Ergodic Invariant Distributions for Non-optimal
Dynamic Economics
Manuel S. Santos
Adrian Peralta-Alva
Department of Economics
University of Miami
Research Department
Federal Reserve Bank of Saint Louis
July 8, 2012
In this paper we are concerned with the simulation of non-optimal dynamic
economies. The equilibrium laws of motion of these economies cannot be characterized by the methods of dynamic programming and may not be described
by continuous policy functions. We prove existence of an invariant distribution
for the equilibrium law of motion, and establish some convergence and accuracy
properties for the simulated moments. We obtain these results without resorting
to artificial randomizations of the equilibrium correspondence or discretizations
of the state space.
KEY WORDS: Markov equilibrium, invariant distribution, computed solution,
simulated moments.
JEL codes: C63, E60.
1
Introduction
In this paper we are concerned with the simulation of non-optimal dynamic economies. Simulations are mechanically implemented in macroeconomics and other disciplines for assessing
models’ predictions [Cooley and Prescott (1965) and Santos and Peralta-Alva (2005)]. But
an obvious question is how these models should be simulated. For many models there is no
clear answer, since we lack a general theory that can justify the validity of these simulations.
1
Available laws of large numbers rely on the ergodic theorem [e.g., Arnold (1998) and Kifer
(1986)]. For many non-linear dynamic models, however, the ergodic theorem cannot be
applied since it is hard to locate their ergodic sets. That is, for an arbitrarily initial point
one cannot insure that such point would belong to an ergodic set, or that the dynamical
system will eventually enter one of its ergodic sets almost surely [Stokey, Lucas and Prescott
(1989, Ch. 11)]. In our earlier work [Santos and Peralta-Alva (2005)] we provide some
generalized laws of large numbers for continuous policy functions that avoid the difficulties
of applying the straightforward version of the ergodic theorem. Continuous policy functions
are encountered in convex optimization problems and representative agent economies where
the techniques of dynamic optimization can be applied.
For more general models, however, the continuity of Markov equilibria does not come out so
naturally. To guarantee existence of an ergodic distribution, some researchers have either resorted to a discretization of the state space [Ericson and Pakes (1995)] or to randomizations
over the equilibrium correspondence so that the dynamical system is a convex-valued correspondence in the space of distributions [Blume (1982) and Duffie et al. (1994)]. As is well
known, discrete state spaces are quite convenient to compute the set of invariant distributions, but these spaces become awkward for the characterization of optimal solutions and the
calibration and estimation of the model. Randomizing over the equilibrium correspondence
may result in an undesirable expansion of the equilibrium set.
Under fairly standard conditions, we present a proof of existence of an invariant distribution
for a wide range of economic models. The existence of an invariant distribution is a long
standing problem in the economics literature of stochastic dynamic models. For instance,
Duffie et al. (1994, p. 758) write:
In order to obtain an ergodic measure, however, it is necessary to convexify g
.... The convexification of g seems quite natural, but on the other hand we
know of no example satisfying our assumptions that does not admit a spotless
time-homogeneous Markov equilibrium with an ergodic measure. We wish to
stress that the question whether such a situation can exist is an open problem
2
of considerable interest: a positive answer would suggest that our methods are
capable of extracting all available general results, so that little remains unsaid,
while a negative answer would, in all likelihood, involve novel and interesting
methods of analysis.
We will offer a negative answer to this conjecture. As discussed below, our strategy of proof
is a Krylov-Bogolyubov type argument approximating an invariant measure by a sequence
of empirical measures.
We also establish a generalized law of large numbers that guarantees convergence of the simulated moments to the population moments of some stationary equilibrium. These stationary
equilibria display some continuity properties over the parameters of the model. Therefore,
laws of large numbers ensure that such continuity properties are extended to the simulated
moments.
The analysis focuses on the simulation of exact equilibria. In practice, however, exact equilibria cannot be simulated, since a researcher may only have access to numerical solutions.
Consequently, we address the convergence of the simulated moments for approximations
of the equilibrium laws of motion. Our results on existence of invariant distributions and
convergence of the simulated moments apply naturally to approximate solutions. Hence,
the simulated moments from a numerical solution approach asymptotically some invariant
distribution of the numerical approximation. Combining these arguments with some convergence results, we establish some accuracy properties for the simulated moments as the
approximation error goes to zero.
These accuracy properties also hold for discretized economies [Ericson and Pakes (1995)].
Thus, as we refine the discretization procedure the simulated moments of a discretized economy will converge in a well defined sense to those of the original economy. Therefore, commonly observed discretization methods can be justified on the grounds that the simulated
moments of these economies will eventually converge to their continuum counterparts.
In summary, our purpose here is to provide a framework for the quantitative study of dynamic
models whose equilibrium solutions cannot be characterized by the techniques of convex dynamic programming. Our results open the door for the simulation, estimation, and testing of
3
general equilibrium models widely used in economics for which the existence of a continuous
Markov equilibrium is not known to exist. We have in mind dynamic models with financial
frictions, economic distortions, asymmetric information, bargaining power, optimal taxation,
and monetary policies.
The paper is structured as follows. Section 2 lays down a simplified setting for the equilibrium
laws of motion. Section 3 presents our main analytical results. Section 4 illustrates our
methods of analysis with an extension of the Brouwer fixed-point theorem over the unit
interval. We conclude in Section 5 with a discussion of some economic models covered by
our analysis.
2
The Analytical Framework
Following Blume (1982) and Duffie et al. (1994), let us assume that the equilibrium law of
motion of the state variables can be specified by a time-invariant dynamical system of the
following form
sn+1 ∈ ϕ(sn , εn+1 ),
n = 0, 1, 2, · · · ,
(2.1)
where ϕ : S × E → S is a correspondence, and n = 0, 1, 2, · · · is the time index. In
applications, state space S is usually conformed by a well chosen subset of economic variables
describing the equilibrium dynamics. More specifically, a vector s ∈ S may include: (i)
exogenous state variables such as the level of total factor productivity or international
market prices, (ii) predetermined state variables such as physical capital and financial
assets, and (iii) endogenous state variables such as consumption, investment, asset prices
and interest rates. In models with multiple equilibria, endogenous state variables may be
needed to sort out a Markov equilibrium [Duffie et al. (1994), Kubler and Schmedders (2003)
and Feng et al. (2011)]. The shock ε follows an iid process under some probability law ν on
a measurable space (E, E).
Assumption 2.1 S is compact subset of some finite-dimensional space Rm , and S is its
Borel σ-algebra. E is a compact metric space.
Assumption 2.2 Correspondence ϕ : S × E → S is upper semicontinuous.
4
These assumptions are completely standard. Most of our results can be extended to the case
in which the domain S × E is an unbounded Borel set, and correspondence ϕ is defined for
ν-almost all ε.
By the measurable selection theorem [e.g., Crauel (2002) and Hildenbrand (1974)], under the
above assumptions there exists a sequence of measurable mappings {ϕ̂j }, ϕ̂j : S × E → S,
such that ϕ(s, ε) = cl{ϕ̂j (s, ε)} for all (s, ε) and all j (cl denoting closure). Let us pick a
measurable selection ϕ̂ ∈ ϕ. Then, transition probability Pϕ̂ (s, A) is defined as
Pϕ̂ (s, A) = ν({ε|ϕ̂(s, ε) ∈ A}).
(2.2)
Note that Pϕ̂ (s, ·) is a probability measure for each s ∈ S, and Pϕ̂ (·, A) is a measurable
function for each A in S.
For an initial probability µ0 on S, the evolution of future probabilities, {µn }, can be specified
by the following operator Tϕ̂∗ that takes the space of probabilities on S into itself
Z
∗
µn+1 (A) = (Tϕ̂ µn )(A) = Pϕ̂ (s, A)µn (ds),
(3.3)
for all A in S and n ≥ 0. An invariant probability measure or invariant distribution µ∗ is a
fixed point of operator Tϕ̂∗ , i.e., µ∗ = Tϕ̂∗ µ∗ .
Let f : S → R be a function of interest. Let C(S) be the space of all continuous realR
valued functions f on S. The integral f (s)µ(ds) or expected value of f over µ will be
denoted by E(f ) whenever distribution µ is clear from the context. The weak topology is the
R
coarsest topology such that every linear functional in the set {µ → f (s)µ(ds), f ∈ C(S)}
is continuous. A sequence {µj } of probability measures on S is said to converge weakly
R
R
to a probability measure µ if f (s)µj (ds) →j f (s)µ(ds) for every f ∈ C(S). The weak
topology is metrizable [e.g., see Billingsley (1968)].
Let Pϕ (s, ·) = {Pϕ̂ (s, ·) : ϕ̂ ∈ ϕ}. Then, s → Pϕ (s, ·) is a correspondence of stochastic
probabilities — often called a multivalued stochastic kernel. We say that s → Pϕ (s, ·) is
an upper semicontinuous correspondence in the weak topology of measures if for every fixed
R
f ∈ C(S) the associated functional s → f (s0 )Pϕ̂ (s, ds0 ), all Pϕ̂ (s, ·) ∈ Pϕ (s, ·), is an upper
semicontinuous correspondence. Under the above assumptions, Blume (1982, Th. 3.1 and
5
Prop. 2.3) shows that the correspondence Pϕ (s, ·) is upper semicontinuous in s in the weak
topology of measures µ on S.
3
Results
Our main goal in this section is to show existence of an invariant distribution. In applications, however, invariant distributions cannot be computed by either analytical or numerical
methods. Hence, a second technical step is to establish equality between the range of variation of the moments obtained from arbitrarily long simulations and the population moments
of the invariant distributions of the model. This equivalence result is quite relevant because
analytical properties of the population moments can thus be extended to the simulated
moments.
EXISTENCE OF AN INVARIANT DISTRIBUTION
There are two basic ways to establish existence of an invariant distribution [e.g., Crauel
(2002)]: (i) Via the Markov-Kakutani fixed-point theorem: An upper semicontinuous convexvalued correspondence in a compact set has a fixed point; and (ii) Via a Krylov-Bogolyubov
type argument: The invariant distribution is found as the limit of a sequence of iterations
over empirical probability measures. Blume (1982) and Duffie et al. (1994) follow (i), and are
required to randomize over the existing equilibria to build a convex-valued correspondence.
We follow (ii), and rely on the upper semicontinuity of this correspondence.
In preparation for our analysis, we define a new probability space comprising all infinite
sequences ω = (ε1 , ε2 , · · · ). Let Ω = E ∞ be the countably infinite cartesian product of
copies of E. Let F be the σ-field in E ∞ generated by the collection of all cylinders A1 × A2 ×
· · · × An × E × E × E × · · · where Ai ∈ E for i = 1, · · · , n. A probability measure λ can be
constructed over these finite-dimensional sets as
λ{ω : ε1 ∈ A1 , ε2 ∈ A2 , · · · , εn ∈ An } =
n
Y
Q(Ai ).
(3.1)
i=1
Measure λ has a unique extension on F. Hence, the triple (Ω, F, λ) denotes a probability
space. Finally, for every initial value s0 and sequence of shocks ω = {εn }, let {sn (s0 , ω, ϕ̂)}
be the sample path generated by function ϕ̂; that is, sn+1 (s0 , ω, ϕ̂) = ϕ̂(sn (s0 , ω, ϕ̂), εn+1 )
for all n ≥ 1 and s1 (s0 , ω, ϕ̂) = ϕ̂(s0 , ε1 ).
6
We are now ready to show our central result on the existence of an invariant probability
measure. As before, we say that Pϕ (s, ·) has an invariant probability µ∗ if there is Pϕ̂ (s, ·) ∈
R
P (s, ·) such that µ∗ (A) = (Tϕ̂∗ µ∗ )(A) = Pϕ̂ (s, A)µ∗ (ds), for all A in S.
Theorem 3.1 (Existence of an invariant probability measure) The transition correspondence Pϕ (s, ·) has an invariant probability µ∗ .
The invariant distribution is found by iteration as a limit of a sequence of empirical measures.
This type of Krylov-Bogolyubov argument [Crauel (2002)] relies on the upper semicontinuity
of the equilibrium correspondence. As shown later, simple examples of non-existence of an
invariant distribution can readily be constructed if the equilibrium correspondence is not
upper semicontinuous.
In applications the equilibium correspondence is usually written as ϕ(·, ·, γ), where γ is
a vector of parameters. Then, under very general conditions [Blume (1982)] the set of
invariant distributions is an upper semicontinuous correspondence in γ. We later provide a
more general convergence result for numerical approximations of correspondence ϕ.
CONVERGENCE OF THE SIMULATED MOMENTS
Besides existence of an invariant distribution, a second important technical step is to show
equality of the range of variation between the moments computed from simulations and
those computed from the invariant distributions of the model. For this equivalence result,
we rely on two basic contributions from probability theory: (i) Kingman0 s subadditive
ergodic theorem [Kingman (1968)], which selects upper and lower bounds for the moments
of the simulated paths (almost surely); and (ii) T he ergodic decomposition theorem [Kifer
(1986, Ch. 1)], which provides an integral representation of an invariant measure via ergodic
measures.
We actually need a simple extension of the ergodic subadditive theorem of Kingman (1968).
Our proof has to deal with certain technical measurability issues as our mapping ϕ̂ is not
continuous. A main advantage of this theorem is that we do not need to know existence
of an invariant measure µ∗ for function ϕ̂. Indeed, function ϕ̂ may not even have such an
invariant measure.
7
Theorem 3.2 (Kingman’s subadditive ergodic theorem) Consider a measurable selection ϕ̂ ∈ ϕ. Let f belong to C(S). Then, under Assumptions 2.1-2.2 there are constants
L(f, ϕ̂) and U (f, ϕ̂) such that for λ-almost all ω,
(i)
(ii)
N
1 X
lim (infs0 ∈S [
f (sn (s0 , ω, ϕ̂))]) = L(f, ϕ̂)
N →∞
N n=1
lim (sups0 ∈S [
N →∞
N
1 X
f (sn (s0 , ω, ϕ̂))]) = U (f, ϕ̂).
N n=1
(3.2a)
(3.2b)
Note that these bounds are the same for λ-almost all ω because of the ergodic nature of the
iid process {εt }.
In what follows I(ϕ̂) denotes the set of invariant measures of a measurable selection ϕ̂ ∈ ϕ and
I(ϕ) denotes the set of invariant measures of correspondence ϕ. An ergodic decomposition of
an invariant measure µ∗ , or more precisely an ergodic decomposition of a ϕ̂-invariant measure
R
µ∗ into its ergodic components Q, is a measure ρ satisfying ρ(I(ϕ̂)) = 1, and µ∗ = S Qρ(dQ).
R
R
R
For every f ∈ C(S) we then have f (s)µ(ds) = I(ϕ̂) ( (f (s)Q(ds))ρ(dQ).
Theorem 3.3 (The ergodic decomposition theorem) Assume that µ∗ is an invariant
R
probability measure for P (s, ·). Then, µ∗ has an invariant decomposition µ∗ = S Qρ(dQ).
This result holds true for every measurable selection ϕ̂. Hence, continuity of mapping ϕ̂ is
not necessary for the ergodic decomposition theorem; see Kifer (1968) and Klunger (1998).
With these two basic results from probability theory in place, we now show that the range
of variation of the sample moments is equal to the range of variation of the population
R
R
moments. Let E inf (f ) = inf µ∗ ∈I(ϕ) f (s)µ∗ (ds), and E sup (f ) = supµ∗ ∈I(ϕ) f (s)µ∗ (ds).
Theorem 3.4 (A generalized law of large numbers) Consider the set of measurable
selections ϕ̂ ∈ ϕ. Let f belong to C(S). Then, under Assumptions 2.1-2.2 for λ-almost
all ω,
(i)
inf L(f, ϕ̂) = E inf (f )
ϕ̂∈ϕ
8
(3.3a)
(ii)
sup L(f, ϕ̂) = E sup (f ).
(3.3b)
ϕ̂∈ϕ
Of course, equalities (3.3a) - (3.3b) hold true over ergodic sets. But Theorem 3.4 applies
to any arbitrary initial condition s0 ∈ S outside ergodic sets, and hence it is not a mere
consequence of the ergodic theorem. Indeed, the method of proof actually shows that the
sequence simulated moments will converge almost surely to the population moments of some
invariant distribution of the model. Therefore, for λ-almost all ω, the range of variation of
the simulated moments is bounded by the range of variation of the population moments of
the model’s invariant distributions.
ACCURACY OF NUMERICAL SIMULATIONS
We now apply the above results to the numerical simulation of stochastic dynamic models.
Assume that a researcher is concerned with the predictions of a stochastic dynamic model
whose equilibrium law of motion can be specified by a correspondence ϕ. Usually, his
Markovian solution ϕ does not admit an analytical representation, and so it is approximated
by numerical methods. Moreover, the invariant measures or stationary solutions of the
numerical approximation cannot be calculated analytically. Hence, population moments are
computed by simulations after appealing to some law of large numbers.
The critical question is whether invariant distributions generated from numerical approximations will approach some invariant distribution of the original model as the approximation error converges to zero. As is typical in the simulation of stochastic models we suppose that the
researcher can draw sequences {b
εn } from a generating process that can mimic the distribution
of the shock process {εn }. Probability measure λ is defined over all sequences ω = (ε1 , ε2 , ...).
Once a numerical approximation ϕ̂j is available, it is generally not so costly to generate
sample paths {sn (s0 , ω, ϕ̂j )} defined recursively as sn+1 (s0 , ω, ϕ̂j ) = ϕ̂j (sn (s0 , ω, ϕ̂j ), εn+1 )
for every n ≥ 0 for fixed s0 and ω. Averaging over these sample paths we get sequences of
P
simulated moments { N1 N
n=1 f (sn (s0 , ω, ϕ̂j ))} as defined by some function of interest f .
Let us consider the following notion of distance from the sup norm in the space of mappings. Assume that | · | is a norm in Rm . For mappings ϕ̂j and ϕ̂, let the distance
kϕ̂j − ϕ̂k = sup(s,ε)∈S×E |ϕj (s, ε) − ϕ̂(s, ε)|. We next show that for a sufficiently good nuP
merical approximation ϕ̂j and for a sufficiently large N the series { N1 N
n=1 f (sn (s0 , ω, ϕ̂j ))}
9
approaches (almost surely) the range of variation of the simulated moments generated by
correspondence ϕ.
Theorem 3.5 (Accuracy of the simulated moments) Assume that in the sup norm a
sequence of measurable functions {ϕ̂j } can approach asymptotically some selection ϕ̂ ∈ ϕ.
Let f belong to C(S). Then, under Assumptions 2.1-2.2 for every η > 0 there are functions
Nj (w) and an integer J such that for all j ≥ J and N ≥ Nj (ω),
N
1 X
inf L(f, ϕ̂) − η <
f (sn (s0 , ω, ϕ̂j )) < sup U (f, ϕ̂) + η
ϕ̂∈ϕ
N n=1
ϕ̂∈ϕ
(3.4)
for all s0 and λ-almost all ω.
If ϕ has a unique invariant probability, then inf ϕ̂∈ϕ L(f, ϕ̂) = supϕ̂∈ϕ U (f, ϕ̂). Hence, in
the case of a unique invariant distribution, Theorem 3.5 implies convergence of the simulated moments from numerical approximations to the simulated moments of the model.
For the case of multiple invariant distributions, we have that the moments from numerical
approximations will eventually be located (up to small error) within the range of variation
of the simulated moments of the original model. As a matter of fact, as is Theorem 3.4 a
strengthening of our method of proof would show that the simulated moments from numerical approximations will eventually converge to the population moments of some invariant
distribution.
4
An Extension of the Brouwer Fixed-Point Theorem in the Space of
Distributions
Let us illustrate our methods of analysis in a simple deterministic example. Let ϕ : [0, 1] →
[0, 1] be an upper semicontinuous, convex-valued correspondence. As is well known, by the
Kakutani fixed-point theorem the correspondence ϕ has a fixed point, i.e., there is some s∗
such that s∗ ∈ ϕ(s∗ ). In other words, a measurable selection ϕ̂ ⊂ ϕ contains a fixed point
s∗ = ϕ̂(s∗ ). A convex-valued correspondence is the natural outcome of concave optimization,
but convexity is lost for equilibrium solutions of many other economic models of interest.
10
As in Figure 1, we now drop this convexity assumption, but we still assume that ϕ is an
upper semicontinuous correspondence. Then, a fixed-point solution s∗ may not longer exist.
Our analysis searches for a fixed point in the space of distributions µ∗ . That is, the invariant
set may not longer be a point s∗ , but it could be a finite cycle or some other limit set which
is the support of some invariant distribution µ∗ . To locate such stationary solutions, we use
a Krylov-Bogolyubov type argument: The invariant distribution µ∗ is obtained as the limit
of a convergent subsequence of empirical distributions.
The idea is to construct N -period paths x1 , x2 , · · · , xN . That is, sequences of points generated under a function ϕ̂ ∈ ϕ. For each path x1 , x2 , · · · , xN we associate a distribution µN
that places the same weight in each point of the path. Note that as we extend this path by
11
one more period distribution µN hardly changes for N large enough. Then, as N goes to ∞,
by the upper semicontinuity of the correspondence the limiting distribution µ∗ must belong
to the image of the correspondence. That is, µ∗ ∈ ϕ(µ∗ ). Let χ∗ be the support of µ∗ . Then,
we have proved:
Theorem 4.1 (An extension of the Brouwer fixed-point theorem) Let ϕ : [0, 1] →
[0, 1] be an upper semicontinuous correspondence. Then, there is a set χ∗ such that χ∗ =
ϕ(χ∗ ).
Therefore, we cannot show existence of a simple fixed point s∗ . But we show existence of a
more general invariant set χ∗ which could be a finite cycle, or a recurrent set characterized
by an invariant distribution.1 Our work in the previous section extends this basic result to
stochastic equilibrium correspondences with iid shocks. Our constructive algorithm iterates
over continuation equilibrium values, and so it may not pick some general stochastic solutions
such as sunspots or arbitrary randomizations of correspondence ϕ.
We actually seek to ascertain the range of variation of the moments. That is, to establish
the equivalence between the upper and lower bounds for the moments of both the simulated
paths and the population moments of the invariant solutions of the model.
Assume that f is a function of interest that may represent a certain moment or some other
statistic. Again, let ϕ̂ ∈ ϕ be a measurable selection. Then, to pick up an upper bound
P
for the sample moments we let supx0 ∈[0,1] N1 N
n=1 f (xn ) where x1 , x2 , · · · , xN is a sequence
generated under function ϕ̂. For simplicity, assume that the sup is always attained. Let µN
be the empirical measure that places the same weight in each point of the path. A limit
point of {µN }N ≥0 may not be an invariant distribution µ∗ for ϕ̂ because ϕ̂ is not a continuous
mapping. However, ϕ is an upper semicontinuous correspondence. Therefore, every limit
point of the sequence {µN }N ≥0 is an invariant distribution µ∗ under the action of some other
selection ϕ̂0 ∈ ϕ.
1
The existence of an invariant set for ϕ can also be established by iterating over this correspondence. Let
ϕ the n-times composition of ϕ. Then, {ϕn ([0, 1])}n≥1 is a non-increasing sequence of compact sets with
non-empty intersection. One readily shows that this non-empty intersection is an invariant set for ϕ. This
argument, however, does not yield existence of an invariant probability measure µ∗ .
n
12
Now, let I(ϕ) be the set of invariant distributions under ϕ. It follows from the previous
R
P
argument that limn→∞ supx0 ∈[0,1] N1 N
f (x)µ∗ (dx), where by conn=1 f (xn ) ≤ maxµ∗ ∈I(ϕ)
struction the limit on the left-hand side does exist. [For stochastic models existence of the
limit (almost surely) follows from Kingman’s subadditive ergodic theorem.] To prove the
reverse equality, we may invoke the ergodic decomposition theorem, which partitions the
domain of an invariant distribution into ergodic subsets with measures Q. Then, at each
R
ergodic component we have equality between the expected value f (x)Q(dx) and the simP
ulated moment limn→∞ N1 N
n=1 f (xn ). Therefore, as N goes to ∞ the range of variation of
the population moments is equal to the range of variation of the simulated moments.
The upper semicontinuity of the correspondence cannot be dropped. For instance, the correspondence in Figure 2 has no stationary solution and all paths converge to point x = 1/2.
P
Hence, all sample statistics N1 N
n=1 f (xn ) converge to f (1/2). Since our focus is on the simulated moments, we could actually consider models in which an invariant distribution µ∗ may
not exist. The existence of an invariant distribution is of independent interest and becomes
instrumental in our accuracy results: Analytical properties of the population moments under
perturbations of the model will be inherited by the simulated moments.
13
5
Concluding Remarks
This paper shows existence of an invariant distribution and a generalized law of large numbers of application to economies with heterogeneous agents, real and financial frictions, and
incentive-compatibility constraints. Equilibria of these economies cannot be characterized
by the techniques of convex dynamic programming. Indeed, the equilibrium laws of motion
may not be summarized by continuous transition functions.
Researchers have long been interested in the computation and simulation of dynamic economies.
Duffie et al. (1994) show existence of an ergodic invariant measure in three types of models:
14
An asset pricing model with incomplete markets, and overlapping generations economy, and
a stochastic game. Kubler and Schmedders (2003) develop a numerical algorithm for the
computation of an asset pricing model with collateral constraints, and Feng et al. (2011)
extend these methods to incomplete-markets economies with endogenous borrowing constraints and taxation as in the international business cycle model of Kehoe and Perri (2003).
In industrial organization, Ericson and Pakes (1995) provide an algorithm for the simulation of a general model with entry and exit. Also, an extensive literature has developed for
the computation of models of optimal taxation, time-inconsistency of monetary and fiscal
policies, and incentive-compatibility constraints.
Quantitative work in this area has attempted the simulation of these economies in various
known ways. First, in econometric work it is typical to assume that the ergodic set expands
over the whole state space S [e.g., Hansen (1982)]. In other cases, this later condition can be
satisfied by perturbing the stochastic process under a mixing condition [Billingsley (1968) and
Doob (1953)] that forces the equilibrium system to visit every state s infinitely often. Second,
the state S can be discretized. As is well known [Stokey, Lucas and Prescott (1989, Ch. 11)],
a finite Markov chain always contains a stationary equilibrium. Of course, a problem would
arise if the moments computed from the discretized state space are quite different from the
population moments of the original model. And third, the equilibrium correspondence can
be convexified. Then, existence of an invariant distribution is guaranteed by standard fixedpoint methods. This type of randomization, however, may expand arbitrarily the equilibrium
set.
In contrast, under standard economic assumptions we show existence of stochastic stationary equilibria for the original formulation of the model. Hence, the volatility of economic
aggregates in a wide class of economies can be well defined over stationary solutions without
resorting to a discretization of the state space or perturbations of the equilibrium correspondence. Moreover, the moments obtained from numerical simulations will approach asymptotically the population moments over these stationary equilibria. We establish such a law of
large numbers using a new method of proof that builds on two basic results from probability
theory: Kingman’s subadditive ergodic theorem and the ergodic decomposition theorem.
The law of large numbers becomes instrumental to establish accuracy properties of the sim-
15
ulated moments since analytical and approximation properties of invariant distributions are
extended to the simulated moments.
6
Appendix
This appendix contains the proofs o our main results in Section 3. As already remarked, our
stated results in Section 2 about the upper semicontinuity of the equilibrium correspondence
in the space of distributions follow from Blume (1982). See also Duffie et al. (1994) for a
related method of proof.
Proof of Theorem 3:1: We begin with a given measurable function s0 : Ω → S of initial
conditions and a measurable selection ϕ
b ∈ ϕ. Starting from each s0 (ω) under the action
of ϕ
b we generate finite sequences {sn (ω)}N
n=1 . For all N = 1, 2, 3, · · · and ω we define the
P
N
empirical measure µN (ω, ·) = N1 n=1 δsn (ω) for each sn ∈ {sn (ω)}N
n=1 , where δsn (ω) (s) = 1 if
R
s = sn (ω) and δsn (ω) (s) = 0 otherwise. Let µN (·) = µN (ω, ·)λ(dω).
We are now ready to apply a Krylov-Bogolyubov type argument [cf., Crauel (2002, p. 87)].
Since S is a compact set, the sequence of measures {µN (·)}N ≥1 must have a convergent
subsequence {µN 0 (·)}N 0 ≥1 . Assume that µ∗ is the limit point of {µN 0 (·)}N 0 ≥1 . Define now the
R
measure µ̃N (A) = (Tϕb∗ µN )(A) = Pϕb(s, A)µN (ds). We claim that the sequence {µ̃N 0 (·)}N 0 ≥1
also converges to µ∗ . If this claim holds true, then we can prove the theorem. Indeed,
µ̃N 0 (·) ∈ Tϕ∗ (µN 0 (·)) for every N 0 . Hence, from Blume (1982, Th. 3.1 and Prop. 2.3) we know
that our correspondence is upper semicontinuous and so it has a close graph. Therefore,
µ∗ ∈ Tϕ∗ (µ∗ ).
It remains to prove our claim that the sequence {µ̃N 0 (·)}N 0 ≥1 also converges to µ∗ . By
definition, measure µ̃N (·) can be constructed in the following way. For each {sn (ω)}N
n=1 , let
P
N
N
+1
1
{ϕ(s
b n (ω))}N
n=1 = {sn (ω)}n=2 . Now, we define the empirical measure µ̃N (ω, ·) = N
n=1 δsn (ω)
+1
for each sn ∈ {sn (ω)}N
n=2 , where δsn (ω) (s) = 1 if s = sn (ω) and δsn (ω) (s) = 0 otherwise. Let
R
µ̃N (·) = µ̃N (ω, ·)λ(dω). Hence, for any set A ∈ S, we get that |µN (A) − µ̃N (A)| ≤ N2 . It fol-
lows that the distance between µN (·) and µ̃N (·) approaches zero as N goes to ∞. Therefore,
the sequences {µN 0 (·)}N 0 ≥1 and {µ̃N (·)}N ≥1 must have the same limit points µ∗ .
Proof of Theorem 3.2: To deal with some measurability issues, we initially restrict the
16
domain S to a countable set Sb ∈ S. Consider the following function:
HSN
b[
b (ω) = sups0 ∈S
N
1 X
f (sn (s0 , ω, ϕ))].
b
N n=1
(6.1)
N
b
Note that HSN
b (ω) is measurable since S is a countable set. Moreover, HS
b (ω) is also subN −T
T
additive. That is, HSN
(ζ T (ω)) for all 1 ≤ T ≤ N , where ζ T defines
b (ω) ≤ HS
b (ω) + HS
b
the T -times shift operator (ε1 , ε2 , ε3 , · · · , εT +1 , εT +2 , · · · ) → (εT +1 , εT +2 , · · · ). Hence, by
Kingman’s subadditive ergodic theorem it follows that HSN
b (ω) converges almost surely to a
constant function HSb. Let us next define
H = supS∈S
b HS
b.
(6.2)
As a matter of fact, the sup can actually be achieved over a countable sequence Sbj so that the
constant bound H is well defined for λ-almost all ω. This proves (3.2b) for initial conditions
b Therefore, U (f, ϕ)
s0 in countable sets S.
b = H. In a similar way, we can also prove equality
(3.2a) for λ-almost all ω.
Proof of Theorem 3.3: Given that the process {εn } is iid, this theorem follows from Kifer
(1986, Ch. 1) and Klunger (1998, Sect. 4).
Proof of Theorem 3.4: We first show (3.3b). We begin with: supϕ∈ϕ
U (f, ϕ)
b ≥ Eϕmax (f ).
b
b ∈ ϕ.
Assume that µ∗ is an invariant probability µ∗ = Tϕb∗ (µ∗ ) for some measurable selection ϕ
Then, by the ergodic decomposition theorem there is no loss of generality to assume that µ∗
is ergodic. It follows that for λ-almost all ω we have
Z
f (s)µ∗ (ds) =
N
X
f (sn (s0 , ω, ϕ))
b ≤ U (f, ϕ).
b
(6.3)
n=1
Since µ∗ has been arbitrarily chosen, we have proved that supϕ̂∈ϕ U (f, ϕ)
b ≥ Eϕmax (f ).
Let us now show the reverse inequality: supϕ∈ϕ
U (f, ϕ)
b ≤ Eϕmax (f ). Consider an arbitrary
b
measurable function of initial conditions s0 (ω) for all ω ∈ Ω. As in the proof of Theorem 3.1
we can generate a sequence of empirical measures {µN (·)}N ≥1 . Moreover, every limit point µ∗
of {µN (·)}N ≥1 is am invariant measure µ∗ = Tϕ (µ∗ ). Therefore, supϕ∈ϕ
U (f, ϕ)
b ≤ Eϕmax (f ).
b
In the same way we can establish (3.3a).
Proof of Theorem 3.5: There is no loss of generality to assume that graph(ϕ
bj ) is closed,
for j = 1, 2, · · · , and hence ϕ
bj could be a correspondence. Then, by Theorem 3.4, it is enough
17
max
min
max
to consider the interval of values [Eϕmin
(f )]. Now,
bj (f ), Eϕ
bj (f )] for each j and [Eϕ (f ), Eϕ
as Tϕ∗ is an upper semicontinuous correspondence, then the set of fixed points I(ϕ) is also
an upper semicontinuous correspondence in ϕ [Blume (1982)]. Hence, for given > 0 and
max
ϕj sufficiently close to ϕ in the sup norm we have: Eϕmin (f ) ≤ Eϕmin
(f ) ≥
bj (f ) + and Eϕ
Eϕmax
bj (f ) − .
18
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