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On competitive cycles and sunspots in productive economies with a positive money stock
Research in Economics 59 (2005) 137–147 www.elsevier.com/locate/yreec
On competitive cycles and sunspots in productive economies with a positive money stock Guido Cazzavillana, Patrick A. Pintusb,* a
b
Department of Economics, University Ca’Foscari di Venezia, and SET, Italy Department of Economics, University de la Mediterranee, and GREQAM, France
This paper has been written in memory of Philippe Michel
Abstract This paper shows that stationary sunspot equilibria may occur in productive, overlappinggenerations economies when the stock of outside money is positive. More precisely, a destabilizing real-balance effect makes sunspots compatible with positive money, substitutable inputs and inelastic labor. Key to the results is the presence of capital externalities associated with learning-bydoing creation of knowledge. q 2005 University of Venice. Published by Elsevier Ltd. All rights reserved.
1. Introduction It is by now well known that monetary overlapping-generations (OLG thereafter) economies may exhibit cyclical and sunspot equilibria (see e.g. Gale 1973; Shell 1977; Cass et al. 1979; Farmer and Woodford 1984; Azariadis 1981; Grandmont 1985; Azariadis and Guesnerie 1986). However, recent contributions focusing on productive economies turn out to rely on stringent assumptions: large income effects (see Grandmont 1986; Jullien 1988), input complementarity and elastic labor supply (see Woodford 1986; Grandmont et al. 1998), negative outside money or public debt (see Farmer 1986; Benhabib and Laroque 1988). The purpose of this paper is to show that sunspots and cycles may occur in a version of the OLG model with productive investment, inelastic labor and large input substitution. Within the very same framework, Benhabib and Laroque (1988) have shown that * Corresponding author. E-mail addresses: [email protected] (G. Cazzavillan), [email protected] (P.A. Pintus).
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the emergence of cyclical equilibria requires both the quantity of outside money to be negative, at the golden rule, and inputs to be complementary. We will demonstrate that neither of the latter conditions is needed when capital externalities are introduced into the economy: stationary sunspots may occur when the quantity of money is positive and inputs are substitutable enough. We focus on external effects through the average capital stock (creating spillovers by learning-by-doing creation of knowledge), which have been repeatedly claimed, since the seminal contributions by Arrow, Shell, Frankel and Romer, to be important to understand growth and business cycles. Although the same model was studied by Benhabib and Laroque (1988), some departures with the present work should be mentioned. First, we emphasize conditions leading to the existence of stationary sunspot equilibria (around steady states or cyclical equilibria), whereas Benhabib and Laroque (1988) focused on cycles. In contrast with the latter authors, we restrict the analysis to the case of inelastic labor, which allows us to deal with a twodimensional dynamical system that can be studied geometrically by applying some results by Grandmont et al. (1998). Moreover, abstracting from labor movements helps to highlight the critical mechanisms that are at work: as in the work of Benhabib and Laroque (1988), sunspots and cycles are originated by a destabilizing real balance effect, when money and capital are assumed to be alternative ways of savings that pay the same rate of return. This may be intuitively described as follows. In the economy with constant returns to scale at the social level (absent the externalities) and negative money, optimistic expectations may be self-fulfilling because two conflicting effects on savings coexist. When the capital stock is, say, higher than its steady state value, the wage rate is higher. On the contrary, the rate of return on capital is lower and this leads, by arbitrage, to a lower real return on money (that is, the price of the consumption good in terms of money goes up). Therefore, this generates two conflicting effects on savings and, therefore, on tomorrow’s capital stock: while the wage effect tends to increase savings and investment in physical capital, the decline in the real value of (negative) money tends to depress savings. Although the wage effect dominates initially, it is eventually overcome by the real balance effect so that the economy may follow cyclical and sunspot equilibria. Now with capital externalities, sunspots occur with positive money. Here again, a growing capital stock tends to increase savings, as the wage rate goes up. However, both the rate of return on capital and the real return on money now go up (that is, the price of the consumption good goes down), which lowers savings in the form of physical capital. This shows that financial mechanisms, although often absent in the most recent literature, may also be important in understanding how endogenous fluctuations occur (see also, on this point, the related literature on bubbles, e.g. Blanchard 1979; Tirole 1985, and on cash-in-advance constraints, e.g. Michel and Wigniolle 2003). In Section 2, the model and dynamics are analyzed, while Section 3 derives and discusses the conditions leading to cyclical and sunspot equilibria. Section 4 gathers some concluding remarks.
2. Model and intertemporal equilibria The economy is composed of agents who work and save in the form or productive capital and outside money while young, consume a unique good while old, and achieve maximal
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utility derived from consumption of the unique good and leisure V2(ctC1)KV1(lt), where ctC1O0 is consumption demand by the old and ltO0 is labor supplied by the young. The budget constraints corresponding to active periods of a representative household born at time tO0 are therefore: ktC1 C MtC1 qt Z ut lt ;
ctC1 Z RtC1 ktC1 C MtC1 qtC1 ;
(1)
where ktC1O0, MtC1O0 denote, respectively, capital and money demands, and qt, ut, RtC1 stand, respectively, for the price of money, the real wage and the gross return on capital (including capital depreciation), all expressed in terms of the produced good which is chosen to be the numeraire. Benhabib and Laroque (1988) assume away any possibility of arbitrage between the two available assets so that the rates of return of capital of money are identical at equilibrium, i.e. RtC1ZqtC1/qt at all dates. Under constant returns to scale, technology is summarized by a production function F that is homogeneous of degree one that may, therefore, be written F(k, l)ZAlf(k/l), where AO0 is total factor productivity and f is a strictly concave function. In contrast with Benhabib and Laroque (1988), we assume that total factor productivity is an increasing 0, i.e. AZ jðkÞ, following, among the others, function of the average capital stock kO Benhabib and Farmer (1994) and Cazzavillan et al. (1998). Accordingly, competitive return on capital and real wage are, in equilibrium, respectively: RZ def def jðkÞf 0 ðk=lÞC 1K dZ Rðk; lÞ and uZ jðkÞ½f ðk=lÞK kf 0 ðk=lÞ=lZ uðk; lÞ, where 0!d!1 is the rate of capital depreciation. Finally, the quantity of money is assumed to be constant through time and equal to M. It is then easily shown that the set of intertemporal equilibria with perfect foresight (equilibria in the sequel) is summarized by a three-dimensional dynamical system that may be expressed, for instance, in terms of the capital stock k, labor l and the price of money q (see Benhabib and Laroque 1988, Prop. I.4, p. 148). Definition 2.1. An intertemporal competitive equilibrium with perfect foresight is a 3 sequence (kt, lt, qt) of RC , tZ1,2,., such that 8 v ðRðktC1 ; ltC1 Þuðkt ; lt Þlt Þ Z v1 ðlt Þ > < 2 ktC1 Z uðkt ; lt Þlt K Mqt (2) > : qtC1 Z RðktC1 ; ltC1 Þqt : The first equality in Eq. (2) expresses the first order condition describing the intertemporal allocation of consumption and leisure, where v2 ðcÞZdef cV20 ðcÞ and def v1 ðlÞZ lV10 ðlÞ. The second and third identities in Eq. (2) are, respectively, the first period budget constraint and the no-arbitrage condition, for an agent born at date t. Under constant returns, standard assumptions on preferences and technology (see Benhabib and Laroque 1988, p. 147) are sufficient to establish uniqueness of the monetary (or golden rule) steady state, i.e. a unique stationary solution of Eq. (2) such that the price of money is strictly positive. In our model with externalities, it is not difficult to adapt the proof of Cazzavillan et al. (1998), Prop. 3.1, p. 68, to establish the existence of a normalized steady state. To save space, this is left to the reader.
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We now study the local dynamics around the monetary steady state, i.e. the behavior of the dynamical system given by the (locally) invertible map (ktC1, ltC1, qtC1)ZG(kt, lt, qt) near the steady state, by studying the linearized map obtained from Eq. (2).1
3. Sunspot equilibria with a positive money stock Benhabib and Laroque (1988) give analytical conditions for the occurrence of flip and Hopf bifurcations (see their Propositions III.1 and 3, pp. 154–5). Although the authors restrict their study to deterministic cycles, an additional interesting result emerges in their diagrams, though not pointed out by the authors: in their Figs. 1–3, two eigenvalues are stable, i.e. have modulus less than one, when both the ratio of capital over consumption is large enough and the elasticity of factor substitution is sufficiently large (see Benhabib and Laroque 1988, pp. 158–9). Since the capital stock is the only predeterminate variable, this means that the steady state is locally indeterminate and, therefore, that one expects stochastic equilibria to exist near the steady state when the elasticity of capital-labor substitution is not too low. However, this requires a negative quantity of outside money. As we shall now see, this result holds true with positive money, provided that capital externalities are present. For simplicity, we will focus on inelastic labor supply. In that case, the model is then summarized by a two-dimensional map, as pointed out by Benhabib and Laroque (1988), Prop. II.1. Accordingly, we can apply previous results to show that sunspot equilibria exist in this economy, when factors are not too complementary. It is therefore expected that stochastic equilibria also occur when labor is not too elastic. In view of the first order condition in Eq. (2), we see that labor supply does not vary with the real wage u if the function v2(c)ZcV 0 2(c) is constant or equivalently if the elasticity of the marginal utility function V 0 2(c*) is equal to K1 at the steady state c* (the utility function is then equivalent to the logarithmic function at this point). Under this assumption, the first equation of (2) yields, under appropriate boundary conditions, a unique (interior) stationary labor supply ltZl*. Accordingly, the last two equalities of Eq. (2) then summarize the dynamics of the economy so that the corresponding dynamical system is two-dimensional. Therefore, in the case of inelastic labor, we need to linearize only the last two equations in (2) around the monetary steady state, under our assumption that ltZl* for all dates, i.e. that 3v2 ðc ÞZ 0, where the notation 3v2 stands for the elasticity of the function v2. In particular, all relevant information regarding the local stability of the golden rule steady state reduce to the trace T and the determinant D of the Jacobian matrix of the map given by the last two equations of (2), i.e. respectively, the sum and the product of the eigenvalues of this matrix, evaluated at the steady state.
1
The dynamics arising in the model without money have been studied by Reichlin (1986) and Cazzavillan and Pintus (2004), among others.
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Fig. 1. Local indeterminacy and bifurcations with constant returns and negative money.
3.1. Local indeterminacy and bifurcations It is easily shown (see Appendix A) that: DZ
u 3 R 0; k u
T Z 1 CD K
Mq 3 ; k R
(3)
with Mq*/k*Zd(1Ks)/sK1, where the starred variables (e.g. k*) are evaluated at the monetary steady state while 3u and 3R denote, respectively, the elasticities or u(k, l) and R(k, l) with respect to k evaluated at the steady state k*, l*. Then the following is shown to hold. Proposition 3.1. (Local Indeterminacy and Local Bifurcations) A necessary condition for local indeterminacy and local (saddle-node, flip or Hopf) dð1KsÞ bifurcation is T!1CD, that is, (Mq*/k*)3RO0, where Mq Z K 1. The latter k s condition requires: 1. a negative quantity of money (M!0) when returns to scale are constant or not too increasing (3R!0), i.e. when 3j!(1Ks)/s, where 3jR0, 0!s!1 and sO0 denote, respectively, the level of capital externalities, the capital share in total income and the elasticity of factor substitution, all evaluated at the monetary steady state. 2. a positive quantity of money (MO0) when returns to scale are sufficiently increasing (3RO0), i.e. when 3jO(1Ks)/s.
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Fig. 2. Local indeterminacy and bifurcations with increasing returns and positive money.
A corollary of Proposition 3.1 may help to restate some results that are implicit in the case of constant returns analyzed by Benhabib and Laroque (1988). In that configuration, it is not difficult to get (see Appendix A.1): T Z 1 C dð1 K sÞD=sR 0
D Z dð1 K sÞ=sR 0;
(4)
An immediate outcome follows from direct inspection of Eq. (4): in the positive orthant of the (T, D) plane (both T and D are positive), the path described by the point (T(s), D(s)) when s is made to vary between CN and zero is a half-line D starting at (1, 0) and with slope given by s/[d(1Ks)] (see Fig. 1). Accordingly, two configurations occur: in the case where the slope of the half-line D is less than one, i.e. when d(1Ks)Os, the steady state is a saddle, while if, on the contrary, the slope of DD is greater than one, i.e. if d(1Ks)!s, the half-line D crosses the segment [AB] and intercepts the trapezoid OABC. This means that, in the former case, there exists a neighborhood of the steady state where no endogenous fluctuations occur. On the contrary, a Hopf bifurcation is generally expected in the latter configuration, when D intersects [AB], while local indeterminacy prevails when s is large enough, that is, when sOsH (see Fig. 1). More precisely, when d(1Ks)!s, endogenous fluctuations are shown to occur: a Hopf bifurcation occurs generally if DZ1, i.e. if sZ dð1K sÞZdef sH , leading to the creation of an invariant closed curve on which the dynamics are either periodic or quasiperiodic,
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whereas the steady state is locally indeterminate when sOsH and, therefore, infinitely many stochastic equilibria exist near the steady state (see Fig. 1). The conditions leading to Hopf bifurcations have already been established by Benhabib and Laroque (1988), Prop. III. 3, who noted that inputs must be complementary for Hopf cycles to emerge, so Proposition 3.2 emphasizes the occurrence of sunspots. Proposition 3.2. (Sunspots with Large Elasticities of Factor Substitution) Assume constant returns to scale (that is, 3jZ0). Moreover, let d(1Ks)!s or, equivalently, M!0. Then the following holds: def
† 0! s! dð1K sÞZ sH : the monetary steady state is unstable (locally determinate), i.e. a source. Therefore, there exists a neighborhood of the steady state in which no nondegenerate stochastic equilibria occur. † sH!s: the monetary steady state is aymptotically stable, i.e. a sink (locally indeterminate). Therefore, there exist infinitely many nondegenerate stochastic equilibria in the neighborhood of the steady state. On the other hand, the monetary steady state is a saddle (locally determinate) when d(1Ks)Os or, equivalently, MO0. Therefore, there exists a neighborhood of the steady state in which no nondegenerate stochastic equilibria occur. Both statements regarding sunspot equilibria follow directly from Grandmont et al. (1998), Theo. 3.2. It is stated there that if, in a two-dimensional dynamical system with one predeterminate variable, the steady state is a sink, then in an arbitrarily small open ball centered at the steady state, one can find a compact convex set I containing the steady state in its interior where infinitely many nondegenerate (truly) stochastic equilibria staying in I for all dates can be constructed. On the contrary, when the steady state is a saddle or a source (locally determinate), it is shown by Grandmont et al. (1998), Theo. 3.2 that, generically, if the open ball centered at the steady state is small enough, the only stationary stochastic equilibria are actually deterministic and converge to (resp. coincide with) the steady state when it is a saddle (resp. a source). The condition that d(1Ks)!s, pointed out by Benhabib and Laroque (1988), Prop. III.3, when considering Hopf bifurcations, is required for indeterminacy to occur when sOsH. It implies a quite low depreciation rate, on yearly basis (less than 4% when the capital share is set at, for example, 0.4). More importantly, this configuration leads to a negative quantity of outside money at the (golden rule) steady state. On the other hand, it appears that, under the previous assumption, sHZd(1Ks)!s is not large: the share of capital s is approximately 0.4 in industrialized countries, so that sH is lower than 0.4 (the latter value is attained when dZs/(1Ks)). Therefore, neither strong input complementarity nor highly elastic labor supply is needed in OLG economies with outside nominal debt, in contrast with most standard real business-cycle models (including OLG models without money). We now show that this result still holds true when the quantity of outside money is positive, provided that capital externalities are present.
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3.2. Stationary sunspot equilibria with a positive money stock We now consider the effect of capital externalities, which create increasing returns to scale at the aggregate level. Moreover, we focus on the case with positive money, at the golden rule steady state, i.e. we assume that d(1Ks)Os. Notice that the latter condition is consistent with the time period associated with the OLG structure of the model, which imposes dz1. Proposition 3.1 shows that local indeterminacy and bifurcations occur only if externalities are so large that 3RO0, or equivalently 3jO(1Ks)/s. Therefore, the following holds (see Appendix A.2). Proposition 3.3. (Sunspots with a Positive Money Stock) Assume returns to scale to be increasing but not too large (that is, 0!3j!s/[d(1Ks)]). Moreover, let d(1Ks)Os or, equivalently, MO0. Then the following holds: def
† 0! s! ð1K sÞ=3j Z sI : the monetary steady state is a saddle. Therefore, there exists a neighborhood of the steady state in which no nondegenerate stochastic equilibria occur. † The monetary steady state is expected to undergo either a transcritical or a pitchfork bifurcation at sZsI. † sI!s: the monetary steady state is aymptotically stable, i.e. a sink. Therefore, there exist infinitely many nondegenerate stochastic equilibria in the neighborhood of the steady state. The main implication of Proposition 3.3 is that sunspot equilibria are compatible with substitutable inputs, inelastic labor and a positive money stock, as long as externalities operating through the capital stock are large enough. For instance, to include the benchmark Cobb-Douglas configuration in the indeterminacy range (sI,N), one has to impose 3jO1Ks, which implies large external effects. However, some recent estimates report elasticities of capital-labor substitution that are larger than one. For instance, (Duffy and Papageorgiou (2000), Table 1, p. 99) compute robust estimates of that are contained in [1.24,3.24] (see also Krusell et al. (2000), Table 3, who find a substitution elasticity between unskilled labor and equipment of 1.67). In view of these values, setting sZ2 (resp. sZ3) implies that 3j has to be larger than (1Ks)/2 (resp. (1Ks)/3) (that is, 3j must exceed 0.3 and 0.2, respectively). 3.3. Interpreting the results The purpose of this section is to provide some intuitive account of the main mechanisms that originate cyclical and sunspot equilibria. Recall that in contrast with the analysis of Benhabib and Laroque (1988), we restrict the analysis to the case of inelastic labor, which allows us to deal with a two-dimensional dynamical system. Moreover, abstracting from labor movements helps to highlight the critical mechanisms that are at work: as in the work of Benhabib and Laroque (1988), sunspots and cycles are due to a destabilizing real balance effect, when money and capital are assumed to be alternative ways of savings that pay the same rate of return. This may be intuitively described as follows. In the economy with constant returns to scale (absent the externalities) and negative money, optimistic
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expectations may be self-fulfilling because two conflicting effects on savings coexist. When the capital stock is, say, higher than its steady state value, the wage rate is higher. On the contrary, the rate of return on capital is lower and this leads, by arbitrage, to a lower real return on money (that is, the price of the consumption good in terms of money goes up); see the last equation in (2). Therefore, this generates two conflicting effects on savings and, therefore, on tomorrow’s capital stock: while the wage effect tends to increase savings and investment in physical capital, the decline in the real value of (negative) money tends to depress savings (as seen from the second equation in (2) with a negative money stock M). Although the wage effect dominates initially, it is eventually overcome by the real balance effect so that the economy may follow cyclical and sunspot equilibria. Now when capital externalities lead to increasing returns to scale, sunspots occur with positive money. Here again, a growing capital stock tends to increase savings, as the wage rate goes up. However, both the rate of return on capital and the real return on money also go up (that is, the price of the consumption good goes down), which lowers savings in the form of physical capital: with positive money held by the young agents as a means to save, an increase in real balances reduces the need to save in the form of productive capital, given wage income (see the second equation in (2), with positive M). Therefore, the interaction of the wage and real-balance effects leads here again to sunspot equilibria.
4. Conclusion In this paper, we have shown that stationary sunspots may occur when the quantity of money is positive, inputs are substitutable and labor supply is inelastic. As emphasized by the analysis of Benhabib and Laroque (1988), a destabilizing real-balance effect is still found to be critical in a version of the model with externalities and increasing returns. However, we have shown that the latter feature makes local indeterminacy compatible with a positive stock of outside money held by agents as an alternative asset to productive capital. Similar results are expected in an extended version of the model with consumption also allowed in the first period of life, building on the analysis of Cazzavillan and Pintus (2004). More importantly, it would be useful to develop the structure of available assets, along the lines of, for instance, Calvet (2001), so as to underline the features of actual financial markets and transactions that may originate expectation-driven volatility.
Appendix A. Deriving trace T and determinant D In this section, we derive the expressions of trace T and determinant D of the dynamical system described by the last two equations of (2), with inelastic labor normalized, without loss of generality, to lZ1. That is, the dynamical system is now written as: ( ktC1 Z uðkt ; 1Þ K Mqt (A1) qtC1 Z RðktC1 ; 1Þqt :
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The (golden rule) steady state is defined by the strictly positive numbers k*, q* solving (A1), that is: ( k Z uðk ; 1Þ K Mq (A2) 1 Z Rðk ; 1Þ: Therefore, the steady state value of k solves R(k*, 1)Z1 while the steady state value of q solves q*Z(u(k*,1)Kk*)/M. Moreover, it is not difficult to show that Mq*/k*Z d(1Ks)/sK1 and that u*/k*ZMq*/k*C1Zd(1Ks)/s (see Eq. (1)). It is then straightforward to compute the Jacobian matrix of (A1) evaluated at k*, q* whose expression is such that: DZ
u 3 R 0; k u
T Z 1 CD K
Mq 3 ; k R
(A3)
where 3u and 3R denote, respectively, the elasticities or u(k,1) and R(k,1) with respect to k evaluated at the steady state value k*.
Appendix A.1. Constant returns to scale In that case, j(k) is constant for all kO0 and one has that 3uZs/s and 3RZKd(1Ks)/s, so that, from (A3): T Z 1 C dð1 K sÞD=sR 0;
D Z dð1 K sÞ=sR 0:
(A4)
The resulting configuration is described in Proposition 3.2 and Fig. 1.
Appendix A.2. Increasing returns to scale In that case, j(k) is an increasing function for all kO0, the elasticity of which is 3j at the steady state k*. Then one has that 3u Z 3j C s=s and 3R Z dð3j ð1K sÞ=sÞ. Therefore, the rate of return on capital is increasing with the capital stock k if and only if 3jO(1Ks)s. Finally, (A3) yields that: D Z dð1 K sÞ½s=s C 3j =sR 0;
T Z 1 C D C L;
L Z ½dð1 K sÞ=s K 1½dð1 K sÞ=s K d3j : The resulting configuration is described in Proposition 3.3 and Fig. 2.
References Azariadis, C., 1981. Self-fulfilling prophecies. J. Econ. Theory 25, 380–396. Azariadis, C., Guesnerie, R., 1986. Sunspots and cycles. Rev. Econ. Stud. 53, 725–737.
(A5)
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Benhabib, J., Farmer, R.E.A., 1994. Indeterminacy and increasing returns. J. Econ. Theory 63, 19–41. Benhabib, J., Laroque, G., 1988. On competitive cycles in productive economies. J. Econ. Theory 45, 145–170. Blanchard, O., 1979. Speculative bubbles, crashes and rational expectations. Econ. Lett. 3, 387–389. Calvet, L., 2001. Incomplete markets and volatility. J. Econ. Theory 98, 295–338. Cass, D., Okuno, M., Zilcha, I., 1979. The role of money in supporting the Pareto optimality of competitive equilibrium in consumption-loan type models. J. Econ. Theory 20, 277–305. Cazzavillan, G., Pintus, P.A., 2004. Robustness of multiple equilibria in OLG economies. Rev. Econ. Dyn. 7, 456–475. Cazzavillan, G., Lloyd-Braga, T., Pintus, P.A., 1998. Multiple steady states and endogenous fluctuations with increasing returns in production. J. Econ. Theory 80, 60–107. Duffy, J., Papageorgiou, S., 2000. A cross-country empirical investigation of the aggregate production function specification. J. Econ. Growth 5, 87–120. Farmer, R.E.A., Woodford, M., 1984. Self-fulfilling prophecies and the business cycle. Mimeo University of Pennsylvania. Farmer, R.E.A., 1986. Deficits and cycles. J. Econ. Theory 40, 77–88. Gale, D., 1973. Pure exchange equilibrium of dynamic economic models. J. Econ. Theory 6, 12–36. Grandmont, J.-M., 1985. On endogenous competitive business cycles. Econometrica 53, 995–1045. Grandmont, J.-M., 1986. Stabilizing competitive business cycles. J. Econ. Theory 40, 57–76. Grandmont, J.-M., Pintus, P.A., de Vilder, R.G., 1998. Capital-labor substitution and competitive nonlinear endogenous business cycles. J. Econ. Theory 80, 14–59. Jullien, B., 1988. Competitive business cycles in an overlapping generations economy with productive investment. J. Econ. Theory 46, 45–65. Krusell, P., Ohanian, L., Rios-Rull, J.-V., Violante, G., 2000. Capital-skill complementarity and inequality: a macroeconomic analysis. Econometrica 68, 1029–1054. Michel, P., Wigniolle, B., 2003. Temporary bubbles. J. Econ. Theory 112, 173–183. Reichlin, P., 1986. Equilibrium cycles in an overlapping generations economy with production. J. Econ. Theory 40, 89–102. Shell, K., 1977. Monnaie et allocation intertemporelle. Communication to the Roy-Malinvaud Seminar, mimeo Paris. Tirole, J., 1985. Asset bubbles and overlapping generations. Econometrica 53, 1499–1528. Woodford, M., 1986. Stationary sunspot equilibria in a finance constrained economy. J. Econ. Theory 40, 128–137.
On competitive cycles and sunspots in productive economies with a positive money stock Guido Cazzavillana, Patrick A. Pintusb,* a
b
Department of Economics, University Ca’Foscari di Venezia, and SET, Italy Department of Economics, University de la Mediterranee, and GREQAM, France
This paper has been written in memory of Philippe Michel
Abstract This paper shows that stationary sunspot equilibria may occur in productive, overlappinggenerations economies when the stock of outside money is positive. More precisely, a destabilizing real-balance effect makes sunspots compatible with positive money, substitutable inputs and inelastic labor. Key to the results is the presence of capital externalities associated with learning-bydoing creation of knowledge. q 2005 University of Venice. Published by Elsevier Ltd. All rights reserved.
1. Introduction It is by now well known that monetary overlapping-generations (OLG thereafter) economies may exhibit cyclical and sunspot equilibria (see e.g. Gale 1973; Shell 1977; Cass et al. 1979; Farmer and Woodford 1984; Azariadis 1981; Grandmont 1985; Azariadis and Guesnerie 1986). However, recent contributions focusing on productive economies turn out to rely on stringent assumptions: large income effects (see Grandmont 1986; Jullien 1988), input complementarity and elastic labor supply (see Woodford 1986; Grandmont et al. 1998), negative outside money or public debt (see Farmer 1986; Benhabib and Laroque 1988). The purpose of this paper is to show that sunspots and cycles may occur in a version of the OLG model with productive investment, inelastic labor and large input substitution. Within the very same framework, Benhabib and Laroque (1988) have shown that * Corresponding author. E-mail addresses: [email protected] (G. Cazzavillan), [email protected] (P.A. Pintus).
1090-9443/$ - see front matter q 2005 University of Venice. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.rie.2005.04.004
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the emergence of cyclical equilibria requires both the quantity of outside money to be negative, at the golden rule, and inputs to be complementary. We will demonstrate that neither of the latter conditions is needed when capital externalities are introduced into the economy: stationary sunspots may occur when the quantity of money is positive and inputs are substitutable enough. We focus on external effects through the average capital stock (creating spillovers by learning-by-doing creation of knowledge), which have been repeatedly claimed, since the seminal contributions by Arrow, Shell, Frankel and Romer, to be important to understand growth and business cycles. Although the same model was studied by Benhabib and Laroque (1988), some departures with the present work should be mentioned. First, we emphasize conditions leading to the existence of stationary sunspot equilibria (around steady states or cyclical equilibria), whereas Benhabib and Laroque (1988) focused on cycles. In contrast with the latter authors, we restrict the analysis to the case of inelastic labor, which allows us to deal with a twodimensional dynamical system that can be studied geometrically by applying some results by Grandmont et al. (1998). Moreover, abstracting from labor movements helps to highlight the critical mechanisms that are at work: as in the work of Benhabib and Laroque (1988), sunspots and cycles are originated by a destabilizing real balance effect, when money and capital are assumed to be alternative ways of savings that pay the same rate of return. This may be intuitively described as follows. In the economy with constant returns to scale at the social level (absent the externalities) and negative money, optimistic expectations may be self-fulfilling because two conflicting effects on savings coexist. When the capital stock is, say, higher than its steady state value, the wage rate is higher. On the contrary, the rate of return on capital is lower and this leads, by arbitrage, to a lower real return on money (that is, the price of the consumption good in terms of money goes up). Therefore, this generates two conflicting effects on savings and, therefore, on tomorrow’s capital stock: while the wage effect tends to increase savings and investment in physical capital, the decline in the real value of (negative) money tends to depress savings. Although the wage effect dominates initially, it is eventually overcome by the real balance effect so that the economy may follow cyclical and sunspot equilibria. Now with capital externalities, sunspots occur with positive money. Here again, a growing capital stock tends to increase savings, as the wage rate goes up. However, both the rate of return on capital and the real return on money now go up (that is, the price of the consumption good goes down), which lowers savings in the form of physical capital. This shows that financial mechanisms, although often absent in the most recent literature, may also be important in understanding how endogenous fluctuations occur (see also, on this point, the related literature on bubbles, e.g. Blanchard 1979; Tirole 1985, and on cash-in-advance constraints, e.g. Michel and Wigniolle 2003). In Section 2, the model and dynamics are analyzed, while Section 3 derives and discusses the conditions leading to cyclical and sunspot equilibria. Section 4 gathers some concluding remarks.
2. Model and intertemporal equilibria The economy is composed of agents who work and save in the form or productive capital and outside money while young, consume a unique good while old, and achieve maximal
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utility derived from consumption of the unique good and leisure V2(ctC1)KV1(lt), where ctC1O0 is consumption demand by the old and ltO0 is labor supplied by the young. The budget constraints corresponding to active periods of a representative household born at time tO0 are therefore: ktC1 C MtC1 qt Z ut lt ;
ctC1 Z RtC1 ktC1 C MtC1 qtC1 ;
(1)
where ktC1O0, MtC1O0 denote, respectively, capital and money demands, and qt, ut, RtC1 stand, respectively, for the price of money, the real wage and the gross return on capital (including capital depreciation), all expressed in terms of the produced good which is chosen to be the numeraire. Benhabib and Laroque (1988) assume away any possibility of arbitrage between the two available assets so that the rates of return of capital of money are identical at equilibrium, i.e. RtC1ZqtC1/qt at all dates. Under constant returns to scale, technology is summarized by a production function F that is homogeneous of degree one that may, therefore, be written F(k, l)ZAlf(k/l), where AO0 is total factor productivity and f is a strictly concave function. In contrast with Benhabib and Laroque (1988), we assume that total factor productivity is an increasing 0, i.e. AZ jðkÞ, following, among the others, function of the average capital stock kO Benhabib and Farmer (1994) and Cazzavillan et al. (1998). Accordingly, competitive return on capital and real wage are, in equilibrium, respectively: RZ def def jðkÞf 0 ðk=lÞC 1K dZ Rðk; lÞ and uZ jðkÞ½f ðk=lÞK kf 0 ðk=lÞ=lZ uðk; lÞ, where 0!d!1 is the rate of capital depreciation. Finally, the quantity of money is assumed to be constant through time and equal to M. It is then easily shown that the set of intertemporal equilibria with perfect foresight (equilibria in the sequel) is summarized by a three-dimensional dynamical system that may be expressed, for instance, in terms of the capital stock k, labor l and the price of money q (see Benhabib and Laroque 1988, Prop. I.4, p. 148). Definition 2.1. An intertemporal competitive equilibrium with perfect foresight is a 3 sequence (kt, lt, qt) of RC , tZ1,2,., such that 8 v ðRðktC1 ; ltC1 Þuðkt ; lt Þlt Þ Z v1 ðlt Þ > < 2 ktC1 Z uðkt ; lt Þlt K Mqt (2) > : qtC1 Z RðktC1 ; ltC1 Þqt : The first equality in Eq. (2) expresses the first order condition describing the intertemporal allocation of consumption and leisure, where v2 ðcÞZdef cV20 ðcÞ and def v1 ðlÞZ lV10 ðlÞ. The second and third identities in Eq. (2) are, respectively, the first period budget constraint and the no-arbitrage condition, for an agent born at date t. Under constant returns, standard assumptions on preferences and technology (see Benhabib and Laroque 1988, p. 147) are sufficient to establish uniqueness of the monetary (or golden rule) steady state, i.e. a unique stationary solution of Eq. (2) such that the price of money is strictly positive. In our model with externalities, it is not difficult to adapt the proof of Cazzavillan et al. (1998), Prop. 3.1, p. 68, to establish the existence of a normalized steady state. To save space, this is left to the reader.
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We now study the local dynamics around the monetary steady state, i.e. the behavior of the dynamical system given by the (locally) invertible map (ktC1, ltC1, qtC1)ZG(kt, lt, qt) near the steady state, by studying the linearized map obtained from Eq. (2).1
3. Sunspot equilibria with a positive money stock Benhabib and Laroque (1988) give analytical conditions for the occurrence of flip and Hopf bifurcations (see their Propositions III.1 and 3, pp. 154–5). Although the authors restrict their study to deterministic cycles, an additional interesting result emerges in their diagrams, though not pointed out by the authors: in their Figs. 1–3, two eigenvalues are stable, i.e. have modulus less than one, when both the ratio of capital over consumption is large enough and the elasticity of factor substitution is sufficiently large (see Benhabib and Laroque 1988, pp. 158–9). Since the capital stock is the only predeterminate variable, this means that the steady state is locally indeterminate and, therefore, that one expects stochastic equilibria to exist near the steady state when the elasticity of capital-labor substitution is not too low. However, this requires a negative quantity of outside money. As we shall now see, this result holds true with positive money, provided that capital externalities are present. For simplicity, we will focus on inelastic labor supply. In that case, the model is then summarized by a two-dimensional map, as pointed out by Benhabib and Laroque (1988), Prop. II.1. Accordingly, we can apply previous results to show that sunspot equilibria exist in this economy, when factors are not too complementary. It is therefore expected that stochastic equilibria also occur when labor is not too elastic. In view of the first order condition in Eq. (2), we see that labor supply does not vary with the real wage u if the function v2(c)ZcV 0 2(c) is constant or equivalently if the elasticity of the marginal utility function V 0 2(c*) is equal to K1 at the steady state c* (the utility function is then equivalent to the logarithmic function at this point). Under this assumption, the first equation of (2) yields, under appropriate boundary conditions, a unique (interior) stationary labor supply ltZl*. Accordingly, the last two equalities of Eq. (2) then summarize the dynamics of the economy so that the corresponding dynamical system is two-dimensional. Therefore, in the case of inelastic labor, we need to linearize only the last two equations in (2) around the monetary steady state, under our assumption that ltZl* for all dates, i.e. that 3v2 ðc ÞZ 0, where the notation 3v2 stands for the elasticity of the function v2. In particular, all relevant information regarding the local stability of the golden rule steady state reduce to the trace T and the determinant D of the Jacobian matrix of the map given by the last two equations of (2), i.e. respectively, the sum and the product of the eigenvalues of this matrix, evaluated at the steady state.
1
The dynamics arising in the model without money have been studied by Reichlin (1986) and Cazzavillan and Pintus (2004), among others.
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Fig. 1. Local indeterminacy and bifurcations with constant returns and negative money.
3.1. Local indeterminacy and bifurcations It is easily shown (see Appendix A) that: DZ
u 3 R 0; k u
T Z 1 CD K
Mq 3 ; k R
(3)
with Mq*/k*Zd(1Ks)/sK1, where the starred variables (e.g. k*) are evaluated at the monetary steady state while 3u and 3R denote, respectively, the elasticities or u(k, l) and R(k, l) with respect to k evaluated at the steady state k*, l*. Then the following is shown to hold. Proposition 3.1. (Local Indeterminacy and Local Bifurcations) A necessary condition for local indeterminacy and local (saddle-node, flip or Hopf) dð1KsÞ bifurcation is T!1CD, that is, (Mq*/k*)3RO0, where Mq Z K 1. The latter k s condition requires: 1. a negative quantity of money (M!0) when returns to scale are constant or not too increasing (3R!0), i.e. when 3j!(1Ks)/s, where 3jR0, 0!s!1 and sO0 denote, respectively, the level of capital externalities, the capital share in total income and the elasticity of factor substitution, all evaluated at the monetary steady state. 2. a positive quantity of money (MO0) when returns to scale are sufficiently increasing (3RO0), i.e. when 3jO(1Ks)/s.
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Fig. 2. Local indeterminacy and bifurcations with increasing returns and positive money.
A corollary of Proposition 3.1 may help to restate some results that are implicit in the case of constant returns analyzed by Benhabib and Laroque (1988). In that configuration, it is not difficult to get (see Appendix A.1): T Z 1 C dð1 K sÞD=sR 0
D Z dð1 K sÞ=sR 0;
(4)
An immediate outcome follows from direct inspection of Eq. (4): in the positive orthant of the (T, D) plane (both T and D are positive), the path described by the point (T(s), D(s)) when s is made to vary between CN and zero is a half-line D starting at (1, 0) and with slope given by s/[d(1Ks)] (see Fig. 1). Accordingly, two configurations occur: in the case where the slope of the half-line D is less than one, i.e. when d(1Ks)Os, the steady state is a saddle, while if, on the contrary, the slope of DD is greater than one, i.e. if d(1Ks)!s, the half-line D crosses the segment [AB] and intercepts the trapezoid OABC. This means that, in the former case, there exists a neighborhood of the steady state where no endogenous fluctuations occur. On the contrary, a Hopf bifurcation is generally expected in the latter configuration, when D intersects [AB], while local indeterminacy prevails when s is large enough, that is, when sOsH (see Fig. 1). More precisely, when d(1Ks)!s, endogenous fluctuations are shown to occur: a Hopf bifurcation occurs generally if DZ1, i.e. if sZ dð1K sÞZdef sH , leading to the creation of an invariant closed curve on which the dynamics are either periodic or quasiperiodic,
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whereas the steady state is locally indeterminate when sOsH and, therefore, infinitely many stochastic equilibria exist near the steady state (see Fig. 1). The conditions leading to Hopf bifurcations have already been established by Benhabib and Laroque (1988), Prop. III. 3, who noted that inputs must be complementary for Hopf cycles to emerge, so Proposition 3.2 emphasizes the occurrence of sunspots. Proposition 3.2. (Sunspots with Large Elasticities of Factor Substitution) Assume constant returns to scale (that is, 3jZ0). Moreover, let d(1Ks)!s or, equivalently, M!0. Then the following holds: def
† 0! s! dð1K sÞZ sH : the monetary steady state is unstable (locally determinate), i.e. a source. Therefore, there exists a neighborhood of the steady state in which no nondegenerate stochastic equilibria occur. † sH!s: the monetary steady state is aymptotically stable, i.e. a sink (locally indeterminate). Therefore, there exist infinitely many nondegenerate stochastic equilibria in the neighborhood of the steady state. On the other hand, the monetary steady state is a saddle (locally determinate) when d(1Ks)Os or, equivalently, MO0. Therefore, there exists a neighborhood of the steady state in which no nondegenerate stochastic equilibria occur. Both statements regarding sunspot equilibria follow directly from Grandmont et al. (1998), Theo. 3.2. It is stated there that if, in a two-dimensional dynamical system with one predeterminate variable, the steady state is a sink, then in an arbitrarily small open ball centered at the steady state, one can find a compact convex set I containing the steady state in its interior where infinitely many nondegenerate (truly) stochastic equilibria staying in I for all dates can be constructed. On the contrary, when the steady state is a saddle or a source (locally determinate), it is shown by Grandmont et al. (1998), Theo. 3.2 that, generically, if the open ball centered at the steady state is small enough, the only stationary stochastic equilibria are actually deterministic and converge to (resp. coincide with) the steady state when it is a saddle (resp. a source). The condition that d(1Ks)!s, pointed out by Benhabib and Laroque (1988), Prop. III.3, when considering Hopf bifurcations, is required for indeterminacy to occur when sOsH. It implies a quite low depreciation rate, on yearly basis (less than 4% when the capital share is set at, for example, 0.4). More importantly, this configuration leads to a negative quantity of outside money at the (golden rule) steady state. On the other hand, it appears that, under the previous assumption, sHZd(1Ks)!s is not large: the share of capital s is approximately 0.4 in industrialized countries, so that sH is lower than 0.4 (the latter value is attained when dZs/(1Ks)). Therefore, neither strong input complementarity nor highly elastic labor supply is needed in OLG economies with outside nominal debt, in contrast with most standard real business-cycle models (including OLG models without money). We now show that this result still holds true when the quantity of outside money is positive, provided that capital externalities are present.
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3.2. Stationary sunspot equilibria with a positive money stock We now consider the effect of capital externalities, which create increasing returns to scale at the aggregate level. Moreover, we focus on the case with positive money, at the golden rule steady state, i.e. we assume that d(1Ks)Os. Notice that the latter condition is consistent with the time period associated with the OLG structure of the model, which imposes dz1. Proposition 3.1 shows that local indeterminacy and bifurcations occur only if externalities are so large that 3RO0, or equivalently 3jO(1Ks)/s. Therefore, the following holds (see Appendix A.2). Proposition 3.3. (Sunspots with a Positive Money Stock) Assume returns to scale to be increasing but not too large (that is, 0!3j!s/[d(1Ks)]). Moreover, let d(1Ks)Os or, equivalently, MO0. Then the following holds: def
† 0! s! ð1K sÞ=3j Z sI : the monetary steady state is a saddle. Therefore, there exists a neighborhood of the steady state in which no nondegenerate stochastic equilibria occur. † The monetary steady state is expected to undergo either a transcritical or a pitchfork bifurcation at sZsI. † sI!s: the monetary steady state is aymptotically stable, i.e. a sink. Therefore, there exist infinitely many nondegenerate stochastic equilibria in the neighborhood of the steady state. The main implication of Proposition 3.3 is that sunspot equilibria are compatible with substitutable inputs, inelastic labor and a positive money stock, as long as externalities operating through the capital stock are large enough. For instance, to include the benchmark Cobb-Douglas configuration in the indeterminacy range (sI,N), one has to impose 3jO1Ks, which implies large external effects. However, some recent estimates report elasticities of capital-labor substitution that are larger than one. For instance, (Duffy and Papageorgiou (2000), Table 1, p. 99) compute robust estimates of that are contained in [1.24,3.24] (see also Krusell et al. (2000), Table 3, who find a substitution elasticity between unskilled labor and equipment of 1.67). In view of these values, setting sZ2 (resp. sZ3) implies that 3j has to be larger than (1Ks)/2 (resp. (1Ks)/3) (that is, 3j must exceed 0.3 and 0.2, respectively). 3.3. Interpreting the results The purpose of this section is to provide some intuitive account of the main mechanisms that originate cyclical and sunspot equilibria. Recall that in contrast with the analysis of Benhabib and Laroque (1988), we restrict the analysis to the case of inelastic labor, which allows us to deal with a two-dimensional dynamical system. Moreover, abstracting from labor movements helps to highlight the critical mechanisms that are at work: as in the work of Benhabib and Laroque (1988), sunspots and cycles are due to a destabilizing real balance effect, when money and capital are assumed to be alternative ways of savings that pay the same rate of return. This may be intuitively described as follows. In the economy with constant returns to scale (absent the externalities) and negative money, optimistic
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expectations may be self-fulfilling because two conflicting effects on savings coexist. When the capital stock is, say, higher than its steady state value, the wage rate is higher. On the contrary, the rate of return on capital is lower and this leads, by arbitrage, to a lower real return on money (that is, the price of the consumption good in terms of money goes up); see the last equation in (2). Therefore, this generates two conflicting effects on savings and, therefore, on tomorrow’s capital stock: while the wage effect tends to increase savings and investment in physical capital, the decline in the real value of (negative) money tends to depress savings (as seen from the second equation in (2) with a negative money stock M). Although the wage effect dominates initially, it is eventually overcome by the real balance effect so that the economy may follow cyclical and sunspot equilibria. Now when capital externalities lead to increasing returns to scale, sunspots occur with positive money. Here again, a growing capital stock tends to increase savings, as the wage rate goes up. However, both the rate of return on capital and the real return on money also go up (that is, the price of the consumption good goes down), which lowers savings in the form of physical capital: with positive money held by the young agents as a means to save, an increase in real balances reduces the need to save in the form of productive capital, given wage income (see the second equation in (2), with positive M). Therefore, the interaction of the wage and real-balance effects leads here again to sunspot equilibria.
4. Conclusion In this paper, we have shown that stationary sunspots may occur when the quantity of money is positive, inputs are substitutable and labor supply is inelastic. As emphasized by the analysis of Benhabib and Laroque (1988), a destabilizing real-balance effect is still found to be critical in a version of the model with externalities and increasing returns. However, we have shown that the latter feature makes local indeterminacy compatible with a positive stock of outside money held by agents as an alternative asset to productive capital. Similar results are expected in an extended version of the model with consumption also allowed in the first period of life, building on the analysis of Cazzavillan and Pintus (2004). More importantly, it would be useful to develop the structure of available assets, along the lines of, for instance, Calvet (2001), so as to underline the features of actual financial markets and transactions that may originate expectation-driven volatility.
Appendix A. Deriving trace T and determinant D In this section, we derive the expressions of trace T and determinant D of the dynamical system described by the last two equations of (2), with inelastic labor normalized, without loss of generality, to lZ1. That is, the dynamical system is now written as: ( ktC1 Z uðkt ; 1Þ K Mqt (A1) qtC1 Z RðktC1 ; 1Þqt :
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The (golden rule) steady state is defined by the strictly positive numbers k*, q* solving (A1), that is: ( k Z uðk ; 1Þ K Mq (A2) 1 Z Rðk ; 1Þ: Therefore, the steady state value of k solves R(k*, 1)Z1 while the steady state value of q solves q*Z(u(k*,1)Kk*)/M. Moreover, it is not difficult to show that Mq*/k*Z d(1Ks)/sK1 and that u*/k*ZMq*/k*C1Zd(1Ks)/s (see Eq. (1)). It is then straightforward to compute the Jacobian matrix of (A1) evaluated at k*, q* whose expression is such that: DZ
u 3 R 0; k u
T Z 1 CD K
Mq 3 ; k R
(A3)
where 3u and 3R denote, respectively, the elasticities or u(k,1) and R(k,1) with respect to k evaluated at the steady state value k*.
Appendix A.1. Constant returns to scale In that case, j(k) is constant for all kO0 and one has that 3uZs/s and 3RZKd(1Ks)/s, so that, from (A3): T Z 1 C dð1 K sÞD=sR 0;
D Z dð1 K sÞ=sR 0:
(A4)
The resulting configuration is described in Proposition 3.2 and Fig. 1.
Appendix A.2. Increasing returns to scale In that case, j(k) is an increasing function for all kO0, the elasticity of which is 3j at the steady state k*. Then one has that 3u Z 3j C s=s and 3R Z dð3j ð1K sÞ=sÞ. Therefore, the rate of return on capital is increasing with the capital stock k if and only if 3jO(1Ks)s. Finally, (A3) yields that: D Z dð1 K sÞ½s=s C 3j =sR 0;
T Z 1 C D C L;
L Z ½dð1 K sÞ=s K 1½dð1 K sÞ=s K d3j : The resulting configuration is described in Proposition 3.3 and Fig. 2.
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(A5)
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