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 equilibrium 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 functions. 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 algorithm 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-GlicksbergFan 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.
1
Introduction
Krusell and Smith (1998) provide a workhorse model with heterogeneous agents, subject to both idiosyncratic and aggregate shocks, and markets are incomplete. Their paper
offers an algorithm to compute a recursive equilibrium, i.e., a sequential competitive equilibrium summarized by a mapping from current wealth distribution to prices and allocations (policy function), and future wealth distribution (transition function). Despite the
increasing popularity of the model1 , the theoretical question whether a recursive equilibrium exists, and if it does what we can say about its properties, is still open. The present
paper makes some progress toward answering this question.
First, I define a generalized recursive equilibrium as a correspondence that maps current
wealth distribution and exogenous aggregate shock to a set of possible prices and allocations (include value functions), as well as possible future wealth distributions. At least
one of the elements in the set satisfies short-run equilibrium conditions. Any sequence
of allocations and prices generated by a generalized recursive equilibrium forms a sequential competitive equilibrium. I then prove that a generalized recursive equilibrium
always exists in Krusell and Smith’s economy. In addition, if starting from any initial
wealth distribution and aggregate shock, there exists no more than one sequential competitive equilibrium; then, from a generalized recursive equilibrium, one can select a standard recursive equilibrium, as computed in Krusell and Smith (1998). The concept and
proof techniques apply equally well to economies with a finite number of agents or with
a continuum of agents.
The existence proof follows the steps in Cao (2010). That is, I first show the existence
of sequential competitive equilibria in finite horizon economies. Second, I show that the
allocations and prices in these economies lie in a compact set. Lastly, I take the limit of the
equilibrium allocation and prices as the horizon tends to infinity.2 Cao (2010) works with
finite-agent economies and assumes that the agents receive strictly positive endowment
of final good every period and in all states. This assumption is not made in Krusell and
Smith (1998). In this present paper, I relax this assumption in the finite-agent economy. I
show that aggregate capital is always bounded away from 0 in any sequential competitive
equilibrium, using the agents’ Euler equation and Jensen’s inequality. This lower bound
on aggregate capital implies a lower bound on labor income if the agents always receive
a strictly positive labor endowment. A strictly positive labor income plays a similar role
to a strictly positive final good endowment assumed in Cao (2010).
More substantively, I extend the proof arguments in Cao (2010) to allow for a continuum of agents exactly as in Krusell and Smith (1998). To do so, I take advantage of the
compactness of the space of probability measures over a bounded interval endowed with
weak topology and the compactness of the space of Lipschitz continuous value functions
1 The
list of papers using Krusell and Smith’s or a similar framework and algorithm is long and fastgrowing; 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.
2 This method of proving existence using the limit of finite-horizon economies is standard in the infinitehorizon pure-exchange incomplete markets literature such as Magill and Quinzii (1994), Araujo, Pascoa and
Torres-Martinez (2002), and more recently Le Van and Pham (2016).
1
and weakly increasing policy functions. These compactness properties enable the application 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 Point
Theorem, standard in game and general equilibrium theory. The existence proof in this
case also implies several important properties of the value and policy functions such that
monotonicity and Lipschitz continuity.
The existence proof is constructive and suggests an algorithm alternative to the one
used in Krusell and Smith (1998). The algorithm involves solving for the global nonlinear solution of finite horizon economies and taking the horizon to infinity, just as in the
existence proof. I implement the algorithm for two-agent economies, in which the agents
differ in either labor productivity or discount factor. I also carry out the simulation exercise as in Krusell and Smith (1998), i.e., regressing future log aggregate capital on current
log aggregate capital over the simulated paths of the economy from the global nonlinear solution. I find that, at least for the two-agent economy with heterogeneous labor
productivity, the “aggregation result,” according to which current log aggregate capital
alone predicts almost perfectly future log aggregate capital, holds only in the simulations.
In the global nonlinear solution, however, wealth distribution, in addition to aggregate
capital, 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. The
existence of a generalized recursive equilibrium implies the existence of a competitive
equilibrium. Miao (2006) is the first to formulate and prove the existence of a competitive
equilibrium in Krusell and Smith-type economies with both idiosyncratic and aggregate
shocks. I borrow many important ingredients from his paper such as the “conditional
no aggregate uncertainty condition” and the concept of recursive equilibrium with value
function as an extended state variable. The existence proof in Miao (2006) relies on the
existence and uniqueness of the value and policy functions (with arguments including
individual capital holding and aggregate wealth distribution) as a solution to a Bellman
equation. However, the proof does not directly apply to cases in which the production
function satisfies the Inada condition at zero aggregate capital. This restriction excludes
the Cobb-Douglas specification used in Krusell and Smith (1998), which is the focus of the
present paper. Because, in these cases, the Bellman operator is not well-defined when the
distribution of capital holdings is a Dirac mass at 0, leading to zero aggregate capital and
consequently 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 that
simple ways to get around the issue with zero aggregate capital, in order to apply Miao’s
machinery, do not work in Krusell and Smith (1998)’s economy.
Therefore, in the present paper, I follow a different route to establish the existence of
a competitive equilibrium by taking the limit of finite horizon economies as described
above. I derive a lower bound on aggregate capital, or equivalently an upper bound on
the rate of return on capital, using the agents’ Euler equation, and hence sidestep the issue
with zero aggregate capital. In my proof, I also characterize several important properties
of the value and policy functions, which do not appear in Miao (2006). Lastly, my proof
allows for unbounded utility functions, i.e., log utility as in Krusell and Smith (1998) or
CRRA 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 Schmedders (2003), Cao (2010), and Feng, Miao, Perata-Alva and Santos (2014). But these papers
only allow for a finite number of agents. Extending the techniques developed therein to
economies with a continuum of agents is not straightforward. As Miao (2006) wrote, “The
key idea of Duffie et al. (1994) is to construct an expectation correspondence, which specifies, for
each possible current state, the transitions that are consistent with feasibility and satisfy short-run
equilibrium conditions. Typically, the expectations correspondence is constructed using the firstorder conditions for all agents. This procedure seems invalid for the continuum agents economies
since there is a continuum of first-order conditions.” In the present paper, I show that the
short-run equilibrium conditions in Krusell and Smith (1998)’s economy can be summarized using the value function and policy function instead of first-order conditions. This
innovation allows me to use standard techniques to establish the existence of a generalized 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 exogenous and endogenous state variables. In the case of Krusell and Smith (1998), I show
that 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 version
of Krusell and Smith (1998) and the existence of a generalized recursive equilibrium in
this environment. Section 3 shows the existence of a generalized recursive equilibrium
in the original version of Krusell and Smith (1998) with a continuum of agents. Section
4 presents the a numerical algorithm based on the existence proofs to compute recursive
equilibria when they exist, as well as several numerical examples. The details of the
proofs 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 proof
for the existence of a generalized recursive equilibrium in this model is simpler than the
one 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 agents
in Section 3.
The environment Consider an endowment, a single consumption (final) good economy
in infinite horizon. Time runs from t = 0 to ∞. The economy is populated by H representative, infinitely-lived agents (households) indexed by:
h ∈ H = {1, 2, . . . , H }
3
Each representative agent represents a continuum of measure H1 of identical agents. The
preferences over the streams of consumption of agent h is given by
"
#
n
o
∞ U
cth (st )
= E0 ∑ Πtt0 =0 βh (st0 ) u(cth )
(1)
t
t
t≥0,s ∈S
t =0
where
c 1− ν − 1
u(c) = lim
ν→σ 1 − 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 level
uncertainties) and aggregate uncertainties. For example, state s can be a vector:
s = ( A, i1 , ..., i H ) ,
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 caveat
with a finite number of agents is that each idiosyncratic shock is by construction an aggregate shock because it changes the aggregates, for example, aggregate labor supply
when ih determines idiosyncratic labor supply. However, when the number of agents is
very large, the effects of each idiosyncratic shock on the aggregates become negligible. In
the limit with a continuum of agents considered in the next section, by the law of large
numbers, an idiosyncratic shock does not have direct aggregate effects.
The exogenous shock follows a first-order Markov process with the transition probabilities π (s, s0 ). Let st denote the history of realizations of shocks up to time t:
s t = ( s0 , s1 , . . . , s t ) ∈ S t .
At time t, state st determines the agents’ endowments, l h (st ) > 0 units of labor for
h ∈ H. We assume that, there exist L, L̄ > 0 such that:
L≤
1
H
∑
l h (st ) ≤ L̄
h∈H
for all s ∈ S . State st also determines the agents’ discount factor, βh (st ). In addition, there
exist 0 < β, β̄ < 1 such that:
β < βh (st ) < β̄.
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 function
through the first argument.
We make the following standard assumptions on F.
4
Assumption 1. F is strictly increasing, strictly concave and has constant returns to scale in K
and L.
This assumption nests the Cobb-Douglas production used in Krusell and Smith (1998)
as a special case
F (s, K, L) = A(s)K α L1−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 Ct
according to
Ct + Kt+1 − (1 − δ)Kt = Yt .
We further assume that:
Assumption 2. There exists K̂ such that
F (s, K, L̄) − δK < 0
for all K ≥ K̂ and for all s ∈ S .
This assumption guarantees that aggregate capital is always bounded above. It is also
satisfied under the Cobb-Douglas production function.
Market Arrangements In each history st , there are rental markets for capital and labor
market. Agents of type h rent out their capital to the representative firm at competitive
rental rate rt (st ) and supply their labor endowment inelastically to the representative firm
at competitive wage rate wt (st ).
We assume that markets are incomplete inter-temporally, i.e., the agents can only hold
capital to insure against idiosyncratic and aggregate shocks. Therefore they face the sequential budget constraints:
cth (st ) + k ht+1 (st ) − (1 − δ)k ht (st−1 ) ≤ rt (st )k ht (st−1 ) + wt (st )l h (st )
(3)
and the borrowing constraints:
k ht+1 (st ) ≥ 0.
(4)
Agent h solves
max U
ch ,k h
n
cth (st )
o
t,st
(5)
subject to (3) and (4).
The representative firm in history st maximizes profit:
Πt (st ) =
max Yt − rt Kt − wt Lt
Yt ,Kt ,Lt ≥0
subject to
Yt ≤ 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 ).
The definition of a competitive equilibrium in this environment is standard.
5
(6)
Definition 1. A competitive equilibrium
givenan initial distribution of capital holdings
h
k0 h∈H consists of an allocation cth , k ht+1 t,st
and {Kt , Lt }t,st and prices {rt , wt }t,st
h∈H
(rt , wt > 0) such that:
1. For each agent h ∈ H, cth , k ht+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 maximization
problem, i.e., (6) is satisfied.
3. Markets for capital, labor, and final good clear in each history st :
and
and
1
H
∑
1
H
∑
k ht (st ) = Kt (st )
h∈H
1
H
∑
l h (st ) = Lt (st )
h∈H
cth (st ) + k ht+1 (st ) − (1 − δ)k ht (st−1 ) = Yt (st ).
h∈H
Let Ω denote a set of wealth distributions, or equivalently of the distributions of capiH:
tal holdings, and is a compact subset of R+
n o
H
Ω=
k ht
⊂ R+
.
h∈H
Following Krusell and Smith (1998), we define a generalized recursive equilibrium as
following.
Definition 2. A generalized recursive equilibrium is a policy correspondence and a transition correspondence:
+2
Q : S × Ω ⇒ R3H
+
and
T : S ×Ω ⇒ Ω
h
h
h
h
c , k + , v h∈H , r, w ∈
and c > 0 such that for all s ∈ S and k h∈H ∈ Ω and
Q s, kh h∈H , we have ch ≥ c and there exists
n
o
h
0
h
0
h
0
0
0
c+ (s ), k ++ (s ), v+ (s )
, r + ( s ), w + ( s )
s0 ∈S
h∈H
that satisfies the following
properties:
h
h
1. k + h∈H ∈ T s, k h∈H
h , kh , vh
0 , kh
2.
c+
,
r
,
w
∈
Q
s
+
+
++ + h∈H
+ h∈H
3. (Market clearing)
1
H
∑h∈H ch +
K=
1
H
∑
1
H
∑h∈H k h+ = F (s, K, L) + (1 − δ)K where
k h > 0 and L =
h∈H
1
H
∑
l h (s) > 0.
h∈H
4. (Firms’ maximization) r = FK (s, K, L) > 0 and w = FL (s, K, L) > 0.
6
5. (Agents’ maximization) For each h ∈ H
u0 (ch ) ≥ βh (s)
∑
0
h
πss0 1 − δ + r+ (s0 ) u0 (c+
(s0 ))
(7)
s ∈S
with equality if k h+ > 0 and
ch + k h+ = (1 − δ + r )k h + wl h
and
vh = u(ch ) + βh (s)
∑
0
h
πss0 v+
( s 0 ).
(8)
(9)
s ∈S
A recursive equilibrium is a generalized recursive equilibrium in Definition 2 with the
correspondences Q, T being single-valued.
The following lemma shows the connection between a generalized recursive equilibrium and competitive equilibrium.
Lemma 1. A sequence of allocations and prices generated by a generalized recursive equilibrium
forms a competitive equilibrium.
Proof. Appendix A.
To show the existence of a generalized recursive equilibrium, we need the following
properties on the production function.
Assumption 3. For any L > 0 and K > 0:
max sup FL (s, K̃, L) < +∞,
s∈S
and
0<K̃ ≤K
max sup FK (s, K, L̃) < +∞.
s∈S
0< L̃≤ L
Lastly, for any L > 0 and s ∈ S
lim F (s, K, L) = 0.
K →0
This assumption assures that a competitive equilibrium exists in the finite horizon
economy.
Assumption 4. There exists K ∗ such that for any 0 < K < K ∗ and L ≤ L ≤ L̄, and s, s0 ∈ S :
1
σ
F (s0 , K, L) + (1 − δ)K
00
β min (1 − δ + FK (s , K, L))
>
.
F (s, K, L̄) − δK
s00 ∈S
This assumption requires that the marginal rate of return on capital is very high when
capital is low. Together with the agents’ Euler equation, it implies a lower bound on
aggregate capital in any competitive equilibrium.
It is easy to verify that the last two assumptions hold for the Cobb-Douglas production
F (s0 ,K,L)
function in (2) since FK (s00 , K, L) → ∞ as K → 0 and F(s,K,L) is bounded above as K → 0.
Armed with the assumptions above, we arrive at the first existence result.
7
Theorem 1. Assume that Assumptions 1, 2, 3, and 4 hold. Given any initial distribution of
capital, H1 ∑h∈H k0h = K0 > n0, there exist 0 < K < K0 < K̄ such that
o a generalized recursive
H : K ≤ 1
h
equilibrium exists with Ω = k h h∈H ∈ R+
H ∑ h∈H k ≤ K̄ .
Proof. We choose K̄ sufficiently large:
K̄ > max K0 , K̂, max max ( F (s, K, L̄) + (1 − δ)K ) , 2 L̄ ,
s∈S 0≤K ≤K̂
(10)
where K̂ 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 equilibrium exists for any finite horizon economy. In addition, the equilibrium variables in a
finite horizon economy always lie in a compact set. Then I take the limit of the horizon to
infinity and construct appropriate correspondences to show the existence of a generalized
recursive equilibrium.
However, Cao (2010) assumes that each agent receives an strictly positive amount of
final good endowment in every period and history of shocks. In this paper, we relax this
assumption. We only require that each agent receives an strictly positive amount of labor
endowment in every period and history of shocks. We show that aggregate capital is
always bounded from below:
Kt,T (st ) ≥ K
for all t and st , where Kt,T (st ) is the aggregate capital at time t and in history st in the
T-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 on
wage rate implies a strictly positive labor income, which plays a similar role to a strictly
positive final good endowment in Cao (2010).
To show that aggregate capital is bounded from below, we use the agents’ Euler equation, (7):
h
i
u0 (cth ) ≥ βht Et (1 − δ + rt+1 ) u0 (cth+1 ) .
This equation implies that if Kt+1 is too small, the rate of return on capital r T +1 is every
high, driving up saving from time t, and in turn, increasing Kt+1 . Assumption 4 then
leads to a contradiction.
The details of the proof are given in Appendix A.
Generalized Recursive Equilibrium and Recursive Equilibrium Can we always select
a recursive equilibrium from a generalized recursive equilibrium? The answer is no. The
definition of a generalized recursive equilibrium involve policy and transition correspondences, Q and T . We can show that Q upper-hemi continuous. Therefore there exists a
8
measurable selection (function) Q0 from Q. Let T 0 denote the transition function that corresponds to the selection Q0 . However, Q0 ,T 0 might not
form a recursive equilibrium.
h
To see this clearly, let us say k + h∈H = T 0 s, k h h∈H and
n
o
h
h
c+
(s0 ), kh++ (s0 ), v+
(s0 )
, r + ( s 0 ), w + ( s 0 )
h∈H
s0 ∈S
such that Conditions 2-5 in Definition 2 are satisfied. Now it is possible that k h+ h∈H =
h
k h∈H and
n o
o
n
n o
h
h
\Q0 s0 , kh+
c+
, k h++ , v+
, r+ , w+ ∈ Q s0 , k h+
h∈H
h∈H
h∈H
for some s0 ∈ S . Therefore at s0 , we would select the “wrong” allocation if we set
n
o
n o
h
h
h
0
0
h
.
c+ , k ++ , v+
, r+ , w+ = Q s , k +
h∈H
h∈H
From the last observation, in general, we cannot always select a recursive equilibrium
from a generalized recursive equilibrium. Therefore, we would need additional conditions to guarantee the existence of a recursive equilibrium. The following result provides
such a sufficient condition for when a generalized recursive equilibrium gives rise to a
recursive equilibrium.
Corollary 1. Assume that the conditions in Theorem
We have:
h 1 are satisfied.
H
1. Starting from any wealth distribution k0 h∈H ∈ R+ 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 k0h h∈H and
aggregate state s, the sequences of allocation and prices generated by a generalized recursive equilibrium is a competitive equilibrium. Theorem 1 guarantees the existence of
a generalized recursive equilibrium. Hence, a competitive
equilibrium exists.
2. Because starting from each s ∈ S and k h h∈H ∈ Ω, there exists no more than one
competitive equilibrium, there exists a unique element
o
n o
n
ch , k h+ , vh
, r, w ∈ Q st , k h
h∈H
h∈H
that satisfies Conditions 1.-5. in Definition 2. Let Q0 denote the mapping from s, k h h∈H
to this element, and T 0 s, k h h∈H = k h+ h∈H . Then Q0 , T 0 forms a recursive equilibrium.
A generalized recursive equilibrium also gives rise to a recursive equilibrium if we
allow 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-
9
butions Ω and the correspondence Q in Definition 2 and Theorem 1, let
h h h
n o n
o
c , k + , v , r, w ∈ Q st , k h
2H h h
h
∈ S ×Ω×R Ξ=
s, k , c , v
for some k h+ and r, w > 0
where x h is a short-cut for x h h∈H .
A recursive equilibrium
an extended state variable can be constructed over Ξ as a mapping
with from from ξ = s, k h , ch , vh to
a. a current capital choices and values k h+ , and current factor prices r, w ;
h h b. next period capital holdings and consumption s0 , k h+ , c+
, v+ s0 ∈S such that
o
n o n
h
h
∈ Ξ for all s0 ∈ S ,
, v+
ξ s+0 = s0 , k h+ , c+
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 = H1 ∑h k h . Therefore, from the agents’ budget constraint, (8), ch
pins down k h+ . Consequently, the values of ch , vh uniquely select the element in Q(s, k h )
and T (s, k h ), i.e. , for any ξ = s, k h , ch , vh ∈ Ξ there exists a unique tuple k h+ , r, w
such that ch , k h+ , vh , r, w ∈ Q st , k h . The selection gives rise to a recursive equilibrium
in the extended state space.
In the following section, we extends the analysis above in this environment with a
finite 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 the
transition matrix πss0 . However, the economy is populated by a continuum of agents
with index h ∈ H̄ = [0, 1] and state s captures purely aggregate shocks. Let φ denote the
Lebesgue measure over H̄.
The agents are subject to idiosyncratic shock i ∈ I . The realizations of the idiosyncratic shocks are independent across agents so that the law of large number applies. We
also 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 idiosyncratic shocks: (st , it ) forms a first-order Markov process with the transition matrix
πss0 ,ii0 :
Pr(st+1 = s0 , it+1 = i0 , st = s, it = i ) = πss0 ,ii0
with the restriction3
∑ πss0 ,ii0 = πss0 ,
i0
3 This condition means that the evolution of the aggregate state is independent of the idiosyncratic states
(but not vice versa):
Pr(st+1 = s0 , it+1 = i0 , st = s, it = i ) = Pr(st+1 = s0 , st = s, it = i ) = πss0 ,
∑
0
i ∈I
for all i ∈ I .
10
for each i0 ∈ I , and s, s0 ∈ S .
As in Miao (2006, Section 3.2), we assume the“conditional no-aggregate uncertainty
condition,” on the random variables st , ith h∈H̄ so that the law of large numbers for
a continuum of agents applies. In particular, the empirical distributions of implied individual random variables are the same as the theoretical distributions from which these
random variables are drawn.
As a consequence, we can assume, as in Krusell and Smith (1998), that, in aggregate
state s, the fraction of agents with idiosyncratic type i is ms (i ), independent of the past
history of aggregate shocks and agents’ idiosyncratic shocks. However, the fractions ms ’s
have to be consistent with the transition matrix πss0 ,ii0 :
πss0 ,ii0
∑ ms (i) πss0 = ms0 (i0 ),
i ∈I
for all s, s0 ∈ S and i0 ∈ 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 uncertainty
condition, the total supply of labor in aggregate state s is
L(s) =
∑ ms (i)l (s, i).
i ∈I
Idiosyncratic shock also determine the agents’ discount factor in state i: β(i ). We make
the following assumptions on the idiosyncratic labor supply l (., .) and discount factor
β (.).
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 < β̄ < 1 such that
β < β(i ) < β̄
for all i ∈ I .
Since S and I have finite elements, we can choose 0 < L, L̄ such that
L≤
∑ ms (i)l (s, i) ≤ L̄
i ∈I
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 assumption on the production function, Assumption 6, guarantees the existence of a competitive equilibrium in the infinite horizon economy, as well as the existence of a generalized recursive equilibrium.
Assumption 6. For any L ≥ L and s ∈ S ,
lim FK (s, K, L) = ∞.
K →0
11
There exists α > 0, such that for all K, L > 0:
LFL (s, K, L)
> α.
F (s, K, L)
For any s, s0 ∈ S :
lim sup
K →0
F (s0 , K, L̄)
< ∞.
F (s, K, L)
This assumption guarantees that aggregate capital always exceeds some strictly positive lower bound in a finite horizon economy (by an argument based on the agents’ Euler
equation as in the proof of Theorem 1). Together with Assumption 2, Assumption 6 implies 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, as
the horizon T goes to infinity to obtain an equilibrium in the infinite horizon economy as
well as a generalized recursive equilibrium.
It is easy to verify that Assumption 6 is satisfied under Cobb-Douglas production
function (2).
Let K̄ be chosen sufficiently large as in (10). In addition, we assume that the agents’
choice of capital is bounded above by k̄ sufficiently large so that:4
k̄ > l¯ + max F (s, 2K̄, 2 L̄).
(12)
s∈S ,i ∈I
The agents’ Euler equation, which is similar to equation (7) in the finite agent economy
and is crucial in deriving a lower bound for aggregate capital, does not hold when the
upper bound binds. Therefore the upper bound has to be large enough to minimize its
effect.
Let i h,t = i0h , i1h , ..., ith denote the history of idiosyncratic shocks for each h and ît =
i h,t h∈H̄ denote the history of idiosyncratic shocks for all h ∈ H̄.
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 ît , agent h
solves
"
#
max
{cth (.),kht+1 (.)}
E0
∞
∑ Πtt0−=10 β(ith0 )u(cth (st , ît ))
(13)
t =0
subject to
cth (st , ît ) + k ht+1 (st , ît ) ≤ (1 − δ + rt (st , ît ))k ht (st−1 , ît−1 ) + wt (st , ît )l (st , ith ),
(14)
and cth (st , i h,t ) ≥ 0 and
0 ≤ k ht+1 (st , ît ) ≤ k̄.
(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 extension of Definition 1 for a continuum of agents.
4 An upper bound on the choice of capital is implicitly assumed by all numerical algorithms since capital
choice is bounded by machine numerical upper bound.
12
Definition 3. A competitive equilibrium
initial
distribution of capital hold given an
h
ings k0 h∈H consists of an allocation
cth , k ht+1 t,st ,ît
and {Kt , Lt }t,st ,ît and prices
h∈H
{rt , wt }t,st ,ît (rt , wt > 0) such that:
1. For each agent h ∈ H, cth , k ht+1 t,st ,î,t solves (13).
2. In each history of aggregate and idiosyncratic shocks st and ît , {Yt , Kt , Lt } solves the
representative firm’s profit maximization problem, which implies (6).
3. Markets for capital, labor, and final good clear in each history st :
Z
H̄
and
k ht (st−1 , ît−1 )φ(dh) = Kt (st , ît )
Z
H̄
and
Z
H̄
l h (st , ith )φ(dh) = Lt (st , ît )
cth (st , ît ) + k ht+1 (st , ît ) − (1 − δ)k ht (st−1 , ît−1 ) φ(dh) = Yt (st , ît ).
Instead of working with the allocations of capital over households h ∈ H̄, it is easier
to work with the distributions over capital holdings (or equivalently,
distribution)
hwealth
and idiosyncratic shocks. For each distribution of asset holding k h∈H̄ , consider the
following probability measure µ defined by
h h
µ( A × I ) = φ h ∈ H : k , i ∈ A × I
(16)
for each A × I ∈ B
0,
k̄ × B(I) - where B denote the Borel σ−algebras. It is immediate thatµ ∈ P 0, k̄ × I , where the later denotes the space of probability
measures
over 0, k̄ × I endowed with the weak topology. It is well-known that P 0, k̄ × I is
compact (see
Bogachev (2000, Theorem 8.9.3)). Let Ω̄ denote a closed sub for example
set
of
P
0,
k̄
×
I
,which
we will define below. Let C denote the set of functions over
0, k̄ × I which are continuous in k. The generalized recursive equilibrium is defined
over the set of distributions Ω̄ similar to Definition 2 with a finite number of agents.
Definition 4. A generalized recursive equilibrium is a policy correspondence and a transition correspondence:
Q : S × Ω̄ ⇒ C 3 × R2+
and
T : S × Ω̄ ⇒ Ω̄S
and
V, V̄, with the following property: For each s ∈ S and µ ∈ Ω̄, and
some bounds
ĉ, k̂, V̂, r, w ∈ Q(s, µ), we have V ≤ V̂ ≤ V̄ and there exist s0 , µ+
0
s s0 ∈S ∈ T ( s, µ ) and
+
+ +
+
ĉ+
,
k̂
,
V̂
,
r
,
w
0
0
0
0
0
s
s
s
s
s
0
s ∈S
such that:
+ +
+ +
+
0
1. For each s ∈ S , ĉs0 , k̂ s0 , V̂s0 , rs0 , ws0 ∈ Q s0 , µ+
.
0
R
R s
2. (Market clearing) ∑i ĉ(k, i )µ(dk, i ) + ∑i k̂ (k, i )µ(dk, i ) = F (s, K, L) + (1 − δ)K
13
where
K=
∑
i ∈I
Z k̄
0
kµ(dk, i ) and L =
∑ ms (i)l (s, i).
i ∈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, k̄ , V̂, V̂ + satisfy the Bellmantype equation:
(17)
V̂ (k, i ) = max u(c) + β(i ) ∑ πss0 ,ii0 V̂s+0 (k0 , i0 )
c,k0
i0 ,s0
s.t. c ≥ 0 and 0 ≤ k0 ≤ k̄ and
c + k0 ≤ (1 − δ + r )k + wl (s, i ).
In addition, ĉ(k, i ), k̂ (k, i ) solves (17).
5.(Distribution Consistency) For each s0 ∈ S , i0 ∈ I and A ∈ B 0, k̄ :
πss0 ,ii0 0
−1
µ+
(
A,
i
)
=
µ
k̂
(
.,
i
)
(
A
)
,
i
.
∑ πss0
s0
i ∈S
(18)
Similar to Lemma 1 for finite-agent economy, the following lemma shows that generalized recursive equilibrium generates competitive equilibrium.
Lemma 2. Starting from an initial distribution of wealth µ0 and aggregate state s0 , sequences of
allocation and prices generated by a generalized recursive equilibrium form a competitive equilibrium.
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 distribution
of capital µ0 with
K0 =
Z
H̄
k0h φ(dh)
=
∑
i ∈I
Z k̄
0
µ0 (dk, i ) > 0,
there exist 0 < K < K0 < K̄, such that a generalized recursive equilibrium exists over
(
)
Z
Ω̄ =
µ:K≤
∑
i ∈I
k̄
0
kµ(dk, i ) ≤ K̄
.
Proof. Given the bounds determined in Lemma 12, let Θ̄ denote the set of
k̂ (., .), V̂ (., .), r, w
such that for each i ∈ I , k̂ (k, i ) is weakly increasing in k and 0 ≤ k̂ (k, i ) ≤ k̄ and V ≤
V̂ (k, i ) ≤ V̄ for all k ∈ [0, k̄ ] and V̂ (k, i ) is weakly increasing and weakly concave in k and
Lipschitz continuous with a Lipschitz constant lV > 0. V, V̄, lV are given in Lemma 12.
In addition 0 < r ≤ r ≤ r̄ and 0 < w ≤ w ≤ w̄ where r, r̄, w, w̄ are also given in Lemma
12. Lemma 11 shows that Θ̄, endowed with the topology of pointwise convergence for
14
k̂ and uniform convergence for V̂ and the standard topology for r and w, is sequentially
compact.5
Let ḡ : S × Ω̄ ⇒ Θ̄ × Θ̄S denote
the following
correspondence:
for each s ∈ S ,
µ ∈ Ω̄, ḡ(s, µ) is the set of θ =
k̂ (., .), V̂ (., .)
+
+
+
k̂+
s0 (., .), V̂s0 (., .), rs0 , ws0 ∈ Θ̄ such that
h∈H
, r, w
∈ Θ̄, and (θs0 )s0 ∈S with θs0 =
r = FK (s, K, L) > 0 and w = FL (s, K, L) > 0,
where
K=
∑
i ∈I
and
L=
Z k̄
0
kµ(dk, i ) > 0
∑ ms (i)l (s, i) = L(s),
i ∈I
and
∑
i
Z
ĉ(k, i )µ(dk, i ) + ∑
Z
k̂(k, i )µ(dk, i ) = F (s, K, L) + (1 − δ)K
i
where ĉ(k, i ) = (1 − δ + r )k + wl (s, i ) − k̂ (k, i ) > 0. In addition ĉ, k̂, V̂ solves the func
tional equation, (17), given V̂s+0 s0 ∈S .
Lemma 11 shows that ḡ is a closed-valued correspondence.
Consider the following mapping G from the set of correspondences V : S × Ω̄ ⇒ Θ̄ to
itself defined as following. For each V , G(V ) is the correspondence W such that, for each
s ∈ S and µ ∈ Ω̄, we have




 θ = k̂, V̂, r, w ∈ Θ̄ : for each s0 ∈ S ,∃θs0 ∈ V s0 , µ+

0
s
+
W (s, µ) =
where µs0 is given


by (18)


and θ, (θs0 )s0 ∈S ∈ ḡ(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 ∈ Ω̄, then W = G(V ) is sequentially compact.
m → θ =
Indeed, assume that {θ m }∞
m=0 ∈ W ( s, µ ), and θ
k̂, V̂, r, w . Since Θ̄ is
sequentially compact, θ ∈ Θ̄. To show that W = G(V ) is sequentially compact, we need
to show that θ ∈ W (s, µ). By the definition of G , for each s0 ∈ S , ∃θsm0 ∈ V s0 , µ+
s0
such that θ m , θsm0
∈ ḡ(s, µ). Since V s0 , µ+
is
sequentially
compact,
we
can
extract
a
s0
ml
+
0
converging subsequence, θs0 → θs0 for some θs0 ∈ V s , µs0 . Because ḡ is a closed-valued
correspondence, θ, (θs0 )s0 ∈S ∈ ḡ(s, µ), which implies θ ∈ W (s, µ). So W is sequentially
compact.
P2. If V ⊂ V 0 in the sense that V (s, µ) ⊂ V 0 (s, µ) for all s ∈ S and µ ∈ Ω̄ then
G(V ) ⊂ G(V 0 ).
P3. Let V 0 denote the complete correspondence: V 0 (s, µ) = Θ̄ for all s ∈ S and µ ∈ Ω̄.
5 In
infinite dimensional spaces, compactness and sequential compactness are not equivalent. For the
current theorem, we need the sequential compactness property.
15
Then G(V 0 ) ⊂ V 0 .
n+1 = G(V n ).
Given V 0 , we construct the sequence of {V n }∞
n=0 recursively using G : V
Then by P1, P2, and P3, we have V n+1 ⊂ V n and is sequentially compact. By the existence
of a competitive equilibrium in (n+1)-horizon economy in Lemma 5, V n+1 is a non-empty
valued correspondence.
Let V ∗ be defined by
n
V ∗ (s, µ) = ∩∞
n=0 V ( 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 ∗ ⊂ V n , so G(V ∗ ) ⊂ G(V n ) = V n+1 for all
n. This implies G(V ∗ ) ⊂ V ∗ .
Now, for each s ∈ S and µ ∈ Θ̄ and θ = k̂, V̂, r, w ∈ V ∗ (s, µ). Since V ∗ ⊂ V n+1 =
+
n
n
n
0
n
G(V ), there exists θs0 ∈ V s , µs0 such that θ, θs0 s0 ∈S ∈ ḡ(s, µ). By the sequennl tial compactness of Θ̄, we can find a converging subsequence {nl }∞
l =0 , θs0 s0 ∈S −→l →∞
compactness of V nl , we have θs0 ∈ V nl s0 , µ+
(θs0 )s0 ∈S . By the sequential
s0 and since ḡ has
+
n
0
0
closed-value, θ, (θs )s0 ∈S ∈ ḡ(s, µ). Moreover, V s , µs0 is a decreasing sequence so
nl s0 , µ+ = V ∗ s0 , µ+ . Therefore, by the definition of G , we have θ ∈ G(V ∗ ).
θ s0 ∈ ∩∞
0
0
l =0 V
s
s
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 ∈ Ω̄, θ = k̂, V̂, r, w ∈
0
0
Q(s, µ), there exists θs0 ∈ Q s0 , µ+
s0 for each s ∈ S such that θ, ( θs )s0 ∈S ∈ ḡ ( s, µ ). We
also define T as
n
o
+
+
T (s, µ) = µs0 s0 ∈S : given ĉ, k̂, V̂, r, w ∈ Q(s, µ), µs0 is determined by (18) .
It is immediate that (Q, T ) defined as such forms a generalized recursive equilibrium for
the 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 exogenous
aggregate state s0 , there exists a competitive equilibrium. If the competitive equilibrium is unique
for every initial distribution of capital holding and aggregate state then there exists a recursive
equilibrium 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 equilibrium, of which the existence is established in Theorem 2, we can construct a recursive
equilibrium 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 Ω̄ and the correspondence Q in Definition 4 and Theorem 2, let
n
o
Ξ = s, µ, V̂ ∈ S × Ω̄ × C : ∃ ĉ, k̂, V̂, r, w ∈ Q(s, µ) for some ĉ, k̂ ∈ C 2 and r, w > 0
16
where C denotes the set of functions over 0, k̄ × I which are continuous in k.
A recursive equilibrium
with an extended state variable can be constructed over Ξ as a mapping
from from ξ = s, µ, V̂ to
a. a current policy function k̂, and current factor prices r, w ; b. next period wealth distributions and value functions µs0 , V̂s+0 s0 ∈S such that
ξ s+0 = s0 , µs0 , V̂s+0 ∈ Ξ for all s0 ∈ S ,
and Conditions 4. and 5. in Definition 4 are satisfied (given the current policy function and current
factor prices).
Formulated in this manner, the existence of this recursive equilibrium is a direct application of
the 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 compute
recursive equilibria, alternative to the one put forth in Krusell and Smith (1998), using the
equilibria in finite horizon economies. The next subsection presents the algorithm and
the 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 in
Definition 2 for finite agent economies, assuming that the equilibrium is indeed a recursive 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:
(
)
K, ω h
: K ≤ K ≤ K̄ and 0 ≤ ω h ≤ 1, ∑ ω h = 1 ,
Ω = [K, K̄ ] × ∆ H =
h∈H
h∈H
kh
where ωth = HKt t . We calculate recursively, for each T ≥ 0, the function ϕ T from Ω,
the set of current wealth distributions, to current prices and allocations, and to future
wealth distributions. Function ϕ T corresponds to the equilibrium mapping, for the ( T +
1)−horizon economy presented in Appendix A, from the initial distribution of capital
holdings and aggregate shock in period 0 to allocation and prices in the period. Indeed,
ϕT :
n
S × [K, K̄ ] × ∆ H ⇒
o n
h h
H
h h
7→ c , k + , λ , v
s ∈ S , K ∈ [K, K̄ ] , ω ∈ ∆
defined as follows.
1. For T = 0:
n
o n
ϕ0 : s ∈ S , K ∈ [K, K̄ ] , ω ∈ ∆ H 7→ ch , k h+ , λh , vh
17
h∈H
h∈H
+2
R4H
+
o
, r, w
, r, w
o
where
r = FK (s, K, L(s)) and w = FL (s, K, L(s))
and for all h ∈ H
ch = (1 − δ + r ) ω h K + wl h (s) and vh = u(ch ),
and
k h+ = 0 and λh = 0.
2. For T > 0, assuming that we have calculated ϕ T −1 , ϕ T is calculated as:
n
o n
o
H
h h
h h
ϕ T : s ∈ S , K ∈ [K, K̄ ] , ω ∈ ∆
7→ c , k + , λ , v
, r, w
h∈H
such that, for each
s0
∈ S,
h
h
h
h
0
h
c+ , k ++ , λ+ , v+ = ϕ T −1 s , K+ , ω+
where
K+ =
1
H
and for each h ∈ H:
h
ω+
=
∑
1
H
∑h∈H ch +
1
H
h∈H
k h+
h∈H
k h+
∑ k h+
and Conditions 2.-5. in Definition 2 are satisfied:
A1.
,
∑h∈H k h+ = 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 ∈ H
u0 (ch ) = βh (s)
∑
0
h
πss0 1 − δ + r+ (s0 ) u0 (c+
(s0 )) + λh
(19)
s ∈S
with
and
and
λh
≥ 0 and
λh k h+ = 0
ch + k h+ = (1 − δ + r )k h + wl h
vh = u(ch ) + βh (s)
∑
0
h
πss0 v+
( s 0 ).
s ∈S
h
Condition A3 is a reformulation of Condition 5 in Definition 2 using themultipliers λ
’s
and the complementary-slackness condition. Notice also that for each K, ω h h∈H ∈
[K, K̄ ] × ∆ H , the conditions in A2. and A3. gives us 4H + 2 equations for 4H + 2 unknowns (including λh ’s). The market clearing, condition A1., is satisfied by summing up
the budget constraint of each agent.
18
Discretization and Approximation We discretize Ω using Ωd :
o n oM
n
d
Ωd = K = K1d < K2d < ... < K dN = K̄ × ωm
(20)
m =1
d ∈ ∆ H for m = 1, ..., M.
where ωm
Let ϕdT denote the discrete approximation of ϕ T over S × Ωd . For each s ∈ S and at
d , we solve for
each Knd , ωm
h h
h h
d
d
d
ϕ T s, Kn , ωm = c , k + , λ , v
such that Conditions A2 and A3 are satisfied. In Conditions A2 and A3, the future values
h , v̂ h are computed using multi-dimensional cubic splines approximation:
ĉ+
+
!
h
k
1
h
h
h
c+
, v+
, λ+
, k h++ = ϕdT −1 s0 , ∑ k h+ , +h .
(21)
H
∑ k+
Fixing a precision ν, the algorithm converges when
d
d
≤ ν.
ϕ T − ϕ T −1 d
S×Ω
4.2
Numerical Results
We present two numerical examples in economies with two agents. When H = 2, we just
need to keep track of the wealth share of agent 1 because ω 2 = 1 − ω 1 . Therefore, in (20),
n oM
n oM
d
1
ωm
= ω̃m
m =1
m =1
where
0 = ω̃11 < ω̃21 < ... < ω̃ 1M = 1.
In the first example, Subsection 4.2.1, the agents differ in labor productivity but have the
same discount factor. In the second example, Subsection 4.2.2, the agents have the same
labor 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
share the same intertemporal expected utility
"
#
E0
1
2
each. The agents
∞
∑ βt log cth
.
t =0
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) and
aggregate 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 and
agent 2 is employed, and vice versa for i = 1.6 The employed agent has 2(1 − υ) L(s) units
6 This 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 unemployment
transfers by the government and is set at 7%.
The parameters are taken from Krusell and Smith (1998, Section 2) in particular, the
discount 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 = [πss0 ii0 ], directly taken from Krusell and Smith (1998, Section 2):7


0.5250 0.3500 0.0312 0.0938
0.0389 0.8361 0.0021 0.1229

π=
0.0938 0.0312 0.2917 0.5833
0.0091 0.1159 0.0243 0.8507
Figure 1 shows next period aggregate capital, K+ as a function of current period aggregate capital, K, in state (b, 0), for two different values of ω: ω = 0 and ω = 1. The
figure shows that future aggregate capital depends on not only current aggregate capital
but also on current wealth share ω of agent 1.
Given the global nonlinear solution for ϕ∞ , we can also simulate forward and carry
out 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 = 0.0438 + 0.9832 log K;
R2 = 0.999223
in good times and
log K 0 = 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 aggregate
capital, 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 parameters above, but in which idiosyncratic shocks are truly idiosyncratic. We obtain the
following regression results:
log K 0 = 0.0906 + 0.9631 log K;
R2 = 0.999999
log K 0 = 0.0807 + 0.9651 log K;
R2 = 0.999999
in good times
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 autoto the approximation in Heaton and Lucas (1995).
7 In the transition matrix, we use the convention {1, 2, 3, 4} correspond to {( b, 0) , ( b, 1) , ( g, 0) , ( g, 1)}
respectively.
20
50
Future Aggregate Capital
45
40
35
30
25
20
(b,0) ω=0
(b,0) ω=1
15
10
5
450
5
10
15
20
25
30
35
40
45
Current Aggregate Capital
Figure 1: Evolution of Aggregate Capital in Bad Times
21
50
50
Future Aggregate Capital
45
40
35
30
25
20
(b,0) ω=0
(b,0) ω=1
15
10
5
450
5
10
15
20
25
30
35
40
45
50
Current Aggregate Capital
Figure 2: Evolution of Aggregate Capital in Bad Times
correlation coefficients for log aggregate capital are lower than those in the two-agent
economy. 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 their
discount rate. The discount factor can be low (β) or high ( β̄), where:
β = 0.9858 and β̄ = 0.9930,
taken from Krusell and Smith (1998, Section 3). As in their paper, the transition from one
to 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 two
agents have the same labor productivity, which varies with the aggregate state, s. The
aggregate productivity and aggregate labor supply are given in (22). The evolution of
the aggregate state is the same as in the previous example. The other aggregate state i
determines the agents’ discount factor (i = 0 agent 1 has low discount factor and agent
2 has high discount factor and vice versa for i = 1). The evolution of aggregate state i is
independent of the evolution of aggregate state s.
Figure 2 shows next period aggregate capital, K+ as a function of current period aggregate capital, K, in state (b, 0), for two different values of ω: ω = 0 and ω = 1. The
figure shows that future aggregate capital depends mostly on current aggregate capital
and 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 periods
dropped), we obtain the following regression results:
22
log K 0 = 0.0916 + 0.9633 log K;
R2 = 0.999999
log K 0 = 0.0789 + 0.9662 log K;
R2 = 0.999999
in good times
and in bad times. Because future aggregate capital depends mostly on current aggregate
capital, the fitness of the linear regressions are very high.
As in the previous example, these regression results are comparable to the ones in
Krusell and Smith (1998)’s model in which the discount rates are truly idiosyncratic:
log K 0 = 0.0871 + 0.9662 log K;
R2 = 0.999981
in good times and
log K 0 = 0.0836 + 0.9670 log K;
R2 = 0.999976
in bad times.
4.3
Discussion of the Algorithm for Many Agents and for Continuum
of Agents
The discretization and approximation method laid out in Subsection 4.1 applies to general
model in Section 2 with many agents. However, when the number of agents is larger than
2, the algorithm suffers from the curse of dimensionality, i.e., it takes many points to
discretize Ω using Ωd (the number of points is approximately N dim(Ω) where N is the
number of points used to discretize each dimension). There are two ways to get around
this problem.
First, notice that from Conditions A1-A3, in Subsection 4.1, we just need to solve for
h
c h∈H as functions of the exogenous and endogenous states (s, K, ω ) ∈ Ω. The idea is
to approximate numerically ch ’s using some basis functions
{ξ 1 , ξ 2 , ..., ξ m }
that is
ch (s, K, ω ) =
m
∑ ĉih ξ (s, K, ω ).
i =1
h∈H
We can then solve for the approximation coefficients ĉih i=1,2,...,,m . The advantage of this
algorithm is that the number of basis functions can be significantly smaller than the number of points to discretize Ω and does not increase fast with the dimension of Ω. The basis 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 requires
knowing the value of ch ’s at a few number of collocation points in Ω to approximate
the whole functions. A comprehensive exposition of the method and applications can be
found in Maliar and Maliar (2014).
Using these ideas, the algorithm in Subsection 4.1 can potentially be applied to the
model in Section 3 with a continuum of agents. However, there are two major difficul23
ties. First, the endogenous state space Ω̄ of probability measures is infinite-dimensional.
Therefore, we need to approximate Ω̄ with a finite-dimensional space. For example, one
can approximate Ω̄ with the set of convex combinations of Dirac masses, i.e., for each
µ ∈ Ω̄ we approximate µ by
M
µ≈
∑ ∑ µ̂i,j D(k j )
i ∈I j=1
where 0 ≤ k1 < k2 < ... < k M ≤ k̄, and ∑i∈I ∑ jM=1 µ̂i,j = 1.8 Second, following Definition
4, for each µ ∈ Ω̄, we need to solve for the value and policy functions, V̂ and k̂ (in contrast
to the case with a finite number of agents, we just need to solve for a vector of current
consumptions and future capital holdings). In other words, we need to solve for the
value and policy functions that depend both on capital holding and wealth distribution,
V̂ (k, i; s, µ) and k̂ (k, i; s, µ). Having approximated Ω̄ with a finite n−dimensional space, V̂
and k̂ become functions over (n + 1) dimensions. They can then be approximated using
basis functions or Smolyak’s sparse grid method.
While these are viable paths to implement the algorithm for many agents or for a
continuum of agents, they would require a significant amount of engineering and thus lie
outside the scope of the present paper.
5
Conclusion
In this paper, I define the concept of generalized recursive equilibrium and show its existence in the neoclassical growth model with both idiosyncratic and aggregate shocks as in
Krusell and Smith (1998). The proof applies equally well to economies with a finite number of agents and with a continuum of agents. I also provide (rather strong) conditions
under which a generalized recursive equilibrium gives rise to a recursive equilibrium. In
general, however, it is still an open question whether a recursive equilibrium exists in
these economies. The question deserves further research given the rising important of
this class of economies.
The proof suggests an algorithm to compute a recursive equilibrium if it exists. The
algorithm is an global (in the space of distributions) alternative to the local ones put forth
by Krusell and Smith (1998) and related algorithms presented in the JEDC’s symposium
Den 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 local
algorithms focus on a subset of the space of distributions around the stationary distribution without aggregate shocks.9 There are also challenges in implementing the suggested
global nonlinear algorithm for economies with a large number or a continuum of agents.
Subsection 4.3 suggests several possible ways to overcome these challenges.
8 Approximating
distributions using Dirac masses is similar the histogram technique used in Young
(2010).
9 For example, the backward algorithm in Reiter (2010) resembles the time iterations presented in Subsection 4.1. But Reiter only uses a small set of aggregate statistics to summarize wealth distributions and
the 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 Lemma
3. 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. Given
prices
rt,T (st ), wt,T (st ) t≤T,st ∈S t
the representative firm solves
max Πt = Yt − rt Kt − wt Lt
Yt, Kt ,Lt
s.t. Yt ≤ F (st , Kt , Lt ). We allow for Πt potentially be different from 0, but we show that in
equilibrium Πt = 0. We also assume that the profits (or losses) are divided equally across
agents.
27
Given prices and the representative firm’s profit, agents solve
"
#
T h
max E0 ∑ Πtt0 =0 βh (st ) u(ct,T
)
h ,k h
ct,T
t+1,T
(23)
t =0
s.t.
h
ct,T
+ kht+1,T ≤ (1 − δ)kht,T + rt,T kht,T + wt,T lth +
1
Πt
H
and
h
ct,T
, k ht+1,T ≥ 0.
A competitive equilibrium is defined similarly as in Definition 1. Lemma 3 show that
a competitive equilibrium exists. Lemma 4 shows that
h
≤ c̄
c ≤ ct,T
0 ≤ k ht,T ≤ k̄
h
v ≤ vt,T
≤ v̄
K ≤ Kt ≤ K̄
r ≤ rt,T ≤ r̄
w ≤ wt,T ≤ w̄
with the bounds appropriately defined.
Lemma 3. Given an initial distribution of capital holding k0h h∈H such that
K0 =
1
H
∑
k0h > 0,
h∈H
a competitive equilibrium exists in the finite horizon economy.
Proof. The proof uses Kakutani’s Fixed Point Theorem as in Cao (2010), which builds
upon Debreu (1959).
To simplify the proof, we switch from choosing the final good as numeraire to the
following normalization:
pct,T (st ) + wt,T (st ) + rt,T (st ) = 1.
The sequential budget constraint of the consumers become:
h
+ kht+1,T − (1 − δ)kht,T ) ≤ rt,T kht,T + wt,T lth +
pct,T (ct,T
1
Πt .
H
The objective function of the representative firms
Πt,T = pct,T Yt,T − rt,T Kt,T − wt,T Lt,T .
(24)
Given a sequence ẽ = {et }tT=0 such that et > 0 for t = 0, 1, ..., T, we impose an additional
restriction on the set of normalized prices:
pct,T (st ) ≥ et > 0.
(25)
This restriction effectively puts an upper bound on marginal rate of returns on capital:
rt,T
1 − et
≤
c
pt,T
et
therefore a lower bound on aggregate capital.
For e > 0, let ∆e denote the subset of R3+ :
n
o
∆e = ( pc , w, r ) ∈ R3+ : pc + w + r = 1 and pc ≥ e > 0 .
28
(26)
For each history st , given our normalization and the additional restriction (25):
pct,T (st ), wt,T (st ), rt,T (st ) ∈ ∆et .
We also denote
T
∆Σ
ẽ
=
n
pct,T , rt,T , wt,T t,st
:
pct,T , rt,T , wt,T
o
∈ ∆ et .
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 restrictions:
Kt,T ≤ 2K̄ and Lt,T ≤ 2 L̄,
t
for all t and s , where K̄ is defined in (10).
Similarly, each consumers maximize (23) subject to
1
h
pct,T ct,T
+ pct,T kht+1,T − (1 − δ)kht,T ≤ rt,T kht,T + wt,T lth + Πt,T
(27)
H
and
h
0 ≤ ct,T
, k ht+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 ) < Π̄
for all t, st . Given the initial distribution of capital holding, the budget constraints (27)
and the exogenous lower bound on pct,T , (25), it is easy to show that there exists ĉ, k̂ > 0
such that
h
< ĉ and kht+1,T < k̂
ct,T
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 :∆ẽΣ
T
⇒R3kΣ k
T
p T ∈ ∆Σ
ẽ
T
7→ x T = (excess demands)
The component of the excess demand in each market corresponds to the component of
the price system in that market:
1
c
h
t
h
t
h t −1
Consumption:xt,T
(st ) =
c
(
s
)
+
k
(
s
)
−
(
1
−
δ
)
k
(
s
)
− Yt,T (st )
t
t,T
t+1,T
H h∑
∈H
k
Capital:xt,T
(st ) = Kt,T (st ) −
Labor:
l
xt,T
(st )
1
H
∑ kht,T (st−1 )
h
t
= Lt,T (s ) − L(st ).
29
It is standard to show that ψx is upper hemi-continuous and compact, convex-valued.10
h , kh
Given that each individual choices ct,T
t+1,T are bounded, ψx is bounded by a closed
cube K x ⊂ RΣ . For example,
T
k
−k̂ ≤ xt,T
(st ) ≤ 2K̄
for all t, st .
Consider the following correspondence:
Ψ : ∆Σ
ẽ × K x
n
o
T
T
p T ∈ ∆Σ
,
x
∈
K
x
ẽ
⇒∆Σẽ × K x
(
T
T
)
7→ arg max p̃ · x T
T
p̃∈∆Σ
ẽ
× ψx ( p T ).
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 guarantees
that Ψ has a fixed point p̄ T , x̄ T . By choosing ẽ appropriately, we can show that p̄ T , x̄ T
corresponds to a competitive equilibrium. The proof is similar to the one in Lemma 6
and Lemma 7 below so we omit the details here. For example, ẽ and {K t }tT=0 are chosen
recursively using Assumption 3 and the agents’ Euler equation (28):
1. e0 and K0 are chosen as in Lemma 7.
2. Given K t−1 , et is chosen sufficiently small such that, for any K such that
1 − et 1 + max0≤K ≤2K̄ FL (s, K, L)
FK (s, K, L) ≥
,
et
for some L ∈ [0, 2 L̄] we have
1
σ
>
πs− s (1 − δ) et + 1 − et 1 + max FL (s, K̃, L
0≤K̃ ≤2K̄
F (s, K, L̄) + (1 − δ)K
.
F ( s − , K t −1 , L ) + (1 − δ ) K t −1 − K
for all s, s− ∈ S .
K t is chosen such that for all s, s− ∈ S , and Kt ≤ K t :
1
( π s − s (1 − δ ) e ) σ >
F (s, K, L̄) + (1 − δ)K
.
F ( s − , K t −1 , L ) + (1 − δ ) K t −1 − K
Lemma 4. Consider a competitive equilibrium with the initial aggregate capital K0 > 0 and
let K, K̄ be defined as in (11) and (10). Then for all t ∈ {0, ..., T } and st ∈ S t , we have K ≤
Kt,T (st ) ≤ K̄ and, for all h ∈ H:
h
0 ≤ ct,T
, k ht,T ≤ H max { F (s, K̄, L(s)) + (1 − δ)K̄ } = c̄ = k̄,
s∈S
and
r = min min FK (s, K̄, L(s)) ≤ rt,T ≤ r̄ = max max FK (s, K, L(s))
s∈S L≤ L≤ L̄
s∈S L≤ L≤ L̄
10 The additional restriction (25) is crucial for the upper hemi-continuity of ψ . Without the restriction ψ
x
x
is not upper hemi-continuous.
30
and
w = min min FL (s, K, L̄) ≤ wt,T ≤ w̄ = max max FL (s, K, L).
s∈S K ≤K ≤K̄
s∈S K ≤K ≤K̄
h ≥ c for all t and st and h ∈ H.
In addition, there exists c > 0 such that ct,T
Proof. First we show by induction that Kt,T (st−1 ) ≤ K̄ for all t and st . At t = 0, this
property is satisfied by the definition of K̄. Assume that the property holds for t, and all
st ∈ S t , we show that it holds for t + 1 and st+1 ∈ S t+1 .
Indeed, from the market clearing conditions, we have
K t +1 ( s t ) =
1
H
∑
k ht+1,T (st )
h∈H
= Yt,T (st ) + (1 − δ)Kt (st−1 ) −
1
H
∑
h
ct,T
(st )
h∈H
t −1
t
≤ Yt,T (s ) + (1 − δ)Kt (s )
= F s t , K t ( s t −1 ), L ( s t ) + (1 − δ ) K t ( s t −1 ).
If Kt (st−1 ) ≥ K̂ then
Kt+1 (st ) = Kt (st−1 ) + F st , Kt (st−1 ), L(st ) − δKt (st−1 )
≤ Kt (st−1 ) ≤ K̄.
If Kt (st−1 ) ≤ K̂ then
K t +1 ( s t ) = F s t , K t ( s t −1 ), L ( s t ) + (1 − δ ) K t ( s t −1 )
≤ max max F (K, L(s), s) + (1 − δ)K ≤ K̄
s∈S 0≤K ≤K̂
So in either case we have Kt+1 (st ) ≤ K̄.
Therefore, by induction, we have Kt,T (st−1 ) ≤ K̄ 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 some
st ∈ S t .
From the first order condition of the agents, we have
h
i
0 h
0 h
u (ct,T ) ≥ βEt (1 − δ + FK (st+1 , Kt+1 , Lt+1 ))u (ct+1,T )
(28)
for all h ∈ H. Therefore
u
0
h
(ct,T
)
h
0
(cth+1,T )
i
≥ min (1 − δ + FK (st+1 , K, Lt+1 )) βEt u
st+1 ∈S
≥ min (1 − δ + FK (st+1 , K, Lt+1 )) βu0 Et [cth+1,T ]
st+1 ∈S
31
where the last inequality comes from the fact that u0 (c) = c−σ is strictly convex.
Consequently,
h
i
1
Et cth+1,T
σ
≥
β
min
(
1
−
δ
+
F
(
s
,
K,
L
))
K
t
+
1
t
+
1
h
st+1 ∈S
ct,T
and11
Et
h
1
h
H ∑ h∈H ct+1,T
1
h
H ∑ h∈H ct,T
i
1
≥
β min (1 − δ + FK (st+1 , K, Lt+1 ))
st+1 ∈S
σ
.
From the market clearing conditions, we have
1
H
∑
cth+1,T ≤ F (st+1 , Kt+1 , Lt+1 ) + (1 − δ)Kt+1 ≤ max F (s, K, L̄) + (1 − δ)K
s∈S
h∈H
and
1
H
∑
h
= F (st , Kt , Lt ) + (1 − δ)Kt − Kt+1 ≥ F (st , K, L̄) − δK.
ct,T
h∈H
Therefore,
Et
h
1
h
H ∑ h∈H ct+1,T
1
h
H ∑ h∈H ct,T
i
≤
maxs∈S F (s, K, L̄) + (1 − δ)K
.
F (st , K, L̄) − δK
So finally, we obtain
maxs∈S F (s, K, L̄) + (1 − δ)K
≥
F (st , K, L̄) − δK
1
β min (1 − δ + FK (st+1 , K, Lt+1 ))
st+1 ∈S
σ
.
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 cth , k ht , rt , wt follow immediately.
Now we show that there exists c > 0 such that cth ≥ c for all t, st , and h. Indeed, from
the agents’ maximization problem, since starting from any history st , an agent can always
consumes her labor endowment, we have
"
#
"
#
0
0
u(cth (st )) + Et
∞
t
∑∏
t+1 t00 =t0
β(st00 )u(cth0 ) ≥ u(wl ) + Et
≥
∞
∑∏
t+1 t00 =t0
1
u(wl ).
1 − β̄
In addition, cth0 ≤ c̄ for all t0 . Therefore
u(cth (st )) +
11 We
use the inequality that m ≤
aj
bj
β
1−β
u(c̄) ≥
for all j implies m ≤
32
1
u(wl ).
1 − β̄
∑ aj
.
∑ bj
t
β(st00 )u(wl )
So
cth (st ) ≥ c = u−1
β
1
u(wl ) −
u(c̄)
1−β
1 − β̄
!
> 0.
Proof of Theorem 1. The steps of the proof are similar to the ones for Theorem 2 presented
in the main paper. With some abuse, we re-use several notations in the proof such as Θ, g
etc.
Given the bounds determined in Lemma 4, let Θ denote the set of
ch , k h+ , vh
, r, w
h∈H
such that c ≤ ch ≤ c̄, 0 ≤ k h+ ≤ k̄, v ≤ vh ≤ v̄, r ≤ r ≤ r̄, and w ≤ w ≤ w̄.
Let g : S × Ω ⇒ Θ × ΘS denote thefollowing correspondence:
for each s ∈ S ,
h
h
h
h
ω = k h∈H ∈ Ω, g(s, ω ) is the set of θ = c , k + , v h∈H , r, w ∈ Θ and (θs0 )s0 ∈S ∈ ΘS
0
0
h
0
h
0
h
0
with θs0 =
c+ (s ), k ++ (s ), v+ (s ) h∈H , r+ (s ), w+ (s ) 0 such that:
s ∈S
1
H
∑
h∈H
ch +
1
H
∑
k h+ = F (s, K, L) + (1 − δ)K
h∈H
where K = H1 ∑h∈H k h > 0 and L = H1 ∑h∈H l h (s) > 0, and r = FK (s, K, L) > 0 and
w = 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, for
h
each s ∈ S and ω = k h∈H ∈ Ω, we have
)
(
θ = ch , k h+ , vh h∈H , r, w ∈ Θ : for each s0 ∈ S ,∃θs0 ∈ V s0 , k h+ h∈H
W (s, ω ) =
and θ, (θs0 )s0 ∈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.
∞
m
m
h
h
h
Indeed, assume {θ }m=0 ∈ W (s, ω ), and θ → θ = c , k + , v h∈H , r, w . Since Θ
is 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 s0 ∈ S , ∃θsm0 ∈ V s0 , k h+ h∈H and θ m , θsm0
∈ g(s, ω ).
m
Since V s0 , k h+ h∈H is compact, we can extract a converging subsequence, θs0 l → θs0 for
some θs0 ∈ V s0 , k h+ h∈H . Because g is a closed valued correspondence, θ, (θs0 )s0 ∈S ∈
g(s, ω ), which implies θ ∈ W (s, ω ).
P2. If V ⊂ V 0 in the sense that V (s, ω ) ⊂ V 0 (s, ω ) for all s ∈ S and ω ∈ Ω then
G(V ) ⊂ G(V 0 ).
P3. Let V 0 denote the complete correspondence: V 0 (s, ω ) = Θ for all s ∈ S and ω ∈ Ω.
Then G(V 0 ) ⊂ V 0 .
33
n+1 = G(V n ).
Given V 0 , we construct the sequence of {V n }∞
n=0 recursively using G : V
Then by P1, P2, and P3, we have V n+1 ⊂ V n and is non-empty, compact valued. Nonempty-ness comes from the existence of competitive equilibrium in the n + 1-horizon
economy proved in Lemma 3.
Let V ∗ be defined by
n
V ∗ (s, ω ) = ∩∞
n=0 V ( s, ω ).
Since V ∗ (s, ω ) is the intersection of decreasing compact sets, V ∗ (s, ω ) is compact and is
non-empty. We show that G(V ∗ ) = V ∗ .
Indeed, by the definition of V ∗ , we also have V ∗ ⊂ V n , to G(V ∗ ) ⊂ G(V n ) = V n+1 , so
G(V ∗ ) ⊂ V ∗ .
Now, for each s ∈ S and ω ∈ Θ and θ =
ch , k h+ , vh h∈H , r, w ∈ V ∗ (s, ω ). Since
V ∗ ⊂ V n , there exists θsn0 ∈ V n s0 , kh+ h∈H and θ, θsn0 s0 ∈S ∈ g(s, ω ). By the comnl 0
,
θ
pactness of Θ, we can find a converging subsequence {nl }∞
s0 s0 ∈S −→l →∞ ( θs )s0 ∈S .
l =0
By the compactness of V nl , we have θs0 ∈ V nl s0 , k h+ h∈H and since g has closed graph,
θ, (θs0 )s0 ∈S ∈ g(s, ω ). Moreover, V n s0 , k h+ h∈H is a decreasing sequence so θs0 ∈
∗
nl s 0 , k h
∗ s0 , k h
∩∞
V
=
V
+ h∈H . So by the definition of G , we have θ ∈ G(V ).
+ h∈H
l =0
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 = k h h∈H ∈ Ω, θ =
h
h
h
0
h
c , k + , v h∈H , r, w ∈ Q(s, ω ), there exists θs0 ∈ Q s , k + h∈H for each s0 ∈ S and
θ, (θs0 )s0 ∈S ∈ g(s, ω ). We also define T as
n o
h
h h
h
h h
T (s, ω ) =
k+
:
c , k+ , v
, r, w ∈ Q(s, ω ) for some c , v
and some r, w .
h∈S
h∈H
h∈H
It is immediate that (P , T ) defined as such forms a generalized recursive equilibrium for
the economy with a finite number of types.
Proof of Lemma 1. Consider sequences of allocation and
generated by a general hprices
ized recursive equilibrium, starting from s0 ∈ S and k0 h∈H ∈ Ω. That is, sequences
h t h
ct (s ), k t+1 (st ), vth (st ) t,st ,h and rt (st ), wt (st ) t,st such that for each t, st
o
n
o
n
, rt (st ), wt (st ) ∈ Q st , k ht (st )
,
cth (st ), k ht+1 (st ), vth (st )
h∈H
and
n
o
cth+1 (st+1 ), k ht+2 (st+1 ), vth+1 (st+1 )
h∈H
h∈H
n
o
, rt+1 (st+1 ), wt (st+1 ) ∈ Q st+1 , k ht+1 (st )
h∈H
and Conditions 3-4 in Definition 2 are satisfied (with the variable without subscript stands
for 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 example
h stands for c
h
h
ch stands for cth , c+
t+1 and c++ stands for ct+2 , etc.).
The market clearing
conditions are
We just need to verify that
satisfied obviously.
given rt (st ), wt (st ) , the allocation cth (st ), k ht+1 (st ) t,st solves agent h’s maximization
34
,
problem, (5). That is for any alternative allocation c̃th (st ), k̃ ht+1 (st ) t,st that satisfies (3)
and (4), we have
"
"
#
#
∞ ∞ E0 ∑ Πtt0−=10 βh (st0 ) u(cth ) ≥ E0 ∑ Πtt0−=10 βh (st0 ) u(c̃th ) .
(29)
t =0
t =0
The proof of this inequality follows closely Duffie et al. (1994). First, we show by induction that for all T ≥ 0:
"
"
"
#
#
#
T ∞ ∞ E0 ∑ Πtt0−=10 βh (st0 ) u(cth ) ≥ E0 ∑ Πtt0−=10 βh (st0 ) u(c̃th ) + E0 ∑ Πtt0−=10 βh (st0 ) u(cth )
t =0
t =0
+ E0
h
T +1
−1 h
0 h
h
h
ΠtT0 =
0 β ( st0 ) u ( c T ) k̃ T +1 − k T +1
i
.
(30)
For T = 0, inequality (30) is
u(c0h ) ≥ u(c̃0h ) + u0 (c0h ) k̃1h − k1h ,
which is true because
u(c0h ) ≥ u(c̃0h ) + u0 (c0h ) c0h − c̃0h ,
from the concavity of u(.) and from
c̃0h + k̃1h ≤ (1 − δ + r0 )k0h = c0h + k1h ,
which implies c0h − c̃0h ≥ k̃1h − k1h .
Now, assume that (30) holds for T, we need to show that it also holds for T + 1, i.e.
#
#
#
"
"
"
T +1 ∞ ∞ E0 ∑ Πtt0−=10 βh (st0 ) u(cth ) ≥ E0 ∑ Πtt0−=10 βh (st0 ) u(c̃th ) + E0 ∑ Πtt0−=10 βh (st0 ) u(cth )
t =0
t =0
+ E0
h
T +2
ΠtT0 =0 βh (st0 )
u
0
(chT +1 )
k̃ hT +2
− khT +2
i
.
(31)
Given (30) holds for T, to show (31), we just need to show:
h
i
i
h
T
h
h
T −1 h
h
h
0 h
0
0
E0 Πt0 =0 β (st ) u(c T +1 ) + E0 Πt0 =0 β (st ) u (c T ) k̃ T +1 − k T +1
h
i
h
i
+1 h
h
T
h
0 h
h
h
0
0
≥ E0 ΠtT0 =
β
(
s
)
u
(
c̃
)
+
E
Π
β
(
s
)
u
(
c
)
k̃
−
k
.
0
0
t
t
T +2
T +2
T +1
T +1
t =0
0
Equivalently,
h
E0 β
h
≥ E0
(s T )u(chT +1 )
h
i
h
0
(chT )
k̃ hT +1
− khT +1
i
+ E0 u
i
h
i
βh (s T )u(c̃hT +1 ) + E0 βh (s T )u0 (chT +1 ) k̃ hT +2 − k hT +2 .
Because of Condition 5. in Definition 2,
u
if k hT +1 > 0, and
0
(chT )
=
βhT (s T )ET
h
(1 − δ + r T +1 ) u
0
(chT +1 )
i
,
h
i
u0 (chT ) ≥ βhT (s T )ET (1 − δ + r T +1 ) u0 (chT +1 ) ,
35
(32)
if k hT +1 = 0, which implies k̃ hT +1 − k hT +1 ≥ 0 . Therefore
i
h
E0 u0 (chT ) k̃ hT +1 − k hT +1
i
h
≥ E0 βhT (s T ) (1 − δ + r T +1 ) u0 (chT +1 ) k̃hT +1 − khT +1 .
From this inequality, we obtain (32) if
u(chT +1 ) + (1 − δ + r T +1 ) u0 (chT +1 ) k̃ hT +1 − k hT +1
≥ u(c̃hT +1 ) + u0 (chT +1 ) k̃hT +2 − khT +2 .
(33)
Since
chT +1 + k hT +2 = (1 − δ + r T +1 ) k hT +1
c̃hT +1 + k̃ hT +2 ≤ (1 − δ + r T +1 ) k̃ hT +1 ,
we have
(1 − δ + r T +1 ) k̃hT +1 − khT +1 ≥ k̃hT +2 − khT +2 + c̃hT +1 − chT +1 .
Plugging this into (33), we obtain the desired inequality if
u(chT +1 ) + u0 (chT +1 ) c̃hT +1 − chT +1 ≥ u(c̃hT +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 k̄ > 0 such that k ht (st ) ≤ k̄ for all h, t, st . Now, from (30), taking
T → ∞ and noticing that
#
"
∞ 1
E0 ∑ Πtt0−=10 βh (st0 ) u(cth ) ≥ β̄ T +1
u (c ) → T →∞ 0
1
−
β̄
T +1
and
i
h
h
0 h
−1 h
0
)
k̃
−
k
β
(
s
)
u
(
c
ΠtT0 =
t
T
T +1
T +1
0
i
h
−1 h
0 h h
≥ −E0 ΠtT0 =
0 β ( s t 0 ) u ( c T ) k T +1
E0
h
≥ − β̄ T u0 (c)k̄ −→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 competitive
equilibrium in a finite horizon economy. Then we show that in any competitive equilibrium, prices and allocations lie in compact sets.
Consider the finite horizon version of the economy in Section 3 with t = 0, 1, ..., T. We
restate 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
max
Yt,T, Kt,T ,Lt,T
Πt,T = Yt,T − rt,T Kt,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 we
show that in equilibrium Πt = 0. We also assume that the profits (or losses) are divided
equally across agents.
Given prices and the representative firm’s profit, the value function of the agents,
V̂t,T (k, i; st ) satisfies the Bellman equation (starting from t = T + 1 with VT +1,T ≡ 0 moving backward):
h
i
V̂t,T (k, i; st ) = max u(c) + β(i )Et V̂t+1,T (k0 , i; st+1 )
(34)
c,k0
subject to
c + k0 − (1 − δ)k ≤ rt,T (st )k + wt,T (st )l (st , it ) + Πt,T (st ),
(35)
and c ≥ 0 and
0 ≤ k ≤ k̄,
with the policy functions ĉt,T
and k̂ t,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 V̂t,T , ĉt,T , k̂ t,T that satisfy (34) and sequences of wealth distribution µt,T (k, i; st ) such that the following identity holds:
(k, i; st )
∑
Z
t
t
ĉt,T (i, k; s )µt,T (dk, i; s ) +
i ∈I
k̂ t,T (i, k; st )µt,T (dk, i; st )
i ∈I
= F (st , Kt,T (s
t −1
where
Kt,T (st−1 ) =
∑
Z
∑
i ∈I
), L(st )) + (1 − δ)Kt,T (st−1 )
Z k̄
0
kµt (dk, i; st ) and L(st ) =
∑ m s t ( i ) l ( s t , i ).
i ∈I
In addition, we have
t
t −1
t
t −1
rt,T (s ) = FK st , Kt,T (s ), L(st ) > 0 and wt,T (s ) = FL st , Kt,T (s ), 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, k̄ :
πss0 ,ii0
t −1
t
0
t
µt,T (k̂ t,T (., i; s )) ( A), i; s .
µt+1,T A, i ; s , st+1 = ∑
πss0
i ∈I
In the proof of Lemma 2, this recursive version of a competitive equilibrium can be
mapped back to the definition using agents’ index, as in Definition 3.
The following lemma establishes the existence of this recursive version of a competitive equilibrium.
Lemma 5. There exists a competitive equilibrium in the finite horizon economy version of the
model in Section 3 with a continuum of agents.
37
Proof. Given a sequence e T = {et }tT=0 such that et > 0 for all t ∈ {0, 1, ..., T }, let us define
∆eΣT = ×(t,st ) ∆et
ΣT
c
c
c
3
: pt + wt + rt = 1 and pt ≥ et ,
= ( pt , wt , rt )t,st ∈ R+
T
where ∆e is defined in (26).
T
Given the prices p T ∈ ∆Σ
. The firms maximize profit and the agents maximize the
eT
inter-temporal expected utility. In particular, in each history st the representative firm
solves
max Πt,T (Kt,T , Lt,T )
(36)
Yt ,Kt ,Lt ≥0
s.t.
Yt,T ≤ F (st , Kt,T , Lt,T )
and
Πt,T = pct,T Yt,T − rt,T Kt,T − wt,T Lt,T .
We also impose two additional constraints:
0 ≤ Lt,T ≤ 2 L̄ and 0 ≤ Kt,T ≤ 2K̄,
where L̄ and K̄ are defined in Assumption 5 and in (10). Given Π T = Πt,T (st ) ,12 the
agents solve the dynamic programing problem: V̂t,T is defined recursively (starting from
t = T + 1 with V̂T +1,T ≡ 0 moving backward) as
h
i
V̂t,T (k, i; st , p T , Π T ) = max u(c) + β(i )Et V̂t+1,T (k0 , i; st , p T , Π T )
(37)
c,k0
subject to c ≥ 0, 0 ≤ k0 ≤ k̄ and
pct,T (st ) c + k0 − (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 k0 , k̂t,T (k, i; st ) is continuous and is
weakly increasing.
Given the policy function k̂t,T , we construct the sequence of measures µ̃ T = µ̃t,T (.; st ) t,st
as following:
1. µ̃0,T = µ0,T
2. For t ≥ 0, for every A ∈ B 0, k̄
−1
t +1
t
µ̃t+1,T ( A, it+1 ; s ) = ∑ Pr(it+1 |it , st , st+1 )µt,T
k̂t,T
( A ), i t ; s .
it ∈I
We denote
Σ
ψµ : ∆Σ
⇒ Ω̄Σ
e T × Ω̄
T
T
T
the correspondence that map the sequence of prices p T and distributions µ T to the sequence of distributions µ̃ T as constructed above.
maximization problem (36) might have many maximizers but the maximized objective Πt,T is
uniquely determined given prices pt,T ∈ ∆et .
12 The
38
We form the excess demand functions
Z k̄ c
t
Consumption:xt,T (s ) =
ĉt,T (k, i; st ) + k̂ t,T (k, i; st ) − (1 − δ)k µt,T (dk; it , st ) − Yt,T (st )
0
k
Capital:xt,T
(st )
t
= Kt,T (s ) −
Z k̄
0
kµt,T (dk, it ; st )
l
Labor: xt,T
(st ) = Lt,T (st ) − L(st ).
We have:
c
(st ) < x̄ c =
x c = −(1 − δ)k̄ − Ȳ < xt,T
where e =
1
2
1−e
1
1−e
L̄ + Ȳ,
k̄ +
e
e
e
min0≤t≤T et . We also have
k
x k = −2k̄ < xt,T
(st ) < x̄ k = 2K̄
and
l
(st ) < x̄ l = 2 L̄
x l = −2 L̄ < xt,T
ΣT
Let K x denote the cube [ x c , x̄ c ] × x k , x̄ k × x l , x̄ l
. We define the correspondence
ΣT
ψx : ∆ Σ
⇒ Kx
e × Ω̄
T
that maps a sequence of prices p T and a sequence of distributions µ T to the excess demand
in every history.
Lastly,
T
ψ p : K x ⇒ ∆Σ
eT
such that
pt,T = arg max p · xt,T .
p ∈ ∆ et
Let Φe denote an operator (which depends on {et }) taking φ p , φµ , φx as components:
Σ
Ψe : ∆Σ
× K x ⇒ ∆ΣeT × Ω̄Σ × K x
(39)
e T × Ω̄
Ψe = ψ p , ψµ , ψx
Lemma 9 shows that Ψe is upper-hemi continuous and is non-empty, compact, and
T
Σ T × K is a compact and convex subset of a locally
convex valued. In addition ∆Σ
x
e × Ω̄
convex Hausdorff space.13 Therefore, by the Kakutani-Glicksberg-Fan Fixed Point Theorem, Φe admits a fixed point. By choosing et appropriately, in Lemma 6, we show that
this fixed point constitutes a competitive equilibrium.
T
T
T
T
Lemma 6. Consider the sequence {et , K t }tT=0 constructed in Lemma 7, and Ψe as defined in (39)
given the sequence {et }. Let
ψ̄ = ( p̄t,T )t,st , (µ̄t,T )t,st , ( x̄t,T )t,st
be a fixed point of Ψe . Then ψ̄ corresponds to a competitive equilibrium.
properties follow directly from the result that the space Ω̄ of probability measures endowed
with weak topology is metrizable, shown in Bogachev (2000, Theorem 8.3.2).
13 These
39
Proof. To show that ψ̄ corresponds to a competitive equilibrium, we need to show that
xt,T (st ) = 0 and rt,T , wt,T > 0 for all t ≤ T and st ∈ S t . To simplify the notations, we omit
the bar notation on variables.
First, we notice that for all t ≤ T and st ∈ S t ,
c
k
l
pt,T · xt,T = pct,T xt,T
(st ) + rt,T xt,T
+ wt,T xt,T
Z k̄ c
ĉt,T (k, i; st ) + k̂ t,T (k, i; st ) − (1 − δ)k µt,T (dk, it ; st ) − pct,T Yt,T
= pt,T ∑
i ∈I
0
+ rt,T Kt,T − rt,T
∑
Z k̄
i ∈I
=
pct,T
∑
i ∈I
Z k̄ 0
0
kµt,T (dk; it , st ) + wt,T Lt,T − wt,T L(st ).
t
ĉt,T (k, i; s ) + k̂ t,T (k, i; s ) − (1 − δ)k µt,T (dk, it ; st )
− Πt,T (Kt , Lt ) − rt,T
∑
i ∈I
t
Z k̄
0
kµt,T (dk; it , s ) − wt,T (s ) ∑
t
t
Z k̄
i ∈I
0
l (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 max p · xt,T
p ∈ ∆ et
So
or
c
0 = pt,T · xt,T ≥ 1 0 0 · xt,T = xt,T
c
xt,T
(st ) ≤ 0 ∀t ≤ T and ∀st ∈ S t .
(40)
st
S t.
Now, we show by induction that xt,T = 0 for all t ≤ T and ∈
In particular, we
show 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 for
all st−1 ∈ S t−1 then xt,T = 0 and rt,T , wt,T > 0 and in addition Kt > K t (Step 2).
c ≤ 0.
Step 1: Starting with t = 0, we have just shown in (40) that x0,T
k
If x0,T
< 0 then r0,T = 0 (since p0,T ∈ arg max p p · x0,T and p0,T · x0,T = 0). The
maximization of the representative firm, (36), at t = 0 implies that K0,T = 2K̄. But then
R
k > 0 since we chose 2K̄ > K = k̄ kµ
0
k
x0,T
0
0,T ( dk; i0 , s ). So x0,T ≥ 0.
0
l
Similarly, if x0,T
< 0 then w0,T = 0. Then, also from the maximization of the reprel
sentative firm, (36), at t = 0, L0,T = 2 L̄. But then x0,T
> 0 since we chose 2 L̄ > L̄ >
l
maxs∈S L(s). So x0,T ≥ 0.
c
c
Now, we show by contradiction that x0,T
= 0. Assume to the contrary that x0,T
< 0.
c
Then p0,T = e0 (since p0,T ∈ arg max p p · x0,T and p0,T · x0,T = 0). Therefore we have
k
l
c
r0,T x0,T
+ w0,T x0,T
= −e0 x0,T
> 0.
If r0,T = 0 then K0,T = 2K̄ and
k
l
x0,T
≥ 2K̄ − K0 > K̄ > 2 L̄ > x0,T
.
40
(41)
But then, because p0,T ∈ arg max p p · x0,T , we have r0,T = 1 − e0 and w0,T = 0, a contradiction with the assumption that r0,T = 0. So we must have r0,T > 0. And since
k ≥ x l . From (41), we have x k > 0.
p0,T ∈ arg max p p · x0,T , it must be that x0,T
0,T
0,T
l
If w0,T = 0 then L0,T = 2 L̄ > 0 and x0,T
= L0,T − L(s0 ) > 0. If w0,T > 0 then since
k . Since we have shown above that
l
≥ x0,T
p0,T ∈ arg max p p · x0,T , it must be that x0,T
l
k ≥ x l , this leads to x k = x l
x0,T
0,T > 0. In either case, we have x0,T > 0.
0,T
0,T
k + K > K and L
l
Therefore K0,T = x0,T
0
0,T = x0,T + L ( s0 ) > L.
0
Now, at time t = 0 and s0 , (because p0,T = e0 ) (Y0,T , K0,T , L0,T ) solves:
max e0 Y − r0,T K − w0,T L
Y,K,L
s.t.
Y ≤ F (s0 , K, L)
and K ≤ 2K̄, L ≤ 2 L̄. Because e0 > 0, Y0,T = F (s0 , K0,T , L0,T ) and
e0 FK (s0 , K0,T , L0,T ) ≥ r0,T
(with equality if K0,T < 2K̄) and
e0 FL (s0 , K0,T , L0,T ) ≥ w0,T
(with equality if L0,T < 2 L̄). Therefore
e0 ( FK (s0 , K0,T , L0,T ) + FL (s0 , K0,T , L0,T )) ≥ r0,T + w0,T = 1 − e0 .
Equivalently,
e0 (1 + FK (s0 , K0,T , L0,T ) + FL (s0 , K0,T , L0,T )) ≥ 1.
Because F is concave and K0,T ≥ K0 ,
(42)
FK (s0 , K0,T , L0,T ) ≤ FK (s0 , K0 , L0,T ) ≤ max FK (s0 , K0 , L),
0≤ L≤2 L̄
where max0≤ L≤2 L̄ FK (s0 , K0 , L) < ∞ by Assumption 3. Similarly, because L0,T ≥ L,
FL (s0 , K0,T , L0,T ) ≤ max FL (s0 , K, L).
0≤K ≤2K̄
Therefore,
e0 (1 + FK (s0 , K0,T , L0,T ) + FL (s0 , K0,T , L0,T ))
< e0 (1 + max FK (s0 , K0 , L) + max FL (s0 , K, L)) < 1,
0≤K ≤2K̄
0≤ L≤2 L̄
where the last inequality comes from the property (52) in Lemma 7. But this contradicts
the earlier inequality, (42).
c = 0. If x k > 0 or x l > 0 then max p · x
So we obtain by contradiction that x0,T
0,T > 0,
0
0
k
l
which contradicts 0 = p0,T · x0,T = max p p · x0,T . Therefore x0 = x0 = 0.
If w0,T = 0, then L0,T = 2 L̄ and x0l > 0 therefore w0,T > 0. If r0,T = 0 then K0,T = 2K̄
k = 2K̄ − K > 0 therefore r
and x0,T
0
0,T > 0.
Step 2: From t − 1 to t.
Since xtc−1,T = xtk−1,T = 0, we have
41
∑
Z k̄ 0
i ∈I
ĉt−1,T (k, i; st−1 ) + k̂ t−1,T (k, i; st−1 ) − (1 − δ)k µt−1,T (dk, i; st−1 ) − F (st−1 , Kt−1 , Lt−1 ) = 0
and
∑
Z k̄
0
i ∈I
kµt−1,T (dk, i; st−1 ) = Kt−1 .
From the definition of µt,T , µ̃t,T and the fixed point property of ψ̄,
∑
Z k̄
i ∈I
0
t
kµt,T (dk, i; s ) =
∑
Z k̄
i ∈I
0
k̂ t−1,T (k, i; st−1 )µt−1,T (dk, i; st−1 ).
Therefore,
∑
i ∈I
Z k̄
0
kµt,T (dk, i; st ) ≤ F (st−1 , Kt−1 , Lt−1 ) + (1 − δ)Kt−1 < K̄,
where the last inequality comes from condition (10) on K̄.
c ( s1 ) ≤
Now, we show that xt,T = 0 and rt,T , wt,T > 0. Indeed, we show in (40) that x1,T
0. The following arguments are similar to the argument in Step 1.
k ( st ) < 0 then r ( s1 ) = 0. Then K ( st ) = 2K̄ but then x k ( st ) > 0 since we have
If xt,T
t,T
t,T
t,T
R k̄
t
k
t
shown that K̄ > ∑i∈I 0 kµt,T (dk; i, s ). So xt,T (s ) ≥ 0.
l ( st ) < 0 then w ( s1 ) = 0. Then L ( st ) = 2 L̄ but then x l ( st ) > 0 since
If xt,T
t,T
t,T
t,T
l
t
2 L̄ > maxs∈S L(s). So xt,T (s ) ≥ 0.
c ( st ) = 0. Assume to the contrary that, x c ( st ) < 0.
We show by contradiction that xt,T
t,T
k , x l ≥ 0). Therefore,
Then pct,T (st ) = et (since pt,T ∈ arg max p∈∆et p · xt,T and xt,T
t,T
c
l
k
(st ) > 0.
(st ) = −et xt,T
(st ) + wt,T (st ) xt,T
rt,T (st ) xt,T
(43)
If rt,T (st ) = 0 then Kt,T (st ) = 2K̄ and
k
(st )
xt,T
= 2K̄ − ∑
i ∈I
Z k̄
0
kµt,T (dk, i; st )
l
> 2K̄ − K̄ > 2 L̄ > xt,T
( s t ).
But then, since pt,T ∈ arg max p p · xt,T , so rt,T (st ) = 1 − et and wt,T (st ) = 0, a contradiction
with the assumption that rt,T (st ) = 0. So we must have rt,T (st ) > 0. Therefore, since
k ( st ) ≥ x l ( st ). From (43), we have x k ( st ) > 0.
pt,T ∈ arg max p p · xt,T , it implies that xt,T
t,T
t,T
l ( st ) = L ( st ) − L ( s ) > 0. If w ( st ) > 0
If wt,T (st ) = 0 then Lt,T (st ) = 2 L̄ > 0 and xt,T
t
t,T
t,T
l
t
k
t
then since pt,T ∈ arg max p p · xt,T , it must be that xt,T (s ) ≥ x0,T (s ). We have just shown
k ( st ) ≥ x l ( st ). This leads to x k ( st ) = x l ( st ) > 0. In either case, we have
above that xt,T
t,T
t,T
t,T
l ( st ) > 0.
xt,T
k (st ) + K
t −1 ) > K
t
l
t
Therefore Kt,T (st ) = xt,T
t−1,T ( s
t−1 and Lt,T ( s ) = xt,T ( s ) + L ( st ) > L.
42
c ( st ) ≤ 0, which by the definition of x c implies:
As we show in (40) above xt,T
t,T
∑
Z k̄ i
0
ĉt,T (k, i; st ) + k̂ t,T (k, i; st ) − (1 − δ)k µt,T (dk; i, st ) − Yt,T (st ) ≤ 0.
Therefore
∑
Z k̄
0
i
ĉt,T (k, i; st )µt,T (dk; i, st ) ≤ (1 − δ) ∑
i
Z k̄
0
kµt,T (dk; i, st ) + Yt,T (st )
≤ (1 − δ)Kt,T (st ) + Yt,T (st ),
(44)
k ( st ) ≥ 0.
where the last inequality comes from xt,T
From the agents’ Euler equation, shown in Lemma 8,
pct−1,T u0 (ĉt−1,T ) ≥ Et−1 [((1 − δ)et + rt,T )u0 (ĉt,T )]
if k̂ t−1,T (k, i; st−1 ) < k̄. In this case
u0 (ĉt−1,T ) ≥
∑
∑ Pr(it |st−1, st , it−1 )((1 − δ)et + rt,T )u0 (ĉ1,T )
π s t −1 s t
st ∈S
it ∈I
≥ πst−1 st ∑ Pr(it |st−1 , st , it−1 )((1 − δ)et + rt,T )u0 (ĉt,T )
it
≥ πst−1 st ((1 − δ)et + rt,T )u0
∑ Pr(it |st−1, st , it−1 )ĉt,T
!
,
it
where the last inequality comes from Jensen’s inequality and the convexity of u0 .
Therefore
1
∑i ∈I πst−1 st ,it−1 it ĉt,T
πst−1 st ((1 − δ)et + rt,T ) σ ≤ t
.
ĉt−1,T
Integrating over µt−,T , and by (44),14 we obtain:
πst−1 st ((1 − δ)et + rt,T )
1
σ
≤
∑i
where χ is the set characteristic function.
Now
Z
k̄
∑
k̄χk̂
0
i ∈I
t−1,T ≥ k̄
R
Yt,T + (1 − δ)Kt,T
,
ĉt−1,T (k, i )χk0 <k̄ µt−1,T (dk, i )
(45)
t−1,T
µt−1,T (dk, i ) ≤ Kt .
Equivalently,
k̄ ∑
Z
i ∈I
14 We
use the inequality that m ≤
aj
bj
χk̂
t−1,T ≥ k̄
µt−1,T (dk, i ) ≤ Kt .
for all j implies m ≤
footnote 11.
43
R
a
R j,
bj
(46)
an integral version of the inequality in
We also have
∑
Z
ĉt,T (k, i )χk̂t,T (i,k)<k̄ µt,T (dk, i )
i
Z
=∑
ĉt,T (k, i )µt,T (dk, i ) − ∑
i
i
For k such that k̂t−1,T
Z
(k, i; st )
ĉt,T (k, i )χk̂t,T ≥k̄ µt,T (dk, i ).
(47)
= k̄ we have
pct−1,T (ĉt−1,T (k, i ) + k̄ − (1 − δ)k)
= rt−1,T k + wt−1,T l (st , i ) + Πt,T
≤ (1 − et−1 )(k̄ + l¯) + Ȳ,
where Ȳ = maxs∈S F (s, 2K̄, 2 L̄). Therefore for these values of k:
2
(1 − et−1 )(k̄ + l¯) + Ȳ
≤
k̄
ĉt−1,T (k, i ) ≤
e t −1
e t −1
since k̄ > l¯ + Ȳ.
So, (46), (47), and (48) yield
∑
Z
i
(48)
ĉt,T (k, i )χk̂t,T <k̄ µt,T (dk, i )
≥∑
Z
≥∑
Z
ĉt,T (k, i )µt,T (dk, i ) −
i
ĉt,T (k, i )µt,T (dk, i ) −
i
2
e t −1
2
e t −1
k̄ ∑
Z
i
χk̂t,T ≥k̄ µt,T (dk, i )
Kt,T .
Therefore, from (45), we get
πst−1 st ((1 − δ)et + rt,T )
1
σ
≤
Yt,T + (1 − δ)Kt,T
F (st−1 , Kt−1,T , Lt−1,T ) + (1 − δ)Kt−1,T − Kt,T −
2
et−1 Kt,T
(49)
From the firm’s problem, (Yt,T , Kt,T , Lt,T ) solves
max et Y − rt,T K − wt,T L
Y,K,L
s.t.
Y ≤ F (s1 , K, L)
and 0 ≤ K ≤ 2K̄, 0 ≤ L ≤ 2 L̄. Since et > 0, we have Yt = F (st , Kt,T , Lt,T ) and since we
established that Kt,T , Lt,T > 0, we have
et FK (st , Kt,T , Lt,T ) ≥ rt,T (st )
and
et FL (st , Kt,T , Lt,T ) ≥ wt,T (st ).
Notice that
FL (st , Kt,T , Lt,T ) ≤ FL (st , Kt,T , L) ≤ max FL (st , K, L).
0≤K ≤2K̄
By Assumption 3, max0≤K ≤2K̄ FL (st , K, L) < +∞.
44
So, since rt,T = 1 − pct,T − wt,T = 1 − et − wt,T ,
et FK (st , Kt,T , Lt,T ) ≥ rt,T ≥ 1 − et (1 + max FL (st , K, L))
0≤K ≤2K̄
(50)
Since, we chose et such that
et FK (st , Kt,T , Lt,T ) < 1 − et (1 + max FL (st , K, L))
0≤K ≤2K̄
for all Kt,T ≥
(1−δ)K t−1
.
2 1+ e 2
(1−δ)K t−1
,
2 1+ e 2
i.e., Property e1. in Lemma 7. Therefore (50) implies that Kt,T ≤
t −1
So (49), together with Kt−1 > K t−1 , yields
t −1
1
πst−1 st ((1 − δ)et + 1 − et (1 + max FL (st , K, L)))
σ
0≤K ≤2K̄
≤
Yt,T + (1 − δ)Kt,T
F (st , Kt,T , 2 L̄) + (1 − δ)Kt,T
≤
.
1
1
(
1
−
δ
)
K
(
1
−
δ
)
K
t −1
t −1
2
2
By the choice of et in Lemma 7 (Property e2.), because Kt,T satisfies:
FK (st , Kt,T , Lt,T ) ≥
1 − et (1 + max0≤K ≤2K̄ FL (st , K, L))
et
we have
1
σ
F (st , Kt,T , 2 L̄) + (1 − δ)Kt,T
πst−1 st ((1 − δ)et + 1 − et (1 + max FL (st , K, L)))
>
.
1
0≤K ≤2K̄
2 (1 − δ ) K t −1
This is a contradiction with the earlier inequality.
c
k
l
So xt,T
= 0. If xt,T
> 0 or xt,T
> 0 then pt,T · xt,T = max p · xt,T > 0. Therefore
k
l
xt,T = xt,T = 0.
l > 0. Therefore w
If wt,T = 0, then Lt,T = 2 L̄ and xt,T
t,T > 0. If rt,T = 0 then Kt,T = 2K̄
k > 0. Therefore r
and xt,T
t,T > 0.
Now we show that Kt,T > K t . Following the derivation of (49), we obtain
πst−1 st ((1 − δ) pct,T + rt,T )
1
σ
≤
Yt,T + (1 − δ)Kt,T
F (st−1 , Kt−1,T , Lt−1,T ) + (1 − δ)Kt−1,T − Kt,T −
2
et−1 Kt,T
(51)
Therefore, if Kt,T ≤ K t ,
Kt,T 1 +
2
e t −1
≤ (1 − δ)K t−1 < (1 − δ)Kt−1,T .
So, because pct,T ≥ et , (51) implies
πst−1 st ((1 − δ)et )
1
σ
<
F (st , Kt,T , 2 L̄) + (1 − δ)Kt,T
F ( s t −1 , K t −1 , L )
which contradicts the definition of K t in Lemma 7. Therefore Kt,T > K t .
45
Lemma 7. We choose the sequence {et , K t } recursively as follow:
1
K0 = ∑
2 i
Z k̄
0
kµ0 (dk, i; s0 ) < K0
and e0 > 0 such that.
e0 <
1
.
1 + max0≤ L≤2 L̄ FK (s0 , K0 , L) + max0≤K ≤2K̄ FL (s0 , K, L)
(52)
For t > 0, given et−1 and K t−1 , we choose et and K t as follow.
There exists et > 0 such that the following properties e1. and e2. are satisfied.
e1. et is sufficiently small such that
et <
1
min
s∈S ,0≤ L≤2 L̄
FK
s,
(1−δ)K t−1
,L
2 1+ e 2
!
(53)
+ 1 + max0≤K̃≤2K̄ FL (s, K̃, L)
t −1
and
e2. for all s, s− ∈ S , and for all K such that
FK (s, K, L) ≥
1 − et (1 + max0≤K̃ ≤2K̄ FL (s, K̃, L))
,
et
for some 0 ≤ L ≤ 2 L̄ , we have
1
σ
F (s, K, 2 L̄) + (1 − δ)K
.
πs− s ((1 − δ)et + 1 − et (1 + max FL (s, K̃, L)))
>
1
(
1
−
δ
)
K
0≤K̃ ≤2K̄
t
−
1
2
Given et−1 , et , and K t−1 , there exists K t <
(1 − δ ) K t −1
1+ e 2
(54)
such that for all s, s− ∈ S and K ≤ K t ,
t −1
1
( π s − s (1 − δ ) et ) σ >
F (s, K, 2 L̄) + (1 − δ)K
.
F ( s − , K t −1 , L )
Proof. The existence of K0 is immediate. By Assumption 3,
max FK (s0 , K, L) < +∞ and
0≤ L≤2 L̄
max FL (s, K̃, L) < +∞,
0≤K̃ ≤2K̄
so there exists e0 > 0 that satisfies (52). Now we construct et and K t recursively.
The right hand side of (53) is finite and is strictly positive. Let et > 0 denote this value.
Let
(
)
1
.
ẽt = min et ,
2 1 + max0≤K̃ ≤2K̄ FL (s, K̃, L))
By Assumption 3, there exists K̂t such that for all s, s− ∈ S :
1
1 σ 1
π s− s
(1 − δ)K t−1 > F (s, K, L̄) + (1 − δ)K,
2
2
46
(55)
for all K ≤ K̂t . Again, by Assumption 3, we can find 0 < et < ẽt such that
max FK (s, K̂t , L) ≤
0≤ L≤2 L̄
1 − et (1 + max0≤K̃ ≤2K̄ FL (s, K̃, L))
.
et
(56)
We show that et defined as such satisfies Properties e1. and e2.
Since et < et , (53) holds, i.e., et satisfies e1. Now we show that et satisfies e2. Indeed,
for all s, s− ∈ S , and for all K such that
FK (s, K, L) ≥
1 − et (1 + max0≤K̃ ≤2K̄ FL (s, K̃, L))
,
et
for some 0 ≤ L ≤ 2 L̄ , by (56), we have K ≤ K̂t . Because
et <
1
2 1 + max0≤K̃ ≤2K̄ FL (s, K̃, L))
we have
1
πs− s ((1 − δ)et + 1 − et (1 + max FL (s, K̃, L)))
0≤K̃ ≤2K̄
σ
>
1
π s− s
2
1
σ
So (55) yields (54).
By Assumption 3 (limK →0 F (s, K, 2 L̄) = 0), there exists K t such that
0 < Kt <
(1 − δ ) K t −1
1 + et2−1
(57)
and for all s, s− ∈ S , and 0 < K < K t ,
1
(πs− s (1 − δ)et ) σ F (s− , K t−1 , L) > F (s, K, 2 L̄) + (1 − δ)K.
(58)
Lemma 8. Consider the value and policy functions defined recursively by the Bellman equations,
(37). We have the following properties:
1. The value functions V̂t,T are continuous, strictly increasing, strictly concave.
2. The corresponding policy correspondence, ĉt,T , k̂ t,T are single-valued, i.e., are functions,
continuous, and k̂ t,T are weakly increasing, and the budget constraints, (38), hold with equality.
3. (Euler Equation) If k0 = k̂ t,T (k, i; st ) < k̄ then
h
i
0
t −1
t +1
0
0 0 t +1
u (ĉt,T (k, i; s )) ≥ β(i )E (1 − δ + rt+1,T (s ))u (ĉt+1,T (k , i ; s ))
with equality if k0 > 0.
Proof. These properties are standard. The single-valued property of the policy function
comes from the fact that u(.) is strictly concave. The monotonicity of k̂ t,T comes from a
standard-single crossing argument.
Indeed, for k1 < k2 , let
k01 = k̂ (k1 , i; st , p T , Π T ) and k02 = k̂(k2 , i; st , p T , Π T )
47
and
R1 = (1 − δ) pct,T (st ) + rt,T (st ) k1 + wt,T (st )l (st , it ) + Πt,T (Kt , Lt )
R2 = (1 − δ) pct,T (st ) + rt,T (st ) k2 + wt,T (st )l (st , it ) + Πt,T (Kt , Lt ).
We show that k02 ≥ k01 .
If k02 ≥ R1 then k02 ≥ R1 ≥ k01 we obtain the desired inequality.
Otherwise, from (37), we have
!
h
i
R1 − k01
0
t T
T
u
V̂
(
k
+
β
(
i
)
E
,
i;
s
,
p
,
Π
)
t
t+1,T 1
pct,T (st )
!
i
h
R1 − k02
t T
T
0
≥u
,
i;
s
,
p
,
Π
)
+
β
(
i
)
E
V̂
(
k
t
t+1,T 2
pct,T (st )
and
u
R2 − k02
pct,T (st )
≥u
!
h
i
+ β(i )Et V̂t+1,T (k01 , i; st , p T , Π T )
R2 − k01
pct,T (st )
!
i
h
+ β(i )Et V̂t+1,T (k02 , i; st , p T , Π T ) .
Adding up the two inequalities side by side and simplify, we obtain
!
!
!
!
R1 − k01
R2 − k01
R1 − k02
R2 − k02
u
−u
≥u
−u
pct,T (st )
pct,T (st )
pct,T (st )
pct,T (st )
which implies k01 ≤ k02 since u is concave.
Lemma 9. We show that the correspondence Ψe constructed in Lemma 5 is upper hemi-continuous,
and is non-empty, compact and convex valued.
Proof. In order to show that Ψe is upper hemi-continuous, we need to show that given
T
T
T
any sequence ( pn , µn , x n ) ∈ ∆eΣT × Ω̄Σ × K x that converges to some ( p, µ, x ) ∈ ∆Σ
×
T
e
Ω̄Σ × K x :
T
( pn , µn , x n ) → ( p, µ, x )
and
and
( p̃n , µ̃n , x̃ n ) → ( p̃, µ̃, x̃ )
( p̃n , µ̃n , x̃ n ) ∈ Ψe ( pn , µn , x n )
then we must have:
Indeed, since
( p̃, µ̃, x̃ ) ∈ Ψe ( p, µ, x ) .
( p̃n , µ̃n , x̃ n ) ∈ Ψe ( pn , µn , x n ) ,
there exists
n
n
Yt,T
(st ), Kt,T
(st ), Lnt,T (st )
48
t,st
that solves (36). Let
and Πn,T = Πnt,T (st )
n
n
n
n
n
Πnt,T = pc,n
t,T Yt,T − rt,T Kt,T − wt,T Lt,T ,
t,st
.
n ( k, i; st , p T,n , Π T,n ) denote the value functions that solves (37) given p T,n and
Let V̂t,T
Π T,n and k̂nt,T (k, i; st , p T,n , Π T,n ) denote the corresponding policy functions.
By choosing converging subsequence, we can assume that there exists
Yt,T (st ), Kt,T (st ), Lt,T (st ) t,st
such that
n
n
Yt,T
(st ), Kt,T
(st ), Lnt,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 p T .
Indeed, for any (Y, K, L) such that
Y ≤ F (st , Kt,T , Lt,T )
and
0 ≤ L ≤ 2 L̄ and 0 ≤ K ≤ 2K̄,
n ( st ), K n ( st ), Ln ( st ) solves (36), we have
since Yt,T
t,T
t,T
n,c n
n
n
n
n
n
n
pn,c
t,T Y − rt,T K − wt,T L ≤ pt,T Yt,T − rt,T Kt,T − wt,T Lt,T .
Taking n → ∞, we obtain
pct,T Y − rt,T K − wt,T L ≤ pct,T Yt,T − rt,T Kt,T − wt,T Lt,T .
Therefore Yt,T (st ), Kt,T (st ), Lt,T (st ) solves (36) given p T .
In addition, from the expression for Πnt,T (st ) and Πt,T (st ), we also have Πnt,T (st ) −→n→∞
Πt,T (st ). Lemma 10 then shows that
k̂nt,T (., i; st , pn,T ) −→n→∞ k̂ t,T (., i; st , p T )
uniformly over 0, k̄ .
From the definition of ψx :
Z c
n
x̃t,T;n
(st ) = ∑
ĉt,T;n (k, i; st ) + k̂ t,T;n (k; i, st ) − (1 − δ)k µnt,T (dk; i, st ) − Yt,T
(st )
i ∈I
k
x̃t,T;n
(st )
=
n
Kt,T
(st ) −
Z
kµnt,T (dk; it , st )
l
x̃t,T;n
(st ) = Lnt,T (st ) − L(st ).
As shown in Lemma 8,
ĉt,T;n (k, i; st ) + k̂ t,T;n (k; i, st ) − (1 − δ)k =
n (st )k + wn (st )l (s , i ) + Πn (K n , Ln )
rt,T
t t
t,T
t,T
t,T t,T
n,c t
pt,T
(s )
49
.
Therefore
c
x̃t,T;n
(st )
=
∑
Z ∑
Z
n
ĉt,T;n (k, i; st ) + k̂ t,T;n (k; i, st ) − (1 − δ)k µnt,T (dk; i, st ) − Yt,T
(st )
i ∈I
=
=
+
n (st )k + wn (st )l (s , i ) + Πn (K n , Ln )
rt,T
t t
t,T t,T
t,T
t,T
n
µnt,T (dk; i, st ) − Yt,T
(st )
t
pn,c
t,T ( s )
i ∈I
Z
n (st )
rt,T
kµnt,T (dk; i, st ) +
n,c t
pt,T (s ) i∈I
n , Ln )
Πnt,T (Kt,T
t,T
n
− Yt,T
( s t ).
t)
pn,c
(
s
t,T
∑
n (st )
rt,T
addition, as we show above,
Because µn → µ,
∑
Z
n ,Ln )
Πnt,T (Kt,T
t,T
n,c t
pt,T (s )
→
kµnt,T (dk; i, st )
n (st )
rt,T
t
pn,c
t,T ( s )
∑
Z
→
rt,T (st )
pct,T (st )
and
n (st )
rt,T
n,c t
pt,T
(s )
→
rt,T (st )
.
pct,T (st )
In
n ( s t ) → Y ( s t ).
and Yt,T
t,T
∑
Z
kµt,T (dk; i, st )
∑
Z
l (st , it )µt,T (dk; i, st ).
i ∈I
l (st , it )µnt,T (dk; i, st )
→
i ∈I
Therefore, for all t and
l (st , it )µnt,T (dk; i, st )
→
Πt,T (Kt,T ,Lt,T )
pct,T (st )
i ∈I
and
∑
Z
n,c t
pt,T
(s ) i∈I
c
Because pn → p, and pn,c
t,T , pt,T > et > 0,
c
x̃t,T;n
(st )
!
i ∈I
st ,
r (st )
→ t,T
pct,T (st ) i∑
∈I
we have:
Z
rt,T (st )
kµt,T (dk; i, s ) + c
pt,T (st ) i∑
∈I
t
Z
l (st , it )µt,T (dk; i, st )
Πt,T (Kt,T , Lt,T )
− Yt,T (st )
pct,T (st )
Z =∑
ĉt,T (k, i; st ) + k̂ t,T (k; i, st ) − (1 − δ)k µt,T (dk; i, st ) − Yt,T (st ).
+
i ∈I
c
c ( st ) . Therefore, for all t and st :
In addition, we also have x̃t,T;n
(st ) → x̃t,T
Z c
x̃t,T = ∑
ĉt,T (k, i; st ) + k̂ t,T (k; i, st ) − (1 − δ)k µt,T (dk; i, st ) − Yt,T (st )
i ∈I
Similarly, we can also show that, for all t and st :
k
x̃t,T
(st )
t
= Kt,T (s ) −
Z
kµt,T (dk; it , st )
l
x̃t,T
(st ) = Lt,T (st ) − L(st ).
Therefore
x̃ ∈ ψx ( p, µ, x ).
Following the same steps, it is also easy to show that p̃ ∈ ψ p ( p, µ, x ). Now we show
that µ̃ ∈ ψµ ( p, µ, x ).
50
0, k̄
−1
t
( A ), i t ; s .
k̂t,T;n
Pr(it+1 |it , st , st+1 )µt,T;n
Indeed, from the definition of ψµ , for every A ∈ B
µ̃t+1,T;n ( A, it+1 ; st+1 ) =
∑
(59)
it ∈I
We need to show that, for every A ∈ B
µ̃t+1,T ( A, it+1 ; s
t +1
)=
0, k̄
∑ Pr(it+1 |it , st , st+1 )µt,T
k̂t,T;n
−1
( A ), i t ; s
t
.
it ∈I
From the construction of the Kantorovich-Rubinshtein norm for the space of measures
in Bogachev
Section 8.3), to show the identity, we just need to show that for all
(2000,
ϕ ∈ Lip1 0, k̄ ,
Z k̄ Z k̄
t +1
t
ϕ(k )µ̃t+1,T (dk, it+1 ; s ) = ∑ Pr(it+1 |it , st , st+1 )
ϕ k̂ t,T (k, i; s ) µt,T (dk; it , st )
0
0
it ∈I
(60)
From (59), we have
Z k̄
0
ϕ(k )µ̃t+1,T;n (dk, it+1 ; s
t +1
)=
∑ Pr(it+1 |it , st , st+1 )
it ∈I
Z k̄
0
t
ϕ k̂ t,T;n (k, i; s ) µt,T;n (dk; it , st ).
Since µ̃n → µ̃,
lim
Z k̄
n→∞ 0
ϕ(k )µ̃t+1,T;n (dk, it+1 ; s
t +1
)=
Z k̄
0
ϕ(k)µ̃t+1,T (dk, it+1 ; st+1 ).
Therefore, to establish (60), we just need to show that:
Z k̄ Z k̄ ϕ k̂ t,T (k, i; st ) µt,T (dk; it , st ).
lim
ϕ k̂ t,T;n (k, i; st ) µt,T;n (dk; it , st ) =
n→∞ 0
0
Indeed,
Z k̄ Z k̄ ϕ k̂ t,T (k, i; st ) µt,T (dk; it , st )
ϕ k̂ t,T;n (k, i; st ) µt,T;n (dk; it , st ) −
0
0
Z
Z k̄ k̄ t
t
=
ϕ k̂ t,T;n (k, i; s ) µt,T;n (dk; it , s ) −
ϕ k̂ t,T (k, i; st ) µt,T;n (dk; it , st )
0
0
Z k̄ Z
k̄ +
ϕ k̂ t,T (k, i; st ) µt,T;n (dk; it , st ) −
ϕ k̂ t,T (k, i; st ) µt,T (dk; it , st 0
0
Z k̄ t
t ≤
ϕ k̂ t,T;n (k, i; s ) − ϕ k̂ t,T (k, i; s ) µt,T;n (dk; it , st )
0
Z k̄ Z k̄ +
ϕ k̂ t,T (k, i; st ) µt,T;n (dk; it , st ) −
ϕ k̂ t,T (k, i; st ) µt,T (dk; it , st .
0
0
(61)
(62)
We first show that
lim
Z k̄ n→∞ 0
t
t ϕ
k̂
(
k,
i;
s
)
−
ϕ
k̂
(
k,
i;
s
)
µt,T;n (dk; it , st ) = 0.
t,T;n
t,T
51
(63)
Indeed, because ϕ ∈ Lip1 0, k̄ ,
Z k̄ ϕ k̂ t,T;n (k, i; st ) − ϕ k̂ t,T (k, i; st ) µt,T;n (dk; it , st )
0
Z k̄ k̂ t,T;n (k, i; st ) − k̂ t,T (k, i; st ) µt,T;n (dk; it , st )
0
≤ sup k̂ t,T;n (k, i; st ) − k̂ t,T (k, i; st ) µt,T;n ( 0, k̄ ; it , st )
≤
0≤k≤k̄
We show in Lemma 10, that k̂ t,T;n → k̂ t,T uniformly, therefore
lim sup k̂ t,T;n (k, i; st ) − k̂ t,T (k, i; st ) = 0,
n →0
0≤k≤k̄
In addition, since µt,T;n ( 0, k̄ ; it , st ) ≤ ∑i∈I µt,T;n ( 0, k̄ ; i, st ) = 1. These two results
imply (63). t
Because ϕ k̂ t,T (k, i; s ) is continuous, we also have:
Z
Z k̄ k̄ t
t
t
t
lim ϕ k̂ t,T (k, i; s ) µt,T;n (dk; it , s ) −
ϕ k̂ t,T (k, i; s ) µt,T (dk; it , s = 0.
n→∞ 0
0
Combining this limit with (63), and (62), we arrive at (61). As argued above, this implies
µ̃ ∈ φµ ( p, µ, x ).
We have just established that ( p̃, µ̃, x̃ ) ∈ Ψe ( p, µ, x ), i.e., Ψe is upper hemi-continuous.
It is standard to show that Ψe is compact and convex valued. The proof is facilitated by
the fact that if
( p̃1 , µ̃1 , x̃1 ) ∈ Ψe ( p, µ, x )
and
( p̃2 , µ̃1 , x̃2 ) ∈ Ψe ( p, µ, x )
then Π1t,T = Π2t,T for all t and st . Therefore by Lemma 8, k̂1t,T ≡ k̂2t,T for all t and st . So
µ̃1 ≡ µ̃2 , i.e., ψµ ( p, µ, x ) is single-valued.
n solves
Lemma 10. Assume that pn,T −→n→∞ p T and Πnt,T −→n→∞ Πt,T . In addition, V̂t,T
(37), given pn,T and Πn,T with the corresponding policy function k̂nt,T and V̂t,T solves (37) given
p T and Πt,T with the corresponding k̂ t,T . Then, for all st ∈ S t and i ∈ I and k ∈ 0, k̄ , we have
n
V̂t,T
(., i; st , pn,T , Πn,T ) −→n→∞ V̂t,T (., i; st , p T , Π T )
pointwise, and
k̂nt,T (., i; st , pn,T , Πn,T ) −→n→∞ k̂ t,T (., i; st , p T , Π T )
uniformly over 0, k̄ .
Proof. We show the results stated in the lemma by induction backward from t = T + 1.
1. At t = T, the result is obvious since
!
n ( s T ) k + w n ( s T ) l ( s , i ) + Π n + (1 − δ ) k
r
T
T,T
T,T
n
V̂T,T
(k, i; s T , pn,T , Πn,T ) = u T,T
T)
pn,c
(
s
T,T
52
and
V̂T,T (k, i; s T , p T , Π T ) = u
r T,T (s T )k + wT,T (s T )l (s T , i ) + Π T,T + (1 − δ)k
pcT,T (s T )
!
and k̂nT,T ≡ 0 and k̂ T,T ≡ 0.
2. Assume that the results in the current lemma hold for t + 1 ≤ T, we show that they
also hold for t.
Indeed, given st ∈ S , i ∈ I and k ≥ 0, we first show that
n
lim inf V̂t,T
(k, i; st , pn,T , Πn,T ) ≥ V̂t,T (k, i; st , p T , Π T ).
n→∞
This is immediate if the right hand side is −∞, which happens if and only if
Πt,T (st ) = wt,T (st ) = k = 0.
Now if the right hand side is finite, for any ν > 0, there exists c ≥ 0 and k0 ∈ 0, k̄ such
that
pct,T (st ) c + k0 − (1 − δ)k < rt,T (st )k + wt,T (st )l (st , it ) + Πt,T (st ),
and
h
i
V̂t,T (k, i; st , p T , Π T ) ≤ u(c) + β(i )Et V̂tn+1,T (k0 , i; st , pn,T , Πn,T ) + ν
Because
pnt,T , Πnt,T → ( pt,T , Πt,T ) ,
there exists N such that for all n ≥ N
n
t
0
n
(st )l (st , it ) + Πnt,T (st ).
(st )k + wt,T
pn,c
(
s
)
c
+
k
−
(
1
−
δ
)
k
≤ rt,T
t,T
Therefore,
i
h
n
(k, i; st , p T , Πn,T ) ≥ u(c) + β(i )Et V̂tn+1,T (k0 , i; st , pn,T , Πn,T )
V̂t,T
and since V̂tn+1,t (k0 , i ) → V̂t+1,T (k0 , i ) by the the induction assumption,
h
i
n
lim inf V̂t,T
(k, i; st , pn,T , Πn,T ) ≥ u(c) + β(i )Et V̂t+1,T (k0 , i; st , pn,T , Πn,T )
n→∞
≥ V̂t,T (k, i; st , p T , Π T ) − ν.
Therefore,
n
lim inf V̂t,T
(k, i; st , pn,T ) ≥ V̂t,T (k, i; st , p T ).
n→∞
(64)
We show by contradiction that
n
lim sup V̂t,T
(k, i; st , pn,T ) ≤ V̂t,T (k, i; st , p T ).
n→∞
(65)
Case 1: V̂t,T (k, i; st , p T ) > −∞. Assume to the contrary that there exists ν > 0 and a
subsequence nm → ∞ such that
nm
V̂t,T
(k, i; st , pn,T ) > V̂t,T (k, i; st , p T ) + ν.
By the definition of V nm , there exists cnm ≥ 0, and k0nm ∈ 0, k̄ , such that
nm t
nm t
pnt,Tm ,c (st ) cnm + k0nm − (1 − δ)k ≤ rt,T
(s )k + wt,T
(s )l (st , it ) + Πnt,Tm ,
53
(66)
(67)
and
h
i
nm
V̂t,T
(k, i; st , p T ) = u(cnm ) + β(i )Et V̂tn+m1,T (k0nm , i; st , pnm ,T ) .
0
By choosing
subsequences,
we can assume that cnm → c∗ and k nm → k∗ for some c∗ ≥ 0
and k∗ ∈ 0, k̄ . From (67), and because
pnt,T , Πnt,T → ( pt,T , Πt,T ) ,
we have
pct,T (st ) (c∗ + k∗ − (1 − δ)k ) ≤ rt,T (st )k + wt,T (st )l (st , it ) + Πt,T .
Since V̂tn+m1,T → V̂t+1,T pointwise and V̂t+1,T is continuous and increasing in k, we obtain15
lim sup V̂tn+m1,T (k0nm , i; st , pnm ,T ) ≤ V̂t+1,T (k∗ , i; st , p T ).
m→∞
Consequently,
h
i
nm
lim sup V̂t,T
(k, i; st , p T ) ≤ u(c∗ ) + β(i )Et V̂t+1,T (k∗ , i; st , p T ) ≤ V̂t,T k, i; st , p T .
m→∞
This contradicts (66). So we obtain (65) by contradiction.
Case 2: V̂t,T (k, i; st , p T ) = −∞. Then
Πt,T (st ) = wt,T (st ) = k = 0.
n , we have
From the budget’s constraint for V̂t,T
n
V̂t,T
(0, i; st , p T , Πn,T ) ≤ u
n (st )l (s , i ) + Πn (st )
wt,T
t t
t,T
n,c t
pt,T
(s )
!
h
i
+ β(i )Et V̂tn+1,T (k̄, i; st , pn,T , Πn,T ) .
Now,
n
lim wt,T
(st ) = wt,T (st ) = 0
n→∞
lim Πnt,T (st ) = Πt,T (st ) = 0,
n→∞
t
and pn,c
t,T ( s ) > et > 0, and u (0) = − ∞. Therefore,
lim u
n→∞
n (st )l (s , i ) + Πn (st )
wt,T
t t
t,T
n,c t
pt,T
(s )
!
= −∞.
In addition, V̂tn+1,T (k̄, i; st , pn,T , Πn,T ) is finite. So
n
(0, i; st , p T , Πn,T ) = −∞ = V̂t,T (k, i; st , p T ).
lim sup V̂t,T
n→∞
We have shown that, in either case, we obtain (65). Combining this inequality, with
15 For
any 0 ≤ k∗ < k̃,
lim sup V̂tn+m1,T (k0nm , i; st , pnm ,T ) ≤ lim V̂tn+m1,T (k̃, i; st , pnm ,T )
m→∞
m→∞
= V̂t+1,T (k̃, i; st , p T ).
Taking the limit k̃ to k∗ , we obtain the desired inequality.
54
(64), we finally get the desired limit:
n
lim V̂t,T
(k, i; st , pn,T , Πn,T ) = V̂t,T (k, i; st , p T , Π T ).
n→∞
Given k ∈ 0, k̄ , we also show by contradiction that
lim k̂n (k, i; st ,
n→∞ t,T
pn,T , Πn,T ) = k̂ t,T (k, i; st , p T , Π T ).
Assume to the contrary. Then, there exists a subsequence {nm } such that
0
k nm = k̂nt,T (k, i; st , pn,T , Πn,T ) → k∗
for some k∗ ∈ 0, k̄ and k∗ 6= k̂ t,T (k, i; st , p T , Π T ). Let cnm be defined such that the budget
constraint, (67) holds with equality. Taking, further subsequence if necessary, we can
assume that cnm → c∗ for some c∗ . As shown above (c∗ , k∗ ) must satisfy the budget
constraint at p T , Π T , and
h
i
nm
V̂t,T k, i; st , p T , Π T = lim V̂t,T
(k, i; st , p T , Πn,T ) = u(c∗ ) + β(i )Et V̂t+1,T (k∗ , i; st , p T , Π T ) .
m→∞
Therefore, k∗ = k̂ t,T (k, i; st , p T , Π T ) (by Lemma 8 the maximizer is unique). This is a
contradiction.
So we have established the pointwise convergence of k̂nt,T to k̂ t,T . Because k̂ t,T and k̂nt,T
are increasing and continuous (by Lemma 8), the convergence is uniform.
Lemma 11. Θ̄, ḡ defined in Theorem 2 satisfy:
1. Θ̄ is sequentially compact.
2. ḡ is a closed-valued correspondence.
Proof. Proof of Part 1: We endow the space of increasing function with pointwise convergence topology and we endow the space of V function with
the sup
norm topology.
M denote the space of monotone functions from 0, k̄ to 0, k̄ . We endow M with
the topology of pointwise-convergence. Then, by Helly’s selection theorem, M is sequentially 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 V̄. We endow L with the
topology of convergence in sup norm. Then, by Ascoli-Arzela theorem, L is sequentially
compact.
Proof of Part 2: We need to show that g(s, µ) is closed for all s ∈ S and µ ∈ Ω̄. That is,
∞
n
n
for any sequence θ , θs0 s0 ∈S
∈ g(s, µ) such that
n =0
∞
θ n , (θsn0 )s0 ∈S n=0 −→n→∞ θ, (θs0 )s0 ∈S
then θ, (θs0 )s0 ∈S ∈ g(s, µ).
By the definition of convergence (topology) in different spaces, we have k̂n → k̂ (pointwise convergence) and V̂ n → V̂ and V̂s+0 n → V̂ (convergence in sup norm).
16 See
Exercise 7.13 in Rudin (1976) for an elementary proof.
55
∞
First, since θ n , θsn0 s0 ∈S
n =0
∈ g(s, µ), we have
V̂ n (k, i ) ≥ u (1 − δ + r n )k + wn l (s, i ) − k0 + β(i ) ∑ πss0 ,ii0 V̂s+0 n (k0 , i0 )
i0 ,s0
for each k0 ∈ 0, k̄ . Taking the limit n → ∞, we have
V̂ (k, i ) ≥ u (1 − δ + r )k + wl (s, i ) − k0 + β(i ) ∑ πss0 ,ii0 V̂s+0 (k0 , i0 )
i0 ,s0
for all
k0
∈ [0, k]. Therefore
V̂ (k, i ) ≥ max u (1 − δ + r )k + wl (s, i ) − k0 + β(i ) ∑ πss0 ,ii0 V̂s+0 (k0 , i0 ).
k0 ∈[0,k̄ ]
i0 ,s0
∞
Now, since θ n , θsn0 s0 ∈S
∈ g(s, µ):
n =0
V̂ (k, i ) = u (1 − δ + r )k + w l (s, i ) − k̂ (k, i ) + β(i ) ∑ πss0 ,ii0 V̂s+0 n (k̂n (k, i ), i0 ).
n
n
n
n
i0 ,s0
We show that
lim V̂s+0 n (k̂n (k, i ), i0 ) = V̂s+0 k̂ (k, i ), i0 .
n→∞
Indeed
+n n
+n n
0 0
+
0 0
+n
V̂s0 (k̂ (k, i ), i ) − V̂s0 k̂(k, i ), i ≤ V̂s0 (k̂ (k, i ), i ) − V̂s0 (k̂ (k, i ), i )
+ V̂s+0 n (k̂(k, i ), i0 ) − V̂s+0 k̂(k, i ), i0 .
The first term goes to zero because V̂s+0 n is Lipschitz continuous and k̂n converges pointwise to k̂ and the second term goes to 0 because of the pointwise convergence of V̂s+0 to
V̂s+0 .
Therefore
V̂ (k, i ) = u (1 − δ + r )k + wl (s, i ) − k̂ (k, i ) + β(i ) ∑ πss0 ,ii0 V̂s+0 (k̂ (k, i ), i0 ).
i0 ,s0
So
V̂ (k, i ) = max u (1 − δ + r )k + wl (s, i ) − k0 + β(i ) ∑ πss0 ,ii0 V̂s+0 (k0 , i0 ),
k0 ∈[0,k̄ ]
i0 ,s0
and k̂ (k, i ) is a maximizer.
Now we find bounds for the endogenous variables.
Lemma 12. There exist 0 < K < K0 , 0 < r < r̄ and 0 < w < w̄, and V < V̄ and lV > 0,
such that in competitive equilibrium in the finite horizon economy, starting with an initial wealth
distribution µ0 (k, i ) and
K0 =
∑
i ∈I
we have, for all t ≤ T and
st
∈
St
Z k̄
0
:
56
µ0 (dk, i )
1.
2.
3.
4.
Kt,T (st ) ≥ K
rt,T (st ) ∈ [r, r̄ ] and wt,T (st ) ∈ [w, w̄]
V̂t,T (k, i; st ) ∈ [V, V̄ ]
0 ( k, i; st ) ≤ l .17
0 ≤ V̂t,T
V
n
K0 , L2
Proof. By Assumption 6, there exists K < min
1.There exists γ > 0, such that, for all K ≤ K,
o
such that:
F (s0 , K, L(s0 ))
< γ,
F (s, K, L(s))
for all s, s0 ∈ S .
2. For all K ≤ K,
FK (s, K, L(s)) > max



1,


2(2− δ ) σ
γ α



β mins,s0 πss0 

(68)
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 ĉt,T , k̂ t,T or rt,T , wt,T . In a
competitive equilibrium, Lt,T (st ) = L(st ) = ∑i∈I m(i, st )l (i, st ), so we write Lt instead of
Lt,T .
From the first order condition, if k̂ t,T (k, i ) < k̄ then
h
i
0
0
u (ĉt,T (i, k )) ≥ β(i )Et (1 − δ + FK (st+1 , Kt+1,T , Lt+1 ))u (ĉt+1,T (i, k̂ t,T (i, k ))) .
Therefore, since Kt+1,T < K, FK (st+1 , Kt+1,T , Lt+1 ) > FK (st+1 , K, Lt+1 ) and the last inequality implies:
πs s ,ii
0
u (ĉt,T (i, k)) ≥ min(1 − δ + FK (K, L(s), s)) β(i )πst st+1 ∑ t t+1 t+1 u0 ĉt+1,T (k̂ t,T (i, k ), it+1 )
π s t s t +1
s∈S
i t +1
!
π
s s ,ii
≥ min(1 − δ + FK (K, L(s), s)) βπst st+1 u0 ∑ t t+1 t+1 ĉt+1,T (k̂ t,T (k, i ), it+1 )
π s t s t +1
s∈S
i
t +1
where the last inequality comes from the fact that u0 (c) = c−σ is strictly convex.
Consequently,
1
min(1 − δ + FK (s, K, L(s))) βπst st+1
σ
s∈S
≤
∑ i t +1
πst st+1 ,iit+1
πst st+1 ĉt+1,T ( k̂ t,T ( k, i ), it+1 )
ĉt,T (k, it )
.
17 V̂ might not be differentiable everywhere because of the borrowing constraint, k 0 ≥ 0, in this case
we use the concept of generalized derivative and the associated Envelope Theorems in Milgrom and Segal
(2002).
57
Therefore18
1
min(1 − δ + FK (K, L(s), s)) βπst st+1
σ
s∈S
≤
∑it ,it+1
πst st+1 ,it it+1
π s t s t +1
R
∑it
k̂ t,T <k̄ ĉt+1,T ( k̂ t+1,T ( k, it+1 ), it+1 ) µt,T ( dk, it )
R
k̂ t,T <k̄ ĉt,T ( k, it ) µt,T ( dk, it )
(69)
Now, we show that this would lead to a contradiction.
Indeed, for k such that k̂t,T (k, i; st ) = k̄ we have
ĉt,T + k̄ − (1 − δ)k
= rt,T k + wt,T l (st , i )
≤ FK (st , Kt,T , Lt )k̄ + FL (st , Kt,T , Lt )l.¯
or
¯
ĉt,T ≤ (−δ + FK (st , Kt,T , Lt ))k̄ + FL (st , Kt,T , Lt )l.
In addition,
K > Kt+1,T ≥
∑
Z
i
Therefore
Kt+t,T
>
k̄
∑
k̂ t,T =k̄
k̂ t,T (k, i )µt,T (dk, i )
Z
i
(70)
k̂ t,T =k̄
µt,T (dk, i ).
Combining this inequality with (70), we obtain
∑
i ∈I
Z
k̂ t,T =k̄
ĉt,T (k, i )µt,T (dk, i )
< Kt+1,T (−δ + FK (st , Kt,T , Lt,T )) + Kt+1,T FL (st , Kt,T , Lt,T )
l¯
k̄
< Kt+1,T (−δ + FK (st , Kt,T , Lt,T )) + Kt+1,T FL (st , Kt,T , Lt,T )
1
< Kt,T FK (st , Kt,T , Lt,T ) − δKt+1,T + Lt,T FL (st , Kt,T , Lt,T ))
(71)
2
where the second inequality comes from (12), which implies k̄ > l¯ and the last inequality
comes from Kt+1,T < K < Kt,T and K < L2
18 We
use the result that
aj
bj
≤ c and a j , b j > 0 then
R
a
R j
bj
58
< c.
Therefore,
∑
Z
k̂ t,T <k̄
i ∈I
=∑
Z
i
ĉt,T (k, i )µt,T (dk, i )
ĉt,T (k, i )µt,T (dk, i ) − ∑
i
Z
k̂ t,T =k̄
= Yt,T + (1 − δ)Kt,T − Kt+1,T − ∑
i
ĉt,T (k, i )µt,T (dk, i )
Z
k0 =k̄
ĉt,T (k, i )µt,T (dk, i )
= Kt,T FK (st , Kt,T , Lt,T ) + Lt,T FL (st , Kt,T , Lt,T ) + (1 − δ)Kt,T − Kt+1,T
−∑
Z
k0 =k̄
i
ĉt,T (k, i )µt,T (dk, i ).
Replacing the last item with (71), we have
∑
i ∈I
Z
k̂ t,T <k̄
ĉt,T (k, i )µt,T (dk, i ) >
1
Lt,T FK (st , Kt,T , Lt,T ) + (1 − δ) (Kt,T − Kt+1,T )
2
> Lt,T FL (st , Kt,T , Lt,T ).
Assumption 6 implies that
Lt,T FL (st , Kt,T , Lt,T ) > αF (st , Kt,T , Lt,T ) > αF (st , Kt+1,T , Lt,T ).
In addition
πst st+1 ,it it+1 Z
ĉt+1,T (k̂ t,T (k, it ), it+1 )µt,T (dk, it )
∑ πs s
k̂ t,T <k̄
t t +1
i
t
=
∑
Z
∑
Z k̄
it+1 k<k̄
<
i t +1 0
ĉt+1,T (k, it+1 )µt+1,T (dk, it+1 )
ĉt+1,T (it+1 , k)µt+1,T (dk, it+1 )
= Yt+1,T + (1 − δ)Kt+1,T − Kt+2,T
< F (st+1 , Kt+1,T , Lt+1 ) + (1 − δ)Kt+1,T .
Therefore (69) becomes
n
o
1
σ
min(1 − δ + FK (s, K, L(s))) βπst st+1
s
F (st+1 , Kt+1,T , Lt+1 ) + (1 − δ)Kt+1,T
<
.
α
F
(
s
,
K
,
L
)
t
t,T
t,T
2
2F (st+1 , Kt+1,T , Lt+1 )
Kt+1,T
<
1 + (1 − δ )
≡ E.
αF (st , Kt+1,T , Lt )
F (st+1 , Kt+1,T , Lt+1 )
Because
Kt+1,T
1
1
<
<
< 1,
F (st+1 , Kt+1,T , Lt+1 )
FK (st+1 , Kt+1 , Lt+1 )
FK (st+1 , K, Lt+1 )
59
and by (68), we have
E<
F (st+1 , Kt+1,T , Lt+1 ) 2(2 − δ)
2(2 − δ )
<γ
.
F (st , Kt+1,T , Lt,T )
α
α
However, this contradicts the definition of K, which satisfies (68). We obtain the desired
contradiction.
So, we have shown by contradiction that, Kt,T ≥ K for all t and st .
Now, for each t and st , we have:
FK (st , K̄, L(st )) ≤ rt,T = FK (st , Kt,T , Lt ) ≤ FK (st , K, L(st ))
Therefore rt,T ∈ [r, r̄ ], where:
0 < r = min FK (s, K̄, L(s)) and r̄ = max FK (s, K, L(s)).
s∈S
s∈S
Similarly, there exist 0 < w < w̄, such that wt,T ∈ [w, w̄] for all t and st .
From the (34), for all k ∈ 0, k̄ , i ∈ I , and t ≤ T and st ∈ S t , we have:
1
1
¯ ≥ V̂t,T (k, i ) ≥
u
(
1
−
δ
+
r̄
)
k̄
+
w̄
l
u (wl ) .
T
−
t
1 − ( β ) T −t
1 − ( β̄)
Let
V̄ =
1
1
u (1 − δ + r̄ )k̄ + w̄l¯ and V = inf
u (wl ) .
T
−
t
t,T:0≤t≤ T 1 − ( β ) T −t
t,T:0≤t≤ T 1 − ( β̄ )
sup
Then
V ≤ V̂t,T (k, i; st ) ≤ V̄
for all k ∈ 0, k̄ , i ∈ I , and t ≤ T and st ∈ S t .
Now
V̂t,T (k, i ) = u (ĉt,T (k, i )) + β(i )Et V̂t+1,T (k0 , i ) .
Therefore
u (ĉt,T (k, i )) > V − β̄V̄.
Since limc→0 u(c) = −∞, there exists c > 0 such that
ĉt,T (k, i ) > c
for all k ∈ 0, k̄ , i ∈ I , and t ≤ T and st ∈ S t .
From the envelope condition
0
V̂t,T
(k, i ) = 1 − δ + rt,T (st ) u0 (ĉt,T (k, i )),
Therefore
0
V̂t,T
(k, i ) ≤ (1 − δ + r̄ ) u0 (c) ≡ lV
for all k ∈ 0, k̄ , i ∈ I , and t ≤ T and
st
∈
(72)
S t.
Proof of Lemma 2. Consider sequences of allocation and prices generated
h by a generalized recursive equilibrium, starting from s0 ∈ S and a distribution k0 h∈H such that
µ0 (k, i ) as defined in (16) belongs to Ω̄. That is, sequences of distributions µt (st ) t,st
n
o
and, policy functions and value functions ĉt (., .; st ), k̂ t (., .; st ), V̂t (., .; st ) t , and prices
t,s
60
such that for each t, st : ĉt , k̂ t , V̂t , rt , wt ∈ Q(st , µt ), and for each st+1 ∈
S , st+1 , µt st , st+1 s ∈S ∈ T (st , µt ) , Conditions 1-5 in Definition (4) are satisfied. For
t +1
convenience, we restate Conditions 1-5 below.
1. For each st+1 ∈ S ,
ĉt+1 ., .; st , st+1 , k̂ t+1 ., .; st , st+1 , V̂t+1 ., .; st , st+1 , rt+1 st , st+1 , rt+1 st , st+1
∈ P s t +1 , µ t +1 s t , s t +1 .
r t ( s t ), w t ( s t )
t,st
2. The following identity holds:
∑
Z
ĉt (i, k; st )µt (dk, i; st ) +
i ∈I
∑
Z
k̂ t (i, k; st )µt (dk, i; st )
i ∈I
t
= F (st , Kt (s ), L(st )) + (1 − δ)Kt (st )
where
t
Kt ( s ) =
∑
Z k̄
0
kµt (dk, i; st ) and L(st ) =
i ∈I
t
( s ), L ( s
∑ m s t ( i ) l ( s t , i ).
i ∈I
3. rt (st ) = FK st , Kt
)
>
0 and wt (st ) = FL st , Kt (st ), L(st ) > 0.
t
4. For each i ∈ I and k ∈ 0, k̄ , V̂t (., .; st ), V̂ ., .; st , st+1 , st+1 ∈ S satisfy the Bellman equation:
V̂t (i, k; st ) = max u(c) + β(i ) ∑ πss0 ,ii0 V̂t+1 i0 , k0 ; st , s0
(73)
c,k0
i0 ,s0
s.t. c ≥ 0 and 0 ≤ k0 ≤ k̄ and
c + k0 ≤ (1 − δ + rt (st ))k + wt (st )l (st , i ).
In addition, ĉt (i, k; st ), k̂ t (i, k; st ) solves (73).
5. For each st+1 ∈ S , and i0 ∈ I and A ∈ B 0, k̄ :
πss0 ,ii0 µt (k̂ t (i, .; st ))−1 ( A), i; st .
µt+1 i0 , A; st , st+1 = ∑
πss0
i ∈I
Let the allocation cth (st , ît ), k ht (st , ît ) h∈H̄ be determined recursively by:
(74)
cth (st , ît ) = ĉt (k ht , ith ; st ) and k ht+1 (st , ît ) = k̂ t (k ht , ith ; st ),
We show by induction that µt (st ) corresponds to the distribution (16) from k ht (st , ît ) h∈H̄ ,
i.e.
µt ( A × I; st ) = φ h ∈ H : k ht (st , ît ), ith ∈ A × I
(75)
for each A × I ∈ B 0, k̄ × B(I).
The identity (75) holds at t = 0 by definition. Now, assume that the identity holds at
t, we show that it holds at t + 1. Indeed,
φ h ∈ H̄ : k ht+1 (st+1 , ît+1 ), ith+1 ∈ A × I
h t t
h t
h
= φ h ∈ H̄ : k̂ t (k t (s , î ), it ; s ), it+1 ∈ A × I .
61
By the “conditional no aggregate uncertainty” condition, the last expression can be written as:
πst st+1 ,ii0 h t t
t
h
∑ ∑ πs s φ h ∈ H : k̂t (kt (s , î ), i; s ) ∈ A, it = i
t t +1
i ∈I i0 ∈ I ⊂I
πst st+1 ,ii0 =∑ ∑
φ h ∈ H : k ht (st , i h,t ) ∈ (k̂ t (i, .; st ))−1 ( A), ith = i
π s t s t +1
i ∈I i0 ∈ I ⊂I
πst st+1 ,ii0 =∑ ∑
µt (k̂ t (i, .; st ))−1 ( A), i; st ,
π s t s t +1
i ∈I i0 ∈ I ⊂I
where the last equality comes from the assumption that (75) holds at t. By (74), the last
expression is equal to
t +1
0
t
A
×
I;
s
.
µ
A,
i
;
s
,
s
=
µ
t +1
t +1
∑ t +1
i0 ∈ I ⊂I
Therefore, we obtain (75) at t + 1.
So by induction (75) holds for all t and st . Consequently,
Z
H̄
k ht (st , ît )φ(dh)
H̄
l
h
(st , ith )φ(dh)
∑
=
∑
i ∈I
Z k̄
0
H̄
and
Z
H̄
kµt (dk, i; st ) = Kt .
l (s, i )µt (dk, i; st ) =
cth (st , ît )φ(dh)
∑ mst (i)l (s, i) = L(st ),
i ∈I
and
Z
0
i ∈I
and
Z
=
Z k̄
=
∑
Z k̄
i ∈I
k ht+1 (st , i h,t )φ(dh)
0
∑
=
ĉt (i, k; st )µt (dk, i; st )
Z
i ∈I
( s t ), w
k̂ t (i, k; st )µt (dk, i; st ).
, let cth (st , ît ), k ht (st , ît ) t,st denote
the allocation generated by the policy functions ĉ, k̂ and let c̃th (st , ît ), k̃ ht (st , ît ) t,st denote
a sequence that satisfies (14), and (15), we show that
"
#
Now given the sequences of prices rt
t (s
t)
∞
∑ Πtt0−=10 β(ith0 )u(cth (st , ît ))
V̂0 (k0h , i0h ; s0 ) = E0
(76)
t =0
"
≥ E0
∞
∑ Πtt0−=10 β(ith0 )u(c̃th (st , ît ))
#
.
(77)
t =0
From the Bellman equation (73), we have
"
V̂0 (k0h , i0h ; s0 ) = E0
T
∑ Πtt0−=10 β(ith0 )u(cth ) + ΠtT0 =0 β(ith0 )V̂T+1 (kht , ith ; st )
t =0
62
#
Now, the second term in the right hand side is bounded (in absolute value) by
β̄ T +1 max {[V, V̄ ]} −→ T →∞ 0.
Therefore taking T → ∞, we obtain (76).
Similarly, from the Bellman equation (73), we have
"
V̂0 (k0h , i0h ; s0 ) ≥ E0
T
∑ Πtt0−=10 β(ith0 )u(c̃th ) + ΠtT0 =0 β(ith0 )V̂T+1 (k̃ht , ith ; st )
#
.
t =0
The second term in the right hand side is bounded below by
min{ β̄ T +1 V, β T +1 V } −→ 0.
Therefore taking T → ∞, we obtain (77).
C
Relation to Miao (2006)
Consider the economy with a continuum of agents in Section 3. Let P̂ denote the set of
probability measures µ over [0, ∞) × I such that
∑
Z ∞
0
i
kµ(dk, i ) ≤ K̄,
where K̄ is defined in (10) and
S
P̂ ∞ = ×∞
t=0 P̂ .
The existence proof in Miao (2006) relies on the fixed point of the following operator:
t
T : C ([0, ∞) , I , S , P ∞ ) → C ([0, ∞) , I , S , P ∞ )
define for each
µ̃ =
µt (st )
t≥0,st ∈S t
∈ P∞
as
u (1 + r ( s0 , µ0 ) − δ ) k + w ( s0 , µ0 ) l ( i ) − k 0
k0 ∈Γ(k,i,s,µ0 )
n
o t +1
+ β(i ) ∑ πs0 s1 ;ii0 V k, i; s1 , µt+1 (s )
TV (k, i; s0 , µ̃) =
max
t ≥0
s1 ∈S ,i0 ∈I
where
(78)
Γ (k, i, s, µ0 ) = k0 : 0 ≤ k0 ≤ (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 Blackwell Theorem). Therefore, T admits a unique fixed point V̂ and the corresponding policy
function is
k̂ (k, i; s0 , µ̃)
We define
Λµ̃ = µ̃˜
63
where µ̃˜ 0 = µ̃0 and
µ̃˜ t+1 (st+1 ) A, i0 =
∑
i ∈I
πst st+1 ,ii0 −1 A, i; st , {µτ (sτ )}τ ≥t .
µ̃t k̂
π s t s t +1
The mapping Λ is continuous in P ∞ . Therefore by the Brouwer-Schauder-Tychonoff
Fixed Point Theorem, Λ admits a fixed point, which corresponds to a sequentially competitive equilibrium.
However this proof does not directly apply to Krusell and Smith (1998)’s model in
which 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 T
is 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 P̄∗∞ where P∗ = P̄ \D(0).
However, the following proposition shows that Λ does not preserve P̄∗∞ , i.e., there
exists µ̃ ∈ P̄∗∞ such that, Λµ̃ ∈
/ P̄∗∞ . The intuition is that if aggregate capital in µ̃ is very
high, the implied marginal rate of return on capital (interest rate) is very low. Together
with a sufficiently low discount factor, the agents will not want to save, leading to zero
aggregate capital in µ̃˜ .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
¯ L (s, K̄, L(s)) > βu0 lFL (s0 , K̄, L(s0 ))
(79)
u0 lF
for all s, s0 ∈ S . Then there exists K ∗ such that, for all K ≤ K ∗ , and
µ̃ = D (K ) , {D (K̄ )}t>0,st ∈S t
we have
Λµ̃ = D(K ), {D(0)}t>0,st ∈S t
∈
/ P̄∗∞ .
Proof. Because limc↓0 u0 (c) = +∞ and limK ↓0 F (s0 , K, L(s0 )) − δK = 0, there exists K ∗ such
that for all K ≤ K ∗ , we have
u0 ( F (s0 , K, L(s0 )) − δK ) ≥ βu0 lFL (s0 , K̄, L(s0 ))
(80)
for all K ≤ K ∗ . Let µ̃ be defined above. Using the agents’ Euler equation, we have, at
k = K, the solution to (78) involves
k t = 0 for all t > 0.
(81)
Indeed, we just have to verify that that
u0 (l (it ) FL (st , K̄, L(st ))) ≥ βEt (1 − δ + FK (st+1 , K̄, L(st+1 ))) u0 (l (it+1 ) FL (st+1 , K̄, L(st+1 )))
(82)
19 The
discount factor has to be sufficiently low to break the precautionary saving motive coming from
uncertain labor income.
64
for all s, s0 and
u0 ( F (s0 , K, L(s0 )) − (1 − δ)K ) ≥ βE0 (1 − δ + FK (s1 , K̄, L(s1 ))) u0 (l (i1 ) FL (s1 , K̄, L(s1 ))) .
(83)
Because for all s ∈ S
F (s, K̄, L(s)) − δK̄ < 0
we have
FK (s, K̄, L(s)) < δ
or
1 − δ + FK (s1 , K̄, L(s1 )) < 1.
So (82) and (83) follow directly from (79) and (80).
From (81), we obtain
Λµ̃ = D(K ), {D(0)}t>0,st ∈S t .
Second of all, P̄∗∞ , endowed with the product topology (of weak topology in P̄∗ ) is
not a a compact set. Therefore one cannot apply the Brouwer-Schauder-Tychonoff Fixed
Point Theorem for continuous functions defined on this set.
In the present paper, I follow a different route to establish the existence of a competitive equilibrium by taking the limit of finite horizon economies as in Appendix B. I derive
a lower bound on aggregate capital using the agents’ Euler equation, hence indirectly rule
out D(0). Another way to put it is that D(0) implies an infinite marginal rate of return r
on capital but in Lemma 5, by restricting prices on ∆e , we impose an upper bound on r. I
show that this bound does not bind in a competitive equilibrium.
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