Competitive Equilibrium Cycles - Google Sites

JOURNAL

OF ECONOMIC

THEORY

35, 284306

Competitive

(1985)

Equilibrium

Cycles*

JESS BENHABIB Department

of Economies,

New

York

University,

New

York.

New

York

10003

AND KAZUO

University

NISHIMURA

Department of Economics, qf Southern California, Los Angeles,

Received

November

16, 1983; revised

Cal(fornia

October

90027

1, 1984

Sufficient conditions on the structure of technology that give rise to robust periodic cycles in stocks, outputs, and relative prices in a stationary, fully competitive economy with an infmitely lived representative agent are given. The results are first derived in the context of a general intertemporal model of accumulation that has been used in the turnpike literature. These results are then applied to a two-sector neoclassical model of production, and sufftcient conditions giving rise to robust periodic cycles are obtained in terms of relative capita! intensities of the two sectors. Journal of Economic Literature Classification Numbers: 021,022,023. i? 1985 Academic

Press. Inc.

1.

INTRODUCTION

Recently there has been a surge of interest in endogenous business cycles that arise in competitive laissez-faire economies. In the context of standard for the existence of overlapping generations economies, conditions equilibrium cycles have been given by Grandmont [ 151 and by Benhabib and Day [4]. Models of the economy with extrinsic uncertainty or “sunspots” that have been developed by Shell [31] and by Cass and Shell [ 1 l] (see also Balasko [3]) can also lead to equilibrium cycles. The relation between sunspot equilibria and the existence of deterministic cycles has been explored in a recent paper by Azariadis and Guesnerie [2]. A search model where beliefs of agents also play a role in generating endogenous cycles is given in a paper by Diamond and Fudenberg [14]. Although the equilibrium cycles in many of the works cited above arise in a * Jess Benhabib’s research was supported Grant SES 8308225, and by the C. V. Starr University. We would like to thank the referees valuable suggestions. We are also grateful to discussions. For any remaining errors, we are

in part by the National Science Foundation Center for Applied Economics at New York for pointing out errors as well as making very Clive Bull and Chuck Wilson for enlightening entirely responsible.

284 0022-053

l/85 $3.00

Copyright 0 1985 by Academic Press, Inc. All rrghts of reproduction m any hrm reserved.

COMPETITIVE

EQUILIBRIUM

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285

competitive framework, they are not necessarily Pareto-efficient and they allow the possibility of policy intervention. In contrast, works by Kydland and Prescott [20] and Long and Plosser [21] develop models with a single infinitely lived representative agent that generate Pareto-optimal equilibria. They then explore the possibility of business fluctuations in these models by using simulation methods. The purpose of this paper is to provide economically interpretable sufficient conditions under which equilibrium cycles arise in a deterministic, perfect foresight model with an infinitely lived reresentative agent and a neoclassical technology. One of our main results, obtained for a standard two-sector model of production, gives sufficient conditions for generating “robust” (see the Remark following the proof of Corollary 1) periodic cycles in outputs, stocks, and relative prices. These sufficient conditions are expressed in terms of the discount rate and differences in the relative factorintensities of the two industries. The main contribution is to give an insight into the structure of general classes of neoclassical technologies that lead to equilibrium cycles in a fully stationary environment. While we obtain sufficient conditions for persistent cycles in a two-sector discrete-time model, similar results hold in multisector continuous time models. Benhabib and Nishimura [7] provide a general method for constructing periodic equilibrium with a multisector trajectories Cobb-Douglas technology in a continuous time framework. The structure of their examples suggests that the existence of cycles can be linked to generalized capital intensity conditions when the rate of time preference is not too close to zero. So far, however, there have not been any sufficient conditions for the existence of cycles given in the literature. In the next section Theorem 1 gives sufficient conditions for the existence of optimal periodic trajectories in an abstract and general setting. Section 3 discusses existing examples of cycles due to Sutherland [32] and due to Weitzman as reported in Samuelson [29], especially as they relate to Theorem 1.’ In Section 4 Theorem 2 gives general conditions under which the equilibrium trajectory is globally monotonic or oscillatory and Theorem 3 gives conditions under which the optimal equilibrium trajectory converges either to a stationary point or to a cycle of period two. We obtain our main results in Section 5 by applying Theorem 1 to a two-sector neoclassical technology. We give capital-intensity conditions that lead to perrsistently cyclical, efficient equilibrium trajectories of outputs, stocks, and relative prices. We also discuss an application to the adjustment-cost model of investment. Finally, in Section 6 we summarize the intuitive explanation for the existence of cyclical equilibria that arise in neoclassical technologies. ’ Although we explore the existence of cyclical trajectories, our framework of analysis in the spirit of turnpike theory. For an excellent survey of turnpike results see McKenzie

is also 1251.

286

BENHABIB

2.

THE

EXISTENCE

AND

AND

NISHIMUBA

STABILITY

OF PERIODIC

CYCLES

A technology set D is defined as a closed convex subset of R: = {xe R* I x >O}, where (x,, x2) ED represents the input stock x1 and a feasible output stock x2. If x2 > x1 and (x,, y,) E D, we assume that there exists a y, > y, such that (x,, yz) E D. Furthermore, assume that there exists stock level X > 0 such that if x > 2 and (x, y) E D, then y < x, and if x>O and xx. Also assume that (x, y) E D implies (x, y’) E D for 0 < y’d y. Let d = {(x, y) ED 10 (1 +6-) I/,,(E(6-), k(C)) (ii) [v2,(k(s+),~(6+))+6+V,,(~(6+), k(6+))] q, = $( q, + 2V,, I/,, + IQ, which in turn gives 0 > (V,, - V,,)‘. This holds only if V,, = V,,. Thus the main restriction imposed by (A4) is that a 6- exists. We use (A4) in establishing the properties of the roots of Eq. (3) (see Lemma 2). Remark.

Requirement

k,) is strongly

Remark. In our one capital good model (A4(ii)) is the same as the dominant diagonal conditions that Araujo and Scheinman [l] use to obtain global stability results in a multisector economy. In this paper, however, we require (A4(i)) and (A4(ii)) to hold only at the steady state, rather than along the optimal path. The result (ii) of Lemma 2 shows that the dominant diagonal condition (A4(ii)) implies the local asymptotic stability of the steady state. In a multisector context, Dasgupta and McKenzie [ 121 have shown that if [I’,,] is symmetric, then the dominant diagonal condition is both necessary and suflicient for local asymptotic stability. For later use we also define the following sets:

Pp = (6 1 Vz2+6Vj1 >(1+6) P+ = {S 1 V2,+6Yl, P”=(61

(1+6)V,,,1,,/1,< -1. -l),l,E(-1,O). If Vz2 + 6V,, k,+l;6)

(6)

.vt+z=k,+l. 3 For higher order systems similar conditions. See Marden 1241.

conditions

can be obtained

by exploiting

Schur-Cohn

COMPETITIVE

EQUILIBRIUM

289

CYCLES

Consider also the following system obtained by expressing ( y, + 1, k, + ,) in terms of (y

(8)

A stationary point of (7) is given by M(6, k, y) = 0. A simple calculation shows that if [G] is the Jacobian of the right side of (6) evaluated at a stationary point of (6), [G]” is the Jacobian of the right side of (7). Thus if 1, and AZ are the roots of the right side of (6) evaluated at the steady state k(S), the roots of the right side of (8) will be given by (1 -A:) and (1 - 1:). Note that AI and i, are also the roots of (3). Consider an interval [IS-,S+], where 6- and 6+ are defined by (A4). We can consider M(6, k, y) as a homotopy on the interior of B over [S -, 6+ 1. (i) Consider first the case where the set Z(6) = {(k, y) 1 M(6, k, y) = 0) does not, for any 6 E [S -, 6 + 1, intersect the boundary of a nonempty, convex, open subset B of D which contains the steady state k(6) in its interior. By construction and by Lemma 2, for any 6 E [S-, S+] n P(where Pp is defined by (2)), the Jacobian determinant of the right side of (8), that is, of M(6, F(6), F(6)), evaluated at the steady state of (6), will be (l-nf(s))(l-Iz(s))>O. Also, for any d~[&,fi+]nP~+, the Jacobian determinant of M(6, F(6), K(6)), evaluated at the steady state of (6) corresponding to 6, will be (1 - ;i,(S)‘)( 1 - 3,,(6)‘) < 0. But the topological degree of M(6, k, y) over the boundary of B is a homotopy invariant. This that if the Jacobian of M(6, k, y) is [J(S)], implies then C,X,?,EZ,6, sign Det[J(6)] is constant over 6 E [S-, S+] (see Milnor [27]). But since Det[J(6)] evaluated at the steady state of (6) changes sign as 6 crosses from [S-, 6’]n Pp to [&, G+]nP+, either M(6, k, y)=O has at least two solutions with Det[J(G)] < 0 (i.e., solutions other than the steadystateof(6))inBforall6E[6-,6+]nP-,orM(k,y;6)=0hasat least two solutions with Det[J(s)] > 0 (i.e., solutions other than the steady state of (6)) in B for all 6 E [S -, 6 + ] n Pt. Since such solutions are stationary points of (7) and periodic paths of (6), the set L in Theorem 1 is either [S-, S+] n P- or [S-, S’] n P+.

290

BENHABIB AND NISHIMURA

(ii) Now consider the case where no matter how we choose a convex open subset of B in d containing the steady state R(8) in its interior, Z(6) intersects the boundary of B for some 6 E [S-, 6 ‘1 for any feasible choice of the interval [S-, S’]. Since (R(6), K(6)) stays in the interior of D by (A2) and (F(6), &(6)) changes continuously with 6 by Lemma 1, it must be the case that Z(S) contains other points than (k(S), E(8)) for some 6 E [S-, 6 + 1, which by construction are periodic solutions of (6). We note that (6) gives the first order conditions for the problem given by (1). However, since V(li,, k, + , ) in problem ( 1) is concave and bounded in B and since the periodic cycles are bounded by B, standard arguments assure the sufficiency of transversality conditions for the periodic paths to be optimal (see Weitzman [33] or the proof of Lemma 16 in Q.E.D. Scheinkman [ 301). We can sharpen Theorem 1 to obtain the corollary add the following assumptions:

below, provided we

(A5) P” (defined by (2)) is a discrete set. (A6) The stationary points of Eq. (7) are isolated bE [S-, S’].

for

each

COROLLARY 1. (i) Under (Al )-(A6) the interval L in Theorem 1 is of positive length. (ii) Either there exist periodic cycles of (6) (i.e., stationary points of are locally unstable (i.e., both roots of the (7))whichfor6~[8~,$+]nP+ Jacobian of the right side of (7), evaluated at the stationary point of (7), are outside the unit circle), or there exist periodic cycles of (6) for sE[$-,8+]nP-, which locally are saddle points (i.e., one root of the Jacobian of the right side of (7), evaluated at the stationary point of (7), is inside, the other outside the unit circle).

Remark. Part (ii) of the above corollary implies that if the periodic cycles of (6) exist for [$-, s^+] n P+, they are locally repelling and if they exist for [s^-, s”] n P-, locally they have a l-dimensional stable manifold, such that initial conditions on this manifold lead to convergence to the cycle. Proof: Under (A5) P” is discrete. So for any 6’~ P” we can choose an [&-, 8+] c [S-, S+] such that 6’ E (6 I [8-, 6’1 n P”}. interval Moreover, as the stationary points of (7) are isolated by (A6) we can choose a convex neighborhood B of (k(6), k(6)) in the interior of b such that stationary points of (7) do not lie on the boundary of B as long as [d-, $‘I is sufficiently small. Therefore, the arguments used in case (i) of the proof of Theorem 1 apply using [8-, 8’ ] and B. Thus, for every

COMPETITIVE

EQUILIBRIUM

CYCLES

291

6 E [s^-, 8’1 n P- or for every 6 E [b-, 8’1 n P+, B contains periodic paths of (6). L= [d-, 8+] n P- or [8-, b+] n P+, both of which are of positive length. This proves part (i) of the corollary. By the proof of Theorem 1 either there exist at least two nonstationary periodic cycles of (6) for BE [8-, 8’1 n P- with Det[J(6)] O, where Det[J(6)] is evaluated at the corresponding nonstationary periodic points. Consider first the case of nonstationary cycles for 6E [S-, S+] n P-. By proof of Theorem 1, Det[J(6)] = (1 - p,)( 1 - pLz)< 0, where p, and pL2 are the roots of the Jacobian of the right side of (7) evaluated at these periodic points. Calculating the determinant and trace of the Jacobian of the right side of (7), say [H(S)], and using the fact that k, + z = k,, k,, 3= k,, , at the periodic points, we obtain Det[W(G)]

= 6-’ > 0

Trace[H(G)]

V,,(kttk,+1) f’,,W,+,3kr+d dV,Jkt+,,k+d+ 6f’,,(k,,k,, 1)1 + V,z(k,,k,+,)+6v,,(k,+,,k,+,) [ ~Vdk,+l,k,,,) 1 V&,+ t, kt+,)+~V,,(k,,k,+,) X

=-

dv,,(k,>

k,,,)

1

6(J’,,(k,,k,+,)V&r,kr+,)- V,kk,k+,)2) = $-4f’,,(k~+,~k~+~) Vzz(k,+,>k,+dUk+~,k,+d2) h2V,,(k r+,,k,+d

V,Ak,,k,+,)

by (Al).

Since Trace[H(G)] > 0 and Det[H(G)] > 0, p,, p2 > 0. Thus, Det[J(G)]= (1 -,~,)(l --p2) 1. Therefore, the fixed points of (7) are saddle points. For 6 E [s’-, 8’1 n P+, Det[J(6)] > 0 at periodic points of (7). This implies that p, > 1, pl> 1 since p, pz = 6 e-2>, 1. Therefore, the periodic points of (7) are unstable. Q.E.D. Remark. We can point out details that the periodic cycles class of all c’ maps F: P x A4 + and A4 is an n-dimensional C’

without getting of the corollary M, where P is a manifold, Y3 2,

into too many technical are generic. Let 9 be the l-dimensional C’ manifold such that for every p E P,

292

BENHABIB

AND

NISHIMURA

F,: M + A4 is a diffeomorphism. The function F in Eq. (6) satisfies these requirements. Then for every F from a residual subset s c 9 (i.e., p1 us the countable intersection of open sets, each of which is dense in 9), (A5) and (A6) are satisfied (see Brunovsky [lo, Theorem 1, (ii) and (iii)]. Furthermore, if P” is discrete as in (A5), under the corollary above cycles will exist for intervals of 6 for which 6 $ PO. In addition, for any such 6, the Jacobian of (6) does not have roots on the unit circle by Lemma 3. For any such b consider any C*-perturbation of F(k, y; 6). Then we can show that cycles will persist in the perturbed system and will be “close” to the cycles of the unperturbed system. (For a proof see Hirsch and Smale [IS], Proposition, p. 305, Chap. 16.)4

3.

DISCUSSION

OF EXAMPLES

There are several examples in the literature of periodic cycles in the setting of problem (1). The example given by Sutherland [32] fits precisely the conditions of our Theorem 1. In the example, V(k,_ , , k,) = V,,= -11, V,,= -l&and V,,= -8 9k:-, - llk,k,_,-4kf+43k,,where for any (k,- I, k,) since V(k,- 1, k,) is quadratic. Sutherland obtains cycles for S=$ For S=$, (V’22+SV1,)/V,2=# 2. Furthermore, there is a unique ho= f such that (~22+~~1,Y~,* ( VZZ+ 6OV,, )/VIZ = 1 + 6’. Thus the set P” in (A5) is discrete. Theorem 1 and its corollary immediately apply. Another example due to Weitzman is reported in Samuelson [29]. The example is Max C;:o 6’k;“( 1 - k,, 1)“, for cr=P=$. The roots of the associated linear system for this example are - 1 and -6-l, so one root is always on the unit circle. Given 6 any pair (x, y) satisfying (1 -X)/X) = 6*( ,Y/(1 - ?I)) qualifies as a periodic cycle. Since for any SE (0, 11 none of the roots of the associated linear system is inside the unit circle, our (A4) fails and Theorem 1 cannot be applied. Note that (A5) also fails since one root is always - 1 and the set P” is a continuum. This suggests that the cycles may not persist under small C’-perturbations (see the Remark following corollary 1). In fact, if we set a, p > 0, SI+/J < 1, then k:( 1 -k,+ ,jp is strictly concave at a steady state and periodic solutions will disappear for 6 sufficiently close to 1 (see Scheinkman [30] and McKenzie [26]). It is, however, possible to use Theorem 1 to obtain cycles if 6 is not close to 1. Consider the expression q( 6) = ( Vz2 f 6 V, 1)/VI, - ( 1 + 6) = ((1 -P)/a)(k(6)/(1 -&(s)))+&(l -a)/j?)((l -@@)/k(6))-(1 +6) for the above model where the steady state k(s) is the solution to 4 Using the same techniques, it is possible to generalize dimensional cases. We will pursue this in further work.

the results

of Theorem

1 to higher

293

COMPETITIVE EQUILIBRIUM CYCLES

(1 - &?))/I+?) = /?/scl. c onsider the cases cc+/?