Recursive competitive equilibrium in dynamic stochastic ...allocations in a representative agent dynamic stochastic production economy, where the transition probabilities between states

Recursive competitive equilibrium in dynamic stochastic economieswith endogenous risk

Lukasz BalbusInstitute of Mathematics, Wrocaw University of Technology, Wrocaw, Poland

email: [email protected]

Kevin ReffettDepartment of Economics, WP Carey School of Business, Arizona State University


Lukasz WoznyDepartment of Theoretical and Applied Economics, Warsaw School of Economics, Warsaw, Poland. Address: al.

Niepodleglosci 162, 02-554 Warszawa, Poland


We show how recursive competitive equilibrium of Prescott, Mehra [35] can be defined to decentralize optimalallocations in a representative agent dynamic stochastic production economy, where the transition probabilitiesbetween states are endogenous. We characterize and prove existence of such equilibrium and hence extend Magill,Quinzii [21] model of a two-period stochastic production ’stakeholders’ economy, to recursive equilibrium withinfinite horizon and uncountable state space.

Key words: recursive competitive equilibrium; stochastic transition; endogenous risk

MSC2000 Subject Classification: Primary: 91B50; Secondary: 49J30, 52A41

OR/MS subject classification: Primary: Economics; Secondary: Mathematics: convexity

1. Introduction Since the seminal papers of Mirman [26], Brock, Mirman [8, 9], and Mirman,Zilcha [27, 29] economists studied optimal allocations in the infinite horizon stochastic economies. Theapplicability of this class of models is wide and includes growth economies, optimal investment andinventory management, labor market search and asset pricing (see also Stokey, Lucas and Prescott [40],chapter 10) among many others.

Simultaneously, many studied dynamic macroeconomic models in the presence of limited commitmentamong economic agents. Such limited commitment frictions arise naturally in diverse fields of economics,with examples including work in: models of stochastic growth without commitment international lendingand sovereign debt, optimal Ramsey taxation or models of savings and asset prices with hyperbolic dis-counting. The fundamental tool, for analyzing such macroeconomic problems with limited commitment,is a class of stochastic games, first introduced by Shapley [39]. From a technical perspective the classof games recently applied is an infinite horizon, stochastic game with (stochastically) convex transitionover uncountable number of states, while the equilibrium concept applied is Markov Stationary Nashequilibrium (MSNE, henceforth).

Both strands of literature (that on optimal decisions in economies and on equilibria in stochasticgames) supported contention that stochastic modeling in economics is more than a mere extension ofthe deterministic realm. The key difference, between both literatures, at least to the motivation of thispaper, lays in the tools used to specify uncertainty. The first one models uncertainty as a random variablei.e. as a map from a state space to the real line, while the other as a probability distribution it induceson the state/outcome space. In the former approach probabilities of states (typically not outcomes) areexogenous, while in the latter probabilities of states/outcomes are endogenous. Technically it can beshown that both descriptions are equivalent, i.e. one can be represented by the other. The assumptionsneeded to guarantee existence of optimal decisions or equilibria (in economies and most importantlyin games), however, are typically weaker for the probability distribution approach. The reason is thatprobability distribution approach uses all the convexity-type properties of the transition probability. Tothe best of our knowledge there have been only few studies that tried to analyze the relations betweenboth methods1. The one that discusses its implications and advocates for a use of probability distributionapproach in modeling optimal allocations in dynamic economies is Amir [3]. He presents examples that

1See also [24] and [15].

1

mailto:[email protected]






2 Balbus, Reffett, Wozny: RCE and endogenous riskMathematics of Operations Research xx(x), pp. xxx–xxx, c©200x INFORMS

compare both approaches in the context of a infinite horizon stochastic growth model and prove usefulresults (e.g. optimal policy smoothness) concerning optimal policies that are hard to obtain under thestandard, random variable toolkit. See also a paper by Karatzas, Sudderth [18]. Recent applicability ofstochastic games to macroeconomic problems also advocates for a thoroughly understanding of relationof both methods in modeling uncertainty.

The applicability of both kind of approaches to model dynamic (macro)economies with risk requires,however, definition and existence of appropriate prices decentralizing efficient or equilibrium allocations.For the random variable approach it has been successfully accomplished in a number of papers, see Peleg[32], Mirman, Zilcha [28] Prescott, Mehra [35] for example. To the best of our knowledge appropriateprices for dynamic economies were not defined for the probability distribution approach, yet. To ac-complish this task, one needs to answer two immediate questions: (i) how to model asset prices in aneconomy with stochastic transition modeled using probability distribution approach and (ii) how to proveexistence of prices taking into account the infinite dimensional nature of the economy (infinite horizonand uncountable number of states). Answering these two questions is necessary, though not sufficient tobring the probability distribution approach to the current macroeconomic research toolkit.

As far as the first is concerned, the first steps toward such decentralization has been already made byMagill and Quinzii [21] or Magill, Quinzii and Rochet[22] in a context of a two-period finite number ofstates economy2. Although the motivation of both papers is focused on studying so called ”stakeholder”equilibrium under incomplete markets, their tools (as we show in the paper) can prove useful the pricing ofoptimal allocations of Amir’s [3] stochastic growth economy. Specifically in this paper we build on Magilland Quinzii [21] idea, and generalize their equilibrium concept to a recursive equilibrium (henceforthRCE) of an infinite horizon economy and (to restore efficiency and information transition through prices)infinite dimensional state space.

Secondly, we need to prove the existence of a RCE with allocations in L∞ and prices in L1. Theproblem is far from trivial and involves choice of an appropriate topology as well as use of Hahn-Banahtheorem (see Bewley [7]). Alternatively one creates prices decentralizing core allocations (building onan idea of (Peleg and Yaari [33]) in Riesz spaces (Aliprantis, Brown, Burkinshaw [2], Mas-Colell, Zame[23]). In this paper we rather use Nigishi [30] approach (see also Jofre, Rockafellar and Wets[16]), whereequilibrium prices are Lagrange multiplier of the appropriate constrained optimization problem (see alsoPapageorgiou [31] and LeVan and Saglam [19]). To show existence of such Lagrange multipliers in infinitedimensional space we use Rockafellar and Wets [37] theorem for stochastic convex programming.

Hence, the contribution of our paper is twofold. We first define appropriate equilibrium prices thatdecentralize optimal allocations of economy with endogenous risk and prove counterpart of the first welfaretheorem. Furthermore we prove existence of such an equilibrium. The rest of the paper is organized asfollows. In section 2 we define an economy under study and characterize its optimal allocation. Then insection 3 we define a RCE and state our main existence result. Section 4 concludes, while existence inthe optimal, stochastic growth section 5 contains proofs of all results in the paper.

2. Optimal allocation in dynamic stochastic economy In this section, following Amir [3], weconsider an economy with a representative household with preferences:

∞∑t=0

βtEt[u(ct, 1− lt)]. (1)

The feasibility constraints give: (∀t ≥ 0) ct + it = f(kt, lt), ct ≥ 0, lt ∈ [0, 1] with 1 > β > 0, where ctstands for consumption, it for investment, kt for capital, lt for labor supply, β denotes discount factor,while u, f denotes utility and production function respectively. The capital (shocks) space is S = [0, S],where S is a maximal attainable level of capital. The transition between states kt and st+1 is Markovand given by a distribution Q(·|it) (with density q(·|it)) as a function of current investment it. We letY = [0, f(S, 1)]. Assume household follows some stationary policy γ, and by Et denote expectationsover realization of state (a random capital level) s ∈ S each period with respect to a unique probabilitydistribution induced on the space of all histories for γ, given by Ionescu-Tulcea’s theorem.

2Prescott and Townsend [36] analysis of prices in private information economies can be also seen as contributing to this

discussion.

Balbus, Reffett, Wozny: RCE and endogenous riskMathematics of Operations Research xx(x), pp. xxx–xxx, c©200x INFORMS 3

We now state our assumptions. Let us mention: they are far from being necessary for the characteri-zation of the optimal solution but prove useful in our the decentralization specification.,

Assumption 2.1 Let:

• u : Y × [0, 1] → R is strictly increasing, strictly concave with respect to each argument, weeklyconcave jointly and continuously differentiable;

• function i →∫SV (s)dQ(s|i) is increasing, concave, continuously differentiable for any given

increasing, bounded and Borel measurable V : Y → R,

• f : S × [0, 1]→ Y is strictly increasing and strictly concave with each argument separately for apositive value of the second, weekly concave jointly, and continuously differentiable with constantreturns to scale,

• liml→1 u′2(c, 1−l) =∞, liml→0 u

′2(c, 1−l) = 0, liml→0 f

′2(k, l) =∞ as well as limc→0 u

′1(c, 1−l) =

∞, limc→S

u′1(c, 1 − l) = 0 and limi→0ddi

∫SV (s)dQ(s|i) = ∞ for a given increasing, bounded and

Borel measurable V : Y → R.

We first characterize the recursive solution to the problem of finding optimal allocations for thiseconomy. The problem is to

sup{ct,lt,it}∞t=0

∞∑t=0

βtEt[u(ct, 1− lt)], (2)

subject to constraints given above. Formulating the problem recursively we obtain the value function:

V (k) = maxc∈[0,f(k,l)],l∈[0,1]

{u(c, 1− l) + β

∫S

V (s)dQ(s|f(k, l)− c)}, (3)

where by standard arguments we know that V : S → R exists, is unique, continuous, concave andcontinuously differentiable (see appropriate theorems in [40]). Under assumptions 2.1, we obtain thefollowing necessary and sufficient condition for a unique, interior argmax c : S → Y and l : S → [0, 1] is:

u′1(c(k), 1− l(k)) = βd

di

∫S

V (s)dQ(s|f(k, l(k))− c(k)),

and

u′2(c(k), 1− l(k)) = βf ′2(k, l(k))d

di

∫S

V (s)dQ(s|f(k, l(k))− c(k)).

Using envelope theorem (see Benveniste, Scheinkman [6]) we conclude that:

V ′(k) = u′1(c(k), 1− l(k))f ′1(k, l(k)).

Let Q′(s|i) ≡ ∂∂iQ(s|i). Using lemma 5.1 from the section 5 we obtain:

u′1(c(k), 1− l(k)) = −β∫S

V ′(s)Q′(s|f(k, l(k))− c(k))ds,

= −β∫S

u′1(c(s), 1− l(s))f ′1(s, l(s))Q′(s|f(k, l(k))− c(k))ds, (4)

and

u′2(c(k), 1− l(k)) = f ′2(k, l(k))u′1(c(k), 1− l(k)). (5)

Conditions (4) and (5) (∀k ∈ S) are necessary and sufficient for the c, l to solve problem (2).

3. Recursive competitive equilibrium In this section we define a recursive competitive equi-librium for this economy. Our approach follows that of Prescott, Mehra [35] but takes a probabilitydistribution representation of uncertainty, generalizing Magill, Quinzii [21] stakeholder equilibrium to thecase of infinite horizon and infinite dimensional state space.


The economy consists of households and two types of firms: one producing consumption goods andsecond possessing (stochastic) capital accumulation technology.

There is a representative household solving:

sup{ct,lt,at}∞t=0

∞∑t=0

βtEt[u(ct, 1− lt)], (6)

where expectations Et with respect to the state s are given by a perceived law of motion Π(s|Kt)parameterized by current level of aggregate capital. As above ct, lt denote choices of consumption andleisure while at choice of Arrow securities. Let pt(s) be a price of an Arrow security, i.e. a commitment toreceive (or deliver) a unit of capital in the next period when state s is realized. The wealth accumulationprocess (budget constraint) is given by:

wtlt + rtkt + πt ≥ ct +

∫S

at(s)pt(s)ds,

where wt and rt stands for a price of labor and capital respectively. Household takes prices pt, profitsπt and aggregate states {Kt}Tt=0 as given. Note that each period we normalize prices of capital (andinvestment) goods to 1 and express prices for capital and Arrow-securities relative to consumption prices.

There is a representative consumption sector firm with deterministic technology given by a productionfunction f , which problem is to maximize profits:

maxkt≥0,lt≥0

f(kt, lt)− rtkt − wtlt.

Observe that since f is constant returns to scale profits of consumption sector firms are 0.

There is a representative investment sector firm transforming current physical goods into next periodcapital. Following Magill, Quinzii [21] we let investment sector firm to decide on investment it in astochastic technology yielding revenue Φ(s,Kt) in state s, i.e. revenue for a sale of s units of capital instate s. Firm maximizes expected profits:

maxit≥0

∫S

Φ(s,Kt)dQ(s|it)− it,

where Q(s|i) is a transition probability of a next period state s parameterized by current investment.Magill, Quinzii [21] shows that by a appropriate choice of a revenue function Φ we can obtain an optimalallocation in this decentralized economy. We will specify this function in the definition of a RCE.

Definition 3.1 A recursive competitive equilibrium of the economy under study is a list of functions(V,Φ,Π, c∗, a∗, l∗, l

∗, k∗, i∗, p∗, r∗, w∗) satisfying (for any k,K ∈ S):

(i) taking prices r∗(K), w∗(k), p∗(·,K) and profits π∗(K) as given c∗(k,K), l∗(k,K) and a∗(·, k,K)

solve:

maxc,l,a

u(c, 1− l) + β

∫S

V ∗(a(s), s)Π(s|K)ds, (7)

under the budget constraint w∗(K)l + r∗(K)k + π∗(K) = c +∫Sa(s)p∗(s,K)ds, c ≥ 0, l ∈

[0, 1], a(s) ≥ 0, where

V ∗(k,K) = {u(c∗(k,K), 1− l∗(k,K)) + β

∫S

V ∗(a∗(s, k,K), s)Π(s|K)ds},

(ii) taking prices as given k∗(K), l∗(K) solves:

maxk≥0,l≥0

f(k, l)− r∗(K)k − w∗(K)l,

(iii) taking p∗(·|K) and Π(·|K) as given i∗(K) solves:

maxi≥0

∫S

Φ(s,K)dQ(s|i)− i, (8)


where

(∀s ∈ S) Φ′1(s,K) =p∗(s,K)

Π(s|K)(9)

and we set

π∗(K) =

∫S

Φ(s,K)dQ(s|i∗(K))− i∗(K),

(iv) markets clear: k = k∗(K) = K, l∗(K) = l∗(K,K), f(K, l∗(K)) = c∗(K,K) + i∗(K), (∀s ∈

S) s = a∗(s,K,K),

(v) perceived and actual law of motion for capital coincide, i.e. (∀K ∈ S) Π(·|K) = q(·|i∗(K)).

Conditions 1,2,3 and 5 are standard (see [35]). Condition 4 for investment sector firm says thatfirm is maximizing expected profits implied by a revenue function Φ, which is defined implicitly by itsderivatives. To obtain the value of Φ firm must integrate an observed ratio of Arrow securities pricesp∗ over probabilities3 Π. Hence we may argue that, all the investment sector firm must observe toproperly indicate their revenue functions, are Arrow securities prices and probabilities q(s|I∗(K)). Otherjustifications for such a profit function for investment sector firms can be found in the work [22].

Observe that our definition directly generalize Magill, Quinzii [21] efficient equilibrium concept to theinfinite horizon, recursive equilibrium. The only technical difference lays in the fact that in our case thenumber of states is uncountable. Apart from the fact that such assumption is commonly used in macroapplications or stochastic games discussed in the introduction, it also plays an important economic role.To know their revenue function Φ investment sector firms must integrate ratio of Arrow-securities pricesand probabilities of tomorrow states assumed by consumers. If the number of states is finite, firms mayonly approximate their revenue functions and hence the equilibrium choice of i may not be optimal.The assumption of continuously many states S is hence critical for efficient information transition inequilibrium (see [21]).

Under the assumption that such an equilibrium exists let us now show its main property, i.e. a coun-terpart of the first welfare theorem.

Theorem 3.1 Assume 2.1 and let (V,Φ,Π, c∗, a∗, l∗, l∗, k∗, i∗, p∗, r∗, w∗) be a RCE. Then (∀K) V (K) =

V (K,K) solves (3) with c(K) := c∗(K,K) and l(K) = l∗(K) being the arguments maximizing the righthand side of (3).

Theorem 3.1 assures that RCE decentralize the optimal allocation to the problem (2) and hence isan appropriate definition of an equilibrium to adopt for the economy with the probability distributionrepresentation of uncertainty.

Intuition behind theorem is simple. Condition 9 require that marginal rate of substitution and trans-formation between periods are equal. Together with other standard conditions, it implies that a RCEallocation is optimal and hence yields the same value function and corresponding choices. The proof ispostponed to section 5.

We have showed so far, a possible scheme to decentralize optimal allocation in the recursive competitiveequilibrium. We have assumed however that appropriate prices exists in the dual to a commodity space.We now show that such prices exists indeed.

Theorem 3.2 Assume 2.1. Then RCE exists.

Our way of proving the above theorem is the following. As our state space S and Y are compact weconduct the proof for a given k,K. First we guess r∗, w∗,Π, i∗ and for a given continuation value V , weconsider a simple (one period) maximization problem under feasibility constraints, yielding the optimal

3Alternatively we may substitute Π(s|K) by q(s|I∗(K)), with I∗ being aggregate investments, indicating that a single

investment firm does not see its impact on relative prices pq

. In the equilibrium one would require that i∗(K) = I∗(K).


allocation c, l, a. Secondly using a version of Kuhn-Tucker theorem for infinite dimensional spaces weconclude the existence of Lagrange multipliers associated with this optimization problem. Third we showthat this Lagrange multipliers can be interpreted as prices (for Arrow-securities and consumption goods)in our one period economy with consumers, production sector firms and fixed i∗. Fourth we construct anoperator T on the space of value functions, and show it has a unique fixed point V ∗. To finish the proofwe claim that under the obtained prices, and V ∗, guessed i∗ is indeed solving a problem of investmentsector firms. The detailed proof follow in section 5.

4. Conclusions In this paper we formulated and proved existence of prices decentralizing optimalallocations in stochastic economy with endogenous risk. In this concluding section we specify and brieflycomment on, what remains to be done to bring stochastic game techniques to the current research toolkitof applied economists. Specifically to accomplish this task one needs to show: (i) how to decentralizeNash equilibrium allocations, rather than optimal ones and (ii) how methods develop in the paper canbe extended to the multi agent economies. We start with the former one.

The problem is not new. Economists decentralized Nash equilibria of strategic market games oranalyzed general equilibrium models with strategic interactions (see e.g. Dubey [10], Dubey, Shubik [11],Gale [12], Schmeidler [38] to name just a few). See also mechanism design literature (Hurwicz and Reiter[14]). In our case this problem can be also attacked using e.g. Lindhal price system. Moreover, in manymacroeconomic applications the representative agent (household/firm) allocation is of interest, henceone commonly analyze games with a single player each period. Observe, that although there is a singleagent in the economy each period, there can still be uncountably many players (one for each period) (seemodels of hyperbolic discounting [4] or paternalistic bequest models [5] for a reference) playing a dynamicgame. As a consequence, one needs also to think of internalizing externalities of the game played betweenperiods. But as stochastic game models applied in macroeconomics aim to describe a (macro) game(e.g. time consistency, self-control) one can leave it as such in decentralization (see Karatzas, Shubikand Sudderth [17], Geanakoplos, Karatzas, Shubik, Sudderth [13] or Phelan, Stacchetti [34] for similartreatment).

Concerning the second question. Before one proceeds to decentralize an allocation in an economy ora game, one needs to specify the stochastic technology Q. Here one can analyze the case, where e.g.every agent has access to technology Q transforming his investment ii into random capital tomorrow si.In such case the state of the economy is a n−dimensional vector (s1, . . . , sn) whose elements are drawnindependently. Alternatively one can consider an economy with public state s ∈ S and shared technol-ogy Q where individual investments ii are summed and determine the stochastic transition technologyQ(·|

∑i ii). More generally one can also analyze the case, where i→ Q(·|i) depends on the whole vector

of i’s, i.e. Q(·|i1, i2, . . . , in). All of these different formulation yield different optimal allocations. Havingstochastic technology given and decentralizing efficient allocation of the stochastic economy, one needsto appropriately define function Φ. In the case of multi-household economy Magill and Quinzii [21] show

that to obtain the optimal allocation one lets: Φc(s) = maxξ≥0

{∑iβiVi(ξi,s)u′i(ci)

:∑i ξi = s

}. That is, Φ is

defined as a sup-convolution of the n utilities functions βiVi weighted by marginal utilities u′i(ci). Suchformulation allows for optimal risk sharing among agents. Both issues needs further research, however.

5. Proofs To proceed with the proofs of the main theorems we state and prove the following lemmas.

Lemma 5.1 Let Φ : S → R be a continuously differentiable function and (∀s ∈ S)Q(s|·) : Y → [0, 1] be acontinuously differentiable function. If (∀i)Q(·|i) is a distribution on S then:

d

di

∫S

Φ(s)dQ(s|i) = −∫S

Φ′(s)Q′(s|i)ds.

Proof of lemma 5.1. Integrating by part we obtain4:∫S

Φ(s)dQ(s|i) = Φ(s)Q(s|i)|S0 −∫S

Φ′(s)Q(ds|i)ds, (10)

= Φ(S)−∫S

Φ′(s)Q(s|i)ds.

4See [3] or [21] for similar arguments.


By Assumption 2.1 ddiQ(s|i) is integrable, hence if we take a derivative in (10) w.r.t. i we obtain a thesis.

�

Now, let’s define

V := {V : R2 → R : v is a Caratheodory function and ∀K≥0v(·,K) is concave}.

For the next lemmas assume, without loss of generality, that S := [0, 1]. Let L∞(S) be a set of allreal valued bounded Borel measurable functions on S endowed with the weak-star topology. Moreover,let v ∈ V, Π(·|K) be a density, λ and p∗(s,K) numbers. Consider:

Jv(s, λ) := arg maxa∈[0,1]

{Pv(s, a, λ)} (11)

where

Pv(s, a, λ) := βv(a, s)Π(s|K)− λa p∗(s,K).

Lemma 5.2 There exist measurable functions aλ : S → [0, 1] and aλS → [0, 1] such that

Jv(s, λ) = [aλ(s), aλ(s)]. (12)

Proof of lemma 5.2. Observe that a → P (s, a;λ) is continuous and concave, hence a set ofmaximizers is nonempty, convex and closed. Thus, we have an existence of the functions aλ and aλsatisfying (12). We need to show that both are measurable in s. To do it observe that a(s) ∈ [0, 1]for all s, hence a constraint correspondence is weakly measurable. Moreover, P (·, ·, λ) is a Caratheodoryfunction. Hence by Measurable Maximum Theorem (18.19 in [1]) Jv(·, λ) is a measurable correspondence.Observe that

aλ(s) = maxa∈Jv(s,λ)

a,

hence again by Measurable Maximum Theorem ([1]) aλ is measurable. Similarly we prove that aλ ismeasurable. �

Lemma 5.3 Consider aλ and aλ from Lemma 5.2. Both aλ and aλ are decreasing in λ.

Proof of lemma 5.3. Clearly for given s the function P (s, a, λ) is supermodular in a and hasdecreasing differences in (a, λ). Hence by Theorem 6.2. in Topkis [41] aλ and aλ are decreasing in λ. �

Lemma 5.4 Let fn, gn, f, g ∈ L∞ and fn∗→ f and gn

∗→ g. Assume that for all n holds fn(s) ≤ gn(s)for L almost all s. Then f(s) ≤ g(s) almost everywhere.

Proof of lemma 5.4. Let E := {s : f(s) > g(s)}. Clearly 1E(s) is integrable. Thus

0 ≤∫E

(f(s)− g(s))ds =

∫E

f(s)ds−∫E

g(s)ds

=

∫S

f(s)1E(s)ds−∫S

g(s)1E(s)ds,= limn→∞

∫S

(fn(s)− gn(s))1E(s)ds,

= limn→∞

∫E

(fn(s)− gn(s))ds ≤ 0,

since fn ≤ gn for a.e. s. Thus L(E) = 0. �

Lemma 5.5 Let v ∈ V. Assume λn → λ, an∗→ a and an(s) ∈ Jv(s, λn). Then a(s) ∈ Jv(s, λ) for almost

all s.


Proof of lemma 5.5. Without loss of generality assume that λn is monotone increasing sequence.By Lemma 5.2 there are measurable functions aλ and aλ such that Jv(s, λ) = [aλ(s), aλ(s)]. Therefore,aλn

(s) ≤ an(s) ≤ aλn(s). By Lemma 5.3 we immediately obtain that aλnand aλn(s) are pointwise

convergent sequences, hence also converge in the weak-star topology. For given s ∈ S and v ∈ V Jv(s, ·)is u.h.c. by Berge Maximum Theorem. Hence

limn→∞

aλn(s) and lim

n→∞aλn(s) ∈ Jv(s, λ).

From Lemma 5.4 and definition of aλ and aλ we have

aλ(s) ≤ limn→∞

aλn(s) ≤ a(s) ≤ lim

n→∞aλn

(s) ≤ aλ(s).

for almost all s. Hence from Lemma 5.2 a(s) ∈ Jv(s, λ). �

Define:J∗v (λ) := {a ∈ L∞(S) : a(s) ∈ Jv(s, λ) for a.a. s ∈ S}.

Lemma 5.6 If λn → λ, an∗→ a and an ∈ J∗v (λ) then a ∈ J∗v (λ). In other words J∗v is u.h.c. correspon-

dence.

Proof of lemma 5.6. It immediately follows from Lemma 5.5. �

Lemma 5.7 Let F0 :n∏i=1

[0,mi]→ R and G0 :n∏i=1

[0,mi]→ R+. Assume that both are continuous functions

and F0 is increasing and G0 is decreasing in all arguments. Let κλ(x) := F0(x) + λG0(x), and xλ be anarbitrary maximizer of κλ. Then limλ→∞ xλ = 0 (where 0 is a zero vector in Rn).

Proof of lemma 5.7. On the contrary suppose that xλ do not converge to 0. Without loss ofgenerality assume that xλ > ε, where ε is some strictly positive vector. Then there are two possibilities

1. λ(G(xλ)−G(ε)) < F (ε)− F (xλ) for sufficiently large λ, or

2. xλ → ε.

Clearly case 1. contradicts definition of xλ. The second case reduce to the case 1, when we replace ε by12ε. �

Lemma 5.8 Let Θ and X be a convex subsets of a vector spaces. Let G : X 7→ R be bounded, concavefunction, and Z : Θ→ X be a correspondence such that

αZ(θ1) + (1− α)Z(θ2) ⊂ Z(αθ1 + (1− α)θ2)∀α ∈ [0, 1] (13)

and θi ∈ Θ i = 1, 2. ThenG(θ) := sup

x∈Z(θ)

{G(x)},

is a convex function.

Proof of lemma 5.8. Let ε > 0 be given. For arbitrary θi ∈ Θ, i = 1, 2 let xεi ∈ Z(θi) satisfyG(θi) ≤ G(xεi ) + ε. By (13) αxε1 + (1− α)xε2 ∈ Z(αθ1 + (1− α)θ2) for arbitrary α ∈ [0, 1]. Therefore

G(αθ1 + (1− α)θ2) ≥ G(αxε1 + (1− α)xε2)

≥ αG(xε1) + (1− α)G(xε2)

≥ αG(θ1) + (1− α)G(θ2)− ε.

Since ε is arbitrary positive value hence G is concave. �

Lemma 5.9 Let Γ : R+ → R be a (nonempty valued) u.h.c. correspondence. Assume additionally, thatthere exists λi > 0 and xi ∈ Γ(λi) i = 1, 2 such that x1x2 < 0. Then there exists λ0 such that 0 ∈ Γ(λ0).


Proof of lemma 5.9. On the contrary suppose that 0 /∈ Γ(λ) for all λ. Then

R = {λ : Γ(λ) ⊂ (0,∞)} ∪ {λ : Γ(λ) ⊂ (−∞, 0)} = Γ−1u (0,∞) ∪ Γ−1

u (−∞, 0),

where Γ−1u means upper inverse image of Γ. As Γ is upper hemicontinuous both Γ−1

u (0,∞) and Γ−1u (−∞, 0)

are open sets. Clearly both are disjoint as ∀λΓ(λ) 6= ∅. Clearly R is a connected set hence Γ−1u (0,∞) = ∅

or Γ−1u (−∞, 0) = ∅. This contradicts assumption of the existence of λi > 0 and xi ∈ Γ(λi) i = 1, 2 such

that x1x2 < 0. �

For given (k,K) ∈ S2 and functions w∗, p∗ let us define a correspondence:

Mu(λ) = arg max(c,l)∈S×[0,1]

{u(c, 1− l) + λ(w∗(K)l + r∗(K)k − c)} (14)

Proposition 5.1 Consider a Markov Decision Problem in (6) with given functions r∗, w∗, π∗, p∗5. Thenthere exists a stationary optimal policy c∗, l∗ and a∗. Moreover, if c∗, l∗ is arbitrary optimal solution then

(i) For all (k,K) ∈ S2 (c∗(k,K), l∗(k,K)) ∈ Mu(λ∗) and for all K ∈ S and a∗ ∈ J∗v∗(λ∗) and

λ∗ = u′1(c∗(k,K), 1− l∗(k,K))

(ii) For given K the function v∗(·,K) is differentable and

v∗′

1 (k,K) = u′1(c∗(k,K), 1− l∗(k,K))r∗(K).

Proof of proposition 5.1. Step 1. By standard arguments (see [40]) the value of the problemsatisfies:

v∗(k,K) = Tv∗(k,K) := sup(c,l,a)

u(c, 1− l) + β

∫S

v∗(a(s), s)q(s,K)ds, (15)

where (c, l, a) ∈ [0, 1]× [0, 1]× L∞[0, 1] satisfy the constraints

H(c, l, a) := r∗(K)k + w∗(K)l + π∗(K)− c−∫S

a(s)p∗(s,K)ds ≥ 0.

Moreover, v∗ = limn→∞

Tnv1 with v1 = 0. By Lemma 5.8 T maps V into itself. Since a limit preserves

all properties of V, v∗ ∈ V. Let k and K be given. By Luenberger [20] theorem problem 15 can berepresented by the saddle point problem of associate Lagrangian

L(c, l, a;λ) = u(c, 1− l) + β

∫S

v∗(a(s), s)q(s,K)ds+ λH(c, l, a).

Define

ML(λ) := arg maxc∈[0,1],l∈[0,1],a∈L∞(S)

Lv(c, l, a;λ),

and

Γ(λ) := {H(cλ, lλ, aλ(·)) : (cλ, lλ, aλ(·)) ∈ML(λ)}Note that ML(λ) = Mu(λ)×J∗v (λ). By Berge Maximum Theorem (Theorem 17.31 in [1]) Mu is u.h.c. ByLemma 5.6 J∗v is u.h.c., if we endow L∞(S) with a weak*-topology. Hence ML is u.h.c. with a respectiveproduct topology. Note that H is a continuous function (w.r.t. product of Euclidian topologies) in (c, l)and (w.r.t. weak*-topology) in a. We show that Γ is an upper hemicontinuous correspondence. Letλn → λ, yn ∈ Γ(λn) and yn → y. We show y ∈ Γ(λ). Since yn ∈ Γ(λn) hence yn = H(cn, ln, an(·)) forsome (cn, ln, an(·)) ∈ ML(λn). Clearly by Lemma 5.4 L∞([0, 1]) is a closed subset of the unit ball inL∞ with weak*-topology. Since by Banach-Alaoglu Theorem this unit ball is compact, hence L∞(S) is

compact. Therefore, without loss of generality we may assume that (cn, ln)→ (c0, l0) and an∗→ a0. Thus

(c0, l0, a0) ∈ML(λ) as we have mentioned before ML is u.h.c., and y = H(c0, l0, a0), since H is continuousw.r.t. a product of Euclidian topologies on R2 and weak*-topology on L∞(S). Hence y ∈ Γ(λ), whichimplies that Γ is u.h.c. Clearly, by Lemma 5.2 Γ is also nonempty set valued.

5That is rt = r∗(Kt), wt = w∗(Kt), πt = π∗(Kt), and pt(s) = p∗(s,Kt). Thus these values depend on the current

aggregate state only.


Step 2. We prove (i). First assume that H(c, l, a) > 0 over all (c, l, a) ∈ ML(λ). Then unique saddlepoint is ((c∗, l∗, a∗), λ∗) with λ∗ = 0, and c = 1, a ≡ 1 and l = 1. Then also c∗(k,K) = 1, l∗(k,K) = 0and a∗(s, k,K) ≡ 1. Clearly λ∗ = u′1(1, 0). Hence in this case (i) holds. Assume that H(c, l, a) < 0 forsome (c, l, a) ∈ ML(λ). We show that there exists (c, l, a) such that H(c, l, a) > 0. Let λ → ∞. FromLemma 5.7 that aλ(s)→ 0 for a.a. s and (cλ, lλ)→ (0, 1), where (cλ, lλ) is some selection of Mu(λ). Thisimplies that

H(cλ, lλ, aλ)→ r∗(K)k + w∗(K) + π∗(K) > 0.

Therefore, by Lemma 5.9 there exists λ∗ > 0 such that H(c∗, l∗, a∗) = 0 and (c∗, l∗, a∗) ∈ ML(λ∗). ByLuenberger [20] theorem (c∗, l∗, a∗) also solves the problem (2). This immediately implies (i). Clearlyλ∗ = u′1(c∗(k,K), 1− l∗(k,K)).

Step 3. We prove (ii). Observe that V ∗(k,K) is a value of the saddle point problem in the previousstep. By Milgrom and Segal (Theorem 4 in Milgrom and Segal [25]) V ∗ satisfies

V ∗(k,K) = V ∗(0,K) + r∗(K)

k∫0

λ∗(s,K)ds, (16)

with λ∗(s,K) := u′1(c∗(s,K), 1− l∗(s,K)). We need to show λ∗(·,K) is continuous. First of all we showthat the Lagrange multiplier is unique. On the contrary suppose there is some other Lagrange multiplierλ0. Observe that a formula in (16) do not depend on the choice of the saddle point. Thus

0 = r∗(K)

k∫0

(λ∗(s,K)− λ0(s,K))ds,

for all k. Therefore, λ∗(s,K) = λ0(s,K) for a.a. s. We show that the function λ∗(·,K) (from S ⊂ R intoR+) has a closed graph (w.r.t. Euclidian topology).

Let kn → k0 and λ∗(kn,K) → y. Let us denote Ln(c, l, a;λ) as a Lagrangian Lv∗(c, l, a;λ) in theprevious step, with k := kn and L as a Lagrangian Lv∗ with k := k0. Let (cn, ln) ∈ Mu(λn) andan ∈ J∗V ∗(λn) be chosen in such a way (cn, ln, an;λn) to be a saddle point of Ln. Since Mu is u.h.c. hence(c0, l0) ∈Mu(λ). J∗v is u.h.c. by Lemma 5.6 hence a0 ∈ J∗v∗(y). Clearly by [20] theorem λn satisfies

λn

r∗(K)kn + w∗(K)l + π∗(K)− cn −∫S

a(s, kn,K)p∗(s,K)ds

= 0.

Hence taking a limit above we obtain that y is a Lagrange multiplier in L0(c, l, a;λ). Hence λ∗(·,K) hasa closed graph and therefore is continuous. unique saddle point. (c, l) ∈ R2 and pointwise

SP (k) = {(k, c∗(k,K), l∗(k,K), a∗(k,K, ·), ) : k ∈ S}has closed graph, hence (c∗(k,K), l∗(k,K)) has closed graph for any fixed K. Therefore, as k ∈ S and Sis compact, hence Hence V ∗(·,K) is differentiable and by (16)

v∗′

1 (k,K) = u′1(c∗(k,K), 1− l∗(k,K))r∗(K).

�

Proof of of theorem 3.1. Let (V ∗,Φ∗,Π∗, c∗, a∗, l∗, l∗, k∗, i∗, p∗, r∗, w∗) be a RCE and K ∈ S

given. We need to show that (c∗(K), l∗(K)) := (c∗(K,K), l∗(K,K)) satisfies equations (5) and it is anoptimal value in the problem (6). By Proposition 5.1 we have

V ∗′

1 (k,K) = r∗(K)u′1(c∗(k,K), 1− l∗(k,K)).

Observe also that (K, l∗(K,K)) solves a maximization problem of the function (k, l)→ f(k, l)−r∗(K)k−w∗(K)l hence r∗(K) = f ′1(K, l∗(K,K)), w∗(K) = f ′2(K, l∗(K,K)) and if k = K then

V ∗′

1 (K,K) = u′1(c∗(K,K), 1− l∗(K,K))f ′1(K, l∗(K,K)). (17)

Clearly V ∗, (c∗(K), l∗(K), s) satisfies Bellman equations

V ∗(K,K) = maxH(c,l,a)≥0

u(c, 1− l) + β

∫S

V ∗(s)Π(s|K)ds

, (18)

= u(c∗(K), 1− l∗(K)) + β

∫S

V ∗(s, s)Π(s|K)ds. (19)


By Proposition 5.1 we obtain that (c, l) ∈ Mu(λ∗) and a(s) = s ∈ MV ∗(λ∗) for some λ∗ > 0. Thus

λ∗ = u′1(c∗(K), 1− l∗(K)) and

βV ∗′(K,K)Q′(s|i∗(K))− λ∗p∗(s,K) (20)

for a.a. s ∈ S. Thus

p∗(s,K) =βV ∗

′(s, s)Q′(s|i∗(K))

u′1(c∗(K), 1− l∗(K))= Φ′1(s,K)Q′(s|i∗(K)).

Now we show that (c∗(K,K), l∗(K,K)) solves problem 2. We show also that v(K) := V ∗(K,K) =K∫0

u′1(c∗(s), 1− l∗(s))f ′1(s, l∗(s))ds is an optimal value. In other words we need to show that for given K

the pair (c∗, l∗) solves maximization problem

U(c, l; v) := u(c, 1− l) + β

∫S

v(s)Q(ds|f(K, l)− c) (21)

subject to f(K, l)− c ≥ 0, c ∈ S, l ∈ [0, 1]. By Lemma 5.1 first order derivatives have a form

u′1(c, 1− l) + β

∫S

v′(s)Q′(s|i)ds = 0, (22)

−u2(c, 1− l)− β∫S

v′(s)Q′(s|i)f2(K, l)ds = 0, (23)

with i = f(k, l)− c. Put (c, l) = (c∗, l∗) in the left side in (22) and we show that it is actually 0.

u′1(c∗(K), 1− l∗(K)) + β

∫S

v′(s)Q′(s|i∗(K))ds =

u′1(c∗(K), 1− l∗(K)) + β

∫S

u′1(c∗(s), 1− l∗(s))f ′1(s, l∗(s))Q′(s|i∗(K))ds =

u′1(c∗(K), 1− l∗(K))

1 +

∫S

Φ′1(s,K)Q′(s|i∗(K))ds

=

u′1(c∗(K), 1− l∗(K))

1− d

di

∫S

Φ(s,K)Q(ds|i) |i=i∗(K)

= 0

(24)

We do the same in (23). Observe that by (24) the equation (23) is equivalent to

u′2(c∗(K), 1− l∗(K))− f ′1(K, l∗(K))u′1(c∗(K), 1− l∗(K)) = 0. (25)

The equation above is true since (c∗, l∗) ∈ Mu(λ∗) with λ∗ = u′1(c∗(K), 1 − l∗(K)). To finish the proofwe need to show that v(K) = U(c∗(K), l∗(K), v). Observe that maximization problem of U(·, ·, v) isequivalent to saddle point problem of the function U(·, ·, v) subject to f(K, l) − c − i ≥ 0, c, i ∈ S,l ∈ [0, 1] and λ ≥ 0. By Theorem 4 in Milgrom and Segal [25] we have

U(c∗(K), l∗(K), v) = maxc,l

U(c, l, v)

=

K∫0

u′1(c∗(s), 1− l∗(s))f ′1(s, l∗(s))ds

= v(K).

�


Proof of of theorem 3.2. We construct RCE knowing the solution to problem 2 (c, l) withoptimal payoff v. Let

r∗(K) := f ′1(K, l(K)), w∗(K) := f ′2(K, l(K))

i∗(K) := f(K, l(K))− c(K), Π(·|K) := q(·|i∗(K))

and

Φ(s,K) :=v(s′)

u′1(c(K), 1− l(K))− ζ(K), (26)

where ζ : S → R will be defined soon. By Proposition 5.1 optimal solution of the problem 6 exists for anygiven function p∗, w∗, r∗ and π∗. Let (c∗(k,K), l∗(k,K), a∗(·, k,K)) be this solution. Step 1. Observethat k := K and l := l(K) solves a problem in 2.

Step 2. For given r∗, w∗ we construct p∗, π∗ such that c∗(K,K) = c(K), l∗(K,K) = l(K) anda∗(s,K,K) = s for all K > 0. Let v ∈ V be given function satisfying v′1(s, s) = v′(s). For given Kconsider a maximization of the function

Uv(c, l, a(·), k) := u(c, 1− l) + β

∫S

v(a(s), s)Π(s|K)ds, (27)

under the constraints

f(k, l)− c− i∗(K) ≥ 0

∀s∈Sa(s)− s ≥ 0 and s− a(s) ≥ 0

a(·) ∈ L1, c ≥ 0, l ∈ [0, 1], k ∈ [0,K].

Let (c0, l0, a0, k0) solve the problem above. Clearly a0(s) = s. Since the objective function and constraintfunctions are concave in all arguments hence by Rockafellar and Wets [37] there exist Lagrange multipliersλvK ≥ 0 and µvK(·) ∈ L1 (not necessarily positive functions) such that maximization problem of (27) isequivalent to maximization of the associated Lagrangian

Lv(c, l, a(·), k;λ, µ(·)) := Uv(c, l, a(·), k) + λ(f(k, l)− c− i∗(K)) (28)

+

∫R+

µ(s)(s− a(s))ds. (29)

Thus k0 = K. Observe that (c(K), l(K)) with k = K, a∗(s) = s satisfy the constraints (28) with equality.By Inada conditions of u we have λK = u′1(c0, 1 − l0) > 0. Observe that λK does not depend on thecontinuation values v as in the Lagrangian (c, l) and v are separated. Further

Lv(c, l, a(·), k;λK , µK(·)) = Uv(c, l, a(·), k) + (30)

λK (f(k, l)− r∗(K)k − w∗(K)l) + (31)

λK

r∗(K)k + w∗(K)l + π∗v(K)− c−∫S

a(s′)p∗v(s′,K)ds′

, (32)

where π∗(K) :=∫S

s′µvK(s′)λK

ds′ − i∗(K) and

p∗v(s′,K) :=

µvK(s′)

λK. (33)

Observe that µvK , π∗ neither pvK depend on v(·) as

µvK(s) = βv′1(s, s)Π(s,K) = βv′(s)Π(s|K). (34)

Hence we can write p∗, π∗ and µ∗K . Observe that a solution of the maximization problem ofmax

k≥0,l∈[0,1]f(k, l) − r∗(K)k − w∗(K)l is k∗ := K and l∗ := l(K). By CRS assumption f(K, l(K)) =

r∗(K)K + w∗(K)l(K). Hence also

r∗(K)K + w∗(K)l(K) + π∗(K)− c(K)−∫S

a∗(s)p∗(s,K)ds′ = 0. (35)


Note that

λK = u′1(c(K), 1− l(K)). (36)

Then we immediately obtain that c∗(K,K) = c(K), l∗(K,K) := l(K), k∗ = K and a∗(s,K,K) = smaximizes a function

Uv(c, l, a(·), k) + λK

r∗(K)k + w∗(K)l + π∗(K)− c−∫S

a(s)p∗(s,K)ds′

.

By Kuhn-Tucker Theorem in Luenberger [20] we obtain that c∗ = c(K), l∗ = l(K), a∗(s) = s is a solutionof the problem:

maxc≥0,l∈[0,1],a(s)∈L+

1

u(c, 1− l) + β

∫S

v(a(s), s)Π(s|K)ds, (37)

over the constraints

r∗(K)K + w∗(K)l + π∗(K) ≥ c+

∫S

a(s)p∗(s,K)ds,

with equality in the expression above.

Step 3. In the previous step it has been shown (c(K), l(K), a∗(·,K,K)) with a∗(s,K,K) = s is anoptimal solution with arbitrary continuation v satisfying v′1(s, s) = v′(s). To show that (c, l, a∗) solvesa problem 6 we need to show that V ∗ ∈ V and V ∗

′

1 (s, s) = v′(s). First is clear by Proposition 5.1. ByProposition 5.1 we also have

V ∗′

1 (k,K) = r∗(K)u′1(c∗(k,K), 1− l∗(k,K)) = u′1(c∗(k,K), 1− l∗(k,K))f ′1(K, l(K)).

Thus if we put k := K, then by previous step we have V ′1(K,K) = v′(K). Therefore, and by previousstep there exists an optimal policy c∗, l∗, a∗ of the problem (6) satisfying c∗(K,K) = c(K), l∗(K,K) =l(K) and a∗(s,K,K) = s.

Step 4. We need to show that p∗(s,K) = Φ′1(s,K)Π(s|K). Observe that combining (33), (34) and

p∗(s,K) =βv′(s)Π(s|K)

u′1(c(K), 1− l(K)), (38)

we have (9) by (26).

Step 5. We need to show that i∗(K) maximizes the problem (8). Observe that by Lemma 5.1 we alsohave

d

di

∫S

Φ(s,K)Q(ds|i)− i = −∫S

Φ′1(s,K)Q′(s|i)ds− 1

= −

∫S

v′(s)Q′(s|i)ds

u′1(c(K), 1− K)− 1

=

ddi

∫S

v(s)Q(ds|i)

u′1(c(K), 1− K)− 1

To finish the proof we just need to put i := i∗(K). Hence RCE exists.

Step 6. Finally we show that

π∗(K) =

∫S

Φ(s,K)Q(ds|i∗(K))− i∗(K). (39)

In (26) we select a function ζ in such a way the expression (39) to be satisfied. Thus Φ is given as in(26) with

ζ(K) = π∗(K) + i∗(K)−∫S

v(s)

u′1(c(K), 1− l(K))Q(ds|i∗(K)).

�


Acknowledgments. We thank Manjira Datta, Jan Werner as well as participants of our sessionduring 11th SAET conference in Faro, Portugal 2011.

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