Money and external habit persistence

Economics Letters 76 (2002) 121–127 www.elsevier.com / locate / econbase

Money and external habit persistence A tale for chaos ´ ` c Stephane Auray a , *, Fabrice Collard b , Patrick Feve a

Gremaq, UMR CNRS 5604, University of Nantes, ( LEN-C3 E), Universite´ des Sciences Sociales, ˆ ` ´ ´ de Brienne, 31000 Toulouse, France F, 2 eme etage , 21 Allee Manufacture des Tabacs, Batiment b University of Toulouse ( CNRS–GREMAQ and IDEI), Toulouse, France c University of Toulouse ( GREMAQ and IDEI), Toulouse, France Received 20 May 2001; received in revised form 18 December 2001; accepted 7 January 2002

Abstract This paper studies the dynamic properties of a simple monetary economy model with external habit formation, in which money is used for transaction motives. We show that large enough (though reasonable) habit can generate deterministic cycles and chaotic equilibria.  2002 Elsevier Science B.V. All rights reserved. Keywords: Monetary economy; External habit; Deterministic cycles; Chaos JEL classification: E32; E4

1. Introduction Modern macroeconomics largely builds on explicitly dynamic models which, in some cases, generate multiplicity of equilibria and complex dynamics. Monetary economies can easily display such dynamic properties (see e.g. Grandmont, 1985; Matsuyama, 1990, 1991; Woodford, 1994; Michener and Ravikumar, 1998). This paper pursues this line of research. We consider a monetary economy in which households’ preferences are characterized by habit persistence, therefore introducing intertemporal complementarity. An important feature of our modeling of habit persistence, is that it is not internalized by the agents. The dynamic properties of the economy are thus summarized by a first order nonlinear difference equation. Our results suggest that deterministic cycles and chaotic motion occur for very plausible values of the habit parameter. For instance, even for very low elasticity of labor supply, the results hold as the critical value for habit persistence lies within point * Corresponding author. Tel.: 133-561-128-560. E-mail address: [email protected] (S. Auray). 0165-1765 / 02 / $ – see front matter PII: S0165-1765( 02 )00035-6

 2002 Elsevier Science B.V. All rights reserved.

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estimates from both individual and aggregate data. The paper is organized as follows. The next section presents a monetary model with external habit. Section 3 characterizes the dynamics of the economy. The last section offers some concluding remarks.

2. The model economy The economy is comprised of a unit mass continuum of identical infinitely lived agents, so that we will assume that there exists a representative household in the economy. This household enters period t with real balances m t /Pt brought into period t from the previous period. These real balances are then used to purchase consumption goods. Therefore, the household faces the following constraint on the goods market: m ]t > c t . Pt

(1)

Money is therefore held purely for transaction motives. The household supplies its hours on the labor market at the real wage Wt . During the period, it also receives a lump-sum transfer from the monetary authorities in the form of cash equal to Nt /Pt . All these revenues are then used to get money balances for the next period, in order to finance future consumption streams. Therefore, the budget constraint is simply written as: Nt m t 11 Wt h t 1 ] > ]]. Pt Pt

(2)

Each household has preferences over consumption and leisure represented by the following intertemporal utility function: h Ob Flog (s ) 2 ]] G 11w 11 w t

`

max

t

t

(3)

t50

where b [ (0, 1) is the discount factor and w $ 0 is the inverse of the labor supply elasticity. h t denotes hours supplied by each household and s t is the consumption index from which the household derives utility. Implicit in this formulation is that we allow for habit persistence in the consumption behavior, and therefore introduce time non-separability in the utility function. We consider external habit specified in difference with one lag in aggregate consumption which is unaffected by any one agent’s decision, therefore joining the catching up with the Joneses literature (Abel, 1990): s t 5 c t 2 u C¯ t21 with u [ (0, 1)

(4)

where C¯ t21 is the economy-wide average consumption. The consequences of such a specification for habit persistence on the dynamic properties of a monetary economy have not been, to our knowledge, investigated, although it may be of importance as it creates some intertemporal complementarity in the money demand behavior. The household determines its optimal money holdings and labor supply plans maximizing (3) subject to (2) and (1). Note that the monotonicity of the utility function, reflecting the nonsatiation

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assumption, insures that the two constraints bind at the household’s optimum. Money demand behavior together with labor supply yields: hw Pt m t11 ]t 5 b ]] ]] 2 u C¯ t Wt Pt11 Pt 11

S

D

21

.

(5)

The technology is described by a constant returns to scale production function Yt 5 h t , such that in equilibrium the real wage is Wt 5 1. Money is exogenously supplied by the central bank according to the following money growth rule Mt11 5 m Mt where m $ 1 is the gross rate of growth of money, such that Nt 5 Mt 11 2 Mt 5 ( m 2 1)Mt . The good and labor market clearing condition implies y t 5 c t 5 h t and the money market clearing requires m t 11 5 Mt 11 5 Mt 1 Nt . Since the two constraints are ] binding—implying that Ct can be expressed in terms of real balances—the equilibrium can be written as:

S D M ]t Pt

11 w

S

Mt b Mt 11 Mt 11 5 ] ]] ]] 2 u ] m Pt11 Pt 11 Pt

D

21

.

(6)

This last equation determines the path of real balances—or equivalently of output—and therefore constitutes the key relationship in the remainder of the paper.

3. Dynamic properties of the economy To study the dynamic properties of this economy, it is more convenient to express (6) in terms of the inverse of real money balance pt 5 Pt /Mt :

S]p1 D t

11 w

1 u 2 ]D S DS]] p p

b 1 5 ] ]] m pt 11

t 11

21

t

which after some manipulation rewrites: pt11 5 (1 /u )pts1 2 ( b /m )p t11 wd ; F( pt ; u, w ).

(7)

We obtain a logistic map when the elasticity of labor supply is infinite (w 5 0): pt11 5 (1 /u )pt s1 2 ( b /m )ptd. As the dynamic behavior of the economy crucially depends on the parameter u, we determine the conditions on this parameter with respect to the elasticity of labor supply such that a three-period cycle appears. ] Proposition 1. There exists a unique value u * [ (u], u ) with 11w ] 11w u] 5 ]]]]] u 5 ]] 21 w / 11 w , 21w (2 1 w ) such that:

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] .p] $ F 3 (p) ] . F 2 (p) ] F(p) ] 5 0. where ]p satisfies F9(p) Proposition 1 establishes conditions on the external habit parameter for which there exists a three-period cycle and chaos in the Li and York sense. When w 5 0, we can use directly the conditions on the logistic map. For 1 / 3 , u , 1, the model admits two fixed points, i.e. 0 and (1 2 u )m /b. The first is unstable, while the second satisfies uF9u , 1. For u 5 1 / 3, this fixed point becomes unstable (F9 5 2 1) and a two-period cycle occurs. It is worth noting that the value of u for which there is a cycle of order three is reasonable (around 0.26). Further, as established by the following proposition, the elasticity of labor supply plays a very important role in the emergence of cycles and chaos. ] Proposition 2. The two bounds of the (u], u ) interval satisfy (i) ] ≠u ≠u] ] . 0 and ] . 0 ≠w ≠w (ii) ] u 5 1. lim u 5wlim →` ]

w →`

Proposition 2 shows that complex dynamics occur when the elasticity of labor supply is sufficiently high. Conversely, when labor supply is inelastic, i.e. w → `, the critical value reaches its upper bound (u * 5 1). With w infinite, any change in current money demand behavior cannot affect directly the labor supply behavior. Since labor supply remains constant, the labor income and thus current consumption cannot increase following for instance higher inflation expectations. Because each individual in the economy has the same labor supply behavior, the aggregate consumption remains constant and external habit has no effect on individual behavior. For illustrative purposes, Table 1 presents the critical values for u with respect to w. The critical value for u increases significantly with w, but remains reasonable. Microeconomic evidence on variations of hours worked suggests an elasticity lower than unity (see Browning et al., 1999). For instance, when w 5 2, the elasticity of labor supply is equal to 0.5 and the critical value for u is 0.50. This value is supported by empirical evidence. For instance, Constantidines and Ferson Table 1 Critical values with respect to w ] w u* u ]u 0 0.25 0.26 0.5 1 0.38 0.41 0.67 2 0.47 0.50 0.75 5 0.62 0.67 0.86 10 0.73 0.78 0.92

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Fig. 1. Bifurcation diagrams.

(1991) and Braun et al. (1993) obtain an estimated value of u that lies within [0.5; 0.9] on macro data. Habit persistence appears to be lower but still significant on micro data. Naik and Moore (1996) report estimates for habit persistence on food consumption data of 0.49. Fig. 1 illustrates the effect of w on the bifurcation diagrams. As can be seen from this figure, higher labor supply elasticities yield endogenous cycles for lower values of the habit persistence parameter.

4. Concluding remarks This paper introduces habit persistence as a device to obtain deterministic cycles and chaotic motion in infinitely-lived agents’ monetary models. We show that the elasticity of the labor supply can affect the dynamic properties of the economy but critical values of habit persistence always remain reasonable compared to point estimates.

Acknowledgements We wish to thank F. Portier, Y. Le Pen, A. Martineau and G. Verdier for helpful comments. All remaining errors are ours.

Appendix A. Proof (Proposition 1) 1 / (11 w ) ) such that F9( p) 5 0, F9( p) . 0 if p ,p] and Firstly, there exists a unique ]p [ (0, ( m /b ) ] F9( p) , 0 if p .p. 1 / (11 w ) ] if u #]u 5 (1 1 w ) / ) and p* $p Secondly, there exists a unique fixed point p* [ (0, ( m /b ) (2 1 w ). Moreover, F( p) . p if p , p* and F( p) , p if p . p*.

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The fixed point p* is locally stable if u [ ((1 1 w ) /(3 1 w ), 1) and becomes unstable for ] .p] and F(p) ] . F( p*). u # (1 1 w ) /(3 1 w ). For u , (1 1 w ) /(2 1 w ), F(p) 2 ] 3 ] 2 ] ] ] ] ] As F9( p) , 0 for p .p, F (p) 5 F(F(p)) ,p. For p ,p, F( p) . p and it follows that F (p) . F (p). We have:

S

S

S

1 ] 11w b 1 11w ] 5 ] F 2 (p) 1 2 ] ] p] ]] 2 p ]] 2 1 w m u 21w u

S

D DD 11 w

D

1 2] b ] 5] ] 11 w . F 3 (p) F (p) 1 2 ] F 2 (p) u m 2 ] ] 5 0. Therefore, a necessary condition for F 2 (p) ] 5 0 and 5 0, then F 3 (p) As ]p ± 0 and w $ 0, if F (p) ] 3 ] 21 w / 11 w 3 ] ] if u 5u 5 (1 1 w ) /(2 1 w ). By . Moreover, F (p) 5 F(p) F (p) 5 0 is u 5u] 5 (1 1 w ) /(2 1 w ) the chain rule differentiation, we have: 3 ] 3 ] 2 ] ] ≠F (p) ≠F (p) ≠F (p) ≠F(p) ]]] 5 ]]] ]]] ]] . ] ] ≠F(p) ≠u ≠u ≠F 2 (p)

] F9( p) . 0 and it follows that ≠F 3 (p) ] / ≠F 2 (p) ] . 0. Moreover, for p .p, ] F9( p) , 0 and For p ,p, 2 ] 3 ] ] ] ] ≠F (p) / ≠F(p) , 0. Finally, ≠F(p) / ≠u 5 2 (1 /u )F(p) , 0. It follows that ≠F (p) / ≠u . 0 and there 3 ] ] 5 F(p). h exists a unique value of u such that F (p) Proof ( Proposition 2) (i) can be simply established by computing the explicit derivatives of the two bounds. Differentiat] ing ]u and u with respect to w, we obtain: ] log(2 1 w ) ≠]u ≠u 1 ] 5 ]]]]]]] , ] 5 ]]]. ≠w (1 1 w )(2 1 w )21 w / 11 w ≠w (2 1 w )2 As w $ 0, these two derivatives are trivially positive. (ii) is immediate. h

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Matsuyama, K., 1991. Endogenous price fluctuations in an optimizing model of a monetary economy. Econometrica 59, 1617–1631. Michener, R., Ravikumar, B., 1998. Chaotic dynamics in a cash-in-advance economy. Journal of Economic Dynamics and Control 22, 1117–1137. Naik, N.Y., Moore, M.J., 1996. Habit formation and intertemporal substitution in individual food consumption. The Review of Economics and Statistics 78 (2), 321–328. Woodford, M., 1994. Monetary policy and price level determinacy in a cash-in-advance economy. Economic Theory 4, 345–380.