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Rethinking Macroeconomic Policy within a Simple Dynamic Model: Simple Dynamic Model to Rethink Standard Policy Article in Metroeconomica · May 2016 DOI: 10.1111/meca.12133

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Francisco Fatas-Villafranca

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Metroeconomica 00:00 (2016) doi: 10.1111/meca.12133

00–00

RETHINKING MACROECONOMIC POLICY WITHIN A SIMPLE DYNAMIC MODEL Isabel Almudi, Francisco Fatas-Villafranca, Gloria Jarne and Julio Sanchez-Choliz* University of Zaragoza (October 2014; revised April 2016)

ABSTRACT We propose a simple macro-dynamic model to rethink standard policy prescriptions. Our model includes exogenous growth, endogenous capital accumulation and debt, demand-driven production with a nonlinear IS curve, a dynamic Phillips curve, and fiscal and monetary policy instruments. It has multiple steady states with different stability properties, and it is analytically tractable to a significant extent. We complete the analytical results with simulations. We find alternative growth patterns, endogenous fluctuations, and demand-driven level effects even in the long-run. For certain steady states the model shows saddle-path type instabilities, which lead us to reflect on fiscal and monetary policy standards.

1. INTRODUCTION

Following a century of debates in Macroeconomics, the start of the 21st century saw a certain consensus around three patterns of macroeconomic policy (Blanchard et al., 2010): (1) The fundamental tool of macroeconomic policy was monetary policyfocused on maintaining a stable and low inflation rate; this was considered an almost-sufficient condition for economic stability. (2) Flexibility policies in the labor market were considered key structural transformations to combine stability with low structural unemployment. (3) Finally, fiscal policy as a stabilizing discretionary instrument was not recommended. * We would like to thank Giancarlo Gandolfo, Giovanni Dosi, Mario Pianta, the Editors and three anonymous referees of Metroeconomica for their helpful comments on previous versions of this work. This work has been supported by the S10 Research Group (DGASpain/FSE). C 2016 John Wiley & Sons Ltd V

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Isabel Almudi et al.

This (simplified) vision of the pre-crisis consensus in macroeconomic policy rested on a theoretical convergence towards dynamic stochastic general equilibrium (DSGE) models (Clarida et al., 2000; Blanchard, 2009). Furthermore, these policy patterns seemed to be backed up by facts. The end of the 20th century (the Great Moderation period) was perceived as a huge success in macroeconomics (Bernanke, 2013). Let us notice that this mainstream consensus was not free of some criticism (Sayer, 1995; Lavoie, 2006; Fontana and Palacio-Vera, 2007) and, nowadays, the fact that the Great Moderation has been abruptly interrupted by the Great Recession (Krugman, 2011; Blanchard et al., 2012; Romer, 2012) has sparked off debates regarding the consensus policy prescriptions and the underlying theoretical frameworks (Solow, 2008; Tamborini, 2010; Stiglitz, 2011). Considering this situation, we propose a simple macro-dynamic model to address the following (old, but reinvigorated) questions (Frydman and Phelps, 2013): (1) Is the objective of stable and reduced inflation a sufficient condition for macroeconomic stability and growth? (2) What are the roles (if any) of fiscal policy in different time horizons? (3) Looking at the Great Recession, can we state that the Great Moderation was an essentially stable period? To analyze these questions, we propose an exogenous growth model (Solow, 1956) in discrete time, in which we incorporate a non-linear IS curve, an expectations-augmented Phillips curve (EAPC), and a Taylor (monetary policy) rule (Taylor, 1999). More precisely, we combine within a Solow growth model the following elements: (1) From the aggregate demand-side (AD): A Blinder and Solow (1973, 1976) type of setting with non-linear capital accumulation, endogenous public deficit and debt, public-debt interests influencing disposable income and consumption, and a Taylor rule; (2) From the aggregate supply-side (AS): An EAPC with natural employment rate and backward-looking expectations (Friedman, 1968). By combining these elements, we arrive at a non-linear aggregative (nonmainstream) model which, nevertheless, may be recognizable even for mainstream economists. A strong point of the model is that it is tractable, transparent, and it allows us to revise some of the (contemporary) mainstream prescriptions at the light of well established (classical, but updated)

C 2016 John Wiley & Sons Ltd V

Simple Dynamic Model to Rethink Standard Policy

3

theoretical elements (i.e. exogenous growth; the Blinder–Solow setting; a dynamic IS curve from structural aggregate functions; the EAPC a la Friedman; but everything combined within a simple non-linear discretetime model, with a Taylor rule (TR)). As we have said, the resulting model may be familiar even for contemporary mainstream economists; in fact, if we consider constant levels of capital, population and technology in our model, and we suppose that there is no public debt, what we obtain is a non-linear IS-AS-TR model with backward-looking (instead of rational) expectations. Clearly, the absence of stochastic elements makes our model more suitable for analytical treatment, and the relatively simple framework allows us to incorporate significant non-linearities (in the IS dynamic curve; see Day and Yang, 2011). Thus, the resulting model is recognizable, transparent, tractable, but, at the same time, open the way to the appearance of complex behavioral patterns such as multiple alternative steady states, and rich emergent dynamics, with interesting policy implications. As we will show, the model allows us to deal with the abovementioned policy questions. Thus, it reveals demand-driven level effects and fiscal policy influences even in the long run; it produces endogenous growth fluctuations, with fresh fiscal and monetary policy implications for short-run and long-run stability; and it generates interesting patterns of apparent stability, which give way to (unexpected) endogenous increases in volatility under certain conditions. We believe that these results add new arguments to previous non-mainstream contributions which have been highlighting, for a long time, the fragility of the pre-crisis mainstream consensus (see Vogt, 1996; Isaac, 2009; Anundsen et al., 2014; Hannsgen, 2014). Obviously, we recognize that a model as stylized as ours—and which does not incorporate stochastic components—will hardly be able to replicate real data (influenced by effects we do not capture here; e.g. international macroeconomic issues). In this sense, we point out that the objective of our simple model is not to reproduce the complexity of real data, but to help us to understand apparently well-known macro-mechanisms (Krugman, 2000), and guide us through the reconsideration of policy prescriptions questioned after the Great Recession. Regarding the theoretical elements of our model, we point out again that we incorporate (in a stylized way) elements which resemble standard contemporary models (Woodford, 2003; Galı and Gertler, 2007): a dynamic IS condition; a dynamic Phillips curve; a Taylor rule. In this sense, our model is directly comparable with DSGE-models, but we avoid some of the assumptions and complexities of this approach widely criticized in the literature (Howitt, 2006; Kirman, 2010). Thus, instead of deducing our central elements from


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assumptions of perfect intertemporal rationality, representative agents, rational expectations or dynamic general equilibrium (with, somehow arbitrary, imperfections), we propose structural functions inspired by classic works (Domar, 1944; Solow, 1956; Friedman, 1968; Blinder and Solow, 1973; Turnovsky, 1977; Benassy, 2011), and by simplifications of recent empirical results (Orphanides, 2003; Stock and Watson, 2008). This way of proceeding generates slightly different versions of the central theoretical axes, and allows us to obtain a tractable transparent model with new implications for economic policy. In addition, our non-mainstream model connects with recent critical contributions arising within the mainstream realm. Thus, the formal analysis of our model shows that there are multiple steady states with different stability/instability properties (along the lines of Farmer, 2010, 2012). On the other side, far from the belief of most mainstream economists that steady states are stable (rational expectations in saddle-path type situations, transversality conditions), our formal analysis reveals plentiful sources of instability (also pointed out by sophisticated mainstream economists; Frydman and Phelps, 2013). Finally, although under certain conditions monetary policy helps to stabilize the economy in our model, we also show that it can become a source of instability. This result leads us to reconsider to what extent monetary authorities can control the interest rate without triggering cumulative destabilizing processes similar to those in Howitt (1992, 2006). Another interesting result in our model (this time a bit further from the mainstream approach) is that fiscal policy plays a key role, not just in the short to mid term, but also when attempting to bring the economy closer to one of the multiple (alternative) steady states. Therefore, fiscal policy in our model is essential even in the long term. Moreover, fiscal policy interacts in our model with other factors—normally considered to be long term factors—technical change, population growth, input substitutability—in influencing the parametric configurations for stability/instability of the steady states, and the specific pattern of dynamic behavior (dampened oscillations, stability or instability, etc.). Finally, as we anticipated above, the stability analysis of the model and the simulations show that there are parametric configurations and steady states for which the system seems to become stable but, suddenly, it enters into a regime of oscillating instability and increasing volatility. This is due to the existence of stable paths in a context of saddle-path type instability. This result leads us to suggest possible re-interpretations of the Great Moderation period, and new ways to analyze possible policies in the context of the Great Recession. We organize our work as follows. In section 2, we present the model. In sections 3 and 4 we study the existence, multiplicity and local stability of



5

steady states, which allow us to characterize important aspects of the system dynamics. In section 5, we strengthen and extend some of the results with simulations. The simulations allow us to better understand the global dynamic characteristics of the system, and lead us to reflect on and pose new conjectures for the origins of the Great Recession. Finally, in section 6 we summarize our conclusions and extract policy implications related to the open issues we have pointed out.

2. THE MODEL

2.1 Overview We propose a non-linear aggregative model of exogenous growth, in discrete time, which displays demand-driven level effects even in the long-run. In our model, production in the short-run adjusts to what demand dictates. Regarding aggregate demand (AD), we depart from a Blinder and Solow (1973, 1976) type of setting—with endogenous public deficit, debt and public interest payments affecting disposable income and consumption—, and we incorporate a non-linear investment function in a dynamic IS-curve. Thus, firms accumulate physical capital through endogenous investment, and we assume full-capacity use, and endogenous employment creation in the labor market. Imperfections and disequilibria in the labor market, together with the assumption of na€ıve expectations, drive the dynamics of our Expectations-Augmented Phillips curve (EAPC). Moreover, inflation in the model evolves following the EAPC—which is compatible with the natural rate hypothesis (Friedman, 1968). This EAPC plays the role of an aggregate supply (AS) curve in the model. Finally, production and inflation in the model condition monetary policy via a Taylor rule (TR) (see Taylor, 1999). This, in turn, affects investment. Regarding the very long-run, we assume exogenous labor efficiencyaugmenting technical change, and population growth at a constant rate (as in Solow, 1956).1 More precisely, we suppose that the population in t is: Nt 5N0 ð11nÞt ;

n 2 ð0; 1Þ

1

We make this supposition for simplicity. Even so, the model leads to a system of six difference equations. In other works we have endogenized technical change (Fatas-Villafranca et al., 2009, 2012; Almudi et al., 2013).



6

and that the technical progress is shown by: At 5A0 ð11dÞt ;

d 2 ð0; 1Þ

From now on we shall use a combined indicator of these growth rates which will be: a5n1d1dn;

a 2 ð0; 1Þ

Furthermore, we shall express real variables per capita, and adjusted by the level of technical progress. Small-case letters will indicate that, in each case, the corresponding variable has been divided by At Nt and we shall refer to these variables as normalized variables. 2.2

Demand and public financing

The aggregate demand (y)—in normalized terms—has three components: consumption c, investment i, and public spending g: yt 5ct 1it 1g;

g>0

(1)

By assuming a constant g (constant normalized public spending), we assume that real public spending is growing at a constant rate 0 a0, which, as we shall see, is the rate at which income grows in the steady states of the model. If we assume proportional taxes (tax rate s 2 ð0; 1Þ), we denote by v > 0 the constant interest rate of public debt, and we define dt as the stock of public debt in normalized units, then we can define the public deficit as2: def t 5g1vdt 2syt The deficit includes both the difference between public spending and tax revenue, and the interest payments for the debt. Then, the dynamic equation for the evolution of public debt in normalized units is (Blinder and Solow, 1973; Turnovsky, 1977; Romer, 2012, p. 586): 2

We could consider a variable interest rate for public debt so that: def t 5g1ðRt 2pt Þ dt 2syt , where Rt is the nominal rate and pt the inflation rate. However, this assumption complicates immensely the analysis of the model. Thus, for the time being, we consider a constant (guaranteed) real interest rate for public debt, which is different from the private interest rate in the economy. The differences between distinct interest rates can be seen (e.g.) in Buetow et al. (2009). In any case, endogenizing the interest rate on public debt is an interesting and highly challenging technical line of advance for future research.


Simple Dynamic Model to Rethink Standard Policy dt 5

1 ½ð11vÞ dt21 1g2syt21 11a

7 (2)

The consumption function has the following expression (in normalized units): bð12sÞ ðyt21 1v dt21 Þ; 11a

ct 5

b 2 ð0; 1Þ

(3)

where b is the propensity to consume,3 and consumption depends on (after-tax) disposable income. The interests paid for public debt appear explicitly in the definition of disposable income. Notice that we are moving around a Blinder and Solow (1973, 1976) type of setting in this regard. We shall suppose that investment depends on the real interest rate rt21 5Rt21 2pt21 , and on the income from the previous period. In normalized units this will respond to: it 5

1 ðb2hrt21 Þyt21 ; h > 0; 0 < b < 1 11a

(4)

We assume that there is no depreciation and that the propensity to invest depends in a negative way on the real interest rate. Substituting (3) and (4) in (1) we obtain equation (5) which is a nonlinear dynamic IS curve: yt 5

b ð12sÞ 1 ðyt21 1 v dt21 Þ1 ½b2hðRt21 2pt21 Þyt21 1g 11a 11a

(5)

2.3 Production We shall suppose that the productive sector adjusts its production to the level laid down by demand in accordance with (5). Furthermore, we shall suppose that the productive sector produces in accordance with a Cobb– Douglas production function with constant returns to scale, which incorporates labor-enhancing technical change, so: Yt 5Ktl ðAt Lt Þ12l ; 3

l 2 ð0; 1Þ

The consumption and investment functions we propose have a long tradition in Macroeconomics (from Allen, 1967 to Gandolfo, 2009). In previous works, we have analyzed different micro-foundations for consumption and savings (see Fatas-Villafranca et al., 2007).



8

Kt is the capital stock and Lt the level of employment at any time. We shall assume full use of the productive capacity and define the employment rate t as et 5 L Nt It is possible to express this function in normalized units, leaving us with: yt 5ktl et12l

(6)

Note that if yt is determined by (5), and the accumulation of kt is determined by (4), then equation (6) allows us to obtain the employment rate 1 12l y . et 5 klt t

2.4

Capital accumulation

We shall obtain the dynamics of the capital in normalized units, kt 5 AKt Nt t . We know that, if depreciation is null, the investment in normalized units is: it 5

Kt11 2Kt A t Nt

Therefore, it is clear that: kt11 5

2.5

it 1kt 11a

(7)

The Phillips curve

We propose that the inflation rate evolves following a dynamic expectationsaugmented Phillips curve (EAPC)-with na€ıve expectations. As in Friedman (1968), this function plays the role of an aggregate supply (AS) curve—compatible with the natural rate hypothesis. To be specific, we suppose that the variation in the inflation rate depends on employment according to a version of the corroborated specifications in Stock and Watson (2008)4: pt 52c1qet21 1pt21 4

q>c>0

(8)

In Fatas-Villafranca et al. (2012, 2014) we analyzed questions related to the labor market, salary formation and their aggregate effects in more detail. On this occasion, we have opted to use a plausible formal treatment but as stylized as possible.



9

2.6 Monetary policy rule Finally, we can suppose that the monetary authority (Central Bank) determines the short-term nominal interest rate in accordance with a version of Taylors rule (Taylor, 1999): (9) 0 < ay < ap Rt 5Rt21 1ap pt21 2p 1ay yt21 2y ;

where p and y are the objectives for inflation and national income. This specification, which incorporates lagged values in the determination of the short-term nominal interest rate, is inspired by the empirical results in Orphanides (2003). In (9), we are not supposing that p* or y* are necessarily the values of a dynamic equilibrium; they are simply reference objectives for monetary policy. In this way, we capture possible misperceptions of steady state values, and cumulative effects such as those analyzed in Howitt (2006).

2.7 The model equations The dynamics of the model can be synthesized, from the previous equations, in the following system of difference equations: 9 1 > > ðkt 1it Þ > > 11a > > > > > b ð12sÞ 1 > ðyt 1 v dt Þ1 ½b2hðRt 2pt Þyt 1g > yt11 5 > > > 11a 11a > > > > 1 > > = it11 5 ½b2hðRt 2pt Þyt 11a > 1 12l > > yt > > pt11 52c1q l 1pt > > > kt > > > > > > Rt11 5Rt 1ap ðpt 2p Þ1ay ðyt 2y Þ > > > > > 1 > ; dt11 5 ½ð11vÞ dt 1g2syt 11a kt11 5

(10)

The system consists of a process of capital accumulation starting out from endogenous investment, a non-linear IS dynamic equation, a dynamic (expectations-augmented) Phillips curve (EAPC) compatible with the natural rate—it plays the rule of an AS-curve, a Taylor monetary policy rule



10

(TR), and an equation for the evolution of public debt. Relevant variables are expressed in normalized terms, since we assume exogenous population growth and technological change. Now, in section 3, we will study the possible steady states of the model. In sections 4 and 5, we will analyze the determinants of greater or lesser stability and instability of the economy, as well as the emergence of endogenous fluctuations. As we shall see, fluctuations are not due to the fact that we have chosen a discrete time version of the model; fluctuations also appear in possible continuous-time specifications of the model. 3. STEADY STATES I: EXISTENCE AND MULTIPLICITY

Any steady state of the previous dynamic system will be characterized by the constant values fk; y; i; p; R; d g which are solutions of the system of equations that we obtain when, in (10), we state kt11 5kt 5k; yt11 5yt 5y; it11 5it 5i; pt11 5pt 5p; Rt11 5Rt 5R; dt11 5dt 5d; that is, any steady state is characterized by the solution values of the system: 9 1 > > ðk1iÞ > > 11a > > > > > b ð12sÞ b ð12sÞ v 1 > y5 y1 d1 ½b2hðR2pÞy1g > > > > 11a 11a 11a > > > > 1 > = i5 ½b2hðR2pÞy 11a > 1 > y 12l > > > 1p p52c1q l > > > k > > > > > R5R1ap ðp2p Þ1ay ðy2y Þ > > > > > > 11v 1 s ; d5 d1 g2 y 11a 11a 11a

k5

3.1

(11)

Real variables in steady state

From the fourth equation of (11), we can deduce: p52c1q

1 y 12l

kl

1 p ) e5


1 y 12l

kl

12l c c 5 ) y5 kl q q

(12)


11

In steady state, the employment rate is the natural rate—for which the inflation rate will remain constant. Furthermore, as seen in (12), production in steady state is determined by the natural rate of employment e and by the capital stock k. This is the natural state of the economy. The sixth equation of (11) also allows us to express debt as a function of capital: 11v 1 s 11v 1 s d1 g2 y ) d 12 5 g2 y d5 11a 11a 11a 11a 11a 11a (13) 12l ! 1 c g2s kl ) d ða 2 vÞ 5 g 2 sy ) d 5 a2v q If we arrange (13), we can obtain the typical Domar (1944) result which states that, in any steady state, tax revenue should cover current government spending and interest payments on government debt. Thus, from (13) we can obtain: sy5g1ðv2aÞd The first equation of (11) allows us to obtain a necessary relationship in steady state between capital and investment: 1 1 1 ðk1iÞ ) k 12 5 i ) i5ka (14) k5 11a 11a 11a This means that investment in steady state is the replacement investment— that is, the necessary investment to maintain normalized capital constant. Moreover, in natural income (in the natural state of the economy), from the third equation of (11), (12) and (14) we obtain: ka5

12l 1 c ½b2h ðR2pÞ kl () 11a q ()

k

ka b2h ðR2pÞ 12l 5 11a l c q

ka b2h ðR2pÞ 5 y 11a

That is, the investment rate in steady state (replacement investment/natural income) must coincide with the propensity to invest. The previous equation and (12) allows us to obtain the steady state value for a real interest rate r5R2p, as a k-decreasing function and certain parameters:



12

l21 ! 1 1 ð11aÞak 1 að11aÞ 12l c b2að11aÞk b2 b2 r5 5 5 h q h y h ðy=kÞ

(15)

The real interest rate in steady state must be lower, the higher the necessary replacement investment, the higher the rate of growth, and the lower the natural income. In addition, r is higher, the higher the average capital productivity in steady state. In (15) it can be seen that r also depends on demand parameters like b and h: We now have all the real variables as functions of k. We must now prove the existence of k—which we will do in section 3.3.

3.2

Nominal variables in steady state

Regarding the nominal variables, taking into account the fifth equation of (11), the following must be verified in the equilibrium: ay ap ðp2pÞ1ay ðy2yÞ50 ) p5p 2 ðy2yÞ ap By also considering (12), we can deduce the value of inflation rate p in steady state: " # ay c 12l l k 2y (16) p5p 2 ap q It is important to note that, according to (16), as the income objective of the Central Bank may differ from the natural income in steady state, the equilibrium inflation may differ from the objective of the Central Bank. Only in the case where monetary policy establishes exactly the steady state value of y as an objective (difficult in uncertain environments), will the steady state inflation coincide with the goal of the Central Bank. On the other hand, from (15) and (16) the steady state value of R is obtained as the sum of the inflation and the real interest rate of equilibrium: " " # l21 # ay c 12l l 1 12l c (17) k 2y 1 b2að11aÞk R5p 2 ap q h q



13

3.3 The k variable in steady state Obtaining the capital variable in the steady state is very important because it allows us to obtain the values of the other (real and nominal) variables in the system [(12), (13), (14), (16) and (17)]. To obtain k in the steady state we start out from the second and third equations of (11) and we consider (12), (13) and (14): b ð12sÞ b ð12sÞ v 1 ½b2hðR2pÞy1g y1 d1 11a 11a 11a b ð12sÞ b ð12sÞ v s b ð12sÞ v y1 y2 g5ka1g ) 12 11a ð11aÞ ða 2 vÞ ð11aÞ ða 2 vÞ 12l b ð12sÞ b ð12sÞv s c b ð12sÞ v 1 g5ka1g ) 12 kl 2 11a ð11aÞ ða 2 vÞ q ð11aÞ ða 2 vÞ

y5

From here, we can obtain the following equation which must be fulfilled at any steady state for variable k: 12

b ð12sÞ b ð12sÞv s 1 11a ð11aÞ ða 2 vÞ l

12l c b ð12sÞ v kl 2 g5ka1g q ð11aÞ ða 2 vÞ

(18)

l

g () Tk 5ak1g1mg () Tk 5ak1~

with: 12l 12l b ð12sÞ b ð12sÞv s c b ð12sÞða 2 v2v sÞ c 1 T5 12 5 12 11a ð11aÞ ða 2 vÞ q ð11aÞ ða 2 vÞ q m5

b ð12sÞ v ð11aÞ ða 2 vÞ

and

g~ 5ð11mÞg

The three addends on the right of (18), ak1g1mg, are the three destinations for savings in steady state (the left part of the equality). The addends on the right are: replacement investment, current public spending, and a public debt-interests component.5 If we look at the left-hand term of the equation, we can define a composition of parameters r:

5

mg5

bð12sÞ sy vd1v 11a a2v



14

b ð12sÞ b ð12sÞ v s b ð12sÞ vs 1 512 12 r512 ð11aÞ ða 2 vÞ 11a 11a a2v which reflects the savings rate of the economy. This interpretation implies the need to impose the condition a > ð11sÞv which guarantees r < 1. If we assume, as a specific case in (18), that there is no public sector and we have full employment, we would have an expression for the steady state in the goods market similar to the Solow (1956) steady state condition. However, this similarity must not mislead us. In our model, as soon as we move slightly away from the steady state, the dynamics are not ruled by the Solow mechanisms, but by the interactions between production, inflation, expectations, public finance, monetary policy and, also, capital accumulation. Later on, we will show that, if g is sufficiently low, the system (under very general conditions) becomes unstable. This is, again, an important difference as compared with the global stability in the Solow model. Returning to the formal analysis, to determine whether (18) has a solution and, in such a case, how many solutions it has (uniqueness or multiplicity of steady states), we shall begin by studying the properties of the function hðkÞ5Tkl . As it is fulfilled that hð0Þ50 ; lim h ðkÞ511 ; h 0 ðkÞ5lTkl21 > 0 ; h 00 ðkÞ5lðl21ÞTkl22 < 0 ; k!11

0

lim h ðkÞ511 ; lim h 0 ðkÞ50 ;

k!01

k!11

we can confirm that h(k) is concave with an infinite slope at the origin which decreases as values move towards infinity, tending to be horizontal, as we can see in figure 1.

Figure 1. Existence and multiplicity of steady states.



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Looking at (18), considering the properties of h(k), and noticing that h ðkÞ5ry; we can see that hðkÞ2ak (the flow-funding capacity of the private sector in steady state), is initially growing until it reaches point kM, and then decreasing, with kS being the upper available k (see figure 1). We can affirm that all the possible k will be some k 2 D5ð0; ks Þ. As can be seen in figure 1, considering (18), there are two k of steady state in the model (which vary according to the parametric configuration) k1 and k2 verifying k1 < kM < k2 : The value ks is easy to calculate: 1 12l T ðk Þ T2ak 50 ) k 5 a 1 1 1 b ð12sÞða 2 v2v sÞ 12l c r12l 12 5 5 e a ð11aÞ ða 2 vÞ q a (19)

s l

s

s

It is also easy to obtain the value kM , for which the function zðkÞ5hðkÞ 2ak5Tkl 2ak reaches its maximum value: 1

z 0 ðkÞ5lTkl21 2a50 ) kM 5 l12l

1 12l 1 T 5l12l ks a

with the maximum value of z(k) as: " # 1 l 1 12l 12l lT lT T lT 21 lT 12l zðk Þ5 Tðk Þ 2ak 5T 2a 5 21 a a a a a a M l

M

5a

M

1 12l M 12l 12l k 5a l kS l l

M Expression a 12l is the maximum (in flow terms) funding capacity of l k the private sector in steady state (after covering its own investment). Thus, from (18), it is logical that: 1 12l M 12l 12l k l ks a l l 5 g b ð12sÞ v b ð12sÞ v 11 11 ð11aÞ ða 2 vÞ ð11aÞ ða 2 vÞ

a

(20)

To sum up, returning to figure 1 we can see two possibilities in terms of number of steady states in the model: (1) in general, there will be two



16

steady states for k, k1 < kM < k2 , for each value of g which verifies inequality (20) in a strict way; or (2) there will be just one steady state for k (kM ) for the very specific case of equality in (20). This throws up an interesting comment: it is very relevant that the level of public spending ends up being an essential factor for the existence and characteristics of the steady states. Thus, fiscal policy may induce demand-driven level effects on the steady states—contrary to what is usually stated in standard macroeconomics.

3.4

To sum up

The expressions of the equations which define a steady state are: 9 12l b ð12sÞða 2 v2v sÞ c b ð12sÞ v > l > g5ka1g > k 2 12 > > ð11aÞ ða 2 vÞ q ð11aÞ ða 2 vÞ > > > > 12l > > > c > l > y5 k > > q > > > > > 12l ! > > 1 c > l > = g2s d5 k a2v q > > > > i 5 ak > > " # > > 12l > > ay c > l > p5p 2 k 2y > > ap q > > > > " # " #> > 12l l21 > > ay c 1 c > > > R5p 2 kl 2y 1 b2að11aÞk12l ; a q h q p

where it must be verified that: 1 ! 1 b ð12sÞða 2 v2v sÞ 12l c k 2 D5 0; 12 a ð11aÞ ða 2 vÞ q

9 > > > > > > > > > =

1 1 12l 12l 1 b ð12sÞða 2 v2v sÞ 12l c > l 12 > l a ð11aÞ ða 2 vÞ q> > > 0 > b ð12sÞ v > > ; 11 ð11aÞ ða 2 vÞ a

and the following must also be fulfilled:



17

" l21 # 9 > 1 c > > r5R2p5 b2að11aÞk12l > = h q > 1 y 12l > c > > ; 5 e5 l k q Note that the steady state situations also verify three other properties: (1) Given that normalized variables are constant in steady state, the model has steady states in which production, investment, consumption, debt and capital all grow at rate a, and, in terms per capita, they grow at the rate of technical progress. These are balanced growth paths which are compatible with the long-run observed trends in modern economies. (2) The dominium D of possible k values for a steady state will grow if the natural employment rate or the tax rate s increase, or if the propensity to consume, b, falls. (3) The dominium D of possible steady states k is smaller when the exogenous growth rate a grows, and it is larger if the public debt-interest S @kS rate increases (see @k @a and @v ). 1 ! 1 b ð12sÞða 2 v2v sÞ 12l c 12 ð11aÞ ða 2 vÞ q a l c 1 1 b ð12sÞða 2 v2v sÞ @ 1 b ð12sÞða 2 v2v sÞ 12l 5 12 12 q 12l a ð11aÞ ða 2 vÞ @a a ð11aÞ ða 2 vÞ l c 1 1 b ða 2 v1 asÞ 12l 12 5 q 12l a ð11aÞ ða 2 vÞ " # 1 b ð12sÞ ð112aÞ b ð12sÞ v s ð2a2v13a2 22avÞ 2 21 2 a a2 ð11aÞ2 a2 ð11aÞ2 ða 2 vÞ2 l c 1 1 b ða 2 v1 asÞ 12l 12 5 q 12l a ð11aÞ ða 2 vÞ " # 2122a2a2 1b ð12sÞ ð112aÞ b ð12sÞ v s ð2a2v13a2 22avÞ 2 a2 ð11aÞ2 a2 ð11aÞ2 ða 2 vÞ2 l c 1 1 b ða 2 v1 asÞ 12l 12 5 q 12l a ð11aÞ ða 2 vÞ " # a2 1ð12b ð12sÞÞ ð112aÞ b ð12sÞ v s ða1a2v1a2 12aða2vÞ 2 2 0 12 q 12l ð 11a Þ ð a 2 v Þ ð 11a Þ 12l ða 2 v Þ2 a

@kS @ 5 @v @v

4. STEADY STATES II: STABILITY ANALYSIS

The dynamic stability of a steady state ðk; yðkÞ; iðkÞ; pðkÞ; RðkÞ; dðkÞÞ of (10) is determined by the eigenvalues of the Jacobian matrix of this system, evaluated in this point. The Jacobian matrix of (10) is given by:

Jðk; y; i; p; R; dÞ5 0 1 0 B 11a B B B B b ð12sÞ b2h ðR2pÞ B 1 0 B 11a 11a B B B b2h ðR2pÞ B 0 B B 11a B B l 1 B 2l y 12l 1 y12l Bq q B 12l k B 12l k B B B 0 ay B B B @ 2s 0 11a

1 11a

1 0

0

0

h h y 2 y 11a 11a

0

h h y 2 y 11a 11a

0

1

0

0

ap

1

0

0

0

0

C C C C b ð12sÞ v C C 11a C C C C C 0 C C C C C C 0 C C C C C 0 C C C 11v A 11a

which, in the steady state corresponding to a k solution of (18) will be:


Simple Dynamic Model to Rethink Standard Policy J5Jðk; yðkÞ; iðkÞ; pðkÞ; RðkÞ; dðkÞÞ 0 1 0 B 11a B B B b ð12sÞ k B 0 1a B B 11a y B B B k B 0 a B B y 5B B B l 1 1 y12l B 2l y12l Bq q B 12l k 12l k B B B 0 ay B B B @ 2s 0 11a

1 11a 0

0

0 0 0

0

0

19

0

1

C C C h h b ð12sÞ v C C y 2 y C 11a 11a 11a C C C C h h C y 2 y 0 C C 11a 11a C C C C C 1 0 0 C C C C ap 1 0 C C C 11v A 0 0 11a

(21) where y5

12l c q

kl.

4.1 The instability of the steady states with k < kM One condition which ensures the (local) asymptotic stability of the steady state ðk; yðkÞ; iðkÞ; pðkÞ; RðkÞ; dðkÞÞ is that all the eigenvalues of the characteristic equation of matrix (21) have a modulus below one. If one of the eigenvalues has a modulus above 1, this is sufficient cause for the steady state to be unstable. Based on this, we can establish the following proposition: Proposition 1: The following is verified: (1) If there are two steady states, the one associated to the lower value of k (k1 < kM ) is an unstable steady state, with at least two stable paths. (2) Condition k < kM is equivalent to the following inequalities: g~ k l b ð12sÞða 2 vð11sÞÞ ry 1 M 12 () > () k < k () < i y a ð11aÞ ða 2 vÞ ak l 12l > l



20

Proof: In Appendix A we develop the characteristic polynomial of matrix (21), QðkÞ5jJ2kI6 j, whose roots are the eigenvalues of this matrix.6 We find that there is always a negative k1 2 ð21; 0Þ, and a posi eigenvalue, 1 : tive eigenvalue lower than 1, k2 2 0; 11a Taking into account the results of Appendix A, it is fulfilled that: 1 y12l 11v b ð12sÞ lq hy 21 21 ap Q3 ð1Þ52 11a 11a ð11aÞ ð12lÞ k 11a l y12l 11v 1 h q ap y 21 21 2 Q6 ð1Þ52 11a 11a ð11aÞ ð12lÞ k sb ð12sÞ v 1 21 k ð121Þ2 50 Q4 ð1Þ52 2 11a ð11aÞ Q1 ð1Þ52

sb ð12sÞ v ð11aÞ2

1 y12l ql hy ap ð11aÞð12lÞ k 11a

Then: 2

ðv2aÞ ðb ð12sÞ212aÞ

3

7 6 1 6 y12l 1 k7 7 6 Qð1Þ5Q3 ð1Þ1Q6 ð1Þ1Q1 ð1Þ52 7 ða2vÞ a ð11aÞ

1 k l y

() ½vb ð12sÞ2ab ð12sÞ2vð11aÞ1að11aÞ1ðsb ð12sÞ vÞ > ða2vÞ a ð11aÞ

1 k l y

k l < ½vb ð12sÞ2ab ð12sÞ2vð11aÞ1að11aÞ1ðsb ð12sÞ vÞ y a ð11aÞða2vÞ ða2vÞb ð12sÞ2ðsb ð12sÞ vÞ 2 ð11aÞða2vÞ ð11aÞða2vÞ b ð12sÞ sb ð12sÞ v 12 1 ð11aÞ ð11aÞða2vÞ

() ða2vÞ ð11aÞ ()

k l < y a

()

k l < y a

Substituting y5

12l c q

kl in the last inequality, we have:

6 Given that we start now the stability analysis, we point out that—as we show in Appendix B—the results regarding dynamics do not vary substantially if we consider continuous time.



Qð1Þ < 0 () k
> 0= > ;

) 9k4 2

1 ;1 11a

so that Qðk4 Þ50

Then, there are at least two eigenvalues with modulus lower than 1 ðk2 and k4 Þ with two associated eigenvectors which define two stable paths converging to the equilibrium. Thus part 1 of the proposition is proven. Part 2 is easily deduced from (12), (18), (19) and the expressions of r and kM : l21 l21 M 12l l k c c 1 12l k < k 5 r () k < k () 5 y q q a l g~ 12l ry 1 ak1ð11mÞg () < () > < i ak l ak l M

䊏

It is interesting to interpret the meaning of these results. The first part shows us that the steady states for k < kM are unstable; therefore, they are essentially irrelevant from an economic point of view. Consequently, when k < kM , we must expect, in general, that a deviance from the steady state will give way to increasing deviances from this. However, it is also possible that, when we are near the stable paths, the deviance takes time to show itself, leading to an apparently stable path which eventually will move away from the equilibrium. On the other hand, part 2 of Proposition 1 shows that the economy being in steady states with (k < kM ) is equivalent to the public spending/



22

private investment ratio being above a certain limit, which is determined by input substitutability; and this means instability. This leads to a policy implication: if the authorities insist on maintaining the public spending/private investment ratio above 12l l , the economy will remain in an instability region, and the equilibria in which k > kM will not be reached. We shall now study this kind of steady states. 4.2

Results of stability regarding steady states where k > kM

As we can infer from the previous section, the steady states associated to k > kM verify that the savings/investment ratio is lower than l1; i.e. the public spending/private investment ratio is lower than 12l l . That is, it can be verified in these steady states that: k > kM ()

g~ 12l ry 1 ð11mÞg 12l < () < () < i ak l i l l

Proposition 2: In the case of a steady state with k > kM it is verified that: (1) For certain ranges of parametric values the steady state is locally asymptotically stable. (2) Two of the eigenvalues of (21) are real; one being negative, and one 1 . Thus, we can state that there is positive with a value lower than 11a at least one stable path. (3) If g is sufficiently close to 0, v sufficiently close to its maximum value a a 11s, and bð12sÞ > 11s s, the steady state is unstable. (4) If we are in a stable steady state, and all the parameters of the model are constant except the sensitivity to inflation in the Taylor Rule, the increase of this sensitivity will eventually make the steady state unstable, due to monetary policy overreactions.

Proof: Part 1 is proven directly by simulations made with Mathematica 10.0, calculating the corresponding eigenvalues of (21). In section 5, we shall present the results of some simulations, confirming that there are parametric configurations for which steady states associated with k > kM are locally asymptotically stable. The properties presented in part 2 of the proposition, are proven in Appendix A. Part 3. In Appendix A, it is proven that:


Simple Dynamic Model to Rethink Standard Policy k1 1k2 1k3 1k4 1k5 1k6 521

23

21v1bð12sÞ k 1a 11a y

() k1 1k2 1k3 1k4 1k5 1k6 521

1 11v bð12sÞ k 1 1 1a 11a 11a 11a y

From equation (18) we obtain: y5

bð12sÞ bð12sÞ bð12sÞ bð12sÞ d i g y1 vd1i1g () 15 1 v 1 1 11a 11a 11a 11a y y y

Bearing in mind (13) and (14), the above equation can be written as: bð12sÞ bð12sÞ g 1 s ak g 1 v 2 1 1 15 11a 11a a2v y a2v y y from where, bð12sÞ ak bð12sÞvs bð12sÞv g 1 511 2 11 11a y ð11aÞða2vÞ ð11aÞða2vÞ y And substituting in the sum of eigenvalues we obtain: 1 11v bð12sÞvs bð12sÞv g 1 1 2 11 k1 1k2 1k3 1k4 1k5 1k6 531 11a 11a ð11aÞða2vÞ ð11aÞða2vÞ y 1 As proven in Appendix A, k1 < 0 and 0 < k2 < 11a , therefore: 11v bð12sÞvs bð12sÞv g 1 2 11 k3 1k4 1k5 1k6 > 31 11a ð11aÞða2vÞ ð11aÞða2vÞ y a , we have: Then if g ! 0 and v ! 11s

k3 1k4 1k5 1k6 31

11v bð12sÞvs 1 1 531 ð11v1bð12sÞÞ 11a ð11aÞvs 11a

a s, we will have 9e > 0 tal que: Supposing that bð12sÞ > 11s

bð12sÞ5

a a e e s1e () bð12sÞ1 2 2 5a 11s 11s 2 2

As we suppose v to be sufficiently close to its maximum value, a 2 2e v. this 2e it is fulfilled that 11s


a 11s,

for


24 Then, we have: bð12sÞ1

a e e e 2 2 5a ) bð12sÞ1v2 a ) bð12sÞ1v > a 11s 2 2 2

Therefore, k3 1k4 1k5 1k6 > 4 We can conclude that an eigenvalue of (21) exists with modulus greater than one and, consequently, the steady state will be unstable. Regarding part 4, multiplying the six eigenvalues (a6 5ð21Þ6 k1 k2 k3 k4 k5 k6 ) we obtain the following expression (see Appendix A): Qð0Þ5a6 52

1 l q b hð12sÞ y yl21 ð11ap Þð11v1vsÞ ð11aÞ4 ð12lÞ k

q b hð12sÞ y Note that the part 2 ðl11a Þ4 ð12lÞ

1 y l21

k

ð11v1vsÞ does not change if all the

parameters of the model are maintained except the sensitivity to inflation in Taylors rule. Therefore, it is verified that as ap grows, there will be a moment at which the product will be, in absolute values, greater than 1. This means that one of the eigenvalues will have a modulus greater than 1 and, therefore, the equilibrium will be unstable. 䊏 Proposition 2 throws up interesting intuitions. First, stable steady states only exist when k > kM ; this result forces us to focus especially on these equilibria. For instance, looking at Proposition 2, we can say that stability requires (as a necessary condition) the public spending/private investment ratio to be below 12l l . Second, Proposition 2 reveals the possibility of apparent stabilities, which could be misleading. Since there may be saddle-path type instability, we can find paths which, though they seem to be stable, are not. Third, Proposition 2 reveals a degree of trade-off in the running of fiscal policy. Thus, we have seen that stability requires, as a necessary condition, < 12l; that is, with public that we be in steady states k > kM () ð11mÞg i l i spending g below a certain level: g < 12l 11m . However, on the other l hand, Part 3 of Proposition 2 shows that levels of g which are too low, contribute to the instability of the steady states above kM. To be specific, if g is close to 0, and the debt interests are high, then the steady state is unstable. As a consequence of the above, a certain level of public spending is seen to be necessary for stability in our model. This all leads us to a normative criteria for fiscal policy: the public spending/investment ratio must be



25

below 12l l , necessary for stability, but not too much lower, or we may face instabilities even in potentially stable regions. Finally, regarding monetary policy, part 4 of Proposition 2 shows another interesting result. Taken with caution, monetary policy acts, in the case of steady states above kM, as an equilibrating factor. However, an excessively forceful use of this policy could even break up previously stable situations, and lead to volatility. In section 5, we will strengthen globally the interpretation of the dynamics of our model through simulations. 5. SIMULATIONS

The results we obtain clearly show that the steady states of the model are not always stable; these states are unstable when associated with k < kM. Furthermore, as we have seen, even though k > kM, the steady state may be unstable. Owing to the difficulty in obtaining more formal results, we use simulations to check the existence of stable steady states associated with values of k 2 kM ; kS : The simulations were carried out using Mathematica 10.0. For the sake of transparency, and given the limitations of space, let us mention just some of the most important simulation results and their implications. 5.1 Simulation results (1) The steady states associated with k > kM can be locally asymptotically stable. To be specific, for a reference scenario such as this: g 5 0.6; v 5 0.0005; s 5 0.2; ap 5 0.06; ay 5 0.015; p* 5 0.02; y* 5 9.9; b 5 0.3; a 5 0.006; h 5 0.2; c 5 0.00084; l 5 0.4; b 5 0.9; q 5 0.001, the steady state ðk2 ; yðk2 Þ; iðk2 Þ; pðk2 Þ; Rðk2 Þ; dðk2 ÞÞ; with k2 > kM is locally asymptotically stable (with the highest modulus of the eigenvalues of (21) being 0.99973). The estimations in Day and Yang (2011), or the values for some parameters that appear in some of the simulations in Heijdra (2009) reveal that, both, our selected reference scenario, and the sensitivity ranges provided below, may reflect sensible economic conditions. In figure 2, we can see the trajectories for some relevant variables, 12l considering as initial conditions: k0 5 40; y0 5 qc k0 l ; i0 5 2; p0 5 0.025; R0 5 0.0251; d0 5 2. It can be seen how the trajectories evolve towards the stable steady state. We decided to represent the variations (not the levels) of the nominal interest rate and the



26

Figure 2. Stable steady state.

inflation rate, because the rates of change of these variables will help us to interpret the results of the other variables. As it can be seen in figure 2, the fact that jDRj > jDpj, would act as a (necessary but not sufficient) mechanism of global stabilization. This is so as, in these conditions, the real interest rate would slow down expansions (the nominal rate rises more than inflation) and would stimulate the economy in times of recession (the nominal rate falls more than inflation). We point out that the real interest rate in our model (not even that of the steady state) is not a natural interest rate in the Wicksellian sense; our rate does not come from any natural underlying profit rate; rather, it is just the real interest rate which assures that the propensity to invest equals replacement investment.



27

Figure 3. Unstable steady state.

(2) If g falls sufficiently, the steady state associated with k > kM can lose its stability. From the abovementioned parametric scenario, if the value of g decreases to, for example, 0.3, the steady state loses its stability, which corroborates that obtained in part 3 of Proposition 2. This is illustrated in figure 3. In this case, although the steady state is not stable for a reduced value of g, the fact (as seen in the figures) that jDRj > jDpj, means that the evolution of the real interest rate avoids (to an extent) an explosive process. (3) It is important to point out that the simulations can give way to misleading situations. This is so since a steady state can be unstable, while being a saddle-path type; this can give way to evolutions which seem to be stable but, in fact, are not. This is what happens, for example, in the conditions of point 1, if g falls to 0.4, thus making the equilibrium unstable.



28

Figure 4. Apparently stable steady state.

Figure 5. Increasing volatility around the steady state for a sufficiently high t.

The system evolves along an apparently stable path coming close to (without reaching) an unstable steady state (figure 4). Suddenly, the dynamics start to gain volatility, moving away from the equilibrium (figure 5). This result could make us rethink episodes such as the Great Moderation and its sudden transformation into the Great Recession. (4) Regarding the robustness of the standard scenario seen in point 1. (i) The value of g 5 0.6 is not the only one compatible with stable equilibria. With the parametric values of point 1, the previous g can be multiplied by, for example, a factor fg僆 (0.7, 1.7) without losing stability. (ii) A similar result can be obtained for b. Setting out from the parametric values of point 1, b can be multiplied by a factor fb僆 (0.1, 3.1) without losing stability. (iii) The value of h also allows for a range of variations. With the parametric values of point 1, parameter h can be multiplied by fh僆 (0.0001, 1.9) without losing stability.



29

(iv) The value of b also allows for a range of variations. With the parametric values of point 1, parameter b can be multiplied by fb僆 (0.9, 1.1) without losing stability. 5.2 Synthesizing the results To sum up, the first conclusion we draw from these simulations is the possible existence of stable equilibria in the model, as stated in Proposition 2. Likewise, it is worth pointing out that these simulations—specifically those associated with figures 4 and 5—prove another of the conclusions deduced from Proposition 2—that is, the existence of apparently stable equilibria which were not actually stable and could mislead economic previsions. Figures 4 and 5 show a clear stability in the mid term, but not in the long term, which can lead us to interpret long periods of economic history as being stable when in fact they are not. We pose the conjecture (which may require much more empirical and theoretical research) that certain episodes—such as the Great Moderation period—could be the result of this type of behavior. Let us note, also, that stability could be reached through dampened oscillations as seen in figure 2. In other words, the cyclical evolution is not anomalous but a usual case—its presence being possible in both stable and unstable situations, as can be seen in figures 2–4 and 5. Finally, as we are analyzing the model in normalized units, the dynamics correspond to fluctuating growth scenarios (in GDP, consumption, capital, etc.). 6. CONCLUDING REMARKS

The Great Recession obliges us to reflect on its causes and consequences, and leads us to rethink some macroeconomic policy issues which had apparently been resolved before the crisis. In this work, we have proposed a tractable model which, starting out from known hypotheses, allows us to analyze the origin of some economic instabilities. Based on this model, we have rethought classic questions: the role of fiscal policy; the role of monetary policy; short-term and long-term interactions; and conditions for stability/instability in industrialized economies. Regarding these questions, we can claim from the model that certain pre-crisis prescriptions of policy were less unquestionable than previously thought. Our formal analysis reveals that there are multiple steady states with different stability characteristics. Far from the usual conviction in mainstream macroeconomics that steady states are stable, our analysis shows many sources of instability. In this sense, it is interesting that, although under certain



30

conditions monetary policy helps to stabilize the dynamics, in other circumstances we show it can be a source of instability. Another interesting result is that fiscal policy plays a very relevant role in our model, not only in the short term, but also in determining the (alternative) steady states. Furthermore, fiscal policy interacts in our model with typical long-term factors (substitutability of inputs, technical change) by determining the ranges of stability/instability for the steady states, and the specific pattern of dynamic behavior (dampened oscillations, saddle-path type instability, etc.). Finally, the formal stability analysis, and the simulations, show that there exist ranges of the parametric space in which the steady states are unstable; as well as other ranges in which the system has trajectories which converge to asymptotically stable equilibria; and others where the system seems to stabilize, suddenly entering a regime of instability and increasing volatility. This is down to the existence of stable paths in a context of saddle-path type instability. These properties lead us to, at least, consider the possibility that the Great Moderation might not have been such a stable period as was previously thought, but more a process of approximation to a steady state which, perhaps having saddle-path type instability, ended up engendering increasing volatility and the Great Recession. The strong implications of this possibility—and the limited space of a single paper—mean we must leave a detailed exploration of this conjecture for future research. APPENDIX A

We calculate the characteristic polynomial of (21), developing QðkÞ5jJ2kI6 j for the sixth file: QðkÞ5jJ2kI6 j

1

11a 2k

0

0

5

1

2l y12l

q

12l k

0

0

0

1 11a

0

b ð12sÞ k 1a 2k 11a y

0

h y 11a

2

h y 11a

2k

h y 11a

2

h y 11a

0

12k

0

ay

0

ap

12k

2s 11a

0

0

0


k y l 1 y12l q 12l k a

0

0

b ð12sÞ v

11a

0

0

0

11v

2k

11a


31

1 1

0 0 0

11a 2k

11a

b ð12sÞ k h h

1a 2k 0 y 2 y

0

11a y 11a 11a

11v

k h h 2k

5 0 a 2k y 2 y

11a

y 11a 11a

l

1 1 y12l

2l y 12l

q

q 0 12k 0

12l k

12l k

0 ap 12k

0 ay

1

11a 2k

0

2s

0 11a

1

2l y12l

q

12l k

0


1 11a

0

0

h y 11a

2k

h y 11a

0

12k

0

ap

0 0

h b ð12sÞ v

2 y 11a 11a

h

5ð Þ1ðÞ 2 y 0

11a

0 0

12k 0


32 2

3

b ð12sÞ k h h

1a 2k y 2 y

0 6

11a y 11a 11a

7 6 7

6

7 6

k h h 7 6 7 y 2 y 7 0 a 6 1

6 y 11a 11a 7

6 11a

7 6

7 l 1 6

7 2l q y 12l q y 12l 6

7 12k 0 6

7 12l k 12l k 6

7 6

7 6 7

ap 12k 7 0 ay 6 6 7 11v 7 2k 6 ðÞ5 6 7 11a

7 6

7 6 1

7 6 0 0 0

11a 2k

7 6

7 6

7 6

7 6 b ð12sÞ k h h

6 1a 2k y 2 y

7 0

6 11a y 11a 11a 7 6 2k

7

7 6

6 7 l 1

2l q y12l

7 q y12l 6

7 6 12k 0

12l k

7 6 12l k

5 4

0 ay ap 12k

3

b ð12sÞ k h h

11a 1a y 2k 11a y 2 11a y 7 6

7 6

7 1

y12l 6 l q

62 k h h 7

6

y 2 y 7 a 6 ð11aÞð12lÞ k 7

y 11a 11a

7 6

6

7 6 7

6 ap 12k 7 ay 6 7 11v 6 7 2k 6 5 7 6 7 11a

6 7

b ð12sÞ k h h 6 7

1a 2k y 2 y

6 7

11a y 11a 11a 6 7

6 7

1 6 7

l 6 2k 7 2k

q y12l

6 7

12k 0 11a 6 7

12l k 4 5

ay ap 12k 2



33

2

3

bð12sÞ

2k 0 0 6 7

11a

6 7

1 6 7

lq y 12l

62 7

k h h 6 ð11aÞð12lÞ k 7

a y2 y

6 7

y 11a 11a 7

6 6 7

11v 6 7

2k 6 5 ap 12k ay 7 6 7 11a 2 3 l y12l 6 7 bð12sÞ k hqa y p 2 6 7 6 77 6 11a 1a y 2k ð12kÞ 2 ð11aÞ ð12lÞ k 6 6 77 1 6 2k 77 2k 6 6 6 77 l y12l 11a 4 4 55 hay y hqy ð12kÞ2 1 ð12kÞ 11a ð11aÞ ð12lÞ k

1 y12l 11v lq b ð12sÞ hy 2k 2k 52 ð11ap 2kÞ 11a ð11aÞ ð12lÞ k 11a 11a 3 b ð12sÞ k 1a 12 k2 2k3 1 7 6 11a y 7 6 7 6 6 l 7 7 6 h a y 2b k h q y y ð12sÞ 12l y 6 k7 7 62 11 11a 12a y 1 11a 2 ð11aÞ ð12lÞ k 7 6 11v 1 7 2 2k 6 2k k 7 6 l 11a 11a 7 6 b ð12sÞ k h q a y y 12l p 7 61 1a 2 7 6 11a 11a 12l y ð Þ ð Þ k 7 6 7 6 7 6 l 5 4 h ay y hqy y 12l 2 1 11a ð11aÞ ð12lÞ k 2

1

11a 2k

0 sb ð12sÞ v

ðÞ5 2

ð11aÞ 2l y12l 1

q

12l k

0


1 11a 2k 0 0

h h

y 2 y

11a 11a

12k 0

12k

ap 0

0


34

3

h h

2k

y 2 y 6 7

11a 11a

6 7

6 7

1 6 7

6 11a 2k 0 7

12k 0 6 7

6 7

6 7

6 7

0

ap 12k 6 7 6 7 6 7 7 sb ð12sÞ v 6 6 7 5 6 2

7 ð11aÞ 6 h h 7

6 7

y 2 y 7 0 6

7 6 11a 11a

7 6

7 6

7 6 1

1 y12l

7 62 2l

7 6 11a q 12k 0

12l k

7 6

7 6

5 4

12k

0 ap 1 y12l sb ð12sÞ v 1 ql h 2 2k k ð12kÞ 2 y ð11ap 2kÞ 2 5 11a ð11aÞð12lÞ k 11a ð11aÞ2 2

Therefore, QðkÞ5jJ2kI6 j5Q3 ðkÞ1Q6 ðkÞ1Q4 ðkÞ1Q1 ðkÞ, with: 1 y12l 11v lq b ð12sÞ hy Q3 ðkÞ52 2k 2k ð11ap 2kÞ 11a ð11aÞ ð12lÞ k 11a 11a

11v 1 2k k 2k Q6 ðkÞ52 11a 11a 2 3 b ð12sÞ k 3 2 2k 1a 12 k 1 6 7 11a y 6 7 6 7 6 7 6 7 l 2b ð12sÞ k h ay y hqy y 12l 7 6 6 2 11 12a 1 2 k7 6 7 11a y 11a ð11aÞ ð12lÞ k 6 7 7 36 6 7 l 6 b ð12sÞ 7 k h q ap y y 12l 61 7 1a 2 6 7 11a y ð11aÞ ð12lÞ k 6 7 6 7 6 7 6 7 l y12l 4 h ay y 5 hqy 1 2 11a ð11aÞ ð12lÞ k


Simple Dynamic Model to Rethink Standard Policy 2 52

11v 1 2k k 2k 11a 11a

Q4 ðkÞ52

Q1 ðkÞ52

sb ð12sÞ v ð11aÞ2 sb ð12sÞ v ð11aÞ2

6 6 6 4

l 3 y12l b ð12sÞ k h q ap y 1a 2k ð12kÞ2 2 7 11a y ð11aÞ ð12lÞ k 7 7 l y12l 5 h ay y hqy ð12kÞ2 ð12kÞ 1 11a ð11aÞ ð12lÞ k

1 2k k ð12kÞ2 11a

1 y12l ql h y ð11ap 2kÞ ð11aÞð12lÞ k 11a

With simple substitution it is easy to see that the following is verified: lim QðkÞ511

k!21

lim QðkÞ511

k!11

Qð0Þ5Q3 ð0Þ1Q6 ð0Þ1Q4 ð0Þ1Q1 ð0Þ5Q3 ð0Þ1Q1 ð0Þ < 0 1 1 1 1 1 Q 5Q3 1Q6 1Q4 1Q1 11a 11a 11a 11a 11a 1 1 vlahy a 1Q1 5 1a 5Q3 p 11a 11a ð11aÞ4 ð12lÞ 11a 1 y12l 12bð12s2 Þ > 0 3 k Applying the Bolzano theorem, we can deduce: ) lim QðkÞ511 k!21 ) 9k1 2 ð21; 0Þ so that Qðk1 Þ50 Qð0Þ < 0 9 Qð0Þ < 0 > = 1 so that Qðk2 Þ50 ) 9k2 2 0; 1 11a Q > 0> ; 11a


35


36

Grouping together the same degree terms in QðkÞ5jJ2kI6 j, we obtain the characteristic equation of (21), written as QðkÞ5k6 1a1 k5 1a2 k4 1a3 k3 1a4 k2 1a5 k1a6 50, where: 21v1bð12sÞ k a1 52 21 1a 11a y

a6 52

1 l q b hð12sÞ y yl21 ð11ap Þð11v1vsÞ ð11aÞ4 ð12lÞ k

And it is verified that a6 5ð21Þ6 k1 k2 k3 k4 k5 k6 , thus, taking into account that a6 < 0, and the signs of k1 < 0 and k2 > 0, we deduce that k3 k4 k5 k6 > 0: It is also verified that a1 52ðk1 1k2 1k3 1k4 1k5 1k6 Þ, thus, k1 1k2 1k3 1k4 1k5 1k6 521

21v1bð12sÞ k 1a : 11a y

APPENDIX B

In this appendix we present the continuous time version of the model—system of equation (10)—and we explain how to check that the essential results do not depend on whether we use discrete time or continuous time treatments. Considering the approximation x_ xt11 2xt in all the variables of the discrete model (10), the following model is obtained in continuous time: 8 1 > _ > ðk1iÞ2k k5 > > > 11a > > > > > bð12sÞ 1 > > _ y5 ðy1vd Þ1 ½b2hðR2pÞy1g2y > > 11a 11a > > > > > 1 > > _ ½b2hðR2pÞy2i < i5 11a (B1) 1 y 12l > > > p52c1q > _ > > kl > > > > > > _ > R5a p2p y2y 1a p y > > > > > > > 1 > : d_ 5 ½ð11vÞd1g2sy2d 11a



37

_ Imposing the condition of steady state for all the variables, x50, we are left with the same system of equations as that which defines the stationary states in the discrete model (see (11)). Therefore, the results regarding the existence and multiplicity of equilibria, as well as the relationships between the variables in the stationary state are maintained as in the discrete case. Regarding the stability of the stationary states, the condition which assures local asymptotic stability is that the eigenvalues of the Jacobian matrix of (B1), evaluated at the equilibrium point, have a real negative part. The Jacobian matrix of (B1) evaluated at the equilibrium point is: 0 1 1 1 21 0 0 0 0 B 11a C 11a B C B b ð12sÞ k h h b ð12sÞ v C B C 0 1a 21 0 y 2 y B C B 11a y 11a 11a 11a C B C B C k h h B C 21 y 2 y 0 0 a B C C y 11a 11a JC 5B B C B C l 1 1 y 12l B 2l y 12l C Bq C q 0 0 0 0 B 12l k C 12l k B C B C 0 ay 0 ap 0 0 B C B C @ A 2s 11v 0 0 0 0 21 11a 11a (B2) It is immediate that if the Jacobian matrix of the discrete system evaluated at the equilibrium point (see (21)) is JD, then it is fulfilled that: JC 5JD 2I6 . Hence, we verify: jJC 2kI6 j5jJD 2I6 2kI6 j5jJD 2ð11kÞI6 j. Labeling the JC eigenvalues as kC and the JD ones as kD, the following relationship is deduced: kD 5kC 11 () kC 5kD 21. From this relationship it can be deduced that the stability results of the steady states are maintained for the continuous case. Thus, the stationary state associated with k < kM is unstable, although it has at least two stable trajectories. With respect to the steady state k > kM , there will be parametric configurations for which the state will be locally asymptomatically stable (see figure A2.1), but there will be other configurations for which it will be unstable (see figure A2.2) (and in this case there will also be at least two stable trajectories). Furthermore, we can deduce that the



38

continuous time model is more stable than the discrete one—that is, there are parametric configurations which make the steady state k > kM unstable in the discrete model, but are asymptomatically stable in the continuous model as seen in the following simulation (B1) for the parametric values of the simulation of the discrete model shown in figure 3.

Figure A2.1. Stable steady state in the continuous model.

In the following figure we show a simulation of our model (B1) in which the steady state k > kM is unstable. The parametric values and initial conditions taken are: g 5 0.1; v 5 0.0005; s 5 0.2; ap 5 12; ay 5 10; p* 5 0.02; y* 5 9.9; b 5 0.3; a 512l 0.006; h 5 0.2; c 5 0.00084; l 5 0.4; k0 l ; i0 5 2; p0 5 0.025; R0 5 0.0251; b 5 0.9; q 5 0.001; k0 5 40; y0 5 qc d0 5 2.

Figure A2.2. Unstable steady state in the continuous model.



39

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Heijdra, B. J. (2009): Foundations of Modern Macroeconomics, 2nd edn, Oxford University Press, Oxford. Howitt, P. (1992): Interest rate control and nonconvergence to rational expectations, Journal of Political Economy, 100, pp. 776–99. Howitt, P. (2006): Monetary policy and the limitations of economic knowledge, in Colander, D. (ed.): Postwalrasian Macroeconomics, Cambridge University Press, New York. Isaac, A. G. (2009): Monetary and fiscal interactions: short-run and long-run implications, Metroeconomica, 60 (1), pp. 197–223. Kirman, A. (2010): The economic crisis is a crisis for economic theory. CESifo Economic Studies, 56, pp. 498–535. Krugman, P. (2000): How complicated does the model have to be?, Oxford Review of Economic Policy, 16, pp. 33–42. Krugman, P. (2011): The profession and the crisis, Eastern Economic Journal, 37, pp. 307–12. Lavoie, M. (2006): A post-Keynesian amendment to the new consensus on monetary policy, Metroeconomica, 57, pp. 165–92. Orphanides, A. (2003): The quest for prosperity without Inflation, Journal of Monetary Economics, 50, pp. 633–63. Romer, D. (2012): Advanced Macroeconomics, McGraw-Hill, New York. Sayer, S. (1995): Macroeconomic policy analysis, Metroeconomica, 46, pp. 290–308. Solow, R. M. (1956): A contribution to the theory of economic growth, Quarterly Journal of Economics, 70, pp. 65–94. Solow, R. M. (2008): The state of macroeconomics, Journal of Economic Perspectives, 22, pp. 243–9. Stiglitz, J. (2011): Rethinking macroeconomics: what failed and how to repair it, Journal of the European Economic Association, 9, pp. 591–645. Stock, J., Watson, M. (2008): Phillips curve inflation forecasts, NBER Working Paper Series, 14322. Tamborini, R. (2010): Monetary policy with investment-saving imbalances, Metroeconomica, pp. 473–509. Taylor, J. B. (ed.) (1999): Monetary Policy Rules, University of Chicago Press, Chicago. Turnovsky, S. J. (1977): Macroeconomic Analysis and Stabilization Policy, Cambridge University Press, Cambridge, MA. Vogt, W. (1996): Accelerating inflation or rising unemployment: is there an alternative?, Metroeconomica, 47, pp. 197–230. Woodford, M. (2003): Interest and Prices, Princeton University Press, New Jersey. Isabel Almudi University of Zaragoza Gran Via 2 50005 Zaragoza Spain E-mail: [email protected]

Francisco Fatas-Villafranca University of Zaragoza Gran Via 2 50005 Zaragoza Spain E-mail: [email protected]

Gloria Jarne University of Zaragoza Gran Via 2 50005 Zaragoza Spain E-mail: [email protected]

Julio Sanchez-Choliz University of Zaragoza Gran Via 2 50005 Zaragoza Spain E-mail: [email protected]


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