Journal

of Economic

GLOBAL

Behavior

and Organization

8 (1987) 429452.

North-Holland

EQUILIBRIUM DYNAMICS WITH STATIONARY RECURSIVE PREFERENCES* J. BENHABIB New York Unicersity, New York, NY 10003, USA

M. MAJUMDAR Cornell University, Ithaca, NY 14053, USA

K. NISHIMURA Kyoto Received

University, Kyoto,

April 1986, final version

Japan

received

October

1986

We study the global dynamics of capital accumulation for a general two-sector model which is not necessarily convex and where preferences of an infinitely-lived agent are stationary but not additively separable. We obtain monotonicity and convergence results for capital under ‘normality’ assumptions on preferences and factor intensity assumptions on technology. We then derive results on oscillatory dynamics under alternative factor-intensity conditions or under the assumption of inferiority of ‘future utilities’. Finally, in an exchange model with two agents we show that utilities will be monotonic or oscillatory depending on the normality or inferiority of the preferences.

1. Introduction In this paper we will study the global dynamics of capital accumulation for a two-sector model of technology which is not necessarily convex and where the preferences of an infinitely-lived agent are stationary but not additively separable.’ In the last section of the paper, we will also study the global dynamics of efficient allocations over time in a pure exchange model of two agents that have additively non-separable preferences. One-sector optimal growth models with non-convex technologies have been studied by Majumdar and Mitra (1982, 1983) and Dechert and Nishimura (1983). Benhabib and Nishimura (1985) gave a characterization of the monotonic and oscillatory dynamics of a two-sector model of growth *Presented at the Workshop on the ‘Advances in the Analysis of Economic Dynamic Systems’, University of Venice, January 7-11, 1986. The generous support from the C.V. Starr Center for Applied Economics for this project is gratefully acknowledged. The project was also supported by N.S.F. Grant 86-055-03. ‘Multisector models with convex technology and additively separable preferences have been extensively studied. See McKenzie (1976), Brock and Scheinkman (1976) Cass and She11 (1976).

0167;2681/87/$3.50

@) 1987, Elsevier Science Publishers

B.V. (North-Holland)

430

J. Benhabib

et al., Global equilibrium dynamics

with a representative agent having additively separable preferences. The dynamics of optimal growth models with stationary but non-separable preferences have been studied by Beals and Koopmans (1969) and Iwai (1971), among others, for one-sector convex technologies. Recently, Lucas and Stokey (1984) and Epstein (1985) also studied the dynamics of capital accumulation in models of many agents that have non-separable utility functions. The paper is organized as follows. The general model is set up in section 2. In section 3 we obtain monotonicity and convergence results for the capital stock under ‘normality’ assumptions on preferences and factor intensity assumptions on technology. In section 4 we obtain results on oscillatory dynamics under alternative factor-intensity conditions or under the assumption of inferiority of ‘future utilities’. Section 5 considers a model where the stock of capital enters the utility function and gives conditions under which it converges to a non-zero steady-state, and also we consider a pure exchange model with two agents and show that utilities over time will be monotonic or oscillatory depending on the normality or inferiority properties of the agent’s preferences.

2. Technology

preferences

Let c= T(y, k) be a social production function where k is the amount of capital stock available, and y and c are the maximal amounts of the capital stock and the consumption good producible. c= T(y, k) may be understood to represent the joint production of outputs c and y from the input k or it may be viewed as an aggregated social production function in a two-sector economy, in which one industry produces a pure consumption good c and the other produces a pure capital good Y.~ In either case, all the variables are normalized by labor. We assume that (PJ) (P.2)

(P.3)

There exists E>O such that for O< k < k, T(y, for k > E, T(y, k) = 0 implies y < k. T(y, k) is continuous on [0, k] x [0, E] and of and Tl ~0, T, > 0, Tl 1
k) =0 implies

y> k, and

C2 class on (0, E) x (0, It) Tl = dT/dy, T2 = aT/dk, ak.

Let k=[O,QK”=K xK x . . . Given (P.2) we can obtain a solution of T(y, k)=O. Then the correspondence F:K+K”

a function y(k) as is defined by

‘If g is the rate of depreciation plus populations growth, we can define y,=k,+, -(l -g)k,. k,). For given k,,dc/dy represents the slope of the Then c=r(k,+, -(I-g)k,,k,)=T(k,+,, production surface. For further analysis, see Benhabib and Nishimura (1984).

J. Benhabib et al., Global equilibrium dynamics

431

which gives the set of feasible capital paths attainable from the initial stock k,. We shall write ,k=(ko, k,, . . .). Let RT=R+ x R, . . . and ,c=(c,,ci,...). Let G:K”+Ry be defined by

%k)=(W,,k,),

(2)

T(k,>k,),. . .),

where ,k E K”. Then the correspondence

of C: K-R:

defined

by (3)

gives the set of feasible consumption paths attainable from k,. We impose the product topology on Ry. It can be shown that C is upper semicontinuous in k,. As K is a compact subset of R,, C(K) = iJkoeK C(k,) is a compact set [Hildebrand and Kirman (1976, p, 192)]. We assume that choice among consumption paths in C(K) can be represented by a preference ordering which is complete, transitive and continuous in the product topology, or equivalently we assume that there exists a continuous utility function U:C(K)-+R such that Oc’>Oc if and only if U(,c’) > U(,c). Below we explicitly state assumptions on this intertemporal utility function. (U.Z) (U.2) (U..?)

(U.4)

U is continuous on C(K) with respect to the product topology. Sensitivity: There exists cO, CL and ic such that U(c& 1c) > U(c,, 1c). Limited non-complementarity: For all ,,c, &E C(K), U(c,, I~) 2 U(c& 1c) implies U(c,, 1c’) 2 U(cb, 1c’), U(c,, 1c) z(c,, 1c’) implies U(& 1c) 2 U(& 14. Stationarity:

For some c0 and all ic, U(c,, 1c) 2 U(c,, 1c’) if and only if

U(,c)ZU(,c’). Since C(K) is a compact,

there exists OF and ,c_ in C(K) such that

~=U(,,cj~U(~c)~U(~~=b

for all

,,cEC(K).

(4)

(1960, Let c=max{c, I(cO,ci,. . .) EC(K)}. Th en it follows from Koopmans pp. 287-295) that on’ C(K) there exists a function V:[O, Zl x [a, b]-+R+ such that U(,c) = V(c,, U( 1c))

for

ICE C(K).

This function V is called the aggregator. properties for this function.

(5) We further

assume

the following

J. Benhabib et al., Global equilibrium dynamics

432 (U.5)

V(c, U)

has positive continuous derivatives Vi and I’, on (0, C) x (a, b) V, = d V/de and V, = dV/du, and limCO,, Vl(c,, u) = 00, V2(c,, u) are bounded.

where liqo,,

Given (5), this assumption implies that U, =aU/&, on (0, F))” and U, = VI, U, = V, x VI. The relation

and

U,SC~CJ/&,

exist

U(,c) = VC,, VC,, . . ., VC,, U(,+ 14). . .) can then be used to prove that U,, 1=XJ(,c)/ac,

exists.

3. Capital accumulation Let

U(G(,k)).

W(ko)=maxokEF(kO)

pondence

is defined

Then

the

optimal

capital

path

corres-

by

@(k,) = {ok E F(k,) 1U(G(,W = Wk,)), and the optimal ‘W,)

consumption

path correspondence

(6) Y: K -+K”

is given by (7)

= G(W,)).

Let @,(k,) and Y,(k,),tzO, be the set of capital stocks in the tth period of optimal capital paths and optimal consumption paths arising from k,. The existence and the upper semi-continuity of optimal paths are easy consequences of the compactness of K” and the continuity of U. Lemma 1. (i) @(k,,)#@for any k,EK. (ii) Qi is upper semi-continuous

on K.

The value function

is defined by

W: K-r R,

(8)

Wk,) = U(@(k,)). It satisfies the following

function

equation:

W(k,) = max U(,c) CFE Wo)

(9)

= max V(c,, U(,c)) ,,cEG(ko) =

max c,=T(k,,ko)

V(c,,

max ,=G(k,)

U(,c))

J. Benhabib et al., Global equilibrium dynamics

=

max

c,,=T(k,,ko)

433

(10)

Uco, WW

Lemma 2. If &E @(I$) and ,E E ‘Y(k,) for I$ > 0, then & >O and ?, >O for all t 2 0, that is, optimal paths from &, > 0 are interior paths. Proof: (i) Suppose that k^,=O. Then given (P.3) ?, =k, =0 and I’(&,, I/(0, a)) is the maximum of V(c,, V(c,, U(,c))) on the set of feasible consumption paths where a= U(O,O,... ). Let g(k,)= V(T(k,,k,), V(T(O,k,),a)). Since k, >O, if k, maximizes g(k,), we must have

lim g’(k,) SO. k, -+O

However,

+

v2(co,U(C,,O,...)I/~(C~,~).T,(O,~,).

(11)

derivatives are evaluated at (I?~,,k,,O,. . .), (c,,,cl,O,O,. . .) and U(O,O,. . .), where co = T(k,, Lo) and cr = T(0, k,). Let k,-+O. Then

a=

lim g’(k,) = V,(e,, a). T,(O, co) + V,(e,, a) lim [Vr(c,, a). 7”(0, k,)].

(12)

The

c -0 k;AO

k,-0

Vl(eo,a), V,(&,a) and T,(O, k,) are bounded by (US) and (P.2). Also, Tz2 ~0 implies that for kl > F > 0, T2(0,kl-&)>T2(0,kl)>0.

Hence lim T,(O, k,) > 0. k,+O

(13)

Also (U.5) implies limCl,, Vl(c,,a)= 00 and the second term in the parenthesis goes to cc. Hence lim,,,g’(k,) > 0, and this is a contradiction. Therefore k. > 0 implies f, > 0. By induction I$~> 0 for all t 2 0. (ii) Suppose that i. =0 and E, > 0. Let h(k,) = V( T(k,, Lo), V( T&, k,), U( ,2))). Note that given ko, k, can be bounded above by the solution to 0 = T(k,, k,) (which is unique if it exists), since co = T(k,, k,) and co 2 0. Therefore, if 6, maximizes h(k,), limkl _f h’(k,) 2 0 should hold. However, h’(k,)=I/,(co,I/(c,,ti)).T,(k,,~o,)

J. Benhabib et al., Global equilibrium dynamics

434

+

1/2(c,,VC,, 2)). r/l(c,, 4. T2&,k,).

(14)

The derivatives are evaluated at (&,, k,, ff,, . . .), (c,,,cI, e2,&, . . .), and z?=u(~~) where co=T(k,,&,) and c,=T(&,k,). Let k,+t),. Then lim h’(k,) = k,

1 1

lim Vi(c,, I/(c,, 9)) Co-O

-+k,

c,4,

+

. Tl(L,,

lim Vz(co, V(c,, a)) Co-O C, 4,

LO)

. V,(Z,, 12).T2(&, k,).

(15)

Since fi, > k,, TI 1 ~0 implies

(16) T,, < 0 implies (17) As V,(&i;) is also bounded, (US) implies lim++, h’(k,)= -co. This is a contradiction. Hence if Lo >O and E, >O, then to > 0 must be true. Next suppose that k^,>O and 2, = E, =O. It is clear that there exist some t such that I?,> 0. Choose the smallest such t, say t’. Then &, _ i >O, I?,,_ i = 0 and E,.> 0. Otherwise, since c, > 0 is feasible for some t > 1 on account of k, > 0, any program with some c, > 0 would dominate a program with c, = 0 for all t. We can apply the same argument as above and reach the contradiction. Hence r?, > 0 must imply E,> 0 for all t 2 0. Q.E.D. Cbrollary

1.

Zf ,ff~ @(k,), o2 E Y(k,) fir k, > 0,

where U= u( ,2).

Proof: W’kl, lo),VW&,U a 1 is maximized at c1 and its first order Q.E.D. condition is satisfied with equality by Lemma 2. We will refer to (18) as the Euler equation. By the principle of optimality, if ok E @(k,), then ,k E @(kI) or *k E @(k,), t 11. We write ,,,(k)=(k, k,...) and call it a constant path. A steady state is defined to be k*E [O,E] such that ,,,(k*) E @(k*). Zero is called a trioial steady state. All the other steady states are called nontrivial steady states. Let K* be the set of steady states. Since it is easy to show that k* -CL, every non-trivial steady state k* satisfies

435

J. Benhabib et al., Global equilibrium dynamics

r/l(c*, v/(c*, U*))Tl(k*, k*) +

iqc*, qc*,u*).

I/l(c*, u*) * T,(k*, k*)

=o, (19)

or Vz(c*, u*) = - T,(k*, k*)/T2(k*, k*),

(194

where c* = T(k*, k*), u* = u(,,,(c*)). It should be noted that we so far did not assume even the quasi-concavity of the utility function. There may be many optimal paths from the single initial stock k, [see Majumdar and Mitra (1982)]. The following assumption assures the convergence of the optimal paths to the steady state:

(Ml

I/,(T(k,, k,), W(k,)).

T,(k,, k,) is strictly

increasing

in k,.

If I/ is twice differentiable and W is differentiable, this condition is easily interpreted. We have t3V,T,/~?k, = I/,,T,T,+ T2V12w)+ I/,T,,. From the first order conditions in the maximization problem given by eq. (10) we also have W’=-1/,T,/1/, so that av,T,/ak,=-T,T,(-I/,,+(I/,/I/,)V,,)+I/,T,,.3 The first term on the right hand side is positive if future utility V(,c)) is normal. This follows because Tl ~0, T2 >O and (- V,, +(V,/V,)V,,) is positive under the normality assumption. Tl 2 is positive in a two-sector model if the consumption good is labor intensive [for a proof, see Benhabib and Nishimura (1985)] and is always positive in a one-sector non-joint production model given by U(c) = U( T(k,, k,)) = U(f(k,) -(l -g)k, - k,), where g is the depreciation rate, f is the concave production function and U is a concave and increasing utility function. We now can give an informal diagrammatic exposition to show the role of assumption M. For expositional simplicity, we assume the concavity of the aggregator function V(cO, w(k,)) in c0 and k,. In fig. 1 we draw indifference curves showing the trade-off between current consumption c0 and future utility W(k,). These isoquants can then be drawn in the cO- k, space since w(k,) is concave and increasing in kl. Similarly, we can also draw the production possibility surface giving the tradeoff in c0 and k, (assuming full depreciation) for k, fixed. In fig. 1 assume that the optimal choice of k,, k^,> ko. This implies that the production possibility curve shifts out. In a one-sector model this shift is a parallel one since c0 + k, =f(k,). Note that whether & is greater than or smaller than ff, depends on whether the current consumption is a ‘luxury good’; & depicts such a situation. On the other hand, & is possible if ‘future consumption’ is not inferior and & is greater than k,, implying a monotonic trajectory. If V(cO, W(k,))=C(c,))+ -‘Strictly speaking, eq. (10) hold only along an optimal path and substituting w’ into the expression for a( V, 7’,)/&, is illegitimate. However, continuity considerations assure that the arguments that follow from this substitution will hold locally, which is all that is required to establish the global monotonicity or oscillation properties of the optimal path. A rigorous proof of the above arguments is given following Theorem 5 in section 4.

436

J. Benhabib et al., Global equilibrium dynamics

Fig. 1

/W(k,), where /I is the discount factor, we have an additivity separable form in q, and k,, which implies that the trajectory is monotonic because both q, and k, will be ‘normal goods’. Even in the additively separable case, however, if we have a two-sector model where the consumption good is capital intensive, oscillatory behavior is possible if T,, is negative. This is shown in fig. 2. Therefore, whether VI ’ T2 is monotonic in k, or not depends on whether or not future consumption is a normal good or not, as well as on the factor intensities of the two-sector model. Note that our assumption on the concavity of the aggregator function V was for the convenience of diagrammatic exposition only. We will not use it for the proofs in this section. We shall prove several lemmas before we formally prove convergence of the optimal trajectory. Lemma 3.

If k* E K*, the @(k*) is a singleton.

Proof. If ,,k* =O, it is clear that (C0,,(O)} = @(k*). Let 0~ k*
431

J. Benhabib et al., Global equilibrium dynamics

k

i+ko

6

Fig. 2

k,=k”,

Oit
k,=kj-,,

tzT+l,

is optimal, since k* is also optimal. Consider then the triplet (kT_ 1, k,, k,, I) E (k*, k*, k;), giving rise to utility V( T(k*, k*), V( T(k;, k*), 12))= V( T(k*, k*), u*) for ii = W(k;). The last equality follows since (k*, k*, k*) is also optimal. Thus we obtain U* = V( T(k;, k*), i;). Using (18) yields Vl(c*,u*).

T,(k*, k*)+

V2(c*,u*).

Vl(T(k;, k*), W(k;)).

T,(k;, k*)=O. (20)

However, (M) implies that (20) has a unique solution k; = k* because it assumes that VI T2 is strictly increasing in k’,. Hence @(k*) consists of a Q.E.D. unique path con(k*). Lemma 4.

If k, = k, for ,k E @(k,), then k, E K* and ,k =,,,(k,,).

438

J. Benhabib et al., Global equilibrium dynamics

Proof If k, = k, and ,kE @(k,), then ,ke @(k,) by the principle of optimality. It follows by induction, using the uniqueness implied by (M) as in Q.E.D. Lemma (3) that cO,(kO)E @(k,), and k, is a steady state. Corollary 2. steady state. Lemma 5.

If kT=kT+l,

for

,kE @(k,),

then k,= k*, t 2 7; where

k*

is a

Let k& k, E K. Given ,k’ E @(kb) and ,k E @(k,),

(i) kO>k, implies k;>=k,, (ii) k, > k, implies k, >=k,. (<)

(5)

ProoJ (i) Suppose that k; < k, holds. Then T(k,, k,)lO and Ti ~0 that T(k;, k,)>O. Hence k; is attainable from k,,. The optimality implies

WhM>

implies of ,k

Wk,))2 VT(k;,k,), Wk;)).

(21)

Also, T(k,,k,)zO, k>,kb>k,zO and T,>O implies attainable from kb. The optimality of ,k’ implies

T(k,,kb)>O.

Hence

k, is

(22) Adding

(21) and (22), we get

VT@,, ko),Wk,)) + VT(k;, kb)),Wk;)) 2 W-(k;,M,

WW+

VT(k,,kbh Wk,)).

That is,

or ~IV,(T(k;,a),W(k;))T,(k;,r)-V,(T(k,,r),W(k,))T,(k,,n)ldrr>n. 0 (23) Note that both k; and k, hypothesis k;
are

attainable

from

UE [k,, kO]. (M) and

the

439

J. Benhabib et al., Global equilibrium dynamics

This together with kb>k, makes the value of the integral in (23) negative. This is a contradiction. Hence k’, zk, must be the case. (ii) If k,
and

ko>,,,kI,

then

k,~cs,k,+I,

Proof Suppose that k,>k,. Then k, 2 k, by Lemma 5. If k, =k,, k, E K* by Corollary 2. If k, > k,, then k, 2 k,. It is clear that induction to the theorem. Q.E.D.

then leads

Theorem 2. Assume that (P.Z)4P.3), (U.&o.5) and (M). For any k,EK optimal path monotonically converges to a steady state.

an

Proof As {k,} is a monotone bounded sequence, it has a limit. Say k”= k,. Consider a sequence of optimal paths. That is, ,,k’, oki, ok2,. . . lim,,, where kg=k,,ki=k,,..., kb = k,, . . . . This sequence converges to ,,,(k) in the product topology. ,,,(@ is an optimal path by the upper semi-continuity of @. Hence I?E K*. Q.E.D. If we assume that K* contains a finite number of elements only), we can write K* = {kz, . . , k,*}, 0 = k8 < kT < .
(steady

states

Theorem 3. If K* has a finite number of elements and k, E Ki, ,k E @(k,), then {k,} converges to k? or ki*, 1. Proof By Theorem 2 the optimal path converges to a steady state monotonically. Let E=lim k,. By Lemma 5, ki*, 1 > k, implies ki*, 12 k,. If this arguk,*,,=k,, the k”=ki*,,. If kF+1> ki, then ki*,, 2 k,. By continuing ment, k*,+ 1 2 k,, t 2 0. Similarly, k, 2 k:, t 2 0, must hold. Since k”~ K*, k”must be either k* or ki*, 1. Q.E.D. Suppose that one optimal path from k, E K converges to a high steady state ky> k, while some other optimal path from k, converges to a lower steady state k:< k,. Then k, is called a critical level of capital stock. We can have the following generalization of Dechert and Nishimura (1983). Corollary 3. Let k, E (k:, kF+I) be a critical level of the capital stock. Then all strictly increasing optimal paths converge to ki*,,, while all of strictly decreasing optimal paths converge to k:. Moreover the critical level in (kf, kT+1) is unique (tf it exists).

440

J. Benhabib

et al., Global equilibrium dynamics

Proof. Let ,k~@(k,) be a strictly increasing path. Since k,
m

I/,(T(k,,

is that we can easily apply the property of optimal paths under

k,), W(k,)) . T2(kl, k,) is strictly

decreasing

in k,.

Again we do not need to use the differentiability of the value function W(k). We do not repeat the proofs of the results under this alternative condition.

J. Benhabib et al., Global equilibrium dynamics 441

0

k*

(i)

(ii) Fig. 3. Bifurcation.

442

J. Benhabib et al., Global equilibrium dynamics

Y

ST 4

s \-

4’ *--------------------

-

\

-

-----\u

-------------_

.------------

1

*m-----m

\\

-

-

\ \

443

J. Benhabib et al., Global equilibrium dynamics

Rather, we discuss next chapter. 4. Normality

the oscillatory

case under

the stronger

condition

in the

assures convergence and inferiority produces oscillations

Though we so far did not assume the concavity of the utility function or the production function (except for the diagrammatic exposition in the previous section), we proved the monotone convergence of the optimal paths. Below we obtain slightly stronger results in terms of strict monotonicity and asymptotic convergence under concavity assumptions. In addition, when the converse of the assumption (M) holds, we show that the optimal trajectory is oscillatory, as suggested in the diagrammatic exposition in section 3. Below we assume the following: (P.4) (U.6) (U.7)

T(y, k) is concave in y and k. U is strictly quasi-concave on C(K). V(c, u) has continuous second derivatives V, 1, VIZ, and V,, on (0,5) x (a, b), where V, 1= d2 V/~L~, VI2 = d2 V/C%C?C and V,, = a2 V/8u2.

It is easy to show that the set of feasible capital paths F(k,) and the set of feasible consumption paths become convex sets. It should be noted that (U.6) does not imply the quasi-concavity of the aggregator V(c,u) with respect to (c, u). Lemma 6.

An optimal path from k, E K is unique.

Proof: Let ok’, ,kF @(k,) and let &, 0c E Y/(k,) be the corresponding consumption paths. Suppose that ,k’#,k. Then we assume k’, #k, loss of generality. But this with T1
c; = T(k;,

k,) # T(k,, k,) =I+.

optimal without

(24)

But for Oc’# Oc, (U.6) implies that a feasible consumption path dOc’+ (1 - 0),c from k, yields a higher utility. This is a contradiction. Hence ok’= ,k. Q.E.D. Now the optimal path @(k,) becomes be shown that W(k,) is strictly increasing W(k,)

is differentiable

at k, E(O, It].

The condition

(M) assumed

in the previous

(U.8)

of k,. It can

a continuous function in k, We assume that

section

is replaced

by

444

J. Benhabib et al., Global equilibrium dynamics

Vi iv, - V, Vi, 50 implies that for the aggregator function V(c, u) u is a non-inferior good. Note that T,, 20 is trivially satistied if T(k,+ i, k,) = U(f(k,)-k,, i), where U is the one-period utility function. On the other hand, if T(y,k) is a social production function in a two-sector economy, T1 is always non-positive and T12 2 0 as long as the pure consumption good sector is not capital intensive [see Benhabib and Nishimura (1985), section 5, and also the diagrammatic exposition following assumption (M) in the previous section]. Theorem 4. Under the assumptions (P.&o.I), optimal paths from k, E K are strictly monotonic state.

(U.I)+U.8) and converge

and (AZ), the to some steady

Proof Let W’=dW(k,)/~k,. Using (U.8) and the fact that if ,kE@(ko) k,E(O, k), V(T(x, k), W(x)) is maximized at k,, we obtain

v,(T(k,>kohWk,))T,(k,,h)+ V,(T(k,,k,), Wk,NW’(k,)=O. Then Let i3H/ak, =(VIITI H(k,,k,)=I/,(T(k,,k,), Wk,))T,(k,,M. V,,W’)T, + VI Tzl. By substituting for W’(k) from (23, (M’) implies

for

(25) +

Assume that Ok’E @(kb), ,k E @(k,) and suppose kb > k,. Let k; = @,(kb). First consider the case where k; #k,. By the uniqueness of optimal paths we have

l CI/,(Wi,4,

Wk;))T,(k;,4-

1/,(T(k,,4,

Wk,))T,(k,>4ld~>0. (27)

The strict inequality comes from the uniqueness of the optimal hypothesis k; #k,. Using double integral, we can rewrite (27).

path and the

kb k’,

where I’= V( T(P,or), W(p)), T = T(fi, a), If kb is sufficiently close to k,, the integrand of (28) has the same sign as (26) for any (8, E) contained in [k,, kO] x [k,, k;]. This is a result of (25) and the continuity of k, =@,(k,). Hence for k, sufkiently close to kb and Ke> k, (28) implies k; >k,. This implies that k, =@,(k,) is increasing in k, for k, l(0,k). The rest of the argument is identical to that of (ii) in Lemma 5. To complete the proof we have to rule out the case where &,> k, implies

445

J. Benhabib et al., Global equilibrium dynamics

k; = k,.

First note that if kb > k, =O, then and ,k satisfy the Euler equation (18). Assuming that k; = k, in (1 S), we have

k; >O= k,.

If kb > k, >O, then

I/,(W,>x)> Yk,)) T,(k,,x)+1/,(T(k,,k,),W(k,)).T,(k,,k,)=O. v,(W,>x)>W(h))

,k’

(29)

(29) is satisfied by x=k& k,. However, (M’) implies that the first term on the left hand side of (29) is increasing in x. Therefore, if KO> k,, we cannot have k, = k’, without violating (18). This rules out the case where k, = k; and Q.E.D. completes the proof. To prove our results for the oscillation case, we maintain all the assumptions except that we reverse the assumption (M’). (M’) below requires either the consumption good to be capital intensive (Ti2 0). If both of these conditions hold, whether M’ is satisfied or not depends on the weighted sum of the two effects.

However, it may not be reasonable to assume we restrict ourselves to the rectangular region O
5.

Let

(P.l)-(P.4),

(U.Z)
and

that (M’) holds globally. So x [c,, d,] where

H = Cc,, d,]

(M’)

hold.

Let

,kc

@(k,),

,k’ E @(kb). (i) (ii)

For (k,, k,) For (k,,k,)

and (kb, k;) E H, if k, > kb, then k, < k; . and (k,,k,)EH, fk,>k,, then k,
Proof (i) Since either (VI 1V, - VI VI,) or - T’i is positive in condition (M’), the same proof as in Theorem 4 is applied to show that k,> kb implies k; # k, for ,kE @(k,), ,,k’E @kb). Choose k, and kb such that ((k& k,),(k,, k,))EH and kb is sufficiently close to k,. We suppose that k, >Ko and k, > k; holds. Then ((I&, k;), (k,, k’J)E H follows from the fact that ((k,, k,),(kb, k;)) is in H. Also T(k,, k,) 20 and r, >O implies that T(k;, kb) > 0. Hence, k; is attainable from k,. Also, since an optimal path from k,>O is an interior path, c0 = T(k,, k,) >O. If kb is sufficiently close to k,, we have T(k,, kb) >0 as well. This is due to the continuity of @(k,). Then the principle of optimality implies

f’(Vk,>ko), Wk,))> W(k;,k,),

Wk;)),

VT(k;, KA Wk;)) > ~(W,, kb),Wk,)).

(30)

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From (30) we can derive the same double integral as given by (28) above. But for kb sufliciently close to k, the sign of the term under the double integral is negative under (M’), as in proof of Theorem 4. Hence we have the contradiction and k; > k, must hold. This shows that @,(k,)= k, is strictly decreasing in k, as long as (k,, @,(k,))cH. Hence for any (k,,@,(k,)), (Fo,@,(kb))~H, k,>kb implies @,(k,)=k, <@,(Ko)=k;. (ii) This follows immediately from part (i) if we let kb = k, and k; = k,. Remark 5. When (a’) holds over the appropriate domain, Q,(k) will be monotonic. Then it is easily shown that both odd and even iterates of k, will be monotonic. This implies that both odd and even iterates converge. If the limits of odd and even iterates are the same we get convergence to a steady state; otherwise we get convergence to a two-period cycle. The proof in the case of additively separable utility also applies to the non-separable case. [See Benhabib and Nishimura (1985), Theorem 3.1 Of course, if (M’) does not hold and Q,(k) is not monotonic, unstable and chaotic behavior can occur [see Boldrin and Montrucchio (1986)]. Remark 6. Under the assumption (M’) over the appropriate region birfurcations of periodic optimal paths are possible. This is so even when production and utility functions share concavity properties provided 6 is small enough. Fig. 5 shows that as 6 decreases the steady state becomes unstable and a periodic path bifurcates. This is the case whose existence is shown in Benhabib and Nishimura (1985).

5. Generalizations

(1) We have so far discussed the Koopmans type intertemporal utility function so that it can be transformed into an aggregator function. However, it is clear from proofs given above that we can deal with any intertemporal utility into the aggregator function !JI = %(k,, k,, . . .) which may be transformed form

Wko,kl, WW).

(31)

This is more general than a Koopmans consider the following example. ‘X = -f cYU(c,_

1,

k,),

0<6<1

type utility

subject

t=1

f(k-k 1+1=c,zo,

~20,

k,

given.

function.

to

For example,

(32)

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I

k k,

k” (i)

k k:,

k,k* (ii)

Fig. 5. Bifurcation

due to decrease

in 6.

Let W(k,) the maximized value of utility. Then it satisfies (33) This differs from the ‘Koopmans’ type intertemporal utility function, since the current utility does not only depend on current consumption, f(k,)k,, but also depend on k,. Rather than discussing the problem (32) in the general context, we shall

J. Benhabib

448

et al., Global equilibrium dynamics

briefly sketch how problem (32) is handled by a method above. We assume that The production function f(k) differentiable at k > 0.

(PI)

f(O)=O,

(P.4 (a.4 (a.4

f’(k)>O,

at kz0

and continuously

f(@=E>O.

The utility function is continuous at c, k 2 0. U,(c, k) >O exists and is continuous at c >O, k>O, U,(c, k) > 0 exists and is continuous at c >=0, k > 0. for k>O, lim,,,U,(c,k)=cc for c>O. lim,,, U,(c,k)=co

(u.3)

The monotonicity (m)

is continuous

similar to that

U,(f(k,)-k,,

of optimal path is implied by the following assumption. k,) is strictly increasing in k,.

Under the assumptions above we can show that the optimal paths are interior and converge to steady states. Again we do not require the concavity of the utility function or the production function. Lemma 7. Let ,$ and ,C be optimal capital and consumption problem (32). If& > 0, I$ > 0 and c, > 0 for t 2 0. Proof:

paths from f, in

(i) Suppose that ff, = 0. Then E, = ff, = 0. Consider (34)

g(k,,~,)=U(f(~~)-k,,k,)+GU(f(k,)-~,,I;,). g(kl,&)

is maximized at k, =El =O. However, lim ag(kl,k^,)ldk,=-U,(f(~,)-k,,k,)+U,(f(l;,)-k,,k,)

k,-0

+dfz

U,U(k,)-x,xl.f’(U.

Since 0 5 lim,,, U,(f(~o)-x,~)O by assumption (m), and since the other terms are positive, ~im+.,oU~(f(ko)k,, k,)= co implies that the limkl +. lim,z,o i3g(k,, k,)/ak, is positive. This is contradiction. Hence ff, > 0. It is obvious that & > 0 for t 2 o must follow. (ii) Let E, = 0, C1> 0. Consider h(k,)=U(S(co)-k,,k,)-Wf(k,)-&&).

If E, =O, (35) is maximized at kl = fi, = f(co).

(35)

Consider

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449

+su,(f(k,)-k;,&,)f’(k,), lim h’(k,)= *I-k,

- lim U,(z,,fC,)+

(36)

U,(O,~,)+6U,(~,,I;,)f’(~,)=

-GC.

Zo*O

(37) This is a contradiction. Hence f, < f(co) if E0 > 0. Next let 2, = P, = 0. Then it is clear from assumptions (u.2) and (u.3) that there must exist some t such that q ~0 and C,>O. We apply the same arguments given above and reach a contradiction. Hence E> 0 must imply E, > 0 for all t 10. Q.E.D. Theorem 6. Let (pl), (p.2), (u.&o.3) and (m) hold. Then any optimal from k. E [0, E] in problem (32) converges monotonically to a steady state.

path

This theorem may be proved exactly in the same way as Theorem 2. We omit the proof. A stronger convergence result may be proved by assuming the twice differentiability of utility function and replacing assumption (nz) as follows: (m’)

U,,--U,,>O

on (O,L)x(O,k).

We do not, however, assume concavity. The stronger convergence result may be proved under much weaker assumptions than those corresponding to Theorem 4. Theorem 7. Let (p.l), (p.2), (u.&o.3) and (m’) hold. Then any optimal path ,k from k, E [0, I?] in problem (34) converges to a steady state. Moreover, {k,} is a strictly monotone sequence for k, $ K*. Proof: implies

Since (m’) implies (m), Theorem 6 holds. Then we show that ko#ko k; #k, as was done in the proof of Theorem 4. Q.E.D.

So far we did not prove whether the origin is a locally stable steady state or not. To guarantee that it is a locally unstable steady state, we assume the following additional properties. (p.3) (u.4)

lim,,, f’(k)>X’, U,,(c,k)
for sufficiently

small c,k>O.

Theorem 8. Let (p.+o.3), (u.+o.4) and (m’) hold. Then any optimal path ok from k, ~(0, k) converges to a non-trivial steady state. Furthermore, (k,} is strictly monotone if k, is not a steady state.

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Proof. We first prove that a non-trivial steady state is bounded away from 0. Consider the Euler equations satisfied at a steady state. (dj”(k*)-

l)U,(c*, k*)+ Uz(c*, k*)=O.

(38)

Since lim,,, iif’ > 1, (38) cannot be satisfied for sufficiently small k > 0. Suppose that an optimal path ,k from k, >O converges to 0 so that {k,) is monotonically decreasing to 0. So k, > k, > k, >O. Consider the Euler equation

-U,(c,,k,)+U,(c,,k,)+Gf’(k,)U,(c,,k,)=O.

(39)

By rewriting eq. (39), we obtain (6f’(k,)-l)U,(c,,k,)+CU,(c,,k,)-U,(c,,k,)l+Uz(co,k,)=O.

(40)

The first and third terms are positive for k, sufficiently close to 0. Hence

U,(c,,k,)-U,(c,,k,)
(41)

Since iJi,, and U,, are non-positive by assumption (u.4), it easily follows from (41) that kl > k, implies cO k, > k, > . . ., co 0, ,k converges to a steady state k* > 0.

In the second part of this section we consider a two-agent, pure exchange economy where the preferences of each agent are subject to assumptions in (u.l)O for all t. Since V’s are monotonic in their arguments, we can invert 1/2 to obtain

c:=c(u,“,u:+l) with c1 = &(u:, u,2+r)/au2 = l/V:, and c2 = &(I$, uf+ r)/&:+ I = - V’.‘/Vf. Substituting into the first agent’s utility, we obtain uA= V’(s-c(u&uf), u:). We would like to maximize the intertemporal utility of one agent subject to a given utility of the other and give a global characterization of the time profile of utilities and consumptions for each agent. This problem has also been studied by Lucas and

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451

Stokey (1984) and we seek to generalize their results. Our problem can be set in a dynamic programming framework as follows:

u:=M(ui)

= max V’(s--c(u& u:), M(ut)),4

where 0 5 c(u& u:) 5 s. We say that u:+ 1 is interior if c(uf, uF+r) ~(0, s). Theorem 9. (Monotonicity) If u;+~ and u:+~ are interior and H(u&u:) = - V,l(s-c(u&u:), M(u~)).c,(u&u:) is strictly increasing in u:, then along an optimal path u: #u:+ 1 implies (u:+ t -u:)(uf+, -uF+ I)>O. Proof:

Noting that (- Vfc,) increasing is the analogue of assumption M for Q.E.D. Lemma 5, the proof is identical to that of Lemma and Theorem 1.

In Theorem 9 we do not need the twice differentiability of the utility possibility frontier M. In order to interpret the condition that H is strictly increasing, we further assume that I/‘, V2 satisfy (U.6), (U.7) and that M(ut) is differentiable. Theorem 10. Along an interior optimal path, the utilities of agents are monotonic if 1 is a ‘normal good’ in I/‘(& uf+ 1) for i= 1,2.

U i+

Proof:

We have to show that the ‘normal good’hypothesis for future utility in the utility functions follows from (- Vic,) increasing in u:. We have

First we evaluate a2ci/aui auf using &$/auf = - V.‘/V,” and &$/au; = l/V:. A simple calculation yields a2cg/&tG au: = l/( Vf)” [ V%Vz - V: VZJ. To calculate dM/du:, maximize V’(s-c(u& u:), M(uf)) with respect to u: and solve the first order conditions to obtain dM/du: = - V~V,‘/V~V,‘. Substituting these terms into aH/au: we obtain

The bracketed expressions in the numerator are positive if future utility is ‘normal’ for V’ and V*. This completes the proof, Q.E.D. Remark.

Note also that if future utility is sufficiently inferior for at least one of the agents, aH /au: may be negative and this will imply oscillations in utilities. 4Note that u; = M(ui) is a utility possibility

frontier.

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References Beals, R. and T.C. Koopmans, 1969, Maximizing stationary utility in a constant technology, SIAM Journal of Applied Mathematics, 17. Benhabib, J. and K. Nishimura, 1985, Competitive equilibrium cycles, Journal of Economic Theory 35, 284306. Boldrin, M. and L. Montrucchio, 1986, The emergence of dynamic complexities in models of optimal growth: The role of impatience, Journal of Economic Theory, forthcoming. Brock, W.A. and J. Scheinkman, 1976, Global asymptotic stability of optimal growth with applications to the theory of economic growth, Journal of Economic Theory 12, 164-190. Cass, D. and K. Shell, 1976, The structure and statility of competitive dynamical systems, Journal of Economic Theory 12, 3 l-70. Dechert, D.W. and K. Nishimura, 1981, A complete characterization of optimal growth paths in an aggregated model with a non-concave production function, Journal of Economic Theory 27. Epstein, L., 1985, A simple dynamic general equilibrium model, working paper no. 8501 (University of Toronto, Toronto). Hildebrand, W. and A.P. Kirman, 1976, Introduction to equilibrium analysis (North-Holland, Amsterdam). Iwai, K., 1972, Optimal economic growth and stationary ordinal utility: A Fisherian approach, Journal of Economic Theory 5. Koopmans, T., 1960, Stationary ordinal utility and impatience, Econometrica 28, 287-309. Lucas, R. and N. Stokey, 1984, Optimal growth with many agents, Journal of Economic Theory 32, 1399171. McKenzie, L.W., 1976, Turnpike theory, Econometrica 44, 841-865. Majumdar, M. and T. Mitra, 1982, Intertemporal allocation with a non-convex technology: The aggregate framework, Journal of Economic Theory, 27. Majumdar, M. and T. Mitra, 1983, Dynamic optimization with a non-convex technology: The case of a linear objective function, Review of Economic Studies 50.