Recursive utility, martingales, and the asymptotic behaviour of optimal processes

EUEVIER Journal of Economic Dynamics and Control

21 (1997) 505-523

Recursive utility, martingales, and the asymptotic behaviour of optimal processes

Sumit Joshi

Department of Economics, The George Washington University, Washington, DC 20052, USA

(Received April 1995; final version received February 1996)

Abstract

This paper develops a new approach to the asymptotic behaviour of optimal programs in aggregate growth models permitting nonstationarities in the description of the technology and the stochastic environment. Added generality is provided by specifying a recursive utility description of preferences. The analysis makes critical use of martingale methods afforded by a stochastic specification and is applicable to growth models with both convex and nonconvex technologies.

Key words: Martingales; Euler equations; Growth; Stochastic; Asymptotics JEL c~assifcation: C60; D90

1. Introduction

It has been observed in both aggregate and multisector convex models of economic growth with time additively separable (TAS) preferences that optimal

programs from different initial stocks converge in an appropriate metric. This property is described as the late (or twisted) turnpike property. In the deterministic case, it was proved in Mitra and Zilcha (1981) in a model permitting

nonstationarities in both preferences and technology. It was subsequently extended to the uncertainty case by Majumdar and Zilcha (1987) in a model allowing nonstationarities in the evolution of the stochastic environment.

I would like to thank George Washington University for a Junior Scholar Incentive Award for

Summer 1993 which made this research possible. I would also like to thank an anonymous referee from another journal for detailed comments on an earlier draft and an anonymous referee from this

journal whose comments have improved the paper. I remain responsible for any errors.

0165-1889/97/$15.00 0 1997 Elsevier Science B.V. All rights reserved

PII SO165-1889(96)00942-6

506 S. Joshi 1 Journal of Economic Dynamics and Control 21 (1997) 505-523

The analysis of Majumdar and Zilcha (1987) is the definitive work on the asymptotic behaviour of optimal programs in nonstationary stochastic aggregate models with TAS preferences. However, it has three limitations. First, since it directly applies the deterministic tools developed in Mitra and Zilcha (1981), the stochastic element remains underutilized and does not play a critical role in the proof of the turnpike property. Second, the method used requires the monotonicity of the optimal consumption program in the initial stock. With nonclassical technologies, while the optimal capital input program does display monotonicity with respect to the initial stock, the optimal consumption program may not. This prevents their technique from being extended to general nonconvex technologies. Third, the production function in their analysis is stationary and satisfies a uniformity condition in the form of a strictly positive lower bound on its degree of concavity. With nonstationary technologies, this extends to the requirement that the degree of concavity of the sequence of time-varying production functions is uniformly bounded from below by a positive constant. This is restrictive since it rules out nonstationary technologies which asymptotically approach the linear case. For instance, in a deterministic framework, consider the sequence of time-varying production functions (1;: IR, -+ R,} wheref;(k) = k1-“(‘+2) for t = 0, 1,2,. . . Then, eachf, is strictly concave for k > 0 and approaches a linear production function as t + co. Alternatively, the degree of concavity of A, given by -f,“(k)k/“(k), is l/(t + 2) and approaches zero as t + co. Further, the uni-

formity condition excludes all nonconcave production functions from their analysis.

In this paper, the stochastic structure does not remain a mere addendum but contributes crucially to the late turnpike analysis by allowing a passage to the rich theory of martingales. In particular, critical to the convergence proof is the martingale process generated by the stochastic Euler equations. The uniformity condition used is also more general in that it does not preclude nonstationary technologies of the linear or nonclassical type.

Two other papers in the same vein which were designed to exploit the stochastic specification to obtain the late turnpike property were Chang (1982) and Fiillmer and Majumdar (1978). The unifying theme in both papers was the choice of the Lyapunov function and the fact that proofs were driven by the transversaIity condition. In the former, an expected value loss condition was formulated in terms of the Lyapunov function and the asymptotic result was proved through the first Borel-Cantelli lemma. In the latter, a value loss condition was coupled with the Doob-Meyer decomposition of the Lyapunov function and the convergence theorem was proved by exploiting the integrability of the increasing process generated by this decomposition. The critical theme was that the Lyapunov function generated a submartingale process which could be bounded in expectation by invoking the transversality condition. It was this boundedness property which ensured the summability requirement of the

S. Joshi / Journal of Economic Dynamics and Control 21 (1997) 505~-523 501

Borel-Cantelli lemma in Chang (1982) and the integrability of the increasing process in Follmer and Majumdar (1978).

Similar to the spirit of the above two papers, the analysis here attempts to show how the passage to martingale methods afforded by a stochastic framework allows a new resolution of the convergence property. However, the reliance on martingale techniques notwithstanding, there are substantive differences between the modus operandi here and the analysis of Chang- Follmer-Majumdar and Majumdar-Mitra-Zilcha. These are outlined next:

(1) The novelty of the approach here is that it is driven by the Euler conditions. In fact, the Euler conditions contain all the relevant properties which are sufficient to derive the turnpike property. This is in contrast to Chang- Fbllmer-Majumdar where two constructs are required - the price characterization of optimal programs (the competitive conditions) and the transoersality condition. The reliance on one construct, and in particular on the Euler conditions, makes for substantial analytical ease. The competitive conditions used in the above-mentioned papers were assumed but not proved. They could be verified using the Hahn-Banach theorem, but the proof would be lengthy, particularly because the characterization required is different from the standard ones of reduced utility maximization and intertemporal profit maximization. In contrast, the Euler characterization of optimal programs is immediately avail- able in aggregate models.

(2) The analytical ease also extends to obtaining the essential martingale boundedness property which drives the turnpike theorem. In Brock and Majumdar (1978, Thm. 1) on which Chang (1982) and Fijllmer and Majumdar (1978) are based, the boundedness in expectation of the submartingale process (generated by the Lyapunov function) via the transversality condition requires some nontrivial steps. In contrast, for the submartingale and supermartingale processes considered here, boundedness in expectation follows in one quick step from the Euler conditions. Use is made of interiority of optimal programs and the fact that a martingale process with finite expectation at period 0 is automati- cally integrable for all t 2 0.

(3) In the context of the aggregate model, the Euler approach has an added degree of flexibility which does not obtain in the multisector analysis of Brock-Chang-Follmer-Majumdar - it is capable of being generalized to nonconuex technologies. The crucial property of optimal programs which is ex- ploited is the Euler characterization and this obtains even in aggregate growth models with nonconvex technologies. In contrast, the analysis of Chang et al. is critically dependent on the convexity of technology since the competitive conditions generating the submartingale process have to be derived via separating hyperplane arguments.

The turnpike theorem in the nonconvex case is proved in Section 4. In this regard, this paper bridges an important gap in the aggregate growth literature. While the nonconvex aggregate model with TAS preferences has been studied in

508 S. Joshi /Journal of Economic Dynamics and Control 21 (1997) 505-523

Amir, Mirman, and Perkins (1991) Dechert and Nishimura (1983), Majumdar and Mitra (1982), Majumdar, Mitra, and Nyarko (1989), and Mitra and Nyarko (1991), none of these papers have analyzed the late turnpike property of optimal programs in a general nonstationary nonconuex growth model with no restriction on the form of nonconvexity.

This paper further extends the late turnpike property to a recursive specification of preferences. The desirability of recursive utility over TAS utility with a constant rate of discount is by now well documented (Epstein and Zin, 1989; Boyd, 1990; Becker and Boyd, 1990; Streufert, 1990). Recursive utility permits partial complementarity in intertemporal consumption in a manner permitting the applicability of dynamic programming techniques. Further, the discount factor is not an arbitrarily specified constant but varies with the underlying consumption sequence thereby permitting flexibility in time preference. Within a stochastic paradigm, the recursive specification permits nonindifference to the temporal resolution of uncertainty as well as a separation between the elasticity of intertemporal substitution and the degree of risk aversion. Given the flexibility inherent in the recursive paradigm, the study of the asymptotic behaviour of optimal programs within such a general specification of preferences is clearly of theoretical interest.’

It may be noted that the definitive analysis of Majumdar-Mitra-Zilcha in the convex case under TAS preferences with a constant rate of discount was also based on the Euler characterization of the optimal programs. As noted earlier, their analysis cannot be extended to the nonclassical framework since convexity requirements entered into their analysis through the uniformity restriction on the degree of concavity of the production function and the monotonicity of the optimal consumption program in the initial stock. It is also difficult to extend their analysis to general recursive utility preferences because of the endoge- nously determined discount factor which varies with the consumption level.

2. The basic model

Let a compact metric space Sz, with Bore1 a-field S, represent the possible states of the environment at date t 2 1 and o,~a, the state at date t 2 1. The stochastic environment is given by the probability space (52,4, v) where

r In fact, as Chew and Epstein (1990) note in the conclusion to their paper: ‘It would clearly bc of interest to exploit the flexibility of. . . the general recursive utilities. . to reexamine standard issues in capita1 theory such as asset pricing or optimal stochastic growth.’ Since the turnpike property of optimal programs does constitute one of the ‘standard issues’ of optimal stochastic growth, this paper addresses this issue in the recursive paradigm by incorporating the additional dimension of nonconvexities in technology.

S. Joshi / Journal of Economic Dynamics and Control 21 (I 997) 505-523 509

Q = fl?Q,, @’ = By&‘, is the a-field on C? generated by the measurable cylin- drical sets and v is the measure on C2, Let h, E fl:sZ, denote the partial history at time t 2 1 and & denote the sub a-field of 9 induced by this partial history till date t. Let F0 denote the coarse a-field { /21, s2).

Technology is given by a sequence of possibly time-varying production functions (5: R, x Q2,+, -+ R,} where, for each t 2 0, ft is continuous on R, x0,+,. Further ~w,+~ES~,+~,~~(O,W~+~)=O,~~(~,W~+~) is twice continu- ously differentiable at k > 0, and ft’ > 0 at k > 0 where fi’ = af,/ak. Given an initial stock s > 0, the real-valued nonnegative {F;,}-adapted process

(k, c) = {(k,, cJ> is a feasible program from s if k0 + c,, I s and k ~CI + c,+~ 5 y, =_L(k,,w,+l ) v-a.s. V’t r 0. Let @(s,(J)) denote the set of all feasible programs from s. From the continuity of ft and the compactness of Q,, there exists a maximum pure accumulation process from s, {e}, which is independent of o and provides a coordinatewise upper bound for any feasible program

{(k,, cl)> in @(s,{f,}):

0 I k,, c, I R, v-a.s., t 2 0.

This paper adopts a general recursive utility description of preferences.2 Let U:lR~-+R+ be a utility function and consider any h, E fllsZ, and consumption

sequence (c,, c, + 1, c,+ 2, . .I consisting of deterministic current consumption, c,, and future stochastic consumption, (c,+ 1, c,+ 2, . .). Let U(c,, c,+ 1, . . 1 h,) denote the utility from this consumption sequence, given the history h,. It is assumed that U satisfies the recursive relationship:

U(c,, G+I, c,+2,. . . Ih) = W

where W: IR: + R + is the aggregator function. This function aggregates current deterministic consumption, ct, with an index of future random utility (generated by future stochastic consumption (c,+ 1,c,+ 2, . . .)) to determine current utility. The specification given above describes the same preferences as a version of the Kreps and Porteus (1978) aggregator:

J’k,, c,+ 1, cc+29 . . . IhJ

g(J’(c,+,, c,+z,. . . Ih+ddw+l 1) > (2)

‘This description follows Epstein (1990), Ozaki (1991), etc. in defining the recursive utility function over the set of feasible consumption programs. Note that the analysis here could also be extended to the relatively more complex case of Epstein and Zin (1989) where the domain of recursive utility is the space of temporu! lotteries - lotteries indexed by the time of resolution of the uncertainty.

510 S. Joshi / Journal of Economic Dynamics and Control 2/ (1997) 505-523

where g: R, -+ R, is some increasing function. To see this, let U = g(V) and W(c, z) = g(w(c, g-‘z)). The existence of the recursive specification given by (1) has been discussed in Epstein and Zin (1989), Boyd (1990), and Becker and Boyd (1990) within the context of the contraction mapping framework, and by Streufert (1990) using the notion of biconvergence. Conse- quently, the issue of existence of a recursive utility specification is not pursued here.

The aggregator function W is assumed to satisfy twice continuous differentia- bility on R:, W(0, 0) = 0, W(c, z) is strictly increasing in both c and z, W(c, z) is strictly concave in c for each z E R +, 3W (c, z)/& -+ + cc as c -+ 0 for each z E R+, and W is supermodular (i.e., i3’ W(c, z)/&53z 2 0). The analysis here accommodates the TAS aggregator W(c, z) = u(c) + 6z, the Koopmans- Diamond-Williamson aggregator (under a monotonic transformation in Streufert, 1990, p. 89) W(c, z) = ce + Slog(l + z), 0 -c e -c 1, the exponential aggregator of Streufert (1990), W(c, z) = ce + (1 + z)& - 1,O < e < 1, and, under suitable restrictions, the Koopmans quadratic aggregator W(c, z) = z + (c - z) x (a - pc + yz).

A program {(kf, G)} in @+,{_A)) is said to be optimal if it satisfies

(3)

The existence of an optimal program follows standard dynamic programming techniques adapted to recursive utility (Epstein and Zin, 1989; Becker and Boyd, 1990; Streufert, 1990). From the Inada condition on W, cf > 0 and k: > 0 for a.e. o Vt L 0. Now define the time-varying value functions:

Jtk ht) = sup w ct, (S u(ct+ 1, ct+z, . . . lh+ Adw+ t > ra,+,

where the supremum is over the set of all real-valued nonnegative (Pt}-adapted process {(k, Ci)>, i 2 t, satisfying k, + ct 5 x and ki+ 1 + Ci+ 1 I yi =f;:(kip Oi+ 1)

3The supermodularity assumption is not required to prove the late turnpike property. The only reason for including it in the list of assumptions on the aggregator is that it extends to the recursive paradigm a well-known result of optimal growth theory on the monotonicity of the optimal path of capital with respect to parameters such as the initial and terminal stocks. The role of supermodularity in obtaining such monotonicity results was first developed in Topkis (1978) and extended in Amir et al. (1991) and Hopenhayn and Prescott (1992).

S. Joshi 1 Journal of Economic Dynamics and Control 21 (I 997) 505.-523 511

v-a.s. V’i 2 t. The stochastic Euler equations for t 2 0 are

xfi’K,~+~)ll~ , I v-a.s., (5)

where W,(c, z) = a W(c, z)/i%, W2(c, z) G aW(c, z)/az, and the dependence on partial history is dropped from the value functions for notational ease. These stochastic Euler equations can be derived by combining the first-order conditions from the maximization of the Bellman equation with the envelope properties of the value functions or by adapting the argument of Majumdar, Mitra, and Nyarko (1989, Thm. 5). Now define the discounted competitiue price process (or, in the nonconvex case, the discounted shadow price process), {pf}, associated with the optimal program {(k:, <)> for t 2 0, as

(6)

(7)

Then, using a well-known property of conditional expectations, the stochastic Euler equations become

PS = ECps+lf,‘(ks,o,+l)l(~*], v-a.s., t 20. (8)

Further, let

Then rc: is &measurable and strictly positive for a.e. w because of interiority of the optimal program. Here {II:} denotes the sequence of future values of a unit of

512 S. Joshi /Journal ofEconomic Dynamics and Control 21 (1997) SOS-523

capital in time period 0. The Euler equations can now be written succinctly for tlOas

WP:+ I n:+ I - pf 7cf I( F,] = 0, v-a.s.

Since p~$ is Ft-measurable, (9) also indicates that the process {p~$} is a martingale with E[pf$] = E[p$n”,] = pi.

3. Asymptotic property of optimal programs: The convex case

In this section, in addition to the assumptions on technology of Section 2, it is also assumed that the technology is convex, i.e.,

J;“(k, w + I ) < 0, for k > 0, Vu,+ 1 EQ,+ 1, Vt 2 0, where f;” = a’S,/ak’.

Now consider an initial stock z > s and let ((kf, cf)} be the optimal program in @(z,(ft}). Adapting the Optimality Equation technique of Dechert and Nishimura (1983, Thm. 1) or the lattice programming techniques of Amir, Mirman, and Perkins (1991), it can be shown that the optimal capital input program satisfies monotanicity with respect to the initial stock,4 i.e., kf I kf, v-a.s., Vlt 2 0. Therefore, using concavity off,, from (1):

PS 2 ECP”+If,‘(k:,w,+l)II~l, v-a.s., (10)

and with a strict inequality if kf > kf because of the strict concavity off; for all o,+ 1 E O,, 1. Multiplying both sides by n:, the above becomes

E CP;+ I R:+ I - pf$ jlPt] I 0, v-a.s.

Now, combining (9) and (ll), it follows that

(11)

E CP;+ I 4+ I - pS+ I 4+ 1 - {PS 4 - ptnf} II *I 2 0, v-u., (12)

with a strict inequality if kf > ki. Let I/, = pf$ - p$rf, t 2 0. Since V, is Ft- measurable, it follows from (12) that the process {V,} is a submartingale. Further, it is bounded in expectation independent of t since

EV,IW$=p”,= W,(c.,,~$,(yb)dwl), t=0,1,2,...

4The monotonicity result is not critical to the proof of the late turnpike property. It just simplifies the presentation of the proof in a convex growth model wheref,‘, t 2 0, is decreasing in k for all CU. Theorem 2 indicates a method of proof which does not exploit the monotonicity property.

S. Joshi /Journal of Economic Dynamics and Control 21 (1997) 505-523 513

The Doob-Meyer decomposition can be used to decompose the submartingale process into a martingale and an increasing process and arguments similar to Follmer and Majumdar (1978) can be utilized. This paper develops an alternative argument based on the first Borel-Cantelli lemma. Critical to this argument, as in the proof of any such convergence theorem, is a uniformity condition. To motivate this condition, consider any E > 0 and let 11 I/ be the metric in terms of which the divergence in the capital stocks is evaluated. Let S: =

1~: Ilk:(w) - K(w)ll > E an note that S: belongs to the sub-a-field Ft since > d the optimal programs are adapted to the filtration {PC}. Now, if a realization o belongs to S:, then from the above discussion, E[ I/,+ r - I/, II &] (w) > 0 for the given o. This implies that for any realization w E SF, a &(a, E) > 0 can be chosen such that E[V,+, - V, 11 Ft] (0) 2 6,(w, E). Here 6,(0, E) denotes a ‘value loss’ if the capital stocks k: and e are more than a-distance apart in terms of the metric II 11 for the realization w. A similar value loss which also, in addition to E, depends on both w and t is derived in Brock and Majumdar (1978, p. 234) and Fiillmer and Majumdar (1978, p. 280) via the competitive conditions. As in these papers, the uniformity condition here requires that a strictly positive ‘value loss’ can be chosen independently ofthe tuple (t, o) for any t 2 0 and any w for which I( k:(o) - k:(w) I/ > E:

Uniformity Assumption 1. For any E > 0 there exists a 6(a) > 0 such that for any t 2 0 and any ~ESZ:

If /k:(w) - kf(o)ll > E, then E[V/,+, - V,IIPt](o) 2 &(a, E) 2 6(a).

(13)

An alternative (but, as noted in Chang, 1982, p. 167, not weaker) way to frame Uniformity Assumption 1 is the ‘expected value loss’ form (Chang, 1982, Assumption A.4’ p. 166). Theorem 1 of this subsection is also valid under this expected value loss assumption. Note that Uniformity Assumption 1, in contrast to the uniformity condition of Majumdar-Mitra-Zilcha, does not a priori rule out linear technologies. The convergence property can now be proved. Since the late turnpike property is concerned with the asymptotic behaviour of optimal programs, the proof will not be affected by letting s; = 52.

Theorem 1. Consider initial stocks s, z where z > s B 0. Under Un$ormity Assumption I and the stated assumptions on technology and preferences, 11 kf - kf 11 + 0, v-a.s.

514 S. Joshi /Journal ofEconomic Dynamics and Control 2I (1997) 505-523

Proof: Letting xA denote the indicator function of a measurable set A, it follows from (13) that

x~~(E) I xs:E[J’,+, - Vtll%l.

Further, since { V,) is a submartingale, the random variable E [V, + 1 - V, llPt ] is almost surely nonnegative. For any A E St, since JAE[ I/,+ 1 - V, (( FJdv = 0 when v(A) = 0, it can be assumed without loss of generality that &? - S: does not contain o belonging to the complement of the set on which E [V,, 1 - Vt II Ft] is nonnegative. Hence:

B(E)V(Sf) I s ED’,+, - V,llEldv + W”,+I - V, I( .FJ dv . s: s n-s:

From a well-known property of conditional expectations, for any A E Ft and any integrable random variable X, JA E [X (1 &]dv = J,Xdv (see Billingsley, 1979, p. 395). Therefore:

~(+V:) I WV,+, - Vtl.

Summing up (and noting that V. = 0):

8(~) i v(Ste) 5 ECV,+,l 5 W,(c”,, EJoW), VT21. t=o

Hence, crv(S:) < co. Then, v(lim SUP~_+~ Se) = 0 from the first Borel-Cantelli f lemma (Billingsley, 1979, Thm. 4.3). W

4. Asymptotic property of optimal programs: The nonconvex case

Only the assumptions of Section 2 are maintained in this section. No restriction is imposed on the sign of$“. Hence, general forms of nonconvex technologies are permitted in the analysis. This section, therefore, extends the aggregate analysis of Amir et al. (1991), Dechert and Nishimura (1983), Majumdar and Mitra (1982), and Mitra and Nyarko (1991) to incorporate the turnpike property in a nonstationary nonconvex stochastic aggregate growth model with recursive utility preferences. The proof will require the following preliminaries:

(i) The first important step in turnpike theory (as also in Section 3) is the determination of a process which gives strictly positive value loss in any period t and for any state o for which the capital stocks g(o) and k:(o) are some critical

S. Joshi /Journal of Economic Dynamics and Control 21 (1997) 505-523 515

distance apart in terms of I/II. The construction of such a process makes critical use of conditions characterizing the optimal programs. For instance, in the convex multisector models of Chang-Follmer-Majumdar, the competitive conditions characterizing the optimal programs were utilized, while in the convex aggregate model of Majumdar-Mitra-Zilcha and Section 3, the Euler conditions were used. Since the Euler conditions also characterize the optimal programs in the nonconvex case, they are used in this section to construct a ‘value loss process’ showing the desired sensitivity to a divergence in capital stocks.

(ii) The construction of the value loss process has also to be conditioned to the peculiar exigencies of the nonconvex case. Let S: = {o: I/ k:(o) - kf(o)II > E) once again. Since the sign off;” is left unspecified, ft’ can display a variety of behaviour on the set SF: for some w in S:,j;‘(kf, w,+ r) >f,‘(kf, co,+ J, whereas for

other cG’(k~, 0, + J < f,‘(K w f + A, and for yet other wf,‘K, w, + A = .h’K, 0, + J. An example of this is the well-known convex-concave production function where, for each w,J’ first increases and then decreases. These possibilities are incorporated into the construction of the value loss process by partitioning the realizations w in S: into two subsets: those w for whichf;’ registers a change to an s-difference in capital stocks and those o for which no change is registered.

Theorem 2. Consider the minimum selection {(kf, cf)} in @(i,(A)), i = s, z and s # z. Under the given assumptions on technology and preferences, there exists a supermartingale process {Z,} which displays the following sensitivity property:

for any E > 0 and any t 2 0, there exists a 6, (which in general will depend on w and E) such that

11 k:(o) - k:(w)11 > E *E[Zl - Z,+, il%l(4 2 &(~s) ’ 0. (14)

Proof Recall the Euler equations characterizing the optimal program {(kf, cf)}, i = s, z, for t 2 0:

of = ECpf+ 1 f,‘(kf, u,+ 1)11~1, v-a.s. (15)

Consider any arbitrary E > 0. There are two possible cases to consider depend- ing on the behaviour off,‘:

Case I. f;’ displays sensitivity to an s-divergence in capital stocks on a measurable set of positive measure. That is, if 11 kf(w) - k:(o)11 > E, then f;‘(e, w,+ 1) #

K(k:, w, + 1 ) for w E Af c SF and v(A:) > 0. For t 2 0, let

It = minCftl(ks,o,+l),ftl(k:,ur+l)l,

516 S. Joshi f Journal of Economic Dynamics and Control 21 (1997) SOS523

where r, is strictly positive for a.e. w and 9, + i-measurable for all t. From the Euler characterization,

PS = ECp:+i_UK, w+~WJ 2 J3I~~+~~,ll~1, v-a.s. (16)

P: = ECp:+I_C(~:,w+I)ll~l 2 EC~;+~~~ll~l, v-a.s. (17)

Now suppose that Ilk: - k:II > E. If J;‘(& wt+i) >f,‘(k:, ot+i) for some WEA:, it follows that for this o the inequality in (16) is strict and (17) holds as an equality throughout. The opposite is true for we A: f;‘(K o,+ i) <f,‘(kf ,q + 1). Consider now the process {rcr} which is follows:5

for which defined as

7cl-J = 1, nt+l=i~ri~ t20.

Then n, + i is strictly positive for a.e. o and S,+ 1 -measurable, t 2 0. Multiplying (16) and (17) by rc, yields

(19)

Note once again that (18) holds as a strict inequality and (19) as an equality for those w E A: for which f,‘(ks, o,+ 1) >f,‘(k;, co,+ 1). The opposite is true for o E A: for which f;‘(kf, co,+ 1) <J’(kf, co,+ 1). Now let W, = pfn, + pfq, t 2 0. It follows by adding (18) and (19) that { W,} is a supermartingale process with EW, < EWo = pi + pi. More importantly, given the sensitivity off;’ on A:, the supermartingale inequality is satisfied strictly by all WEA:.

Case II. ft’ displays insensitivity to an e-divergetice in the capital stocks on a measurable set of positive measure. That is, if IIkf(o) - kf(o)I( > E, then

J’(K,o~r+1) =L’(k:,or+1 ) for o E Bf c Sf where v(g) > 0. The resolution of this case is based on the observation that, iffy’ is insensitive

on a set g, then kf,’ will show sensitivity on B:. Let w = max{s, .a}. From

’ In the convex case, if z > s, it was known that x: is the minimum in each t 2 0 of $ and 7~:. This is no longer true in the nonconvex case and, therefore, {n:} and {z:} are replaced with {n,} denoting the coordinatewise minimum of the sequence of future values of a unit of capital in period 0 with respect to the minimum selection programs from initial stocks s and .z.

S. Joshi 1 Journal of Economic Dynamics and Con1rol21 (I 997) 505-523 517

interiority and the bounds imposed by the maximum pure accumulation process, 0 < k#” = @ I 1 and 0 < k:/k; = @ I 1 v-a.s., for all t 2 0. Now let

qr = minC@L’K, wt+ A @_L’K, w+ dl, t 20.

Then6 qt is positive v-a.s. and 9,+ ,-measurable, t 2 0. Further, for a.e. w and i=s,z,

.L’(kf> w+d 2 $f,‘(kf,w+,) r qt.

Utilizing the Euler equations,

P: 2 UP:+ &;:fi’(k:, a+ I)> II%;,1 2 Wd+ ~qtllEl, v-as. (21)

The second inequality in (20) is strict (and the second inequality in (21) is an equality) if K(w) > k:(w) and w E Bf. Similarly, the second inequality in (21) is strict (and the second inequality in (20) is an equality) if K(w) < k:(o) and w E Bf. Now let p0 = 1 and pl+l = nbqi, t 2 0. Then, pt+ 1 is strictly positive for a.e. w and 9, + ,-measurable, t 2 0. It then follows that v-a.s. for all t 2 0,

p:pt 2 ECp:+ lpL,+ 1 lIFtI

Letting Y, = pfpt + pfp,, t 2 0, the process { Yt} is a supermartingale with E Y, I E Y0 = pi + ~‘0. Further, the supermartingale inequality holds strictly for all oeZ$.

Note that if I/K(o) - k:(w)11 > E for some realization w, then there are only two possibilities: either f,‘(kf, co,+ J #f,‘(k:, CD,+ 1) or f,‘(kf, w,+ 1) =J’(kf, w,+ 1) for the given o. To incorporate both possibilities, let 2, = W, + Y,, t 2 0, and note that {Z,} is also a supermartingale. Combining Cases I and II, if II G(w) - k:(w)11 > E for some realization o E 52, then:

ECW,- W,+Ill%l(~)~O, WY,- Yt+IIl~tl(~)20 if o~$,

E[W, - Wt+,llFt](o) = 0, - E[Y, - Y,+,l19t](o) > 0 if WEB:.

6 The term $fi’(ki, UI,+ I) can be roughly interpreted as (normalized) total return to capital, i.e., the product of capital (in some normalized units) and the per unit return to capital. The total return to capital is strictly increasing in the capital input for those w for which the per unit retum,f;‘, is constant. Hence, the analysis in Case II turns to total return to capital to demonstrate sensitivity to an s-divergence in capital stocks in contrast to the convex case, as well as Case I, where the per unit return to capital could signal such a divergence.


Therefore, E [Z, - Z, + 1 l/PC] (0) > 0 for all realizations w E S: = A:LJ B:. Now, for any w E SF, a 6,(0, E) can be selected such that E[Z, - Z,, 1 IIFt](o) 2 &(w, E) > 0. n

The supermartingale process {Z,}, taking the particularities of the nonconvex environment into account, gives a ‘generalized value loss’ of 6,(0, E) > 0 in any period t and for any realization o for which the capital stocks e(o) and k:(w) are more than s-distance apart in the metric I[ 11. A general value loss result can now be proved which extends Fijllmer and Majumdar (1978, Thm. 3.1) to the nonconvex case with recursive utility preferences. This result shows that for any II > 0, {Z,} will almost surely leave the set on which generalized value loss exceeds 1 after at most a finite number of visits.

Theorem 3. For any ;I > 0 consider the A-value loss set, A, = {(t, w): &(w, E) 2 A>, and define the total time spent by {Z,} in /iA for any given CIJ as T,(o) = Cyx(&)(t, CD). Under the stated assumptions on technology and preferences, ETA < cc .

Proof: For any N 2 1, let the N-truncation of Tl be T:(w) = C:= Ix(AA)(t, w). From Theorem 2,

W&)(4 4 = 0 I &(o, E) 5 ECZ, - Z,+ill~,](o) if (t, m)&A,.

It now follows that

A s s

T:dv = $ Ax(A,)(t, o)dv r=1

I i EC4 - Z,+lWrldv r=1 s

= [Zr - Z+Jdv from Billingsley (1979, p. 395)

< s

I’edv = 2cpS0 + pi].

S. Joshi 1 Journal of Economic Dynamics and Control 21 (1997) 505-523 519

Since the above holds for all N 2 1, it follows from the Monotone Convergence Theorem that

The convergence of the optimal programs in the metric /I I/ will once again require a strengthening of the sensitivity of the value loss process via the strong restriction of:

Uniformity Assumption 2. For any E > 0 there exists a b(s) > 0 such that for any t 2 0 and any WEQ:

If IIkf(w) - kf(o)II > E, then E[Z, - Z,+ 1 II%](w) 2 &(a, a) 2 6(s).

Remark. Similar to the convex case, the purpose of the uniformity assumption is to ensure that there exists some critical minimum strictly positive value loss of b(s) for any time period t and for any realization w for which 11 K(o) - k:(o)11 > E. Such an assumption is admittedly a strong restriction, but uniformity assumptions in turnpike theory are invariably strong. Hence, citing some instances, they have been labeled ‘strong uniformity’ in Brock and Majumdar (1978, (A.4) and subsequent discussion, p. 234) or as a ‘strong value loss assumption’ in Follmer and Majumdar (1978, p. 281).

Example. An example is now constructed which satisfies Uniformity Assump- tion 2. The technique is drawn from Theorem 2. Letft(k, o) zj(k, w) = k2u/2, WE&& - [cc, /?I, 0 < 01 < /3 < 03, and W be any aggregator function satisfying the assumptions of Section 2. Let z > s and {(kf, ~$1 be the minimum selection from the set of optimal programs in @(i,f), i = s, z. From standard arguments, kf 2 k:, v-as. This example will measure the divergence between capital stocks kf and ks by using a variation of the relative distance metric of Majumdar- Mitra-Zilcha. This metric is defined for any w E Sz as

I W4 - kX4 I /I kfk4 - 44 11 = maxllcs(o), k;(o)) =

664 - W4 W4 ’

It is readily verified that 11 II is a valid distance function on lR+ - (0). Note that sincef,’ is strictly increasing in k > 0 V’w, the event in Case II of Theorem 2 holds on a set of probability measure zero.


Since 71: is Y,-measurable, the Euler equations given by (8) characterizing the optimal program {(k:, c:)} can be written for t 2 0 as

p:$ = E[p:+ l~Sk:~,+ 1 II .Ft], v-as. (22)

In particular, using the fact that k: 2 kf, v-a.s., it follows from (22) that

~36 2 ECpf+ 14+ 1 II%], v-a.s., t 2 0, (23)

indicating that the process {p:~$} is a supermartingale. Further, pfnf is strictly positive for a.e. o, t 2 0, from interiority. Now consider any A E 9,. Using the definition of S:, it follows from (22) that

2 s P:+ I$K + Mw,+ 1 dv + SfnA s P:+ I<+ &

(R-S:)nA

(24)

Since (24) holds for all A E Pt and xs: is F,-measurable, it follows that for a.e. o,

(25)

where the equality is a consequence of (22). By the mean value theorem, there exists a random variable h, such that for a.e. o, ECp:+ i$+ 1 II9J(o) 5 h,(o) I p:(o)a@) and

Using Jensen’s inequality for conditional expectations (see Billingsley, 1979 p. 399) on (23),

h.dpM 2 log(EM+ 1 nf+ 1 11=%1) 2 EClogW+ 1 zS+ dll%l, v-a.s., (27)

S. Joshi / Journal of Economic Dynamics and Control 21 (1997) 505-523 521

indicating that the process {log(p:7c:)} is also a supermartingale. Further, from Jensen’s inequality, E[log(p:Q] 5 log(E[p:n:]) I log(Wi(c’,, EJ,(y’,))) for all t 2 0. Combining (29, (26) and (27), along with the fact that h, I p:r& v-a.s., then yields for a.e. o:

log(p:$’ - ECWp:+ I $+I Wtl 2 1~~W~~) - log(ECp:+ I n;+ I lIEI)

2 EXs: . (28)

Let Z, = log(p:$). From (28), given any E > 0, there exists a d(s) = E > 0 such that for any t 2 0 and WEQ if Ilk:(w) - kf(w)ll > E, then E[Z, - Z,+ 1 IISt](ll(w) > B(E). n

The convergence property can now be proved using the supermartingale process {Z,}:

Theorem 4. Consider the minimum selection {(kf ,c:)} in @(i,(J)), i = s,z and s # z. Under Uniformity Assumption 2 and the stated assumptions on technology and preferences, llg - kfll + 0, v-a.s., as t + 03 .

Proof From Uniformity Assumption 2,

Ij(E)V(X) 5 s EC& - Z+,ll%ldv. s:

Further, from the supermartingale properties of (Z,}, it can be assumed without loss of generality that

s EC&-Z,+,lW]dv 2 0. R - s:

It now follows that

s(E)v(s:) 5 s El?, - Z,+ 1 ll%ldv + s

EC& - &+lllZldv X n-s;

= s ECZ, - Z+,ll~ldv. R


Using Billingsley (1979, p. 395) from the above:

WvW i ECZ - Z,+ 11.

Summing up, and noting that Z, is strictly positive for a.e. w:

49 i: 4s;) I i EC-G - Z,+Ll < ECZol = 2CpsO +&I, T= 1,2,... r=o t=o

Hence, 1,” v(S:) < co yielding the result via the first Borel-Cantelli lemma. n

5. Conclusion

This paper developed a general methodology by which the late turnpike property could be recovered in both convex and nonconvex aggregate growth models and for both time-additively separable preferences and recursive utility preferences. An important area for future research is to develop the analytical tools by which the late turnpike property can also be obtained in nonconvex multisector growth models. This would extend the work of Brock-Chang-Fiillmer-Majumdar on the asymptotic behaviour of optimal programs in convex multisector growth models along the dimensions of nonclassical technologies and recursive utility preferences. The extension is nontrivial, however, because a nonconvex technology renders inapplicable the classical method of constructing value loss processes. As noted earlier, the value loss process in Brock-Chang-Fiillmer-Majumdar was constructed using the competitive conditions. The derivation of these conditions require separation arguments such as the Hahn-Banach theorem which rely critically on the convexity of technology. The problem then, in the general nonconvex multisector case, is to find suitable conditions characterizing the optimal programs of capital accumulation which do not hinge on the convexity of technology and are malleable enough to construct value loss processes satisfying the twin properties of sensitivity to a divergence of capital stocks and integrability (or boundedness). While the Euler equations met the above conditions in the one-sector nonconvex model, corresponding characterization results for the multisector nonclassical model remain an open question.

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Recursive utility, martingales, and the asymptotic behaviour of optimal processes

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