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.

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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
5 YV(k,, k,,,) t=0

subject to

(k,tk,+t)EB O
given

t > 0,

(1)

0<6<1 (Al) V is continuous and concave on D, of class C* on the interior of B, with V,,, V,, < 0, V, , V,, - V:, 2 0 and 1V(x, y)] < co for (x, y) E 6. It can be shown that under (Al), problem (1) has an optimal solution (see Brock [9], Majumdar [22]). A path is an interior Euler path if it satisfies V,(k,+,,k,)+dV,(k,, k,+,)=O and f>k,>O for all t. A steady state k(8) is a solution of solution of V,(k, k) + dV,(k, k) = 0. (A2) A steady state k(6) exists on (0, ,Ic) for 6 E [S, 11, where 1>6>0. (A3) V,,(x,, x2) < 0 for (x,, x2) E interior D.’ LEMMA

1.

The steady state k(6) is a dij$erentiable function

on [S, 11.

Proof: Consider H(k, k; 6) = V,(k, k) + 6V,(k, k) on d x (6, 1). For any 6, dH/dk < 0 by (Al)-(A3) and thus for any 6 E (6, 1) there is a unique E(6) such that H(6(6), S) = 0, The differentiability follows from the implicit

function theorem.

Q.E.D.

‘This assumption can be relaxed to hold only on some relevant subset of b, but the additional notation need would be cumbersome. For a class of models where (0, y)~ d implies v = 0 (the impossibility of the Land of Cockaigne), V,, < 0 may hold only in the interior of D as in the two-sector model of Section 5 below, or in other models only on some subset of b. Note that for (k,, k,,,) in the neighborhood of (k(X),IF(6-)), (A4) implies Viz < 0 and the steady state is locally unique. If we choose the set B in the proof of Theorem 1 small enough to exclude other steady states that may exist when (A3) does not hold, we can dispense with (A3) in proving Theorem 1.

COMPETITIVE

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287

Assumptions (A3) and (A4) (which follows) will be interpreted in Section 5 in terms of capital intensity conditions for a neoclassical economy. (A4)

There exists 6-, 6’ with [S-, S+] c [IS, l] such that

(i) [V22(1;(~-),k(~-))+8~C/11(~(8-), > (1 +6-) I/,,(E(6-), k(C)) (ii) [v2,(k(s+),~(6+))+6+V,,(~(6+), k(6+))]<(1+6+) v,,(k(s+),E(s+)).

/t(F))]

(A4)(ii) that a 6’ exists is only mildly restrictive. concave, V,, + V,, < 2V,, and setting 6 + sufIf Vk,+l, ficiently close or equal to 1 satisfies (A4(ii)). If V(k,+ i, k,) is only concave 6 + exists if VI1 # V,,. To see this, note that if Vi, + I/,, = 2 I/,,, the concavity of I/ implies V,, I’,, > 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

<(I

I/,, at E(6); kid}

+S)V,,

V2,+6V,,=(1+6)V,,at

at k(6); k(&)eB)

(2)

k(8); k(B)Eb}.

The roots of the Jacobian of an Euler path evaluated at a steady state will be the solutions of

6v,,A2+(Vz,+6V,,)a+ v,,=o.

(3)

It is easily shown that both roots of Eq. (3) are real and that I’,, < 0 implies that both are negative. We will need the following lemmas.

288

BENHABIB

LEMMA

(i) (ii) (iii)

AND

NISHIMURA

2. Let (Al )-( A3) hold and A, and R, be the roots of Eq. (3). IfV,,+6V,,>(1+6)V,,,1,,/1,< -1. -l),l,E(-1,O). If Vz2 + 6V,, <(1+6)v,~,~‘~(-co, IfV,,+bV,, =(I +6)V,,,2, = --l/6, I.,= -1.’

Proof Dividing Eq. 3,,+1”,= -(V22+8V,,)(~V,1)-’ cavity of V, and remembering

note that (3) by dV2, we and 2,&=6-‘. Therefore, by the conthat V,, , Vz2< 0,

i,+~~+2=(6V1*)-‘[-V~z-6V1*+26Vlz] 6(6v,z)-1[-v~,-6v,,-2(6v,,

v2*)“2]

=(6v,,)~‘[((v*,~)1’2-((6v,,()“2]2~o. Also, we have (i*+

l)(&$l)=

1 +jb,2,+;1, =6-‘[l

Combining

(4)

+A,

+6-(v,2)-‘(v?2+svI,)].

(5)

(4) and (5) the results of the lemma follow immediately. Q.E.D.

LEMMA 3. There e.xists a differentiable function F: interior D + R which satisfies k, + z = F(k,, k, + , ; 6) along an interior Euler path.

Proof Let (k,) be an interior Euler path. Then it satisfies 0 = Since dSldk,+2<0 S(k I+23 k,+,, k,;b)= V,(k,,k,+,)+6V,(k,+,,k,+,). by (Al)-(A3), along an Euler pat k1+2 is unique for each (k,, r, k,) and 6 E (6, 1) such that S= 0. The differentiability follows from the implicit Q.E.D. function theorem. Remark. Note that F, z aF(k,, k,,,; o)/ak, (TV&,+,> k+J-I, F, = Wk,, k,+l; hW,+, + dV,,(k,+l, k,+,)N~V,,(k,+,t k+d-‘.

= -VZl(k,, = -(V22(kr,

k,,,) k,,,)

THEOREM 1. Under (Al )-(A4), there exists a nonemptv set L c [S -, 6 + ] such that for 6 EL, problem (1) has optimal solution trajectories which are cycles of period two.

Proof:

We can convert Fin Lemma 3 to a first-order system as follows: k ,+2=F(.vl+l>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<, k,), using Eq. (7) and substituting back in (6): k r+z=F(k,,F(y,,k,;6),6)

~r+~=J’(yr,

(7)

k,; 6).

Clearly the fixed points of (6) are fixed points of (7) but not vice versa. The fixed points of (7) that are not fixed points of (6) correspond to periodic points of (6) of period two. We will prove that such fixed points of (7) exist. Consider the vector function on the interior of B: M(& k, y) =

k-W,F(y,k;6)) k;6) i y-Fly,

>

(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

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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)<0 implies that p, < 1, 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=$, (V’22+SV1,)/V,2=#<1 +$, but for 6=1, = g> 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+/?2(crfi)-”‘-4 = 4((a+p) - (yw -P1”)2)-‘-4 > 4((a+/?-‘120. Thus, q(l)>0 and (A4) is satisfied. Furthermore, dq(6)/d6 = ( 1 - 2/3)/b > 0 and there is a unique 6’ = j(2cr - 1 )/a( 1 - 2a) > 0 such that q(S”) = 0. Thus Theorem 1 applies to the above cases.

4. CONDITIONS FOR MONOTONIC

AND OSCILLATORY

TRAJECTORIES

Let W(k,)=Max~~~,6’V(k,,k,+,) for (k,,k,+,)~D for all t. We can show that under (A3) the optimal path from k, is unique. We will use the following lemma in proving Theorem 2. 4. Let (Al)-(A3)

LEMMA

hold. Then the optimal path (k,} from given k,

is unique. Proof. Since V(k,, k, + 1) is concave W(k,) is concave. Consider W(k,) = Max,, V(k,, k,) + 6W(k,). Since V,, < 0, V(k,, k,) + 6W(k,) is strictly concave in k, and the choice of k, that maximizes it is unique. This

follows because if there were two optimizing k,'s their convex combinations would be feasible and yield a higher value for V(k,, k, ) + 6 W(k, ). Since the argument holds for any k, + , given k,, the optimal path is unique. Q.E.D. THEOREM

let (Al),

2. Let (k,} he an optimal path. Let (k,, k,, ,) E interior D and

(A2) hold. Then

(i) ifV’,,(x, y)>Ofor all (s, y)EinteriorD, k, k, + 2 (i.e., k l+,>k,+?. - lf(k,+,, any interior segment qf an optimal path is oscillatory). ProoJ Let W(ko)=Maxx:,“=06’V(k,, k,,,) with (k,, k,+,)ED for all t 3 0. Let (k,} and {k;} be optimal paths from k, and kb, where kb > k,. These paths are unique from Lemma 4. Suppose that (k,, k,) E int D. Then for kb sufficiently close to k,, (k,, k’,)E int 6. This follows from the continuity of optimal stock in initial stocks. For the same reason we have (kb, k, ) E int D. By the principle of optimality WV,) = V(k,, k) + sW(k,) z V’(k,, k;) + sW(k;) WG)=

W,,

k;)+6W(k;)>

V(k;, k,)+GW(k,).

294

BENHABIB

AND

NISHIMURA

Adding, we obtain

By the convexity of B, [k,, kb] x [k;, k,] c 4. Then we have

and V(kh,k,)=jk~‘(V,(k;,s,)+

Substituting

V(kb,k;))ds,

these above, we obtain

The above inequality implies k’, k, if V,,>O, provided kb is sufficiently close to k,. But the choice of k, was arbitrary. Hence this local property may be extended to the whole domain (0, -?). It follows that for any kb and k, in (0, X), kb > k, implies k’, 6 k, if I’,, < 0 and k;>k, if V,,>O. Now consider an optimal path (k,) and let kb = k, and k; = kZ. The and V,,>O above result shows that if k, > k,, V,? ~0 implies k26k, implies k2 3 k,. This proves the first parts of (i) and (ii) in Theorem 2. To obtain the results with strict inequalities we use the first order conditions or the Euler equations. We prove the results by contradiction. Suppose k’,=k,. Then (k,,k,,kz...) and (kb,k,, k,...} are both optimal paths. If (k,, k,) and (k,, k2) are in the interior of D the Euler equation holds with equality and Vl(.u,k,)+6V,(kl,k,)=O

holds for x = ko, kb. But if I’,, # 0 on the interior of D this cannot be true and a contradiction follows. Thus k, # kb implies k, # k;. This proves the Q.E.D. second parts of (i) and (ii) of Theorem 2. Remark. We used Lemma 4 in proving Theorem 2 where we assumed that V,, < 0. If V,, is allowed to be zero the optimal choice kl from k. may not be unique but will be a closed convex set. We can define S(k,) =

COMPETITIVEEQUILIBRIUM

CYCLES

(k, ( k, =ArgmaxR, V(k,, k,)+6W(k,)j. Theorem 2 still goes through is, provided we assume that k,x S(k,) and S(k,) x S(k,+,) are in interior of D for all k,, 1E S(k,). Part (i) of Theorem 2 (monotonicity) can be proved under a weaker of assumptions. We can drop the assumption of the convexity of B and concavity of V and replace (Al) and (A3) with the following:

(Al’) V,>O and (A3’) a countable

295

as the set the

V is continuous on 4 and of class C2 in the interior of D with V,
The following theorem generalizes a theorem by Dechert and Nishimura [13]. For a slightly different formulation proving the monotonicity of the optimal path see Majumdar [23]. 2’. Assume that (Al’) and (A3’) hold. Then for an optimal path {k,}, k, > k,,, implies k,,, > k,,,. Further, if (k,,k,+,), ( k,,, implies k,,, > k,,,. Wr+,,k,+2)Eintd, (<) (<) THEOREM

Proof: Let W(k,)=MaxC;“=,G’V(k,, k,,,) and define k;=k,+l, t>O. Then kb is an optimal path from kb = k,, Assume that kb > k, and k, > k;, Then (k,, k,) E B implies that (k,, k;) E d for k’, k,. Hence again using the principle of optimality, as in the proof of Theorem 2, we obtain

W,,k,)+

Wb,k;)>,

W&k;)+

W&k,).

Since [k,, kb] x [k’,, k,] ED by assumption and the above hypotheses,

The above inequality implies that k’, 2 k, under (A3’). To get the strict inequality, we use the Euler equation as in the proof of Theorem 2. Q.E.D. The intuitive idea behind Theorems 2 and 2’ is not hard to grasp from the structure of the proof. If an increase in k, decreases the marginal benefit of k,+l, that is, if I/,, is negative, then the optimal adjustment is to decrease k, + ,. As the optimal choice of k in any period depends only on the value of k in the preceding period, if k,, i is greater than k,, k,,, will then be less than k, + 1. We will discuss this point further in the next section in terms of a neoclassical two-sector model.

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The above theorems raise the question of whether cycles of perodicity higher than two or more complicated chaotic dynamics is possible in our model. Unlike overlapping generation models (see Benhabib and Day [4] or Grandmont [15]) it is very difficult to construct examples which generate chaotic dynamics for infinite-horizon models that have concave utility functions. The difficulty in constructing such examples arises because the range of parameter values that lead to chaotic behavior also violate the concavity of the function V(x, y). 5,6 For nonconcave functions it should be possible to construct chaotic optimal paths although we do not know of any particular economic example that has been worked out. In the context of our model we can rule out complicated dynamics or periodic cycles of order greater than two. In Theorem 3 below we show that if VJx, y) is of uniform sign over 0, any interior optimal path must converge either to a steady state or to a cycle of period two. In the next section (see Theorem 6) we will show that for a standard two-sector neoclassical technology, a uniform sign for VIZ(s, y) restricts differences in the relative factor intensities of the two industries and rules out factor intensity reversals. THEOREM 3. Let jk,} he an interior optimal path so that (k,, k,+,)Einterior D and let (Al), (A2) hold. If V,,(.x, y)#O for all (s, y) E interior 0, {k,} converges either to a stationary point or to a cycle of period two.

Proqf: If V,2(.x, v) > 0 for all (x, y) E interior D, by Theorem 2 the trajectory (k,} is monotonic and bounded and there can be no cycles, (k,} must converge to a point in D. So consider the case V,,(x, y) < 0 for all (x, ~1) interior B. In the proof of Theorem 2 we showed that kb > k, implies k’, k, implies k, < k,. Suppose k2 > k,. If we define k2 = kb, it follows by the same logic that the optimal choice from k2, that is k,, is less than the optimal choice from k,, that is k,. But if k3 < k, , then by the same argument k, > k,. Therefore k, > k, implies k, > k,. Similarly, the converse also holds so that kz < k, implies k, < k2. Thus the even iterates are monotonic and since the optimal path is bounded they must converge to a point in D, say Z. The same applies to odd iterates and they must also converge to a point in D, say Z. Thus any interior path either converges to a period two cycle (5, z) or to a single Q.E.D. point if 5 = z.

’ Ray Deneckere of Northwestern University and S. Pelikan of the University have recently constructed a particular example that does not violate concavity. 6 The well-known Henon map, for example, generates chaotic dynamics ranges that violate the concavity of V(x, y).

of Cincinnati for

parameter

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We will now apply the results of the previous sections to a model in which a representative agent maximizes the sum of discounted conis characterized by a two-sector sumption and where production neoclassical nonjoint technology. Before applying the results of the previous section we will start by giving a detailed exposition and discussion of the model. Let the per capita stock at time t be k,. The per capita output of the capital good, y,, represents the gross accumulation of capital, given as - (1 - g)k,, where g is the rate of depreciation with g > 0. The I’r=k,+, efficient per capita output of the consumption good can be obtained as a function of the per capita stock and the output of the capital good as follows. For Y, Y, L >, 0 consider Max C( I, k) (LkI subject to F(L-1,

K-k)3

Y

O
where C( . , . ), F( . , . ) are standard neoclassical production functions relating the inputs of capital and labor to the outputs of the consumption good and of the capital good, respectively, and where Y is the minimum output of the capital good, K is the stock of capital, and L is the stock of labor. It is easily shown that there is a function r: R’+ + R, such that C = T( Y, K, L) > 0. Assuming constant returns to scale in both industries, we can normalize by labor. Define c = C/L, f = F/L, k = K/L, and y = Y/L. Expressing all quantities in per capita terms we have c, = r( yt, k,, 1) = T(Y,, k,) = W,+, - (1 - g) k,, k,) > 0. Under standard differentiability and quasi-concavity assumptions on the underlying production functions, it can also be shown that T is of class C* and concave (see Benhabib and Nishimura [7, Appendix AI]). For given k,, T( yr, k,) describes the production possibility frontier between c, and y,. We will assume that there exists a F> 0 such that for y satisfying T(y,k)=O, y+(l-g)kli and y+(l-g)k>k if k < & Thus it is not possible to maintain stocks above K. The technology set D given in Section 2 here will be given by D = {(k,, k,, ,)I0 < k, O, T(y,,k,)>O} for t=O,l,.... The consumer’s problem is to maximize Cp”=O d’T(k,+ 1- (1 - g) k,, k,) given k,, where 6 is the discount factor. First-order conditions for an interior solution are given by T,(k,+ I - (l-g)k,,k,)+6T*(kr+*-(l-g)k,+l,k,+,) -6(l-g)T,(k,+2-(l-g)kl+,,k,+l)=O

(9)

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where T1 = iYT(y, k)/ay and T2 = dT( y, k)/dk. It is well known that the solution to this problem can be viewed as a competitive intertemporal equilibrium, where the rate of return on capital is equal to the discount rate, where factors earn their marginal products and the production of outputs is determined according to relative (shadow) prices. Furthermore, we can specify that T,( y, k) = -p and T2( y, k) = w, where p can be taken as the price (or the slope of the production possibility frontier) and w as the rental of the capital good expressed in terms of the price of the consumption good. Setting 6 = l/( 1 + r), where r is the discount rate, the firstorder conditions (9) then can be written as (9’)

which states that one plus the discount rate equals the rental return plus the capital gain on the depreciated stock as a percentage of the cost. It is of course possible to start with the function c = T( y, k) describing efficient production and allocation of resources. The first-order conditions (9) then describe competitive intertemporal allocation without using the representative agent. The additional restriction that the representative agent imposes (via the transversality conditions) is to rule out the nonoptimal explosive or implosive (“bubble”) paths of accumulation. While the analysis of Section 2 and Theorem 1 operates on the first-order conditions directly, a bounded cyclical solution or the paths converging to it will automatically satisfy the transversality conditions. To apply the theorems of the previous two sections and give them economic interpretations in terms of our competitive model we now turn to establish some results concerning the technology and the function T( y, k). The details and proofs of the relationships and results used below are given for the general multi-sector case in Benhabib and Nishimura [6]. For completeness we give a heuristic as well as a diagrammatic exposition. First we note that we can express prices as functions of factor prices alone, as p = p(w’, w), where w” is wage rate for labor, the primary input. Note that we can express w” in terms of the technology as well. We set

+r(;,;, wo=

I)] aL

+T(f,g] =

aL

=T(y,k)+p(y,k).y-w(y,k).k,

where w” is the marginal product of labor in terms of the price of the consumption good and can be expressed as an function of per capita variables (y, k). Since T12 = Tzl, we obtain Tz2 = dw/dk, T,, =

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299

-(a~/%) = -(+/&v)(~w/%) = -4(&v/X), and T,, = -(@/c~w)(~w/$Y) = -(@/a~)( T,,) = b2(8w/X), where 6 = (+/dw) is a weighted relative capital intensity difference (invariant to redefinitions of units) given by bzao,

a,,-%?

and where a,,(6) and ao1(6) are steady state per unit capital and labor inputs to the capital good and alo and am(S) are the steady state per unit capital and labor inputs to the consumption good. This result follows directly from the envelope theorem applied to cost minimization under a neoclassical technology and is also known as Shepard’s lemma. The reason b is not simply an input coefficient is that p is a relative price. To get b, note that from Shepard’s lemma (dp, 0) = (dw’, dw)A, where A is the input coefficient matrix for factor prices (w’, w). The O-element in (dp, 0) is due to the fact that the price of the consumption good is fixed at unity. Eliminating dw, we can solve for dp/dw = b. Note that b is fully determined by relative factor prices. At a steady state, however, we have from the nonsubstitution theorem that all relative prices as well as all the input coefficients aii are functions of the discount factor S. We can now illustrate diagrammatically how p responds to a change in k. Since T,, = (awl%) is negative the sign of sign of T,, = (ap/i?k)= -b(aw/%) depends on b.’ Since T, = -p we see from Fig. 1 that the change in p when the capital-labor ratio k changes and y remains fixed depends on how the production possibility surface shifts in response to an increase in k. If the consumption good is capital intensive, that is, if b is negative, the surface shifts outward favoring the production of c. This implies that at a given y, p gets steeper and (ap/ak) is negative. This is illustrated in Fig. 1, as the production point shifts from x0 to x,. Alternatively, if k declines at constant relative prices, the output of y increases. To maintain a fixed ~1,p must decline, implying that (apji3k) is negative. Note that a decline in k results in a higher y at constant prices because the production of c declines sufficiently and factors released from the production of c flow into the production of y. This is possible for a marginal change in k as long as c is produced by a positive amount, that is, as long as (k,, k,, ]) E interior 4. We now adapt the consumer’s optimization problem for our model to problem (1) of Section2. Define V(k,,k+,)=T(k,+I-(l-g)k,,k,), was defined in Section 2. To apply Theorems 1 and 2 we where VW,, k,,,) have to establish the sign of I’,* on the interior of 6. We have V,, = ’ Concavity implies T12 < 0. However, products imply Tz2 < 0. For a discussion Nishimura [6].

it can be shown that strictly diminishing marginal in the case of a multisector model see Benhabib and

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7

.

FIGURE 1

-( 1 - g)T,, + T,, = b(dw/ak)[ - 1 - (1 - g)b], provided y, k, T( y, k) > 0. Since Tz2= aw/ak < 0 by the concavity of T (see footnote 7 as well), we have V,, < 0 if h E (- (1 - g)- ‘, 0) and (k,, k,, ,) E interior D. The latter implies that JJ, k, T( y, k) > 0. As discussed above, under constant returns to scale in both industries input coefhcients and therefore b depends only on relative factor prices (w/w’). Furthermore, as shown above, T,( y,, k,) = ~1~ and T(y,,k,)+p(y,,k,)-w(~~,k,)k,=wy for (k,,k,+l)Einterior D where, for all t, y, = k,, , - (1 - g)k,. Then relative factor prices (w/w’) and therefore b depends on (k,, k, + , ). We can now state an assumption which restricts 6, or the differences between the capital labor ratios in the two industries: (Bl ) For the two-sector technology described above, for all (k, , k,) E interior B, b E (- (I - g) -- ‘, 0). Of course if for all factor-price ratios (ul/w,), cost minimizing factor intensities are such that b E (- (1 - g))‘, 0), (Bl) will be automatically satisfied. Note that for the special technology of this section, (Al) holds and (Bl) implies that (A2) and (A3) of Section 2 also hold. THEOREM

4.’

Let (Bl ) hold f or the two-sector technology described

’ For a generalization of this theorem to a stochastic economy with an endogenous labor supply see Benhabib and Nishimura [S-l.

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above. Then along an optimalpath (k,+,-k,+,)(k,+,-k,)
Theorem 3 follows directly from Theorem 2.

301

if k,#k,., Q.E.D.

As noted at the end of Section 4, a negative V,, implies that the marginal benefit of k,, 1 declines with k, and the optimal sequence of k’s can then be shown to be oscillatory. We can intepret this result in the context of our competitive equilibrium model. An increase in k, in our two-sector model results in two opposing forces. When the consumption good is capital intensive the trade-off in production becomes more favorable to consumption goods. For a given y,, this is reflected in the steeper price line at X, than at -‘iO in Fig. 1. This tends, like a substitution effect, to lower investment and the capital stock in the next period. On the other hand, the capital intensity differences affect not only the shift but the shape of the production possibility frontier. For a given k, + , , an increase in k, reduces the required gross investment !I, = k,, , - (1 - g)k,, and, like a wealth effect, makes it possible to produce more c. However, y cannot be transformed into c at a constant rate. When the consumption good is substantially more capital intensive than the investment good the slope of the production possibility is more sensitive to changes in the composition of output. A reduction in ))r changes the tradeoff in production in favor of the investment good. The combined change in p due to a change in k can be seen in Fig. 1 as a switch from -x0to .Y~. Thus V,, < 0 if the slope -p is steeper at x2 than at x0. This requires that b E (- (1 - g)- I, O), that is, the consumption good to be more capital intensive than the investment good but not too strongly so. However, with full depreciation g = 1 and oscillations occur if b < 0. In this case k, fully depreciates and y, = k, + , . This implies that the second effect described above, the movement from X, to .x~ along the production possibility frontier, is absent. So far the above discussion concerns the existence of oscillations but not that of persistent cycles. Persistent cycles require a further restriction of the capital-intensity differences between consumption and investment goods. For cycles to be sustained, the oscillations in relative prices must not present intertemporal arbitrage opportunities. Thus possible gains from postponing consumption from periods when the marginal rate of transformation between consumption and investment is high to periods when it is low must not be worth it. Whether this is the case or not depends on the discount rate as well as the slope of the production possibility frontier. Thus the existence of cycles depends not only on h but also on 6, the discount factor. As shown rigorously below, b must lie in an interval defined by the discount factor and the rate of depreciation. To establish cycles we have to investigate the conditions of Theorem 1, so that the set P- defined in Theorem 1 is not empty. That is, for some 6,

302

BENHABIB

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NISHIMURA

the condition VI2 + 6 I’,, > (1 + 6)V,, has to be satisfied. In terms of technology, this reduces to .4b2 + Hb + 6 < 0, where A = (1 + 6( 1 - g)2 + (1 + +a)( -g)), H=(l +6(3-2g)). This inequality is satisfied if beM= -6(1+6(1-g))-‘), which is empty if 6=1. At the (-(1+(1-g))-‘, same time, for 6 = 1, it is easily shown that V,, + V22- 2 Vi, = (1 - b(2 - g))2T22 < 0, which implies that Vi, + V,, < 2Vi2. Therefore, there exists some 6< 1 such that 6 E (6, I] c P+. Thus, P+ is not empty and there exists a 6 + as required in (A4). Then the following assumption covers (A4) of Theorem 1. (B2) There exists a 6* such that at the steady state value of b at 6*, 6(6*)~M, where M=(-(1+(1-g))-‘, -6*(1+6*(1-g))-‘)c (- 190). Note that 6 = 0 implies M= ( - (1 + (1 - g))) ‘, 0). (B2) assures that there exists a 6- as required by (A4). Note also that with g= 1 (full depreciation) we have M= (-1, -6*) and (B2) reduces to b(d*)E(-l, -6*). THEOREM 5. Let (Bl) and (B2) hold. The consumer’s intertemporal optimization problem with the two-sector technology T described above has optimal solution trajectories of the capital stock k,, outputs y,, c, and proces pr and w,, which are cycles of period 2 for 6 in some nonempty set [S -, 6 + 1.

Proof:

Theorem 4 follows directly from Theorem 1.

Q.E.D.

The results of Corollary 1 also apply to the two-sector technology of this section. Applying Theorem 3, we can rule out periodic cycles of higher order and more complicated dynamic behavior in our two-sector model if we impose conditions on the relative factor intensities. We have shown that V,, = b(dw/dk)( - 1 - (1 - g)b) and that b and (aw/ak) can be expressed in terms of (k,, k, + , ). The following assumption rules out Vi2 = 0, as required by Theorem 3. (B3) For all (k,, k,+,)Einterior B, b#O and b# -(l-g)-‘. Since the sign of b determines whether the consumption or the capital good is capital intensive, b # 0 implies that there are no factor intensity reversals for factor price ratios which correspond to (k,, k,, 1) ED. If the consumption good is capital intensive, that is, if b < 0, the requirement of (B3) that b # -( 1 - g) ~’ prevents the consumption good from becoming too capital-intensive relative to the capital good. Note that as g approaches 1 (full depreciation), this last requirement becomes less and less binding and disappears in the limit. THEOREM

6.

Let (B3) holdfor

the two-sector

technology described above.

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303

Then an interior optimal path converges either to a steady state or to a cycle of period two. Proof. Since (B3) assures that VI2 #O for (k,, k,+,)Einterior result follows immediately from Theorem 3.

D, the

Remark 1. The

above theorems can be extended to include an endogenous labor supply which depends on the wage rate. The technology function can be defined as T( y, k, 1), where 1 is the labor input. The model can then be closed by specifying an inverse labor supply function w,(l), since labor demand is given by aT( y, k, l)/aZ= wO(y, k, I), where w0 is the real wage in terms of the price of the consumption good. Solving for I as a function of ( y, k), we have T( y, k, I( y, k)).9 Then the above theorems can be directly applies to this function and, if the assumptions hold, employment will behave in a cyclical manner as well. Of course, there is no involuntary unemployment in such a model. However, if the equilibrium trajectory is cyclical, it may be plausible to introduce frictions preventing the continuous flow of factors between industries and obtain some unemployment of factors. Remark 2. The results of this section merely illustrate how the structure of technology can lead to efficient cycles. The capital-intensity relations will be more complicated in a multi-sector model. The initial accumulation of a particular type of machine can lead to its subsequent decumulation if it causes a shift in production which favors other capital good and thereby generates cycles. Furthermore, it can be seen from the examples of Benhabib and Nishimura [7] that generalized capital-intensity conditions play a role in producing cycles in their continuous time model as well. In a continuous time framework, however, at least two distinct capital goods are required for cycling. As one capital stock crosses its steady state level the other capital stock, also on its optimal path, will be away from its steady state value. Remark 3. It

is also possible to interpret the technology function (1 g) k,, k,) as the output of a firm subject to adjustment costs, T(k,+ 1 as in the standard adjustment-cost literature. The first argument of T( ., . ) can represent gross investment, which affects output directly. By Theorem 2, the optimal path would be oscillatory if - T,,( 1 - g) + T,, < 0. To show that periodic cycles exist we have to show that (A4) holds and the crucial condition is the existence of some 6* such that ( V,, + 6* VI 1)/VI, < 1 + 6* at a stationary point. This will hold for some 6* 9 Equivalently, we could specify a utility v(/,) is the disutility of work. The first ik( [,)/al, for all t.

function V(k,, k,, , , I,) = T( y,, k,, I,) - u(l,), where order conditions would imply dT( y,, k,, L,)/dl,=

304

BENHABIB

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if at the stationary point corresponding to 6*, AT,, - HT,, + 6*T,, > 0, where A=(1+6*(1-g)‘+(1+6*)(1-g)) and H=(1+6*(3-2g)). Since T,, , T,, < 0, the inequality is likely to hold for small 6 provided T12 is sufficiently negative. A negative value of T,, implies an adjustment cost function for which investment has a negative effect on the marginal productivity of the existing stock. Thus, a firm with a large enough discount rate and negative T1? is likely to have an alternating level of capital stock, with gross investment falling below and overshooting” depreciation in alternating periods. If a secondary market for capital goods exists we could also observe the firm buying and selling capital equipment in alternating periods, even though it incurs positive adjustment costs whenever it is buying or selling the equipment.

6.

CONCLUSION

The basic reason why the oscillatory behavior of economic variables can persist in the stationary environment of our model is because the assumulation of capital assets changes the trade-offs in the production of different goods. If the process of accumulation of a certain capital good changes the slope of the production possibility frontier in favor of other goods or of consumption goods, that particular capital good may then be decumulated. Then as this capital good is decumulated the process will be reversed. As such a process results in oscillations of relative prices in a competitive equilibrium context, the possibility of intertemporal arbitrage may arise. Thus persistent cycles in prices require some degree of discounting of the future. These anticipated relative price changes will not necessarily be dampened or eliminated by arbitrage since when the future is discounted relative prices can change across periods without generating profitable trading or storing opportunities. As has been shown in Section 5, it is the relation between the capital intensity differences in the production of different goods (which determine the curvature of the production possibility frontier) and the rate of time preference (or discount rate) which determines whether or not the competitive solution will exhibit persistent and “robust” cycles in outputs, stocks, relative prices, and employment. A potentially interesting issue that we do not address here is the possibility of friction and resistance (possibly due to retraining costs or informational difficulties) in the movement of factors across industries in reponse to the cyclical changes in the equilibrium relative prices.” ” For some empirical evidence of such overshooting behavior see Nadiri and Rosen [28]. ” According to Von Hayek, cyclical changes in the relative prices of consumption and investment goods lead to business cycles by generating a tendency to the reallocation of fac-

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tors of production that cannot easily retrain or tion” [ 161 Von Hayek argues that a Wicksellian cyclical changes in relative prices. In “Profits. relative prices cycle because an expansion in goods at the expense of capital goods lowers production of consumer and investment goods expansion of consumption and the contraction

move across industries, In “Prices and Producelastic money supply may be responsible for Interest and Investment” [17] he argues that the demand for and production of consumer real wages and changes the tradeotT in the in favor of investment. This reverses the initial of investment.

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