Existence of Generalized Recursive Equilibrium in Krusell and Smith (1998) * Dan Cao Georgetown University September 2016 Abstract In this paper, I define and show the existence of a generalized recursive equilib- rium in Krusell and Smith (1998)’s economy with both aggregate and idiosyncratic shocks. With a continuum of agents, my proof relies on the compactness of the space of probability measures over a bounded interval endowed with weak topology and of the space of Lipschitz continuous value functions and weakly increasing policy func- tions. These properties enable the use of the Kakutani-Glicksberg-Fan Fixed Point Theorem for infinite dimensional spaces. The existence proof suggests a numerical algorithm to compute a global nonlinear recursive equilibrium. I implement the algo- rithm and analyze its numerical solutions for two-agent economies. Keywords: Neoclassical Growth Models; Incomplete Markets; Heterogeneous Agents; Aggregate and Idiosyncratic Shocks; Recursive Equilibrium Existence; Kakutani-Glicksberg- Fan Fixed Point Theorem * I would like to thank Wenlan Luo for his excellent computational assistance. For useful comments and discussions, I thank Mark Huggett, Tom Krebs, Roger Lagunoff, Son Le, and Jianjun Miao.
Transcript
Existence of Generalized Recursive Equilibrium inKrusell and Smith (1998)∗
Dan CaoGeorgetown University
September 2016
Abstract
In this paper, I define and show the existence of a generalized recursive equilib-rium in Krusell and Smith (1998)’s economy with both aggregate and idiosyncraticshocks. With a continuum of agents, my proof relies on the compactness of the spaceof probability measures over a bounded interval endowed with weak topology and ofthe space of Lipschitz continuous value functions and weakly increasing policy func-tions. These properties enable the use of the Kakutani-Glicksberg-Fan Fixed PointTheorem for infinite dimensional spaces. The existence proof suggests a numericalalgorithm to compute a global nonlinear recursive equilibrium. I implement the algo-rithm and analyze its numerical solutions for two-agent economies.
Keywords: Neoclassical Growth Models; Incomplete Markets; Heterogeneous Agents;Aggregate and Idiosyncratic Shocks; Recursive Equilibrium Existence; Kakutani-Glicksberg-Fan Fixed Point Theorem
∗I would like to thank Wenlan Luo for his excellent computational assistance. For useful comments anddiscussions, I thank Mark Huggett, Tom Krebs, Roger Lagunoff, Son Le, and Jianjun Miao.
1 Introduction
Krusell and Smith (1998) provide a workhorse model with heterogeneous agents, sub-ject to both idiosyncratic and aggregate shocks, and markets are incomplete. Their paperoffers an algorithm to compute a recursive equilibrium, i.e., a sequential competitive equi-librium summarized by a mapping from current wealth distribution to prices and alloca-tions (policy function), and future wealth distribution (transition function). Despite theincreasing popularity of the model1, the theoretical question whether a recursive equilib-rium exists, and if it does what we can say about its properties, is still open. The presentpaper makes some progress toward answering this question.
First, I define a generalized recursive equilibrium as a correspondence that maps currentwealth distribution and exogenous aggregate shock to a set of possible prices and alloca-tions (include value functions), as well as possible future wealth distributions. At leastone of the elements in the set satisfies short-run equilibrium conditions. Any sequenceof allocations and prices generated by a generalized recursive equilibrium forms a se-quential competitive equilibrium. I then prove that a generalized recursive equilibriumalways exists in Krusell and Smith’s economy. In addition, if starting from any initialwealth distribution and aggregate shock, there exists no more than one sequential com-petitive equilibrium; then, from a generalized recursive equilibrium, one can select a stan-dard recursive equilibrium, as computed in Krusell and Smith (1998). The concept andproof techniques apply equally well to economies with a finite number of agents or witha continuum of agents.
The existence proof follows the steps in Cao (2010). That is, I first show the existenceof sequential competitive equilibria in finite horizon economies. Second, I show that theallocations and prices in these economies lie in a compact set. Lastly, I take the limit of theequilibrium allocation and prices as the horizon tends to infinity.2 Cao (2010) works withfinite-agent economies and assumes that the agents receive strictly positive endowmentof final good every period and in all states. This assumption is not made in Krusell andSmith (1998). In this present paper, I relax this assumption in the finite-agent economy. Ishow that aggregate capital is always bounded away from 0 in any sequential competitiveequilibrium, using the agents’ Euler equation and Jensen’s inequality. This lower boundon aggregate capital implies a lower bound on labor income if the agents always receivea strictly positive labor endowment. A strictly positive labor income plays a similar roleto a strictly positive final good endowment assumed in Cao (2010).
More substantively, I extend the proof arguments in Cao (2010) to allow for a contin-uum of agents exactly as in Krusell and Smith (1998). To do so, I take advantage of thecompactness of the space of probability measures over a bounded interval endowed withweak topology and the compactness of the space of Lipschitz continuous value functions
1The list of papers using Krusell and Smith’s or a similar framework and algorithm is long and fast-growing; it includes Krusell and Smith (1997), Mukoyama and Sahin (2006), Storesletten, Telmer and Yaron(2007), Chang and Kim (2007), Krusell, Mukoyama and Sahin (2010), Krusell, Mukoyama and Smith (2011),Bachmann and Bai (2013), Vavra (2013), Krueger, Mitman and Perri (2016), and many others.
2This method of proving existence using the limit of finite-horizon economies is standard in the infinite-horizon pure-exchange incomplete markets literature such as Magill and Quinzii (1994), Araujo, Pascoa andTorres-Martinez (2002), and more recently Le Van and Pham (2016).
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and weakly increasing policy functions. These compactness properties enable the appli-cation of the Kakutani-Glicksberg-Fan Fixed Point Theorem (Fan, 1952 and Glicksberg,1952), which is an extension to infinite dimensional spaces of the Kakutani Fixed PointTheorem, standard in game and general equilibrium theory. The existence proof in thiscase also implies several important properties of the value and policy functions such thatmonotonicity and Lipschitz continuity.
The existence proof is constructive and suggests an algorithm alternative to the oneused in Krusell and Smith (1998). The algorithm involves solving for the global nonlin-ear solution of finite horizon economies and taking the horizon to infinity, just as in theexistence proof. I implement the algorithm for two-agent economies, in which the agentsdiffer in either labor productivity or discount factor. I also carry out the simulation exer-cise as in Krusell and Smith (1998), i.e., regressing future log aggregate capital on currentlog aggregate capital over the simulated paths of the economy from the global nonlin-ear solution. I find that, at least for the two-agent economy with heterogeneous laborproductivity, the “aggregation result,” according to which current log aggregate capitalalone predicts almost perfectly future log aggregate capital, holds only in the simulations.In the global nonlinear solution, however, wealth distribution, in addition to aggregatecapital, is required to predict future aggregate capital.
The present paper is an extension of Cao (2010), allowing for a continuum of agents.Such extension includes the economy in Krusell and Smith (1998) as a special case. Theexistence of a generalized recursive equilibrium implies the existence of a competitiveequilibrium. Miao (2006) is the first to formulate and prove the existence of a competitiveequilibrium in Krusell and Smith-type economies with both idiosyncratic and aggregateshocks. I borrow many important ingredients from his paper such as the “conditionalno aggregate uncertainty condition” and the concept of recursive equilibrium with valuefunction as an extended state variable. The existence proof in Miao (2006) relies on theexistence and uniqueness of the value and policy functions (with arguments includingindividual capital holding and aggregate wealth distribution) as a solution to a Bellmanequation. However, the proof does not directly apply to cases in which the productionfunction satisfies the Inada condition at zero aggregate capital. This restriction excludesthe Cobb-Douglas specification used in Krusell and Smith (1998), which is the focus of thepresent paper. Because, in these cases, the Bellman operator is not well-defined when thedistribution of capital holdings is a Dirac mass at 0, leading to zero aggregate capital andconsequently an infinite marginal rate of return on capital due to the Inada condition.Hence the Contraction Mapping Theorem does not apply. In Appendix C, I show thatsimple ways to get around the issue with zero aggregate capital, in order to apply Miao’smachinery, do not work in Krusell and Smith (1998)’s economy.
Therefore, in the present paper, I follow a different route to establish the existence ofa competitive equilibrium by taking the limit of finite horizon economies as describedabove. I derive a lower bound on aggregate capital, or equivalently an upper bound onthe rate of return on capital, using the agents’ Euler equation, and hence sidestep the issuewith zero aggregate capital. In my proof, I also characterize several important propertiesof the value and policy functions, which do not appear in Miao (2006). Lastly, my proofallows for unbounded utility functions, i.e., log utility as in Krusell and Smith (1998) orCRRA utility with the risk-aversion coefficient strictly greater than 1, while Miao (2006)
2
requires bounded utility functions.The definition and existence proof for a generalized recursive equilibrium are similar
to those in Duffie, Geanakoplos, Mas-Colell and McLennan (1994), Kubler and Schmed-ders (2003), Cao (2010), and Feng, Miao, Perata-Alva and Santos (2014). But these papersonly allow for a finite number of agents. Extending the techniques developed therein toeconomies with a continuum of agents is not straightforward. As Miao (2006) wrote, “Thekey idea of Duffie et al. (1994) is to construct an expectation correspondence, which specifies, foreach possible current state, the transitions that are consistent with feasibility and satisfy short-runequilibrium conditions. Typically, the expectations correspondence is constructed using the first-order conditions for all agents. This procedure seems invalid for the continuum agents economiessince there is a continuum of first-order conditions.” In the present paper, I show that theshort-run equilibrium conditions in Krusell and Smith (1998)’s economy can be summa-rized using the value function and policy function instead of first-order conditions. Thisinnovation allows me to use standard techniques to establish the existence of a general-ized recursive equilibrium. This innovation also validates the application of Duffie et al.(1994)’s important results, for example, on the existence of an invariant measure over ex-ogenous and endogenous state variables. In the case of Krusell and Smith (1998), I showthat the state variables can be chosen as exogenous aggregate shock, wealth distribution,and value function.
The rest of the paper is organized as follows. Section 2 presents the finite agent versionof Krusell and Smith (1998) and the existence of a generalized recursive equilibrium inthis environment. Section 3 shows the existence of a generalized recursive equilibriumin the original version of Krusell and Smith (1998) with a continuum of agents. Section4 presents the a numerical algorithm based on the existence proofs to compute recursiveequilibria when they exist, as well as several numerical examples. The details of theproofs are presented in the Appendix.
2 Infinite Horizon Economy with Finite Agents
In this section, I present the finite agent version of Krusell and Smith (1998). The prooffor the existence of a generalized recursive equilibrium in this model is simpler than theone for the original Krusell and Smith’s model but the mechanics of the proofs are similar.Hence it serves as a good illustration of a more complex proof with a continuum of agentsin Section 3.
The environment Consider an endowment, a single consumption (final) good economyin infinite horizon. Time runs from t = 0 to ∞. The economy is populated by H represen-tative, infinitely-lived agents (households) indexed by:
h ∈ H = 1, 2, . . . , H
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Each representative agent represents a continuum of measure 1H of identical agents. The
preferences over the streams of consumption of agent h is given by
U(
cht (s
t)
t≥0,st∈S t
)= E0
[∞
∑t=0
(Πt
t′=0βh(st′))
u(cht )
](1)
where
u(c) = limν→σ
c1−ν − 11− ν
and the discount factor βht depends on the aggregate state st. We require σ ≥ 1 so that in
equilibrium, consumption is bounded from below.In each period t, there are S (finite) possible exogenous states (shocks)
s ∈ S = 1, 2, . . . , S .
The shocks capture both idiosyncratic uncertainties (or more precisely household leveluncertainties) and aggregate uncertainties. For example, state s can be a vector:
s = (A, i1, ..., iH) ,
where A is the aggregate productivity and ih’s are idiosyncratic shocks capturing agents’labor productivity and/or discount rate. As pointed out in Den Haan (2001), one caveatwith a finite number of agents is that each idiosyncratic shock is by construction an ag-gregate shock because it changes the aggregates, for example, aggregate labor supplywhen ih determines idiosyncratic labor supply. However, when the number of agents isvery large, the effects of each idiosyncratic shock on the aggregates become negligible. Inthe limit with a continuum of agents considered in the next section, by the law of largenumbers, an idiosyncratic shock does not have direct aggregate effects.
The exogenous shock follows a first-order Markov process with the transition proba-bilities π (s, s′). Let st denote the history of realizations of shocks up to time t:
st = (s0, s1, . . . , st) ∈ S t.
At time t, state st determines the agents’ endowments, lh(st) > 0 units of labor forh ∈ H. We assume that, there exist L, L > 0 such that:
L ≤ 1H ∑
h∈Hlh(st) ≤ L
for all s ∈ S . State st also determines the agents’ discount factor, βh(st). In addition, thereexist 0 < β, β < 1 such that:
β < βh(st) < β.In each state s, there is a representative firm that produces the final output from capital
and labor using an aggregate production function that employs capital and labor as input:
Y = F(s, K, L).
The aggregate state determines the productivity of the aggregate production functionthrough the first argument.
We make the following standard assumptions on F.
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Assumption 1. F is strictly increasing, strictly concave and has constant returns to scale in Kand L.
This assumption nests the Cobb-Douglas production used in Krusell and Smith (1998)as a special case
F(s, K, L) = A(s)KαL1−α. (2)We also assume that capital depreciates at rate δ ∈ (0, 1) in each period. The final
output at time t can be transformed into future capital, Kt+1, and current consumption Ctaccording to
Ct + Kt+1 − (1− δ)Kt = Yt.We further assume that:
Assumption 2. There exists K such that
F(s, K, L)− δK < 0
for all K ≥ K and for all s ∈ S .
This assumption guarantees that aggregate capital is always bounded above. It is alsosatisfied under the Cobb-Douglas production function.
Market Arrangements In each history st, there are rental markets for capital and labormarket. Agents of type h rent out their capital to the representative firm at competitiverental rate rt(st) and supply their labor endowment inelastically to the representative firmat competitive wage rate wt(st).
We assume that markets are incomplete inter-temporally, i.e., the agents can only holdcapital to insure against idiosyncratic and aggregate shocks. Therefore they face the se-quential budget constraints:
cht (s
t) + kht+1(s
t)− (1− δ)kht (s
t−1) ≤ rt(st)kht (s
t−1) + wt(st)lh(st) (3)
and the borrowing constraints:kh
t+1(st) ≥ 0. (4)
Agent h solves
maxch,khU(
cht (s
t)
t,st
)(5)
subject to (3) and (4).The representative firm in history st maximizes profit:
Πt(st) = maxYt,Kt,Lt≥0
Yt − rtKt − wtLt
subject toYt ≤ F(st, Kt, Lt).
Since F has constant returns to scale, in equilibrium, we must have Πt(st) = 0 and
Yt = F(st, Kt, Lt) and rt = FK(st, Kt, Lt) and wt = FL(st, Kt, Lt). (6)
The definition of a competitive equilibrium in this environment is standard.
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Definition 1. A competitive equilibrium given an initial distribution of capital holdingskh
0
h∈H consists of an allocation(
cht , kh
t+1
t,st
)h∈H
and Kt, Ltt,st and prices rt, wtt,st
(rt, wt > 0) such that:1. For each agent h ∈ H,
ch
t , kht+1
t,st maximizes the intertemporal expected utility (1)subject to the sequential budget constraints, (3) and borrowing constraints, (4).
2. In each history st, Yt, Kt, Lt solves the representative firm’s profit maximizationproblem, i.e., (6) is satisfied.
3. Markets for capital, labor, and final good clear in each history st:
1H ∑
h∈Hkh
t (st) = Kt(st)
and1H ∑
h∈Hlh(st) = Lt(st)
and1H ∑
h∈H
(ch
t (st) + kh
t+1(st)− (1− δ)kh
t (st−1)
)= Yt(st).
Let Ω denote a set of wealth distributions, or equivalently of the distributions of capi-tal holdings, and is a compact subset of RH
+ :
Ω =(
kht
)h∈H
⊂ RH
+ .
Following Krusell and Smith (1998), we define a generalized recursive equilibrium asfollowing.
Definition 2. A generalized recursive equilibrium is a policy correspondence and a tran-sition correspondence:
Q : S ×Ω ⇒ R3H+2+
andT : S ×Ω ⇒ Ω
and c > 0 such that for all s ∈ S and
khh∈H ∈ Ω and
(ch, kh
+, vhh∈H , r, w
)∈
Q(
s,
khh∈H
), we have ch ≥ c and there exists(
ch+(s′), kh
++(s′), vh
+(s′)
h∈H, r+(s′), w+(s′)
)s′∈S
that satisfies the following properties:1.
kh+
h∈H ∈ T
(s,
khh∈H
)2.(
ch+, kh
++, vh+
h∈H , r+, w+
)∈ Q
(s′,
kh+
h∈H
)3. (Market clearing) 1
H ∑h∈H ch + 1H ∑h∈H kh
+ = F(s, K, L) + (1− δ)K where
K =1H ∑
h∈Hkh > 0 and L =
1H ∑
h∈Hlh(s) > 0.
4. (Firms’ maximization) r = FK(s, K, L) > 0 and w = FL(s, K, L) > 0.
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5. (Agents’ maximization) For each h ∈ H
u′(ch) ≥ βh(s) ∑s′∈S
πss′(1− δ + r+(s′)
)u′(ch
+(s′)) (7)
with equality if kh+ > 0 and
ch + kh+ = (1− δ + r)kh + wlh (8)
andvh = u(ch) + βh(s) ∑
s′∈Sπss′vh
+(s′). (9)
A recursive equilibrium is a generalized recursive equilibrium in Definition 2 with thecorrespondences Q, T being single-valued.
The following lemma shows the connection between a generalized recursive equilib-rium and competitive equilibrium.
Lemma 1. A sequence of allocations and prices generated by a generalized recursive equilibriumforms a competitive equilibrium.
Proof. Appendix A.
To show the existence of a generalized recursive equilibrium, we need the followingproperties on the production function.
Assumption 3. For any L > 0 and K > 0:
maxs∈S
sup0<K≤K
FL(s, K, L) < +∞,
andmaxs∈S
sup0<L≤L
FK(s, K, L) < +∞.
Lastly, for any L > 0 and s ∈ SlimK→0
F(s, K, L) = 0.
This assumption assures that a competitive equilibrium exists in the finite horizoneconomy.
Assumption 4. There exists K∗ such that for any 0 < K < K∗ and L ≤ L ≤ L, and s, s′ ∈ S :(β min
s′′∈S(1− δ + FK(s′′, K, L))
) 1σ
>F(s′, K, L) + (1− δ)K
F(s, K, L)− δK.
This assumption requires that the marginal rate of return on capital is very high whencapital is low. Together with the agents’ Euler equation, it implies a lower bound onaggregate capital in any competitive equilibrium.
It is easy to verify that the last two assumptions hold for the Cobb-Douglas productionfunction in (2) since FK(s′′, K, L)→ ∞ as K → 0 and F(s′,K,L)
F(s,K,L) is bounded above as K → 0.Armed with the assumptions above, we arrive at the first existence result.
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Theorem 1. Assume that Assumptions 1, 2, 3, and 4 hold. Given any initial distribution ofcapital, 1
H ∑h∈H kh0 = K0 > 0, there exist 0 < K < K0 < K such that a generalized recursive
equilibrium exists with Ω =(
kh)h∈H ∈ RH
+ : K ≤ 1H ∑h∈H kh ≤ K
.
Proof. We choose K sufficiently large:
K > max
K0, K, maxs∈S
max0≤K≤K
(F(s, K, L) + (1− δ)K) , 2L
, (10)
where K is defined in Assumption 2, and K sufficiently small:
K < min K0, K∗ , (11)
where K∗ is defined in Assumption 4.The proofs follow closely the steps in Cao (2010). I first show that a competitive equi-
librium exists for any finite horizon economy. In addition, the equilibrium variables in afinite horizon economy always lie in a compact set. Then I take the limit of the horizon toinfinity and construct appropriate correspondences to show the existence of a generalizedrecursive equilibrium.
However, Cao (2010) assumes that each agent receives an strictly positive amount offinal good endowment in every period and history of shocks. In this paper, we relax thisassumption. We only require that each agent receives an strictly positive amount of laborendowment in every period and history of shocks. We show that aggregate capital isalways bounded from below:
Kt,T(st) ≥ Kfor all t and st, where Kt,T(st) is the aggregate capital at time t and in history st in theT-period economy. Therefore wage rate is bounded from below:
wt,T(st) = FL(st, Kt,T(st), Lt,T(st)) ≥ w
for some w > 0. Together with a strictly positive labor endowment, the lower bound onwage rate implies a strictly positive labor income, which plays a similar role to a strictlypositive final good endowment in Cao (2010).
To show that aggregate capital is bounded from below, we use the agents’ Euler equa-tion, (7):
u′(cht ) ≥ βh
t Et
[(1− δ + rt+1) u′(ch
t+1)]
.
This equation implies that if Kt+1 is too small, the rate of return on capital rT+1 is everyhigh, driving up saving from time t, and in turn, increasing Kt+1. Assumption 4 thenleads to a contradiction.
The details of the proof are given in Appendix A.
Generalized Recursive Equilibrium and Recursive Equilibrium Can we always selecta recursive equilibrium from a generalized recursive equilibrium? The answer is no. Thedefinition of a generalized recursive equilibrium involve policy and transition correspon-dences, Q and T . We can show that Q upper-hemi continuous. Therefore there exists a
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measurable selection (function)Q0 fromQ. Let T 0 denote the transition function that cor-responds to the selectionQ0. However,
(Q0, T 0)might not form a recursive equilibrium.
To see this clearly, let us say
kh+
h∈H = T 0
(s,
khh∈H
)and(
ch+(s′), kh
++(s′), vh
+(s′)
h∈H, r+(s′), w+(s′)
)s′∈S
such that Conditions 2-5 in Definition 2 are satisfied. Now it is possible that
kh+
h∈H =
khh∈H and(
ch+, kh
++, vh+
h∈H
, r+, w+
)∈ Q
(s′,
kh+
h∈H
)\Q0
(s′,
kh+
h∈H
)for some s′ ∈ S . Therefore at s′, we would select the “wrong” allocation if we set(
ch+, kh
++, vh+
h∈H
, r+, w+
)= Q0
(s′,
kh+
h∈H
).
From the last observation, in general, we cannot always select a recursive equilibriumfrom a generalized recursive equilibrium. Therefore, we would need additional condi-tions to guarantee the existence of a recursive equilibrium. The following result providessuch a sufficient condition for when a generalized recursive equilibrium gives rise to arecursive equilibrium.
Corollary 1. Assume that the conditions in Theorem 1 are satisfied. We have:1. Starting from any wealth distribution
kh
0
h∈H ∈ RH+ and exogenous state s0 ∈ S , there
exists a competitive equilibrium.2. In addition if the competitive equilibrium is unique for every initial wealth distribution and
exogenous state, there exists a recursive equilibrium.
Proof. 1. By Lemma 1, starting from any distribution of capital holdings
kh0
h∈H andaggregate state s, the sequences of allocation and prices generated by a generalized re-cursive equilibrium is a competitive equilibrium. Theorem 1 guarantees the existence ofa generalized recursive equilibrium. Hence, a competitive equilibrium exists.
2. Because starting from each s ∈ S and
khh∈H ∈ Ω, there exists no more than one
competitive equilibrium, there exists a unique element(ch, kh
+, vh
h∈H, r, w
)∈ Q
(st,
kh
h∈H
)that satisfies Conditions 1.-5. in Definition 2. LetQ0 denote the mapping from
(s,
khh∈H
)to this element, and T 0
(s,
khh∈H
)=
kh+
h∈H. Then
(Q0, T 0) forms a recursive equi-
librium.
A generalized recursive equilibrium also gives rise to a recursive equilibrium if weallow for more (endogenous) state variables in addition to the agents’ capital holdings.This point is emphasized more generally in Duffie et al. (1994).
Corollary 2 (Recursive Equilibrium with Extended State Variables). Given the set of distri-
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butions Ω and the correspondence Q in Definition 2 and Theorem 1, let
Ξ =
(s,
kh
,
ch, vh)∈ S ×Ω×R2H
∣∣∣∣ (ch, kh+, vh , r, w
)∈ Q
(st,
kh)for some
(kh+
)and r, w > 0
where
xh is a short-cut for
xh
h∈H.A recursive equilibrium with an extended state variable can be constructed over Ξ as a mapping
from from ξ =(s,
kh ,
ch, vh) toa. a current capital choices and values
kh+
, and current factor prices r, w ;
b. next period capital holdings and consumption(s′,
kh+
,
ch+, vh
+
)s′∈S such that
ξ+s′ =(
s′,
kh+
,
ch+, vh
+
)∈ Ξ for all s′ ∈ S ,
and Conditions 3.− 5. in Definition 4 are satisfied.We notice that from the firms’ maximization problem (Condition 4), r and w are pinned down
by the current states s and K = 1H ∑h kh. Therefore, from the agents’ budget constraint, (8), ch
pins down kh+. Consequently, the values of
ch, vh uniquely select the element in Q(s,
kh)
and T (s,
kh), i.e. , for any ξ =(s,
kh ,
ch, vh) ∈ Ξ there exists a unique tuple(kh+
), r, w
such that(
ch, kh+, vh , r, w
)∈ Q
(st,
kh). The selection gives rise to a recursive equilibriumin the extended state space.
In the following section, we extends the analysis above in this environment with afinite number of (representative) agents to the environment with a continuum of agents.
3 Infinite Horizon Economy with a Continuum of Agents
The structure of aggregate shocks s ∈ S is defined as in the previous section with thetransition matrix πss′ . However, the economy is populated by a continuum of agentswith index h ∈ H = [0, 1] and state s captures purely aggregate shocks. Let φ denote theLebesgue measure over H.
The agents are subject to idiosyncratic shock i ∈ I . The realizations of the idiosyn-cratic shocks are independent across agents so that the law of large number applies. Wealso assume that I has a finite number of states. In addition, following Krusell and Smith(1998), we make the following restrictions on the joint dynamics of aggregate and id-iosyncratic shocks: (st, it) forms a first-order Markov process with the transition matrixπss′,ii′ :
Pr(st+1 = s′, it+1 = i′, st = s, it = i) = πss′,ii′
with the restriction3
∑i′
πss′,ii′ = πss′ ,
3This condition means that the evolution of the aggregate state is independent of the idiosyncratic states(but not vice versa):
∑i′∈I
Pr(st+1 = s′, it+1 = i′, st = s, it = i) = Pr(st+1 = s′, st = s, it = i) = πss′ ,
for all i ∈ I .
10
for each i′ ∈ I , and s, s′ ∈ S .As in Miao (2006, Section 3.2), we assume the “conditional no-aggregate uncertainty
condition,” on the random variables(
st,(iht)
h∈H
)so that the law of large numbers for
a continuum of agents applies. In particular, the empirical distributions of implied indi-vidual random variables are the same as the theoretical distributions from which theserandom variables are drawn.
As a consequence, we can assume, as in Krusell and Smith (1998), that, in aggregatestate s, the fraction of agents with idiosyncratic type i is ms(i), independent of the pasthistory of aggregate shocks and agents’ idiosyncratic shocks. However, the fractions ms’shave to be consistent with the transition matrix πss′,ii′ :
∑i∈I
ms(i)πss′,ii′
πss′= ms′(i′),
for all s, s′ ∈ S and i′ ∈ I .Together with the aggregate shock, idiosyncratic shock determines the labor supply of
the households in state (s, i): l(s, i). Applying the conditional no-aggregate uncertaintycondition, the total supply of labor in aggregate state s is
L(s) = ∑i∈I
ms(i)l(s, i).
Idiosyncratic shock also determine the agents’ discount factor in state i: β(i). We makethe following assumptions on the idiosyncratic labor supply l(., .) and discount factorβ(.).
Assumption 5. There exist 0 < l < l such that
l < l(s, i) < l
for all s ∈ S and i ∈ I .There exist 0 < β < β < 1 such that
β < β(i) < β
for all i ∈ I .
Since S and I have finite elements, we can choose 0 < L, L such that
L ≤ ∑i∈I
ms(i)l(s, i) ≤ L
for all s ∈ S .Assumption 3 on the aggregate production function and Assumption 5 guarantee the
existence of a competitive equilibrium in the finite horizon economy. An additional as-sumption on the production function, Assumption 6, guarantees the existence of a com-petitive equilibrium in the infinite horizon economy, as well as the existence of a general-ized recursive equilibrium.
Assumption 6. For any L ≥ L and s ∈ S ,
limK→0
FK(s, K, L) = ∞.
11
There exists α > 0, such that for all K, L > 0:
LFL(s, K, L)F(s, K, L)
> α.
For any s, s′ ∈ S :
lim supK→0
F(s′, K, L)F(s, K, L)
< ∞.
This assumption guarantees that aggregate capital always exceeds some strictly posi-tive lower bound in a finite horizon economy (by an argument based on the agents’ Eulerequation as in the proof of Theorem 1). Together with Assumption 2, Assumption 6 im-plies that the aggregate capital is bounded above and below in a finite horizon economy.Therefore we can take the limit of the equilibria in finite horizon T-period economy, asthe horizon T goes to infinity to obtain an equilibrium in the infinite horizon economy aswell as a generalized recursive equilibrium.
It is easy to verify that Assumption 6 is satisfied under Cobb-Douglas productionfunction (2).
Let K be chosen sufficiently large as in (10). In addition, we assume that the agents’choice of capital is bounded above by k sufficiently large so that:4
k > l + maxs∈S ,i∈I
F(s, 2K, 2L). (12)
The agents’ Euler equation, which is similar to equation (7) in the finite agent economyand is crucial in deriving a lower bound for aggregate capital, does not hold when theupper bound binds. Therefore the upper bound has to be large enough to minimize itseffect.
Let ih,t =(ih0, ih
1, ..., iht)
denote the history of idiosyncratic shocks for each h and it =(ih,t)
h∈H denote the history of idiosyncratic shocks for all h ∈ H.As in the previous section with a finite number of agents, given interest rate and wage
as functions of the history of aggregate shocks st and idiosyncratic shocks it, agent hsolves
maxch
t (.),kht+1(.)
E0
[∞
∑t=0
Πt−1t′=0β(ih
t′)u(cht (s
t, it))
](13)
subject to
cht (s
t, it) + kht+1(s
t, it) ≤ (1− δ + rt(st, it))kht (s
t−1, it−1) + wt(st, it)l(st, iht ), (14)
and cht (s
t, ih,t) ≥ 0 and0 ≤ kh
t+1(st, it) ≤ k. (15)
The firms’ problem is exactly as in the previous section with a finite number of agents.The following definition of competitive equilibrium is standard, which is a direct exten-sion of Definition 1 for a continuum of agents.
4An upper bound on the choice of capital is implicitly assumed by all numerical algorithms since capitalchoice is bounded by machine numerical upper bound.
12
Definition 3. A competitive equilibrium given an initial distribution of capital hold-ings
kh
0
h∈H consists of an allocation(
cht , kh
t+1
t,st,it
)h∈H
and Kt, Ltt,st,it and prices
rt, wtt,st,it (rt, wt > 0) such that:1. For each agent h ∈ H,
ch
t , kht+1
t,st,i,t solves (13).
2. In each history of aggregate and idiosyncratic shocks st and it, Yt, Kt, Lt solves therepresentative firm’s profit maximization problem, which implies (6).
3. Markets for capital, labor, and final good clear in each history st:∫H
kht (s
t−1, it−1)φ(dh) = Kt(st, it)
and ∫H
lh(st, iht )φ(dh) = Lt(st, it)
and ∫H
(ch
t (st, it) + kh
t+1(st, it)− (1− δ)kh
t (st−1, it−1)
)φ(dh) = Yt(st, it).
Instead of working with the allocations of capital over households h ∈ H, it is easierto work with the distributions over capital holdings (or equivalently, wealth distribution)and idiosyncratic shocks. For each distribution of asset holding
kh
h∈H, consider thefollowing probability measure µ defined by
µ(A× I) = φ(
h ∈ H :(
kh, ih)∈ A× I
)(16)
for each A × I ∈ B([
0, k])× B(I) - where B denote the Borel σ−algebras. It is imme-
diate that µ ∈ P([
0, k]× I
), where the later denotes the space of probability measures
over[0, k]× I endowed with the weak topology. It is well-known that P
([0, k]× I
)is
compact (see for example Bogachev (2000, Theorem 8.9.3)). Let Ω denote a closed sub-set of P
([0, k]× I
),which we will define below. Let C denote the set of functions over[
0, k]× I which are continuous in k. The generalized recursive equilibrium is defined
over the set of distributions Ω similar to Definition 2 with a finite number of agents.
Definition 4. A generalized recursive equilibrium is a policy correspondence and a tran-sition correspondence:
Q : S × Ω ⇒ C3 ×R2+
andT : S × Ω ⇒ ΩS
and some bounds V, V, with the following property: For each s ∈ S and µ ∈ Ω, and(c, k, V, r, w
)∈ Q(s, µ), we have V ≤ V ≤ V and there exist
(s′, µ+
s′)
s′∈S ∈ T (s, µ) and(c+s′ , k+s′ , V+
s′ , r+s′ , w+s′
)s′∈S
such that:1. For each s′ ∈ S ,
(c+s′ , k+s′ , V+
s′ , r+s′ , w+s′
)∈ Q
(s′, µ+
s′).
2. (Market clearing) ∑i∫
c(k, i)µ(dk, i) + ∑i∫
k(k, i)µ(dk, i) = F(s, K, L) + (1 − δ)K
13
where
K = ∑i∈I
∫ k
0kµ(dk, i) and L = ∑
i∈Ims(i)l(s, i).
3. (Firms’ maximization) r = FK (s, K, L) > 0 and w = FL (s, K, L) > 0.4. (Agents’ maximization) For each i ∈ I and k ∈
[0, k], V, V+ satisfy the Bellman-
type equation:V(k, i) = max
c,k′u(c) + β(i) ∑
i′,s′πss′,ii′V+
s′ (k′, i′) (17)
s.t. c ≥ 0 and 0 ≤ k′ ≤ k and
c + k′ ≤ (1− δ + r)k + wl(s, i).
In addition,(
c(k, i), k(k, i))
solves (17).
5.(Distribution Consistency) For each s′ ∈ S , i′ ∈ I and A ∈ B([
0, k])
:
µ+s′ (A, i′) = ∑
i∈S
πss′,ii′
πss′µ((
k(., i))−1(A), i
). (18)
Similar to Lemma 1 for finite-agent economy, the following lemma shows that gener-alized recursive equilibrium generates competitive equilibrium.
Lemma 2. Starting from an initial distribution of wealth µ0 and aggregate state s0, sequences ofallocation and prices generated by a generalized recursive equilibrium form a competitive equilib-rium.
Proof. Appendix B.
Now we arrive at the second existence theorem.
Theorem 2. Assume that Assumptions 1, 2, 3, 5, and 6 hold. Starting from an initial distributionof capital µ0 with
K0 =∫H
kh0φ(dh) = ∑
i∈I
∫ k
0µ0(dk, i) > 0,
there exist 0 < K < K0 < K, such that a generalized recursive equilibrium exists over
Ω =
µ : K ≤ ∑
i∈I
∫ k
0kµ(dk, i) ≤ K
.
Proof. Given the bounds determined in Lemma 12, let Θ denote the set of(k(., .), V(., .), r, w
)such that for each i ∈ I , k(k, i) is weakly increasing in k and 0 ≤ k(k, i) ≤ k and V ≤V(k, i) ≤ V for all k ∈ [0, k] and V(k, i) is weakly increasing and weakly concave in k andLipschitz continuous with a Lipschitz constant lV > 0. V, V, lV are given in Lemma 12.In addition 0 < r ≤ r ≤ r and 0 < w ≤ w ≤ w where r, r, w, w are also given in Lemma12. Lemma 11 shows that Θ, endowed with the topology of pointwise convergence for
14
k and uniform convergence for V and the standard topology for r and w, is sequentiallycompact.5
Let g : S × Ω ⇒ Θ × ΘS denote the following correspondence: for each s ∈ S ,µ ∈ Ω, g(s, µ) is the set of θ =
((k(., .), V(., .)
)h∈H
, r, w)∈ Θ, and (θs′)s′∈S with θs′ =(
k+s′ (., .), V+s′ (., .), r+s′ , w+
s′
)∈ Θ such that
r = FK(s, K, L) > 0 and w = FL(s, K, L) > 0,
where
K = ∑i∈I
∫ k
0kµ(dk, i) > 0
andL = ∑
i∈Ims(i)l(s, i) = L(s),
and
∑i
∫c(k, i)µ(dk, i) + ∑
i
∫k(k, i)µ(dk, i) = F(s, K, L) + (1− δ)K
where c(k, i) = (1− δ + r)k + wl(s, i)− k(k, i) > 0. In addition(
c, k, V)
solves the func-
tional equation, (17), given(V+
s′)
s′∈S .Lemma 11 shows that g is a closed-valued correspondence.Consider the following mapping G from the set of correspondences V : S × Ω ⇒ Θ to
itself defined as following. For each V , G(V) is the correspondenceW such that, for eachs ∈ S and µ ∈ Ω, we have
W(s, µ) =
θ =
(k, V, r, w
)∈ Θ : for each s′ ∈ S ,∃θs′ ∈ V
(s′, µ+
s′)
where µ+s′ is given by (18)
and(θ, (θs′)s′∈S
)∈ g(s, µ)
From the definition of G, we have the following properties P1-P3:P1. If V is sequentially compact, in the sense that V(s, µ) is sequentially compact for
all s ∈ S and µ ∈ Ω, thenW = G(V) is sequentially compact.Indeed, assume that θm∞
m=0 ∈ W(s, µ), and θm → θ =(
k, V, r, w)
. Since Θ is
sequentially compact, θ ∈ Θ. To show thatW = G(V) is sequentially compact, we needto show that θ ∈ W(s, µ). By the definition of G, for each s′ ∈ S , ∃θm
s′ ∈ V(s′, µ+
s′)
such that(θm,(θm
s′))∈ g(s, µ). Since V
(s′, µ+
s′)
is sequentially compact, we can extract aconverging subsequence, θ
mls′ → θs′ for some θs′ ∈ V
(s′, µ+
s′). Because g is a closed-valued
correspondence,(θ, (θs′)s′∈S
)∈ g(s, µ), which implies θ ∈ W(s, µ). SoW is sequentially
compact.P2. If V ⊂ V ′ in the sense that V(s, µ) ⊂ V ′(s, µ) for all s ∈ S and µ ∈ Ω then
G(V) ⊂ G(V ′).P3. Let V0 denote the complete correspondence: V0(s, µ) = Θ for all s ∈ S and µ ∈ Ω.
5In infinite dimensional spaces, compactness and sequential compactness are not equivalent. For thecurrent theorem, we need the sequential compactness property.
15
Then G(V0) ⊂ V0.Given V0, we construct the sequence of Vn∞
n=0 recursively using G: Vn+1 = G(Vn).Then by P1, P2, and P3, we have Vn+1 ⊂ Vn and is sequentially compact. By the existenceof a competitive equilibrium in (n+1)-horizon economy in Lemma 5, Vn+1 is a non-emptyvalued correspondence.
Let V∗ be defined byV∗(s, µ) = ∩∞
n=0Vn(s, µ).Since V∗(s, µ) is the intersection of decreasing, non-empty, sequentially compact sets,V∗(s, µ) is sequentially compact and is non-empty. We show that G(V∗) = V∗.
Indeed, by the definition of V∗, we have V∗ ⊂ Vn, so G(V∗) ⊂ G(Vn) = Vn+1 for alln. This implies G(V∗) ⊂ V∗.
Now, for each s ∈ S and µ ∈ Θ and θ =(
k, V, r, w)∈ V∗(s, µ). Since V∗ ⊂ Vn+1 =
G(Vn), there exists θns′ ∈ V
n (s′, µ+s′)
such that(
θ,(θn
s′)
s′∈S
)∈ g(s, µ). By the sequen-
tial compactness of Θ, we can find a converging subsequence nl∞l=0,
(θ
nls′)
s′∈S −→l→∞
(θs′)s′∈S . By the sequential compactness of Vnl , we have θs′ ∈ Vnl(s′, µ+
s′)
and since g hasclosed-value,
(θ, (θs′)s′∈S
)∈ g(s, µ). Moreover, Vn (s′, µ+
s′)
is a decreasing sequence soθs′ ∈ ∩∞
l=0Vnl(s′, µ+
s′)= V∗
(s′, µ+
s′). Therefore, by the definition of G, we have θ ∈ G(V∗).
Thus V∗ ⊂ G(V∗).Since G(V∗) ⊂ V∗ ⊂ G(V∗), it implies that G(V∗) = V∗.Let Q = V∗. Since G(Q) = Q, for each s ∈ S and each µ ∈ Ω, θ =
(k, V, r, w
)∈
Q(s, µ), there exists θs′ ∈ Q(s′, µ+
s′)
for each s′ ∈ S such that(θ, (θs′)s′∈S
)∈ g(s, µ). We
also define T as
T (s, µ) =(
µ+s′)
s′∈S : given(
c, k, V, r, w)∈ Q(s, µ), µ+
s′ is determined by (18)
.
It is immediate that (Q, T ) defined as such forms a generalized recursive equilibrium forthe economy with a continuum of agents.
There are two immediate implications of Theorem 2.
Corollary 3. Starting from any initial distribution of capital holdings, µ0(k, i) and exogenousaggregate state s0, there exists a competitive equilibrium. If the competitive equilibrium is uniquefor every initial distribution of capital holding and aggregate state then there exists a recursiveequilibrium as defined in Krusell and Smith (1998).
Proof. The proof of this corollary is exactly the same as the proof of Corollary 1.
Following Duffie et al. (1994) and Miao (2006), from the generalized recursive equi-librium, of which the existence is established in Theorem 2, we can construct a recursiveequilibrium if we enlarge the state space with the value function.
Corollary 4 (Recursive Equilibrium with Value Function as an Additional State Variable).Given the set of distributions Ω and the correspondence Q in Definition 4 and Theorem 2, let
Ξ =(
s, µ, V)∈ S × Ω× C : ∃
(c, k, V, r, w
)∈ Q(s, µ) for some
(c, k)∈ C2 and r, w > 0
16
where C denotes the set of functions over[0, k]× I which are continuous in k.
A recursive equilibrium with an extended state variable can be constructed over Ξ as a mappingfrom from ξ =
(s, µ, V
)to
a. a current policy function k, and current factor prices r, w ;b. next period wealth distributions and value functions
(µs′ , V+
s′)
s′∈S such that
ξ+s′ =(s′, µs′ , V+
s′)∈ Ξ for all s′ ∈ S ,
and Conditions 4. and 5. in Definition 4 are satisfied (given the current policy function and currentfactor prices).
Formulated in this manner, the existence of this recursive equilibrium is a direct application ofthe existence of the generalized recursive equilibrium shown in Theorem 2.
4 Numerical Algorithm and Examples
The existence proofs in Section 2 and Section 3 also suggest an algorithm to computerecursive equilibria, alternative to the one put forth in Krusell and Smith (1998), using theequilibria in finite horizon economies. The next subsection presents the algorithm andthe one following presents two numerical examples for two-agent economies.
4.1 Numerical Algorithm
I propose an algorithm to compute the generalized recursive equilibrium as defined inDefinition 2 for finite agent economies, assuming that the equilibrium is indeed a recur-sive equilibrium. That is, we seek to compute the functions (instead of correspondences)Q and T defined over S ×Ω.
Notice that Ω can be re-parametrized as:
Ω = [K, K]× ∆H =
(K,(
ωh)
h∈H
): K ≤ K ≤ K and 0 ≤ ωh ≤ 1, ∑
h∈Hωh = 1
,
where ωht =
kht
HKt. We calculate recursively, for each T ≥ 0, the function ϕT from Ω,
the set of current wealth distributions, to current prices and allocations, and to futurewealth distributions. Function ϕT corresponds to the equilibrium mapping, for the (T +1)−horizon economy presented in Appendix A, from the initial distribution of capitalholdings and aggregate shock in period 0 to allocation and prices in the period. Indeed,
ϕT : S × [K, K]× ∆H ⇒ R4H+2+
s ∈ S , K ∈ [K, K] , ω ∈ ∆H7→(
ch, kh+, λh, vh
)h∈H
, r, w
defined as follows.1. For T = 0:
ϕ0 :
s ∈ S , K ∈ [K, K] , ω ∈ ∆H7→(
ch, kh+, λh, vh
)h∈H
, r, w
17
wherer = FK(s, K, L(s)) and w = FL(s, K, L(s))
and for all h ∈ Hch = (1− δ + r)ωhK + wlh(s) and vh = u(ch),
andkh+ = 0 and λh = 0.
2. For T > 0, assuming that we have calculated ϕT−1, ϕT is calculated as:
ϕT :
s ∈ S , K ∈ [K, K] , ω ∈ ∆H7→(
ch, kh+, λh, vh
)h∈H
, r, w
such that, for each s′ ∈ S ,(ch+, kh
++, λh+, vh
+
)= ϕT−1
(s′, K+,
(ωh+
)h∈H
)where
K+ =1H ∑
h∈Hkh+
and for each h ∈ H:
ωh+ =
kh+
∑ kh+
,
and Conditions 2.-5. in Definition 2 are satisfied:
A1. 1H ∑h∈H ch + 1
H ∑h∈H kh+ = F(s, K, L(s)) + (1− δ)K.
A2. r = FK(s, K, L(s)) > 0 and w = FL(s, K, L(s)) > 0
A3. For each h ∈ Hu′(ch) = βh(s) ∑
s′∈Sπss′
(1− δ + r+(s′)
)u′(ch
+(s′)) + λh (19)
with λh ≥ 0 andλhkh
+ = 0and
ch + kh+ = (1− δ + r)kh + wlh
andvh = u(ch) + βh(s) ∑
s′∈Sπss′vh
+(s′).
Condition A3 is a reformulation of Condition 5 in Definition 2 using the multipliers λh’sand the complementary-slackness condition. Notice also that for each
(K,(ωh)
h∈H
)∈
[K, K] × ∆H, the conditions in A2. and A3. gives us 4H + 2 equations for 4H + 2 un-knowns (including λh’s). The market clearing, condition A1., is satisfied by summing upthe budget constraint of each agent.
18
Discretization and Approximation We discretize Ω using Ωd:
Ωd =
K = Kd1 < Kd
2 < ... < KdN = K
×
ωdm
M
m=1(20)
where ωdm ∈ ∆H for m = 1, ..., M.
Let ϕdT denote the discrete approximation of ϕT over S ×Ωd. For each s ∈ S and at
each(Kd
n, ωdm), we solve for
ϕdT
(s, Kd
n, ωdm
)=(
ch, kh+, λh, vh
)such that Conditions A2 and A3 are satisfied. In Conditions A2 and A3, the future valuesch+, vh
+ are computed using multi-dimensional cubic splines approximation:(ch+, vh
+, λh+, kh
++
)= ϕd
T−1
(s′,
1H ∑ kh
+,kh+
∑ kh+
). (21)
Fixing a precision ν, the algorithm converges when∥∥∥ϕdT − ϕd
T−1
∥∥∥S×Ωd
≤ ν.
4.2 Numerical Results
We present two numerical examples in economies with two agents. When H = 2, we justneed to keep track of the wealth share of agent 1 because ω2 = 1−ω1. Therefore, in (20),
ωdm
M
m=1=
ω1m
M
m=1
where0 = ω1
1 < ω12 < ... < ω1
M = 1.In the first example, Subsection 4.2.1, the agents differ in labor productivity but have thesame discount factor. In the second example, Subsection 4.2.2, the agents have the samelabor productivity but differ in their discount factor.
4.2.1 Heterogeneous Income
There are two representative agents h ∈ 1, 2 in the economy of mass 12 each. The agents
share the same intertemporal expected utility
E0
[∞
∑t=0
βt log cht
].
In each period, the exogenous aggregate state of the economy is a pair of states (s, i)where s ∈ b, g and i ∈ 0, 1. State s determines the aggregate productivity A(s) andaggregate labor supply L(s). The aggregate production function is Cobb-Douglas, (2).
State i determines which agent is employed. If i = 0 then agent 1 is unemployed andagent 2 is employed, and vice versa for i = 1.6 The employed agent has 2(1− υ)L(s) units
6This approximation of a fully idiosyncratic income process using a two agent income process is similar
19
of labor and the unemployed agent has 2υL(s) units of labor. υ stands for unemploymenttransfers by the government and is set at 7%.
The parameters are taken from Krusell and Smith (1998, Section 2) in particular, thediscount rate and the production parameters are:
β = 0.99 δ = 0.025 α = 0.36.
The aggregate productivity and aggregate labor supply are:[A(b) A(g)
]=[0.99 1.01
]and
[L(b) L(g)
]=[0.2944 0.3140
], (22)
with the transition matrix π = [πss′ii′ ], directly taken from Krusell and Smith (1998, Sec-tion 2):7
Figure 1 shows next period aggregate capital, K+ as a function of current period ag-
gregate capital, K, in state (b, 0), for two different values of ω: ω = 0 and ω = 1. Thefigure shows that future aggregate capital depends on not only current aggregate capitalbut also on current wealth share ω of agent 1.
Given the global nonlinear solution for ϕ∞, we can also simulate forward and carryout a regression exercise as in Krusell and Smith (1998). From 10, 000-period simulation(with the first 1000 periods dropped), we obtain the following regression results:
log K′ = 0.0438 + 0.9832 log K; R2 = 0.999223
in good times and
log K′ = 0.0167 + 0.9923 log K; R2 = 0.997372
in bad times. These regression results tell us that, in the simulated paths of the economy,current aggregate capital seems to be a sufficient state variable to forecast future aggregatecapital, which Krusell and Smith call an “approximate aggregation” property. However,Figure 1 tells us that this property does not hold globally.
As a comparison, we also solve the Krusell and Smith’s model, with the exact pa-rameters above, but in which idiosyncratic shocks are truly idiosyncratic. We obtain thefollowing regression results:
log K′ = 0.0906 + 0.9631 log K; R2 = 0.999999
in good times
log K′ = 0.0807 + 0.9651 log K; R2 = 0.999999
and in bad times.The approximate evolution of aggregate capital is not too different in the two-agent
economy compared to the Krusell and Smith’s economy. But we observe that the auto-
to the approximation in Heaton and Lucas (1995).7In the transition matrix, we use the convention 1, 2, 3, 4 correspond to (b, 0) , (b, 1) , (g, 0) , (g, 1)
respectively.
20
5 10 15 20 25 30 35 40 45 505
10
15
20
25
30
35
40
45
50
Current Aggregate Capital
Fu
ture
Aggre
gate
Cap
ital
(b,0) ω=0
(b,0) ω=1
450
Figure 1: Evolution of Aggregate Capital in Bad Times
21
5 10 15 20 25 30 35 40 45 505
10
15
20
25
30
35
40
45
50
Current Aggregate Capital
Fu
ture
Ag
gre
ga
te C
ap
ita
l
(b,0) ω=0
(b,0) ω=1
450
Figure 2: Evolution of Aggregate Capital in Bad Times
correlation coefficients for log aggregate capital are lower than those in the two-agenteconomy. The R2 are also slightly higher than in the two-agent economy.
4.2.2 Heterogenous Betas
In this example, we assume that the agents face idiosyncratic shocks that determine theirdiscount rate. The discount factor can be low (β) or high (β), where:
β = 0.9858 and β = 0.9930,
taken from Krusell and Smith (1998, Section 3). As in their paper, the transition from oneto the other is determined such that the average duration for individual β is 50 years,which corresponds to agents’ lifetime. To simplify the exercise, we assume that the twoagents have the same labor productivity, which varies with the aggregate state, s. Theaggregate productivity and aggregate labor supply are given in (22). The evolution ofthe aggregate state is the same as in the previous example. The other aggregate state idetermines the agents’ discount factor (i = 0 agent 1 has low discount factor and agent2 has high discount factor and vice versa for i = 1). The evolution of aggregate state i isindependent of the evolution of aggregate state s.
Figure 2 shows next period aggregate capital, K+ as a function of current period ag-gregate capital, K, in state (b, 0), for two different values of ω: ω = 0 and ω = 1. Thefigure shows that future aggregate capital depends mostly on current aggregate capitaland does not vary visibly with the current wealth share ω of agent 1.
As in the previous example, from 10, 000-period simulation (with the first 1000 periodsdropped), we obtain the following regression results:
22
log K′ = 0.0916 + 0.9633 log K; R2 = 0.999999
in good times
log K′ = 0.0789 + 0.9662 log K; R2 = 0.999999
and in bad times. Because future aggregate capital depends mostly on current aggregatecapital, the fitness of the linear regressions are very high.
As in the previous example, these regression results are comparable to the ones inKrusell and Smith (1998)’s model in which the discount rates are truly idiosyncratic:
log K′ = 0.0871 + 0.9662 log K; R2 = 0.999981
in good times and
log K′ = 0.0836 + 0.9670 log K; R2 = 0.999976
in bad times.
4.3 Discussion of the Algorithm for Many Agents and for Continuumof Agents
The discretization and approximation method laid out in Subsection 4.1 applies to generalmodel in Section 2 with many agents. However, when the number of agents is larger than2, the algorithm suffers from the curse of dimensionality, i.e., it takes many points todiscretize Ω using Ωd (the number of points is approximately Ndim(Ω) where N is thenumber of points used to discretize each dimension). There are two ways to get aroundthis problem.
First, notice that from Conditions A1-A3, in Subsection 4.1, we just need to solve for(ch)
h∈H as functions of the exogenous and endogenous states (s, K, ω) ∈ Ω. The idea isto approximate numerically ch’s using some basis functions
ξ1, ξ2, ..., ξmthat is
ch (s, K, ω) =m
∑i=1
chi ξ(s, K, ω).
We can then solve for the approximation coefficients(ch
i)h∈H
i=1,2,...,,m. The advantage of thisalgorithm is that the number of basis functions can be significantly smaller than the num-ber of points to discretize Ω and does not increase fast with the dimension of Ω. The ba-sis functions can be polynomials as in Judd (1992) and Gaspar and Judd (1997). Second,we can use Smolyak (1963)’s sparse-grid collocation method. The method only requiresknowing the value of ch’s at a few number of collocation points in Ω to approximatethe whole functions. A comprehensive exposition of the method and applications can befound in Maliar and Maliar (2014).
Using these ideas, the algorithm in Subsection 4.1 can potentially be applied to themodel in Section 3 with a continuum of agents. However, there are two major difficul-
23
ties. First, the endogenous state space Ω of probability measures is infinite-dimensional.Therefore, we need to approximate Ω with a finite-dimensional space. For example, onecan approximate Ω with the set of convex combinations of Dirac masses, i.e., for eachµ ∈ Ω we approximate µ by
µ ≈ ∑i∈I
M
∑j=1
µi,jD(k j)
where 0 ≤ k1 < k2 < ... < kM ≤ k, and ∑i∈I ∑Mj=1 µi,j = 1.8 Second, following Definition
4, for each µ ∈ Ω, we need to solve for the value and policy functions, V and k (in contrastto the case with a finite number of agents, we just need to solve for a vector of currentconsumptions and future capital holdings). In other words, we need to solve for thevalue and policy functions that depend both on capital holding and wealth distribution,V(k, i; s, µ) and k(k, i; s, µ). Having approximated Ω with a finite n−dimensional space, Vand k become functions over (n + 1) dimensions. They can then be approximated usingbasis functions or Smolyak’s sparse grid method.
While these are viable paths to implement the algorithm for many agents or for acontinuum of agents, they would require a significant amount of engineering and thus lieoutside the scope of the present paper.
5 Conclusion
In this paper, I define the concept of generalized recursive equilibrium and show its exis-tence in the neoclassical growth model with both idiosyncratic and aggregate shocks as inKrusell and Smith (1998). The proof applies equally well to economies with a finite num-ber of agents and with a continuum of agents. I also provide (rather strong) conditionsunder which a generalized recursive equilibrium gives rise to a recursive equilibrium. Ingeneral, however, it is still an open question whether a recursive equilibrium exists inthese economies. The question deserves further research given the rising important ofthis class of economies.
The proof suggests an algorithm to compute a recursive equilibrium if it exists. Thealgorithm is an global (in the space of distributions) alternative to the local ones put forthby Krusell and Smith (1998) and related algorithms presented in the JEDC’s symposiumDen Haan, Judd and Juillard (2010) as well as more recent algorithms in Gordon (2011),Mertens and Judd (2013), Childers (2015), Sager (2016), and Winberry (2016). These localalgorithms focus on a subset of the space of distributions around the stationary distribu-tion without aggregate shocks.9 There are also challenges in implementing the suggestedglobal nonlinear algorithm for economies with a large number or a continuum of agents.Subsection 4.3 suggests several possible ways to overcome these challenges.
8Approximating distributions using Dirac masses is similar the histogram technique used in Young(2010).
9For example, the backward algorithm in Reiter (2010) resembles the time iterations presented in Sub-section 4.1. But Reiter only uses a small set of aggregate statistics to summarize wealth distributions andthe statistics are discretized around the stationary distribution without aggregate shocks.
24
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Appendix
A Finite Agents and Finite Horizon Economy
To prove Theorem 1, first we show the existence of a competitive equilibrium in Lemma3. The proof of this lemma uses Kakutani’s Fixed Point Theorem.
We consider a finite horizon economy that lasts for T + 1 periods, t = 0, 1, ..., T. Givenprices
rt,T(st), wt,T(st)
t≤T,st∈S t
the representative firm solves
maxYt,Kt,Lt
Πt = Yt − rtKt − wtLt
s.t. Yt ≤ F(st, Kt, Lt). We allow for Πt potentially be different from 0, but we show that inequilibrium Πt = 0. We also assume that the profits (or losses) are divided equally acrossagents.
27
Given prices and the representative firm’s profit, agents solve
maxch
t,T ,kht+1,T
E0
[T
∑t=0
(Πt
t′=0βh(st))
u(cht,T)
](23)
s.t.ch
t,T + kht+1,T ≤ (1− δ)kh
t,T + rt,Tkht,T + wt,Tlh
t +1H
Πt
andch
t,T, kht+1,T ≥ 0.
A competitive equilibrium is defined similarly as in Definition 1. Lemma 3 show thata competitive equilibrium exists. Lemma 4 shows that
c ≤ cht,T ≤ c 0 ≤ kh
t,T ≤ k v ≤ vht,T ≤ v
K ≤ Kt ≤ K r ≤ rt,T ≤ r w ≤ wt,T ≤ w
with the bounds appropriately defined.
Lemma 3. Given an initial distribution of capital holding
kh0
h∈H such that
K0 =1H ∑
h∈Hkh
0 > 0,
a competitive equilibrium exists in the finite horizon economy.
Proof. The proof uses Kakutani’s Fixed Point Theorem as in Cao (2010), which buildsupon Debreu (1959).
To simplify the proof, we switch from choosing the final good as numeraire to thefollowing normalization:
pct,T(s
t) + wt,T(st) + rt,T(st) = 1.
The sequential budget constraint of the consumers become:
pct,T(c
ht,T + kh
t+1,T − (1− δ)kht,T) ≤ rt,Tkh
t,T + wt,Tlht +
1H
Πt.
The objective function of the representative firms
Πt,T = pct,TYt,T − rt,TKt,T − wt,T Lt,T. (24)
Given a sequence ε = εtTt=0 such that εt > 0 for t = 0, 1, ..., T, we impose an additional
restriction on the set of normalized prices:
pct,T(s
t) ≥ εt > 0. (25)
This restriction effectively puts an upper bound on marginal rate of returns on capital:
rt,T
pct,T≤ 1− εt
εt
therefore a lower bound on aggregate capital.For ε > 0, let ∆ε denote the subset of R3
+:
∆ε =(pc, w, r) ∈ R3
+ : pc + w + r = 1 and pc ≥ ε > 0
. (26)
28
For each history st , given our normalization and the additional restriction (25):(pc
t,T(st), wt,T(st), rt,T(st)
)∈ ∆εt .
We also denote
∆ΣT
ε =(
pct,T, rt,T, wt,T
)t,st :
(pc
t,T, rt,T, wt,T)∈ ∆εt
.
Given the prices, the representative firm maximizes (24) subject to
0 ≤ Yt,T, Kt,T, Lt,T and Yt,T ≤ F(st, Kt, Lt).
To ensure the compactness of the maximization problem, we impose additional restric-tions:
Kt,T ≤ 2K and Lt,T ≤ 2L,for all t and st, where K is defined in (10).
Similarly, each consumers maximize (23) subject to
pct,Tch
t,T + pct,T
(kh
t+1,T − (1− δ)kht,T
)≤ rt,Tkh
t,T + wt,Tlht +
1H
Πt,T (27)
and0 ≤ ch
t,T, kht+1,T
for all t and st.Because the representative firms’ choices are restricted on a compact set. Their profits
Πt are bounded above:Πt,T(st) < Π
for all t, st. Given the initial distribution of capital holding, the budget constraints (27)and the exogenous lower bound on pc
t,T, (25), it is easy to show that there exists c, k > 0such that
cht,T < c and kh
t+1,T < k
for all h, t, st.Let ψx denote the correspondence that maps each set of prices(
pt,T(st), rt,T(st), wt,T(st))
sT∈ΣT
to the excess demand in each market in each history:
ψx :∆ΣT
ε ⇒R3‖ΣT‖
pT ∈ ∆ΣT
ε 7→xT = (excess demands)
The component of the excess demand in each market corresponds to the component ofthe price system in that market:
Consumption:xct,T(s
t) =1H ∑
h∈H
(ch
t,T(st) + kh
t+1,T(st)− (1− δ)kh
t (st−1)
)−Yt,T(st)
Capital:xkt,T(s
t) = Kt,T(st)− 1H ∑
hkh
t,T(st−1)
Labor: xlt,T(s
t) = Lt,T(st)− L(st).
29
It is standard to show that ψx is upper hemi-continuous and compact, convex-valued.10
Given that each individual choices cht,T, kh
t+1,T are bounded, ψx is bounded by a closed
cube Kx ⊂ RΣT. For example,
−k ≤ xkt,T(s
t) ≤ 2K
for all t, st.Consider the following correspondence:
Ψ : ∆ΣT
ε ×Kx ⇒∆ΣT
ε ×KxpT ∈ ∆ΣT
ε , xT ∈ Kx
7→
arg maxp∈∆ΣT
ε
p · xT
× ψx(pT).
It is also standard to show that Ψ is a upper hemi-continuous, non-empty, compact,and convex valued correspondence. Kakutani’s Fixed Point Theorem then guaranteesthat Ψ has a fixed point
(pT, xT). By choosing ε appropriately, we can show that
(pT, xT)
corresponds to a competitive equilibrium. The proof is similar to the one in Lemma 6and Lemma 7 below so we omit the details here. For example, ε and Kt
Tt=0 are chosen
recursively using Assumption 3 and the agents’ Euler equation (28):1. ε0 and K0 are chosen as in Lemma 7.2. Given Kt−1, εt is chosen sufficiently small such that, for any K such that
FK(s, K, L) ≥1− εt
(1 + max0≤K≤2K FL(s, K, L)
)εt
,
for some L ∈ [0, 2L] we have(πs−s
((1− δ) εt + 1− εt
(1 + max
0≤K≤2KFL(s, K, L
))) 1σ
>F(s, K, L) + (1− δ)K
F(s−, Kt−1, L) + (1− δ)Kt−1 − K.
for all s, s− ∈ S .Kt is chosen such that for all s, s− ∈ S , and Kt ≤ Kt:
Lemma 4. Consider a competitive equilibrium with the initial aggregate capital K0 > 0 andlet K, K be defined as in (11) and (10). Then for all t ∈ 0, ..., T and st ∈ S t, we have K ≤Kt,T(st) ≤ K and, for all h ∈ H:
0 ≤ cht,T, kh
t,T ≤ H maxs∈SF(s, K, L(s)) + (1− δ)K = c = k,
andr = min
s∈Smin
L≤L≤LFK(s, K, L(s)) ≤ rt,T ≤ r = max
s∈Smax
L≤L≤LFK(s, K, L(s))
10The additional restriction (25) is crucial for the upper hemi-continuity of ψx. Without the restriction ψxis not upper hemi-continuous.
30
andw = min
s∈Smin
K≤K≤KFL(s, K, L) ≤ wt,T ≤ w = max
s∈Smax
K≤K≤KFL(s, K, L).
In addition, there exists c > 0 such that cht,T ≥ c for all t and st and h ∈ H.
Proof. First we show by induction that Kt,T(st−1) ≤ K for all t and st. At t = 0, thisproperty is satisfied by the definition of K. Assume that the property holds for t, and allst ∈ S t, we show that it holds for t + 1 and st+1 ∈ S t+1.
Indeed, from the market clearing conditions, we have
Kt+1(st) =1H ∑
h∈Hkh
t+1,T(st)
= Yt,T(st) + (1− δ)Kt(st−1)− 1H ∑
h∈Hch
t,T(st)
≤ Yt,T(st) + (1− δ)Kt(st−1)
= F(
st, Kt(st−1), L(st))+ (1− δ)Kt(st−1).
If Kt(st−1) ≥ K then
Kt+1(st) = Kt(st−1) + F(
st, Kt(st−1), L(st))− δKt(st−1)
≤ Kt(st−1) ≤ K.
If Kt(st−1) ≤ K then
Kt+1(st) = F(
st, Kt(st−1), L(st))+ (1− δ)Kt(st−1)
≤ maxs∈S
max0≤K≤K
F(K, L(s), s) + (1− δ)K ≤ K
So in either case we have Kt+1(st) ≤ K.Therefore, by induction, we have Kt,T(st−1) ≤ K for all t and st.Now we show by induction that
Kt,T(st−1) ≥ K
for all t and st.By the definition of K, K0 ≥ K. Now assume that Kt(st−1) ≥ K for all st ∈ S t, we show
by contradiction that Kt+1(st) ≥ K. Assume to the contrary, i.e. Kt+1(st) < K for somest ∈ S t.
From the first order condition of the agents, we have
u′(cht,T) ≥ βEt
[(1− δ + FK(st+1, Kt+1, Lt+1))u′(ch
t+1,T)]
(28)
for all h ∈ H. Therefore
u′(cht,T) ≥ min
st+1∈S(1− δ + FK(st+1, K, Lt+1))βEt
[u′(ch
t+1,T)]
≥ minst+1∈S
(1− δ + FK(st+1, K, Lt+1))βu′(
Et[cht+1,T]
)
31
where the last inequality comes from the fact that u′(c) = c−σ is strictly convex.Consequently,
However, this contradicts the inequality in Assumption 4. Therefore we must have Kt+1(st) ≥K. So by contradiction, Kt(st−1) ≥ K for all t and st.
The other inequalities for cht , kh
t , rt, wt follow immediately.Now we show that there exists c > 0 such that ch
t ≥ c for all t, st, and h. Indeed, fromthe agents’ maximization problem, since starting from any history st, an agent can alwaysconsumes her labor endowment, we have
u(cht (s
t)) + Et
[∞
∑t+1
t′
∏t′′=t′
β(st′′)u(cht′)
]≥ u(wl) + Et
[∞
∑t+1
t′
∏t′′=t′
β(st′′)u(wl)
]
≥ 11− β
u(wl).
In addition, cht′ ≤ c for all t′. Therefore
u(cht (s
t)) +β
1− βu(c) ≥ 1
1− βu(wl).
11We use the inequality that m ≤ ajbj
for all j implies m ≤ ∑ aj∑ bj
.
32
So
cht (s
t) ≥ c = u−1
(1
1− βu(wl)−
β
1− βu(c)
)> 0.
Proof of Theorem 1. The steps of the proof are similar to the ones for Theorem 2 presentedin the main paper. With some abuse, we re-use several notations in the proof such as Θ, getc.
Given the bounds determined in Lemma 4, let Θ denote the set of((ch, kh
+, vh)
h∈H, r, w
)such that c ≤ ch ≤ c, 0 ≤ kh
+ ≤ k, v ≤ vh ≤ v, r ≤ r ≤ r, and w ≤ w ≤ w.Let g : S × Ω ⇒ Θ × ΘS denote the following correspondence: for each s ∈ S ,
ω =(kh)
h∈H ∈ Ω, g(s, ω) is the set of θ =((
ch, kh+, vh)
h∈H , r, w)∈ Θ and (θs′)s′∈S ∈ ΘS
with θs′ =(
ch+(s′), kh
++(s′), vh+(s′)
h∈H , r+(s′), w+(s′)
)s′∈S
such that:
1H ∑
h∈Hch +
1H ∑
h∈Hkh+ = F(s, K, L) + (1− δ)K
where K = 1H ∑h∈H kh > 0 and L = 1
H ∑h∈H lh(s) > 0, and r = FK(s, K, L) > 0 andw = FL(s, K, L) > 0. In addition, (7), (8), and (9) are satisfied.
It is easy to show that g is a closed-valued correspondence.Consider the following mapping G from the set of correspondences V : S ×Ω ⇒ Θ ⊂
R3H+2 to itself as following. For each V , G(V) is the correspondence W such that, foreach s ∈ S and ω =
(kh)
h∈H ∈ Ω, we have
W(s, ω) =
θ =
((ch, kh
+, vh)h∈H , r, w
)∈ Θ : for each s′ ∈ S ,∃θs′ ∈ V
(s′,(kh+
)h∈H
)and
(θ, (θs′)s′∈S
)∈ g(s, ω)
From the definition of G, we have the following properties P1-P3:P1. If V is compact in the sense that V(s, ω) is compact for all s ∈ S and ω ∈ Ω, then
W = G(V) is compact.Indeed, assume θm∞
m=0 ∈ W(s, ω), and θm → θ =((
ch, kh+, vh)
h∈H , r, w)
. Since Θis compact, θ ∈ Θ. To show that G(V) is compact, we need to show that θ ∈ W(s, ω).By the definition of G, for each s′ ∈ S , ∃θm
s′ ∈ V(
s′,(kh+
)h∈H
)and
(θm,(θm
s′))∈ g(s, ω).
Since V(
s′,(kh+
)h∈H
)is compact, we can extract a converging subsequence, θ
mls′ → θs′ for
some θs′ ∈ V(
s′,(kh+
)h∈H
). Because g is a closed valued correspondence,
(θ, (θs′)s′∈S
)∈
g(s, ω), which implies θ ∈ W(s, ω).P2. If V ⊂ V ′ in the sense that V(s, ω) ⊂ V ′(s, ω) for all s ∈ S and ω ∈ Ω then
G(V) ⊂ G(V ′).P3. Let V0 denote the complete correspondence: V0(s, ω) = Θ for all s ∈ S and ω ∈ Ω.
Then G(V0) ⊂ V0.
33
Given V0, we construct the sequence of Vn∞n=0 recursively using G: Vn+1 = G(Vn).
Then by P1, P2, and P3, we have Vn+1 ⊂ Vn and is non-empty, compact valued. Non-empty-ness comes from the existence of competitive equilibrium in the n + 1-horizoneconomy proved in Lemma 3.
Let V∗ be defined byV∗(s, ω) = ∩∞
n=0Vn(s, ω).Since V∗(s, ω) is the intersection of decreasing compact sets, V∗(s, ω) is compact and isnon-empty. We show that G(V∗) = V∗.
Indeed, by the definition of V∗, we also have V∗ ⊂ Vn, to G(V∗) ⊂ G(Vn) = Vn+1, soG(V∗) ⊂ V∗.
Now, for each s ∈ S and ω ∈ Θ and θ =(
ch, kh+, vh
h∈H , r, w)∈ V∗(s, ω). Since
V∗ ⊂ Vn, there exists θns′ ∈ V
n(
s′,(kh+
)h∈H
)and
(θ,(θn
s′)
s′∈S
)∈ g(s, ω). By the com-
pactness of Θ, we can find a converging subsequence nl∞l=0,
(θ
nls′)
s′∈S −→l→∞ (θs′)s′∈S .
By the compactness of Vnl , we have θs′ ∈ Vnl
(s′,(kh+
)h∈H
)and since g has closed graph,(
θ, (θs′)s′∈S)∈ g(s, ω). Moreover, Vn
(s′,(kh+
)h∈H
)is a decreasing sequence so θs′ ∈
∩∞l=0Vnl
(s′,(kh+
)h∈H
)= V∗
(s′,(kh+
)h∈H
). So by the definition of G, we have θ ∈ G(V∗).
Therefore V∗ ⊂ G(V∗).Since G(V∗) ⊂ V∗ ⊂ G(V∗), it implies that G(V∗) = V∗.Let Q = V∗. Since G(Q) = Q, for each s ∈ S and each ω =
(kh)
h∈H ∈ Ω, θ =((ch, kh
+, vh)h∈H , r, w
)∈ Q(s, ω), there exists θs′ ∈ Q
(s′,(kh+
)h∈H
)for each s′ ∈ S and(
θ, (θs′)s′∈S)∈ g(s, ω). We also define T as
T (s, ω) =(
kh+
)h∈S
:((
ch, kh+, vh
)h∈H
, r, w)∈ Q(s, ω) for some
(ch, vh
)h∈H
and some r, w
.
It is immediate that (P , T ) defined as such forms a generalized recursive equilibrium forthe economy with a finite number of types.
Proof of Lemma 1. Consider sequences of allocation and prices generated by a general-ized recursive equilibrium, starting from s0 ∈ S and
kh
0
h∈H ∈ Ω. That is, sequencesch
t (st), kh
t+1(st), vh
t (st)
t,st,h and
rt(st), wt(st)
t,st such that for each t, st(ch
t (st), kh
t+1(st), vh
t (st)
h∈H, rt(st), wt(st)
)∈ Q
(st,
kht (s
t)
h∈H
),
and(ch
t+1(st+1), kh
t+2(st+1), vh
t+1(st+1)
h∈H
, rt+1(st+1), wt(st+1))∈ Q
(st+1,
kh
t+1(st)
h∈H
),
and Conditions 3-4 in Definition 2 are satisfied (with the variable without subscript standsfor the variable at time t, the variables with subscript + stands for the variables at time t+1 and the variables with subscript ++ stands for the variables at time t + 2, for examplech stands for ch
t , ch+ stands for ct+1 and ch
++ stands for cht+2, etc.).
The market clearing conditions are satisfied obviously. We just need to verify thatgiven
rt(st), wt(st)
, the allocation
ch
t (st), kh
t+1(st)
t,st solves agent h’s maximization
34
problem, (5). That is for any alternative allocation
cht (s
t), kht+1(s
t)
t,st that satisfies (3)and (4), we have
E0
[∞
∑t=0
(Πt−1
t′=0βh(st′))
u(cht )
]≥ E0
[∞
∑t=0
(Πt−1
t′=0βh(st′))
u(cht )
]. (29)
The proof of this inequality follows closely Duffie et al. (1994). First, we show by induc-tion that for all T ≥ 0:
E0
[∞
∑t=0
(Πt−1
t′=0βh(st′))
u(cht )
]≥ E0
[T
∑t=0
(Πt−1
t′=0βh(st′))
u(cht )
]+ E0
[∞
∑T+1
(Πt−1
t′=0βh(st′))
u(cht )
]+ E0
[(ΠT−1
t′=0βh(st′))
u′(chT)(
khT+1 − kh
T+1
)]. (30)
For T = 0, inequality (30) is
u(ch0) ≥ u(ch
0) + u′(ch0)(
kh1 − kh
1
),
which is true becauseu(ch
0) ≥ u(ch0) + u′(ch
0)(
ch0 − ch
0
),
from the concavity of u(.) and from
ch0 + kh
1 ≤ (1− δ + r0)kh0 = ch
0 + kh1,
which implies ch0 − ch
0 ≥ kh1 − kh
1.Now, assume that (30) holds for T, we need to show that it also holds for T + 1, i.e.
E0
[∞
∑t=0
(Πt−1
t′=0βh(st′))
u(cht )
]≥ E0
[T+1
∑t=0
(Πt−1
t′=0βh(st′))
u(cht )
]+ E0
[∞
∑T+2
(Πt−1
t′=0βh(st′))
u(cht )
]+ E0
[(ΠT
t′=0βh(st′))
u′(chT+1)
(kh
T+2 − khT+2
)]. (31)
Given (30) holds for T, to show (31), we just need to show:
E0
[(ΠT
t′=0βh(st′))
u(chT+1)
]+ E0
[(ΠT−1
t′=0βh(st′))
u′(chT)(
khT+1 − kh
T+1
)]≥ E0
[(ΠT+1
t′=0βh(st′))
u(chT+1)
]+ E0
[(ΠT
t′=0βh(st′))
u′(chT+1)
(kh
T+2 − khT+2
)].
Equivalently,
E0
[βh(sT)u(ch
T+1)]+ E0
[u′(ch
T)(
khT+1 − kh
T+1
)]≥ E0
[βh(sT)u(ch
T+1)]+ E0
[βh(sT)u′(ch
T+1)(
khT+2 − kh
T+2
)]. (32)
Because of Condition 5. in Definition 2,
u′(chT) = βh
T(sT)ET
[(1− δ + rT+1) u′(ch
T+1)]
,
if khT+1 > 0, and
u′(chT) ≥ βh
T(sT)ET
[(1− δ + rT+1) u′(ch
T+1)]
,
35
if khT+1 = 0, which implies kh
T+1 − khT+1 ≥ 0 . Therefore
E0
[u′(ch
T)(
khT+1 − kh
T+1
)]≥ E0
[βh
T(sT) (1− δ + rT+1) u′(chT+1)
(kh
T+1 − khT+1
)].
From this inequality, we obtain (32) if
u(chT+1) + (1− δ + rT+1) u′(ch
T+1)(
khT+1 − kh
T+1
)≥ u(ch
T+1) + u′(chT+1)
(kh
T+2 − khT+2
). (33)
Since
chT+1 + kh
T+2 = (1− δ + rT+1) khT+1
chT+1 + kh
T+2 ≤ (1− δ + rT+1) khT+1,
we have(1− δ + rT+1)
(kh
T+1 − khT+1
)≥ kh
T+2 − khT+2 + ch
T+1 − chT+1.
Plugging this into (33), we obtain the desired inequality if
u(chT+1) + u′(ch
T+1)(
chT+1 − ch
T+1
)≥ u(ch
T+1),
which is true because u(.) is concave.Having established (30), we are ready to show (29). First we observe that, because Ω
is compact, there exists k > 0 such that kht (s
t) ≤ k for all h, t, st. Now, from (30), takingT → ∞ and noticing that
E0
[∞
∑T+1
(Πt−1
t′=0βh(st′))
u(cht )
]≥ βT+1 1
1− βu(c)→T→∞ 0
and
E0
[(ΠT−1
t′=0βh(st′))
u′(chT)(
khT+1 − kh
T+1
)]≥ −E0
[(ΠT−1
t′=0βh(st′))
u′(chT)k
hT+1
]≥ −βTu′(c)k −→T→∞ 0,
we obtain (29).
B Continuum of Agents and Finite Horizon Economy
As in the case with finite number of agents, we first show the existence of competitiveequilibrium in a finite horizon economy. Then we show that in any competitive equilib-rium, prices and allocations lie in compact sets.
Consider the finite horizon version of the economy in Section 3 with t = 0, 1, ..., T. Werestate the competitive equilibrium in terms of wealth distributions as following.
36
Given prices rt,T(st), wt,T(st)
t=0,...,T;st∈S t ,
the representative firm solves
maxYt,T,Kt,T ,Lt,T
Πt,T = Yt,T − rt,TKt,T − wt,T Lt,T
s.t. Yt,T ≤ F(st, Kt,T, Lt,T). We allow for Πt,T to be potentially different from 0, but weshow that in equilibrium Πt = 0. We also assume that the profits (or losses) are dividedequally across agents.
Given prices and the representative firm’s profit, the value function of the agents,Vt,T(k, i; st) satisfies the Bellman equation (starting from t = T + 1 with VT+1,T ≡ 0 mov-ing backward):
with the policy functions ct,T(k, i; st) and kt,T(k, i; st).A competitive equilibrium consists of prices
rt,T(st), wt,T(st)
, aggregate capital Kt,T(st),
value and policy functions Vt,T, ct,T, kt,T that satisfy (34) and sequences of wealth distri-bution µt,T(k, i; st) such that the following identity holds:
∑i∈I
∫ct,T(i, k; st)µt,T(dk, i; st) + ∑
i∈I
∫kt,T(i, k; st)µt,T(dk, i; st)
= F(st, Kt,T(st−1), L(st)) + (1− δ)Kt,T(st−1)
where
Kt,T(st−1) = ∑i∈I
∫ k
0kµt(dk, i; st) and L(st) = ∑
i∈Imst(i)l(st, i).
In addition, we have
rt,T(st) = FK
(st, Kt,T(st−1), L(st)
)> 0 and wt,T(st) = FL
(st, Kt,T(st−1), L(st)
)> 0.
And lastly, the sequences of wealth distributions are consistent with the policy func-tions:For each st+1 ∈ S , and A ∈ B
([0, k])
:
µt+1,T(
A, i′;(st, st+1
))= ∑
i∈I
πss′,ii′
πss′µt,T
((kt,T(., i; st))−1(A), i; st
).
In the proof of Lemma 2, this recursive version of a competitive equilibrium can bemapped back to the definition using agents’ index, as in Definition 3.
The following lemma establishes the existence of this recursive version of a competi-tive equilibrium.
Lemma 5. There exists a competitive equilibrium in the finite horizon economy version of themodel in Section 3 with a continuum of agents.
37
Proof. Given a sequence εT = εtTt=0 such that εt > 0 for all t ∈ 0, 1, ..., T, let us define
∆ΣT
εT = ×(t,st)∆εt
=
(pc
t , wt, rt)t,st ∈(
R3+
)ΣT
: pct + wt + rt = 1 and pc
t ≥ εt
,
where ∆ε is defined in (26).Given the prices pT ∈ ∆ΣT
εT . The firms maximize profit and the agents maximize theinter-temporal expected utility. In particular, in each history st the representative firmsolves
maxYt,Kt,Lt≥0
Πt,T(Kt,T, Lt,T) (36)
s.t.Yt,T ≤ F(st, Kt,T, Lt,T)
andΠt,T = pc
t,TYt,T − rt,TKt,T − wt,T Lt,T.We also impose two additional constraints:
0 ≤ Lt,T ≤ 2L and 0 ≤ Kt,T ≤ 2K,
where L and K are defined in Assumption 5 and in (10). Given ΠT =(Πt,T(st)
),12 the
agents solve the dynamic programing problem: Vt,T is defined recursively (starting fromt = T + 1 with VT+1,T ≡ 0 moving backward) as
Vt,T(k, i; st, pT, ΠT) = maxc,k′
u(c) + β(i)Et
[Vt+1,T(k′, i; st, pT, ΠT)
](37)
subject to c ≥ 0, 0 ≤ k′ ≤ k and
pct,T(s
t)(c + k′ − (1− δ)k
)≤ rt,T(st)k + wt,T(st)l(st, i) + Πt,T. (38)
In Lemma 8, we show that the policy function for k′, kt,T(k, i; st) is continuous and isweakly increasing.
Given the policy function kt,T, we construct the sequence of measures µT =(µt,T(.; st)
)t,st
as following:1. µ0,T = µ0,T2. For t ≥ 0, for every A ∈ B
([0, k])
µt+1,T(A, it+1; st+1) = ∑it∈I
Pr(it+1|it, st, st+1)µt,T
((kt,T
)−1(A), it; st
).
We denoteψµ : ∆ΣT
εT × ΩΣT⇒ ΩΣT
the correspondence that map the sequence of prices pT and distributions µT to the se-quence of distributions µT as constructed above.
12The maximization problem (36) might have many maximizers but the maximized objective Πt,T isuniquely determined given prices pt,T ∈ ∆εt .
38
We form the excess demand functions
Consumption:xct,T(s
t) =∫ k
0
(ct,T(k, i; st) + kt,T(k, i; st)− (1− δ)k
)µt,T(dk; it, st)−Yt,T(st)
Capital:xkt,T(s
t) = Kt,T(st)−∫ k
0kµt,T(dk, it; st)
Labor: xlt,T(s
t) = Lt,T(st)− L(st).
We have:xc = −(1− δ)k− Y < xc
t,T(st) < xc =
1− ε
εk +
1− ε
εL +
1ε
Y,
where ε = 12 min0≤t≤T εt. We also have
xk = −2k < xkt,T(s
t) < xk = 2K
andxl = −2L < xl
t,T(st) < xl = 2L
Let Kx denote the cube([xc, xc]×
[xk, xk]× [xl, xl])ΣT
. We define the correspondence
ψx : ∆ΣT
ε × ΩΣT ⇒ Kx
that maps a sequence of prices pT and a sequence of distributions µT to the excess demandin every history.
Lastly,ψp : Kx ⇒ ∆ΣT
εT
such thatpt,T = arg max
p∈∆εt
p · xt,T.
Let Φε denote an operator (which depends on εt) taking φp, φµ, φx as components:
Ψε : ∆ΣT
εT × ΩΣT ×Kx ⇒ ∆ΣT
εT × ΩΣT ×Kx (39)
Ψε =(ψp, ψµ, ψx
)Lemma 9 shows that Ψε is upper-hemi continuous and is non-empty, compact, and
convex valued. In addition ∆ΣTε × ΩΣT ×Kx is a compact and convex subset of a locally
convex Hausdorff space.13 Therefore, by the Kakutani-Glicksberg-Fan Fixed Point The-orem, Φε admits a fixed point. By choosing εt appropriately, in Lemma 6, we show thatthis fixed point constitutes a competitive equilibrium.
Lemma 6. Consider the sequence εt, KtTt=0 constructed in Lemma 7, and Ψε as defined in (39)
given the sequence εt. Let
ψ =(( pt,T)t,st , (µt,T)t,st , (xt,T)t,st
)be a fixed point of Ψε. Then ψ corresponds to a competitive equilibrium.
13These properties follow directly from the result that the space Ω of probability measures endowedwith weak topology is metrizable, shown in Bogachev (2000, Theorem 8.3.2).
39
Proof. To show that ψ corresponds to a competitive equilibrium, we need to show thatxt,T(st) = 0 and rt,T, wt,T > 0 for all t ≤ T and st ∈ S t. To simplify the notations, we omitthe bar notation on variables.
First, we notice that for all t ≤ T and st ∈ S t,
pt,T · xt,T = pct,Txc
t,T(st) + rt,Txk
t,T + wt,Txlt,T
= pct,T ∑
i∈I
∫ k
0
(ct,T(k, i; st) + kt,T(k, i; st)− (1− δ)k
)µt,T(dk, it; st)− pc
t,TYt,T
+ rt,TKt,T − rt,T ∑i∈I
∫ k
0kµt,T(dk; it, st) + wt,T Lt,T − wt,T L(st).
= pct,T ∑
i∈I
∫ k
0
(ct,T(k, i; st) + kt,T(k, i; st)− (1− δ)k
)µt,T(dk, it; st)
−Πt,T(Kt, Lt)− rt,T ∑i∈I
∫ k
0kµt,T(dk; it, st)− wt,T(st) ∑
i∈I
∫ k
0l(st, it)µt,T(dk, it; st).
Since pct > 0, (38) holds with equality for each k, i. Therefore the last expression equal to
0. So pt,T · xt,T = 0 for all t, st.From the definition of a fixed point, we have
pt,T ∈ arg maxp∈∆εt
p · xt,T
So0 = pt,T · xt,T ≥
(1 0 0
)· xt,T = xc
t,T
orxc
t,T(st) ≤ 0 ∀t ≤ T and ∀st ∈ S t. (40)
Now, we show by induction that xt,T = 0 for all t ≤ T and st ∈ S t. In particular, weshow that, x0,T = 0 and r0,T, w0,T > 0 (Step 1) and if xt−1,T = 0 and rt−1,T, wt−1,T > 0 forall st−1 ∈ S t−1 then xt,T = 0 and rt,T, wt,T > 0 and in addition Kt > Kt (Step 2).
Step 1: Starting with t = 0, we have just shown in (40) that xc0,T ≤ 0.
If xk0,T < 0 then r0,T = 0 (since p0,T ∈ arg maxp p · x0,T and p0,T · x0,T = 0). The
maximization of the representative firm, (36), at t = 0 implies that K0,T = 2K. But then
xk0,T > 0 since we chose 2K > K0 =
∫ k0 kµ0,T(dk; i0, s0). So xk
0,T ≥ 0.Similarly, if xl
0,T < 0 then w0,T = 0. Then, also from the maximization of the repre-sentative firm, (36), at t = 0, L0,T = 2L. But then xl
0,T > 0 since we chose 2L > L >
maxs∈S L(s). So xl0,T ≥ 0.
Now, we show by contradiction that xc0,T = 0. Assume to the contrary that xc
0,T < 0.Then pc
0,T = ε0 (since p0,T ∈ arg maxp p · x0,T and p0,T · x0,T = 0). Therefore we have
r0,Txk0,T + w0,Txl
0,T = −ε0xc0,T > 0. (41)
If r0,T = 0 then K0,T = 2K and
xk0,T ≥ 2K− K0 > K > 2L > xl
0,T.
40
But then, because p0,T ∈ arg maxp p · x0,T, we have r0,T = 1− ε0 and w0,T = 0, a con-tradiction with the assumption that r0,T = 0. So we must have r0,T > 0. And sincep0,T ∈ arg maxp p · x0,T, it must be that xk
0,T ≥ xl0,T. From (41), we have xk
0,T > 0.If w0,T = 0 then L0,T = 2L > 0 and xl
0,T = L0,T − L(s0) > 0. If w0,T > 0 then sincep0,T ∈ arg maxp p · x0,T, it must be that xl
0,T ≥ xk0,T. Since we have shown above that
xk0,T ≥ xl
0,T, this leads to xk0,T = xl
0,T > 0. In either case, we have xl0,T > 0.
Therefore K0,T = xk0,T + K0 > K0 and L0,T = xl
0,T + L(s0) > L.Now, at time t = 0 and s0, (because p0,T = ε0) (Y0,T, K0,T, L0,T) solves:
maxY,K,L
ε0Y− r0,TK− w0,T L
s.t.Y ≤ F(s0, K, L)
and K ≤ 2K, L ≤ 2L. Because ε0 > 0, Y0,T = F(s0, K0,T, L0,T) and
where the last inequality comes from condition (10) on K.Now, we show that xt,T = 0 and rt,T, wt,T > 0. Indeed, we show in (40) that xc
1,T(s1) ≤
0. The following arguments are similar to the argument in Step 1.If xk
t,T(st) < 0 then rt,T(s1) = 0. Then Kt,T(st) = 2K but then xk
t,T(st) > 0 since we have
shown that K > ∑i∈I∫ k
0 kµt,T(dk; i, st). So xkt,T(s
t) ≥ 0.If xl
t,T(st) < 0 then wt,T(s1) = 0. Then Lt,T(st) = 2L but then xl
t,T(st) > 0 since
2L > maxs∈S L(s). So xlt,T(s
t) ≥ 0.We show by contradiction that xc
t,T(st) = 0. Assume to the contrary that, xc
t,T(st) < 0.
Then pct,T(s
t) = εt (since pt,T ∈ arg maxp∈∆εtp · xt,T and xk
t,T, xlt,T ≥ 0). Therefore,
rt,T(st)xkt,T(s
t) + wt,T(st)xlt,T(s
t) = −εtxct,T(s
t) > 0. (43)
If rt,T(st) = 0 then Kt,T(st) = 2K and
xkt,T(s
t) = 2K−∑i∈I
∫ k
0kµt,T(dk, i; st)
> 2K− K > 2L > xlt,T(s
t).
But then, since pt,T ∈ arg maxp p · xt,T, so rt,T(st) = 1− εt and wt,T(st) = 0, a contradictionwith the assumption that rt,T(st) = 0. So we must have rt,T(st) > 0. Therefore, sincept,T ∈ arg maxp p · xt,T, it implies that xk
t,T(st) ≥ xl
t,T(st). From (43), we have xk
t,T(st) > 0.
If wt,T(st) = 0 then Lt,T(st) = 2L > 0 and xlt,T(s
t) = Lt,T(st)− L(st) > 0. If wt,T(st) > 0then since pt,T ∈ arg maxp p · xt,T, it must be that xl
t,T(st) ≥ xk
0,T(st). We have just shown
above that xkt,T(s
t) ≥ xlt,T(s
t). This leads to xkt,T(s
t) = xlt,T(s
t) > 0. In either case, we havexl
t,T(st) > 0.
Therefore Kt,T(st) = xkt,T(s
t) + Kt−1,T(st−1) > Kt−1 and Lt,T(st) = xlt,T(s
t) + L(st) > L.
42
As we show in (40) above xct,T(s
t) ≤ 0, which by the definition of xct,T implies:
∑i
∫ k
0
(ct,T(k, i; st) + kt,T(k, i; st)− (1− δ)k
)µt,T(dk; i, st)−Yt,T(st) ≤ 0.
Therefore
∑i
∫ k
0ct,T(k, i; st)µt,T(dk; i, st) ≤ (1− δ)∑
i
∫ k
0kµt,T(dk; i, st) + Yt,T(st)
≤ (1− δ)Kt,T(st) + Yt,T(st), (44)
where the last inequality comes from xkt,T(s
t) ≥ 0.From the agents’ Euler equation, shown in Lemma 8,
For t > 0, given εt−1 and Kt−1, we choose εt and Kt as follow.There exists εt > 0 such that the following properties e1. and e2. are satisfied.e1. εt is sufficiently small such that
εt < mins∈S ,0≤L≤2L
1
FK
(s, (1−δ)Kt−1
2(
1+ 2εt−1
) , L
)+ 1 + max0≤K≤2K FL(s, K, L)
(53)
ande2. for all s, s− ∈ S , and for all K such that
FK(s, K, L) ≥1− εt(1 + max0≤K≤2K FL(s, K, L))
εt,
for some 0 ≤ L ≤ 2L , we have(πs−s((1− δ)εt + 1− εt(1 + max
0≤K≤2KFL(s, K, L)))
) 1σ
>F(s, K, 2L) + (1− δ)K
12(1− δ)Kt−1
. (54)
Given εt−1, εt, and Kt−1, there exists Kt <(1−δ)Kt−1
1+ 2εt−1
such that for all s, s− ∈ S and K ≤ Kt,
(πs−s(1− δ)εt)1σ >
F(s, K, 2L) + (1− δ)KF(s−, Kt−1, L)
.
Proof. The existence of K0 is immediate. By Assumption 3,
max0≤L≤2L
FK(s0, K, L) < +∞ and max0≤K≤2K
FL(s, K, L) < +∞,
so there exists ε0 > 0 that satisfies (52). Now we construct εt and Kt recursively.The right hand side of (53) is finite and is strictly positive. Let εt > 0 denote this value.
Let
εt = min
εt,
12(1 + max0≤K≤2K FL(s, K, L))
) .
By Assumption 3, there exists Kt such that for all s, s− ∈ S :(πs−s
12
) 1σ 1
2(1− δ)Kt−1 > F(s, K, L) + (1− δ)K, (55)
46
for all K ≤ Kt. Again, by Assumption 3, we can find 0 < εt < εt such that
max0≤L≤2L
FK(s, Kt, L) ≤1− εt(1 + max0≤K≤2K FL(s, K, L))
εt. (56)
We show that εt defined as such satisfies Properties e1. and e2.Since εt < εt, (53) holds, i.e., εt satisfies e1. Now we show that εt satisfies e2. Indeed,
for all s, s− ∈ S , and for all K such that
FK(s, K, L) ≥1− εt(1 + max0≤K≤2K FL(s, K, L))
εt,
for some 0 ≤ L ≤ 2L , by (56), we have K ≤ Kt. Because
εt <1
2(1 + max0≤K≤2K FL(s, K, L))
)we have (
πs−s((1− δ)εt + 1− εt(1 + max0≤K≤2K
FL(s, K, L)))) 1
σ
>
(πs−s
12
) 1σ
So (55) yields (54).By Assumption 3 (limK→0 F(s, K, 2L) = 0), there exists Kt such that
Lemma 8. Consider the value and policy functions defined recursively by the Bellman equations,(37). We have the following properties:
1. The value functions Vt,T are continuous, strictly increasing, strictly concave.2. The corresponding policy correspondence, ct,T, kt,T are single-valued, i.e., are functions,
continuous, and kt,T are weakly increasing, and the budget constraints, (38), hold with equality.3. (Euler Equation) If k′ = kt,T(k, i; st) < k then
Proof. These properties are standard. The single-valued property of the policy functioncomes from the fact that u(.) is strictly concave. The monotonicity of kt,T comes from astandard-single crossing argument.
We show that k′2 ≥ k′1.If k′2 ≥ R1 then k′2 ≥ R1 ≥ k′1 we obtain the desired inequality.Otherwise, from (37), we have
u
(R1 − k′1pc
t,T(st)
)+ β(i)Et
[Vt+1,T(k′1, i; st, pT, ΠT)
]≥ u
(R1 − k′2pc
t,T(st)
)+ β(i)Et
[Vt+1,T(k′2, i; st, pT, ΠT)
]and
u
(R2 − k′2pc
t,T(st)
)+ β(i)Et
[Vt+1,T(k′1, i; st, pT, ΠT)
]≥ u
(R2 − k′1pc
t,T(st)
)+ β(i)Et
[Vt+1,T(k′2, i; st, pT, ΠT)
].
Adding up the two inequalities side by side and simplify, we obtain
u
(R1 − k′1pc
t,T(st)
)− u
(R2 − k′1pc
t,T(st)
)≥ u
(R1 − k′2pc
t,T(st)
)− u
(R2 − k′2pc
t,T(st)
)which implies k′1 ≤ k′2 since u is concave.
Lemma 9. We show that the correspondence Ψε constructed in Lemma 5 is upper hemi-continuous,and is non-empty, compact and convex valued.
Proof. In order to show that Ψε is upper hemi-continuous, we need to show that givenany sequence (pn, µn, xn) ∈ ∆ΣT
εT × ΩΣT × Kx that converges to some (p, µ, x) ∈ ∆ΣT
εT ×ΩΣT ×Kx:
(pn, µn, xn)→ (p, µ, x)and
( pn, µn, xn)→ ( p, µ, x)and
( pn, µn, xn) ∈ Ψε (pn, µn, xn)
then we must have:( p, µ, x) ∈ Ψε (p, µ, x) .
Indeed, since( pn, µn, xn) ∈ Ψε (pn, µn, xn) ,
there exists Yn
t,T(st), Kn
t,T(st), Ln
t,T(st)
t,st
48
that solves (36). LetΠn
t,T = pc,nt,TYn
t,T − rnt,TKn
t,T − wnt,T Ln
t,T,
and Πn,T =(
Πnt,T(s
t))
t,st.
Let Vnt,T(k, i; st, pT,n, ΠT,n) denote the value functions that solves (37) given pT,n and
ΠT,n and knt,T(k, i; st, pT,n, ΠT,n) denote the corresponding policy functions.
By choosing converging subsequence, we can assume that there existsYt,T(st), Kt,T(st), Lt,T(st)
t,st
such that (Yn
t,T(st), Kn
t,T(st), Ln
t,T(st))−→n→∞
(Yt,T(st), Kt,T(st), Lt,T(st)
)for all t, st.
First, we show that for all t and st,(Yt,T(st), Kt,T(st), Lt,T(st)
)solves (36) given pT.
Indeed, for any (Y, K, L) such that
Y ≤ F(st, Kt,T, Lt,T)
and0 ≤ L ≤ 2L and 0 ≤ K ≤ 2K,
since(
Ynt,T(s
t), Knt,T(s
t), Lnt,T(s
t))
solves (36), we have
pn,ct,TY− rn
t,TK− wnt,T L ≤ pn,c
t,TYnt,T − rn
t,TKnt,T − wn
t,T Lnt,T.
Taking n→ ∞, we obtain
pct,TY− rt,TK− wt,T L ≤ pc
t,TYt,T − rt,TKt,T − wt,T Lt,T.
Therefore(Yt,T(st), Kt,T(st), Lt,T(st)
)solves (36) given pT.
In addition, from the expression for Πnt,T(s
t) and Πt,T(st), we also have Πnt,T(s
t) −→n→∞
Πt,T(st). Lemma 10 then shows that
knt,T(., i; st, pn,T) −→n→∞ kt,T(., i; st, pT)
uniformly over[0, k].
From the definition of ψx:
xct,T;n(s
t) = ∑i∈I
∫ (ct,T;n(k, i; st) + kt,T;n(k; i, st)− (1− δ)k
)µn
t,T(dk; i, st)−Ynt,T(s
t)
xkt,T;n(s
t) = Knt,T(s
t)−∫
kµnt,T(dk; it, st)
xlt,T;n(s
t) = Lnt,T(s
t)− L(st).
As shown in Lemma 8,
ct,T;n(k, i; st) + kt,T;n(k; i, st)− (1− δ)k =rn
t,T(st)k + wn
t,T(st)l(st, it) + Πn
t,T(Knt,T, Ln
t,T)
pn,ct,T(s
t).
49
Therefore
xct,T;n(s
t) = ∑i∈I
∫ (ct,T;n(k, i; st) + kt,T;n(k; i, st)− (1− δ)k
)µn
t,T(dk; i, st)−Ynt,T(s
t)
= ∑i∈I
∫ (rnt,T(s
t)k + wnt,T(s
t)l(st, it) + Πnt,T(K
nt,T, Ln
t,T)
pn,ct,T(s
t)
)µn
t,T(dk; i, st)−Ynt,T(s
t)
=rn
t,T(st)
pn,ct,T(s
t) ∑i∈I
∫kµn
t,T(dk; i, st) +rn
t,T(st)
pn,ct,T(s
t) ∑i∈I
∫l(st, it)µ
nt,T(dk; i, st)
+Πn
t,T(Knt,T, Ln
t,T)
pn,ct,T(s
t)−Yn
t,T(st).
Because pn → p, and pn,ct,T, pc
t,T > εt > 0,rn
t,T(st)
pn,ct,T(s
t)→ rt,T(st)
pct,T(s
t)and
rnt,T(s
t)
pn,ct,T(s
t)→ rt,T(st)
pct,T(s
t). In
addition, as we show above,Πn
t,T(Knt,T ,Ln
t,T)
pn,ct,T(s
t)→ Πt,T(Kt,T ,Lt,T)
pct,T(s
t)and Yn
t,T(st)→ Yt,T(st).
Because µn → µ,
∑i∈I
∫kµn
t,T(dk; i, st)→ ∑i∈I
∫kµt,T(dk; i, st)
and
∑i∈I
∫l(st, it)µ
nt,T(dk; i, st)→ ∑
i∈I
∫l(st, it)µt,T(dk; i, st).
Therefore, for all t and st, we have:
xct,T;n(s
t)→ rt,T(st)
pct,T(s
t) ∑i∈I
∫kµt,T(dk; i, st) +
rt,T(st)
pct,T(s
t) ∑i∈I
∫l(st, it)µt,T(dk; i, st)
+Πt,T(Kt,T, Lt,T)
pct,T(s
t)−Yt,T(st)
= ∑i∈I
∫ (ct,T(k, i; st) + kt,T(k; i, st)− (1− δ)k
)µt,T(dk; i, st)−Yt,T(st).
In addition, we also have xct,T;n(s
t)→ xct,T(s
t) . Therefore, for all t and st:
xct,T = ∑
i∈I
∫ (ct,T(k, i; st) + kt,T(k; i, st)− (1− δ)k
)µt,T(dk; i, st)−Yt,T(st)
Similarly, we can also show that, for all t and st:
xkt,T(s
t) = Kt,T(st)−∫
kµt,T(dk; it, st)
xlt,T(s
t) = Lt,T(st)− L(st).
Thereforex ∈ ψx(p, µ, x).
Following the same steps, it is also easy to show that p ∈ ψp(p, µ, x). Now we showthat µ ∈ ψµ(p, µ, x).
50
Indeed, from the definition of ψµ, for every A ∈ B([
0, k])
µt+1,T;n(A, it+1; st+1) = ∑it∈I
Pr(it+1|it, st, st+1)µt,T;n
((kt,T;n
)−1(A), it; st
). (59)
We need to show that, for every A ∈ B([
0, k])
µt+1,T(A, it+1; st+1) = ∑it∈I
Pr(it+1|it, st, st+1)µt,T
((kt,T;n
)−1(A), it; st
).
From the construction of the Kantorovich-Rubinshtein norm for the space of measuresin Bogachev (2000, Section 8.3), to show the identity, we just need to show that for allϕ ∈ Lip1
([0, k])
,∫ k
0ϕ(k)µt+1,T(dk, it+1; st+1) = ∑
it∈IPr(it+1|it, st, st+1)
∫ k
0ϕ(
kt,T(k, i; st))
µt,T(dk; it, st)
(60)From (59), we have∫ k
0ϕ(k)µt+1,T;n(dk, it+1; st+1) = ∑
it∈IPr(it+1|it, st, st+1)
∫ k
0ϕ(
kt,T;n(k, i; st))
µt,T;n(dk; it, st).
Since µn → µ,
limn→∞
∫ k
0ϕ(k)µt+1,T;n(dk, it+1; st+1) =
∫ k
0ϕ(k)µt+1,T(dk, it+1; st+1).
Therefore, to establish (60), we just need to show that:
We show in Lemma 10, that kt,T;n → kt,T uniformly, therefore
limn→0
sup0≤k≤k
∣∣∣kt,T;n(k, i; st)− kt,T(k, i; st)∣∣∣ = 0,
In addition, since µt,T;n([0, k]
; it, st) ≤ ∑i∈I µt,T;n([0, k]
; i, st) = 1. These two resultsimply (63).
Because ϕ(
kt,T(k, i; st))
is continuous, we also have:
limn→∞
∣∣∣∣∣∫ k
0ϕ(
kt,T(k, i; st))
µt,T;n(dk; it, st)−∫ k
0ϕ(
kt,T(k, i; st))
µt,T(dk; it, st
∣∣∣∣∣ = 0.
Combining this limit with (63), and (62), we arrive at (61). As argued above, this impliesµ ∈ φµ(p, µ, x).
We have just established that ( p, µ, x) ∈ Ψε (p, µ, x), i.e., Ψε is upper hemi-continuous.It is standard to show that Ψε is compact and convex valued. The proof is facilitated bythe fact that if
( p1, µ1, x1) ∈ Ψε (p, µ, x)and
( p2, µ1, x2) ∈ Ψε (p, µ, x)then Π1
t,T = Π2t,T for all t and st. Therefore by Lemma 8, k1
t,T ≡ k2t,T for all t and st. So
µ1 ≡ µ2, i.e., ψµ(p, µ, x) is single-valued.
Lemma 10. Assume that pn,T −→n→∞ pT and Πnt,T −→n→∞ Πt,T. In addition, Vn
t,T solves(37), given pn,T and Πn,Twith the corresponding policy function kn
t,T and Vt,T solves (37) givenpT and Πt,T with the corresponding kt,T. Then, for all st ∈ S t and i ∈ I and k ∈
Assume to the contrary. Then, there exists a subsequence nm such that
k′nm = kn
t,T(k, i; st, pn,T, Πn,T)→ k∗
for some k∗ ∈[0, k]
and k∗ 6= kt,T(k, i; st, pT, ΠT). Let cnm be defined such that the budgetconstraint, (67) holds with equality. Taking, further subsequence if necessary, we canassume that cnm → c∗ for some c∗. As shown above (c∗, k∗) must satisfy the budgetconstraint at pT, ΠT, and
Vt,T
(k, i; st, pT, ΠT
)= lim
m→∞Vnm
t,T (k, i; st, pT, Πn,T) = u(c∗)+ β(i)Et
[Vt+1,T(k∗, i; st, pT, ΠT)
].
Therefore, k∗ = kt,T(k, i; st, pT, ΠT) (by Lemma 8 the maximizer is unique). This is acontradiction.
So we have established the pointwise convergence of knt,T to kt,T. Because kt,T and kn
t,Tare increasing and continuous (by Lemma 8), the convergence is uniform.
Lemma 11. Θ, g defined in Theorem 2 satisfy:1. Θ is sequentially compact.2. g is a closed-valued correspondence.
Proof. Proof of Part 1: We endow the space of increasing function with pointwise conver-gence topology and we endow the space of V function with the sup norm topology.M denote the space of monotone functions from
[0, k]
to[0, k]. We endowM with
the topology of pointwise-convergence. Then, by Helly’s selection theorem,M is sequen-tially compact.16
L denote the space of Lipschitz continuous function with the Lipschitz constant lV ,defined in (72), and bounded below by V and bounded above by V. We endow Lwith thetopology of convergence in sup norm. Then, by Ascoli-Arzela theorem, L is sequentiallycompact.
Proof of Part 2: We need to show that g(s, µ) is closed for all s ∈ S and µ ∈ Ω. That is,
for any sequence(
θn,(θn
s′)
s′∈S
)∞
n=0∈ g(s, µ) such that(
θn, (θns′)s′∈S
)∞n=0 −→n→∞
(θ, (θs′)s′∈S
)then
(θ, (θs′)s′∈S
)∈ g(s, µ).
By the definition of convergence (topology) in different spaces, we have kn → k (point-wise convergence) and Vn → V and V+n
s′ → V (convergence in sup norm).
16See Exercise 7.13 in Rudin (1976) for an elementary proof.
55
First, since(
θn,(θn
s′)
s′∈S
)∞
n=0∈ g(s, µ), we have
Vn(k, i) ≥ u((1− δ + rn)k + wnl(s, i)− k′
)+ β(i) ∑
i′,s′πss′,ii′V+n
s′ (k′, i′)
for each k′ ∈[0, k]. Taking the limit n→ ∞, we have
V(k, i) ≥ u((1− δ + r)k + wl(s, i)− k′
)+ β(i) ∑
i′,s′πss′,ii′V+
s′ (k′, i′)
for all k′ ∈ [0, k]. Therefore
V(k, i) ≥ maxk′∈[0,k]
u((1− δ + r)k + wl(s, i)− k′
)+ β(i) ∑
i′,s′πss′,ii′V+
s′ (k′, i′).
Now, since(
θn,(θn
s′)
s′∈S
)∞
n=0∈ g(s, µ):
Vn(k, i) = u((1− δ + rn)k + wnl(s, i)− kn(k, i)
)+ β(i) ∑
i′,s′πss′,ii′V+n
s′ (kn(k, i), i′).
We show thatlim
n→∞V+n
s′ (kn(k, i), i′) = V+s′
(k(k, i), i′
).
Indeed ∣∣∣V+ns′ (kn(k, i), i′)− V+
s′
(k(k, i), i′
)∣∣∣ ≤ ∣∣∣V+ns′ (kn(k, i), i′)− V+n
s′ (k(k, i), i′)∣∣∣
+∣∣∣V+n
s′ (k(k, i), i′)− V+s′
(k(k, i), i′
)∣∣∣ .
The first term goes to zero because V+ns′ is Lipschitz continuous and kn converges point-
wise to k and the second term goes to 0 because of the pointwise convergence of V+s′ to
V+s′ .
Therefore
V(k, i) = u((1− δ + r)k + wl(s, i)− k(k, i)
)+ β(i) ∑
i′,s′πss′,ii′V+
s′ (k(k, i), i′).
SoV(k, i) = max
k′∈[0,k]u((1− δ + r)k + wl(s, i)− k′
)+ β(i) ∑
i′,s′πss′,ii′V+
s′ (k′, i′),
and k(k, i) is a maximizer.
Now we find bounds for the endogenous variables.
Lemma 12. There exist 0 < K < K0, 0 < r < r and 0 < w < w, and V < V and lV > 0,such that in competitive equilibrium in the finite horizon economy, starting with an initial wealthdistribution µ0(k, i) and
for all s ∈ S .We show that if for some t and st ∈ S t, Kt,T(st) ≥ K then Kt+1,T(st, s) ≥ K for all s ∈ S .Assume to the contrary that Kt+1,T
(st+1) < K. We will show that this leads to a
contradiction.To simplify the exposition, we use the notation zt,T(k, i) as shorthand for zt,T(k, i; st),
where zt,T can be the value, policy, or pricing functions, Vt,T or ct,T, kt,T or rt,T, wt,T. In acompetitive equilibrium, Lt,T(st) = L(st) = ∑i∈I m(i, st)l(i, st), so we write Lt instead ofLt,T.
From the first order condition, if kt,T(k, i) < k then
Therefore, since Kt+1,T < K, FK(st+1, Kt+1,T, Lt+1) > FK(st+1, K, Lt+1) and the last inequal-ity implies:
u′(ct,T(i, k)) ≥
mins∈S
(1− δ + FK(K, L(s), s))
β(i)πstst+1 ∑it+1
πstst+1,iit+1
πstst+1
u′(
ct+1,T(kt,T(i, k), it+1))
≥
mins∈S
(1− δ + FK(K, L(s), s))
βπstst+1u′(
∑it+1
πstst+1,iit+1
πstst+1
ct+1,T(kt,T(k, i), it+1)
)where the last inequality comes from the fact that u′(c) = c−σ is strictly convex.
Consequently,(mins∈S
(1− δ + FK(s, K, L(s)))
βπstst+1
) 1σ
≤∑it+1
πstst+1,iit+1πstst+1
ct+1,T(kt,T(k, i), it+1)
ct,T(k, it).
17V might not be differentiable everywhere because of the borrowing constraint, k′ ≥ 0, in this casewe use the concept of generalized derivative and the associated Envelope Theorems in Milgrom and Segal(2002).
57
Therefore18 (mins∈S
(1− δ + FK(K, L(s), s))
βπstst+1
) 1σ
≤∑it,it+1
πstst+1,it it+1πstst+1
∫kt,T<k ct+1,T(kt+1,T(k, it+1), it+1)µt,T(dk, it)
∑it
∫kt,T<k ct,T(k, it)µt,T(dk, it)
(69)
Now, we show that this would lead to a contradiction.Indeed, for k such that kt,T(k, i; st) = k we have
Similarly, there exist 0 < w < w, such that wt,T ∈ [w, w] for all t and st.From the (34), for all k ∈
[0, k], i ∈ I , and t ≤ T and st ∈ S t, we have:
11− (β)T−t u
((1− δ + r)k + wl
)≥ Vt,T(k, i) ≥ 1
1− (β)T−t u (wl) .
Let
V = supt,T:0≤t≤T
11− (β)T−t u
((1− δ + r)k + wl
)and V = inf
t,T:0≤t≤T
11− (β)T−t u (wl) .
ThenV ≤ Vt,T(k, i; st) ≤ V
for all k ∈[0, k], i ∈ I , and t ≤ T and st ∈ S t.
NowVt,T(k, i) = u (ct,T(k, i)) + β(i)Et
[Vt+1,T(k′, i)
].
Thereforeu (ct,T(k, i)) > V − βV.
Since limc→0 u(c) = −∞, there exists c > 0 such that
ct,T(k, i) > c
for all k ∈[0, k], i ∈ I , and t ≤ T and st ∈ S t.
From the envelope condition
V′t,T(k, i) =(1− δ + rt,T(st)
)u′(ct,T (k, i)),
ThereforeV′t,T(k, i) ≤ (1− δ + r) u′(c) ≡ lV (72)
for all k ∈[0, k], i ∈ I , and t ≤ T and st ∈ S t.
Proof of Lemma 2. Consider sequences of allocation and prices generated by a general-ized recursive equilibrium, starting from s0 ∈ S and a distribution
kh
0
h∈H such thatµ0(k, i) as defined in (16) belongs to Ω. That is, sequences of distributions
µt(st)
t,st
and, policy functions and value functions
ct(., .; st), kt(., .; st), Vt(., .; st)
t,st, and prices
60
rt(st), wt(st)
t,st such that for each t, st:
(ct, kt, Vt, rt, wt
)∈ Q(st, µt), and for each st+1 ∈
S ,(st+1, µt
(st, st+1
))st+1∈S
∈ T (st, µt) , Conditions 1-5 in Definition (4) are satisfied. Forconvenience, we restate Conditions 1-5 below.
1. For each st+1 ∈ S ,(ct+1
(., .;(st, st+1
)), kt+1
(., .;(st, st+1
)), Vt+1
(., .;(st, st+1
)), rt+1
(st, st+1
), rt+1
(st, st+1
))∈ P
(st+1, µt+1
(st, st+1
)).
2. The following identity holds:
∑i∈I
∫ct(i, k; st)µt(dk, i; st) + ∑
i∈I
∫kt(i, k; st)µt(dk, i; st)
= F(st, Kt(st), L(st)) + (1− δ)Kt(st)
where
Kt(st) = ∑i∈I
∫ k
0kµt(dk, i; st) and L(st) = ∑
i∈Imst(i)l(st, i).
3. rt(st) = FK(st, Kt(st), L(st)
)> 0 and wt(st) = FL
(st, Kt(st), L(st)
)> 0.
4. For each i ∈ I and k ∈[0, k], Vt(., .; st), V
(., .;(st, st+1
)), st+1 ∈ S satisfy the Bell-
man equation:
Vt(i, k; st) = maxc,k′
u(c) + β(i) ∑i′,s′
πss′,ii′Vt+1(i′, k′;
(st, s′
))(73)
s.t. c ≥ 0 and 0 ≤ k′ ≤ k and
c + k′ ≤ (1− δ + rt(st))k + wt(st)l(st, i).
In addition,(
ct(i, k; st), kt(i, k; st))
solves (73).
5. For each st+1 ∈ S , and i′ ∈ I and A ∈ B([
0, k])
:
µt+1(i′, A;
(st, st+1
))= ∑
i∈I
πss′,ii′
πss′µt
((kt(i, .; st))−1(A), i; st
). (74)
Let the allocation
cht (s
t, it), kht (s
t, it)
h∈H be determined recursively by:
cht (s
t, it) = ct(kht , ih
t ; st) and kht+1(s
t, it) = kt(kht , ih
t ; st),
We show by induction that µt(st) corresponds to the distribution (16) from
kht (s
t, it)
h∈H,i.e.
µt(A× I; st) = φ(
h ∈ H :(
kht (s
t, it), iht
)∈ A× I
)(75)
for each A× I ∈ B([
0, k])×B(I).
The identity (75) holds at t = 0 by definition. Now, assume that the identity holds att, we show that it holds at t + 1. Indeed,
φ(
h ∈ H :(
kht+1(s
t+1, it+1), iht+1
)∈ A× I
)= φ
(h ∈ H :
(kt(kh
t (st, it), ih
t ; st), iht+1
)∈ A× I
).
61
By the “conditional no aggregate uncertainty” condition, the last expression can be writ-ten as:
∑i∈I
∑i′∈I⊂I
πstst+1,ii′
πstst+1
φ(
h ∈ H :(
kt(kht (s
t, it), i; st))∈ A, ih
t = i)
= ∑i∈I
∑i′∈I⊂I
πstst+1,ii′
πstst+1
φ(
h ∈ H : kht (s
t, ih,t) ∈ (kt(i, .; st))−1(A), iht = i
)= ∑
i∈I∑
i′∈I⊂I
πstst+1,ii′
πstst+1
µt
((kt(i, .; st))−1(A), i; st
),
where the last equality comes from the assumption that (75) holds at t. By (74), the lastexpression is equal to
∑i′∈I⊂I
µt+1(
A, i′;(st, st+1
))= µt+1
(A× I; st+1
).
Therefore, we obtain (75) at t + 1.So by induction (75) holds for all t and st. Consequently,∫
Hkh
t (st, it)φ(dh) = ∑
i∈I
∫ k
0kµt(dk, i; st) = Kt.
and ∫H
lh(st, iht )φ(dh) = ∑
i∈I
∫ k
0l(s, i)µt(dk, i; st) = ∑
i∈Imst(i)l(s, i) = L(st),
and ∫H
cht (s
t, it)φ(dh) = ∑i∈I
∫ k
0ct(i, k; st)µt(dk, i; st)
and ∫H
kht+1(s
t, ih,t)φ(dh) = ∑i∈I
∫kt(i, k; st)µt(dk, i; st).
Now given the sequences of prices
rt(st), wt(st)
, let
cht (s
t, it), kht (s
t, it)
t,st denotethe allocation generated by the policy functions c, k and let
ch
t (st, it), kh
t (st, it)
t,st denote
a sequence that satisfies (14), and (15), we show that
V0(kh0, ih
0; s0) = E0
[∞
∑t=0
Πt−1t′=0β(ih
t′)u(cht (s
t, it))
](76)
≥ E0
[∞
∑t=0
Πt−1t′=0β(ih
t′)u(cht (s
t, it))
]. (77)
From the Bellman equation (73), we have
V0(kh0, ih
0; s0) = E0
[T
∑t=0
Πt−1t′=0β(ih
t′)u(cht ) + ΠT
t′=0β(iht′)VT+1(kh
t , iht ; st)
]
62
Now, the second term in the right hand side is bounded (in absolute value) by
βT+1 max [V, V] −→T→∞ 0.
Therefore taking T → ∞, we obtain (76).Similarly, from the Bellman equation (73), we have
V0(kh0, ih
0; s0) ≥ E0
[T
∑t=0
Πt−1t′=0β(ih
t′)u(cht ) + ΠT
t′=0β(iht′)VT+1(kh
t , iht ; st)
].
The second term in the right hand side is bounded below by
minβT+1V, βT+1V −→ 0.
Therefore taking T → ∞, we obtain (77).
C Relation to Miao (2006)
Consider the economy with a continuum of agents in Section 3. Let P denote the set ofprobability measures µ over [0, ∞)× I such that
∑i
∫ ∞
0kµ(dk, i) ≤ K,
where K is defined in (10) andP∞ = ×∞
t=0PSt.
The existence proof in Miao (2006) relies on the fixed point of the following operator:
T : C ([0, ∞) , I ,S ,P∞)→ C ([0, ∞) , I ,S ,P∞)
define for eachµ =
(µt(st)
t≥0,st∈S t
)∈ P∞
as
TV (k, i; s0, µ) = maxk′∈Γ(k,i,s,µ0)
u((1 + r (s0, µ0)− δ) k + w(s0, µ0)l(i)− k′
)+ β(i) ∑
s1∈S ,i′∈Iπs0s1;ii′V
(k, i; s1,
µt+1(st+1)
t≥0
)(78)
whereΓ (k, i, s, µ0) =
k′ : 0 ≤ k′ ≤ (1 + r (s0, µ0)− δ) k + w(s0, µ0)l(i)
.
Miao shows that operator T is a contraction mapping (as an application of the Black-well Theorem). Therefore, T admits a unique fixed point V and the corresponding policyfunction is
k (k, i; s0, µ)
We defineΛµ = ˜µ
63
where ˜µ0 = µ0 and
˜µt+1(st+1)(
A, i′)= ∑
i∈I
πstst+1,ii′
πstst+1
µt
(k−1
(A, i; st, µτ(sτ)τ≥t
)).
The mapping Λ is continuous in P∞. Therefore by the Brouwer-Schauder-TychonoffFixed Point Theorem, Λ admits a fixed point, which corresponds to a sequentially com-petitive equilibrium.
However this proof does not directly apply to Krusell and Smith (1998)’s model inwhich the production function satisfies the Inada condition at zero aggregate capital.Most importantly because of the following two reasons.
First of all, because of the Inada condition on the production function, the operator Tis not well-defined when µ0 = D(0), where D(x) is the Dirac mass at x because
r(s0,D(0)) = +∞.
Therefore V must be defined over P∞∗ where P∗ = P\D(0).
However, the following proposition shows that Λ does not preserve P∞∗ , i.e., there
exists µ ∈ P∞∗ such that, Λµ /∈ P∞
∗ . The intuition is that if aggregate capital in µ is veryhigh, the implied marginal rate of return on capital (interest rate) is very low. Togetherwith a sufficiently low discount factor, the agents will not want to save, leading to zeroaggregate capital in ˜µ.19 The following proposition formalizes this intuition.
Proposition. Assume that β(i) = β for all i ∈ I and let β ∈ (0, 1) sufficiently small such that
u′(lFL(s, K, L(s))
)> βu′
(lFL(s′, K, L(s′))
)(79)
for all s, s′ ∈ S . Then there exists K∗ such that, for all K ≤ K∗, and
µ =(D (K) , D (K)t>0,st∈S t
)we have
Λµ =(D(K), D(0)t>0,st∈S t
)/∈ P∞
∗ .
Proof. Because limc↓0 u′(c) = +∞ and limK↓0 F(s0, K, L(s0))− δK = 0, there exists K∗ suchthat for all K ≤ K∗, we have
u′ (F(s0, K, L(s0))− δK) ≥ βu′(lFL(s′, K, L(s′))
)(80)
for all K ≤ K∗. Let µ be defined above. Using the agents’ Euler equation, we have, atk = K, the solution to (78) involves
So (82) and (83) follow directly from (79) and (80).From (81), we obtain
Λµ =(D(K), D(0)t>0,st∈S t
).
Second of all, P∞∗ , endowed with the product topology (of weak topology in P∗) is
not a a compact set. Therefore one cannot apply the Brouwer-Schauder-Tychonoff FixedPoint Theorem for continuous functions defined on this set.
In the present paper, I follow a different route to establish the existence of a competi-tive equilibrium by taking the limit of finite horizon economies as in Appendix B. I derivea lower bound on aggregate capital using the agents’ Euler equation, hence indirectly ruleout D(0). Another way to put it is that D(0) implies an infinite marginal rate of return ron capital but in Lemma 5, by restricting prices on ∆ε, we impose an upper bound on r. Ishow that this bound does not bind in a competitive equilibrium.
