Recursive utility and optimal growth under uncertainty

ELSEVIER Journal of Mathematical Economics 24 (1995) 601-617

Recursive utility and optimal growth under uncertainty

Sumit Joshi Department of Economics, The George Washington Unioersity, 2201 G Street NW,

Washington DC 20052, USA

Received January 1993; accepted September 1994

Abstract

This paper extends the theory of optimal growth under uncertainty to a general class of recursive utility preferences. With this general description of preferences, and in a framework which incorporates time-varying productive technologies and non-stationarities in the evolution of the stochastic environment, the monotonicity properties of optimal programs with respect to the various parameters of the model is demonstrated and then utilized to develop the turnpike theorems.

Keywora!s: Recursive utility; Uncertainty; Monotonicity; Turnpike theorems

JEL classification: D90

1. Introduction

The two main objectives of this paper are to integrate the theory of optimal growth under uncertainty with a broad class of recursive utility preferences and to develop within this general framework the turnpike theorems corresponding to productive technologies (those which do not admit a maximum sustainable capital stock) and non-stationarities in the evolution of the stochastic environment. As regards the first objective, stochastic growth models have traditionally been cast in the framework of time additively separable (TAS) preferences (an exception being Epstein (1983) where a specific non-additive functional form was the object of study). The axiomatic structure of TAS preferences, however, is based on the

0304-4068/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0304-406&X(94)00703-9

602 S. Joshi / Journal of Mathematical Economics 24 (1995) 601-617

stringent requirement of complete independence in intertemporal consumption. Further, the rate of time preference under TAS preferences is an arbitrarily specified constant. Recursive utility functions were first introduced by Koopmans (1960) to rigorously formulate Irving Fisher’s notion of impatience. Their first desirable characteristic was that they accommodated partial complementarity in intertemporal consumption in a manner which retained the applicability of dynamic programming techniques. Therefore, analytical ease was not compromised in moving to the more general recursive specification. Further, recursive utility permitted a flexible rate of time preference determined endogenously by the underlying consumption stream. This made it possible to capture Fisher’s conjecture that an increase in consumption in the initial periods or an overall increase in consumption levels would decrease impatience.

If a stochastic environment is included into the analysis, the primitive structure of recursive utility has two additional dimensions of flexibility which do not obtain in the TAS case. First, recursive utility allows non-indifference to the temporal resolution of uncertainty. An elegant characterization of the form of non-indifference is available in Chew and Epstein (1990). In contrast, TAS preferences under expected utility implies indifference to the temporal resolution of uncertainty. Kreps and Porteus (1978) have argued that in intertemporal analysis, the von Neumann-Morgenstern axiom regarding reduction of compound lotteries may not be valid and individuals may have definite attitudes towards the temporal resolution of uncertainty. These attitudes towards the timing of uncertainty can be incorporated in the general recursive case but not in the TAS case.

Second, recursive utility achieves a separation between risk aversion (behaviour towards risk) and the degree of intertemporal substitution (ranking of deterministic consumption programs) if there is non-indifference to the temporal resolution of uncertainty (Chew and Epstein, 1990). Heuristically, risk aversion can be in- creased without changing the ranking of deterministic consumption programs by keeping the same recursive utility function (or aggregator) and changing the certainty equivalent function required to evaluate future random utility. This is not possible in the TAS case and hence two distinct aspects of preferences remain intertwined within the TAS framework. The implications of this separation for intertemporal consumption behaviour is analysed in Epstein and Zin (1989) (hereafter EZ).

Therefore, recursive utility permits treating dynamic issues with much greater generality than the TAS case. Since the neo-classical optimal growth model provides the basic framework for applied dynamic analysis in a wide range of areas such as macroeconomics, international trade and public finance, and since recursive utility can have important implications for standard results in these areas (as Epstein and Hynes (1983) demonstrate with the particular case of the Uzawa functional) it is important to develop a comprehensive theory of optimal growth in a stochastic neo-classical model with recursive utility. This is the primary aim of the paper.

S. Joshi / Journal of Mathematical Economics 24 (I 995) 601-617 603

In addition to the TAS specification, stochastic growth theory up to this point has confined itself exclusively to expected utility theory. In contrast, the analysis here encompasses a broad range of theories of choice under uncertainty by not restricting the certainty equivalent functional. Similar to EZ, the model can include as special cases the expected utility framework, the dynamic choice theory of Kreps and Porteus (1978) and the Chew-Dekel non-expected utility framework. Further, similar to Majumdar and Zilcha (1987) and Mitra and Nyarko (1991) the model explicitly permits non-stationarities in the law governing the evolution of the stochastic environment. This generalizes models where the environment is represented by a sequence of independent and identically distributed random variables (Brock and Mirman, 1972) or a stationary stochastic process (Radner, 1973; Dana, 1974). Moreover, given the flexible time preference, the analysis here is an extension to the stochastic framework of the deterministic analysis in Mitra (1979). While the discount factor varied arbitrarily in Mitra, its dynamics in the recursive framework are dictated by the underlying consumption stream.

The second objective of this paper is to provide a general approach to turnpike theorems in the case of technologies for which no maximum sustainable capital stock exists. Turnpike theorems compare the dynamic properties of optimal programs with different values of the underlying parameters. The definitive analysis of McKenzie (1976) identified three main kinds of theorems. The early turnpike theorem states that a finite horizon optimal program for a sufficiently long horizon lies close in the initial periods to the infinite horizon optimal program from the same initial stock and deviates only in the later periods to meet the terminal stock condition. The middle turnpike theorem claims that a finite horizon optimal program stays close to an infinite horizon optimal program from a different initial stock for all but finitely many time periods independent of the length of the horizon. The late turnpike theorem compares two infinite horizon optimal programs from different initial stocks and states that they will lie close together for all but finitely many time periods.

The turnpike theorems are developed here by exploiting the monotonicity properties of optimal programs in aggregate models. These monotonicity properties describe how an optimal program responds to a change in the parameters of the model. A subsidiary contribution of this paper is to develop these monotonicity properties for general stochastic recursive utility. Monotonicity results in the stochastic case were first obtained by Brock and Mirman (1972) in the TAS case by manipulating the Euler equations. However, the Euler equation approach would require the certainty equivalent functionals used to evaluate stochastic future utility to be sufficiently smooth. This would restrict the class of certainty equivalent functionals being studied. More fundamentally, the Euler equations in the recursive case are not as manipulable as in the TAS case in order to show monotonicity.

An alternative approach to monotonicity for recursive utility was developed by Beals and Koopmans (1969) and Magi11 and Nishimura (1984) by putting restric-


tions on the flexible rate of time preference. These techniques were developed for a deterministic environment and were dependent on the convexity of the technology. The method followed here combines the optimality equation approach of Dechert and Nishimura (1983) with lattice programming techniques based on the work of Topkis (1978). Similar techniques have been used most recently in Hopenhayn and Prescott (1992). An advantage of this method is that it does not rely on the convexity of the technology and is therefore useful for future extensions of this model to non-classical technologies.

Monotonicity results are also of independent interest in dynamic models for they yield important comparative dynamic properties of optimal programs. For instance, the analysis of dynamic incidence of a capital income tax in the TAS case in Becker (1985) was based on monotonicity with respect to the discount factor. The examination of alternative definitions of increase in impatience in Dutta (1987) in the TAS case was based on monotonicity with respect to the discount factor and the length of the planning horizon. The ergodic theorem in Brock and Mirman (1972) and Majumdar, Mitra and Nyarko (1989) for the TAS case was similarly based on monotonicity with respect to the initial stock. Monotonicity results presented here can be used to study similar issues for recursive utility. Moreover, these monotonicity results would generalize those models which restrict attention to steady state comparisons (for instance, the Danthine and Donaldson (1985) analysis on the incidence of a capital income tax on the steady state distribution of capital or the Epstein and Hynes (1983) analysis on the non-invariance of steady state capital to money supply) by allowing a consideration of the entire dynamic path.

It may be noted that monotonicity as the basis for the middle and late turnpike theorems was also used in Majumdar and Zilcha (1987), but their manipulation with the stochastic Euler equations is hard to adapt to recursive utility. Further, their analysis required a uniform lower bound on the degree of concavity of the production function. This ruled out linear production functions and, more gener- ally, endogenous growth models. The analysis here does not require such a bound on the technology and easily accommodates linear technologies and endogenous growth by employing an alternative uniformity condition. Since the method also does not depend upon the convexity of the technology, it opens the possibility of extending the turnpike theorems to non-classical environments.

The paper is organized as follows. The model is described in Section 2. Monotonicity properties of optimal programs are developed in Section 3 and used in Section 4 to prove the turnpike theorems. Section 5 contains the conclusion.

2. The model

This paper employs the standard neo-classical growth model where a central planner maximizes intertemporal utility subject to the constraints imposed by the

S. Joshi / Journal of Mathematical Economics 24 (1995) 601-617 605

stochastic environment and the technological possibilities. These elements of the growth model are outlined next.

The possible states wI of the environment at date t 2 1 is given by a compact metric space R, endowed with the Bore1 a-algebra 8’(. The stochastic environment is given by the measure space (0, 9, & where 0 = lJ;fi, is the space of all sequences w = (w,) such that o, E n,Vt 2 1, 9 = Sy& is the a-algebra on 0 generated by the measurable cylindrical sets and p is the measure on a. Ft denotes the o-algebra induced by partial history up to time f.

Technology is represented by a sequence of possibly time-varying production functions f,:W + X a,+ 1 -+ R + which satisfy for each t 2 0:

CT.11 f, is continuous on I%+X a,,,. Further, for each c++ 1 E LZ,, 1:

0.2) f&O, q+ 1> = 0 given ol+ 1

0.3) f,(k, q+ 1) is strictly increasing in k E R, given wt+ 1

CT.41 f,(k, ol+r) is concave in k E R, given w,+ r (T.5) f,< k, w,+ I) is differentiable in k E R, given mt+ r The commodity space consists of capital input processes k = (k,) and consump

tion processes c = (c,) which satisfy certain feasibility restrictions. These conditions are now defined. For some N < CQ, let (0, N] be the set of initial stocks. If s E (0, N], the real-valued (Y&adapted capital input process k = (k,) and consumption process c = (c,) are called feasible programs if

k, + c,, I s (1)

k,,, +c~+~ ~y,=f,(k,,q+~) p-as., t20 (2)

k,, c, 2 0 CL-a.s., t r 0 (3)

The infinite horizon model is parametrized by S. Given s E (0, N], the set F(s) X C(S) contains all feasible capital input and consumption programs. The finite horizon model has the initial stock S, the terminal stock K and the planning horizon T as parameters. The finite horizon programs are feasible if they satisfy (11, (21, (3) and the terminal stock constraint k, 2 K p-a.s.

From (T.l) and the compactness of L?,, t 2 1, there exists a maximum pure accumulation process from s, {ks], which is independent of w and defined as

k;=s, k;+I =f,(k;), t>O

where the function f,: R, + R, is defined for each t 2 0 as

f,(k) = sup f,(k,o,+,) s.t- ~r+l E Q+,

This process provides a coordinate wise upper bound for all feasible programs in P(S) x C(s):

0 I k,, c, I kf p-a.s., t 2 0 (4)

As a consequence of (4), the commodity space can be assumed to be a subset of


l-w,<a q, CL) x I-p&4 Ft, p ) where L,(0, FO, CL) = Iw,. Recursive utility functions in Boyd (1990), EZ and Streufert (1990) have been constructed on a bounded domain. These bounds are obtained by restricting the rate of growth of feasible programs. Therefore:

Assumption 1. There exists p 2 1 s.t. the process (ki) from any s E (0, N] belongs to the principal ideal I( p) generated by ( /3 ‘>, t r 0, where Z( p> is defined as the set of adapted processes (x3 satisfying

Z( p) = {x = ( xt) E ll3; : 3 E R, s. t. x, I hfi’ j.ka.s. Vt 2 0)

Assumption 1 implies in particular that for any s E (0, N], F(s) x C(s) GZ( p) x I( PI.

The domain for recursive utility functions consisting of temporal consumption lotteries has been constructed in EZ. The construction is complex for two reasons. First, consumption lotteries have to be indexed by the time of resolution of the uncertainty. Second, the recursivity property of recursive utility functions requires that the domain too have a recursive (or stationary) structure. The technical details of the construction are long and have been excellently presented in EZ and Chew and Epstein (1990). To avoid repetition, and for reasons of space, this paper provides only a heuristic description of the construction, referring the reader to the original papers for the details.

The first step is the construction of the sequence {D,) where D, consists of consumption lotteries in which all uncertainty resolves at or before time t. In particular, the set D, consists of elements of the form d, = (c,, m,) where ca is the initial non-random consumption and m, is the atemporal measure on future consumption. Similarly, d, = (co, m,) ED, where m, is a measure on future consumption which resolves by period t. In this manner, the timing of uncertainty is incorporated into the lottery. The domain for recursive utility is the subset D( p) of ll;D, consisting of temporal lotteries in which all uncertainty resolves at infinity and in which the atemporal measure m, on future consumption has compact support. The recursivity of the domain follows from the result of EZ (theorem 2.2) that D( p ) is homeomorphic to Iw + X M( D( p )> where M( D( /3 1) is the set of probability measures on D( p> in which the atemporal measure has compact support. In words, an element of D( /3) is an infinite horizon probability tree emanating from the node c0 in period 0. There is also an infinite horizon probability tree emanating from each node in period 1 but, since the uncertainty in period 1 is yet unresolved, there is a measure m (which may depend on c,,) on the set of t = 1 nodes. This m can therefore be identified with an element of M(D( /3)) and each element of D( p) can be identified with an element (ca, m). Once period 1 uncertainty represented by m is realized, the infinite horizon probability tree from (the now non-random) node c1 has the same structure as that from the node c,,. In EZ (appendix 4) it is then shown that any infinite horizon feasible program (k, c) from s can be associated with a temporal lottery in D( /3 1.


Preferences are given by a measurable utility function U: D( /3) -+ Iw +. Let

S+ denote the Bore1 u-algebra on Iw +. For d = cc,, m) E D( /3 1, ~[ml is a measure on future utility defined by

U[m](B) =rn({dED( P):U(d) EB}) VBES+ (5)

Let M(X) denote the set of probability measures on the space X. Random future

utility is measured using certainty equivalent functions 7: M(IW+) -+ Iw, satisfying:

(1) First degree stochastic dominance: Vm’,m* E M(R+ ), Vx E [w,, if

m’([O, xl> I m*([O, xl) then $m’) 2 v(m*>. (2) ~(f!&,)=x forall nE[W+, 8, being the measure degenerate at x.

It is assumed in addition that the certainty equivalent functions are continuous:

Assumption 2. If /gdU[ m”] -+ jgdU[ m ] f or all g E C([w + ), the space of bounded

continuous functions on Iw +, then v(U[ m”] + v(U[ m].

The utility function U is recursive if there exists an aggregator function

w:Iw+x [w++ IW+s.t.

u(c,, m) = W(c,, q(U[m])) Vd= (co, m) ED( P) (6)

Recursive utility permits time consistent planning in the sense of Johnsen and

Donaldson (1985). With certainty, Koopmans (1960) case follows:

(7) U(c,, Cl, c*,...) =w(c,, U(c,, c,,...))

The aggregator W and the utility function U are assumed to satisfy:

(W.1) W(0, 0) = 0 (W.2) W is continuous on [w, X IF!+ and U is continuous on D( p> (W.3) W(c, x) is strictly increasing in both c E Iw, and x E (w, (W.4) W(c, x) is strictly concave in c E [w, for each x E [w,

(W.5) W is supermodular (W.6) W(c, x) is differentiable in c E Iw, and x E [w,. Further, aW/ac -+ CC

as c-0.

It is implicitly assumed that the aggregator is bounded below. Hence (W.l) is a

useful normalization. Aggregators unbounded below such as WCC, xl = log c +

6x, 6 E (0, 1) is a discount factor, can also be accommodated by suitably modifying the methods of this paper. From Boyd (1990), these aggregators satisfy:

W is increasing in c and x, U.S.C. on Iw, X (w,, continuous for c > 0 and x > - 03

and WCC, - 03) = W(0, y) = - ~0 Vc, x E R,. The use of dynamic programming techniques for existence and monotonicity follows from Schal (1975).

(W.2) and (W.3) can be discussed in the context of the existence of a recursive specification given by (6). One approach due to Koopmans (1960), Chew and Epstein (1990) and Ozaki (1991) takes preferences on D( p) as given and then shows that if preferences are continuous, risk separable and stationary, then an

608 S. Joshi/ Journal of Mathematical Economics 24 (1995) 601-617

aggregator W and utility function U satisfying (6) and assumptions (W.2) and (W.3) exist. The alternative approach of Boyd (1990) takes W as primitive and shows that (W.l)-(W.3) and a Lipschitz condition with respect to the second argument of W are sufficient for the existence of a continuous utility function U satisfying (6).

For the TAS aggregator W(c, x) = u(c) + Sx, (W.4) (i.e. the strict concavity of U) is sufficient for monotonicity results (Dechert and Nishimura, 1980). With recursive utility, both (W.4) and (W.5) are required. Note that W is supermodular if W(CI, X1) - W(C1, X2) 2 W(Cz, X,) - W(Cz, X2) V(Ci, Xi> E Iw+x Iw+, i = 1,2 where (cl, x1) 2 (Q, x,). If W is twice differentiable, it is equivalent to WI, 2 0. The importance of this condition for monotonicity stems from the work of Topkis (1978).

The TAS aggregutor W(c, x) = u(c) + 6x satisfies (W-l)-(W.6) under suitable assumptions on u. The Koopmans-Diamond-Williamson aggregator, under a monotonic transformation given in Streufert, 1990, p, 89), is given by WCC, x) = ce + 6 log(x + 1) where e is an elasticity parameter and satisfies (W.l)-(W.6) if 0 < e < 1. The exponential aggregator of Streufert (1990) is given by WCC, x)

ce +(x + l)‘- 1 and satisfies (W.l)-(W.6) if 0 < e < 1. The Koopmans iadratic aggregator W(c, x) = x + (c - xXa - PC + yx) also satisfies (W.l)- (W.6) under suitable restrictions.

Given the initial stock s and A” E I%+ (from Assumption 11, the planner is restricted to

D( j3;A”) = (d E D( p) : d, = ( c,,,mI) s.t. c0 E [O,s] and

ml E mrCt( 4))

The planner’s maximization problem is given by

sup ~(%rlmd)) s.t. (c,,,m) E D( /3;A”)

If the supremum W * is finite, the temporal lottery d * = cc,‘, m’ ) in D( @;A”) attaining W * is the optimal temporal lottery and the programs (k * , c * ) E F(s) X C(s) associated with d * are the optimal programs. The existence of optimal lotteries and corresponding programs follows standard arguments and is therefore not addressed here. Note that the optimal program will be interior as a consequence of assumption (W.6).

3. Monotonicity properties of optimal programs

This section develops the monotonicity properties of finite horizon optimal programs with respect to the initial stock s, the terminal stock K and the time horizon T. The domain of recursive utility now becomes

4,d PiA”) = dED( P;A”): d, = (c,,m,) s.t. c,, E [0, s],

k, 2 K p-a.s., IIrc,(s)Xfila T+l 11


i.e. temporal lotteries generated by the sequence (c,, . . . , cT, 0,. . . ) s.t. the terminal stock constraint is satisfied. Define the time-varying value functions (which may depend on K and T) as

J,( 2) = sup W(% 77vJbl)) (8) where the supremum is over the set of temporal lotteries (c,, m) such that 0 I c, I z and m is generated by a finite horizon process satisfying (l),(2),(3) and the terminal stock constraint. Note that J,(z) = sup W(c,,O) s.t. 0 I cr I z - K.

From the maximum theorem, J, is continuous. From the recursivity of the domain, the measure m in the temporal lottery

Cc,, m> is on the space of temporal lotteries d = (c,+~, m). There is a temporal lottery from each node in period t + 1 and, since the uncertainty in period t + 1 is unresolved, m captures this uncertainty. Once w,+ 1 is realized, m is degenerate at some Cc,+ 1, ml. Letting yt =fJz - c,, qfl , ) define the measure J,+I(y,>lm] on R, as

The validity of dynamic programming with recursive utility has been shown by Becker and Boyd (1990) and Streufert (1990) in the deterministic framework and Ozaki (1991) in the stochastic framework. The optimal@ equation can be written as

JJz) =SUP W(c,, 77(Jt+I(y,)[ml)) s.t. Olc,lz (9)

If Wt+1 is known then J,+ i( y,)[ m] concentrates all mass at a single point iu 88,. If the uncertainty in the first t periods is realized and the optimal consumption levels are (c,, . . . , c,), then

J,(s) = w(c,,w(c,,... I++ 77(4+dy,Nml))-.)) (10)

Proposition 1. The value functions are increasing.

Proof. Given non-random X, z where n I t, let ((ki, ci)} be optimal for (x, K, T)

starting from period t. Define a new consumption program as _c, = z - k,, _ci = ci for i = t + 1,. . . , T. Since _c, 2 c, =x - k,, it follows from (W.3) that J,(z) 2 Wb,, VU,+ ,(y,XmlN 2 Wet, q(J,+ ,( y,hlN = J,(x). 0

Theorem 1. Let (k”, cx) and (k’, c’) be the optimal programs for (x, K, T) and (z, K, T) respectively where x I z. Then k: < k: p-a.s. for t = 0, 1, . . . , T.

Proof. The proof is by contradiction. Therefore, assume (without loss of generality) to the contrary that k,” > k,‘. Since production functions are strictly increasing in k VW E 0 (Assumption T.3), it follows that y$ = f,(k,“, ol) > fO(kl, wl) = y&

610 S. Joshi / Journal of Mathematical Economics 24 (I 995) 601-617

It is then claimed that kc > kf on a set A,, p(AI) > 0. ’ If not, then kf s kf

p-a.s. First let k; < kf for some w and define

c;=y,“-k;, cf=yo’-k;, _c;= y;-k; and _cf=yi-k;

Since yi > yi and k; < kf, it is true that

c; > _c; > c;, c: > _c; > c; and cr + c; = _c; + _c;

Therefore there exists 0 < T < 1 such that

_c;=rc;+(l-r)c; and _c;=(l-~)c;i+rc;

From the optimality principle:

JI( Y,x) = w(c;, 77(J2( Y3)) 2 W(_cL r)(J2( Y3)) (11)

J,(Y,“) = w(c;, s(Jz(Y;))) 2 w(_c,I, 77(4(Y,x))) (12)

From the strict concavity of W in its first argUUXUt:

q_c;, 77 (J2( y;))) > +;, 77(-q Y:))) + Cl- T)W(G dJdYf)))

(13)

w(_c;, ?@*( y;))) > (I- 7)W(c;, rl(J*(Y;))) + TW(G 7)(JdY3))

(14)

Adding the above two inequalities and collecting separately terms with coefficient

r yields

w(_c;7 rl (A( Y;))) + W(G $A( Y3))

> w(c;, @z(Y;))) + w(c,‘, Q(JdY3))

+ 7[w(c;, 77 (J*( Yf))) + W(cL rl(JdY,x)))

-w(c;7 77(4(y;))) - W(G rl(W:)))l (15)

Now recall from the supermodularity condition (AssUmptioU w.5) that WPI, 41)

+ w(p,, q2) - W(pl, q2) - W(p,, ql) 2 0 if (13, 41) 2 by da Since the value function is increasing (from Proposition 11, J,(Y;) 2 J~(Y$ for a.e. 0 and

hence J,(y;x[O, q)) SJ,(Y;X[O, 4)). F rom first degree stochastic dominance, &,( Y;)) 2 &T~( Y;)). Further, by construction, c; > cf. Now letting

pr = c;, p2 = cf, 41 = 77(J2( Y:))p q2 = 77(J2( Y3)

1 The proof of this claim follows Dechert and Nishimura (1983, theorem 1) adapted suitably to

recursive utility. The supermodularity condition of Topkis (1978) plays a crucial role in extending the

proof from the time additively separable case to the general recursive case.


it follows that the term within parentheses on the RHS in (15) is non-negative from supermodularity. Therefore, it follows from (15) that

IQ;, 77(J2( Yl”))) + w(_c;, 77(&( Yl”)))

> I+:, 77 (J,( Y;))) + W(cf, +,(Y;))) (16)

But (16) contradicts the sum of (11) and (12). Now suppose that k; = k; for some w. Note that in this case, c; > c; since

y,” > y,“. Note that a given w = co,>, i.e. a given sequence of realizations of the stochastic environment, generates a path through the probability tree consisting of non-random nodes at each t. Hence, the optimality equation for a given w = (tit) becomes .Tr(y$ = W<cf, J,(yf)), i = x, z. Since the optimal programs are interior, letting Wl(c, x) = aW/Ck and Wz(c, x) = aW/ax, it follows that

w*(c;J,(Y;)) = K(cX(Y,“))J;t Y;)fI(k;J%) (17)

wr(cX(Y;)) = w,(cf,52(Y;))J;(Yf)fl(k;,0*) (18)

where the differentiability of J, in convex environments follows standard arguments as in Majumdar, Mitra and Nyarko (1989, theorem 16). Note that y; = y; since k; = k,‘. Then, from supermodularity, W&c;, J&y;)) 2 W2(c;, Jz(y$) and the RHS of (17) is greater than the RHS of (18). But, from strict concavity, wrcc;, J*(y;)) < I+;, J,( yf)) and the LHS of (17) is strictly less than the LHS of (18). Hence, there is a contradiction.

Hence the claim that k; > k; on a set A, of strictly positive measure holds. It is now claimed that k; > k; on a set A, GA,, p(AJ > 0. Assume to the contrary that kc I k; for a.e. o EAT. Define CT, ci as before and note that an optimal program is also optimal for all realizations in the set A,. Letting x(A,) denote the indicator function of A,:

JdY3 = W(& ‘I(X(AdJdY,X))) 1 W(_cL 17(X(A*MY,‘)))

Jd Y;) = W(c& 77( X(AIM Y;)>) 2 W(_cL 77( X(AlM Yz*)))

Using the same arguments as before again shows a contradiction. Continuing inductively, kg > k; on a set A,, p(AT) > 0. However, optimality

dictates that ki = k$ = K p-a.s. since WCC, x) is strictly increasing in c (Assump- tion W.3). An optimal program cannot have a terminal stock which is strictly greater than K because reducing the capital stock in period T to K and corre- spondingly increasing consumption would increase utility. Therefore, the assumption to the contrary that k,” > k,’ sets into motion an inductive process given by the above claims which concludes in a contradiction. This proves the theorem. 0

The proof is easily modified for stochastic initial stocks. Note that the Uzuwu- Epstein-Hynes aggregator WCC, x) = ( - 1 + y)exp{ -u(c)) does not satisfy (W.4) or (W.5) and hence the above proof does not apply. However, sensitivity in initial

612 S. Joshi / Journal of Mathematical Economics 24 (I 995) 601-617

stocks for this case has been proved independently in Epstein (1983). The next theorem gives monotonicity in terminal stocks.

Theorem 2. Let (kc, cl> and (k5, co, 5s 5 p-a.s., be optimal for (s, 5, T) and (s, 5, T) respectively. Then, k,! I kf p-a.s., t = O,l, . . . , T.

Proof Assume without loss of generality that k,$ > k,$. Following arguments as in Theorem 1 yields k$ > k{ on some set of strictly positive measure contradicting kj= 5s e= k$ p-a.s. •I

Theorem 3. Let (kT, $1 and (kT+l, cTfl) be optimal for (s, 0, T) and (s, 0, T + 1) respectively. Then, k: 5 krf ’ CL-a.s., t = O,l, . . . , T.

Proof. Note that (kT+l, cTfl) is optimal for (s, kF+l, T). Since the optimal program is non-negative, kF+’ 2 ki = 0 p-a.s. and the result follows from Theo- rem 2. 0

From now on, an optimal program for (s, K, T) is denoted by (k”*“TT, cS,K,T) and the infinite horizon optimal program from s by (k”, c’>. Using (T.11, (4) and Theorem 3, define the process

k’*” = Lim I ks,O,r

T+m f , cf,” = Lim, -rm [f,_~(k~~iT,O~)-kk:,o~T] Vt>O

Since F(s) x C(s) is closed in the topology of almost sure convergence (from the continuity of f,), (k’Fs, cl,‘) is a feasible program and is referred to as the limit program from s.

Theorem 4. The limit program from s is the infinite horizon optimal program from s.

Proof. Let dT = (I$‘*~, mT> be optimal in D,,,( P;h”) and generated by

(k ,c G’,r s,“,T). Let _d = (c _o, _m) E D( p; A”) be any temporal lottery generated by (k, _c). Let _dT = (co, _mT) E Do T( p; A”) be generated by (_kT, of (_k, _c) to the first (T + 1) periods. Note from optimality that

_c~>, the restriction

w( c;*OJ, q(U[mTl)) 2 W(_co7~(u[-mT])) VT20

From standard arguments, D( /3; AS) is compact. Hence, 3(T(n)) s.t. (iTo”, _c~(“)) + (k, _c) and _dTCn) -P _d From (W.2) and Assumption 2:

w( c;s, q( U[ ml)) = Lim,W( c~‘“*T(n), q(U[ mT(“)]))

2 Lim,W(_c,, 77(U[mT(“)]))

= W(_co, ~(mE1))

S. Joshi/Journal of Mathematical Economics 24 (1995) 601-617 613

Using standard arguments it can be shown that d = (c$‘, m) corresponds to ( k’yS, c’,~). Since (_c,,, _m> and (_k, _c) are arbitrary, the conclusion follows. 0

4. Turnpike theory

A critical element in turnpike theorems is the reachability of a terminal stock from a program or from an initial stock. McKenzie’s (1976) multisector analysis used value loss methods which depended critically on the boundedness of capital values. These bounds required strong notions of reachability. In contrast, turnpike theory in the aggregate model exploits the monotonicity of the optimal programs and hence the reachability condition can be kept very general.

As in Majumdar and Zilcha (1987), a terminal stock K 2 0 is reachable from an initial stock s (given horizon T) if the process {k,) defined as _k, = s and

_1+1 =.f& Wt+1 9 k ) t 2 0, satisfies _kr 2 K p-a.s. In similar spirit, a terminal stock K 2 0 is reachable from a program (k”, c’) if for each t 2 0 there exists y < CC

and a program &,, k,, 1,. . . , _k,+ ,,} such that _k, = kf and _k,+ y = K EL-a.s. From this definition, if3hepath from t 2 0 reaches K in y periods, then it also reaches K in y + i periods, i 2 0. Further, if K is reachable from (k”, c’), and the program (k”, cx) is such that k: 2 ki p-a.s. for all t 2 0, then K is also reachable from (k”, c”).

Theorem 5. (The Early Turnpike). Let K 2 0 be a terminal stock reachable from the optimal program (k”, c’), s E (0, N]. Then for all E > 0 and y < UJ there exists T(E) < CO such that for all T 2 T(E):

1 kf - kS*K*TI < E p-a.s. t = 0, 1,. . . , y

Proof. Fix E > 0. It is first claimed that Vy < 4T’ 2 y s.t. VT 2 T’, k; 2 k;‘rT CL-a.s. If not, then 3 t 2 0 and a sequence T(l), T(2), . . . of time horizons, T(i) 2 t, s.t. ks < kS,K,T(i), i = 1,2,. . . , on a set A of positive measure. Select T(i) large enough so that from reachability there exists at least one program starting from ys = f,( ks, w,+~) and reaching K and let (kyvKYTCi), cYsK,T(i)) be optimal for (y:, K, T(i)). It is asserted that 38 GA, p(B) > 0 s.t. k/;KiTCi) < kS,K,T(i) on B. If not then k,,, 1+ 1 yvK9T(i) 2 kS;KiTCi) for a.e. w EA. Note from (T.3) that y; < y;K,r(i) on A. Mimic the proof of Theorem 1 to show a contradiction. Continuing inductively in this way gives K = kiiTjTCi) < k:cjTCi) = K on a set of positive measure, a contradiction.

Since K 2 0, from Theorem 2, ks,OvT I kS,K.T I ks CL-a.s. for 0 I t I y and T 2 T’. From Theorem 4, ks,‘yT T k; p-a.s. as T + m, 0 I t I T - 1. Hence, for each 0 I t I y 3T(t) < 00 s.t. I ki - ks,OFTI < E p-a.s. VT 2 T(t). Letting T# = max o ~ rlvT(t) and T(E) = max(T’, T#), 1 kf - ks,‘TTI < E CL-a.s. for 0 I t I y and T 2 T(E). Thus, I k; - kS,K3T I I I ks - ks,‘sT I < E p-a.s. for 0 I t I y and T> T(e).0


For any E > 0, the middle turnpike theorem characterizes the time spent by a finite horizon optimal program from a given initial stock within the e-interval of an infinite horizon optimal program from a different initial stock. Therefore, it has to be first ensured that for any E > 0, the optimal finite horizon program (k X,K,T ) CX,K,r), x E (0, N], will almost surely be in the e-interval of the optimal infinite horizon program (k”, c’), s E (0, N], for some t 2 0 provided K is reachable from (k”, ~‘1. The next result shows that this is true provided the horizon is sufficiently long. A long horizon is needed in order to satisfy reachability.

Proposition 2. (The Visit Lemma). Let K 2 0 be a terminal stock reachable from the optimal program (k”, ~‘1, s E (0, ~1, and from the initial stock x E (O,N]. Thenforall l >Othereexists [<wandT(E)<~suchthat Ik;-k;+,TI<~ for all T 2 T(E).

Proof. First let x 2 s and note that kfyopT 2 ks,OrT p-a.s. Vt, T. Letting T + m, it follows from Theorem 4 that k: 2 kfVt 2 0. It is now claimed that 3T’ and 6 < T’, s.t. k; f k;*“,T p-a.s. VT 2 T’. If not, then there exists a sequence {T(i)} s t k” < kxgos “‘Vt < T(i) on a set of positive measure. Proceeding as in Theorem *. t 1 yields d I kjci, < kg;qiT(” = 0 o n a set of positive measure, a contradiction. Hence, k; 2 k; 2 k;,“,T p-a.s. VT 2 T’. Since K is reachable from (k”, c’), it is also reachable from (k”, c’). From Theorem 5, given 5, 3T( 5) s.t. kf 2 kf+,T p-a.s. for 0 I t I 5VT 2 T( 5). Hence kc”TT I kisKyT I k; and I ki - k;,K,TI I 2 1 k; - k;,“,TI CL-a.s. VT 2 max{T( t), T’). Since k;,OsT t k;, the result follows from Theorem 4 by choosing T large enough. If x I s, note again that IT’ and 5 < T’ s.t. k;,‘sT 2 ks,‘rT CL-a.s.VT 2 T’. From Theorem 5, kl 2 k$K,T CL-a.s. for large T. Hence kiyoaT I k;*KsT I kS,K*T 5 ki p-a.s. and I k[ - k;sKvT 1 5 1 k; - k;posTI EL-a.s. for large T. The result follows by choosing T large enough. 0

Note that the time of visit 5 when kX9K,T enters the e-interval of k” is independent of the length of the horizon, i.e. the same 5 suffices for all T 2 T(E). The middle turnpike theorem shows that the number of time periods the program k X,KyT spends outside the c-interval of k” is bounded from above independently of T. In the turnpike literature, some kind of a uniformity condition is critical to the proof of the middle (and late) turnpike theorem. It arises in the process of constructing a function which is sensitive to a divergence between kX+sT and k”. In the aggregate growth literature, the turnpike issue has been addressed in Majumdar and Zilcha (1987). The uniformity condition there took the form of a strictly positive lower bound on the degree of concavity of the production function thereby ruling out linear technologies (and endogenous growth). The uniformity condition employed here does not place any such restrictions on the production function.

S. Joshi/Journal of Mathematical Economics 24 (1995’) 601-617 615

For any T 2 1, letting y = l/p, define, for 0 I t s T

~#$=max,~~~,y’k~, ~:,K,T=max,.i.Ty’k:,K,T

From Assumption 1, 3 < ~0 such that c$:, c$:,~,~ s A independent of T. It is also true that (y’/h)I kf - k:,r*T I E [0, 11. Now, for 0 5 t s T - 1, let

qr = min{(-y’/h)l k: - k:,“~Tl,(y’/A)e}, r0 = 1, rrl+l = i’Ihqi

It then follows that

and similarly for c#I:‘“‘~. Note that 4: > 4,“t1qt and +:“‘T > +:;yTql if ( ks - k;>&T I > E. Multiplying both sides by II-,, #rr, > c$,+ r’rr,+ 1 and 4:‘“‘T?r, L

X.KJ 4 rTTt+1. r+1 Once again, the inequality is strict if I ks - kji,K,TI > E. Therefore, letting V, = @rI + 4, “,“sTrr,, 0 I t I T - 1, it follows that V, - V,, r 2 0 and for any E > 0 there exists SzT > 0 (which may depend on w and E) such that 1 k; - k:,K,TJ > E for some w implies V, - V,, , L 8: for the given w. The uniformity assumption, as is standard in the literature, now requires that 8,’ be independent of the tuple (t, T, w), i.e. 8,‘~ 6 > 0.

Uniformity Assumption. For any E > 0, there exists a 6 > 0 such that

1 k;(w) - k:,K,T( w)I > E * y( w) - V,+,( w) 2 6

Theorem 6. (The Middle Turnpike). Let K 2 0 be a terminal stock reachable from the optimal program (k”, I?), s E (0, N], and from the initial stock x E (0, NJ. Then, under the uniformity assumption, for all E > 0 there exists T(E) < CQ and R(E) < m s.t. I k; - k:*KST I I 6 CL-a.s. for all T 2 T(e) and for all except (at most) R(E) values of t where R(E) is independent of T.

Proof. For any E > 0, select T(E) from Proposition 2. For any T L T(E), define the random variables K,, t 2 0, as follows

K, = 1 if ( kf - k:3KSTJ > E

K, = 0 if ) kf - k:,K*TJ I E

From Proposition 2, there will be some t s.t. K, = 1 if T 2 T(E). Let V(E) denote the total number of time periods the program kx,K.T spends outside the e-interval of k”. Then V(E) = C,TK,. It now follows from the properties of {V,) and the uniformity condition that

V,+6KT>V,-VT+SKT

=CoT-l(~-~+l)+SKT

=c,T-‘K,(V,-V,+,)+SKT+C,T-l(l-KJ(~-~+l)

2 COT- 1 K,(V,-V,+1)+6KT2SCoTK,=SV(E)

616 S. Joshi/ Journal of Mathematical Economics 24 (1995) 601-617

Note that q0 I max{e, 1) and recall #, dQK’r I htlt. Therefore, V(E) I [max{ l , 1) + A + 61/S = R(E) which is independent of T. 0

Theorem 7 (The Late Turnpike). Consider the optimal programs (k”, c’) and (k”, c’), s,x E (0, N]. Then, under the uniformity assumption, for all E > 0, I ks - kfJ I E p-a.s. for all but finitely many t.

Proof. Note k” is the almost sure limit of the programs kx,OyT and K = 0 is trivially reachable from (k”, ~3). From Theorem 6, for all E > 0 and sufficiently long horizons, 1 kf - k:*‘pT I< E CL-a.s. for all but finitely many t and independently of T. The result now follows by letting T --) ~0. 0

5. Conclusion

The generality provided by recursive utility has not been fully exploited. It seems a likely conjecture that the flexible rate of time preference embodied in the recursive specification should influence the rate of convergence of optimal paths. That is, recursive utility could enrich the classical turnpike theorems by relating rate of convergence of optimal paths to the variability in time preference. The analysis also needs to be extended to multisector models by suitably adapting value loss techniques to the recursive case.

Acknowledgements

I am extremely grateful to Professors R.A. Becker, R.N. Bhattacharya, F.R. Chang and P. Streufert for their numerous comments and suggestions. I would like to thank two anonymous referees from another journal and two anonymous referees from this journal for suggesting many improvements in the text of the paper. Parts of the paper were also presented at the Midwest Mathematical Economics Conference at the University of Illinois in Urbana Champaign, Fall 1990. I am grateful to the participants for their comments. I retain responsibility for any errors.

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