Many dynamic economic models rely on the assumption that preferences are representedby a functional which is additive over time and discounts future rewards at a constantrate. The additively separable hypothesis has long been recognized as special, but hasdominated research in economics mainly because of its technical convenience. In the last few
� This paper was completed when Cuong Le Van visited CORE and IRES in April 2002.∗ Corresponding author.E-mail addresses:[email protected](C. Le Van),[email protected](Y. Vailakis).
188 C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209
years, some criticism has been directed towards this assumption challenging its economicplausibility.
An important implication that stems from the axiomatic structure of time-additivelyseparable preferences, is that the rate of impatience is a specified constant independentof the consumption stream. Obviously, this is very restrictive since it excludes plausiblesituations in which changes in consumption in the initial periods or an overall increase inconsumption profile would affect impatience.A second and more severe complication arisesin heterogeneous agents models. In this case, additive separability leads to a rather strongconclusion: unless all the agents have the same discount factor, the most patient agent endsup with a positive consumption in the long run while all other agents consume nothing.A third complication arises when uncertainty is introduced. Aversion to risk and aversionto intertemporal substitution are two conceptually distinct attributes of a consumer’s tastesand should be treated independently of each other. Time-additively separable preferencesunder expected utility do not allow this separation.1
Recursive utility provides a way to escape these problems by allowing a flexible rate oftime preference determined endogenously by the underlying consumption stream. Roughlyspeaking, there are two approaches to the construction of recursive utility functions. Thefirst follows Koopman’s[10] early work and is concerned with the axiomatics of preferencesleading to a recursive representation of utilities. The second takes an aggregator functionas the fundamental expression of tastes and then tries to recover the utility function fromthe assumed properties of the aggregator. The first paper that made the aggregator approachfeasible was Lucas and Stokey[13]. Becker and Boyd[2] provide an excellent expositionof these two approaches.
This paper rests on the second approach. Its primary aim is to explore conditions underwhich an aggregator could determine a unique utility function. However, since optimalgrowth models provide the basic framework for applied dynamic analysis in a wide rangeof areas in economics, and since recursive utility has important implications for standardresults in those areas, it is important to extend the study of optimal growth to a general classof recursive utility preferences. This is the second objective of the paper.
When the aggregator function is assumed to be bounded, it is always possible to associatewith it a unique utility function.The proof of this result is well known and relies on exploitinga contraction argument (see[4,13]). Unbounded aggregators create some difficulties sincecontraction methods cannot apply directly. However, this problem can be partially resolvedif one introduces a weaker notion of boundedness. This approach has been followed by Boyd[3] and Duran[6] to deal with aggregators which are bounded from below but unboundedfrom above. Their argument relies on relaxing boundedness by considering functions thatobey a growth condition. It is in this new space of functions that they obtain the contractionproperty for the recursion operator. Another very interesting approach has been proposedby Streufert[17]. His idea is based on the notion of biconvergence, a limiting conditionensuring that returns of any feasible path are sufficiently discounted from above (upperconvergence) and sufficiently discounted from below (lower convergence).
1 In this paper we are not dealing with stochastic programs. For a discussion on the construction of recursivebut not necessarily expected utility preferences, as well as, on the underlying stochastic dynamic programmingarguments, refer to Epstein[8], Epstein and Zin[9] and Duran[7].
C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209 189
For aggregators that allow−∞ as a value, no general existence and uniqueness theoremhas been established. Several authors have followed different methods to overcome thisproblem with the one proposed by Boyd[3] being, to our knowledge, the most general one.Boyd combined a weighted contraction argument with apartial sumtechnique to constructthe utility functions.
Our paper introduces a unified treatment of the aggregator approach, covering situationsin which the recursion operator is bounded or unbounded (below/above). Our assumptionsare easy to check and apply to a wide class of aggregators including the standard examplesstudied in the literature. Following our approach, the recoverability of a utility functiondoes not rely on a contraction mapping technique. Instead, our proof rests on importantinsights derived directly from the assumptions made on the aggregator function. Intuitively,one expects to obtain the utility function by recursive substitution as the pointwise limit ofpartial sums of returns. Under our assumptions, a utility function constructed in that way isalways well defined and recursive. Moreover, it is upper semicontinuous and satisfies a kindof transversalitycondition. It will turn out that these two properties of the return functionare fundamental to establish its uniqueness as a reasonable solution.
Contraction methods have also been used to obtain a fixed point for the maximizingoperator in models with recursive preferences. Dynamic programming arguments are wellestablished in models with bounded returns (see[5,13,16]). Moreover, when the returns areunbounded from above (but bounded from below), it is always possible to use a weightedcontraction argument to escape the problem. This is the case in Becker and Boyd[2] andDuran[6].
For returns unbounded from below, the best-known results have been established for thecase of time-additively separable preferences. In a first attempt to address the problem ofunbounded returns, Alvarez and Stokey[1] study a wide class of homogeneous problems.They show that the principle of optimality applies to problems of this sort. Their methodof proof is based on finding restrictions that bound the growth rates of state variables fromabove along any feasible path, and, for the case where the utility may attain−∞, frombelow along at least one feasible path. In their analysis both decreasing and increasingreturns technologies are excluded. Another recent approach has been proposed by Rincon-Zapatero and Rodriguez-Palmero[14,15].Their argument is mainly based on the contractiontechnique. But instead of considering the usual normed space of functions in which theBellman operator fails to be a contraction, they focus on metric spaces which are differentdepending on the characteristics of the problem. They also apply their method to deal withrecursive preferences, but they only consider the case where returns are unbounded fromabove but bounded from below.
Le Van and Morhaim[11] introduce a synthetic frame to the study of dynamic pro-gramming problems with time-additively separable objectives and bounded or unbounded(below/above) returns. Their argument does not depend on a contraction mapping techniquebut rather builds on important insights derived from the assumptions imposed on the re-turn functions. In this paper, we go a step further and develop an analogous argument forrecursive models. Our approach is general, allowing for both bounded and unbounded (be-low/above) returns without imposing additional restrictions on the technology apart fromthe usual ones. Many recursive economic models, including the standard examples studiedin the literature, are particular cases of our setting.
190 C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209
The paper is organized as follows. In Section 2, we introduce the aggregator and showhow a utility function can be recovered from it. We also present some applications of ourtechnique by considering specific forms for the aggregator function. In Section 3, we proveexistence of optimal paths and establish the properties of the value function. It turns outthat the value function is the unique solution to the Bellman equation within the class offunctions that satisfy these properties. Furthermore, we show that the operator defined bythe Bellman equation provides an algorithm to find the value function.
2. Recursive preferences
An infinite sequence of elements inRn will be denoted byx, i.e. x = (xt )∞t=0, with
xt ∈ Rn ∀t � 0. The space of these sequences is denoted by(Rn)∞.DefineX(�) = {x ∈ (Rn+)∞ : supt
‖xt‖�t<∞} for � � 1 andX(�,M) = {x ∈ (Rn+)∞ :
supt‖xt‖�t<M} for � � 1 andM>0. Note thatX(�) = ∪
M>0X(�,M). Let � denote
the first coordinate projection function andSdenote the shift operator, i.e.�x = x0 andSx = (x1, x2, . . . , xt , . . .).
Definition 1. A functionW : X × Y → Y ∪ {−∞}, X ⊆ Rn+, Y ⊆ R with 0 ∈ Y, is anaggregator if it satisfies the following properties:
(W1) There exists a setD ⊂ Rn+ such that(a) D �= X,(b) ∀(x, z) ∈ D × Y,W(x, z) = −∞,(c) If D �= ∅, then lim
z→−∞ W(x, z) = −∞ ∀x ∈ X,
(d) ∀(x, z) /∈ D × Y,W(x, z) ∈ R.(W2) The functionW is nondecreasing in its second argument.(W3) The functionW is upper semicontinuous in its first argument.(W4) There exists�>0 such that
∣∣W(x, z)−W(x, z′)∣∣ � �∣∣z− z′∣∣ ,
for all x /∈ D and allz, z′ ∈ Y.(W5) Let � � 1 andM>0. There exists�(�) ∈ (0,1) and�(M,�)>0 such that∀x ∈
X(�,M), ∀N � 0,
�NW(xN,0) � �(�)N�(M,�).
(W6) Let� � 1. There existsx ∈ X(�) such that
limN→∞ W(x0,W(x1, . . . ,W(xN,0)...))> − ∞.
C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209 191
2.1. Comments on the assumptions
(1) Let� � 1. Starting from an aggregator functionW, one can define an operatorRoverall extended real-valued functionsU onX(�) as follows:
RU(x) = W(�x, U(Sx)).
We are interested in utility functions that are fixed points ofR. Intuitively, we expect torecover the utility function by recursive substitution as the limit of
RN0(x) =W(x0,W(x1, . . . ,W(xN,0)...)).
Therefore,∀x ∈ X(�) define
U0(x) = limN→∞ RN0(x) = lim
N→∞ W(x0,W(x1, . . . ,W(xN,0)...)). (2.1)
(2) Because of (W1), our set of admissible aggregators is allowed to contain aggregatorsthat are unbounded from below. For such aggregators,U0 may not be the only solution ofR sinceU(x) = −∞ satisfies the recursion too. In W1(c) we requireD �= ∅ to includeconstant aggregators.
In exploring conditions under which an unbounded aggregator could determine a uniqueutility function, Boyd[3] and Duran[6] face a similar problem. For aggregators that arebounded from below they are able to provide a general existence and uniqueness result. Theirapproach is based on finding a positive and continuous real-valued function�, defined ona setA ⊂ (Rn+)∞, such thatW(�x,0) is �-bounded, i.e. supx∈A
|W(�x,0)|�(x) < + ∞, and
�‖�◦S‖�<1, i.e.� supx∈A�(Sx)�(x) <1. In Boyd,A = X(�), while in Duran,A is the set of
feasible action plans endowed with the product topology. If such a function exists, it can beshown that the recursive operatorR is a contraction onC�(A) (the space of�-bounded andcontinuous functions onA). SinceC�(A) endowed with the norm‖f ‖� = supx∈A
|f (x)|�(x)
is a Banach space,R has a unique fixed pointU∗. Moreover, since limN→∞ RN0 = U∗, it
follows thatU∗ = U0.Boyd [3] deals also with aggregators that are unbounded from below. To circumvent the
problem posed by paths that permitU(x) = −∞, he considers first a region that excludesthem from the set of admissible paths. The utility function is initially defined on the set ofsequences with growth rates bounded from above and from below. As before, a contractionargument applies on this set, yielding a unique�-bounded utility functionU∗. Next, byusing a limiting argument analogous to partial summation, he extendsU∗ to the utility func-tionU0 defined on allX(�). It turns out thatU0 is the only recursive upper semicontinuousextension ofU∗ toX(�).
The main weakness of the weighted bounded approach is that in practice it may not beeasy to find a function� with the above-mentioned properties. This problem becomes moresevere when dealing with aggregators that are unbounded from below. Another issue con-cerns the modification of the usual dynamic programming arguments (i.e. the relationshipbetween the original recursive program and its associated Bellman equation) to includemodels with unbounded aggregators. Boyd[3] is not addressing this issue, while Duran[6]shows how weighted contraction methods can be also applied to obtain a contraction for
192 C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209
the maximizing operator only for the case where aggregators are unbounded from abovebut bounded from below.
(3)Assumption (W2) is weaker than that imposed in Boyd[3] (he requires the aggregatorto be increasing with respect to both variables). Boyd makes use of the increasingness ofWwith respect to its first argument to obtain the upper semicontinuity ofU0.
(4) Assumption (W3) is weaker than that imposed in Boyd[3]. Boyd requiresW to becontinuous at any(x, z) with x /∈ D (observe that in BoydD = {0}) andz> − ∞.
(5) Observe that (W4) implies continuity ofW with respect to its second argument atanyz> − ∞. The fact that we do not impose�<1 in (W4) it actually implies that we donot exclude undiscounted or upcounted models. We will see below that for a large class ofaggregators we can allow for� � 1, but at the expense of imposing a stronger requirementin (W5).
(6) Observe that assumptions (W3) and (W4) imply that ifxn → x, then lim supnW(xn, zn) � W(x, lim supn zn). It follows that lim supn W(xn, zn) � W(x, z), when(xn,zn) → (x, z). In case wherex ∈ D, we haveW(xn, zn) → −∞.
(7) Assumption (W5) is a crucial one. It is essential to establish the meaningfulness ofthe limit in (2.1), as well as, its uniqueness as a reasonable solution ofR. Note that Boyd[3] and Duran[6] implicitly impose a stronger requirement than (W5).
In Boyd [3], requiring the recursion operator to be�-bounded, it actually implies that∀x ∈ X(�) there existsm>0 such that|W(xN ,0)|�(SNx) <m ∀N � 0. Therefore, the followingrelation is true:
�N |W(xN,0)| � ��(SNx)
�(SN−1x)× �
�(SN−1x)�(SN−2x)
× ...× ��(Sx)�(x)
�(x)m.
For anyx ∈ X(�), �(x) = �(‖x‖�) where� is a continuous and increasing function. Inaddition,� is assumed to satisfy�‖� ◦S‖� � �<1. Let� � 1 andM>0. It follows that∀x ∈ X(�,M), ∀N � 0,
�N |W(xN,0)| � �(�)N�(M,�),
where�(�) = � and�(M,�) = m�(M).In Duran[6],U0 is defined on feasible action paths. For a given initial statey0, let(y0)
and(y0) denote, respectively, the collection of all feasible fromy0 state and action paths.Assume that there exists� � 1 and a continuous, positive function�(y0), such that2
y ∈ (y0) �⇒ supt
‖yt‖�t
� �(y0),
x ∈ (y0) �⇒ supt
‖xt‖�t
� �(y0).
Observe that for any feasible pair of action and state paths(x, y) we havex, y ∈ X(�,M)withM = �(y0).
2 Although this assumption is not explicitly imposed by Duran[6], it is naturally satisfied by a large class ofrecursive problems that arise in economics. For instance, letx, y denote the feasible consumption and capital pathsin an optimal growth model. It is natural to assume that bothx, y have the same bound on their growth rates. Ifnot, there will be a disequilibrium in the long run.
C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209 193
Duran further requires|W(x,0)| � h(x)‖R0‖h, where‖R0‖h = supx∈D|W(x,0)|h(x)
=m< + ∞, h : D → R++ is a continuous function, andD is the set of feasible one periodaction plans. The link between this upper bound to one period returns and technology isgiven by some other continuous functiong : Rn → R++, such thath(x) � g(y) for anyx feasible fromy. Along any feasible accumulation pathy (i.e. along anyy with yt+1 ∈�(yt )) the functiong is assumed to satisfy� supyt+1∈�(yt )
g(yt+1)g(yt )
� �<1. It follows that∀x, y ∈ X(�,M), ∀N � 0,
�N |W(xN,0)| �m�Nh(xN)�m�Ng(yN)
�m�g(yN)
g(yN−1)× · · · × �
g(y1)
g(y0)g(y0).
= �(�)N�(M,�),
where�(�) = � and�(M,�) = mmax{g(y0) : ‖y0‖ � M}.There is a broad class of aggregators that verify assumption (W5). Clearly (W5) is satisfied
whenW is nonpositive. Another application is to aggregators that satisfy 0� W(x,0) �A ‖x‖� + B, with A andB be nonnegative constants,�>0 and���<1. In this case,∀x ∈ X(�,M), ∀N � 0,
�NW(xN,0)�A�N ‖xN‖� + �NB
�(���)N
(AM� + B).
(8) We mentioned above that our analysis encompasses undiscounted or upcounted mod-els (� � 1) provided that one imposes a stronger requirement in (W5). We introduce thisassumption below:
(W5′) W satisfies assumption(W5)and there exists at least onex ∈ X(�,M) such that foranyN � 0,
�N |W(xN,0)| � �(�)N�(M,�).
There is a broad class of aggregators that verify this assumption. Clearly, (W5) implies(W5′) for aggregators that satisfyW(x,0) � 0. Another application is to aggregators thatsatisfy−bx� � W(x,0) � 0, whereb>0, �<0 and���<1. In this case��<1, so onecan permit� � 1. Consider the sequencex = (M,�M, . . . ,�tM, ...). For anyN � 0,
�N |W(xN,0)| = �N∣∣∣W(�NM,0)
∣∣∣ � (���)NM�b.
(9) Assumption (W6) implies the existence of an action planx ∈ X(�) such thatU0(x)> − ∞.
194 C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209
2.2. The existence of recursive utility
Lemma 1. Let � � 1 andM>0. Assume(W1), (W2), (W4)and (W5). Then, for anyx ∈ X(�,M):(i) lim
N→∞ RN0(x) exists inR ∪ {−∞}.(ii) If �<1, then for anyy ∈ R,
limN→∞ W(x0,W(x1, . . . ,W(xN, y)...)) = lim
N→∞ RN0(x).
(iii) Assume also(W3).Then, limN→∞ RN0(x) is uniformly bounded from above.
Proof. (i) Let x ∈ X(�,M). Assume first thatxt ∈ D for somet . In this case, (W1) impliesthatW(x0,W(x1, . . . ,W(xN,0)...)) = −∞ ∀N � t . Therefore,
C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209 195
Remark 1. We know that for everyx ∈ X(�), there existsM>0, such that,x ∈ X(�,M).Based on the previous lemma, we conclude thatU0(x) = lim
N→∞RN0(x) is always mean-
ingful onX(�).
Proposition 1. Let� � 1.Assume(W1)–(W5).Then:
(i) The return functionU0 is a fixed point of the recursive operatorR.(ii) For anyx ∈ X(�), lim
N→∞ �N [U0(SNx)]+ = 0.More precisely, for anyM>0, for any
ε >0, there exists an integerN0, such that, for anyx ∈ X(�,M), for anyN � N0,wehave�N [U0(SNx)]+ � ε.
(iii) LetM>0.The return functionU0 is upper semicontinuous onX(�,M) for the producttopology.
(iv) The return functionU0 is �-upper semicontinuous onX(�).
Proof. (i) Let x ∈ X(�). Observe that ifx0 ∈ D or U0(Sx) = −∞, (W1) impliesW(�x, U0(Sx)) = −∞ = U0(x). Otherwise, the continuity ofWwith respect to its secondargument (assumption (W4)) implies
U0(x)= limN→∞ W(x0,W(x1, . . . ,W(xN,0)...))
=W(x0, limN→∞ W(x1, . . . ,W(xN,0)...))
=W(�x, U0(Sx)).
(ii) Let x ∈ X(�). If xt ∈ D for somet , then[U0(SNx)]+ = 0 ∀N � t and the claim istrue. Assume thatxt /∈ D ∀t � 0. Assumptions (W2), (W4) imply that
Since�(�)<1, for everyε >0, one can always findN0, such that, ifN � N0, then�(�)N
1−�(�)�(M,�) � ε. It follows that limN→∞ �N [U0(SNx)]+ = 0.
(iii) Let (xn) ∈ X(�,M) be a sequence, such that,xn → x ∈ X(�,M) for the producttopology. We distinguish three cases.Case1: Assume thatxnt /∈ D ∀n, ∀t � 0.
196 C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209
Fix N and chooset >N . From (W2), (W4) and (W5) we obtain
Case2: Assume that for anyn, there existst , such that,xnt ∈ D.
C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209 197
In this case,
lim supnU0(xn) = −∞ � U0(x).
Case3: Assume that there exists a subsequence(nk), such that, for anyk, there existstwith xnkt ∈ D.
Observe thatU0(xnk ) = −∞. Let(nk′)be a subsequence such thatxnk′t /∈ D ∀k′, ∀t � 0.
We know (see Case 1) that lim supk′ U0(xnk′ ) � U0(x). It follows that
lim supnU0(xn) = lim sup
k′U0(xnk′ ) � U0(x).
(iv) Recall thatU0 is meaningful onX(�). Let (xn) ∈ X(�) be a sequence such thatxn → x ∈ X(�) for the �-topology. Forn large enough, there existsM>0, such that,xn ∈ X(�,M) andx ∈ X(�,M). Given statement (iii), the claim is true. �
We have the following theorem.
Theorem 1. Let� � 1.Denote byH the set of upper semicontinuous functions onX(�)that satisfy: If U ∈ H ,
(a) there exists at least onex ∈ X(�) such thatU(x)> − ∞(b) lim
N→∞ �N [U(SNx)]+ = 0∀x ∈ X(�).
Assume(W1)–(W6)and impose�<1 in (W4).Then, the return functionU0 defined in(2.1)is the only fixed point of the recursion operatorR in H .
Proof. LetU be another�-upper semicontinuous recursive utility function that satisfies (a)and (b). Letx ∈ X(�). If xt ∈ D for somet , thenU(x) = U0(x) = −∞ and the claim istrue. Assume thatxt /∈ D ∀t � 0. In this case, (W2) and (W4) imply
198 C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209
The upper semicontinuity ofU together with Lemma 1(ii) imply
U(x)� lim supN
U(xN)
= limN→∞ W(x0,W(x1, . . . ,W(xN−1, y)...))
= limN→∞ W(x0,W(x1, . . . ,W(xN−1,0)...))
=U0(x).
We conclude the proof. �
Remark 2. Since Theorem 1 requires�<1 in (W4) , it is not applicable to undiscountedor upcounted models. In Theorem 2 (see below) we show that our uniqueness result is stillvalid in cases where� � 1, provided that (W5) is replaced by (W5′). Observe that (W5′)implies (W6). Indeed, under (W5′), there existsx ∈ X(�,M), such that, for anyN � 0 thefollowing relation is true:
∣∣ � 11−�(�)�(M,�), in which case (W6) is verified. In addition,
limN→∞ �N
∣∣U0(SNx)∣∣ = 0. We have the following result.
Theorem 2. LetH ′ denote the set of functions U that belong to H and in addition satisfy:
∃ x ∈ X(�) such that limN→∞ �N |U(SNx)| = 0.
Assume(W1)–(W5′).Then, the return functionU0 defined in(2.1) is the only fixed point ofthe recursion operatorR in H ′.
Proof. Let U be another�-upper semicontinuous recursive utility function inH ′. Letx ∈ X(�). If xt ∈ D for somet , thenU(x) = U0(x) = −∞ and the claim is true. Assumethatxt /∈ D ∀t � 0. The proof ofU(x) � U0(x) is identical to the one presented in Theorem1. We next show thatU(x) � U0(x).
SinceU belongs toH ′, there existsx′ ∈ X(�), such that, limN→∞ �N∣∣U(SNx′)
∣∣ = 0.LetM be such that bothx andx′ belong toX(�,M). Consider the sequence(xN) defined asfollows:xNt = xt , for t = 0, . . . , N−1 andxNt = x′
t , for t � N . Note that(xN) ∈ X(�,M)andxN → x for the product topology. We have
There are many aggregators studied previously in the literature that verify our sufficientconditions (assumptions (W1)–(W6) or (W1)–(W5′)). Consider the following example.3
Example 1 (Uzawa–Epstein–Heynes aggregator). Consider the aggregatorW(x, z) =(u(x) + z)e−v(x), x ∈ R+, z ∈ R. The functionsu andv are assumed to be continuous,satisfyingv � 0, v′>0 andu<0, u′>0.
Assumptions (W1)–(W3) and (W5) are obviously satisfied. Whenv(0)>0, (W4) issatisfied for� = e−v(0) <1. It is also easy to see that assumption (W6) is satisfied. Takex ∈ X(�) with xt /∈ D ∀t � 0. We have
limN→∞ RN0(x)=
∞∑t=0
u(xt )exp[−t∑
=0
v(x )]
>
∞∑t=0
u(0)exp[−t∑
=0
v(0)].
= u(0)
1 − e−v(0) .A special case appears whenu(x) = −1 andv(x) = x. Observe that in this casev(0) = 0,so (W4) is satisfied for� = 1. It is also easy to see that (W5′) is verified. Indeed, for
3 For additional examples the interested reader may refer to our corresponding discussion paper[12].
200 C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209
Remark 3. In the above example the aggregator function satisfies our sufficient conditions(assumptions (W1)–(W6) or (W1)–(W5′)). Our existence and uniqueness results applydirectly in such cases, giving rise to a unique recursive utility functionU0 defined as in(2.1). Consider now a monotonic transformation� to this recursive utility function. Thereare three important questions that naturally arise. First, does the new utility functionU =� ◦ U0 have a recursive representation? Or equivalently, does exist an aggregator functionW such thatU(x) = W (x, U(Sx)) ∀x ∈ X(�)? Second, given thatU0 is unique, doesUis the only recursive utility function associated with the aggregatorW? Finally, given thatan aggregatorW exists, does it satisfy our sufficient conditions?
The answer to the first question is immediate. SinceU = � ◦ U0, it follows that
U (x) = � ◦W(x,�−1 ◦ U (Sx)).
Hence, the new aggregator is given by
W (x, z) = � ◦W(x,�−1 ◦ z).
The answer to the second question is also obvious. Since� is one-to-one andU0 is theunique recursive utility associated withW, it follows thatU is unique.
The answer to the third question is not always an affirmative one. Below we display twoexamples where the functionU = � ◦ U0 is associated with an aggregatorW whichmayormay not satisfy our sufficient conditions.
Example 2. Consider the aggregatorW(x, z) = x� + �z, x ∈ R+, z ∈ R+, with �, � ∈(0,1).
Provided that���<1, this aggregator function satisfies assumptions (W1)–(W6). The
associated recursive utility function is given byU0(x) =∞∑t=0
�t x�t . Consider the following
monotonic transformation for this recursive utility function:
�(z) = z1� , �>0.
In this case,U(x) = U0(x)1� and
W (x, z)= � ◦W(x,�−1 ◦ z)= [x� + �z�] 1
� .
C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209 201
When� = �, W has the CES form. Aggregators of this form have been used in stochasticmodels to generate a form of nonexpected utility (see[9]). One can easily check that forthis aggregator function assumption (W4) is violated.4
Example 3. Consider the aggregatorW(x, z) = x�
� + z, x ∈ R+, z ∈ R, with �<0.One can easily check thatWsatisfies assumptions (W1)–(W5′). The associated recursive
utility function is given byU0(x) =∞∑t=0
x�t
� . Consider the following monotonic transforma-
tion for this recursive utility function:
�(z) = ez + 1.
Observe that� ◦ U0 = eU0(x) + 1 is associated with the aggregator,
W (x, z)= � ◦W(x,�−1 ◦ z)= (−1 + z)e x
�� + 1.
One can easily check thatW satisfies assumptions (W1)–(W5′).
3. Dynamic programming with recursive utility 5
Consider an economy where at each periodt there exists a vector of capital stocks on hand,denoted bykt ∈ R
p+. There arem consumption goods, all consumed or freely disposed of in
the period they are produced. Letct ∈ Rm+ denote the vector of these consumption goods inevery period. Technological possibilities are described by a correspondence� : R
p+ → R
p+.
An accumulation path is a sequencek = (kt )∞t=0.Whenk satisfieskt+1 ∈ �(kt ) for all t � 0,
we say that it is feasible from some initial conditionk0. Let (k0) denote the collection ofall feasible fromk0 capital sequences. Feasible consumption sequences are described by acorrespondence� : R
p+ × R
p+ → Rm+. A consumption sequencec = (ct )
∞t=0 is said to be
feasible from somek0 when there is at least onek ∈ (k0), such that,ct ∈ �(kt , kt+1) forall t � 0. Let(k0) denote the collection of all feasible consumption sequences fromk0.
We next specify the assumptions imposed on preferences and technology.Preferences are assumed to be represented by a utility functionU that is assumed to be
recursive, generated by an aggregatorW that satisfies the assumptions (W1)–W6) of theprevious section (in the case of upcounted or undiscounted models assumption (W5) has tobe replaced by assumption (W5′)). Define
U : c ∈ (k0) → U(c) = limN→∞ W(c0,W(c1, . . . ,W(cN,0))...)).
4 The Lipschitz bound for the aggregatorW is given by�′ = supx,z W2(x, z). It is easy to check that when�<1, �′ = ∞.
5Although in this section we borrow some terminology from optimal growth theory, this is only for expo-sitional purposes. It should be clear that all results apply to a broader class of recursive problems.
202 C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209
Regarding technology, the following assumptions are typically made in this context:
(T1): The correspondence� is nonempty, compact-valued and continuous for anyk ∈ Rp+.
(T2): The correspondence� is nonempty, compact-valued and continuous for any(k, y) ∈Rp+ × R
p+
(T3): There exists� � 1 and a continuous, positive function�(k0), such that, for anyk0 ∈ R
p+,
k ∈ (k0) �⇒ supt
‖kt‖�t
� �(k0),
c∈ (k0) �⇒ supt
‖ct‖�t
� �(k0).
Observe that∀k ∈ (k0), ∀c ∈ (k0) we have(k, c) ∈ X(�,M) withM = �(k0).
Lemma 2. Assume(T1)–(T3).Then
(i) For everyk0 ∈ Rp+, (k0) and(k0) are compact for the product topology.
(ii) The correspondences and are upper hemicontinuous for the product topology.
Proof. It is standard. �
3.1. Value function-Bellman equation
Given some initial conditionk0 ∈ Rp+, we define the value function as
V (k0) = maxc∈(k0)
U(c). (3.1)
A planc is said to be optimal fromk0 whenc ∈ (k0) andV (k0) = U(c).A solution tothe Bellman equation can be seen as a fixed point of the maximizing operator
Tf (k) = max{W(c, f (y)) : c ∈ �(k, y), y ∈ �(k)}, (3.2)
over all extended real-valued functionsf on Rp+.
There is a close connection between the solution of the Bellman equation and the valuefunction. In what follows we study this connection. In particular, we would like the valuefunction to solve the Bellman equation and a feasible program(k, c) to be optimal, if andonly if, it verifiesV (kt ) = W(ct , V (kt+1)) ∀t � 0.
Proposition 2. Assume(T1)–(T3)and(W1)–(W6).Then
(i) There exists an optimal solution.(ii) The value function V is upper semicontinuous.
(iii) The value function V satisfies the Bellman equation, i.e.,
∀k ∈ Rp+, V (k) = max{W(c, V (y)) : c ∈ �(k, y), y ∈ �(k)}.
C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209 203
Proof. (i) Recall thatU is upper semicontinuous for the product topology onX(�,M),whereM = �(k0). Since(k0) is compact, the claim is true.
(ii) SinceU(c) may be equal to−∞ for somec, we cannot use the maximum theoremto prove the statement. For that reason, we provide a direct proof.
Let (kn0) be a sequence that converges tok0. Let (knm0 ) be a subsequence such thatlim supn V (k
n0) = limm V (k
nm0 ). From (i), it follows that for everyknm0 , there exists a
consumption plancnm ∈ (knm0 ), such that,
V (knm0 )=U(cnm0 , c
nm1 , c
nm2 , ...)
=W(cnm0 , U(Scnm)).
SinceWsatisfies assumptions (W2) and (W4), we have
W(cnm0 , U(Sc
nm))=W(cnm0 ,W(cnm1 , . . . ,W(c
nmN−1, U(S
Ncnm))...))
�W(cnm0 ,W(cnm1 , . . . ,W(c
nmN−1, U
+(SNcnm))...))� �NU+(SNcnm)+W(cnm0 ,W(c
nm1 , . . . ,W(c
nmN−1,0)...)).
Since is upper hemicontinuous, the subsequence(knm0 ) can be chosen such that the
corresponding consumption sequence(cnm) converges to somec ∈ (k0). In this case,assumptions (W3) and (W4) imply6
C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209 205
Remark 4. It is obvious that the function which is equal to−∞ on Rp+ is a solution to
the Bellman equation. But observe that such a solution does not belong to the setF since itclearly violates condition (b).
Theorem 3. Assume(T1)–(T3)and(W1)–(W6).Then, the value function V is the uniquesolution in F to the Bellman equation.
Proof. Let V ∈ F be another solution to the Bellman equation.We first show thatV (k0) � V (k0) ∀k0 ∈ R
p+. Assume first that′(k0) = ∅. In this case,
V (k0) = −∞ or equivalently,U(c) = −∞ ∀c ∈ (k0). We claim thatV (k0) = −∞.Assume the contrary. Then, there existsk ∈ (k0) andc ∈ (k0) with ct /∈ D ∀t � 0,such that
Taking the limits on both sides asT → ∞, givesV (k0) � U(c). By the definition ofV ,we haveV (k0) � U(c). It follows that(k, c) is an optimal program.
The proof of the reverse way is standard.�
3.2. Algorithm to find the value function
LetTbe the mapping which associates with any u.s.c functionf from Rp+ into R ∪{−∞}
the u.s.c. functionTf defined as follows:
Tf (k) = max{W(c, f (y)) : c ∈ �(k, y), y ∈ �(k)}.
Theorem 5. Assume(T1)–(T3)and(W1)–(W6).Then
(i) T maps F into F.(ii) For everyk0 ∈ RP+,V (k0) = limn T
nf (k0),where f is any function inF . In particular,V (k0) = limn T
n0(k0).
Proof. (i) It is clear thatTf is upper semicontinuous. Let us show thatT mapsF into F.We first show thatTf satisfies property (a) in the definition ofF.
Sincef is upper semicontinuous, for everyt � 0, there existsct ∈ �(kt , kt+1), kt+1 ∈�(kt ), such that,Tf (kt ) = W(ct , f (kt+1)). Observe that
W(ct , f (kt+1)) � W+(ct ,0)+ �f+(kt+1).
We have
�t (Tf )+(kt )� �tW+(ct ,0)+ �t+1f+(kt+1)
� �(�)t�(M,�)+ �t+1f+(kt+1).
Since�(�)<1 andf satisfies property (a), it follows that, givenε >0, �(�)t�(M,�) +�t+1f+(kt+1) � ε, for anyt large enough.
We next show thatTf satisfies property (b) in the definition ofF. Take(k, c), such that,k ∈ ′(k0) andc ∈ (k0) with ct /∈ D ∀t � 0. We have:
W(c0,W(c1, . . . ,W(ct−1,W(ct , f (kt+1))...))
−W(c0,W(c1, . . . ,W(ct−1,0)...))
C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209 207
� W(c0,W(c1, . . . ,W(ct−1, Tf (kt ))...))
−W(c0,W(c1, . . . ,W(ct−1,0)...))
� �t (Tf )+(kt ).Taking the limits on both sides we get
(ii) Let k0 ∈ Rp+ and denote byl a cluster point of{T nf (k0)} : l = lim� T
�f (k0). Wefirst show thatl � V (k0).
If ′(k0) = ∅, thenV (k0) = −∞, which in turn impliesl � V (k0).Assume′(k0) �= ∅.Take(k, c), such thatk ∈ ′(k0) andc ∈ (k0) with ct /∈ D ∀t � 0. Observe that
T �f (k0) � W(c0,W(c1, . . . ,W(c�−1, f (k�))...)).
Sincef satisfies property (b), we getl � U(c). Thus,l � V (k0).Let us show thatl � V (k0). SinceT �f is u.s.c., there existsk�1 ∈ �(k0), k�2 ∈ �(k�1), . . . ,
k�� ∈ �(k��−1) andc�0 ∈ �(k0, k�1), c
�1 ∈ �(k�1, k
�2)..., c
��−1 ∈ �(k��−1, k
��), such that
T �f (k0) = W(c�0,W(c�1, . . . ,W(c
��−1, f (k
��))...)).
Take a sequence(k��+1, k��+2, ...) ∈ (k��) and(c��, c
��+1, ...) ∈ (k��). The sequencek� =
(k0, k�1, . . . , k
��, k
��+1, ...) belongs to(k0), so we can assume that converges to some
208 C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209
k ∈ (k0). The corresponding consumption sequencec� = (c�0, c�1, . . . , c
��−1, c
��, ...)
belongs to(k0), so we can also assume that converges to somec ∈ (k0). Note that
T �f (k0)−W(c�0,W(c�1, . . . ,W(c��−1,0)...))
= W(c�0,W(c�1, . . . ,W(c
��−1, f (k
��))...))−W(c�0,W(c�1, . . . ,W(c��−1,0)...))
� W(c�0,W(c�1, . . . ,W(c
��−1, f
+(k��))...))−W(c�0,W(c�1, . . . ,W(c��−1,0)...))
� ��f+(k��).
We obtain
T �f (k0) � W(c�0,W(c�1, . . . ,W(c
��−1,0)...))+ ��f+(k��).
Fix N � 1 and choose�>N . A similar argument as in the proof of Proposition 1(iii) (seeCase 1) shows that
W(c�0,W(c�1, . . . ,W(c
��−1,0)...))
� W(c�0,W(c�1, . . . ,W(c
�N−1,0)...))+ ��−1W+(c��−1,0)+ ...+ �NW+(c�N,0)
� W(c�0,W(c�1, . . . ,W(c
�N−1,0)...))+
�(�)N
1 − �(�)�(M,�).
Sincef ∈ F , givenε >0, for any� large enough, we have��f+(k��) � ε.Let � → +∞. Assumptions (W3) and (W4) imply7
l � U(c)+ε.Since the above inequality holds for anyε >0, we havel � U(c) � V (k0). �
Acknowledgments
We are grateful to Rabah Amir for reading a first draft and for making valuable sugges-tions. Thanks are also due to Raouf Boucekkine and Jorge Duran for additional helpfuldiscussions. This paper has benefited substantially from the comments of an associate ed-itor and two anonymous referees. Yiannis Vailakis acknowledges the financial support ofthe Belgian Ministry of Scientific Research (Grant ARC 99/04-235, “Growth and IncentiveDesign”) and of the Belgian Federal Government (Grant PAI P5/10, “Equilibrium Theoryand Optimization for Public Policy and Industrial Regulation”).
7 The argument is analogous to the one we have used in the proof of Proposition 1(iii).
C. Le Van, Y. Vailakis / Journal of Economic Theory 123 (2005) 187–209 209
References
[1] F. Alvarez, N. Stokey, Dynamic programming with homogeneous functions, J. Econ. Theory 82 (1998)167–189.
[2] R.A. Becker, J.H. Boyd III, Capital Theory, EquilibriumAnalysis and Recursive Utility, Blackwell Publishers,Oxford, 1997.
[3] J.H. Boyd III, Recursive utility and the Ramsey problem, J. Econ. Theory 50 (1990) 326–345.[4] R.A. Dana, C. Le Van, Structure of Pareto optima in an infinite-horizon economy where agents have recursive
preferences, J. Optim. Theory Appl. 64 (1990) 269–291.[5] R.A. Dana, C. Le Van, Optimal growth and Pareto optimality, J. Math. Econ. 20 (1991) 155–180.[6] J. Duran, On dynamic programming with unbounded returns, Econ. Theory 15 (2000) 339–352.[7] J. Duran, Discounting long run average growth in stochastic dynamic programs, Econ. Theory 22 (2003)
395–413.[8] L.G. Epstein, Stationary cardinal utility and optimal growth under uncertainty, J. Econ. Theory 31 (1983)
133–152.[9] L.G. Epstein, S.E. Zin, Substitution, risk aversion and the temporal behavior of consumption and asset returns:
a theoretical framework, Econometrica 57 (1989) 937–969.[10] T.C. Koopmans, Stationary ordinal utility and impatience, Econometrica 28 (1960) 287–309.[11] C. Le Van, L. Morhaim, Optimal growth models with bounded or unbounded returns: a unifying approach,
J. Econ. Theory 105 (2002) 158–187.[12] C. Le Van, Y. Vailakis, Recursive utility and optimal growth with bounded or unbounded returns, CORE
Discussion Paper 2002/55.[13] R.E. Lucas Jr., N. Stokey, Optimal growth with many consumers, J. Econ. Theory 32 (1984) 139–171.[14] J.P. Rincon-Zapatero, C. Rodriguez-Palmero, Existence and uniqueness of solutions to the Bellman and
Koopmans equations in the unbounded case, Working Paper, University of Valladolid, 2002.[15] J.P. Rincon-Zapatero, C. Rodriguez-Palmero, Existence and uniqueness of solutions to the Bellman equation
in the unbounded case, Econometrica 71 (2003) 1519–1555.[16] N. Stokey, R.E. Lucas Jr., E.C. Prescott, Recursive Methods in Economic Dynamics, Harvard University
Press, Cambridge, MA, 1989.[17] P.A. Streufert, Stationary recursive utility and dynamic programming under the assumption of biconvergence,
