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Global sunspots in OLG models
Journal of Macroeconomics 28 (2006) 27–45 www.elsevier.com/locate/jmacro
Global sunspots in OLG models Theory and computational analysis Gianluca Gazzola a, Alfredo Medio
b,*
a
b
Research Group on Nonlinear Dynamical Systems, Department of Statistical Sciences, University of Udine, Udine, Italy Department of Statistical Sciences, University of Udine, Via Treppo 18, 33100 Udine, Italy Received 5 October 2005; accepted 30 October 2005
Abstract This paper discusses sunspots equilibria in a context that is general in the sense that: (i) the evolution of the system takes place in a general state space (i.e., a space which is not necessarily finite or even countable); and (ii) the orbits of the unperturbed, deterministic component of the system converge to subsets of the state space which can be more complicated than a stationary state or a periodic orbit, i.e., they can be aperiodic or chaotic. This problem is represented mathematically as a system of stochastic difference equations the invariant probability distributions of which correspond to stationary sunspots equilibria. The conditions for stochastic stability are recalled and the theoretical results are applied to a model of overlapping generations with individuals living three periods. A computational analysis of this model is provided, covering the basic different cases suggested by the theory. 2005 Published by Elsevier Inc. JEL classification: C-10; C-60 Keywords: Stochastic stability; Overlapping generations; Sunspots
*
Corresponding author. Tel.: +39 0432 249584; fax: +39 0432 249595. E-mail addresses: [email protected] (G. Gazzola), [email protected] (A. Medio). URL: http://www.dss.uniud.it/~medio/medio.htm (A. Medio).
0164-0704/$ - see front matter 2005 Published by Elsevier Inc. doi:10.1016/j.jmacro.2005.10.003
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1. Introduction and motivation A considerable part of recent economic literature deals with the relevance of agents’ beliefs. A basic issue discussed in that literature is under what conditions those beliefs can be considered rational. In this respect, a crucial role is played by the concept of ‘‘sunspot equilibrium (SE)’’. Broadly speaking, this can be defined as an intertemporal equilibrium in which the state variables are perfectly correlated with certain exogenous random variables that agents believe to determine the course of events, although they do not affect the fundamentals of the economy – essentially tastes and technology. Thus, although agents’ beliefs may be based on ‘‘bad theory’’, observation of actual data corroborates and justifies them. Formally, we can describe this situation by a discrete-time stochastic dynamical system with the canonical form xtþ1 ¼ T ðxt ; ntþ1 Þ;
ð1Þ
where the nt (t = 1, 2, . . .) are i.i.d. random vectors with values in W, an open subset of Rm ; the state variables xt take values in X, an open subset of Rn ; the initial vector x0 is a random vector taking values in X, arbitrary but independent of nt for t P 1. Thus, xt is independent of ntþi , for all i P 1. The index parameterizes the level of n-perturbations. T is a measurable function mapping X · W to X Rn . The fact that xt+1 is conditionally independent of xt1, xt2,. . ., given xt, ensures that (1) has the Markovian property. In words, this means that the present value of the state variable x contains all the information from its past history relevant for the prediction of its future. System (1) generates a collection X ¼ fxt gjt 2 Zþ of random variables, which is called a Markov chain or process. Heuristically, all this can be interpreted as follows. The optimal choice of the value of the variable x at time t depends on its expected value at time t + 1, through some constrained maximizing decision. Agents believe that, for any given xt, xt+1 depends on some exogenous random variables whose distribution is known and they choose accordingly. Eq. (1) describes the situation that obtains when agents’ expectations are systematically fulfilled. In this case, the Markovian chain described by (1) is called a time-independent, sunspot equilibrium (SE). The evolution in time of the distribution of the (random) state variable of (1) can be described as follows: ptþ1 ðAÞ ¼ P½pt ðAÞ; Z Z vA ½T ðx; n Þm ðdnÞ pðdxÞ A 2 BðX Þ; PðpÞ ¼ X
ð2Þ
W
where BðX Þ denotes the Borel r-algebra on X; vA is the indicator function of the set A, i.e., 1 for x 2 A; vA ðxÞ ¼ 0 for x 62 A and m is the probability measure (identical for all t): m ðBÞ ¼ probðnt 2 BÞ
for B 2 BðW Þ.
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
29
is invariant with respect to P (or, equivalently, is preserved by P), if p is a We say that p ¼ Pð is fixed point of the map, namely p pÞ. In other words: a probability measure p and a perturbation seinvariant if, given initial conditions x0 distributed according to p quence {nt}, t P 1 with distribution m, the probability distribution of the state variables x is not changed by the action of T. In the economic applications that we are considering, an invariant probability distribu is interpreted as a stationary sunspot equilibrium (SSE). tion p An interesting special case of (1) obtains when we have fnt g ¼ f^n; ^n; ^n; . . .g, where ^n is a vector of constants and therefore the difference Eq. (1) is deterministic. In what follows, we assume that for = 0, n0t ¼ ^ n for all t, where ^ n corresponds to a certain ‘‘normal’’ configuration of parameters, which, without loss of generality, can be normalized to zero. Let us now define: F ðxÞ ¼ T ðx; 0Þ: \deterministic core"; Gðx; nÞ ¼ T ðx; nÞ F ðxÞ: \perturbation term" and assume that: (i) for all n, kG(x, n)k is bounded (uniformly for x in the relevant subset of X), i.e., for each P 0 there exists a L < 1, such that sup kGðx; n Þk 6 L
a.s.
and, (ii): L ! 0 as ! 0. Thus, system (1) can be de-composed as follows: xtþ1 ¼ T ðxt ; ntþ1 Þ ¼ F ðxt Þ þ Gðxt ; ntþ1 Þ.
ð3Þ
By reducing , we can make the perturbation level as small as we please and, in the limit for ! 0, the stochastic process degenerates to its deterministic core. As we shall soon see, in the applications considered in this paper, the deterministic core of the system corresponds to the case in which agents foresee the future perfectly. The voluminous literature on ‘‘sunspot equilibria’’1 has been mostly concentrated on the question of existence of SSE, and in particular of SSE with finite support, i.e., a support including only a finite number of states. Moreover, the relations between the properties of perfect foresight equilibria and sunspots have been studied almost always with regards to the simpler cases of steady states (fixed points) or periodic solutions of the deterministic dynamics. One of the present authors has recently proved a result (Medio, 2004, Theorem 1) establishing conditions for the existence and stochastic stability of SSE in a context that is general in a twofold sense. First of all, the support of SSE is not necessarily finite or even countable. Secondly, the relations between the asymptotic states of the deterministic, perfect foresight dynamical systems and the associated SSE is discussed in the general case in which, for appropriate initial conditions, the attractors of those systems may be more complex than fixed or periodic points, namely they can be
1 Excellent overviews of this topic are provided by Farmer and Woodford (1997), originally circulated in 1984 as a CARESS W.P., and Chiappori and Guesnerie (1991), with rich bibliography.
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G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
aperiodic or chaotic sets.2 For a detailed discussion of the many difficult technical details involved in the assumptions and proof of the theorem in question, we refer the interested reader to the quoted article and the bibliography therein. Since the purpose of this paper is experimental rather than theoretical, here it will suffice to provide a broad, non-technical description of the main assumptions and conclusions. Suppose that Eq. (3) describes a model of sequential market equilibria, such as an OLG model, with F representing the perfect foresight solution and G the stochastic perturbation determined by agents’ beliefs. Apart from some regularity conditions concerning the structural functions of the model and the random perturbations which we shall not discuss here, the existence of SSE in such a model is proved under the following conditions: 1. The ‘‘deterministic core’’ defined by the function F must possess an asymptotically stable attracting set K. 2. The perturbation, represented by the state-dependent term G, must be sufficiently small with respect to size of the basin of the attracting set and the strength of its attractiveness. When the regularity conditions and condition 1 above hold, uniqueness of SSE depends on the ‘‘degree of dynamical indecomposability’’ of the attracting set, in the following sense: 1. If F is strongly indecomposable (strongly topologically transitive3) then, for any given perturbation fnt g (associated with agents’ beliefs), the SSE is unique. This condition is verified (trivially) when the deterministic attractor K is a fixed point. It is also satisfied when the dynamics of map F are chaotic and mixing on K. It may also be verified when the attractor is quasiperiodic (aperiodic but not chaotic). 2. When the deterministic map is topologically transitive on K but not strongly so, we have a situation that we describe as ‘‘weakly indecomposable deterministic attractor’’. In this case, the Markov process X generated by (1)–(3) is not aperiodic, the iterations of the operator P defined by (2) need not converge to a unique distribution and there need not be a unique SSE. This situation occurs obviously when K is a periodic orbit and, less obviously, when K consists of two or more chaotic sets mapped into each other cyclically by F – the so-called ‘‘periodic, or non-mixing chaos’’. However, this is not as serious a drawback as it seems, and this case can be transformed into the aperiodic case 1, simply changing the time unit and considering instead the map Td, i.e. the dth iterate of T (corresponding to Pd , the dth iterate of the operator P), where the integer d denotes the periodicity of the Markov process X. The Markov process generated by the map Td and the associated operator Pd will be aperiodic and, as n ! 1, Pnd ðp0 Þ 2 Notice that, although the result in question was originally motivated by the study of SE, its relevance is by no means limited to that problem. On the contrary, it can be applied directly to any economic problem which can be represented by a system of discrete-time stochastic equations like (1), for example optimal growth with stochastic perturbations of fundamentals. 3 A map f : K ! K is said to be topologically transitive on K if, for any two open sets U, V K, there exists an integer n such that f n(U) \ V 5 B. Transitivity implies that orbits generated by the map f starting from any arbitrarily small open neighborhood visit any other arbitrarily small open neighborhood in K in finite time. Thus, the set K is dynamically indecomposable and must be studied as one piece. f is said to be strongly topologically transitive on K if for any integer m > 0 the map f m is topologically transitive on K.
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
31
will converge to one of d probability measures pi, i = 1, . . . , d corresponding to d SSEs, the choice among them depending on the initial conditions defined by p0 (of course, the d distributions may well be identical). 3. A third case occurs when the deterministic map F is not topologically transitive on an attracting set K, for example, when there exist multiple attractors for the map F. In this case the Theorem in question does not hold over the entire state space X. However, under fairly general conditions, there exists a finite decomposition of X into absorbing sets4 (plus possibly some probabilistically negligible transient set), such that for each absorbing set there is a unique invariant probability distribution, which is stochastically stable for appropriate initial conditions. In terms of our present application, this of course means that in this case, the dynamics of the map T, restricted to one or the other absorbing set, still generates a unique, stochastically stable SSE. In what follows, we provide a numerical–graphical illustration of the general theoretical results mentioned above, making use of a model of pure exchange overlapping generations with three generations, both in the deterministic (perfect foresight) and in the stochastic versions.5 The setup of the model strikes a compromise between tractability and complexity. The model is simple enough to yield explicit stochastic equations like (1), but generates dynamics sufficiently rich to yield the most interesting cases covered by the theoretical results. We would like to point out that, in the numerical–graphical examples that follow, the structural functions and parameter configurations were not chosen because we believe they are in any sense ‘‘typical’’ for the model in question, but because the resulting dynamics provide insightful ‘‘experimental’’ realizations of the basic kinds of behavior predicted by the theory. 2. The basic model I: the deterministic, perfect foresight case Consider an economy with a constant population where each individual lives three periods (‘‘youth’’, ‘‘maturity’’ and ‘‘old age’’). Since, apart from age, all individuals are assumed to be identical, we can refer to the ‘‘young’’ (‘‘mature’’, ‘‘old’’) representative agent. There is no production but at each period fixed, non-negative amounts of a single, perishable consumption good are allocated to the three classes of individuals, and the good is traded at a price pt, t denoting time. Agents’ preferences take the form of concave utility functions, one for each stage of life. The representative agent is assumed to be ‘‘rational’’ in the sense that he maximizes utility over his three-period life, subject to an intertemporal budget constraint, i.e., the total consumption expenditure planned by agents over their lifespan must be equal to the total values of their endowments. In the deterministic version of the model, agents have perfect foresight, i.e. the future prices used in
We say that a set A X is absorbing if, whenever xt 2 A, T ðxt ; ntþ1 Þ 2 A with probability one, where the properties of T and n are as in Eq. (1) above. 5 In his classic treatment of pure exchange intertemporal equilibrium, Samuelson (1958) considered first the three-generation case and then a simplified two-generation variation. A basic contribution to the analysis of the latter case was provided by Gale (1973). Benhabib and Day’s (1982) pioneering investigation showed that the dynamics of a two-generation, pure exchange model can be very complicated, even chaotic. In his well-known article on endogenous business cycles, Grandmont (1985) considered a two-generation, leisure-consumption OLG model essentially equivalent to that of pure exchange. 4
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formulating their consumption plans are systematically and exactly realized. Moreover, markets are assumed to clear at all times, i.e., for each period t, total demand (=sum of the three coexisting agents’ consumption) equals total supply (=sum of endowments allocated in that period). The notation is as follows: cyt cm t cot wy wm wo pt u1 ðcyt Þ u2 ðcm t Þ u3 ðcot Þ
young agent’s consumption at time t mature agent’s consumption at time t old agent’s consumption at time t young agent’s endowment mature agent’s endowment old agent’s endowment price of the consumption good at time t utility function for the first period consumption utility function for the second period consumption utility function for the third period consumption
The young agent’s maximization problem can now be described mathematically as follows: max s.t.
o fu1 ðcyt Þ þ u2 ðcm tþ1 Þ þ u3 ðctþ2 Þg
pt cyt
þ
ptþ1 cm tþ1
þ
ptþ2 cotþ2
¼ pt wy þ ptþ1 wm þ ptþ2 wo ;
ð4Þ o cyt ; cm t ; ct
P0
8t
ð5Þ
and the market clearing condition is o cyt þ cm t þ ct ¼ wy þ wm þ wo
8t
ð6Þ
(notice that, whereas in (4) and (5) we have consumption of the same agent, over the threeperiod lifetime, in (6) we have consumption of three different agents). From the first order conditions (FOC) of problem (4) and (5) (which in view of concavity of utility functions are sufficient as well as necessary), we obtain u01 ðcyt Þ p ¼ t ; u02 ðcm Þ p tþ1 tþ1 u01 ðcyt Þ p ¼ t ; u03 ðcotþ2 Þ ptþ2
ð7Þ
u02 ðcm p tþ1 Þ ¼ tþ1 . u03 ðcotþ2 Þ ptþ2 Notice that the price ratios on the right hand side of Eq. (7) are interest factors (=interest rate + 1). Defining pt/pt+1 = (1 + r1); pt+1/pt+2 = (1 + r2), Eq. (7) can be re-written as 0 y u02 ðcm tþ1 Þð1 þ r1 Þ ¼ u1 ðct Þ;
u03 ðc0tþ2 Þð1 þ r1 Þð1 þ r2 Þ ¼ u01 ðcyt Þ.
ð8Þ
Making use of (7) the budget Eq. (5) can be transformed into: m 0 o o y m 0 u01 ðcyt Þ½cyt wy þ u02 ðcm tþ1 Þ½ctþ1 wm þ u3 ðctþ2 Þ½ctþ2 wo ¼ H ðct ; ctþ1 ; ctþ2 Þ ¼ 0.
ð9Þ
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
33
Our strategy is now to use Eqs. (4)–(9) to construct a dynamical system of two first-order y m o difference equations in any two of the three state variables ðcyt ; cm t ; ct Þ, for example ðct ; ct Þ, as follows: cytþ1 ¼ f1 ðcyt ; cm t Þ; y m cm tþ1 ¼ f2 ðct ; ct Þ.
ð10Þ
The choice of variables is a matter of expediency because, for any given set of endowments, once we have determined the dynamics of two variables, that of the third can always be derived from the market clearing Eq. (6). In order to make the problem tractable, we will introduce some simplifying assumptions and specifications. First, we assume that only the young and the mature agents receive non-zero endowments, namely wy, wm > 0; wo = 0. Second, we adopt simple utility functions for the second and third period consumption, i.e., we put u2(cm) = km ln cm and u3(co) = ko ln co, whence u02 ðcm Þ ¼ k m =cm and u03 ðco Þ ¼ k o =co , where km, ko are positive constants. For the moment, we leave the function u1 ðcyt Þ unspecified. This allows us to put Eq. (9) in the simpler form: 0 y y m H ðcyt ; cm tþ1 Þ ¼ u1 ðct Þ½ct wy þ k m þ k o k m wm =ctþ1
ð11Þ
and to derive immediately the first of the two difference equations of system (10), namely y cm tþ1 ¼ f1 ðct Þ ¼
k m wm . u01 ðcyt Þðcyt wy Þ þ k m þ k o
ð12Þ
Next, from (6) we have o cytþ1 ¼ wy þ wm ðcm tþ1 þ ctþ1 Þ.
ð13Þ
Thus, to obtain the second difference equation of the dynamical system (10), it is sufficient to express cotþ1 as a function of cyt and cm t . To do that, let us consider the maximizing problem of the young agent at time t 1 (since, apart from age, agents are all identical, problem (4) and (5) is independent of time). From the FOC of that problem and Eq. (7), we obtain u03 ðcotþ1 Þ ptþ1 u02 ðcm Þ ¼ ¼ 0 tþ1 y 0 m pt u2 ðct Þ u1 ðct Þ
ð14Þ
whence the desired equation for cotþ1 , i.e. y 0 y cotþ1 ¼ ðk o =k 2m Þcm t f1 ðct Þu1 ðct Þ.
ð15Þ
From (14) and (15), we can finally derive the second difference equation of system (10), as follows: 2 y m 0 y cytþ1 ¼ f2 ðcyt ; cm t Þ ¼ wy þ wm f1 ðct Þ½1 þ ðk o =k m Þct u1 ðct Þ.
ð16Þ
3. The basic model II: the stochastic case The essential difference from the deterministic case is that at each time t young agents do no not know future prices with certainty, but plan their present and future consumption on the basis of beliefs about those prices, mathematically represented by a probability
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measure. For each generation, all the decisions are taken in the youth and cannot be changed. In this paper, we are concerned with self-fulfilling beliefs, such that the agents’ decisions based on those beliefs determine a course of events validating them. To keep complications to the minimum compatible with our present purposes, we assume here that at each time t, the young agent observes the price pt, foresees the price pt+1 perfectly, but believes that the price pt+2 depends on a random variable xtþ2 whose distribution, defined by the probability measure m, is known. The distribution m is the same for all t with expected value Eðx Þ ¼ 1. The sequence fxt g is an i.i.d. stochastic process. For each t, the state variables cyt , cm t are independent of xtþi , "i > 0. The young agent’s maximizing problem now takes the form o max E½u1 ðcyt Þ þ u2 ðcm tþ1 Þ þ u3 ðctþ2 Þ o s.t. pt cyt þ ptþ1 cm tþ1 þ p tþ2 ctþ2 ¼ p t wy þ p tþ1 wm
ð17Þ
(remember, we assumed wo = 0!). E denotes the expectation operator. Expectations are taken with respect to the probability measure m and are conditional on the information set available to the agent at time t (i.e., they are conditional on (pt, pt+1) or, equivalently, on ðcyt ; cm tþ1 Þ). Therefore, at each time t, the expected value of u3 ðcotþ2 Þ is a function of those variables. We would now like to derive two stochastic difference equations structurally analogous to (1). Making use of the budget constraint, the FOC of problem (17) can be written as p E u03 ðcotþ2 Þ t þ u01 ðcyt Þ ¼ 0; ð18Þ ptþ2 p E u03 ðcotþ2 Þ tþ1 þ u02 ðcm ð19Þ tþ1 Þ ¼ 0. ptþ2 Recalling that u03 ðcÞ ¼ k o =c; u02 ðcÞ ¼ k m =c, and using (18) and (19) and the budget constraint, we can obtain y 0 k o cm tþ1 u1 ðct Þ E m 0 y ð20Þ ¼ u01 ðcyt Þ. Þ ½ctþ1 u1 ðct Þðcyt wy Þ þ k m ðwm cm tþ1 Eq. (20) is satisfied, for example, by 0 y k o cm tþ1 u ðct Þ ¼ u01 ðcyt Þxtþ2 ; wy Þ þ k m ðwm cm tþ1 Þ
y 0 y ½cm tþ1 u ðct Þðct
ð21Þ
where xtþ2 is a random variable distributed according to the measure m, independent of cyt and with E½xtþ2 ¼ 1. (Of course, we might have to choose sufficiently small not to violate the semi-positivity constraint on the state variables.) From (21) we can promptly derive the first stochastic difference equation we are looking for, namely y cm tþ1 ¼ T 1 ðct ; xtþ2 Þ;
ð22Þ
where T 1 ðcyt ; xtþ2 Þ ¼
k m wm xtþ2 . k o þ ½k m þ ðcyt wy Þu01 ðcyt Þxtþ2
ð23Þ
To determine the second stochastic equation, we first recall the market clearing condition requiring that:
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
35
o cytþ1 ¼ wy þ wm ðcm tþ1 þ ctþ1 Þ.
Next, we consider the maximizing problem of the young agent at time t 1 and repeat the procedure followed to determine T1. The FOC are pt1 0 o E u3 ðctþ1 Þ ð24Þ þ u01 ðcyt1 Þ ¼ 0; ptþ1 p E u03 ðcotþ1 Þ t þ u0t ðcm ð25Þ t Þ ¼ 0. ptþ1 This is satisfied by cotþ1 ¼
m k o u01 ðcyt Þcm t ctþ1 k 2m xtþ1
hence cytþ1 ¼ T 2 ðcyt ; cm t ; xtþ1 ; xtþ2 Þ;
ð26Þ
where ! y m 0 k c u ðc Þ o y t 1 t T 2 ðcyt ; cm ; t ; xtþ1 ; xtþ2 Þ ¼ wy þ wm T 1 ðct ; xtþ2 Þ 1 þ k 2m xtþ1
ð27Þ
which is the desired equation. 2 Putting xt ¼ ðcyt ; cm t Þ 2 Rþ ; and nt ¼ nt ¼ xtþ1 2 Rþ , "t P 1, the stochastic dynamical system (22)–(26) has the same form as (1) with univariate random perturbations.6 The decomposition of the map T = (T1, T2) into a deterministic core and a state dependent perturbation term described by (2), can be promptly obtained by putting f1 F ¼ f2 and G¼
T 1 f1 ; T 2 f2
where (f1, f2) and (T1, T2) are defined, respectively, by (12)–(16) and (23)–(27). Consider that the dynamical system (22)–(26) should not be interpreted as a mathematical description of the agents’ decision process. It is instead a dynamical system whose output is a Markov process describing sequences of (random) consumption values, compatible with the young agents’ optimal decisions in the sense of (17) and with the market clearing condition.
6 Notice that, in view of the logic of the model and the assumptions made at the beginning of Section 3, at each time t uncertainty concerns only the value of xt+2. Only once, at the beginning of the process, we need a bivariate random perturbation with independent elements.
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4. Computational analysis To perform the numerical exercises of this section we need further specifications of our model. First of all, we want to select a utility function for the first period consumption such that the resulting model has the most interesting dynamics compatible with tractability. The literature on OLG models suggests that the utility functions of the CARA type (constant absolute risk aversion) are suitable candidates.7 In what follows, we use one such function, namely the exponential utility function u1(c) = rebc, where r, b are positive constants, and the absolute risk aversion is RA(c) = b (hence RR(c) = bc). Secondly, we must specify the properties of the stochastic perturbations. Omitting time indexes, the perturbations take the form of a random variable x = 1 + n, with n = l 0.5; EðnÞ ¼ 0. We assume that the distribution of l belongs to the beta family, with parameters (a, b) and support [0, 1]. Two basic cases are considered, namely: (i) a = b = 1 which is equivalent to the uniform distribution and will be denoted by l U, and (ii) a = b = 5, denoted by l Be(5, 5), i.e., a beta distribution with a unimodal, symmetrical and fairly peaked density function. In both cases, EðlÞ ¼ 0:5. Considering that, in the logic of the model, it must be x > 0, we take 2 [0,2]. As a matter of fact, in most of our simulations (thousands of them), the maximum value of compatible with stochastic stability was well within that interval. Not surprisingly, the maximum admissible support (the largest admissible value of ) is systematically larger for the unimodal beta distribution than it is for the uniform one. The deterministic (perfect foresight) OLG model with three generations is much more complex than the corresponding two-generation model. As is well known, for the latter we have two basic types of economic behavior: the first one, nicknamed ‘‘the classical case’’ in which the young agent is impatient and borrows from the old agent and, when old, saves to pay back his debt; the opposite situation, nicknamed ‘‘the Samuelson case’’, obtains when the young agent is thrifty and saves in order to be able to dissave when old. It is evident that the presence of three generations allows many more combinations of saving/ dissaving behaviors compatible with the intertemporal budget constraint. Schematically, we have the following six possibilities: (S S D); (S D S); (S D D); (D S S); (D S D); (D D S), where each triplet refers to the three periods of the representative agent’s life (and of course saving = endowment consumption). Our numerical ‘‘experiments’’ are restricted to the D S D case: the representative agent dissaves when is young, saves during his maturity and dissaves again when old. This variety of the OLG model, which we nickname ‘‘D S D’’, describes a not altogether implausible situation and, at any rate, yields all the dynamical complications we want to discuss. For different configurations of the parameters, the attractor of the deterministic ‘‘D S D’’ model may be a fixed point, a periodic orbit, a quasiperiodic orbit, a chaotic set made of a single piece or of a finite number of subsets. The bifurcation analysis of the model (that we shall not discuss here) shows some of the canonical ‘‘routes to chaos’’: flip bifurcations and subsequent period-doubling cascades, Neimark–Sacker bifurcations with 7 These utility functions are characterized by the fact that substitution effects are stronger that income effects for low levels of consumption but the opposite is true for higher value. As is well known, this fact is associated with fluctuating intertemporal equilibria. Remember that for a utility function u(c) the relative risk aversion coefficient is RR(c) = u00 (c)c/u 0 (c) and the absolute risk aversion coefficient is RA(c) = RR/c.
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
37
OL3G 40
cMt
35
30
25
12
14
16
18
20
cYt Fig. 1. Perturbed fixed point (l Be(5, 5); = 0.54).
subsequent periodic/quasiperiodic dynamics and possibly ‘‘torus-breaking’’ and chaos; ‘‘catastrophic’’ local/global bifurcations yielding a jump from simple (periodic) dynamics to chaos. Most often, the attractor appears to be unique, but occasionally multiple attractors exist. In each of these cases, we have been able to find numerical evidence of stochastic attractors for the perturbed dynamical system (22)–(26), which we know can be interpreted as stationary sunspot equilibria.8 We are now ready to present examples of each of the cases discussed in the theoretical part of the paper, according to the classification of Section 1. 4.1. Strongly indecomposable deterministic attracting set: fixed point In this case, the deterministic attractor is a fixed point, a steady state perfect foresight intertemporal equilibrium. The unperturbed attractor and the corresponding SSE generated by stochastic perturbations are illustrated by Fig. 1. 4.2. Strongly indecomposable deterministic attracting set: quasiperiodic dynamics The equilibrium dynamics are aperiodic but not chaotic. The attractor and the corresponding SSE are depicted in Fig. 2.
8 A detailed discussion of the different dynamical phenomena mentioned above is out of the question here. The interested reader may consult, for example, Medio and Lines (2001).
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OL3G 20
18
16
cM t
14
12
10
8
14
16
18
20
22
24
cYt
Fig. 2. Perturbed quasiperiodic attractor (l Be(5, 5); = 0.24).
4.3. Weakly indecomposable deterministic attracting set: periodic dynamics The selected example is a perturbed period-4 deterministic attractor shown by Fig. 3.
OL3G
cMt
40
30
20
10
10
15
20
25
cYt
Fig. 3. Perturbed period-4 attractor I (l Be(5, 5); = 0.31).
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
39
OL3G 21.5
21.0
cMt
20.5
20.0
19.5
19.0
18.5
23.7
23.8
23.9
24.0
24.1
cYt
Fig. 4. Perturbed period-4 attractor II: iterations of T4 (l U; = 0.02).
Keeping the same parameter configurations, Fig. 4 shows one of the four possible corresponding SSEs, obtained by using iterations of the map T4 (where T = (T1, T2)), i.e., the 4th iterate of system (22)–(26), and appropriately selecting initial conditions, as explained in Section 1. 4.4. Weakly indecomposable deterministic attracting set: many-piece chaotic set A perturbed 4-piece deterministic chaotic attractor is shown in Fig. 5. To depict one of the four possible SSEs, in Fig. 6 we again employ the 4th iterate of the map T, with appropriate initial conditions. 4.5. Decomposable attracting set I: multiple periodic attractors In Fig. 7, we show the basins of attraction of three different periodic deterministic attractors (the fixed point is considered periodic of period 1). In Fig. 8 we show that, if we select initial conditions sufficiently close to one of the attractors (in this case the attracting fixed point) and the perturbation is sufficiently small with respect to the size of its basin, a stochastic attractor corresponding to a SSE can be obtained. 4.6. Decomposable attracting set II: periodic and chaotic attractors Fig. 9 depicts two coexisting deterministic attractors (a period-3 orbit and an one-piece chaotic set), with their basins.
40
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45 OL3G 35 30
cMt
25 20 15
10 5
10
20
30
40
cYt
Fig. 5. A perturbed 4-piece deterministic chaotic attractor I (l Be(5, 5); = 0.1).
OL3G 14
12
cMt
10
8
6
4
2 28
30
32
34
36
38
40
cYt Fig. 6. A perturbed 4-piece chaotic attractor II: iteration of T4 (l U; = 0.03).
Fig. 10 instead shows the stochastic attractor of the perturbed system, corresponding to a SSE.
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
Fig. 7. Multiple attractor I.
OL3G
cMt
35
30
25
16
17
18
19
20
21
22
cYt Fig. 8. Perturbed fixed point (l Be(5, 5); = 0.2).
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G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
Fig. 9. Multiple attractor II.
OL3G 55
50 45
40
cMt
42
35
30
25
20 5
10
15
20
25
cYt Fig. 10. Perturbed one-piece chaotic set (l Be(5,5); = 0.3).
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
43
5. Concluding remarks Before concluding our paper, we must briefly mention a conceptual problem concerning the choice of initial conditions in OLG models. We discuss this point with regard to the deterministic, perfect foresight case, but the conclusions can be easily extended to the stochastic case. Moreover, here we only consider OLG models yielding a dynamical system for which there exists a unique orbit forward in time for each initial point, as is the case for our Eqs. (12)–(16).9 In dynamical systems usually discussed in mathematical and physical sciences, initial conditions are chosen at random out of an admissible set, and their future values are then determined by the equations of motion. This is also the case, for example, in economic models of optimal growth where the initial conditions (typically, the capital stock) are pre-determined. However, in the logic of the OLG model the values of the state variables at each time t, are justified only by the existence of a sequence of perfectly anticipated future values such that for each successive pair of time instants (t, t + 1), the difference equations of the model are satisfied, and this rule, strictly speaking, should apply to the initial conditions as well. Accordingly, in the case of our model, the initial conditions o 10 ðcm should be interpreted as optimal consumption levels of the mature and old agent, 0 ; c0 Þ planned at time t 1 and t 2, respectively (and perfectly realized). This reasoning could be repeated backward in time without limit. In other words, strictly speaking, the story told by the OLG model should have no beginning and no end. In the economic literature, there exist two basic approaches to this conceptual difficulty that are relevant for our discussion. The first of them – nicknamed here ‘‘unlimited horizon hypothesis (UHH)’’ – is to consider sequences of intertemporal equilibria stretching indefinitely both in the future and in the past. A moment’s reflection suggests that adopting the UHH hypothesis implies a severe restriction of the admissible orbits. For example, for a given initial point there might be few infinite backward in time sequences of the state variables satisfying the requirements of the model (utility constrained maximization, market clearing and nonnegativity), or there might be none. Thus, the UHH is logically impeccable but its realism and fruitfulness can be questioned (cf. Benhabib and Day’s (1982) comments on this question). The second approach – nicknamed ‘‘single shock hypothesis (SSH)’’ – introduces the ad hoc assumption that, at some instant of time t = 0, the economy is perturbed by an unexpected random shock, whose precise nature depends on the details of the model, and that no further disturbances occur. The SSH is adopted, for example, by Kehoe and Levine (1985). The SSH allows one to treat initial conditions as random (within certain constraints), at the price of violating, for the initial period only, the hypothesis of y o agents’ perfect foresight. In our case, for example, if at t = 0 ðcm 0 ; c0 Þ (and therefore c0 ) are chosen at random, in general the expectations of the young agents at times t = 1 and t = 2 will not be fulfilled. All in all, we believe that, in the present context, the SSH is a more fruitful alternative and we implicitly adopt it throughout the paper.
9 A thorough discussion of the difficult problems arising when an OLG model yields a dynamical system moving backward in time, can be found in Medio and Raines (forthcoming). y m 10 o Remember that, in view of the market clearing condition, fixing ðcm 0 ; c0 Þ is equivalent to fixing ðc0 ; c0 Þ.
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G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
Acknowledgements The authors would like to thank Cees Diks of the University of Amsterdam for acting as a discussant of this paper. His very detailed comments and criticisms were exceedingly helpful and prompted the authors to correct a number of mistakes and improve the presentation. All remaining errors and misunderstandings are entirely the authors’ responsibility. Appendix A All the computations have been performed by means of the specialized software iDMC, somewhat elaborated using the ‘‘R’’ software. iDMC can be downloaded from the website: http://www.dss.uniud.it/nonlinear/. Below, we are listing the values of the initial conditions and parameters used to produce the figures appearing in the main text. Fig. 1: cy0 ¼ 16;
cm 0 ¼ 32;
k m ¼ 4:46;
wy ¼ 1;
k o ¼ 1;
wm ¼ 55;
r ¼ 50;
b ¼ 0:3;
wm ¼ 41;
r ¼ 100;
b ¼ 0:3;
wm ¼ 55;
r ¼ 200;
b ¼ 0:3;
wm ¼ 55;
r ¼ 200;
b ¼ 0:3;
wm ¼ 55;
r ¼ 220;
b ¼ 0:3;
¼ 0:54.
Fig. 2: cy0 ¼ 20;
cm 0 ¼ 10;
wy ¼ 1;
k m ¼ 1:4;
k o ¼ 1;
¼ 0:24.
cy0 ¼ 22;
cm 0 ¼ 15;
wy ¼ 1;
k m ¼ 11;
k o ¼ 1;
¼ 0:31.
cy0 ¼ 24;
cm 0 ¼ 20;
wy ¼ 1;
k m ¼ 11;
k o ¼ 1;
¼ 0:02.
cy0 ¼ 20;
cm 0 ¼ 35;
wy ¼ 1;
k m ¼ 2;
k o ¼ 1;
Fig. 3:
Fig. 4:
Fig. 5:
¼ 0:1.
Fig. 6: cy0 ¼ 40;
cm 0 ¼ 5;
k m ¼ 2;
k o ¼ 1;
wy ¼ 1;
wm ¼ 55;
r ¼ 220;
b ¼ 0:3;
¼ 0:03.
Fig. 7: cy0 2 ½0; 55; k m ¼ 4:46;
cm 0 2 ½0; 55; k o ¼ 1;
wy ¼ 1;
¼ 0.
wm ¼ 55;
r ¼ 150;
b ¼ 0:3;
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
45
Fig. 8: wy ¼ 1; wm ¼ 55; cy0 ¼ 20; cm 0 ¼ 30; k m ¼ 4:46; k o ¼ 1; ¼ 0:2.
r ¼ 150;
b ¼ 0:3;
Fig. 9: cy0 2 ½0; 60; cm wy ¼ 1; 0 2 ½0; 60; k m ¼ 30; k o ¼ 1; ¼ 0.
wm ¼ 58:5;
r ¼ 200;
b ¼ 0:3;
Fig. 10: cy0 ¼ 25;
cm 0 ¼ 25;
k m ¼ 30;
k o ¼ 1;
wy ¼ 1;
wm ¼ 58:5;
r ¼ 200;
b ¼ 0:3;
¼ 0:3.
References Benhabib, J., Day, R., 1982. A characterization of erratic dynamics in the overlapping generations model. Journal of Economic Dynamics and Control 4, 37–55. Chiappori, P.A., Guesnerie, R., 1991. Sunspot equilibria in sequential markets models. In: Hildenbrand, W., Sonneschein, H. (Eds.), Handbook of Mathematical Economics, vol. IV. Elsevier, pp. 1683–1762. Farmer, R.E.A., Woodford, M., 1997. Self-fulfilling prophecies and the business cycle. Macroeconomic Dynamics 1, 740–770. Gale, D., 1973. Pure exchange equilibrium of dynamic economic models. Journal of Economic Theory 6, 12–36. Grandmont, J.-M., 1985. On endogenous competitive business cycles. Econometrica 53, 995–1045. Kehoe, T.J., Levine, D.K., 1985. Comparative statics and perfect foresight in infinite horizon economies. Econometrica 53, 433–453. Medio, A., 2004. Invariant probability distributions in economic models: a general result. Macroeconomic Dynamics 8, 162–187. Medio, A., Lines, M., 2001. Nonlinear Dynamics: A Primer. Cambridge University Press, Cambridge. Medio, A., Raines, B., forthcoming. Backward dynamics in economics. The inverse limit approach. Samuelson, P.A., 1958. An exact consumption-loan model of interest with or without the social contrivance of money. Journal of Political Economy 66 (6), 467–482.
Global sunspots in OLG models Theory and computational analysis Gianluca Gazzola a, Alfredo Medio
b,*
a
b
Research Group on Nonlinear Dynamical Systems, Department of Statistical Sciences, University of Udine, Udine, Italy Department of Statistical Sciences, University of Udine, Via Treppo 18, 33100 Udine, Italy Received 5 October 2005; accepted 30 October 2005
Abstract This paper discusses sunspots equilibria in a context that is general in the sense that: (i) the evolution of the system takes place in a general state space (i.e., a space which is not necessarily finite or even countable); and (ii) the orbits of the unperturbed, deterministic component of the system converge to subsets of the state space which can be more complicated than a stationary state or a periodic orbit, i.e., they can be aperiodic or chaotic. This problem is represented mathematically as a system of stochastic difference equations the invariant probability distributions of which correspond to stationary sunspots equilibria. The conditions for stochastic stability are recalled and the theoretical results are applied to a model of overlapping generations with individuals living three periods. A computational analysis of this model is provided, covering the basic different cases suggested by the theory. 2005 Published by Elsevier Inc. JEL classification: C-10; C-60 Keywords: Stochastic stability; Overlapping generations; Sunspots
*
Corresponding author. Tel.: +39 0432 249584; fax: +39 0432 249595. E-mail addresses: [email protected] (G. Gazzola), [email protected] (A. Medio). URL: http://www.dss.uniud.it/~medio/medio.htm (A. Medio).
0164-0704/$ - see front matter 2005 Published by Elsevier Inc. doi:10.1016/j.jmacro.2005.10.003
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1. Introduction and motivation A considerable part of recent economic literature deals with the relevance of agents’ beliefs. A basic issue discussed in that literature is under what conditions those beliefs can be considered rational. In this respect, a crucial role is played by the concept of ‘‘sunspot equilibrium (SE)’’. Broadly speaking, this can be defined as an intertemporal equilibrium in which the state variables are perfectly correlated with certain exogenous random variables that agents believe to determine the course of events, although they do not affect the fundamentals of the economy – essentially tastes and technology. Thus, although agents’ beliefs may be based on ‘‘bad theory’’, observation of actual data corroborates and justifies them. Formally, we can describe this situation by a discrete-time stochastic dynamical system with the canonical form xtþ1 ¼ T ðxt ; ntþ1 Þ;
ð1Þ
where the nt (t = 1, 2, . . .) are i.i.d. random vectors with values in W, an open subset of Rm ; the state variables xt take values in X, an open subset of Rn ; the initial vector x0 is a random vector taking values in X, arbitrary but independent of nt for t P 1. Thus, xt is independent of ntþi , for all i P 1. The index parameterizes the level of n-perturbations. T is a measurable function mapping X · W to X Rn . The fact that xt+1 is conditionally independent of xt1, xt2,. . ., given xt, ensures that (1) has the Markovian property. In words, this means that the present value of the state variable x contains all the information from its past history relevant for the prediction of its future. System (1) generates a collection X ¼ fxt gjt 2 Zþ of random variables, which is called a Markov chain or process. Heuristically, all this can be interpreted as follows. The optimal choice of the value of the variable x at time t depends on its expected value at time t + 1, through some constrained maximizing decision. Agents believe that, for any given xt, xt+1 depends on some exogenous random variables whose distribution is known and they choose accordingly. Eq. (1) describes the situation that obtains when agents’ expectations are systematically fulfilled. In this case, the Markovian chain described by (1) is called a time-independent, sunspot equilibrium (SE). The evolution in time of the distribution of the (random) state variable of (1) can be described as follows: ptþ1 ðAÞ ¼ P½pt ðAÞ; Z Z vA ½T ðx; n Þm ðdnÞ pðdxÞ A 2 BðX Þ; PðpÞ ¼ X
ð2Þ
W
where BðX Þ denotes the Borel r-algebra on X; vA is the indicator function of the set A, i.e., 1 for x 2 A; vA ðxÞ ¼ 0 for x 62 A and m is the probability measure (identical for all t): m ðBÞ ¼ probðnt 2 BÞ
for B 2 BðW Þ.
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
29
is invariant with respect to P (or, equivalently, is preserved by P), if p is a We say that p ¼ Pð is fixed point of the map, namely p pÞ. In other words: a probability measure p and a perturbation seinvariant if, given initial conditions x0 distributed according to p quence {nt}, t P 1 with distribution m, the probability distribution of the state variables x is not changed by the action of T. In the economic applications that we are considering, an invariant probability distribu is interpreted as a stationary sunspot equilibrium (SSE). tion p An interesting special case of (1) obtains when we have fnt g ¼ f^n; ^n; ^n; . . .g, where ^n is a vector of constants and therefore the difference Eq. (1) is deterministic. In what follows, we assume that for = 0, n0t ¼ ^ n for all t, where ^ n corresponds to a certain ‘‘normal’’ configuration of parameters, which, without loss of generality, can be normalized to zero. Let us now define: F ðxÞ ¼ T ðx; 0Þ: \deterministic core"; Gðx; nÞ ¼ T ðx; nÞ F ðxÞ: \perturbation term" and assume that: (i) for all n, kG(x, n)k is bounded (uniformly for x in the relevant subset of X), i.e., for each P 0 there exists a L < 1, such that sup kGðx; n Þk 6 L
a.s.
and, (ii): L ! 0 as ! 0. Thus, system (1) can be de-composed as follows: xtþ1 ¼ T ðxt ; ntþ1 Þ ¼ F ðxt Þ þ Gðxt ; ntþ1 Þ.
ð3Þ
By reducing , we can make the perturbation level as small as we please and, in the limit for ! 0, the stochastic process degenerates to its deterministic core. As we shall soon see, in the applications considered in this paper, the deterministic core of the system corresponds to the case in which agents foresee the future perfectly. The voluminous literature on ‘‘sunspot equilibria’’1 has been mostly concentrated on the question of existence of SSE, and in particular of SSE with finite support, i.e., a support including only a finite number of states. Moreover, the relations between the properties of perfect foresight equilibria and sunspots have been studied almost always with regards to the simpler cases of steady states (fixed points) or periodic solutions of the deterministic dynamics. One of the present authors has recently proved a result (Medio, 2004, Theorem 1) establishing conditions for the existence and stochastic stability of SSE in a context that is general in a twofold sense. First of all, the support of SSE is not necessarily finite or even countable. Secondly, the relations between the asymptotic states of the deterministic, perfect foresight dynamical systems and the associated SSE is discussed in the general case in which, for appropriate initial conditions, the attractors of those systems may be more complex than fixed or periodic points, namely they can be
1 Excellent overviews of this topic are provided by Farmer and Woodford (1997), originally circulated in 1984 as a CARESS W.P., and Chiappori and Guesnerie (1991), with rich bibliography.
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G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
aperiodic or chaotic sets.2 For a detailed discussion of the many difficult technical details involved in the assumptions and proof of the theorem in question, we refer the interested reader to the quoted article and the bibliography therein. Since the purpose of this paper is experimental rather than theoretical, here it will suffice to provide a broad, non-technical description of the main assumptions and conclusions. Suppose that Eq. (3) describes a model of sequential market equilibria, such as an OLG model, with F representing the perfect foresight solution and G the stochastic perturbation determined by agents’ beliefs. Apart from some regularity conditions concerning the structural functions of the model and the random perturbations which we shall not discuss here, the existence of SSE in such a model is proved under the following conditions: 1. The ‘‘deterministic core’’ defined by the function F must possess an asymptotically stable attracting set K. 2. The perturbation, represented by the state-dependent term G, must be sufficiently small with respect to size of the basin of the attracting set and the strength of its attractiveness. When the regularity conditions and condition 1 above hold, uniqueness of SSE depends on the ‘‘degree of dynamical indecomposability’’ of the attracting set, in the following sense: 1. If F is strongly indecomposable (strongly topologically transitive3) then, for any given perturbation fnt g (associated with agents’ beliefs), the SSE is unique. This condition is verified (trivially) when the deterministic attractor K is a fixed point. It is also satisfied when the dynamics of map F are chaotic and mixing on K. It may also be verified when the attractor is quasiperiodic (aperiodic but not chaotic). 2. When the deterministic map is topologically transitive on K but not strongly so, we have a situation that we describe as ‘‘weakly indecomposable deterministic attractor’’. In this case, the Markov process X generated by (1)–(3) is not aperiodic, the iterations of the operator P defined by (2) need not converge to a unique distribution and there need not be a unique SSE. This situation occurs obviously when K is a periodic orbit and, less obviously, when K consists of two or more chaotic sets mapped into each other cyclically by F – the so-called ‘‘periodic, or non-mixing chaos’’. However, this is not as serious a drawback as it seems, and this case can be transformed into the aperiodic case 1, simply changing the time unit and considering instead the map Td, i.e. the dth iterate of T (corresponding to Pd , the dth iterate of the operator P), where the integer d denotes the periodicity of the Markov process X. The Markov process generated by the map Td and the associated operator Pd will be aperiodic and, as n ! 1, Pnd ðp0 Þ 2 Notice that, although the result in question was originally motivated by the study of SE, its relevance is by no means limited to that problem. On the contrary, it can be applied directly to any economic problem which can be represented by a system of discrete-time stochastic equations like (1), for example optimal growth with stochastic perturbations of fundamentals. 3 A map f : K ! K is said to be topologically transitive on K if, for any two open sets U, V K, there exists an integer n such that f n(U) \ V 5 B. Transitivity implies that orbits generated by the map f starting from any arbitrarily small open neighborhood visit any other arbitrarily small open neighborhood in K in finite time. Thus, the set K is dynamically indecomposable and must be studied as one piece. f is said to be strongly topologically transitive on K if for any integer m > 0 the map f m is topologically transitive on K.
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
31
will converge to one of d probability measures pi, i = 1, . . . , d corresponding to d SSEs, the choice among them depending on the initial conditions defined by p0 (of course, the d distributions may well be identical). 3. A third case occurs when the deterministic map F is not topologically transitive on an attracting set K, for example, when there exist multiple attractors for the map F. In this case the Theorem in question does not hold over the entire state space X. However, under fairly general conditions, there exists a finite decomposition of X into absorbing sets4 (plus possibly some probabilistically negligible transient set), such that for each absorbing set there is a unique invariant probability distribution, which is stochastically stable for appropriate initial conditions. In terms of our present application, this of course means that in this case, the dynamics of the map T, restricted to one or the other absorbing set, still generates a unique, stochastically stable SSE. In what follows, we provide a numerical–graphical illustration of the general theoretical results mentioned above, making use of a model of pure exchange overlapping generations with three generations, both in the deterministic (perfect foresight) and in the stochastic versions.5 The setup of the model strikes a compromise between tractability and complexity. The model is simple enough to yield explicit stochastic equations like (1), but generates dynamics sufficiently rich to yield the most interesting cases covered by the theoretical results. We would like to point out that, in the numerical–graphical examples that follow, the structural functions and parameter configurations were not chosen because we believe they are in any sense ‘‘typical’’ for the model in question, but because the resulting dynamics provide insightful ‘‘experimental’’ realizations of the basic kinds of behavior predicted by the theory. 2. The basic model I: the deterministic, perfect foresight case Consider an economy with a constant population where each individual lives three periods (‘‘youth’’, ‘‘maturity’’ and ‘‘old age’’). Since, apart from age, all individuals are assumed to be identical, we can refer to the ‘‘young’’ (‘‘mature’’, ‘‘old’’) representative agent. There is no production but at each period fixed, non-negative amounts of a single, perishable consumption good are allocated to the three classes of individuals, and the good is traded at a price pt, t denoting time. Agents’ preferences take the form of concave utility functions, one for each stage of life. The representative agent is assumed to be ‘‘rational’’ in the sense that he maximizes utility over his three-period life, subject to an intertemporal budget constraint, i.e., the total consumption expenditure planned by agents over their lifespan must be equal to the total values of their endowments. In the deterministic version of the model, agents have perfect foresight, i.e. the future prices used in
We say that a set A X is absorbing if, whenever xt 2 A, T ðxt ; ntþ1 Þ 2 A with probability one, where the properties of T and n are as in Eq. (1) above. 5 In his classic treatment of pure exchange intertemporal equilibrium, Samuelson (1958) considered first the three-generation case and then a simplified two-generation variation. A basic contribution to the analysis of the latter case was provided by Gale (1973). Benhabib and Day’s (1982) pioneering investigation showed that the dynamics of a two-generation, pure exchange model can be very complicated, even chaotic. In his well-known article on endogenous business cycles, Grandmont (1985) considered a two-generation, leisure-consumption OLG model essentially equivalent to that of pure exchange. 4
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G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
formulating their consumption plans are systematically and exactly realized. Moreover, markets are assumed to clear at all times, i.e., for each period t, total demand (=sum of the three coexisting agents’ consumption) equals total supply (=sum of endowments allocated in that period). The notation is as follows: cyt cm t cot wy wm wo pt u1 ðcyt Þ u2 ðcm t Þ u3 ðcot Þ
young agent’s consumption at time t mature agent’s consumption at time t old agent’s consumption at time t young agent’s endowment mature agent’s endowment old agent’s endowment price of the consumption good at time t utility function for the first period consumption utility function for the second period consumption utility function for the third period consumption
The young agent’s maximization problem can now be described mathematically as follows: max s.t.
o fu1 ðcyt Þ þ u2 ðcm tþ1 Þ þ u3 ðctþ2 Þg
pt cyt
þ
ptþ1 cm tþ1
þ
ptþ2 cotþ2
¼ pt wy þ ptþ1 wm þ ptþ2 wo ;
ð4Þ o cyt ; cm t ; ct
P0
8t
ð5Þ
and the market clearing condition is o cyt þ cm t þ ct ¼ wy þ wm þ wo
8t
ð6Þ
(notice that, whereas in (4) and (5) we have consumption of the same agent, over the threeperiod lifetime, in (6) we have consumption of three different agents). From the first order conditions (FOC) of problem (4) and (5) (which in view of concavity of utility functions are sufficient as well as necessary), we obtain u01 ðcyt Þ p ¼ t ; u02 ðcm Þ p tþ1 tþ1 u01 ðcyt Þ p ¼ t ; u03 ðcotþ2 Þ ptþ2
ð7Þ
u02 ðcm p tþ1 Þ ¼ tþ1 . u03 ðcotþ2 Þ ptþ2 Notice that the price ratios on the right hand side of Eq. (7) are interest factors (=interest rate + 1). Defining pt/pt+1 = (1 + r1); pt+1/pt+2 = (1 + r2), Eq. (7) can be re-written as 0 y u02 ðcm tþ1 Þð1 þ r1 Þ ¼ u1 ðct Þ;
u03 ðc0tþ2 Þð1 þ r1 Þð1 þ r2 Þ ¼ u01 ðcyt Þ.
ð8Þ
Making use of (7) the budget Eq. (5) can be transformed into: m 0 o o y m 0 u01 ðcyt Þ½cyt wy þ u02 ðcm tþ1 Þ½ctþ1 wm þ u3 ðctþ2 Þ½ctþ2 wo ¼ H ðct ; ctþ1 ; ctþ2 Þ ¼ 0.
ð9Þ
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
33
Our strategy is now to use Eqs. (4)–(9) to construct a dynamical system of two first-order y m o difference equations in any two of the three state variables ðcyt ; cm t ; ct Þ, for example ðct ; ct Þ, as follows: cytþ1 ¼ f1 ðcyt ; cm t Þ; y m cm tþ1 ¼ f2 ðct ; ct Þ.
ð10Þ
The choice of variables is a matter of expediency because, for any given set of endowments, once we have determined the dynamics of two variables, that of the third can always be derived from the market clearing Eq. (6). In order to make the problem tractable, we will introduce some simplifying assumptions and specifications. First, we assume that only the young and the mature agents receive non-zero endowments, namely wy, wm > 0; wo = 0. Second, we adopt simple utility functions for the second and third period consumption, i.e., we put u2(cm) = km ln cm and u3(co) = ko ln co, whence u02 ðcm Þ ¼ k m =cm and u03 ðco Þ ¼ k o =co , where km, ko are positive constants. For the moment, we leave the function u1 ðcyt Þ unspecified. This allows us to put Eq. (9) in the simpler form: 0 y y m H ðcyt ; cm tþ1 Þ ¼ u1 ðct Þ½ct wy þ k m þ k o k m wm =ctþ1
ð11Þ
and to derive immediately the first of the two difference equations of system (10), namely y cm tþ1 ¼ f1 ðct Þ ¼
k m wm . u01 ðcyt Þðcyt wy Þ þ k m þ k o
ð12Þ
Next, from (6) we have o cytþ1 ¼ wy þ wm ðcm tþ1 þ ctþ1 Þ.
ð13Þ
Thus, to obtain the second difference equation of the dynamical system (10), it is sufficient to express cotþ1 as a function of cyt and cm t . To do that, let us consider the maximizing problem of the young agent at time t 1 (since, apart from age, agents are all identical, problem (4) and (5) is independent of time). From the FOC of that problem and Eq. (7), we obtain u03 ðcotþ1 Þ ptþ1 u02 ðcm Þ ¼ ¼ 0 tþ1 y 0 m pt u2 ðct Þ u1 ðct Þ
ð14Þ
whence the desired equation for cotþ1 , i.e. y 0 y cotþ1 ¼ ðk o =k 2m Þcm t f1 ðct Þu1 ðct Þ.
ð15Þ
From (14) and (15), we can finally derive the second difference equation of system (10), as follows: 2 y m 0 y cytþ1 ¼ f2 ðcyt ; cm t Þ ¼ wy þ wm f1 ðct Þ½1 þ ðk o =k m Þct u1 ðct Þ.
ð16Þ
3. The basic model II: the stochastic case The essential difference from the deterministic case is that at each time t young agents do no not know future prices with certainty, but plan their present and future consumption on the basis of beliefs about those prices, mathematically represented by a probability
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G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
measure. For each generation, all the decisions are taken in the youth and cannot be changed. In this paper, we are concerned with self-fulfilling beliefs, such that the agents’ decisions based on those beliefs determine a course of events validating them. To keep complications to the minimum compatible with our present purposes, we assume here that at each time t, the young agent observes the price pt, foresees the price pt+1 perfectly, but believes that the price pt+2 depends on a random variable xtþ2 whose distribution, defined by the probability measure m, is known. The distribution m is the same for all t with expected value Eðx Þ ¼ 1. The sequence fxt g is an i.i.d. stochastic process. For each t, the state variables cyt , cm t are independent of xtþi , "i > 0. The young agent’s maximizing problem now takes the form o max E½u1 ðcyt Þ þ u2 ðcm tþ1 Þ þ u3 ðctþ2 Þ o s.t. pt cyt þ ptþ1 cm tþ1 þ p tþ2 ctþ2 ¼ p t wy þ p tþ1 wm
ð17Þ
(remember, we assumed wo = 0!). E denotes the expectation operator. Expectations are taken with respect to the probability measure m and are conditional on the information set available to the agent at time t (i.e., they are conditional on (pt, pt+1) or, equivalently, on ðcyt ; cm tþ1 Þ). Therefore, at each time t, the expected value of u3 ðcotþ2 Þ is a function of those variables. We would now like to derive two stochastic difference equations structurally analogous to (1). Making use of the budget constraint, the FOC of problem (17) can be written as p E u03 ðcotþ2 Þ t þ u01 ðcyt Þ ¼ 0; ð18Þ ptþ2 p E u03 ðcotþ2 Þ tþ1 þ u02 ðcm ð19Þ tþ1 Þ ¼ 0. ptþ2 Recalling that u03 ðcÞ ¼ k o =c; u02 ðcÞ ¼ k m =c, and using (18) and (19) and the budget constraint, we can obtain y 0 k o cm tþ1 u1 ðct Þ E m 0 y ð20Þ ¼ u01 ðcyt Þ. Þ ½ctþ1 u1 ðct Þðcyt wy Þ þ k m ðwm cm tþ1 Eq. (20) is satisfied, for example, by 0 y k o cm tþ1 u ðct Þ ¼ u01 ðcyt Þxtþ2 ; wy Þ þ k m ðwm cm tþ1 Þ
y 0 y ½cm tþ1 u ðct Þðct
ð21Þ
where xtþ2 is a random variable distributed according to the measure m, independent of cyt and with E½xtþ2 ¼ 1. (Of course, we might have to choose sufficiently small not to violate the semi-positivity constraint on the state variables.) From (21) we can promptly derive the first stochastic difference equation we are looking for, namely y cm tþ1 ¼ T 1 ðct ; xtþ2 Þ;
ð22Þ
where T 1 ðcyt ; xtþ2 Þ ¼
k m wm xtþ2 . k o þ ½k m þ ðcyt wy Þu01 ðcyt Þxtþ2
ð23Þ
To determine the second stochastic equation, we first recall the market clearing condition requiring that:
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
35
o cytþ1 ¼ wy þ wm ðcm tþ1 þ ctþ1 Þ.
Next, we consider the maximizing problem of the young agent at time t 1 and repeat the procedure followed to determine T1. The FOC are pt1 0 o E u3 ðctþ1 Þ ð24Þ þ u01 ðcyt1 Þ ¼ 0; ptþ1 p E u03 ðcotþ1 Þ t þ u0t ðcm ð25Þ t Þ ¼ 0. ptþ1 This is satisfied by cotþ1 ¼
m k o u01 ðcyt Þcm t ctþ1 k 2m xtþ1
hence cytþ1 ¼ T 2 ðcyt ; cm t ; xtþ1 ; xtþ2 Þ;
ð26Þ
where ! y m 0 k c u ðc Þ o y t 1 t T 2 ðcyt ; cm ; t ; xtþ1 ; xtþ2 Þ ¼ wy þ wm T 1 ðct ; xtþ2 Þ 1 þ k 2m xtþ1
ð27Þ
which is the desired equation. 2 Putting xt ¼ ðcyt ; cm t Þ 2 Rþ ; and nt ¼ nt ¼ xtþ1 2 Rþ , "t P 1, the stochastic dynamical system (22)–(26) has the same form as (1) with univariate random perturbations.6 The decomposition of the map T = (T1, T2) into a deterministic core and a state dependent perturbation term described by (2), can be promptly obtained by putting f1 F ¼ f2 and G¼
T 1 f1 ; T 2 f2
where (f1, f2) and (T1, T2) are defined, respectively, by (12)–(16) and (23)–(27). Consider that the dynamical system (22)–(26) should not be interpreted as a mathematical description of the agents’ decision process. It is instead a dynamical system whose output is a Markov process describing sequences of (random) consumption values, compatible with the young agents’ optimal decisions in the sense of (17) and with the market clearing condition.
6 Notice that, in view of the logic of the model and the assumptions made at the beginning of Section 3, at each time t uncertainty concerns only the value of xt+2. Only once, at the beginning of the process, we need a bivariate random perturbation with independent elements.
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4. Computational analysis To perform the numerical exercises of this section we need further specifications of our model. First of all, we want to select a utility function for the first period consumption such that the resulting model has the most interesting dynamics compatible with tractability. The literature on OLG models suggests that the utility functions of the CARA type (constant absolute risk aversion) are suitable candidates.7 In what follows, we use one such function, namely the exponential utility function u1(c) = rebc, where r, b are positive constants, and the absolute risk aversion is RA(c) = b (hence RR(c) = bc). Secondly, we must specify the properties of the stochastic perturbations. Omitting time indexes, the perturbations take the form of a random variable x = 1 + n, with n = l 0.5; EðnÞ ¼ 0. We assume that the distribution of l belongs to the beta family, with parameters (a, b) and support [0, 1]. Two basic cases are considered, namely: (i) a = b = 1 which is equivalent to the uniform distribution and will be denoted by l U, and (ii) a = b = 5, denoted by l Be(5, 5), i.e., a beta distribution with a unimodal, symmetrical and fairly peaked density function. In both cases, EðlÞ ¼ 0:5. Considering that, in the logic of the model, it must be x > 0, we take 2 [0,2]. As a matter of fact, in most of our simulations (thousands of them), the maximum value of compatible with stochastic stability was well within that interval. Not surprisingly, the maximum admissible support (the largest admissible value of ) is systematically larger for the unimodal beta distribution than it is for the uniform one. The deterministic (perfect foresight) OLG model with three generations is much more complex than the corresponding two-generation model. As is well known, for the latter we have two basic types of economic behavior: the first one, nicknamed ‘‘the classical case’’ in which the young agent is impatient and borrows from the old agent and, when old, saves to pay back his debt; the opposite situation, nicknamed ‘‘the Samuelson case’’, obtains when the young agent is thrifty and saves in order to be able to dissave when old. It is evident that the presence of three generations allows many more combinations of saving/ dissaving behaviors compatible with the intertemporal budget constraint. Schematically, we have the following six possibilities: (S S D); (S D S); (S D D); (D S S); (D S D); (D D S), where each triplet refers to the three periods of the representative agent’s life (and of course saving = endowment consumption). Our numerical ‘‘experiments’’ are restricted to the D S D case: the representative agent dissaves when is young, saves during his maturity and dissaves again when old. This variety of the OLG model, which we nickname ‘‘D S D’’, describes a not altogether implausible situation and, at any rate, yields all the dynamical complications we want to discuss. For different configurations of the parameters, the attractor of the deterministic ‘‘D S D’’ model may be a fixed point, a periodic orbit, a quasiperiodic orbit, a chaotic set made of a single piece or of a finite number of subsets. The bifurcation analysis of the model (that we shall not discuss here) shows some of the canonical ‘‘routes to chaos’’: flip bifurcations and subsequent period-doubling cascades, Neimark–Sacker bifurcations with 7 These utility functions are characterized by the fact that substitution effects are stronger that income effects for low levels of consumption but the opposite is true for higher value. As is well known, this fact is associated with fluctuating intertemporal equilibria. Remember that for a utility function u(c) the relative risk aversion coefficient is RR(c) = u00 (c)c/u 0 (c) and the absolute risk aversion coefficient is RA(c) = RR/c.
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
37
OL3G 40
cMt
35
30
25
12
14
16
18
20
cYt Fig. 1. Perturbed fixed point (l Be(5, 5); = 0.54).
subsequent periodic/quasiperiodic dynamics and possibly ‘‘torus-breaking’’ and chaos; ‘‘catastrophic’’ local/global bifurcations yielding a jump from simple (periodic) dynamics to chaos. Most often, the attractor appears to be unique, but occasionally multiple attractors exist. In each of these cases, we have been able to find numerical evidence of stochastic attractors for the perturbed dynamical system (22)–(26), which we know can be interpreted as stationary sunspot equilibria.8 We are now ready to present examples of each of the cases discussed in the theoretical part of the paper, according to the classification of Section 1. 4.1. Strongly indecomposable deterministic attracting set: fixed point In this case, the deterministic attractor is a fixed point, a steady state perfect foresight intertemporal equilibrium. The unperturbed attractor and the corresponding SSE generated by stochastic perturbations are illustrated by Fig. 1. 4.2. Strongly indecomposable deterministic attracting set: quasiperiodic dynamics The equilibrium dynamics are aperiodic but not chaotic. The attractor and the corresponding SSE are depicted in Fig. 2.
8 A detailed discussion of the different dynamical phenomena mentioned above is out of the question here. The interested reader may consult, for example, Medio and Lines (2001).
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G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
OL3G 20
18
16
cM t
14
12
10
8
14
16
18
20
22
24
cYt
Fig. 2. Perturbed quasiperiodic attractor (l Be(5, 5); = 0.24).
4.3. Weakly indecomposable deterministic attracting set: periodic dynamics The selected example is a perturbed period-4 deterministic attractor shown by Fig. 3.
OL3G
cMt
40
30
20
10
10
15
20
25
cYt
Fig. 3. Perturbed period-4 attractor I (l Be(5, 5); = 0.31).
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
39
OL3G 21.5
21.0
cMt
20.5
20.0
19.5
19.0
18.5
23.7
23.8
23.9
24.0
24.1
cYt
Fig. 4. Perturbed period-4 attractor II: iterations of T4 (l U; = 0.02).
Keeping the same parameter configurations, Fig. 4 shows one of the four possible corresponding SSEs, obtained by using iterations of the map T4 (where T = (T1, T2)), i.e., the 4th iterate of system (22)–(26), and appropriately selecting initial conditions, as explained in Section 1. 4.4. Weakly indecomposable deterministic attracting set: many-piece chaotic set A perturbed 4-piece deterministic chaotic attractor is shown in Fig. 5. To depict one of the four possible SSEs, in Fig. 6 we again employ the 4th iterate of the map T, with appropriate initial conditions. 4.5. Decomposable attracting set I: multiple periodic attractors In Fig. 7, we show the basins of attraction of three different periodic deterministic attractors (the fixed point is considered periodic of period 1). In Fig. 8 we show that, if we select initial conditions sufficiently close to one of the attractors (in this case the attracting fixed point) and the perturbation is sufficiently small with respect to the size of its basin, a stochastic attractor corresponding to a SSE can be obtained. 4.6. Decomposable attracting set II: periodic and chaotic attractors Fig. 9 depicts two coexisting deterministic attractors (a period-3 orbit and an one-piece chaotic set), with their basins.
40
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45 OL3G 35 30
cMt
25 20 15
10 5
10
20
30
40
cYt
Fig. 5. A perturbed 4-piece deterministic chaotic attractor I (l Be(5, 5); = 0.1).
OL3G 14
12
cMt
10
8
6
4
2 28
30
32
34
36
38
40
cYt Fig. 6. A perturbed 4-piece chaotic attractor II: iteration of T4 (l U; = 0.03).
Fig. 10 instead shows the stochastic attractor of the perturbed system, corresponding to a SSE.
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
Fig. 7. Multiple attractor I.
OL3G
cMt
35
30
25
16
17
18
19
20
21
22
cYt Fig. 8. Perturbed fixed point (l Be(5, 5); = 0.2).
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G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
Fig. 9. Multiple attractor II.
OL3G 55
50 45
40
cMt
42
35
30
25
20 5
10
15
20
25
cYt Fig. 10. Perturbed one-piece chaotic set (l Be(5,5); = 0.3).
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
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5. Concluding remarks Before concluding our paper, we must briefly mention a conceptual problem concerning the choice of initial conditions in OLG models. We discuss this point with regard to the deterministic, perfect foresight case, but the conclusions can be easily extended to the stochastic case. Moreover, here we only consider OLG models yielding a dynamical system for which there exists a unique orbit forward in time for each initial point, as is the case for our Eqs. (12)–(16).9 In dynamical systems usually discussed in mathematical and physical sciences, initial conditions are chosen at random out of an admissible set, and their future values are then determined by the equations of motion. This is also the case, for example, in economic models of optimal growth where the initial conditions (typically, the capital stock) are pre-determined. However, in the logic of the OLG model the values of the state variables at each time t, are justified only by the existence of a sequence of perfectly anticipated future values such that for each successive pair of time instants (t, t + 1), the difference equations of the model are satisfied, and this rule, strictly speaking, should apply to the initial conditions as well. Accordingly, in the case of our model, the initial conditions o 10 ðcm should be interpreted as optimal consumption levels of the mature and old agent, 0 ; c0 Þ planned at time t 1 and t 2, respectively (and perfectly realized). This reasoning could be repeated backward in time without limit. In other words, strictly speaking, the story told by the OLG model should have no beginning and no end. In the economic literature, there exist two basic approaches to this conceptual difficulty that are relevant for our discussion. The first of them – nicknamed here ‘‘unlimited horizon hypothesis (UHH)’’ – is to consider sequences of intertemporal equilibria stretching indefinitely both in the future and in the past. A moment’s reflection suggests that adopting the UHH hypothesis implies a severe restriction of the admissible orbits. For example, for a given initial point there might be few infinite backward in time sequences of the state variables satisfying the requirements of the model (utility constrained maximization, market clearing and nonnegativity), or there might be none. Thus, the UHH is logically impeccable but its realism and fruitfulness can be questioned (cf. Benhabib and Day’s (1982) comments on this question). The second approach – nicknamed ‘‘single shock hypothesis (SSH)’’ – introduces the ad hoc assumption that, at some instant of time t = 0, the economy is perturbed by an unexpected random shock, whose precise nature depends on the details of the model, and that no further disturbances occur. The SSH is adopted, for example, by Kehoe and Levine (1985). The SSH allows one to treat initial conditions as random (within certain constraints), at the price of violating, for the initial period only, the hypothesis of y o agents’ perfect foresight. In our case, for example, if at t = 0 ðcm 0 ; c0 Þ (and therefore c0 ) are chosen at random, in general the expectations of the young agents at times t = 1 and t = 2 will not be fulfilled. All in all, we believe that, in the present context, the SSH is a more fruitful alternative and we implicitly adopt it throughout the paper.
9 A thorough discussion of the difficult problems arising when an OLG model yields a dynamical system moving backward in time, can be found in Medio and Raines (forthcoming). y m 10 o Remember that, in view of the market clearing condition, fixing ðcm 0 ; c0 Þ is equivalent to fixing ðc0 ; c0 Þ.
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Acknowledgements The authors would like to thank Cees Diks of the University of Amsterdam for acting as a discussant of this paper. His very detailed comments and criticisms were exceedingly helpful and prompted the authors to correct a number of mistakes and improve the presentation. All remaining errors and misunderstandings are entirely the authors’ responsibility. Appendix A All the computations have been performed by means of the specialized software iDMC, somewhat elaborated using the ‘‘R’’ software. iDMC can be downloaded from the website: http://www.dss.uniud.it/nonlinear/. Below, we are listing the values of the initial conditions and parameters used to produce the figures appearing in the main text. Fig. 1: cy0 ¼ 16;
cm 0 ¼ 32;
k m ¼ 4:46;
wy ¼ 1;
k o ¼ 1;
wm ¼ 55;
r ¼ 50;
b ¼ 0:3;
wm ¼ 41;
r ¼ 100;
b ¼ 0:3;
wm ¼ 55;
r ¼ 200;
b ¼ 0:3;
wm ¼ 55;
r ¼ 200;
b ¼ 0:3;
wm ¼ 55;
r ¼ 220;
b ¼ 0:3;
¼ 0:54.
Fig. 2: cy0 ¼ 20;
cm 0 ¼ 10;
wy ¼ 1;
k m ¼ 1:4;
k o ¼ 1;
¼ 0:24.
cy0 ¼ 22;
cm 0 ¼ 15;
wy ¼ 1;
k m ¼ 11;
k o ¼ 1;
¼ 0:31.
cy0 ¼ 24;
cm 0 ¼ 20;
wy ¼ 1;
k m ¼ 11;
k o ¼ 1;
¼ 0:02.
cy0 ¼ 20;
cm 0 ¼ 35;
wy ¼ 1;
k m ¼ 2;
k o ¼ 1;
Fig. 3:
Fig. 4:
Fig. 5:
¼ 0:1.
Fig. 6: cy0 ¼ 40;
cm 0 ¼ 5;
k m ¼ 2;
k o ¼ 1;
wy ¼ 1;
wm ¼ 55;
r ¼ 220;
b ¼ 0:3;
¼ 0:03.
Fig. 7: cy0 2 ½0; 55; k m ¼ 4:46;
cm 0 2 ½0; 55; k o ¼ 1;
wy ¼ 1;
¼ 0.
wm ¼ 55;
r ¼ 150;
b ¼ 0:3;
G. Gazzola, A. Medio / Journal of Macroeconomics 28 (2006) 27–45
45
Fig. 8: wy ¼ 1; wm ¼ 55; cy0 ¼ 20; cm 0 ¼ 30; k m ¼ 4:46; k o ¼ 1; ¼ 0:2.
r ¼ 150;
b ¼ 0:3;
Fig. 9: cy0 2 ½0; 60; cm wy ¼ 1; 0 2 ½0; 60; k m ¼ 30; k o ¼ 1; ¼ 0.
wm ¼ 58:5;
r ¼ 200;
b ¼ 0:3;
Fig. 10: cy0 ¼ 25;
cm 0 ¼ 25;
k m ¼ 30;
k o ¼ 1;
wy ¼ 1;
wm ¼ 58:5;
r ¼ 200;
b ¼ 0:3;
¼ 0:3.
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