trends across countries, and tries to explain the very different growth experiences of countries around the world. The second studies why economies experience fluctuations from trend, generally known as business cycles. Neoclassical growth model
In this chapter we will focus on the former. We will consider what our simple model economy teaches us about economic growth and see how it applies to the growth experience countries around the world. When applied to the issue of economic growth, our simple model of the economy is often referred to as the neoclassical growth model. Hence, in this introductory chapter on economic growth, we will focus on the neoclassical theory of economic growth. Let's start off with the startling fact that is the main topic of growth theory. This fact is that, since the Industrial Revolution, many industrialized countries have seen historically unprecedented increases in real GDP per capita, i.e. output per person. The U.S. has been leading this growth spurt, taking over the U.K. as the country with the highest level of real GDP per capita around the First World War and not looking back. Figure 5.1 plots the time series for U.S. real GDP per capita, both in level and in logarithms. The reason that we most often consider the logarithm of time series is that the change between one period and another in logarithms is approximately the growth rate. This and the other data used in this chapter are available in the file Chapter5.xls.
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Figure 5.1: Level and logarithm of U.S. real GDP per capita (1990 dollars)
Three things stand out about this time series. First of all, it shows how great the great depression was and how large the war effort during the Second World War. Secondly, it shows how, besides the great depression, the economy has tended to fluctuate around a trend and that these fluctuations became less volatile after the war relative to before the war. Most importantly, though, and the focus of this chapter, it shows how real GDP per capita has been trending up steadily during the last 130 years.
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The reason that economists consider this trend important is because real GDP per capita is seen as a good proxy for the standard of living of a country. This view is not uncontroversial, though. Therefore, consider the following exercise. Exercise 5.1: Real GDP as a measure of the standard of living
Real GDP per capita is arguably a very limited proxy for the standard of living. Can you come up with additional variables that would augment the real GDP per capita measure? If so, which ones. As it turns out, what drives economic growth is technological progress, i.e. a trend in total factor productivity. Therefore, in this chapter we consider a simplified version of our model economy for the case in which a trend in TFP, i.e. Zt, drives economic growth. We consider the equilibrium dynamics of this economy and apply the main prediction of this model to study the different growth experiences of countries over time. We will do all this in three parts. The first part of this chapter contains the derivation of the Neoclassical Growth Model, its long-run behavior, and its transitional dynamics. The second part considers the main facts about economic growth and the datasets that are often used to illustrate them. Finally, in the third part, we put the neoclassical growth model to the test and illustrate which facts it can and which facts it can not explain.
5.1 Neoclassical Growth Model In this section we will include a trend to technological change in our model economy. Technological change that we will consider is simply a trend in total factor productivity. It turns out to be convenient to consider an augmented form of total factor productivity, namely Z t = X t1−α . Let g be the growth rate of Xt, then we will assume that for all t,
X t +1 = (1 + g )X t
(5.1)
ln X t +1 = ln (1 + g ) + ln X t
(5.2)
Or, equivalently, in logarithms
This assumption implies that the growth rate of total factor productivity is constant over time. This is a simplifying assumption that we will use because we are solely interested in the trend properties of our macroeconomic variables in this chapter. We will look at deviations from this trend in the next chapter when we consider business
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cycles. The reason that we use Xt rather than Zt here is because Xt will end up representing exactly the trend that we are interested in. Since economic growth is a long-run concept, we will ignore parts of our model that are mainly used to explain short-run fluctuations. Most notably, employment fluctuations are considered a very important part of the business cycle but are not considered to be very relevant for long-run growth. Therefore, we will assume, throughout this chapter, that labor is supplied inelastically. In terms of Chapter 2, this means that we will assume that θ=1 and thus Lt=1 for all t. Many growth models are developed assuming some type of population growth, which we abstract from here. This assumption does not change any of the main results discussed in this chapter. Dropping it would simply unnecessarily complicate the notation. For the rest, we will simply study the economy that we have studied before. Given the perfectly inelastic labor supply, the equations that determine the equilibrium dynamics simplify to 1 − ⎛ ⎞ −1 Y Ct σ = β ⎜⎜1 + α t +1 − δ ⎟⎟Ct +σ1 K t +1 ⎝ ⎠ Yt = Ct + I t
(5.3)
Yt = Z t K tα = X t1−α K tα K t +1 = (1 − δ )K t + I t
The problem with this dynamic system is that it does not have a steady state, because Zt is trending upwards. In order to calculate the steady state, we have to consider the variables in deviation from their trend. The trend correction that turns out to be appropriate is to divide all variables by the trend variable Xt. When we do so we obtain that 1 1 − ⎡ ⎤ σ σ ⎛ ⎞⎛ C ⎞ ⎛ Ct ⎞ ⎛ X ⎞ Y X ⎜⎜ ⎟⎟ = ⎢⎜⎜ t ⎟⎟ β ⎥⎜⎜1 + α t +1 t +1 − δ ⎟⎟⎜⎜ t +1 ⎟⎟ ⎢ ⎥ X K t +1 X t +1 ⎠⎝ X t +1 ⎠ ⎝ Xt ⎠ ⎢⎣⎝ t +1 ⎠ ⎥⎦⎝ Yt C I = t + t Xt Xt Xt −
1
σ
α
Yt ⎛ K t ⎞ =⎜ ⎟ X t ⎜⎝ X t ⎟⎠ ⎛ X ⎞⎤ K ⎛ X ⎞ I K t +1 ⎡ = ⎢(1 − δ )⎜⎜ t ⎟⎟⎥ t + ⎜⎜ t ⎟⎟ t X t +1 ⎣ ⎝ X t +1 ⎠⎦ X t ⎝ X t +1 ⎠ X t
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(5.4)
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When we denote the trend corrected variables with Yˆt = Yt X t , as well as define
βˆ =
β
(1 + g )
1/ σ
^, such that for example
1−δ g + δ = and δˆ = 1 − 1+ g 1+ g
(5.5)
then we can rewrite the above system of equations as 1 − ⎛ σ ˆ Ct = βˆ ⎜⎜1 + α ⎝ Yˆ = Cˆ + Iˆ
t
t
⎞ ˆ − σ1 Yˆt +1 − δ ⎟⎟Ct +1 Kˆ t +1 ⎠
t
Yˆt = Kˆ tα
(5.6)
( )
1 ˆ Kˆ t +1 = 1 − δˆ Kˆ t + It 1+ g
Apart from the factor, 1/(1+g), which premultiplies investment and reflects the fact that one unit of current trend corrected investment will only yield 1/(1+g) units of trend corrected capital in the next period because the trend grows over time, this system of equations is identical to the dynamic system of equations that described the transitional dynamics of our model in case of a perfectly inelastic labor supply. Hence, around the trend, the transitional dynamics of our macroeconomy here is virtually identical to that of the trendless macroeconomy of the previous chapter. The difference here is that the transitional dynamics are defined in detrended variables such that the steady state of this economy is one in which the economic variables relative to trend are constant. Balanced growth path
This is best seen through an example. In steady state Yˆt = Yˆ is constant over time. This implies that in steady state Yt = X t Yˆt = X t Yˆ . Hence, in steady state output grows at the same constant rate as Xt. This is not only the case for output. Since the steady state in this model implies that all detrended variables are constant, all these variables grow at the same rate g in this steady state. Because in this steady state all variables grow at a constant rate, this type of steady state is often referred to as a balanced growth path. So, how does this economy behave on its way to this balanced growth path? What drives the dynamics of this economy is the Euler equation, i.e. the intertemporal optimality condition that determines savings behavior. That is,
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Cˆ t +1 ⎡ ˆ ⎛ = ⎢ β ⎜⎜1 + α Cˆ t ⎣⎢ ⎝
⎞⎤ Yˆt +1 − δ ⎟⎟⎥ Kˆ t +1 ⎠⎦⎥
σ
(5.7)
In the steady state, the right hand side of this equation is one and detrended consumption is constant over time. Consumption growth outside this steady state is determined by
α Yˆt +1 Kˆ t +1 = αKˆ tα+−11
(5.8)
which is next period's marginal product of capital and, furthermore, is decreasing in the level of capital relative to trend. The lower the capital level, the higher this marginal product and thus the higher the returns to the household's savings. Consequently, when the capital stock is low, households save more. This is reflected in a lower level of detrended consumption today relative to tomorrow, i..e. Cˆ t +1 Cˆ t is relatively high. The increased savings yield an increase in the capital stock and over time a transition to the balanced growth path. Crucial for this transition is that the marginal product of capital is decreasing in the capital stock, reducing the incentive to save as the economy becomes richer (relative to trend). We will get back to this implication about the transitional dynamics of the model in much greater detail when we apply this model the data and consider its implication that countries should converge over time in real GDP per capita levels in the final section of this chapter. First, we will look at which data are often used to study economic growth.
5.2 Measuring economic growth Kaldor growth facts
Some facts on economic growth are remarkably robust across most economies in the world. These facts were first documented by Kaldor (1961) and have become known as the Kaldor stylized growth facts. These facts are A. Output per worker or per capita, Yt/Lt , exhibits continual relatively constant growth. B. The capital labor ratio, Kt/Lt , exhibits continual relatively constant growth. C. The real interest rate, ~ rt , is roughly constant over time. D. The capital output ratio, Kt/Yt , is roughly constant over time.
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~ ~ E. Factor shares, Rt K t / Yt and Wt Lt / Yt , are roughly constant over time Exercise 5.2: Kaldor’s growth facts and the BGP
Show that the balanced growth path of the Neoclassical Growth Model that we considered in Section 1 is consistent with the five stylized facts on growth above. Since Kaldor’s groundbreaking research, much work has been done on the construction of consistent cross-country datasets that allow for the comparison of real GDP per capita and related variables across countries. The construction of such datasets is not trivial because, in order to get comparable ‘real’ quantity measures, it requires the comparison of a large number of prices across countries. Penn World Tables
The dataset that has the most extensive coverage of a broad sample of countries over time and also covers probably the most extensive set of comparable variables is the Penn World Tables, constructed by Robert Summers and Alan Heston at the University of Pennsylvania. Since their first publication in the early 1980’s, the Penn World Tables have led to an extensive literature on cross-country growth experiences. The Penn World Tables are easily accessible and free. They can be downloaded in the form of a simple computer application that can be used to access the data. They also contain a comprehensive documentation of the meaning of each variable. Why not use them for the following exercise. Exercise 5.3: Kaldor in practice
Download the Penn World Tables to your computer and choose a particular country (preferable not the U.S. since the U.S. is extensively covered in the rest of our data exercises). For that country, check whether Kaldor’s facts A, B, and D hold.
5.3 Application: Convergence Hypothesis In this section, we will consider one of the most often-studied implications of the neoclassical growth model, known as the convergence hypothesis. This hypothesis basically reads:
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“Suppose that all countries have access to the same technology, such that the capital output elasticity, α, the depreciation rate, δ, as well as the technological trend, Xt, are the same across countries. Suppose furthermore, that households in all countries have the same preferences, i.e. β and σ are the same across countries. In that case, poorer countries are poor because they have a low capital level relative to trend. However, the neoclassical growth model suggests that they have an increased incentive to save, which will lead to more capital deepening in poor countries than in rich countries. Consequently, real GDP per capita should grow faster in poor countries than in rich ones due to the increased savings incentive and poor countries would ultimately catch up with rich ones.” Let’s first consider how the convergence hypothesis works in our theoretical model. We will do so in the following computer exercise Exercise 5.4: Transitional dynamics and convergence
The worksheet ‘convergence’ in the file Chapter5.xls contains a simulation of the neoclassical growth model. It allows you to compare the predicted paths of real GDP per capita for two economies. They both have the same preference and technology parameters, which you can set under the header ‘model parameters’. However, they differ in their deviation of their trend corrected capital level from its steady state. Choose different percentage deviations from their steady state for both economies and compare the predicted paths of GDP. (i)
Do you observe convergence?
(ii)
Try this a few times and choose different values for the model parameters. Is the convergence result robust?
(iii)
How does the speed of convergence depend on the intertemporal elasticity of substitution? Does it increase or decrease when σ increases?
One of the first studies to illustrate the convergence hypothesis in the data was Baumol’s (1986) ‘Productivity Growth, Convergence, and Welfare: What the LongRun Data Show’. In this study Baumol showed that for the richest industrialized countries in the World we observe that the ones that were the poorest in 1870 grew the fastest in the subsequent century. This is exactly what the convergence hypothesis predicts An updated version of the data that Baumol used is included in the worksheet ‘Baumol’ in Chapter5.xls.
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Baumol’s result that the relatively poorer industrialized countries tended to catch up with the richer ones over the last century was very thought provoking. However, a big chunk of the world population does not live in these countries. What about them? Is the convergence result robust when we consider a bigger set of countries? In order to answer this question, consider the following exercise Exercise 5.5: Convergence for a bigger set of countries
The worksheet ‘Kraay data’ in Chapter5.xls contains data on real GDP per capita for a broad sample of countries for both 1960 and 1994. These data are similar to the Penn World Tables and have been compiled by Aart Kraay at the World Bank. (i)
Which is the poorest country in Kraay's dataset in 1960 and which in 1994?
(ii)
How many times richer are people in the U.S. than people in the poorest country in Kraay's dataset in 1994?
(iii)
Which are the three countries that grew the fastest between 1960-1994. They are generally referred to as ‘growth miracles’.
(iv)
Consider the plot in the worksheet ‘kraay figure’. Do we observe convergence for all economies in the world? If so, how does this figure imply so. If not, how is convergence contradicted by this figure?
The observation that, even though for the industrialized countries we observe that the poor countries tend to catch up with the rich ones, some poor nations do not converge to richer ones at all has spurred an extensive literature on what is known as ‘conditional convergence’. The idea of conditional convergence is that technologies, and possibly preferences, are not the same across the World and depend on observable characteristics for the various countries. Because of these differences, countries tend to converge not to each other but to their own steady state, which is conditional on the factors affecting its technology. The convergence hypothesis has also been criticized for another reason. The neoclassical growth model that we have considered in this chapter assumes that all capital is created domestically and that there are no capital flows between countries. In practice, however, this is a rather unrealistic assumption. If capital would flow completely unrestrictedly between countries, then it would flow to countries with the highest returns on capital. If countries would use the same technologies then this would imply that capital would flow from rich to poor countries. In a celebrated article
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in the American Economic Review, Robert Lucas (1990) asked the question why this is not the case. The following exercise asks you to repeat his calculations. Exercise 5.6: Why doesn’t capital flow from rich to poor?
This question consists of three parts. In the first part of this question, you are asked to derive why the neoclassical growth model predicts that capital flows from rich to poor countries. In the second part, you are asked to derive the implied differences in the rates of return to capital between the U.S. and India in a similar way that Lucas did. In the final part you are asked to answer what this would imply for possible differences in the level of TFP between the U.S. and India. Throughout this problem we will consider two countries, for which we will specifically use data for the U.S. and India, and we will assume that each of these countries uses a Cobb-Douglas technology of the form
Yi = Z i K iα L1i−α
(5.9)
where i=1,2 indexes the country. (i)
Show that the above production function implies that the marginal product of capital can be expressed as
⎛Y ∂Y MPK i = i = αZ i1 / α ⎜⎜ i ∂K i ⎝ Li
⎞ ⎟⎟ ⎠
⎛ 1−α ⎞ −⎜ ⎟ ⎝ α ⎠
(5.10)
(ii)
The value α=0.3 is often used as a reasonable estimate of the capital elasiticity for the U.S.. What empirical fact is this estimate based on?
(iii)
Use the result for the marginal product of capital above to show that the marginal product of capital, and thus the return to capital is decreasing in the level of output per capita (or worker) in a country.
(iv)
Suppose that India and the U.S. both had access to the same technology, such that their TFP levels as well as their capital output elasticities were identical. Then derive the ratio of the marginal products of capital of India and the U.S. as a function of the ratio of their per capita output levels. That is, derive the ratio MPKIndia/MPKUS as a function of the ratio (YIndia/LIndia)/(YUS/LUS).
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(v)
The 2000 levels of real GDP per capita in 1996 US Dollars in India and the U.S. were $2479 and $33293 respectively. This is according to the Penn World Tables, version 6.1. Use the value of the capital output of part (ii) and these per capita GDP levels to calculate the implied ratio of marginal products of capital for India and the U.S., assuming both countries have access to the same technologies.
(vi)
Why would it be likely that capital will flow towards countries with the highest marginal product of capital? If the U.S. and India had access to the same Cobb-Douglas technology, would capital flow towards India or towards the U.S.?
(vii)
Our conclusion is thus that it is unlikely that India and the U.S. have access to the same Cobb-Douglas technology. Suppose that the difference between them is in the level of TFP. What ratio of TFP levels between the U.S. and India would equalize the marginal products of capital in 2000, taking as given the capital output elasticity that you have used above?
Lucas’ analysis suggests that there might be many country specific factors that affect the productivity levels across the World. A large part of recent growth theory has focused on what these factors are. The identification of these factors is beyond the scope of this chapter, however. One final thing on the neoclassical growth model is worth noting at the end of this chapter. That is it is not really a growth model. It is a model of capital accumulation. However, it does not explain why total factor productivity grows steadily over time. It just takes the growth of total factor productivity as exogenous. A large part of recent growth theory has been addressing this issue. The part of economic theory that tries to explain why Xt grows over time is known as ‘endogenous growth theory’.
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Chapter
Real business cycles The neoclassical growth model provides a theory of how economies move towards a balanced growth path in the long run and is used to explain the long run economic achievements of countries in the World. However, it turns out that economies show large deviations from and fluctuations around that balanced growth path in the short run. These fluctuations are called business cycles. In this chapter we will consider a theory of business cycles that explains them as the efficient response of the economy’s competitive equilibrium allocation to fluctuations of productivity growth around its trend. This theory is known as real business cycle theory. Business cycles
In the previous chapter we considered the long run growth trend in real GDP per capita. In this and the following chapters we will consider the deviations of the economy from its trend. These deviations are often called business cycles. They occur with major regularity in the economy. Much of economic policy is aimed at reducing these fluctuations. In order for policies to be effective in doing so, we need a theory of these cycles and in order to have a theory of business cycles, we need to have an understanding of what they are caused by. Though most economists believe that there are many sources of business cycles, their beliefs about the relative importance of the various sources differs extensively. In this chapter we consider a theory that attributes the bulk of business cycles to exogenous fluctuations in technology and productivity and assumes that markets themselves are efficient. Later on, we consider theories of business cycles that attribute the prolonged effects of business cycles to various kinds of imperfections in the market mechanism in the economy and consider sources of business cycles beyond technological fluctuations.
Real business cycle theory (RBC)
Because the theory that we consider here considers shocks to real economic variables, and not to nominal variables like money, as the main source of business cycles, the theory that we consider here is often called real business cycle (RBC) theory. RBC theory is basically a modern revival of the economic theory and philosophy that was
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dominant before the Great Depression, known as classical economics. This school in economics emphasizes the efficiency with which the markets allocate resources and interprets business cycles as a natural response of the economy to exogenous shocks that affect the efficient allocation of resources.
The reason that RBC theory is the first theory of cycles that we consider is because RBC theory is the natural extension of the neoclassical growth theory that we considered in the previous chapter. Furthermore, much of the current toolkit that macroeconomists use to study economic cycles, was developed in the last twenty five years in the context of RBC models. The structure of this chapter is as follows. In the first section, the main toolkit used in business cycle analysis is introduced. This is the toolkit that we will use throughout most of the next chapters. This toolkit enables us to both derive the stylized facts about business cycles that we would like our theory to explain as well as a way to compare the theoretical predictions of our models with the empirical facts in the data. In the second section we will then consider the RBC-hypothesis. Finally, in the final section we will apply our toolkit to compare the theoretical cycles our model generates under the RBC hypothesis with those observed in the data. We consider what parts of the empirical cycles are well and what parts are not so well explained by the RBC hypothesis.
6.1 Measuring business cycles Detrending Correlation Calibration Impulse response function (IRF)
There are three tools that are important for understanding our analysis of business cycles. The first is the tool that we use to distinguish between the trend and the cycle in economic time series, known as detrending. The second is the statistics that we use to describe the main properties of business cycles, i.e. correlations. The final is the way we compare the cycles generated by our model economy with those observed in the data. We do so using two concepts called calibration and impulse response functions. These will be introduced in the next subsection. Detrending: Distinguishing trends and cycles
Since much of macroeconomic theory and policy is concerned with the short-run fluctuations of the economy, we need a way to filter out these short-run fluctuations from the long run trend that we considered in the previous chapter on economic growth. The technique applied to do so is called a detrending method. There is no unique way to detrend a time series. In fact, there is an extensive discussion among economists what would be the appropriate detrending method for specific time series. An alternative approach would be to simply focus on facts that are robust to the detrending method that is applied. This will be our approach in this chapter. The specific filter that we will apply here is called the Hodrick Prescott filter (HP-filter).
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What this filter does precisely is beyond the scope of this class. However, what we obtain when we apply this filter will be the focus of this chapter. RBC theorists generally take into account that their model does not contain a government and foreign economies and adjust their measurement of output accordingly. We, however, will not do so in this chapter, because it is more important for us to know the facts about commonly used time series that reflect the behavior of the whole U.S. economy than about adjusted times series that are specifically adjusted for the application of the RBC model that we consider in this chapter. The time series that we will mainly focus on are output, measured as real GDP, personal consumption expenditures, hours worked in the non-farm business sector, and fixed private non-residential investment. These data and the results that we consider throughout this chapter are provided in the Excel file chapter6.xls. The data are described in the ‘Data description’ worksheet in the file. The file contains a more extensive set of data than what we will focus on in the text here. National Bureau of Economic Research (NBER)
Let's first consider the cyclical behavior of output. Figure 6.1 depicts the cyclical component of GDP obtained using the HP-filter. The shaded quarters are those that are designated as recession quarters by the National Bureau of Economic Research (NBER). The NBER has a committee that officially dates, most often with a few months delay, the recessions of the U.S. economy. A few things that one can notice from the figure: (i) If output falls more than 1.5% below trend, then we have always been in a recession, (ii) Output fluctuations have decreased since the beginning of the 1980's, (iii) The 1992 recession was less ‘deep’ than the other postwar recessions but took longer to recover from, and (iv) Recessions generally do not last more than a year. The most important fact to take away from Figure 6.1 is that output fluctuates a lot around its trend.
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Figure 6.1: Real GDP in deviation from its trend
Knowing the cyclical component of output is not enough, however, if we would like to understand the economic mechanism that underlies these fluctuations. Figure 6.2 depicts the cyclical components of output, consumption and investment. A few things are immediately obvious from looking at this graph. The first is that investment is far more volatile than output, which is a fact that we would like our model to explain. The second is that consumption is smoother than output. Finally, investment seems to peak and bottom out most often about one quarter later than output does. That is, investment tends to lag output by about 1 quarter. Furthermore, though fluctuations in output seem to have been less volatile since the mid-1980's, this does not seem to be the case for investment. In fact, the most recent recession has seen an unprecedented slump in investment, falling more than 12% below trend.
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Figure 6.2: Real GDP, real personal consumption expenditures, and fixed private investment in deviation from trend
Finally, it is useful to consider the relationship between employment and output. This is depicted in figure 6.3, which depicts the cyclical components of output and hours. Just like investment, hours tend to lag output because it takes time for firms to adjust their employment levels in a response to unexpected changes in demand. Employment also turns out to be slightly more volatile than output.
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Figure 6.3: Detrended hours and output
Correlations Standard deviation Correlation
The detrended time series of the previous subsection show some of the properties of business cycles. However, we need a more formal way to assess the average magnitude and the behavior of business cycles over time. For this, we use some basic statistics, namely standard deviations and correlations. We will not study the details of the concepts of standard deviations and correlations in detail in these notes. They are explained in virtually all introductory undergraduate statistics textbooks. The standard deviation is a measure of spread in a variable. That is, the bigger the standard deviation of the cyclical component of a variable the more the variable fluctuates on average over the cycle. The bigger the correlation between two variables, the bigger is the share of fluctuations that two variables have in common. When the correlation is negative, then the fluctuations that two variables have in common go in opposite directions. In order to consider the average properties of business cycles for the us economy for the postwar period, economists generally consider the correlations between several detrended economic time series. The worksheet ‘correlations’ in the Excel file chapter6.xls contains the standard deviations correlations that many economists refer to when they discuss the `stylized facts' of U.S. business cycles. For example, an important observation about U.S. business cycles is that consumption is much less volatile than output. How would we be able to distill this from the reported standard deviations and correlations? Well, we see that the standard deviation
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of detrended consumption expenditures is 1.31%, which is much smaller than the 1.77% for GDP. What is the economic explanation for this? One of the important properties of the households savings decision is that it implies that households would like to smooth consumption relative to their income. This is exactly what we observe in the data. A second example is that investment in structures and equipment (fixed private nonresidential investment) lags output by about a quarter. How do we observe this in the correlations? We see this from the 0.81 correlation of current fixed private nonresidential investment today with GDP 1 quarter ago, which is higher than the correlation of fixed private non-residential investment with GDP for all other leads and lags reported, i.e. the maximum correlation reported in the ‘I_NR’ row in the correlation table. This lag is probably due to firms having to commit to their future investment levels a bit more than three months before the expenditures are reported, i.e. there is time to plan and build required for investment. If this is the case, then investment expenditures will only respond to a current surprise increase in output one quarter from now. The following exercise follows up on the above two examples Exercise 6.1: Find the stylized facts
Use the correlations table in the worksheet ‘correlations’ in the Excel file chapter6.xls to argue why there is evidence in favor of the following stylized facts on the U.S. postwar business cycles: (i)
Hours worked in the non-farm business sector are slightly more volatile than output.
(ii)
Investment is much more volatile than output.
(iii)
Deviations from the trend are persistent, i.e. if output below trend today than output is probably below trend tomorrow.
(iv)
Durable goods consumption is much more volatile than consumption of non-durables and services.
6.2 RBC Hypothesis RBC models come in many shapes and forms and many perturbations, each meant to adjust the model to match some of the stylized features of the data. We will look that the most rudimentary RBC model here, which is essentially the neoclassical growth model that we developed in the previous chapters applied to cycles. No matter what
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type of RBC model one considers, at the heart of real business cycle analysis are two main assumptions. The first, and most important, is that markets tend to allocate resources efficiently and that cycles are simply the realization of shocks that work through the economy. Then the question is: ‘What is the source of these shocks?’. In its most ‘pure’ version, RBC theory assumes that the main source of these shocks is fluctuations around the trend of productivity growth. Hence, the most basic version of the RBC hypothesis is that business cycles represent the efficient fluctuations in the allocation of resources in response to shocks to productivity. The aim of this section is to have a shot at considering in how far productivity shocks generate realistic cycles in our theoretical model. We will do this in the following manner. We will assume that the economy is in its steady state and then look at its response to a shock to total factor productivity, Zt. In order to be able to solve for the response of the economy, we will have to define the full future path of total factor productivity. We will assume that total factor productivity in the long run will return to its trend, but that in the short run productivity growth deviates from it. Most business cycles take the trend out of the model. Hence, Zt→1 is essentially equivalent to Zt returning to its long run trend. Just like in the detrended data that we considered in the previous section, we will consider percentage deviations from trend for our model. That is, the steady state is essentially the long run trend outcome implied by the growth model of the previous chapter, and the cycles that we generate in this chapter are percentage deviations from this trend outcome. We will thus also consider the percentage deviation of total factor productivity from its steady state outcome. That is, define the percentage deviation of Zt from 1, as
Z −1 Zˆ t = t = Zt −1 1
(6.1)
then we will assume that, for a given initial percentage deviation from the steady state ∞ Zˆ , Zˆ follows the following linear first order difference equation t
{ }
t + s s =0
Zˆ t = ρZˆ t −1 where ρ < 1
(6.2)
The parameter ρ reflects the degree of persistence of a productivity shock. That is, the higher ρ the longer it takes for a productivity shock to die out. Just convince yourself in the following exercise.
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Exercise 6.2: Persistence of productivity shock
{ } {Zˆ }
Suppose that Zˆ t = 0.01 , solve for Zˆ t + s (iii) ρ=0. Plot the paths of parameters in one graph. Impulse response function
10 s =1
10
t + s s =0
for (i) ρ=0.9, (ii) ρ=0.5, and
for these three persistence
Our aim is thus to see how our model economy will respond to an impulse that causes an initial deviation from trend productivity growth, that is to an impulse that shocks productivity such that Zˆ 0 = 0.01 . This shock to productivity will cause the economy to respond and we will study the response of the equilibrium variables in our economy to this impulse. The resulting graph that plots the path of the percentage deviations of consumption, output, investment, the capital stock, and hours from the steady state is appropriately called an impulse response function. The worksheet ‘impulse responses’ in the Excel file chapter6.xls contains the computer simulation of these impulse response functions. It allows you to choose the model parameters as well as the ‘initial productivity shock’, which is the value of Zˆ 0 . The graph then shows the paths of the equilibrium variables measured as percentage deviations from their steady state.
6.3 Application: Bringing the model to the data Calibration
In order to compare the simulated cycles with the data, we have to choose specific values for the parameters of the model. A popular methodology for doing so in the RBC literature is called calibration. Calibration chooses parameter values of the model based on certain evidence from the data and then uses these parameter values to compare what the model generates with respect to other empirical facts in the data. For example, previously we had derived that the model implies that the capital elasticity of output, i.e.α, equals one minus the labor share. Since we observed the share of labor in GDP is fairly constant around about 70%, it is a common practice to choose α to be close to 0.3. The depreciation rate, δ, is generally taken directly from data on capital stocks compiled by the bureau of economic analysis. Evidence on structures and equipment suggests that their capital aggregate depreciates at an annual rate that is approximately 10%. This implies a quarterly depreciation rate of about 2.5%, which is the one chosen in the model parameter calibration in the ‘impulse responses’ worksheet. Because of the increased importance of computers in investment and their relatively high depreciation rate, the depreciation rate of the structures and equipment aggregate has actually been increasing over time.
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Besides using outside evidence to calibrate parameters, economists also often calibrate data on the basis of the model's steady state properties. The idea is that if we would like to consider the transitional behavior of the model to its steady state, then we'd better be sure that the steady state to which the economy returns is empirically plausible. One parameter that is often calibrated on the basis of steady state properties is the discount rate, β. In the steady state, the discount rate satisfies
β=
1 ≈ 1− ~ r ~ 1+ r
(6.3)
where ~ r is the steady state real interest rate. Empirical evidence suggests that the longrun real interest rate is about 5% per annum. Therefore, it is often assumed that the annual discount factor is 0.95, such that the quarterly discount factor is (0.95)1/4≈0.9875 which is what is chosen in the benchmark calibration in the worksheet. Evidence on the intertemporal elasticity of substitution, σ, the Frisch elasticity of the labor supply, θ, as well as the degree of persistence of productivity shocks, ρ, is much less conclusive and these parameters are often just varied to see whether the impulse responses are sensitive to their values. Now that we have chosen the parameters, the next step is to generate the associated impulse response functions and see whether they adhere to the facts that we distilled from the data in the previous section. We will do part of this analysis here in the main text. The subsequent analysis is left for an exercise. For the analysis here in the main text, we will refer to the benchmark calibration of the parameters that is included in the Excel file. Our first observation in the data was that consumption is much less volatile than output. Does our model also generate cycles in which this is the case? The answer is yes. The impulse responses plotted are the responses to a one percent positive deviation of productivity from its trend. As you can see, output increases much more in response to the shock than consumption. The increase in consumption only exceeds that in output in the long-run. This is exactly what consumption smoothing implies. A positive shock to productivity will result in a steep increase in output but a relatively smooth increase in consumption. Investment absorbs the difference. Investment is used as a buffer to save current output for consumption in the future. Another observation was that investment is much more volatile than output in the data. This is also true in our theoretical model. As you can see, the initial response of investment is about 5 times as big as that of output. Again, this is because investment is used as a buffer to accommodate consumption smoothing. The general mechanism of the impulse response plotted is as follows. A positive productivity shock increases both the productivity of labor as well as capital. This both increases the real wage as well as the incentive to save. Since consumers want to
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smooth consumption, the increased output due to the productivity shock is not immediately all consumed but is instead for a large part saved. Because of this saving the capital stock increases to a level that exceeds its steady state, causing persistence in the deviation of output from its steady state because the increased future capital input will lead to increased output in the future. In the long run this additional capital depreciates and the productivity shock dissipates because of which the economy returns to its steady state. Income effect Wealth effect
The response of the labor supply to a positive productivity shock turns out to depend on the relative magnitude of two effects. First of all, there is the income effect. Because a positive productivity shock increases the current marginal product of labor it increases the real wage in the short run. This increase in wages unambiguously increases labor supply. The induced increase in the labor supply due to the short run increase in the marginal product of labor is known as the income effect on the labor supply. However, the income effect is not the only effect that affects the labor supply. The second effect is known as the wealth effect on the labor supply. Because a productivity shock increases the present discounted value of lifetime income the household will partially consume this increase in wealth. However, it will also use this increase in wealth to simply work less (it substitutes leisure for increased consumption). This negative effect on the labor supply is known as the wealth effect. The widely held, though not unanimously held, belief is that in the data the income effect dominates the wealth effect. However, it is definitely possible to come up with parameter combinations and situations in which the wealth effect would dominate.
Exercise 6.3: Impulse responses and stylized facts
We have confirmed that our benchmark simulation suggests that consumption is more smooth than output while investment is more volatile in our benchmark economy. Now it is time for you to experiment with the simulations. Use the program to figure out whether the model is able to generate the following stylized facts that we discussed above. (i)
According to our model economy, are the hours worked slightly more volatile than output, as the data suggest, or are they less volatile than output? How does the Frisch elasticity of the labor supply affect the volatility of the labor supply?
(ii)
Are the cyclical fluctuations in output persistent? Suppose you would reduce the persistence parameter of the productivity shock, i.e. ρ, what happens to the persistence of the cycles
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generated by the model? Do you have any intuition why this is the case? (iii)
Does investment trail output in our model economy? If not, can you change the parameters such that it does? Try a few parameter combinations.
6.4 Merits and shortcomings of RBC theory So, what have we learned from the RBC model that we considered here. Well, besides posing the thought-provoking hypothesis that most of economic fluctuations can be explained as efficient responses of the economy to exogenously given shocks to productivity, RBC theory has given us a set of tools that we can use to study business cycles. We have used detrending, correlations, calibration, and impulse response functions. If anything, RBC models have been at the root of the development of these modern tools of macroeconomics. Though the methodological contribution of RBC theory is interesting, as economists we are more interested in which facts we managed to explain using the RBC hypothesis. We found that the consumption smoothing mechanism in our RBC model was able to match the relative volatilities of output, investment, and consumption in the data. RBC models, however, are often criticized for three main reasons. First of all, there are several facts that RBC models have a hard time explaining. In exercise 6.3 we found that our RBC model could not properly match the volatility of hours worked. This turns out to be a major problem of RBC models. They simply can not generate a labor supply function that is elastic enough to generate the observed fluctuations in employment. This partly due to the fact that RBC models generally assume that the labor market is in equilibrium and that there is no unemployment. Allowing for unemployment can possibly generate more volatility in employment. Furthermore, we found that our model does not match the fact that investment tends to trail output by about a quarter. This failure turns out to be relatively easy to solve by assuming more than the one quarter of time to build that we have assumed in our model calibration. Finally, the model can only generate enough persistence in cycles when we assume that the productivity shock that drives these cycles itself is very persistent. This brings us to the second major criticism of RBC models. The productivity shocks that drive fluctuations according to the RBC hypothesis have to be very volatile as well as very persistent in order to match the data. This invites the question whether there are mechanisms that our simple RBC model ignores that both magnify as well as persistently transmit these productivity shocks in the economy. A large part of the
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RBC literature is about extensions of the basic model that we considered here that are aimed at explaining exactly this. The biggest problem with the ‘productivity shock’ explanation, however, that it is just hard to come up with a story about what constitutes these productivity shocks. The RBC hypothesis implies that the economy is frequently hit by shocks that cause productivity to decline. In its ‘purest’ interpretation this would essentially mean that we loose the knowledge to produce things as efficiently as before. Now, it is true that technological knowledge sometimes gets lost over time. For example, we are not able to figure out how the Inca's managed to construct their buildings in the way they did. However, it is hard to believe that such cases of technological regression occur at the frequency that the RBC hypothesis implies. This would imply that the economy suffers from some type of technological amnesia. One possible explanation for these negative productivity shocks comes from the growth literature. Recent theories in economic growth suggest that it is sometimes worthwhile to sacrifice current levels of productivity to learn about a new technology that reaps benefits in the future. For example, when someone first uses a computer, they are probably going to be less productive than they were before they used it. This is worth it however, because when they learn how to use it in the long-run they will be more productive then. Mixing theories of economic growth and cycles is not very common though. For historical reasons, macroeconomic theory has developed a major dichotomy between growth theory and theories of business cycles. The final criticism of RBC models is that they offer no room for money and in fact imply that money is irrelevant for the realizations of the real equilibrium variables in the economy. Hence, they do not teach us anything about the importance of money in the economy. This means that they are virtually useless for the study and quantitative analysis of monetary policy. In fact, there is no need for policy anyway in the RBC model because fluctuations are modeled as efficient responses of the competitive equilibrium allocation to fluctuations of productivity growth around its trend.
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Chapter
Money demand and supply In our previous study of the neoclassical growth model we ignored the existence of money and the possible influence of money on the allocation of resources in the macroeconomy. In equilibrium, the real interest rate and real wage rate were determined, while the price level was not. We simply normalized the latter to one. In this chapter we take our first steps towards considering what determines the price level in the economy. We do so by adding money to the neoclassical model. We focus on the demand for money by the households and the supply of money by the central bank. We consider a model in which households hold money to facilitate their consumption purchases, known as a cash-in-advance model, and determine what would be the optimal money supply rule that could be implemented by a central bank that supplies money in this economy. Central bank Monetary aggregates Friedman Rule CPI PCE deflator
This chapter focuses on four questions that are central in the study of the effect of money on the macroeconomy. First and foremost, we ask ourselves why it is that people hold money and how they use it. Second, we consider who actually supplies money in the economy. For this purpose we introduce the central bank as an additional agent to our model economy. Third, we consider the measurement of money and prices in the economy. Measures of money in the overall economy are often referred to as monetary aggregates. The two most commonly used measures of the prices faced by households are known as the Consumer Price Index (CPI) and the Personal Consumption Expenditures (PCE) deflator. Finally, as an application, we ask ourselves what would be the optimal supply of money in the economy we develop in this chapter. This optimal money supply problem will yield a rule that is called the Friedman Rule.
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7.1 Fiat money Fiat money
In most economies in the world some kind of money is used to facilitate transactions. The most common type of money used is called fiat money. Fiat money is money that a government has declared to be legal tender, despite the fact that it has no intrinsic value and is not backed by reserves. The fact that some kind of fiat money is accepted virtually anywhere in the world is in itself amazing. It implies that, for most economic transactions, the buyer pays the supplier of a good or service which has intrinsic value in terms of some pieces of paper, i.e. bills, or metal, i.e. coins, that have essentially no intrinsic value of their own. The only value that these bills and coins have for the supplier is the expectation that he or she can spend them subsequently to also buy goods and services. Most macroeconomic theories of money introduce money as an additional asset in the household’s nominal budget constraint. In the theory of the household that we considered in Chapter 2, we introduced the following budget constraint
At = (1 + rt )At −1 + Wt Lt − Pt Ct
(7.1)
Here, At denotes the nominal assetholdings at the end of the period, rt is the nominal interest rate, Wt is the nominal wage rate, Pt is the price of the consumption good, Lt is the labor supply, and Ct is the consumption level. When we add money to this budget constraint, the important distinction between money and the assetholding is that holding money between two periods doesn’t yield interest while holding assets does. From this point of view, the physical money you have in your wallet is just as much money as the balance in your non-interest paying checking account. However, your balance in your savings account, which pays interest, is considered part of your assetholdings. In practice, measuring the total amount of money in the economy, i.e. constructing a monetary aggregate, turns out to be complicated because the line between interest yielding and non-interest yielding financial instruments turns out to be rather blurry. We will get back to this measurement issue later on in this chapter. To continue with the theory, we denote the amount of money held at the end of period t by Mt.. Adding money to the budget constraint (7.1) yields that the sum of asset- and moneyholdings at the end of the current period equals their sum at the beginning of this period plus interest and labor income and minus consumption expenditures. That is,
At + M t = (1 + rt )At −1 + M t −1 + Wt Lt − Pt Ct
(7.2)
Just like the nominal budget constraint introduced in Chapter 2, we can also write this budget constraint in terms of real variables, i.e. units of the consumption good, rather
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than in nominal dollar terms. In order to do so, we introduce real money balances at the end of period t as ~ M t = M t Pt
(7.3)
Using this notation, we can rewrite the budget constraint in real terms as
~ ~ ~ At + M t = (1 + ~ rt )At −1 +
1 ~ ~ M t −1 + Wt Lt − Ct 1+ π t
(7.4)
From this representation, we can make one important observation. We do so in the following exercise. Exercise 7.1: Returns on money and assetholdings
The purpose of this exercise is to compare the real gross returns that the household obtains on its assetholdings and money balances and to derive what these returns imply for the household’s portfolio decision. (i) Identify the real gross return on assetholdings in (7.4). (ii) Identify the real gross return on money balances in (7.4). (iii) Show that if the nominal interest rate is non-negative, i.e. rt≥0, then the real gross return on assetholdings always exceeds that on money balances. (iv) Hence, if rt>0, will the household decide to hold any money? (v) Reversely, if rt0, which is what we will assume throughout the rest of this chapter, then the only reason why a household facing a CIA constraint holds money is to facilitate its consumption expenditures. Therefore, the household will not hold more money at the end of period t-1 than it needs to finance its consumption in period t. This means that, under our assumption of perfect foresight, the CIA constraint will always be binding9 such that
Pt Ct = M t −1 and Ct =
1 ~ M t −1 1+ π t
(7.7)
Substituting this restriction into the real budget constraint, i.e. (7.4), yields that ~ ~ ~ At = (1 + ~ rt )At −1 + Wt Lt − (1 + π t +1 )Ct +1
(7.8)
We will focus on the household in our theory of money demand here. One can also generate money demand for firms by assuming that they need cash-in-advance in order to pay for the costs of their inputs.
8
Although not directly relevant for our further analysis, it is worthwhile to note that under uncertainty this constraint might not be binding because the household might decide to hold some precautionary money balances as a buffer stock to finance unexpected increases in consumption.
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Note the difference between this budget constraint and the real version of (7.1). The difference is that current period expenditures are essentially next period’s consumption. That is, in the current period, the household decides to set aside a certain amount of money balances to finance next period’s consumption. Because the money for next period’s consumption level is set aside in the current period and only spent one period later, the household incurs an inflation cost (1+πt+1) that reflects the change in purchasing power of its money balances between the two periods. The derivation of (7.8) allows us to set up the representative household’s utility maximization problem with the CIA constraint. Given this constraint the household ∞ ~ ~ chooses Ct + s , M t + s , At + s , Lt + s s =0 to maximize the present discounted value of its stream of utility
{
}
∞
∑ β U (C s
s =0
t+s
, Lt + s )
(7.9)
~ where it takes as given the initial real money balances M t −1 , the initial assetholdings ~ At −1 , the paths of real wages, the real interest rate, as well as inflation. It maximizes this objective subject to the real budget constraint and the real CIA constraint. These are, respectively,
~ ~ ~ At + s + M t + s = (1 + ~ rt + s )At + s −1 +
1 1 + π t+s
~ ~ M t + s −1 + Wt + s Lt + s − Ct + s
(7.10)
and
Ct + s ≤
1 1 + π t+S
~ M t + s −1
(7.11)
and have to hold for all s=0,…,∞. As explained above, the CIA constraint, (7.11), will be binding for all s=1,…,∞. We will also assume it holds for s=010. Hence, the two constraints the household faces can be collapsed into (7.8). Collapsing this eliminates the path of the real money balances from the household’s optimization problem, since it is implied, through the CIA constraint, by the chosen path of consumption. In principle, we consider a household that optimizes in period t and could thus potentially have chosen ~ in such a way that, when it reoptimizes in period t, constraint (7.11) is not binding in period t, i.e. for M t −1 ~ s=0. However, we will assume that the household also optimized before and chose M t −1 bearing in mind the optimal consumption choice Ct. That is, we will assume that (7.11) is also binding for s=0.
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Just like in the previous chapters, we tackle this optimization problem with our fourstep approach. In order to identify the costs and benefits of the household’s savings and labor supply decision, it is this time around easiest to write the first three terms of the household’s present discounted value of its stream of utility separately from the rest. That is, to write Step 1: Identify the costs and benefits
∞
∑ β U (C s
s =0
t +s
, Lt + s ) = U (Ct , Lt ) + βU (Ct +1 , Lt +1 ) + β 2U (Ct + 2 , Lt + 2 ) (7.12)
∞
+ ∑ β U (Ct + s , Lt + s ) s
s =3
The real budget constraint (7.8) implies that consumption can be written as
Ct =
[
1 (1 + ~rt −1 )A~t −2 + W~t −1 Lt −1 − A~t −1 (1 + π t )
]
(7.13)
Substituting this back into the first three terms of the objective function, we can write ∞
∑ β U (C s
s =0
t +s
, Lt + s ) = U (Ct , Lt )
[
]
⎛ ⎞ 1 (1 + ~rt )A~t −1 + W~t Lt − A~t , Lt +1 ⎟⎟ + βU ⎜⎜ ⎝ (1 + π t +1 ) ⎠
[
]
⎛ ⎞ 1 (1 + ~rt +1 )A~t + W~t +1 Lt +1 − A~t +1 , Lt +2 ⎟⎟ + β 2U ⎜⎜ ⎝ (1 + π t + 2 ) ⎠
(7.14)
∞
+ ∑ β sU (Ct + s , Lt + s ) s=2
This representation allows us to more easily identify the costs and benefits of both the ~ labor supply decision, i.e. Lt, as well as the households savings decision, i.e. At . The costs and benefits of the labor supply decision in the CIA model that we consider here are slightly different from those in the dynamic utility maximization problem considered in Chapter 2. This is because current labor income cannot be used towards today’s consumption, but instead can at most be set aside in the form of money holdings to be used for consumption one period later. That is, the benefit of the labor supply decision is the contribution of labor income to tomorrow’s consumption. The costs are identical to the basic model here. That is, the cost of the labor supply is the reduction in current utility due to the loss of leisure time. Costs and benefits of the labor supply decision
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Mathematically, these costs and benefits show up in (7.14) as follows ⎛ ⎞ ⎜ ⎟ ∞ ⎜ ⎟ s β U (Ct + s , Lt + s ) = U ⎜ Ct , Lt ∑ ⎟ { s =0 cost of labor supply ⎟ ⎜ due to loss of current ⎟ ⎜ leisure time ⎝ ⎠ ⎛ ⎞ ⎤ ⎡ ⎜ ⎟ ⎥ ⎢ ⎜ ⎟ ⎢ 1 ~ ~⎥ ~ (7.15) ~ ⎜ Lt + βU − At ⎥, Lt +1 ⎟ ⎢(1 + rt )At −1 + Wt { ⎜ (1 + π t +1 ) ⎢ ⎟ ⎥ benefit of labor ⎜ ⎟ supply decision ⎥ ⎢ through additional ⎜ ⎟ ⎥ ⎢ consumptio n tomorrow ⎦ ⎣ ⎝ ⎠
[
]
⎛ ⎞ 1 (1 + ~rt +1 )A~t + W~t +1 Lt +1 − A~t +1 , Lt +2 ⎟⎟ + β 2U ⎜⎜ ⎝ (1 + π t + 2 ) ⎠ ∞
+ ∑ β sU (Ct + s , Lt + s ) s =2
As you can see from this representation, contrary to the basic problem the benefits and costs of the labor supply are now incurred in different periods. Hence, in the CIA model, the labor supply decision is intertemporal, rather than intratemporal. The costs and benefits of the ~ household’s savings decision, i.e. its choice of At , are now realized at times t+1 and t+2 respectively. On the cost side, incurred at t+1, we find that if the household in the current period decides to save more in terms of assets it cannot hold these assets in terms of moneyholdings that can be used to buy next period’s consumption. Hence, current savings come at the sacrifice of next period’s consumption. On the benefit side, obtained at t+2, the household obtains interest on its current savings which it can then use towards the moneyholdings in next period which are used to buy consumption in period t+2. Costs and benefits of the savings decision
Mathematically, these costs and benefits show up as follows
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∞
∑ β U (C s
s =0
t+s
A N D
S U P P L Y
, Lt + s ) = U (C t , Lt ) ⎛ ⎞ ⎡ ⎤ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ 1 ~ ~ ~ ~ At , Lt +1 ⎟ + βU ⎜ ⎢(1 + rt )At −1 + Wt Lt − ⎥ { ⎜ (1 + π t +1 ) ⎢ ⎟ current savings reduce ⎥ ⎜ ⎟ current moneyholdi ngs ⎥ ⎢ and, through CIA, ⎜ ⎟ ⎢ ⎥ tomorrow' s consumptio n ⎣ ⎦ ⎝ ⎠ ⎛ ⎞ ⎤ ⎡ ⎜ ⎟ ⎥ ⎢ ⎜ ⎟ ⎥ ⎢ ⎜ ⎟ ⎥ ⎢ 1 ~ ~ ~ 2 ⎜ ~ ⎥ ⎢ ( ) + β U⎜ 1 + rt +1 A + Wt +1 Lt +1 − At +1 , Lt + 2 ⎟⎟ {t ( 1 + π t +2 ) ⎢ ⎥ current savings can ⎜ ⎟ be used towards to ⎥ ⎢ morrow's moneyhol⎜ ⎟ ⎥ ⎢ dings which are used ⎜⎜ ⎟⎟ to buy consumptio n ⎥ ⎢ two periods from now ⎦ ⎣ ⎝ ⎠ ∞
+ ∑ β sU (C t + s , Lt + s )
(7.16)
s=2
Thus, the imposed delay between saving and consumption through the CIA constraint means that the costs and benefits of saving in the CIA model are each shifted by one period. Now the costs and benefits of both the labor supply and savings decisions have been identified, it is again time to bring out the partial derivatives to determine the marginal costs and marginal benefits. Step 2: Calculate the marginal costs and marginal benefits
The marginal benefits of the labor supply decision are made up of the additional consumption that the household can buy in period t+1 when it works an hour more in period t. This can be written as Marginal costs and marginal benefits of the labor supply
β
∂U ∂Ct +1 ∂U ⎛ 1 ⎞ ~ ⎜ ⎟Wt =β ∂Ct +1 ∂Lt ∂Ct +1 ⎜⎝ 1 + π t +1 ⎟⎠
(7.17)
Let’s focus on the more detailed version of the marginal benefits on the right hand side of the equality sign. It consists of four terms. The first is the discount factor that reflects the fact that the benefits of current labor supply are obtained one period from now. The second is the marginal utility of consumption tomorrow and converts the additional units of consumption that can be bought with the labor income into units of additional utility. The final two terms reflect the additional utility that can be bought in period t+1 with the additional labor income earned from working an additional hour in period t. It equals the amount of period t consumption that can be bought with the wages earned for working an additional hour
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~ in time t, i.e. the real wage Wt , times the loss in purchasing power of these wage earnings between periods t and t+1 when they are held as moneyholdings. The latter is the inflation factor 1/(1+πt+1).
The marginal costs of the labor supply are the same as in the basic dynamic optimization problem for the household that we considered in Chapter 2. It consists of the marginal disutility of working due to the loss of leisure. Mathematically, this boils down to
−
∂U ∂Lt
(7.18)
The marginal benefits of the savings decision consist of the additional consumption in period t+2 that a unit of savings in period t yields. This is given by Marginal costs and marginal benefits of savings decision
β2
∂U ⎛ 1 ⎞ ⎜⎜ ⎟⎟(1 + ~ rt +1 ) ∂Ct + 2 ⎝ 1 + π t + 2 ⎠
(7.19)
Just like the marginal benefits of the labor supply, this expression consists of four terms. The first is again the discount factor. In this case, the benefits are doubly discounted because the benefits are received two periods after the savings decision is made. The second is marginal utility of consumption in period t+2, which transforms the additional units of consumption that can be bought with the savings in period t+2 into additional units of utility that constitute the household’s objective. The third term again reflects the effect that inflation has on the purchasing power of the nominal savings of the household. That is, if the household saves a dollar in period t, adds it to is moneyholdings in period t+1, and spends it in period t+2, then it pays the inflation cost on the moneyholdings between periods t+1 and t+2. The final term is the real interest return that the household would obtain on its savings between periods t and t+1. The marginal costs of an additional unit of savings in period t consist of the sacrificed period t+1 consumption that this unit of savings requires. Using partial derivatives, we can write this as
β
∂U ⎛ 1 ⎞ ⎜ ⎟ ∂Ct +1 ⎜⎝ 1 + π t +1 ⎟⎠
(7.20)
The three terms of this equation have essentially the same interpretation as the first three terms in the previous one. Step 3: Equate marginal costs and marginal benefits Equating the marginal benefits
and costs of the labor supply, we obtain the optimality condition
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β
S U P P L Y
∂U ⎛ 1 ⎞ ~ ∂U ⎜⎜ ⎟⎟Wt = − ∂Ct +1 ⎝ 1 + π t +1 ⎠ ∂Lt
(7.21)
The big difference between this optimality condition and that of the basic dynamic optimization problem is that the CIA constraint implies that the consumption benefits from labor earnings are delayed by one period and that, during this one period delay, these earnings are susceptible to losses in purchasing power due to inflation. For the savings decision, we obtain that the household’s path of consumption and savings over time must be consistent in every period with the following condition
β
∂U ⎛ 1 ⎞ ∂U ⎜⎜ ⎟⎟(1 + r~t ) = ∂Ct +1 ⎝ 1 + π t +1 ⎠ ∂Ct
⎛ 1 ⎜⎜ ⎝1+ πt
⎞ ⎟⎟ ⎠
(7.22)
where, compared to equations (7.19) and (7.20), we have divided both the marginal benefits and costs by the common discount factor β and we have reduced the time indexation by one period. Solving the optimality conditions in detail is not necessary in this case. The previous derivation already emphasized the main intuition about the added inflation cost that the CIA constraint imposes on the household. Step 4: Solve the optimality conditions
Monetarism
It turns out that the solution to the household’s problem with a cash-in-advance constraint yields a ‘result’ that is relevant for a particular theory of the effect of money on the economy, called monetarism.
Quantity equation of money
At the heart of the monetarist view on money and the macroeconomy is the quantity equation of money. This equation basically says that the total dollar amount of transactions that take place in the economy should equal the total amount of money that is spent. The total amount of transactions is simply equal to the price level, which we denote by Pt, times the amount sold, which we denote by Yt. The total amount of money spent in the economy is the amount of money held in the economy, i.e. Mt, times the number of times in each period that a dollar is spent. The latter measure is often referred to as the velocity of money, and we will denote it by Vt. The reason that this is called velocity is that it essentially the speed with which money circulates between agents in the economy.
Velocity of money
The above description translates, mathematically, into the following equation
Pt Yt = M tVt
(7.23)
Hence, besides its description above, this equation simply defines the velocity of money as
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Vt =
S U P P L Y
Pt Yt total amount of transactions = Mt moneybalance
(7.24)
Because, in our model, we denote Mt-1 as the money balances that the household uses for transactions in period t and because the total amount of transactions the household does at time t equals its consumption expenditures, PtCt, in our simple model velocity is defined slightly differently. Namely, it equals
vt =
Pt Ct M t −1
(7.25)
For the simple cash-in-advance model that we have studied so far, vt is determined by the CIA constraint. This constraint and the cost of holding money imply that the household will hold just enough money to finance its consumption purchases such that vt=1 for all t. That is, velocity is constant and is not influenced by any other macroeconomic variables, most notably prices and interest rates. The assumption that the velocity of money is (approximately) constant is one of the two main staples of monetarism. The second staple is the assumption that money has little effect on output, i.e. Yt. First difference operator
The quantity equation implies that the growth rate of output plus the growth rate of the price level, being the inflation rate, equals the growth rate of money plus the growth rate of velocity. That is, when take logarithms on both sides of (7.23) and apply the first difference operator, ∆, we find that ∆ ln Pt + ∆ ln Yt = ∆ ln M t + ∆ ln Vt 123
(7.26)
πt
Now, monetarists claim that the response of the growth rate of output is rather subdued compared to the response of the inflation rate. Therefore, the monetarist suggestion to reduce inflation is to simply reduce the growth rate of money. Some of the long-run evidence in the data suggests that there is indeed a strong relationship between the growth rate of money, i.e. ∆lnMt and inflation πt. The following exercise involves an empirical exercise that is very similar to McCandless and Weber (1995).
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900% 800% 700%
inflation
600% 500% 400% 300% 200% 100% 0% 0%
100%
200%
300% 400% money growth
500%
600%
700%
Figure 7.1: Long run relationship (1961-2005) between average money growth and inflation
Exercise 7.2: Money growth and inflation in the long run
This exercise uses the Excel file chapter7.xls for its calculations. Upload this file to your computer and open it in Excel. Click on the worksheet called ‘money growth and inflation’. It both contains the raw data used in the exercise as well as a set of two figures. The first figure is the same as figure 7.1. It plots the average money growth rate versus the average inflation rate for the countries in the sample. The averages are taken over the period 1960-2005. The line in the plot is the 45° line. (i)
What would this line look like, according to the quantity equation of money, if both the average growth rate of output was zero as well as velocity constant?
(ii)
Do we observe this in the data? What is the difference between the observed relationship between money growth and inflation in the data with that described in part (i)?
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(iii)
For what type of country is the growth rate of money a better predictor of inflation, for countries with a high or a low money growth rate? Can you use equation (7.23) to explain why?
(iv)
The Excel file contains a second figure, besides figure 7.1. This figure plots average money growth versus inflation for the low money growth countries. Why do most of these observations lie below the 45° line?
(v)
What is true for these countries in terms of their variations in output growth rates versus their variations in money growth rates? If you want, you can look up the output growth rates in the Penn World Tables, discussed in Chapter 5.
The conclusion of the above exercise is thus that for countries with high money growth and inflation, reducing the money growth rate will most likely temper inflation. However, for many advanced industrialized countries the relationship between money growth and inflation is not as clear-cut. Reverse causation
Some evidence, including that in McCandless and Weber (1995), seems to suggest that for the advanced industrialized countries there is a positive correlation between output growth and money growth. Does this mean that putting on the money presses would increase growth in these economies? Probably not. It is important to bear in mind that statistical correlations do not tell us anything about causation. In fact, in this case it is most likely that countries that had higher output growth had higher money growth to accommodate it, rather than cause it. This is often referred to as the reverse causation of money growth on output growth. Reverse causation is an econometric term for the case in which the causation goes from the dependent variable to the explanatory variables, rather than the other way around as is assumed in the regression model. We ended up looking at monetarism and its implications because monetarists assume that velocity is relatively constant, which is also true in the simple cash-in-advance model that we have studied so far. Is it actually true that velocity is relatively constant or are there economic variables with which velocity is actually correlated? The following exercise is intended to make you address this question.
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Exercise 7.3: Is U.S. velocity constant?
This exercise uses the Excel file chapter7.xls for its calculations. Upload this file to your computer and open it in Excel. Click on the worksheet called ‘U.S. velocity’. It both contains the raw data used in the exercise as well as a set of three figures. Each of these figures contains a measure of velocity in the United States. The measures of velocity depend on which assets they include in the measure of money in the denominator, i.e. the monetary aggregate. These aggregates are numbered M1 through M3 and we will look at them in a lot more detail in section 7.4. Each of these charts plots a particular measure of velocity versus the nominal Federal Funds Rate, which is the interest rate at which a bank lends immediately available funds (balances at the Federal Reserve) to another bank overnight.
Federal Funds Rate
(i)
Is velocity constant in these charts? If so, why? If not, why not?
(ii)
Does velocity depend on the nominal interest rate, as reflected in the Federal Funds rate? If so, do these two variables tend to have a positive or a negative correlation?
(iii)
What do you think underlies the persistent shifts between velocity and the nominal interest rate that we observe in the data, especially for M1?
(iv)
Make a plot of V1 over time. How does the behavior of V1 compare with the deviation of output from trend that we considered in Chapter 6?
In the above exercise, the nominal Federal Funds Rate is used as an approximate measure, also known as a ‘proxy’, of the nominal interest rate that the households can get over their non-monetary assets. As the previous exercise suggests, velocity tends to be increasing in the nominal interest rate and is definitely not constant. In order to illustrate what drives this variation in velocity, we develop a two good version of the simple cash-in-advance model that we considered above. There is actually a second reason that we consider this two good model. This model allows us to consider CES preferences that we will use much more extensively in the next chapter. In this sense, this two good model is a nice warm-up for some of the mathematical derivations that are to come.
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We first consider the two goods model without the cash-in-advance constraint. The two reasons that we do this is that this familiarizes us with the CES preferences that we will use more intensively later and, more importantly, that it provides us with an important benchmark case for comparison of the model with the CIA constraint. After this derivation, we then add the cash-in-advance constraint to the model and derive what it implies for relationship between the nominal interest rate and moneyholdings/velocity. In the simple dynamic utility maximization problem that we considered for the household, the assumption was that the household consumed a ‘representative’ consumption good. We denoted the amount of this good consumed by the household as Ct. In this two good version of the model we assume that Ct itself is actually a utility composite, determined by the consumption levels of two distinct goods, C1t and C2t. CES preferences
The amount of consumption-utility derived from the consumption of the combination of C1t and C2t is given by
[
Ct = C1φt + C 2φt
]
1/ φ
(7.27)
This mapping of Ct as a function of C1t and C2t is known as CES preferences and is a functional form that is fequently used in both micro- as well as macroeconomic theory. Given this two good setup of the utility function, the household decides on the path of
{A
t+s
, C t + s , C1,t + s , C 2,t + s , Lt + s }s = 0 ∞
(7.28)
that maximizes the present discounted value of its stream of utility, i.e. ∞
∑ β U (C s
t+s
, Lt + s )
(7.29)
s =0
subject to the budget constraint
At = (1 + rt )At −1 + Wt Lt − P1t C1t − P2t C 2t
(7.30)
Here, P1r and P2t are the prices of goods 1 and 2 respectively. Note that for this version of the two-good model without the CIA constraint we have left money out of the budget constraint for simplicity. The second constraint that the household faces is that the consumption composite Ct depends on C1t and C2t according to (7.27). We will again solve this dynamic utility maximization using the four step approach that we have used throughout.
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In order to identify the costs and benefits of the household’s savings and labor supply decision, we write out the first two terms of the household’s present discounted value of its stream of utility separately from the rest. Before doing so, we substitute in the restrictions. Step 1: Identify the costs and benefits
First of all, from the budget constraint, we know that
C1t =
1 [(1 + rt )At −1 + Wt Lt − At − P2t C 2t ] P1t
(7.31)
Substituting this into utility composite for Ct , we obtain that φ ⎡⎧ 1 ⎤ ⎫ Ct = ⎢⎨ [(1 + rt )At −1 + Wt Lt − At − P2t C 2t ]⎬ + C 2φt ⎥ ⎢⎣⎩ P1t ⎥⎦ ⎭
1/ φ
(7.32)
When we write out the first two terms of the objective function, this yields ⎛⎡ ⎜ ⎧⎪ 1 U ⎜ ⎢⎨ ⎜⎜ ⎢⎪⎩ P1t ⎝ ⎣⎢
φ ⎤ ⎡ ⎤ ⎫⎪ φ ⎥ ⎢(1 + rt )At −1 + Wt Lt − At − P2t C 2t ⎥ ⎬ + C 2t { { 123 ⎥ {⎥ ⎢⎣ BL CA BC 2 CC 2 ⎦ ⎪ ⎭ ⎥⎦
⎛⎡ ⎜ ⎧⎪ 1 β U ⎜ ⎢⎨ ⎜⎜ ⎢⎪⎩ P1t +1 ⎝ ⎢⎣ s
s=2
⎞ ⎟ , Lt ⎟ + {⎟ CL ⎟ ⎠
φ ⎤ ⎡ ⎤ ⎫⎪ φ ⎥ ⎢(1 + rt +1 )At + Wt +1 Lt +1 − At +1 − P2t +1C 2t +1 ⎥ ⎬ + C 2t +1 43 ⎥ ⎢⎣ 142 ⎥⎦ ⎪⎭ BA ⎥⎦
∞
∑ β U (C
1/ φ
t+s
1/φ
⎞ ⎟ , Lt +1 ⎟ + ⎟⎟ ⎠
(7.33)
, Lt + s )
In this equation, the costs and benefits of the three decisions the household makes are already indicated. Let’s start by considering the costs and benefits of the labor supply decision. These are exactly the same as in the one good model and are indicated by CL and BL, respectively. The costs and benefits of savings are also similar to the one good model and indicated by CA and BA. Finally, there are the costs and benefits of consuming the second good. The costs are that spending more on the second good would reduce the amount spend on and bought of the first good, this is denoted by the CC2 term in the equation above. The benefits are that consuming more of the second good increases the consumption composite Ct. This happens through the BC2 term. Because we have rewritten the objective such that labor and capital income affect consumption of good 1 through the budget constraint, the optimality conditions for the labor supply and savings decisions are the same as in the one good model, with the exception that the marginal utility of consumption is now written in terms of the marginal utility of consumption of good 1.
Step 2: Calculate the marginal costs and marginal benefits
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That is for the optimal labor supply decision, we find that the marginal benefits of supplying another hour of labor are represented by the marginal utility value of an additional unit of consumption of good 1, times the number of units of good 1 the household can buy with the wages earned during this additional hour of work. Mathematically, this can be written as Marginal costs and marginal benefits of the labor supply
⎡ ∂U ∂Ct ⎤ ∂C1t ⎡ ∂U ∂Ct ⎤ Wt =⎢ ⎢ ⎥ × ⎥ ∂ ∂ ∂ C C L t 1 t t ⎣14243⎦ ⎣ ∂Ct ∂C1t ⎦ P1t
(7.34)
marginal utility of consumption of good 1
The marginal costs of supplying more labor are the same as in the one good model. They are simply composed of the reduction in utility that is caused by the reduction in leisure that this additional hour of work induces. That is, mathematically
−
∂U ∂Lt
(7.35)
Marginal costs and marginal benefits of the savings decision The marginal benefits
of the savings decision are made up of the additional amount of consumption of good 1 that can be financed out of the gross interest income on an additional dollar of savings during the next period. That is, these marginal benefits equal
β
∂U ∂Ct +1 ∂C1t +1 ⎡ ∂U ∂Ct +1 ⎤ (1 + rt +1 ) × =⎢ ⎥ ∂Ct +1 ∂C1t +1 ∂At ⎣ ∂Ct +1 ∂C1t +1 ⎦ P1t +1
(7.36)
The marginal costs of saving consist of the current utility value that is foregone because of the sacrifice of a dollar of consumption expenditure on good 1. This equals −
∂U ∂Ct ∂C1t ⎡ ∂U ∂Ct ⎤ 1 × =⎢ ⎥ ∂Ct ∂C1t ∂At ⎣ ∂Ct ∂C1t ⎦ P1t
(7.37)
Again, these costs and benefits are very similar to the one good model, except for the fact that marginal utility is now measured in terms of good 1, rather than the consumption composite. The optimal labor supply and savings conditions above basically do not contain any terms that relate to the level of consumption of good 2. The consumption of good 2 comes back into the problem through the optimality condition of consuming good 2. Essentially, for each dollar the household spends on consumption, it will spend it on the good that yields the highest marginal utility for this dollar. The optimal spending decision is such that the marginal utility values of these two options are the same. Marginal costs and marginal benefits of consuming good 2
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A N D
S U P P L Y
Formally, the marginal benefit of spending a dollar more on good 2 equals the marginal utility of consumption for good 2 times the amount of good 2 that can be bought with this dollar of additional expenditures. It turns out to be convenient, however, to not write the optimality conditions in terms of nominal spending but rather in terms of additional units of consumption of good 2. In that case, the marginal benefit of consuming a unit of good 2 more is simply its instantaneous marginal utility, which is given by
∂U ∂Ct ∂Ct ∂C 2t
(7.38)
On the other hand, the costs of consuming an addition unit of good 2 consist of the utility sacrificed by not using the spending on this item on good 1. Mathematically, this marginal cost equals
∂U ∂Ct × ∂Ct ∂C1t
∂C1t ∂C 2t {
=
∂U ∂Ct P2,t ∂Ct ∂C1t P1,t
(7.39)
the number of goods of type 1 that can be bought for each good of type 2. This is the relative price.
which completes the calculation of the marginal costs and benefits in this setup. Step 3: Equate marginal costs and marginal benefits Equating the marginal benefits
and costs of the labor supply, we find that the optimal labor supply decision is such that ∂U ∂Ct
⎡ ∂Ct 1 ⎤ ∂U ⎢ ⎥Wt = − ∂Lt ⎣ ∂C1t P1t ⎦
(7.40)
For the optimal savings decision we find, when we equate marginal costs and benefits, that
β
∂U ⎡ ∂Ct +1 1 ⎤ ∂U ⎢ ⎥ (1 + rt +1 ) = ∂Ct +1 ⎣ ∂C1t +1 P1t +1 ⎦ ∂Ct
⎡ ∂Ct 1 ⎤ ⎢ ⎥ ⎣ ∂C1t P1t ⎦
(7.41)
Finally, for the optimal consumption decision we obtain the marginal utility obtained from an extra dollar spent on good 1 equals that of a dollar spent on good 2. That is,
∂U ∂Ct
⎡ ∂Ct 1 ⎤ ∂U ⎢ ⎥= ⎣⎢ ∂C1t P1,t ⎦⎥ ∂Ct
⎡ ∂Ct 1 ⎤ ⎢ ⎥ ⎣⎢ ∂C 2t P2,t ⎦⎥
(7.42)
Our main focus here will be on solving the optimality condition for consumption, i.e. (7.42). From this equation we obtain that, Step 4: Solve the optimality conditions
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when we divide both sides by the marginal utility of the consumption composite, Ct, that
⎡ ∂Ct 1 ⎤ ⎡ ∂Ct 1 ⎤ 1 ⎢ ⎥=⎢ ⎥= ⎢⎣ ∂C1t P1,t ⎥⎦ ⎢⎣ ∂C 2t P2,t ⎥⎦ Pt
(7.43)
where we have just defined the variable Pt to satisfy this equation in the optimum. We discuss the interpretation of Pt in a lot more detail below. Using the above equation, we find that
⎛ Ct ⎜⎜ ⎝ Cit
⎞ ⎟⎟ ⎠
1−φ
=
∂Ct Pit for i=1,2 = ∂Cit Pt
(7.44)
Because Ct is homogenous of degree one in C1t and C2t, we can apply Euler’s theorem and use the fact that
Ct =
∂C ∂C C1t + C 2t ∂C 2t ∂C1t
(7.45)
Combining the above two equations, we find that
Ct =
P P ∂C 1 ∂C C1t + C 2t = 1t C1t + 2t C 2t = [P1t C1t + P2t C 2t ] ∂C 2t ∂C1t Pt Pt Pt
(7.46)
Hence, for the optimal consumption choice we obtain that C1t and C2t are chosen such that the demand functions are
⎛P Cit = ⎜⎜ t ⎝ Pit
1
⎞ 1−φ ⎟⎟ Ct for i=1,2 ⎠
(7.47)
and
Pt Ct = P1t C1t + P2t C 2t
(7.48)
Filling in the demand functions into the CES preferences, i.e. (7.27), we obtain that
[
φ
]
φ 1/ φ
Ct = C1t + C 2t
φ
= Ct Pt
1−φ
⎡ ⎢⎛⎜ 1 ⎢⎜⎝ P1t ⎢⎣
When we solve this for Pt, we obtain that
169
φ φ ⎤ ⎞ 1−φ ⎛ 1 ⎞ 1−φ ⎥ ⎟⎟ + ⎜⎜ ⎟⎟ ⎠ ⎝ P2t ⎠ ⎥⎥ ⎦
(7.49)
M O N E Y
D E M A N D
A N D
S U P P L Y
⎡ ⎛ 1 Pt = ⎢⎜⎜ ⎢⎝ P1t ⎢⎣
⎞ ⎟⎟ ⎠
φ 1−φ
⎛ 1 + ⎜⎜ ⎝ P2t
⎞ ⎟⎟ ⎠
φ 1−φ
⎤ ⎥ ⎥ ⎦⎥
−
1−φ
φ
(7.50)
Hence, if at prices P1t and P2t, the household’s consumption expenditures are PtCt, then it obtains a value of the consumption aggregate equal to Ct. Therefore, Pt can be interpreted as the price of the consumption aggregate. By using this notation, we can actually rewrite this two good model in the form of a one good model. That is, because this representation of the solution to the consumption decision problem allows us to rewrite the household’s problem as it maximizing ∞
∑ β U (C s
s =0
t+s
, Lt + s )
(7.51)
subject to the budget constraint
At = (1 + rt )At −1 + Wt Lt − Pt Ct
(7.52)
which is identical to the problem that we solved in Chapter 2. So, does our solution to the two-good problem then also yield the same optimality conditions for the labor supply and savings decisions? Yes it does. This can be easiest seen when one realizes that the solution to the consumption choice problem implies that
⎡ ∂Ct 1 ⎤ 1 ⎢ ⎥= ⎢⎣ ∂C1t P1,t ⎥⎦ Pt
(7.53)
When we combine this result with the optimality conditions (7.40) and (7.41) we obtain that it implies that ∂U ⎡ ∂Ct 1 ⎤ ∂U Wt ∂U ~ ∂U = Wt = − ⎢ ⎥Wt = ∂Ct ⎣ ∂C1t P1t ⎦ ∂Ct Pt ∂Ct ∂Lt
(7.54)
which is the standard labor supply decision we derived in chapter 2. For the optimal savings decision we find that substituting the result that
β
∂U 1 (1 + rt +1 ) = ∂U ∂Ct +1 Pt +1 ∂Ct
⎡1⎤ ⎢ ⎥ ⎣ Pt ⎦
which yields the Euler equation from Chapter 2, because it implies that
170
(7.55)
M O N E Y
D E M A N D
A N D
S U P P L Y
∂U Pt ∂U (1 + rt +1 ) = β ∂U (1 + ~rt +1 ) =β ∂Ct ∂Ct +1 Pt +1 ∂Ct +1
(7.56)
The bottom line is that in this case the two good model actually turns out to boil down to the one good version of the household’s dynamic utility maximization problem. We will use this property of the CES preferences again in the next chapter. Because the CES functional form is commonly used in macroeconomic theory, it is worthwhile to have another close look at it. Exercise 7.4: CES in a bit more detail
In this exercise, we will look at a few more details of the derivation of the consumption demand decision for the two good model above. First of all, CES is an acronym for Constant Elasticity of Substitution. The elasticity of substitution is defined as the minus of the percentage change in the relative demand for good 1, i.e. the percentage change of C1t/C2t, due to a percentage change in the relative price of good 1. Mathematically, this definition translates to
elasticity of substitution = −
∂ ln (C1t C 2t ) ∂ ln(P1t P2t )
(7.57)
The elasticity of substitution is interesting because it measures how quickly demand shifts away from goods whose relative price increases. (i)
Use the demand functions, (7.47), to show that the elasticity of substitution for the CES preferences equals 1/(1-φ).
We could have solved the two good model above in a different way, which is the way it is often solved in academic research. We could first solve what is known as the intratemporal consumption problem. That is, we could first ask ourselves, if the household decides to spend a certain amount, say Et, on consumption in period t. What is the maximum amount of the consumption aggregate Ct that is attainable for the household at prices P1t and P2t at this expenditure level? Secondly, if we define the price level as Pt=Et/Ct, we can then replace the two goods and prices by Ct and Pt into the problem. This will then yield a one good version of the dynamic problem that is easily solved.
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M O N E Y
(ii)
Cash good Credit good
D E M A N D
A N D
S U P P L Y
Consider a household that chooses C1t and C2t to maximize Ct, which depends on C1t and C2t according to (7.27), subject to the budget constraint Et=P1tC1t+ P2tC2t. Show that the solution to this problem yields that Et/Ct=Pt, where Pt is as defined in (7.50).
So, we managed to solve the two-good model in the case the household is not holding any money and does not face a cash-in-advance constraint. Now it is time to reintroduce money and the cash-in-advance constraint. The twist in this two good model is that cash-in-advance is not needed for all consumption goods. Instead , we will assume it is only needed to buy good 2. In this, and similar, model setups good 1 is often called a credit good, because the household can use its non-money holdings to buy it, while good 2 is a cash good11. Thus, the mathematical version of the cash-in-advance constraint in this two good model is
P2t C 2t ≤ M t −1
(7.58)
The budget constraint in this case reads
At + M t = (1 + rt )At −1 + M t −1 + Wt Lt − P1t C1t − P2t C 2t
(7.59)
Again, the household maximizes the present discounted value of its stream of utility, i.e. ∞
∑ β U (C s
s =0
t+s
, Lt + s )
(7.60)
Now, however, this objective is optimized subject to the above two constraints, as well as the definition of the consumption composite, i.e. (7.27). Just like in the one good cash-in-advance model, the costs and benefits of each of the choices are easiest identified when the first terms of the summation of the objective function are written out explicitly. Step 1: Identify the costs and benefits
Before we do so, we first substitute the budget constraint, (7.59), and the CES preferences, (7.27), into the utility function. Doing so yields that φ ⎤ ⎡⎧ 1 ⎫ Ct = ⎢⎨ [(1 + rt )At −1 + Wt Lt + M t −1 − At − M t − P2t C 2t ]⎬ + C 2φt ⎥ ⎥⎦ ⎢⎣⎩ P1t ⎭
11
This model is due to Lucas and Stokey (1987).
172
1/ φ
(7.61)
M O N E Y
D E M A N D
A N D
S U P P L Y
However, just like in the one-good cash-in-advance model, we know that the sole reason for the household to hold money is the CIA constraint. Hence, at the end of period t, the household will not hold more money than P2t+1C2t+1. That is, in each period the CIA constraint is binding such that Mt-1= P2tC2t. Substituting this into the above expression eliminates moneyholdings from the household’s problem and results in φ ⎤ ⎡⎧ 1 ⎫ C t = ⎢⎨ [(1 + rt )At −1 + Wt Lt − At − P2t +1C 2t +1 ]⎬ + C 2φt ⎥ ⎥⎦ ⎢⎣⎩ P1t ⎭
1/φ
(7.62)
This enables us to write out the first two terms from the objective function, such that the objective reads ⎛⎡ ⎜ ⎧⎪ 1 U ⎜ ⎢⎨ ⎜⎜ ⎢⎪⎩ P1t ⎝ ⎢⎣
φ ⎤ ⎡ ⎤ ⎫⎪ φ ⎥ ⎢(1 + rt )At −1 + Wt Lt − At − P2t +1C 2t +1 ⎥ ⎬ + C 2t { { 1 424 3⎥ ⎥ ⎢⎣ BL CA CC 2 ⎦ ⎪⎭ ⎥⎦
⎛⎡ ⎜ ⎧⎪ 1 β U ⎜ ⎢⎨ ⎜⎜ ⎢⎪⎩ P1t +1 ⎝ ⎣⎢ s
s=2
⎞ ⎟ , Lt ⎟ + {⎟ CL ⎟ ⎠
φ ⎤ ⎡ ⎤ ⎫⎪ ⎢(1 + rt +1 )At + Wt +1 Lt +1 − At +1 − P2t + 2 C 2t + 2 ⎥ ⎬ + C 2φt +1 ⎥ 43 ⎥ ⎢⎣ 142 ⎥⎦ ⎪⎭ { BC 2 BA ⎥⎦
∞
∑ β U (C
1/ φ
t+s
1/φ
⎞ ⎟ , Lt +1 ⎟ + ⎟⎟ ⎠
(7.63)
, Lt + s )
Just like in the two good model without a CIA constraint that we dealt with before, the costs and benefits are identified already in this expression. The costs and benefits of the labor supply and the savings decisions are basically identical to those in the standard two good model. The major difference is in the costs and the benefits of consuming good 2. This is because the CIA constraint implies that expenditures on good 2 in period t already have to be set aside (in the form of moneyholdings) in period t-1. Therefore, the costs of consuming good 2, denoted by CC2, show up in the first term, while the benefits, denoted by BC2, show up one period later. Because the costs and benefits of the labor supply decision as well as the savings decision are essentially identical in this model to the one without a cash-in-advance constraint, the marginal costs and benefits of these two decisions are as well. Step 2: Calculate the marginal costs and marginal benefits
Thus, the marginal benefits of the labor supply equal
173
M O N E Y
D E M A N D
A N D
S U P P L Y
⎡ ∂U ∂Ct ⎤ Wt ⎢ ⎥ ⎣ ∂Ct ∂C1t ⎦ P1t
(7.64)
while the marginal costs are
−
∂U ∂Lt
(7.65)
The marginal benefits of saving are ⎡ ∂U ∂Ct +1 ⎤ (1 + rt +1 ) ⎢ ⎥ ⎣ ∂Ct +1 ∂C1t +1 ⎦ P1t +1
(7.66)
while the marginal costs equal ⎡ ∂U ∂Ct ⎤ 1 ⎢ ⎥ ⎣ ∂Ct ∂C1t ⎦ P1t
(7.67)
The difference between the optimality conditions of this model and those of the two-good model without a CIA constraint are all due to the differences in the marginal benefits and costs of consuming good 2. Marginal costs and marginal benefits of consuming good 2
For this purpose, we consider the marginal costs and benefits of choosing C2t+1. Remember that, at time t, C2t is already implicitly chosen through the choice of money balances in the previous period. The marginal costs of consuming an additional unit good 2 in period t+1 consist of the sacrificed units of good 1 in period t. This can be written mathematically as
∂U ∂Ct P2t +1 ∂U ∂Ct P2t +1 P2t ∂U ∂Ct ∂C1t = = ∂Ct ∂C1t ∂C 2t +1 ∂Ct ∂C1t P1t ∂Ct ∂C1t P2t P1t
(7.68)
The marginal benefits of an additional unit of consumption of good 2 in period t+1 equal the present discounted marginal utility of good 2 in that period. This equals
β
∂U ∂Ct +1 ∂Ct +1 ∂C 2t +1
(7.69)
Step 3: Equate marginal costs and marginal benefits Equating the marginal benefits
and costs of the labor supply, we find again that the optimal labor supply decision is such that
174
M O N E Y
D E M A N D
A N D
S U P P L Y
∂U ∂Ct
⎡ ∂Ct 1 ⎤ ∂U ⎢ ⎥Wt = − ∂Lt ⎣ ∂C1t P1t ⎦
(7.70)
For the optimal savings decision we find again, when we equate marginal costs and benefits, that
β
∂U ⎡ ∂Ct +1 1 ⎤ ∂U ⎡ ∂Ct 1 ⎤ ⎢ ⎥ (1 + rt +1 ) = ⎢ ⎥ ∂Ct +1 ⎣ ∂C1t +1 P1t +1 ⎦ ∂Ct ⎣ ∂C1t P1t ⎦
(7.71)
However, the optimal consumption condition is changed by the CIA constraint. It now reads
∂U ∂Ct 1 ∂U ∂Ct +1 1 =β ∂Ct ∂C1t P1t ∂Ct +1 ∂C 2t +1 P2t +1
(7.72)
Again, we mainly focus on the optimal consumption decision problem. When we combine the last two optimality conditions, we obtain that
Step 4: Solve the optimality conditions
∂U ∂Ct 1 + rt ∂U ∂Ct 1 ∂U ∂Ct −1 1 =β =β ∂Ct ∂C 2t P2t ∂Ct −1 ∂C1t −1 P1t −1 ∂Ct ∂C1t P1t
(7.73)
This implies that
∂Ct 1 ∂Ct 1 1 = * = ∂C1t P1t ∂C 2t (1 + rt )P2t Pt
(7.74)
Hence, what is happening here is that consuming a unit of good 2 means having to hold money for one period and foregoing the interest income over that money. This is the reason that the price of good 2 in this equation is multiplied by the gross nominal interest rate. The above conditions can be rewritten into the demand functions 1
⎛ ⎛ P * ⎞ 1−φ Pt* C1t = ⎜⎜ t ⎟⎟ Ct and C 2t = ⎜⎜ ⎝ P1t ⎠ ⎝ (1 + rt )P2t
1
⎞ 1−φ ⎟⎟ Ct ⎠
(7.75)
Similar to the two good model without the CIA constraint, we can apply Euler’s theorem to the CES preferences and solve for Pt*. Doing so, we get ⎡ ⎛ 1 Pt = ⎢⎜⎜ ⎢⎝ P1t ⎢⎣ *
⎞ ⎟⎟ ⎠
φ 1−φ
⎛ 1 + ⎜⎜ ⎝ (1 + rt )P2t
175
⎞ ⎟⎟ ⎠
φ 1−φ
⎤ ⎥ ⎥ ⎥⎦
−
1−φ
φ
(7.76)
M O N E Y
D E M A N D
A N D
S U P P L Y
Furthermore, using the definition of Pt* we can rewrite the optimality conditions for the labor supply and savings decisions as ∂U ⎡ ∂Ct 1 ⎤ ∂U Wt ∂U =− ⎢ ⎥Wt = * ∂Ct ⎣ ∂C1t P1t ⎦ ∂Ct Pt ∂Lt
(7.77)
∂U ∂U ⎡ Pt* ⎤ =β ⎢ ⎥ (1 + rt +1 ) ∂Ct ∂Ct +1 ⎣ Pt*+1 ⎦
(7.78)
and
In many respects, the variable Pt* acts as the price level in this economy. However, it is not a proper price level because it also contains an adjustment factor, in terms of the nominal interest rate, which reflects the cost of the cash-in-advance constraint. So, how does velocity in this economy depend on the nominal interest rate? Let’s first do the math and then consider the economic intuition. In this economy, because of the cash-in-advance constraint, let’s define velocity as the ratio of current consumption expenditures over the money balances at the end of last period. If we denote velocity by Vt, then this definition turns into the following equation
Vt =
P1t C1t + P2t C 2t P1t C1t + P2t C 2t = M t −1 P2t C 2t
(7.79)
Where the last part follows from the fact that the cash-in-advance constraint is binding. Essentially, velocity here equals the inverse of the share of cash-goods in total consumption expenditures. Solving this, using the demand functions, we can write
Vt = 1 + (1 + rt )
1 1−θ
⎛ P2t ⎜⎜ ⎝ P1t
θ
⎞ 1−θ ⎟⎟ ⎠
(7.80)
Hence, because of the CIA constraint, an increase in the nominal interest causes an increase in the relative price of the cash-good relative to the credit good. Consequently, if the elasticity of substitution between the two goods is large enough, i.e. θ1/2 a decrease in consumption can only be compensated for by a bigger increase in leisure than in the case where θ=1/2. Therefore, at any point in the consumption/leisure space, the indifference curve for θ >1/2 must be less downward sloping than that for θ=1/2. The reverse applies for θ0 instead of the original input levels Kt and Lt, then the total output level would be Yt′ = Z t (λK t ) (λLt ) α
1−α
= λYt
(A.81)
Hence, changing the inputs of the factors of production, capital and labor, by the same factor λ changes the level of output by the same factor. (xxxvii) If the firm would choose input levels λKt and λLt for λ>0 instead of the original input levels Kt and Lt, then total profits would equal
Π ′t = Pt λYt − Wt λLt − Rt λK t = λ [Pt Yt − Wt Lt − Rt K t ] = λYt
(A.82)
Hence, when all inputs are changed by a factor λ, then profits also change by a factor λ. (xxxviii) that
Filling in the production function in the profit function, we obtain
Π t = Pt Z t K tα L1t−α − Wt Lt − Rt K t
(A.83)
Dividing both sides of this equation by Lt yields the level of profits per worker, which is
⎛K Πt = Pt Z t ⎜⎜ t Lt ⎝ Lt
α
⎞ ⎛K ⎟⎟ − Wt − Rt ⎜⎜ t ⎠ ⎝ Lt
which is what we are asked to derive.
293
⎞ ⎟⎟ ⎠
(A.84)
S O L U T I O N S
(xxxix) If the firm could choose a capital labor ratio such that it would make positive profits per worker, then it would like to hire as many workers as possible, i.e. an infinite number of them essentially. This is because for each worker it makes a positive profit. Hence, the more workers the firm hires the higher its profit level. Because of this the demand for workers will raise the wage rate in this case. (xl)
If the wage rate is so high that the best the firm can do is make negative profits per worker, then the firm will simply decide not to hire any workers and not to produce. Because of this, the lack of demand for workers will lead to a lower wage rate.
Answer to Exercise 3.5: Estimating the output elasticity of capital
(xli)
In part (i) exercise 3.3, we derived that, no matter what the levels of the wage rate, Wt, prices Pt, and the rental price of capital, Rt, the firm that uses a Cobb-Douglas technology will always choose its labor input such that
Wt Lt = (1 − α ) Pt Yt
(A.85)
(xlii)
The figure suggests that the average labor share, WtLt/PtYt, for the U.S. during the postwar period is about 0.7. Furthermore, consistently with the Cobb-Douglas technology, it seems to be almost constant over time.
(xliii)
Combining parts (i) and (ii), we obtain that they imply that (1-α)≈0.7. This implies that the output elasticity of capital in the U.S., α, is approximately 0.3.
Answer to Exercise 3.6: Capital stock and past investment levels
When we decrease all indices in the capital accumulation equation by one period, then we obtain, through recursive substitution, that K t = (1 − δ )K t −1 + I t −1
= (1 − δ )[(1 − δ )K t − 2 + I t − 2 ] + I t −1 = I t −1 + (1 − δ )I t − 2 + (1 − δ ) K t − 2 2
=K k
= ∑ (1 − δ ) I t − s + (1 − δ ) K t − k s −1
k
s =1
Now, let k→∞ such that (1-δ)k→0, then we are left with
294
(A.86)
S O L U T I O N S
∞
K t = ∑ (1 − δ ) I t − s s −1
(A.87)
s =1
which is what we are supposed to show. Answer to Exercise 3.7: Optimal capital stock, Kt
(xliv)
Since the equation has to hold at any time t, we can reduce the indices by one. This yields
Pt MPK t = (1 + rt )qt −1 − (1 − δ )qt
(A.88)
This equation can be rewritten as MPK t =
(xlv)
⎡ ⎤ qt [(1 + rt )qt −1 − (1 − δ )qt ] = qt ⎢(1 + rt ) qt −1 − (1 − δ )⎥ Pt Pt ⎣ qt ⎦
(A.89)
This result follows from the fact that ⎡q − q
⎤
= t −1 π tq = ⎢ t t −1 ⎥ ⇒ (1 + π tq ) = t ⇒ q qt −1 (1 + π t ) qt ⎣ qt −1 ⎦ q
q
1
(A.90)
such that MPK t =
(xlvi)
qt Pt
⎤ ⎡ ⎤ q ⎡ (1 + rt ) qt −1 − (1 − δ )⎥ = t ⎢ − (1 − δ )⎥ ⎢(1 + rt ) q qt ⎣ ⎦ Pt ⎣ 1 + π t ⎦
(
)
(A.91)
We already derived this in the answer to part (ii) of exercise 3.3
(xlvii) Combining the above results, we find that
⎛K αZ t ⎜⎜ t ⎝ Lt
⎞ ⎟⎟ ⎠
α −1
=
⎤ qt ⎡ (1 + rt ) − (1 − δ )⎥ ⎢ q Pt ⎣ 1 + π t ⎦
(
Such that
295
)
(A.92)
S O L U T I O N S
⎛ Kt ⎜⎜ ⎝ Lt
⎞ ⎟⎟ ⎠
⎛ Kt ⎜⎜ ⎝ Lt
⎞ ⎟⎟ ⎠
α −1
1−α
⎤ qt ⎡ (1 + rt ) − (1 − δ )⎥ ⎢ q αZ t Pt ⎣ 1 + π t ⎦ ⎧ ⎫ ⎪ ⎪ αZ t Pt ⎪ ⎪ =⎨ ⎬ ⎪ q ⎡ (1 + rt ) − (1 − δ )⎤ ⎪ ⎥⎪ ⎪ t ⎢⎣ 1 + π tq ⎦⎭ ⎩
=
(
)
(
(A.93)
)
⎧ ⎫ ⎪ ⎪ ⎛ Kt ⎞ ⎪ αZ t Pt ⎪ ⎜⎜ ⎟⎟ = ⎨ ⎬ ⎝ Lt ⎠ ⎪ q ⎡ (1 + rt ) − (1 − δ )⎤ ⎪ ⎥⎪ ⎪ t ⎢⎣ 1 + π tq ⎦⎭ ⎩
(
1 1−α
)
(xlviii) The optimal capital labor ratio is increasing in productivity, i.e. Zt. The reason for this is that the higher Zt, the higher the marginal product of capital and thus the higher the return to investing. If δ increases, this is implicitly an increase in the cost of investment and thus will lead to a lower capital labor ratio. Same thing for rt. For πqt it is the opposite. The higher πqt, the higher the resale value of the capital goods and thus the lower the usercost of capital and the higher the capital labor ratio. Answer to Exercise 3.8: Dropping time to build
(xlix)
Under this alternative capital accumulation equation, the firm chooses the paths of its capital and labor inputs, as well as the implied paths of output ∞ and investment, i.e. {K t + s , I t + s , Lt + s , Yt + s }s =0 , to maximize the present discounted value of its profits ∞ ⎡ s ⎛ 1 Π t + ∑ ⎢∏ ⎜ ⎜ s =1 ⎢ ⎣ j =1 ⎝ 1 + rt + j
⎞⎤ ⎟⎥ Π t + s ⎟⎥ ⎠⎦
(A.94)
subject to the definition of profits
Π t + s = Pt + sYt + s − Wt + s Lt + s − qt + s I t + s
(A.95)
the production function
Yt + s = Z t + s K tα+ s L1t−+αs and the capital accumulation equation
296
(A.96)
S O L U T I O N S
K t + s = (1 − δ )K t + s −1 + I t + s
(A.97)
Doing so, the firm takes as given the paths of the price level, the nominal interest rate, the investment price, the wage rate and of TFP. (l)
Before we apply the four-step approach, we first substitute the definition of the profit function, the production function and the capital accumulation equation into the objective function. That is,
Π t + s = Pt + s Z t + s K tα+ s L1t−+αs − Wt + s Lt + s − qt + s (K t + s − (1 − δ )K t + s −1 )
(A.98)
We will consider the optimality condition for the choice of Kt. In order to derive it, we will write out the first two terms of the firm’s objective. That is ⎧ ⎛ ⎞⎫ ⎜ ⎟⎪ ⎪ α 1−α − (1 − δ )K t −1 ⎟⎬ Kt Lt − Wt Lt − qt ⎜ Kt ⎨ Pt Z t { { ⎜ reduction ⎟⎪ ⎪ contribution of capital in profits to current revenue ⎝ due to investment ⎠⎭ ⎩ ⎧ ⎛ ⎞⎫ ⎜ ⎟⎪ ⎪ ⎛ 1 ⎞⎪ ⎜ ⎟⎪ α 1−α ⎟⎟⎨ Pt +1 Z t +1 K t +1 Lt +1 − Wt +1 Lt +1 − qt +1 ⎜ K t +1 − (1 − δ )K t ⎟⎬ + ⎜⎜ 1 4 2 4 3 ⎝ 1 + rt +1 ⎠⎪ ⎜ increase in profits ⎟ ⎪ due to the resale ⎟ ⎪ ⎜ ⎪ value ⎝ ⎠⎭ ⎩ ∞ ⎡ s ⎛ 1 ⎞⎟⎤ + ∑ ⎢∏ ⎜ Π ⎜ ⎟⎥ t + s s = 2 ⎢ j =1 ⎝ 1 + rt + j ⎠ ⎥ ⎣ ⎦
(A.99)
Step I: The cost of capital is the reduction in current profits due to investment.
Just like the case in the main text, the benefits of capital consist of two things. The first is the contribution of capital to output and revenue and the second is the resale value of capital. The difference is that, because there is not time to build, the revenue benefit accrues in the period in which the investment takes place, not in the next period. The marginal cost of capital consist of the reduction in profits due to an increase in Kt, i.e. it equals qt. Step II:
The marginal benefit of capital consists of two parts. The first part is the increase in current revenue induced by an additional unit of the capital stock. This is ⎛ ∂Y Pt ⎜⎜ t ⎝ ∂K t
⎞ Y ⎟⎟ = Pt MPK t = αPt t Kt ⎠
297
(A.100)
S O L U T I O N S
where the last part of this equation is obtained by substituting the marginal product of capital for the Cobb-Douglas production function into the equation. The second part of the marginal benefits is the same as in the case in the main text. It is the present discounted value of the resale value of the depreciated marginal unit of capital. That is ⎛ 1− δ ⎞ ⎜⎜ ⎟⎟qt +1 ⎝ 1 + rt +1 ⎠
(A.101)
Equating the marginal costs with the sum of the two parts of the marginal benefits, we obtain the optimal capital demand condition. This reads
Step III:
⎛ 1− δ ⎞ ⎟⎟qt +1 qt = Pt MPK t + ⎜⎜ ⎝ 1 + rt +1 ⎠
(A.102)
Reshuffling the above optimality condition, we obtain an expression for the marginal revenue of capital. This is the alternate version of the user cost equation of capital under this assumption about time to build. It reads Step IV:
⎛ 1− δ ⎞ ⎟⎟qt +1 Pt MPK t = qt − ⎜⎜ r 1 + t +1 ⎠ ⎝
(li)
(A.103)
The difference between this equation and the user cost equation derived in the text is that in this equation the marginal revenue of capital is not discounted. The reason is that in this case the marginal revenue of the investment is obtained instantaneously rather than in the next period.
Answer to Exercise 3.9: Tobin’s Q
(lii)
This is similar to the answer to the exercise about the no time to build assumption. Before we apply the four-step approach, we first substitute the definition of the profit function, the production function and the capital accumulation equation into the objective function. That is,
Π t + s = Pt + s Z t + s K tα+ s L1t−+αs − Wt + s Lt + s − qt + s (K t + s − (1 − δ )K t + s −1 )
(A.104)
We will consider the optimality conditions for the choice of Kt and Lt. In order to derive it, we will write out the first two terms of the firm’s objective. That is
298
S O L U T I O N S
⎧ ⎛ ⎞⎫ ⎜ ⎟⎪ ⎪ 1−α α − Wt − qt ⎜ − (1 − δ )K t −1 ⎟⎬ Kt Lt Lt Kt ⎨ Pt Z t { { { { ⎜ reduction ⎟⎪ ⎪ contribution of capital contribution of labor reduction in profits in profits to current revenue to current revenue due to labor due to investment ⎝ ⎠⎭ ⎩ ⎧ ⎛ ⎞⎫ ⎜ ⎟⎪ ⎪ ⎛ 1 ⎞⎪ ⎜ ⎟⎪ 1−α α ⎟⎟⎨ Pt +1 Z t +1 K t +1 Lt +1 − Wt +1 Lt +1 − q t +1 ⎜ K t +1 − (1 − δ )K t ⎟⎬ + ⎜⎜ 1 4 2 4 3 ⎝ 1 + rt +1 ⎠⎪ ⎜ increase in profits ⎟ ⎪ due to the resale ⎟ ⎪ ⎜ ⎪ value ⎝ ⎠⎭ ⎩ ∞ ⎡ s ⎛ 1 + ∑ ⎢∏ ⎜ ⎜ s = 2 ⎢ j =1 ⎝ 1 + rt + j ⎣
(A.105)
⎞⎤ ⎟⎥ Π t + s ⎟⎥ ⎠⎦
Step I: The cost of capital is the reduction in current profits due to investment.
The benefits of capital consist of two things. The first is the contribution of capital to output and revenue and the second is the resale value of capital. Because there is no time to build, the revenue benefit accrues in the period in which the investment takes place, not in the next period. The costs of labor are is the wage bill paid to the workers. The benefit is labor’s contribution to output and thus revenue. The marginal cost of capital consist of the reduction in profits due to an increase in Kt, i.e. it equals qt. Step II:
The marginal benefit of capital consists of two parts. The first part is the increase in current revenue induced by an additional unit of the capital stock. This is ⎛ ∂Y Pt ⎜⎜ t ⎝ ∂K t
⎞ Y ⎟⎟ = Pt MPK t = αPt t Kt ⎠
(A.106)
where the last part of this equation is obtained by substituting the marginal product of capital for the Cobb-Douglas production function into the equation. The second part of the marginal benefits is the same as in the case in the main text. It is the present discounted value of the resale value of the depreciated marginal unit of capital. That is ⎛ 1− δ ⎞ ⎟⎟qt +1 ⎜⎜ r 1 + t +1 ⎠ ⎝
(A.107)
The marginal costs of labor equal the wage rate Wt and the marginal benefits equal the additional revenue generated by a additional unit of labor, given by
299
S O L U T I O N S
the price level, which the firm takes as given, times the marginal product of labor. That is, ⎛ ∂Y Pt ⎜⎜ t ⎝ ∂Lt
(liii)
⎞ Y ⎟⎟ = Pt MPLt = Pt (1 − α ) t Lt ⎠
(A.108)
We answer this question in two parts. Equating the marginal costs and the marginal benefits for the labor demand decision yields.
Wt = (1 − α )Pt
Yt which implies Wt Lt = (1 − α )Pt Yt Lt
(A.109)
This is the first part of the answer. The second part follows from
⎛K Y W P Wt = (1 − α )Pt t where reshuffling yields t t = Z t ⎜⎜ t (1 − α ) ⎝ Lt Lt
⎞ ⎟⎟ ⎠
α
(A.110)
as part of this derivation, we substituted the production function for Yt. Solving the above equation for Lt, we obtain that
⎡ (1 − α )Z t ⎤ Lt = ⎢ ⎥ ⎣ Wt Pt ⎦
1/ α
(A.111)
Kt
Substituting this result into the production function gives
Yt = Z t K tα L1t−α
⎡ (1 − α )Z t ⎤ = Zt ⎢ ⎥ ⎣ Wt Pt ⎦
1−α
α
Kt
(A.112)
which is what we are asked to show. (liv)
We derived the marginal costs and benefits of the investment decision in part (i). Equating them yields ⎛ 1− δ ⎞ Y ⎛ 1− δ ⎞ ⎟qt +1 ⎟⎟qt +1 = Ptα t + ⎜⎜ qt = Pt MPK t + ⎜⎜ K t ⎝ 1 + rt +1 ⎟⎠ ⎝ 1 + rt +1 ⎠
(113)
Substituting in the result derived in part (ii) gives us what we are asked to derive. That is
300
S O L U T I O N S
⎡ (1 − α )Z t ⎤ Y ⎛ 1− δ ⎞ ⎟⎟qt +1 = PtαZ t ⎢ qt = Pt α t + ⎜⎜ ⎥ K t ⎝ 1 + rt +1 ⎠ ⎣ Wt Pt ⎦
1−α
α
⎛ 1− δ ⎞ ⎟⎟qt +1 + ⎜⎜ ⎝ 1 + rt +1 ⎠
(114)
(lv)
The firm will invest until the investment price, which is the marginal cost of a unit of capital, equals the marginal revenue that this unit generates in the current period plus the discounted resale value of the depreciated unit of capital in the next period.
(lvi)
This is easiest to see by first considering the flow profits that result from the firm’s optimal input decision. That is, Π t = Pt Yt − Wt Lt − qt I t
= Pt Yt − Wt Lt − qt (K t − (1 − δ )K t −1 ) ⎡ Y ⎛ 1− δ ⎞ ⎤ ⎟⎟qt +1 ⎥ K t + (1 − δ )qt K t −1 = Pt Yt − (1 − α )Pt Yt − ⎢ Ptα t + ⎜⎜ K r 1 + t t +1 ⎠ ⎝ ⎦ ⎣
(115)
⎛ 1− δ ⎞ ⎟⎟qt +1 K t = (1 − δ )qt K t −1 − ⎜⎜ r 1 + t +1 ⎠ ⎝
Thus, we can write the value of the firm as ∞ ⎡ s ⎛ 1 Vt = (1 + rt )∑ ⎢∏ ⎜ ⎜ s =0 ⎢ ⎣ j =0 ⎝ 1 + rt + j = (1 − δ )qt K t −1
⎞⎤⎛ ⎞ ⎛ ⎞ ⎟⎥⎜ (1 − δ )qt + s K t + s −1 − ⎜ 1 − δ ⎟qt + s +1 K t + s ⎟ ⎜ ⎟ ⎟ ⎟⎥⎜ ⎝ 1 + rt + s +1 ⎠ ⎠ ⎠⎦⎝
⎡ s +1 ⎛ 1 ⎞⎤ ⎟⎥ ((1 − δ )qt + s +1 K t + s − (1 − δ )qt + s +1 K t + s ) + (1 + rt )∑ ⎢∏ ⎜ ⎟ ⎜ 1 r + s =0 ⎣ = 0 j ⎢ ⎝ t+ j ⎥ ⎦ 44244444444444 144444 444⎠4 3 ∞
(116)
=0
⎡ k ⎛ 1 − lim ⎢∏ ⎜ ⎜ k →∞ ⎣⎢ j =1 ⎝ 1 + rt + j
⎞⎤ ⎟⎥ (1 − δ )qt + k K t + k −1 ⎟⎥ ⎠⎦
The last term in this equation is the limit of the discounted depreciated resale value of capital. This term has to go to zero. If not, then the investment price grows so fast that the resale value of the firm’s capital grows so fast that the firm’s objective function is unbounded. Here (1-δ)qtKt-1 is the replacement value of the capital stock that the firm owns at the beginning of period t. James Tobin originally derived the result above and proposed to consider
301
S O L U T I O N S
Qt =
Vt (1 − δ ) qt Kt −1
(117)
(lvii)
The theory predicts that Qt should be constant over time and equal one. Basically, the only thing that profits reflect in this model is the value of the capital that the firm owns. It is essentially the present discounted value of the stream of revenue that the firm will get back for incurring the fixed cost of buying its capital stock.
(lviii)
No, it is not. It turns out that Qt is not constant over time. In fact, it fluctuates a lot and there are times when it is either persistently lower, as well as persistently higher than one.
(lix)
There are many reasons why Tobin’s Q might deviate from one. Below are a few popular ones: When there is imperfect competition the value of the firm does not only reflect the stream of revenue generated by the firm’s capital stock which it needs to earn back the fixed cost of buying it. Instead, the value of the firm also reflects the profits it makes due to the market power it has.
Market power:
We basically only measure the replacement value of the firm’s physical capital stock. However, firms might own types of capital that are not captured in this measure, like goodwill and intellectual capital. In that case the denominator is underestimating the replacement value of the firm’s capital stock and we observe Qt as higher than one. Unmeasured capital:
Another popular theory is that the usercost equation of capital misrepresents the firm’s optimality condition because the firm faces adjustment costs, beyond qt, to its capital stock. If that is the case, the firm needs to make profits to earn these adjustment costs back and hence Qt would be bigger than one. Adjustment costs to capital:
If the capital stock becomes obsolete more rapidly than reflected by the depreciation rate δ, then the measured replacement value of the capital stock is actually lower than the actual replacement value. In that case our empirical measure of Qt is likely to overestimate the denominator, yielding an observed Qt that is smaller than one. This is argued to have been the case in the 1970’s when the oilcrisis made energy inefficient machinery obsolete much more rapidly than reflected in our measures of depreciation. Time varying depreciation:
302
S O L U T I O N S
Answer to Exercise 3.14: Consumer durables
(lx)
In order to identify the costs and benefits of both non durables and durable goods consumption, we substitute both the law of motion of capital and the budget constraint in the first two terms of the objective function. This yields that
CtD = K tD − (1 − δ )K tD−1
(A.118)
~ ~ ~ CtND = Wt + (1 + ~ rt )At −1 − At − qt (K tD − (1 − δ )K tD−1 )
(A.119)
and
such that the objective function can be rewritten as ⎡
⎞⎞ ⎛ σ ⎢ ⎛⎜ ~ ~ ~ ~ At − qt ⎜ K tD − (1 − δ )K tD−1 ⎟ ⎟ ⎢ Wt + (1 + rt )At −1 − { {
σ − 1 ⎢ ⎜⎝
CND
⎣
⎜ ⎝
⎟⎟ ⎠⎠
CD
σ −1 σ
⎛ ⎞ + ⎜ K tD ⎟ ⎜{⎟ ⎝ BD1 ⎠
σ −1 σ
⎤ ⎥ ⎥ ⎥ ⎦
σ −1 ⎡ σ ⎞ ⎛ ⎞ ⎢ ⎛⎜ ~ ~ ~ At − At +1 − qt +1 ⎜ K tD+1 − (1 − δ )K tD ⎟ ⎟ + K tD+1 β ⎢ Wt +1 + (1 + r~t +1 ) { + {⎟⎟ ⎜ σ − 1 ⎢ ⎜⎝ BND BD 2 ⎠ ⎠ ⎝ ⎣
σ
+
σ
(
σ −1
σ −1
⎡ β ⎢ (C ) σ + (K ) σ ∑ σ −1 ∞
s
s=2
⎣
ND t+ s
D t +s
σ −1 σ
)
⎤ ⎥ ⎥ ⎥ ⎦
(A.120)
⎤ ⎥ ⎦
~ The benefits of savings, At , are the non-durables consumption the household can buy with the saving in period t+1, denoted by BND in the above equation. The costs of savings consist of the sacrificed non-durables consumption, denoted by CND. The costs of durable goods capital consist of the investment in it that needs to be done in period t, denoted by CD. The benefits consist of two parts. The first is the utility that is derived at time t from having the durables capital stock KtD. This is denoted by BD1 in the above equation. The second part consists of the resale value of the consumer durables allowing for more consumption of non durables in period t+1. This is denoted by BD2 in the equation above. Step I:
Parts (ii) and (iii) ask us to do steps II and III for the savings decision. Parts (iv) and (v) ask us to do steps II and III for the durable goods capital stock decision. (lxi)
Step II (Savings decision): The marginal costs of savings consist of the decrease in non-durables consumption at time t due to increase in savings times the marginal utility of non durables consumption at time t. Mathematically, these marginal costs can be written
303
S O L U T I O N S
−
(
) (
∂CtND ∂U CtND , K tD = CtND ~ ND ∂Ct ∂At
)
−
1
(A.121)
σ
The marginal benefits of saving at time t consist of the additional non-durable consumption that can be bought at time t+1 with an additional unit of savings at time t times the one period discounted marginal utility of non-durable goods consumption at time t+1. Mathematically, this boils down to
(
)
D ∂CtND ∂U CtND +1 +1 , K t β = (1 + ~ rt +1 )β CtND ~ +1 ND ∂ C ∂At t +1
(
)
−
1
(A.122)
σ
Step IV (Savings decision): Equating the marginal costs and benefits for the savings decision, we obtain the non-durables consumption Euler equation, which yields
(C )
1 ND − σ t
(
= β (1 + ~ rt +1 ) CtND +1
)
−
1
(A.123)
σ
(lxii)
This is exactly the same Euler equation as we derived for consumption in Chapter 2.
(lxiii)
Step II (durable good capital stock): The marginal costs of the durable goods capital stock is the loss in current non-durables consumption due to an increase in KtD times the marginal utility of non-durables consumption in the current period t. This equals
−
(
)
∂CtND ∂U CtND , K tD = qt CtND ∂K tD ∂CtND
(
)
−
1
(A.124)
σ
The marginal benefits of KtD consists of two parts. The first is the utility derived from an additional unit of KtD at time t. That is,
(
) ( )
∂U CtND , K tD = K tD ∂K tD
−
1
(A.125)
σ
The second part is the additional utility obtained from the increase in nondurables good consumption in period t+1 due to the resale for the consumer durables. In terms of math, this is
(
)
D ∂CtND ∂U CtND +1 +1 , K t β = (1 − δ )qt +1 β CtND +1 ∂K t ∂CtND +1
304
(
)
−
1
σ
(A.126)
S O L U T I O N S
Equating the marginal costs and benefits of the durable goods capital stock decision yields the optimality condition
(
qt CtND
(lxiv)
)
−
1
σ
( )
= K tD
−
(
1
+ (1 − δ )qt +1 β CtND +1
σ
)
−
1
σ
(A.127)
This is essentially a user cost equation of consumer durables. This optimality condition equates the utility cost of a unit of consumer durables to the marginal utility of consumer durables plus the discounted utility benefit of the resale value of an additional unit of consumer durables. Note the similarity of this equation with that derived in exercise 3.14. In that exercise, the firm invested until the point at which the profit decrease of an additional unit of capital equaled the marginal profit of capital in the current period plus the discounted increase in profits in the next period due to the additional unit of capital’s resale value.
Answer to Exercise 3.15: DIY growth accounting
In terms of the notation of the notes, the productivity data given imply
Y2004 L2004 95 Z 2004 100 = , = Y1949 L1949 28 Z1949 51
(A.128)
Furthermore, the Cobb-Douglas production function implies that
Y2004 L2004 ⎛ Z 2004 ⎞⎛ K 2004 L2004 ⎞ ⎟⎜ ⎟ =⎜ Y1949 L1949 ⎜⎝ Z1949 ⎟⎠⎜⎝ K1949 L1949 ⎟⎠ ⎛ K 2004 L2004 ⎞ ⎡⎛ Y2004 L2004 ⎞ ⎜⎜ ⎟⎟ = ⎢⎜⎜ ⎟⎟ ⎝ K1949 L1949 ⎠ ⎣⎝ Y1949 L1949 ⎠
α
⎛ Z 2004 ⎞⎤ ⎜⎜ ⎟⎟⎥ ⎝ Z1949 ⎠⎦
1
(A.129)
α
The labor share of 70% implies that α=0.3. See exercise 3.5 for a derivation of this result. Filling in the data into the equation above, we find that 1
⎛ K 2004 L2004 ⎞ ⎡⎛ 95 ⎞ ⎛ 100 ⎞⎤ 0.3 ⎜⎜ ⎟⎟ = ⎢⎜ ⎟ ⎜ ⎟⎥ = 6.21 K L ⎝ 1949 1949 ⎠ ⎣⎝ 28 ⎠ ⎝ 51 ⎠⎦
(A.130)
The average growth rate of the capital labor ratio over these 50 years is given by 1 / 55
⎛ K 2004 L2004 ⎞ ⎜⎜ ⎟⎟ ⎝ K1949 L1949 ⎠
305
− 1 = 0.034
(A.131)
S O L U T I O N S
Hence the average annual growth rate of the capital labor ratio, i.e. our measure of capital deepening, is 3.4% for the U.S. during the postwar period. Because
⎛ K 2004 L2004 ⎞ ⎜⎜ ⎟⎟ ⎝ K1949 L1949 ⎠
0.3
Y2004 L2004 = 0.51 Y1949 L1949
(A.132)
Capital deepening accounts for about 51% of postwar labor productivity growth in the U.S.. Answer to Exercise 3.16: Determining capital service flows
(lxv)
This is just a matter of plugging in the numbers from the table into the equation. The capital service flows of structures equal [0.060-0.0247+0.030] × 6027=0.0653 × 6027=354 Hence, the capital service flows of structures were about 350 billion dollars in 1996. Similarly, the capital service flows for equipment are [0.060+0.0262+0.130] × 3959=0.2162 × 3959=856 Thus, the capital service flows of equipment were about 865 billion dollars in 1996.
(lxvi)
The ratio of the capital service flow to the replacement value of capital is approximately given by
[
Pt MPK t K t = rt − π tq + δ qt K t
]
(A.133)
which is 0.0653 for structures and 0.2162 for equipment. The difference between these two is approximately 0.15. The largest part of this is the 0.10 accounted for by the difference in depreciation rates. The other 0.05 is accounted for the fact that equipment prices dropped by about 2.5%, while those of structures increased by about the same amount. Answer to Exercise 4.1: Rental versus ownership of capital in equilibrium
(lxvii)
The left hand side measures the real marginal benefit of renting capital, it is the amount of output produced with an additional unit of capital, the right hand side is the real marginal cost, i.e. the real rental price of a unit of capital. The firm chooses its optimal capital rental decision such that the marginal costs equal the marginal benefits.
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(lxviii) The reason that the interest rate and the rental rate of capital are related is because they represent the returns to two options that households have for dealing with their assets. Because there are many households, households can, in principle, lend their assets to other households. If they do so, for every unit of assets that they lend to another household they obtain a return of ~ rt . If they rent out their assets to firms on the other hand, then for every unit they rent ~ out, they get Rt units of return in terms of the rental price. However, if the firm uses the assets as capital input, they also depreciate at the rate δ. ~ Consequently, the net return of renting out a unit of capital are Rt − δ . In equilibrium, the net returns to lending capital as well as renting out capital should be the same. This is known as a no-arbitrage condition. (lxix)
This follows from the fact that if Pt-1/Pt=1/1=1 then
P ~ rt = (1 + rt ) t −1 − 1 = (1 + rt ) − 1 = rt Pt
(A.134)
Hence, because the real interest rate is the nominal interest rate corrected for inflation, if there is no inflation, then the real and nominal interest rates must be the same (lxx)
If qt=Pt=1 for all t then the user cost equation implies that ⎛ 1 ⎞⎛ ∂Yt +1 ⎞ ⎛ 1 − δ ⎞ ⎟⎟⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ 1 = ⎜⎜ ⎝ 1 + rt +1 ⎠⎝ ∂K t +1 ⎠ ⎝ 1 + rt +1 ⎠
(A.135)
multiplying both sides of this equation by (1+rt+1) yields that ⎛ ∂Y ⎞ ⎛ ∂Y ⎞ 1 + rt +1 = ⎜⎜ t +1 ⎟⎟ + 1 − δ ⇒ ⎜⎜ t +1 ⎟⎟ = rt +1 + δ = ~ rt +1 + δ ⎝ ∂K t +1 ⎠ ⎝ ∂K t +1 ⎠
where the last step uses the result from part (iii) of this question. (lxxi)
No matter whether capital is owned or rented by firms, the equilibrium dynamics of this economy are going to be the same because in both cases the real interest rate in equilibrium will be the marginal product of capital minus the depreciation rate. The difference between the two cases is that in each case the firm equates different margins to come to the same result.
Answer to Exercise 4.2: Invisible Hand
(lxxii) From the resource constraint, we get that
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S O L U T I O N S
Ct = Yt − I t
(A.136)
The capital accumulation equation implies that gross investment is given by
I t = K t +1 − (1 − δ )K t
(A.137)
Output Yt is given by the production function. Substituting the production function and the above equation into (A.136) yields that
Ct = Z t K tα L1t−α − [K t +1 − (1 − δ )K t ]
(A.138)
(lxxiii) The costs and benefits of the capital accumulation decision and the labor allocation decision are denoted in the equation below ⎛ ⎡ ⎤ ⎞ U ⎜ Z t K tα L1t−α − ⎢ K t +1 − (1 − δ )K t ⎥, Lt ⎟ + { { {⎟ ⎜ BL ⎣⎢ CK ⎦⎥ CL ⎠ ⎝ ⎛
⎡
⎤
⎞
⎥⎦
⎟ ⎠
βU ⎜ Z t +1 { K tα+1 L1t−+α1 − ⎢ K t + 2 − (1 − δ )K t +1 ⎥, Lt +1 ⎟ + { ⎜ ⎝
∞
∑ β U (C s
s =2
⎢⎣
BK 1
t+s
BK 2
(A.139)
, Lt + s )
The term BL reflects the benefits to the social planner of allocating labor to production. It represents the contribution of labor to output. The term CL reflects the utility loss due to the sacrificed leisure incurred because of the choice Lt. It is the cost of the labor allocation decision. Labor allocation decision:
Capital accumulation decision: The terms BK1 and BK2 reflect the two components of the benefits of capital accumulation of the choice of Kt+1. BK1 represents Kt+1 contribution to output in period t+1. BK2 represents the contribution to Kt+2 of the depreciated value of Kt+1. It reflects the benefits of the durability of the capital stock to the planner. CK is the cost of accumulating Kt+1, which consists of the reduction in current consumption.
(lxxiv) The marginal cost of the labor allocation decision can be calculated by considering the minus of the decrease in utility due to an increase in the Lt in the term CL, holding consumption constant. This can be expressed as
−
∂ U (Ct , Lt ) ∂Lt
(A.140)
The marginal benefit of the labor allocation works through consumption and output. They are the increase in utility due to the increase in consumption that
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becomes attainable because of the increase in output resulting from an increase in Lt. Mathematically, this can be written as ⎡ ∂ ⎤ ⎡ ∂C ⎤ ⎡ ∂Y ⎤ ⎡ ∂ ⎤ ⎡ ∂Y ⎤ U (Ct , Lt )⎥ ⎢ t ⎥ ⎢ t ⎥ = ⎢ U (Ct , Lt )⎥ ⎢ t ⎥ ⎢ Yt ⎦ ⎣ ∂Lt ⎦ ⎣ ∂Ct ⎣ ∂Ct ⎦1 ⎣ ∂2 ⎦ ⎣ ∂Lt ⎦ 3
(A.141)
=1
(lxxv) The marginal cost of the capital accumulation decision, i.e. the marginal effect of CK, works through consumption. They consist of the decrease utility due a reduction in consumption required for an increase in capital accumulation, i.e. an increase in Kt+1. That is, mathematically −
⎡ ∂Ct ⎤ ∂ ∂ U (Ct , Lt )⎢ U (Ct , Lt ) ⎥= ∂Ct ∂K t +1 ⎦ ∂Ct ⎣1 424 3
(A.142)
= −1
The marginal benefits of capital accumulation consist of two parts, i.e. the marginal effects of BK1 and BK2. Both of these work through next period’s consumption. The marginal effect of BK1 is the additional utility that is derived from the extra consumption in the next period that is attainable due to the additional output that can be produced with an additional unit of Kt+1. Because this utility is attained in the next period, it is discounted by the factor β. Mathematically, this boils down to ⎡ ∂ ⎤ ⎡ ∂C ⎤ ⎡ ∂Y ⎤ ⎡ ∂ ⎤ ⎡ ∂Y ⎤ U (Ct +1 , Lt +1 )⎥ ⎢ t +1 ⎥ ⎢ t +1 ⎥ = β ⎢ U (Ct +1 , Lt +1 )⎥ ⎢ t +1 ⎥ (A.143) ∂Yt +1 ⎦ ⎣ ∂K t +1 ⎦ ⎣ ∂Ct +1 ⎦1 ⎣4 ⎣ ∂Ct +1 ⎦ ⎣ ∂K t +1 ⎦ 24 3
β⎢
=1
The marginal effect of BK2 is the additional utility the is derived from the extra consumption in the next period that is attainable because a higher current capital stock lowers the required level of gross investment in the next period (for a given path of future capital stocks). Hence, holding Kt+2 fixed, an increase in Kt+1 reduces It+1 and thus increases next periods consumption level, yielding additional utility. In terms of partial derivatives, this reads ⎡ ∂ ⎤ ⎡ ∂C ⎤ ⎡ ∂I ⎤ ⎡ ∂ ⎤ U (Ct +1 , Lt +1 )⎥ ⎢ t +1 ⎥ ⎢ t +1 ⎥ = β ⎢ U (Ct +1 , Lt +1 )⎥ (1 − δ ) (A.144) ∂I t +1 ⎦ ⎣ ∂K t +1 ⎦ ⎣ ∂Ct +1 ⎦ ⎣1 ⎣ ∂Ct +1 ⎦ 42 4 31 424 3
β⎢
= −1
= − (1−δ )
Adding both the marginal effects of BK1 and BK2 we obtain the marginal benefits of capital accumulation, which equal
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S O L U T I O N S
⎡ ∂ ⎤⎛ ∂Y ⎞ U (Ct +1 , Lt +1 )⎥⎜⎜ t +1 + 1 − δ ⎟⎟ ⎣ ∂Ct +1 ⎦⎝ ∂K t +1 ⎠
β⎢
(A.145)
(lxxvi) Equating the marginal costs and benefit of the labor allocation decision, we find that ⎡ ∂ ⎤ ⎡ ∂ ⎤ ⎡ ∂Y ⎤ U (C t , Lt )⎥ = ⎢ U (C t , Lt )⎥ ⎢ t ⎥ −⎢ ⎣ ∂Lt ⎦ ⎣ ∂C t ⎦ ⎣ ∂Lt ⎦
(A.146)
such that −
∂U (Ct , Lt ) ∂Lt ⎡ ∂Yt ⎤ =⎢ ⎥ ∂U (Ct , Lt ) ∂Lt ⎣ ∂Lt ⎦
(A.147)
This can now be solved by using the functional forms for the utility and production functions. These imply that
−
⎡ ∂Y ⎤ ∂U (Ct , Lt ) ∂Lt 1 − θ Ct Y = and ⎢ t ⎥ = (1 − α ) t θ 1 − Lt Lt ∂U (Ct , Lt ) ∂Lt ⎣ ∂Lt ⎦
(A.148)
Substituting these into the above equation, we get (4.17) from the main text. That is, we obtain the planner will choose the labor allocation to satisfy
(1 − θ ) θ
Ct Y = (1 − α ) t 1 − Lt Lt
(A.149)
(lxxvii) Equating the marginal costs and benefits for the capital accumulation decision yields ⎤ ⎡ ∂ ⎤ ⎡ ∂Y ⎡ ∂ ⎤ U (Ct , Lt )⎥ = β ⎢ U (Ct +1 , Lt +1 )⎥ ⎢ t +1 + (1 − δ )⎥ ⎢ ⎣ ∂Ct +1 ⎦ ⎣ ∂K t +1 ⎦ ⎣ ∂Ct ⎦
(A.150)
which is identical to equation (4.15) from the main text. (lxxviii) The five equations (4.25), (4.26), (4.27), (4.15) and (4.17) are exactly identical to the five equations that describe the equilibrium dynamics of the decentralized economy, i.e. equations (4.15), (4.16), (4.17), (4.18), and (4.22). (lxxix) The social planner will choose exactly the same path for the equilibrium ∞ variables {Yt + s , K t + s , Ct + s , Lt + s , I t + s }s =0 as that the decentralized equilibrium
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outcome would be. Hence, the decentralized equilibrium outcome is ‘optimal’ or ‘efficient’ in that it coincide with the best outcome attainable to a planner that has all the relevant information. The powerful thing is that, in the decentralized equilibrium, no one needs to have all the information. Firms and households just have to do what is best for themselves. Answer to Exercise 4.3: Comparative statics in steady state
(lxxx) Economy A will have a higher output level because in economy B workers will substitute away part of the higher consumption level associated with economy A's output level for leisure. (lxxxi) Economy B will have a higher steady state capital labor ratio because an increase in the depreciation rate increases the cost of capital and therefore decreases the steady state level of capital used per hour. (lxxxii) Economy A will have a higher steady state employment level. Part of the employment motive in steady state is to produce output that can be saved for future consumption. That is, part of the steady state tradeoff is the substitution between future consumption and current leisure. The higher the discount rate, the higher the return to future consumption. Hence in country A in steady state workers will substitute current leisure for future consumption more than in economy B. Consequently, economy A must have a higher steady state employment level. (lxxxiii) Economy A must have a higher investment to capital ratio. In the steady state net investment must equal zero. This implies that investment must exactly offset capital depreciation. This implies that the country with the higher depreciation rate must have a higher investment to capital ratio in steady state. Answer to Exercise 5.1: Real GDP as a measure of the standard of living
There are many additional variables that are considered when people look for an index of living standards that is broader than real GDP per capita. Probably the most widely quoted alternative index of living standards is the Human Development Index, published by the UNDP. Human Development Index (HDI) is a composite Index which measures average achievement in three basic dimensions of human development - a long and healthy life (measured by life expectancy at birth), knowledge (measured by adult literacy rate and gross enrollment ratio) and a decent standard living (measured by GDP per capita). Answer to Exercise 5.2: Kaldor’s growth facts and the BGP
On the balanced growth path
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S O L U T I O N S
Yt = X t Yˆ , K t = X t Kˆ , and Lt = 1
(A.151)
On the balanced growth path, Yt Lt = X t Yˆ and grows at the same constant rate as Xt. Thus the BGP is consistent with Kaldor’s first fact. Fact A:
On the balanced growth path, K t Lt = X t Kˆ and also grows at the same constant rate that Xt does. Hence, the BGP is consistent with fact B. Fact B:
Fact C: On the balanced growth path, the real interest rate equals the marginal product
of capital minus the depreciation rate. That is,
⎡ ∂Y ⎤ Y ~ rt = ⎢ t ⎥ − δ = α t − δ = α Kt ⎣ ∂K t ⎦
Yˆ −δ = ~ r ˆ K
(A.152)
Thus, the real interest rate along the balanced growth path is constant over time. Fact D: That the capital output ratio is constant along the growth path was already part
of the answer to fact C. In our model where we have assumed a Cobb-Douglas technology the capital and labor shares are constant, no matter whether the economy is on the balanced growth path or not. It turns out that in models with other technology specifications with constant returns to scale the factor shares are also constant along the BGP. Fact E:
Answer to Exercise 5.3: Kaldor in practice
I have chosen to answer this question for the Netherlands. The data related to this answer can be found in the worksheet ‘Kaldor for NLD’ in the file Chapter5.xls This worksheet contains three figures. The first contains the growth rate of real GDP per worker for the Netherlands. As you can see real GDP per worker was high in the years after the Second World War and growth slowly stagnated in the subsequent decades. If anything, what we observe is a convergence phenomenon. Kaldor’s fact A only seems to hold in the latter part of the sample. The second figure contains the growth rate capital labor ratio for the postwar period. This is very high at first and then stabilizes at around 2% at the end of the sample. Again Kaldor’s fact B., is more applicable to the end of the sample than to the beginning. Finally, the capital output ratio increases until the beginning of the 1980’s after which it is steady, as Kaldor’s fact C would predict. Answer to Exercise 5.4: Transitional dynamics and convergence
(lxxxiv) Yes, no matter where we start the two countries, both of their per capita income levels tend towards the same long-run trend.
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S O L U T I O N S
(lxxxv) The occurrence of convergence does not depend on the model parameters. It is robust. (lxxxvi) The speed of convergence increases when the intertemporal elasticity of substitution increases. In fact, in the case where the intertemporal elasticity of substitution is infinite, convergence will be instantaneous. The reason for this is as follows. For a country that starts off below the BGP capital level, convergence requires saving more in the short-run. This will be happening because a country with a low capital stock will have a high marginal product of capital and thus a high real interest rate. In principle, if the productive capacity of the country is enough it could immediately make the jump to the steady state capital level. This however would require a one-time big sacrifice in terms of current consumption. Households with a high intertemporal elasticity of substitution would be willing to make most of this jump and thus close the gap to the steady state capital stock quickly and converge to the balanced growth path quickly. Households with a low intertemporal elasticity of substitution, however, will not be willing to make this jump in consumption. Instead, they would like to choose a savings path in which consumption is more smooth, thus slowing down the convergence process. Answer to Exercise 5.5: Convergence for a bigger set of countries
(lxxxvii) The poorest country in Kraay’s dataset in 1960 is Ethiopia, whose per capita GDP level was 3.3% of that of the U.S. at that time. The poorest country in Kraay’s dataset in 1994 was the Democratic Republic of Congo (former Zaire) with a per capita GDP level that was only 1.4% of that of the U.S. (lxxxviii) In 1960 the people in the U.S. were 30 times richer (on average) than the people in the poorest country in the World. In 1994 this ratio was 71. (lxxxix)The three countries that had the highest average annual growth rates of per capita GDP over the 1960-1994 period are Korea (6.6%), Singapore (6.4%), and Oman (6.3%). Not far behind, ranked four and five, are Taiwan (6.1%) and Hong Kong (6.1%). This compares to an average growth rate of per capita GDP of 1.8% for the U.S.. (xc)
No, we do not observe convergence for all countries in the World. The negative correlation between initial per capita income levels and subsequent growth rates still seems to be present for the rich countries in the World. However, many poor countries actually saw growth rates that were lower than the U.S. over this period. These countries are represented
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by the cloud of observations in the lower left-hand corner of the scatterplot. Answer to Exercise 5.6: Why doesn’t capital flow from rich to poor?
(xci)
For the marginal product of capital we obtain that
⎛K Y MPK i = α i = αZ i ⎜⎜ i Ki ⎝ Li
⎞ ⎟⎟ ⎠
− (1−α )
(A.153)
The Cobb-Douglas production function implies that output per capita equals
⎛K Yi = Z i ⎜⎜ i Li ⎝ Li
⎞ ⎟⎟ ⎠
α
(A.154)
This equation can be used to solve for the capital labor ratio as a function of output per capita. Doing so yields that
⎛ Ki ⎜⎜ ⎝ Li
⎞ ⎟⎟ ⎠
−(1−α )
⎛Y L = ⎜⎜ i i ⎝ Zi
⎞ ⎟⎟ ⎠
⎛ 1−α ⎞ −⎜ ⎟ ⎝ α ⎠
(A.155)
Combining this result with the first equation, we obtain that
⎛K MPK i = αZ i ⎜⎜ i ⎝ Li
⎞ ⎟⎟ ⎠
−(1−α )
⎛Y L = αZ i ⎜⎜ i i ⎝ Zi
⎞ ⎟⎟ ⎠
⎛ 1−α ⎞ −⎜ ⎟ ⎝ α ⎠
⎛Y = αZ i1 α ⎜⎜ i ⎝ Li
⎞ ⎟⎟ ⎠
⎛ 1−α ⎞ −⎜ ⎟ ⎝ α ⎠
(A.156)
which is what we were supposed to derive. (xcii)
This is based on the observation of an approximately constant U.S. labor share of 70%.
(xciii) This is easiest done by showing that the partial derivative of the marginal product of capital with respect to output per capita is negative. Taking this partial derivative, we obtain that
⎛Y ∂MPK i = −(1 − α )Z i1 α ⎜⎜ i ∂ (Yi Li ) ⎝ Li
⎞ ⎟⎟ ⎠
⎛ 1− 2α ⎞ −⎜ ⎟ ⎝ α ⎠
0, of purchasing power of the households money balances between period t-1 and t because of inflation. That is, if inflation is positive and prices increase then a dollar in money balances today buys you more than that same dollar tomorrow.
(iii)
If the nominal interest rate is non-negative, i.e. rt≥0, then we obtain that
(1 + r~t ) = (1 + rt ) (1 + π t ) ≥ 1 (1 + π t )
(A.165)
where the left-hand-side of this equation equals the real gross return on assets and the right-hand-side is the real gross return on money. (iv)
Hence, if rt>0 then the real gross returns on assets exceed those on money and the household will decide not to hold any money.
(v)
In the opposite case in which rt0 and velocity is constant and equals v, then the quantity equation of money implies that
π t = − g + ∆ ln Vt + ∆ ln M t
(A.166)
Hence, this line would look like 45-degree line that doesn’t cross the intercept, but instead crosses the y-axis at –g + the average change in velocity (which is basically zero). (ii)
For countries with high money growth rate, this seems to be approximately true. For countries with low money growth rate there seems to be much less of a relationship between money growth and inflation.
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(iii)
Money growth is thus a better predictor of inflation for countries where it is high. For countries where money growth is low, variations in velocity and output growth blur the relationship between money growth and inflation. For countries with high money growth the latter dominates variations in output growth and velocity.
(iv)
In those countries, output growth dominates and causes the intercept of the equation above to be negative. Therefore, most observations lie below the 45degree line.
(v)
In low money growth countries, variations in output growth dominate those of money growth.
Answer to Exercise 7.3: Is US velocity constant?
(i)
No, it does not seem to be constant. It varies between about 0.975 and 1.425 over the sample period.
(ii)
It does seem to depend on the nominal interest rate. When the interest rate goes up, velocity tends to increase as well. Consequently, there seems to be a positive correlation between the two of them.
(iii)
These are probably shifts in the relative demand for money due to technological advances. For example, velocity went up a lot in the 1990’s when information technology gave people easier and quicker access to funds, like their mutual fund and stock holdings, that are not considered part of money. Consequently, people substituted away from conventional liquid assets and therefore volatility increased.
(iv)
This is for yourself to do… The general consensus is that there is not much of a relationship between V1 and the business cycle.
Answer to Exercise 7.4: CES in a bit more detail
(i)
The demand functions that result from the CES preferences are
⎛P Cit = ⎜⎜ t ⎝ Pit
1
⎞ 1−φ ⎟⎟ Ct for i=1,2 ⎠
(A.167)
Taking the ratio of both demand functions, yields that 1
⎛P ⎛C ⎞ C1t ⎛ P2t ⎞ 1−φ 1 ⎟⎟ such that ln⎜⎜ 1t ⎟⎟ = − = ⎜⎜ ln⎜⎜ 1t 1 − φ ⎝ P2t C 2t ⎝ P1t ⎠ ⎝ C 2t ⎠
318
⎞ ⎟⎟ ⎠
(A.168)
S O L U T I O N S
Consequently,
elasticity of substitution = − (ii)
∂ ln(C1t C 2t ) ⎛ 1 ⎞ =⎜ ⎟ ∂ ln(P1t P2t ) ⎜⎝ 1 − φ ⎟⎠
(A.169)
The intratemporal problem here is to maximize the amount of the consumption aggregate, Ct, obtained for a given amount of expenditures, Et. This problem can be written as the household choosing C1t and C2t to maximize
[
]
1/ φ
Ct = C1φt + C 2φt
(A.170)
subject to the budget constraint
Et = P1t C1t + P2t C 2t
(A.171)
We can again solve this using our four step approach. We do so by rewriting the budget constraint and substituting it into the objective to obtain ⎡ ⎛E PC Ct = ⎢C1φt + ⎜⎜ t − 1t 1t P2t ⎢⎣ ⎝ P2t
⎞ ⎟⎟ ⎠
φ 1/ φ
⎤ ⎥ ⎥⎦
(A.172)
Step 1: Identify the costs and benefits. The cost is that for every unit if C1t the household buys, it foregoes buying (P1t/P2t) units of C2t. The benefit is the utility obtained from consuming C1t. Step 2: Calculate the marginal costs and the marginal benefits. The marginal costs of consuming C1t is the utility value of the foregone (P1t/P2t) units of C2t. That is
∂Ct P1t ⎛ Ct ⎞ ⎟ =⎜ ∂C 2t P2t ⎜⎝ C 2t ⎟⎠
1−φ
P1t P2t
(A.173)
The marginal benefits consist of the additional utility obtained from consuming an additional unit of C1t. That is,
∂Ct ⎛ Ct ⎞ ⎟ =⎜ ∂C1t ⎜⎝ C1t ⎟⎠
1−φ
(A.174)
Step 3: Equate marginal costs and marginal benefits. Doing so, we find that the optimal consumption choice has to satisfy
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S O L U T I O N S
⎡ ∂Ct 1 ⎤ ⎡ ∂Ct 1 ⎤ 1 ⎥= ⎥=⎢ ⎢ ⎣⎢ ∂C1t P1,t ⎥⎦ ⎣⎢ ∂C 2t P2,t ⎦⎥ Pt
(A.175)
where we have just defined the variable Pt to satisfy this equation in the optimum. We will discuss the interpretation in a lot more detail below. Using the above equation, we find that
⎛ Ct ⎜⎜ ⎝ Cit
⎞ ⎟⎟ ⎠
1−φ
=
∂Ct Pit for i=1,2 = ∂Cit Pt
(A.176)
Because Ct is homogenous of degree one in C1t and C2t, we can apply Euler’s theorem and use the fact that
Ct =
∂C ∂C C1t + C 2t ∂C1t ∂C 2t
(A.177)
Combining the above two equations, we find that
Ct =
P P ∂C ∂C 1 C 2t = 1t C1t + 2t C 2t = Et C1t + ∂C1t ∂C 2t Pt Pt Pt
(A.178)
Hence, for the optimal consumption choice we obtain that C1t and C2t are chosen such that the demand functions are
⎛P ⎞ Cit = ⎜⎜ t ⎟⎟ ⎝ Pit ⎠
1 1−φ
Ct for i=1,2
(A.179)
and ⎡ ⎛ 1 Et Ct = , where Pt = ⎢⎜⎜ ⎢⎝ P1t Pt ⎢⎣
⎞ ⎟⎟ ⎠
φ 1−φ
⎛ 1 + ⎜⎜ ⎝ P2t
⎞ ⎟⎟ ⎠
φ 1−φ
⎤ ⎥ ⎥ ⎥⎦
−
1−φ
φ
(A.180)
which is what we were asked to derive. Answer to Exercise 7.7: CPI and PCE inflation
(i)
You are on your own on this one…
(ii)
CPI inflation is generally higher than PCE inflation. A part of this gap is the substitution bias to which the CPI is more subject than the PCE and which makes the CPI tend to overestimate inflation.
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S O L U T I O N S
(iii)
Core inflation does not include food and energy prices, which tend to fluctuate a lot and often blur medium and long run inflation trends.
Answer to Exercise 7.8: Optimal money supply: the 2-good case
(i)
This follows from the fact that if
(1 + π t ) =
1 1+ ~ rt
(A.181)
1+ π t 1 = ~ 1 + rt 1 + rt
(A.182)
then because
it must be that rt=0. (ii)
What we will have to show is that if rt=0 then the optimality conditions for the 2-good model with a CIA constraint simplify to those of the 2-good model without the CIA constraint. The optimality conditions of the model with the CIA constraint are
∂Ct 1 ∂Ct 1 1 = * = ∂C1t P1t ∂C 2t (1 + rt )P2t Pt
(A.183)
∂U ∂U ⎡ Pt* ⎤ =β ⎥ (1 + rt +1 ) ⎢ ∂Ct ∂Ct +1 ⎣ Pt*+1 ⎦
(A.184)
∂U ∂Ct
⎡ ∂Ct 1 ⎤ ∂U Wt ∂U =− ⎢ ⎥Wt = * ∂Ct Pt ∂Lt ⎣ ∂C1t P1t ⎦
(A.185)
and P*t is defined as ⎡ ⎛ 1 * Pt = ⎢⎜⎜ ⎢⎝ P1t ⎢⎣
⎞ ⎟⎟ ⎠
φ 1−φ
⎛ 1 + ⎜⎜ ⎝ (1 + rt )P2t
⎞ ⎟⎟ ⎠
φ 1−φ
When rt=0, then these conditions boil down to
321
⎤ ⎥ ⎥ ⎦⎥
−
1−φ
φ
(A.186)
⎡ ⎛ 1 * Pt = ⎢⎜⎜ ⎢⎝ P1t ⎢⎣
⎞ ⎟⎟ ⎠
φ 1−φ
⎛ 1 + ⎜⎜ ⎝ P2t
⎞ ⎟⎟ ⎠
φ 1−φ
⎤ ⎥ ⎥ ⎦⎥
−
1−φ
φ
= Pt
(A.187)
and
∂Ct 1 ∂Ct 1 1 = = ∂C1t P1t ∂C 2t P2t Pt
(A.188)
∂U ∂U ⎡ Pt ⎤ ∂U (1 + ~rt +1 ) =β ⎢ ⎥ (1 + rt +1 ) = β ∂Ct ∂Ct +1 ⎣ Pt +1 ⎦ ∂Ct +1
(A.189)
∂U ∂Ct
⎡ ∂Ct 1 ⎤ ∂U Wt ∂U ~ ∂U Wt = − = ⎢ ⎥Wt = ∂Ct Pt ∂Ct ∂Lt ⎣ ∂C1t P1t ⎦
(A.190)
which are exactly the optimality conditions for the household without the cash-in-advance constraint. (iii)
In this case the money supply rule that leads to a zero nominal interest rate is optimal because it will equate the resource allocation chosen by the household with the CIA constraint to the efficient allocation chosen by the unconstraint household.
Answer to Exercise 8.1: Evidence on sticky prices
The good for which prices change the least frequently is for coin-operated apparel laundry and drycleaning. 50% of the prices of this good last longer than 79.9 months (median). (cii)
The median length between price changes is 4.3 months.
(ciii)
The answer to this question is subjective. However, when answering it, it is important what criteria you use to determine whether you consider this a high or low frequency.
Answer to Exercise 8.2: Monetary policy in the NKM: Graphically
(civ)
This is done in figure A.5.
(cv)
This will lead to a decrease in equilibrium output. Equilibrium output is determined by the value of Y for which the AD and AS curves intersect in the lower left-hand diagram in the figure. As you can see the inward shift
322
of the policy rule shifts the AD demand curve inward along the AS curve, thus reducing equilibrium output and inflation. (cvi)
You can do this yourself in the diagram. What you will obtain is an upward sloping aggregate demand curve. The intuition for this result is as follows. In case of a passive policy rule, the central bank does not increase the real interest rate in response to inflation, thus moving down along the IS curve when inflation increases, rather than up. Consequently, an increase in inflation in such an economy will lead to an increase in aggregate demand not a decrease as is the case in part (i).
~ r
~ r IS-curv e
New policy rule (PR)
Old p olicy rule (PR)
π−π*
Y π−π*
π− π*
Old AD -curv e
AS-curve
45 o
Ne w AD-curve
Y
π−π*
Figure A.5: Monetary policy in the New Keynesian model.
Answer to Exercise 8.3: Marginal utility and logarithmic preferences
When the intertemporal elasticity of substitution is one, i.e. σ=1, then preferences are given by
U (Ct + s , Lt + s ) = θ ln Ct + s + (1 − θ )ln (1 − Lt + s ) which means that the marginal utility of consumption equals
323
(A.191)
∂ θ U (Ct + s , Lt + s ) = ∂Ct + s Ct + s
(A.192)
and does not depend on the labor supply, Lt+s. This is relevant, because this will mean that Lt+s will not show up in the consumption Euler equation which depends on the relative marginal utilities of consumption in the current and next period. Answer to Exercise 8.4: Demand curve under monopolistic competition
This follows from ⎛P Cit = ⎜⎜ t ⎝ Pit
1
⎛ 1 ⇒ ⎜⎜ ⎝ Pit
(cvii)
1
⎛ P ⎞ ⎛C ⎛ P ⎞ 1−φ C ⎞ 1−φ ⎟⎟ Ct ⇒ ⎜⎜ t ⎟⎟ = it ⇒ ⎜⎜ t ⎟⎟ = ⎜⎜ it Ct ⎝ Pit ⎠ ⎝ Ct ⎝ Pit ⎠ ⎠ ⎞ 1 ⎛ Cit ⎞ ⎟⎟ ⎟⎟ = ⎜⎜ ⎠ Pt ⎝ Ct ⎠
1−φ
⎛C ⇒ Pit = Pt ⎜⎜ t ⎝ Cit
⎞ ⎟⎟ ⎠
⎞ ⎟⎟ ⎠
1−φ
1−φ
(A.193)
This is easiest to see by taking logarithms, such that we obtain
ln Pit = ln Pt + (1 − φ )ln Ct − (1 − φ )ln Cit
(A.194)
Hence, the larger φ, the less the price depends on the individual demand for the good, i.e. Cit, and the flatter the demand curve. (cviii)
Since the elasticity of substitution is increasing in φ, this implies that the higher the elasticity of substitution the flatter the demand curve. That is, the more demand moves in response to a change in the price.
(cix)
The case of perfect competition depicted in the figure is the one where the demand curve is flat. This is the case when φ is one.
(cx)
In this case the elasticity of substitution in infinite. If the elasticity of substitution is infinite then the consumption goods are perfect substitutes and consumers will simply buy the type that is the cheapest.
Answer to Exercise 8.5: The U.S. monetary policy rule
(cxi)
No, Clarida, Gali, and Gertler claim that these parameters have varied substantially over time. Most notably, they claim to find a very important and significant difference between the monetary policy rule in the era before Paul Volcker became chairman of the Federal Reserve and the rule that they fit for the time Alan Greenspan was at the helm.
324
(cxii)
This answer is in Table II on page 157. For the pre-Volcker era γπ=0.83 and γy=0.27. Under Greenspan it they estimate these parameters to have been γπ=2.15 and γy=0.93.
(cxiii) They claim that monetary policy was more effective in fighting inflation under Greenspan. Their focus is on γπ Under Greenspan γπ>1. This means that an increase in (expected) inflation is offset by an even higher increase in the nominal interest rate. Hence, the policy rule is such that the real interest rate is increasing in (expected) inflation and thus monetary policy will act to reduce output and inflation in response to higher (expected) inflation. This is known as an active monetary policy rule. Before Volcker, the policy rule was passive, according to the estimates in Clarida, Gali, and Gertler, in the sense that γπ0 and any (x1,…,xn) for which the function is defined, it satisfies
f (λx1 ,K, λxn ) = λf ( x1 ,K, xn )
(B.1)
The property of homogeneity is assumed in many different contexts in economics. Probably, the most common application of the assumption is that of constant returns to scale. This assumption just implies that the production function is homogenous of degree one in capital and labor. That is, if the capital and labor inputs are both changed by a factor λ, then so will output. Euler’s theorem
If a function f(x1,…,xn) is homogenous of degree one, it has the property that for all (x1,…,xn) for which the function is defined, it satisfies n ⎡ ∂ ⎤ f ( x1 ,K, xn ) = ∑ ⎢ f ( x1 ,K, xn )⎥ xi i =1 ⎣ ∂xi ⎦
(B.2)
This is result is known as Euler’s theorem and we will use it extensively in chapter 3 when we consider growth accounting methods. Logarithm
The natural logarithm, ln(x), is a function that represents the inverse of the exponential function ex, where e is the constant 2.718…. That is, if x=ey, then y=ln(x). Because e0=1, ln(1)=0. The (natural) logarithm has the following useful properties Logarithm:
329
Table B.3: properties of ln(x)
1. 2. 3. 4. 5.
property ln(xa)=aln(x) ln(ax) = ln(a)+ln(x) ln(a/x) = ln(a)-ln(x) dln(x)/dx = 1/x exp(ln(x))=x
The logarithmic function is often used because it is useful to do calculations in terms of percentage changes. The percentage change in a variable x is the change in the variable, i.e. dx, expressed as a fraction of the level of the variable. That is, the percentage change in x is generally written as dx/x. From property 4. listed in Table B.3, we obtain that the change in the logarithm is equals the percentage change in x for small changes in x. That is, this property implies that d ln x = dx x
(B.3)
For example, let Yt be the level of output at time t, then, for relatively small changes in Yt, i.e. ∆Yt=Yt-Yt-1 small, the percentage change in output in period t is often approximated by
∆ ln Yt = ln Yt − ln Yt −1 ≈ (Yt − Yt −1 ) Yt −1 = ∆Yt Yt
(B.4)
Hence, for small changes in x, such that ∆x=dx, we obtain that the percentage change in a variable can be approximated by the change in the logarithm of that variable. Optimization
Optimization:
We will briefly review the main parts of optimization theory and will do so in two steps. In the first step, we consider the optimization of a function of onevariable. In the second step we consider the optimization of a function of multiple variables. Suppose we were asked to find the value of x for which the function f(x) reaches its maximum. Let f(x) be as depicted in Figure B.1. The x at which f(x) reaches its maximum is such that the slope is zero, i.e. where the function is ‘flat’. This is the condition that we will always use in these notes to find maximizers. Not always is it the case that the point where the slope of a function is equal to zero is the point at which a function reaches its maximum. Think about the bottom of a valley for example, rather than the top of the hill. However, in all cases that we consider in the notes here the slope equal to zero will be a sufficient criterion to find the maximizer. Mathematically, this is called the first order necessary condition. It implies that maximizing the function f(x) with respect to x requires finding the x* for which
0=
( )
d f x* dx
330
(B.5)
It is illustrative to consider why the point x’ in Figure B.1 is not the maximizer. Note that the derivative of f(x) in point x’ is positive. This means that when we increase x a little bit when we are in x’, by the amount dx the function will change by df(x’)>0. Hence, at x’’=x’+dx, f(x’’)=f(x’)+df(x’)>f(x’). Thus x’ is not the point that maximizes the function f(x). f( x)
df(x ’) dx
x*
x’
x
Figure B.1: One-variable optimization
Suppose we would like to maximize the function f(x1,…,xn) with respect to the variables (x1,…,xn). That is, we would like to find the (x*1,…, x*n) for which f(x1,…,xn) reaches its maximum. Then for (x*1,…, x*n) it must be the case that when we change any x*k for k=1,…,n a bit, by dxk, we should not be able to increase f(x1,…,xn). Well, the change induced in f(x1,…,xn) by a small change in xk, i.e. dxk, is (approximately) the partial derivative of f(x1,…,xn) with respect to xk times the change in xk, i.e. dxk. Hence, if (x*1,…, x*n) maximizes the function f(x1,…,xn), then it must be the case that for all dxk ⎡ ∂ f x1* ,K, xn* ⎢ ⎣ ∂xk
(
)⎤⎥ dx ⎦
k
(B.6)
can not be bigger than zero. However, if the partial derivative is negative, then choosing dxkmacro menu.
Figure C.2: Set the macro security level to medium (recommended) or low.
337
E X C E L
Figure C.3: Choose enable macros in the security warning window to activate the macros.
The exercises do not require a very advanced knowledge of Excel. If you are not familiar with Excel, one of the many introductory books about it should get you up to speed to the level you need to do the exercises.
.
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References Abel, Andrew, and Ben Bernanke (2001), Macroeconomics, 4th edition, New York: Addison Wesley. Anderson, Richard G., and Kenneth A. Kavajecz (1994), “A historical perspective on the Federal Reserve's monetary aggregates: definition, construction and targeting”, The Federal Reserve Bank of St. Louis Review, 76-2, 1-31. Arrow; Kenneth J., and Gerard Debreu (1954) “Existence of an Equilibrium for a Competitive Economy”, Econometrica, 22, No. 3, 265-290. Barro, Robert J. (1974), “Are Government Bonds Net Wealth?”, The Journal of Political Economy, 82, 1095-1117. Baumol, William J. (1986), “Productivity Growth, Convergence, and Welfare: What the Long-Run Data Show”, American Economic Review, 76, 1072-1086. Bils, Mark, and Peter J. Klenow (2002), “Some Evidence on the Importance of Sticky Prices”, NBER Working Paper 9069, NBER. Boivin, Jean, and Marc Giannoni (2002), “Assessing Changes in the Monetary Transmission Mechanism: A VAR Approach”, Economic Policy Review, 8, 92111. Brynjolfsson, Erik, and Lorin M. Hitt (2000), “Beyond Computation: Information Technology, Organizational Transformation and Business Performance”, Journal of Economic Perspectives, 14, No. 4, 23–48. Chari, V.V. (1998), “Nobel Laureate Robert E. Lucas Jr.: Architect of Modern Macroeconomics”, Journal of Economic Perspectives, 12, 171-186. Chiang, Alpha (1984), Fundamental Methods of Mathematical Economics, 3rd edition, New York: McGraw-Hill. Clarida, Richard, Jordi Gali, and Mark Gertler (2000), “Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory”, Quarterly Journal of Economics, 147-180. Federal Reserve Bank of New York, “U.S. Monetary Policy and Financial Markets” Gordon, Robert J. (2000), “Does the ‘New Economy’ Measure up to the Great Inventions of the Past?”, Journal of Economic Perspectives, 14, No. 4, 49–74.
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R E F E R E N C E S
Grossman, Sanford J., and Robert J. Shiller (1981), “The Determinants of the Variability of Stock Market Prices”, American Economic Review, 71, No. 2, 222-227. Hicks, John R. (1937) “Mr. Keynes and the "Classics"; A Suggested Interpretation”, Econometrica, 5, 147-159. Kaldor, N. (1961), “Capital Accumulation and Economic Growth”, in F.A. Lutz and D.C. Hague (eds.), The Theory of Capital, New York: St. Martin’s Press. Keynes, John M. (1936), “The General Theory of Employment, Interest and Money” King, Robert G. (2000), “The New IS-LM Model: Language, Logic, and Limits”, Federal Reserve Bank of Richmond Economic Quarterly, 86, 45-103. Kiyotaki, Nobu, and Randall D. Wright (1989), “On Money as a Medium of Exchange”, Journal of Political Economy, 97, 927-954. Krugman, Paul (1998), “Japan’s Trap” Kydland, Finn E., and Edward C. Prescott (1977), “Rules Rather than Discretion: The Inconsistency of Optimal Plans”, Journal of Political Economy, 85, 473-492. Lucas, Robert E. Jr (1973), “Some International Evidence on Output-Inflation Tradeoffs” , American Economic Review, 63, 326-334. Lucas, Robert E. Jr, and Nancy L. Stokey (1987), “Money and Interest in a Cash-InAdvance Economy”, Econometrica, 55, 491-513. Lucas, Robert E., Jr (1990), “Why Doesn't Capital Flow from Rich to Poor Countries?”, American Economic Review, 80, Papers and Proceedings of the Hundred and Second Annual Meeting of the American Economic Association., 92-96. McCandless George T., Jr, and Warren E. Weber (1995), “Some Monetary Facts”, Federal Reserve Bank of Minneapolis Quarterly Review, Vol. 19, No. 3, Summer 1995, pp. 2–11 Oliner, Stephen D., and Daniel E. Sichel (2000), “The Resurgence of Growth in the Late 1990s: Is Information Technology the Story?”, Journal of Economic Perspectives, 14, No. 4, 3-22. Orphanides, Athanasios (2002), “Monetary Policy Rules and the Great Inflation”, American Economic Review – Papers and Proceedings, 92, 115-120. Phillips, A. W. (1958), “The Relation Between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom, 1861-1957”, Economica, 283-299.
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R E F E R E N C E S
Prescott; Edward C., and Rajnish Mehra (1980), “Recursive Competitive Equilibrium: The Case of Homogeneous Households”, Econometrica, 48, No. 6., 1365-1379. Rogers, R. Mark (1998), Handbook of Key Economic Indicators, 2nd edition, New York: McGraw-Hill. Sims, Christopher (1992), “Interpreting the Macroeconomic Time Series Facts: The Effect of Monetary Policy”, European Economic Review, 975-1011. Smith, Adam (1776), An Inquiry into the Nature and Causes of the Wealth of Nations, London: Penguin Classics (1986 Reprint). Taylor, John B. (1993), “Discretion versus Policy Rules in Practice”, Carnegie-Rochester Series on Public Policy, 39, 195-214.
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Index Classical dichotomy · 112 Congressional Budget Office · 246 Constant Elasticity of Substitution · See CES preferences Constant Relative Risk Aversion · 14 Constant returns to scale · 73 Consumer Price Index · See CPI Consumption smoothing · 39 Continuation value · 255 Correlation · 138, 142 Council of Economic Advisers · 246 CPI · 150, 181 Credit good · 172 CRS · See Constant returns to scale Cyclical unemployment · 251
‘ ‘Monetary policy’ shocks · 217 1 12-month growth rate · 45 A
D
AD-curve · 195 Annualized growth rate · 45 Appropriation account · 124 Asset pricing model · 53
Decentralized equilibrium · 114 Depreciation · 77 Depreciation rate of capital · 77 Derivative · 329 Detrending · 138 DGE · See Dynamic general equilibrium Discount factor · 28 Discount window · 177 Discounting
B Balanced budget · 232 Balanced growth path · 130 Bellman equation · 256 Budget deficit · 226 Budget surplus · 226 Bureau of Economic Analysis · 44 Bureau of Labor Statistics · 44
Utility · 28
Disposable personal income · 46 Distortionary tax · 229 Double coincidence · 153 Dynamic budget constraint · 29 Dynamic general equilibrium · 104
Handbook of Methods · 49
Business cycles · 137
definition · 108
Dynamic programming · 253 C E
Calibration · 138, 145 Capital consumption · 77 Capital deepening · 93 Capital service flow · 100 Capital share · 100 Cash good · 172 Cash-in-advance · 153 CBO · See Congressional Budget Office CEA · See Council of Economic Advisers Census Bureau · 44 Central bank · 150 CES preferences · 165 Chain rule · 21 CIA · See Cash-in-advance
Elasticity · 38 Employment Situation Report · 46, 88 Equity premium puzzle · 60 Euler equation · 35 Euler’s theorem · 98, 329 Excess volatility puzzle · 60 F Federal Funds Rate · 164 Federal Reserve System · 177
342
I N D E X
Log-linearization · 122 lump sum tax · 228
Fiat money · 151 first difference operator · 329 First difference operator · 98, 161 First Welfare Theorem · 115 Fiscal year · 246 Fisher identity · 18 Flow of Funds Accounts · 125 Frictional unemployment · 251 Friedman Rule · 150, 189 Fundamental value of a share · 58
M Macroeconomic aggregates · 2 Marginal analysis · 9 Marginal benefit · 9 Marginal cost · 9 Marginal disutility of working · 333
G
Marginal product of capital · 333 of labor · 333
GDP · See Gross Domestic Product, See Gross Domestic Product General equilibrium · 5 Gross Domestic Product · 3, 87 Gross investment · 77 Gross job flows · 253 Growth accounting · 92
Marginal product of capital · 72 Marginal product of labor · 71 Marginal rate of substitution · 22 Marginal revenue of capital · 72 Marginal revenue of labor · 71 Marginal utility of consumption · 332 of leisure · 333
Markup · 209 Menu costs · 206 MFP · See Multi factor productivity Microfoundations · 4 Monetarism · 160 Monetary aggregates · 150 Money in the utility function · 153 Money stock measures · 179 Multi factor productivity · 94
H Homogeneity of degree one · 329 Homogenous of degree one · 98 Hyperinflation · 240 I Impulse response function · 138, 145 Income effect · 147 Indifference curve · 14 Inertia · 262 Inflation rate · 17 Inflation targeting · 222 Inflation tax · 240 Intertemporal · 28, 34 Intertemporal elasticity of substitution · 14, 279 Intratemporal · 15, 33 Invisible hand · 114 IRF · See Impulse response function IS-LM model · 193 Isoquant · 67
N NAICS · See North American Industry Classification System National Bureau of Economic Research · 139 National Income and Product Accounts · 44, 87 Natural rate of interest · 212 NBER · See National Bureau of Economic Research Neoclassical growth model · 126 Net investment · 77 New economy hypothesis · 96 New Keynesian Model · 193 NIPA · See National Income and Product Accounts Nominal rigidities · 192 non-distortionary tax · 229 Nonfarm business sector · 86 North American Industry Classification · 88
J Job offer distribution · 254
O OECD · See Organization for Economic Cooperation and Development Office of Management and Budget · 246 OLG · See Overlapping Generations OMB · See Office of Management and Budget One-sector economy · 105 Open market operations · 177 Optimality condition · 11 Optimization · 330 Organization for Economic Cooperation and Development · 245 Output elasticity of capital · 67 Output gap · 212 Overlapping Generations · 238
K Kaldor growth facts · 131 L Labor force participation rate · 49 Labor productivity · 92 Laffer curve · 234 Limited participation puzzle · 61 Liquidity trap · 224 Logarithm · 25, 329 Logarithmic utility · 25
343
I N D E X
SIC · See Standard Industry Classification Single Coincidence · 153 Stagflation · 196 Standard deviation · 142 Standard Industry Classification · 88 Statistical discrepancy · 107 Steady state · 104, 117 Structural unemployment · 251 Structural VAR · 217 Stylized fact · 1 substitution bias · 181 Summation operator · 29, 333 Supply-side economics · 235
P Partial derivative · 332 Partial derivatives · 20 Payroll employment · 88 PCE deflator · 150, 181 Penn World Tables · 132 Perfect foresight · 27 Perfectly inelastic labor supply · 20 Period to period growth rate · 45 Perpetual inventory method · 77 Personal consumption expenditures · 47 Personal Income and Outlays Report · 46 Phillips Curve · 212 Ponzi Scheme · 31, 227 Present discounted value of utility · 29 Price earnings ratio · 63 Price index · 181 Product operator · 30, 333 Production account · 123 Production function · 66 Productivity slowdown · 93 Public finance · 246
T Tax base · 234 Taylor approximation · 334 TFP · See Total factor productivity time consistency · 236 Time consistency · 221 time series · 335 Time to build · 82 Total factor productivity · 67, 92, 94 Transitional dynamics · 104, 117, 121
Q Quantity equation of money · 160
U
R
Unemployment rate · 50 Unemployment spell · 252 Usercost equation of capital · 76, 81 Usercost of capital · 81 Utility function · 14
RBC · See Real business cycle theory Real budget constraint · 16 Real business cycle theory · 137 Real flow budget constraint · 30 Real interest rate · 17 Real rigidities · 251 Recursive substitution · 276 Reference month · 46 Replacement value of capital · 100 Representative household · 13 Resale value of capital stock · 79 Reserve requirements · 178 Reverse causation · 163 Ricardian Equivalence · 236
V Velocity of money · 160 W Wealth effect · 147
S
Y
S&P 500 · 59 s,S-model · 262 Savings and investment account · 124 Search theory of money · 153 Seasonal adjustment · 45 Seignorage revenue · 239 Separation rate · 254 Shopping time · 153
Year over year growth rate · 45 Z Zero nominal bound · 224
344
