#  Uniformly Self-Justified Equilibria††thanks:  We thank Ken Judd, Filipe
Martins-da-Rocha, Tom Sargent, Xinyang Wang, Bob Wilson, Thomas Winberry, as
well as three anonymous referees, the editor, and seminar participants at
various universities and conferences for useful conversations and comments.

Felix Kubler
DBF, University of Zurich
and Swiss Finance Institute
<EMAIL_ADDRESS>Simon Scheidegger
Department of Economics, HEC Lausanne
and Enterprise for Society (E4S)
<EMAIL_ADDRESS>
###### Abstract

We consider dynamic stochastic economies with heterogeneous agents and
introduce the concept of uniformly self-justified equilibria (USJE)—temporary
equilibria for which forecasts are best uniform approximations to a selection
of the equilibrium correspondence. In a USJE, individuals’ forecasting
functions for the next period’s endogenous variables are assumed to lie in a
compact, finite-dimensional set of functions, and the forecasts constitute the
best approximation within this set. We show that USJE always exist and develop
a simple algorithm to compute them. Therefore, they are more tractable than
rational expectations equilibria that do not always exist. As an application,
we discuss a stochastic overlapping generations exchange economy and provide
numerical examples to illustrate the concept of USJE and the computational
method.

Keywords: Dynamic General Equilibrium, dynamically simple equilibrium.

JEL Classification: C63, C68 , D50, D52.

##  1 Introduction

Dynamic stochastic general equilibrium models with heterogeneous agents and
incomplete financial markets play an important role in macroeconomics and
public finance (see, e.g., Krueger et al. (2016)). Unfortunately, for many
model specifications, the computation of rational expectations equilibria is
not possible. A recursive equilibrium does not always exist (see Kubler and
Schmedders (2002)), and even when it does, no regularity properties (beyond
measurability) can be established for the equilibrium policy functions (see
Brumm et al. (2017)). Computational methods typically focus on computing
rational expectation $\epsilon$-equilibria, that is, allocations and prices
that clear markets and satisfy agents’ optimality conditions (i.e., Euler
equations) up to some “small” $\epsilon\in\mathbb{R}_{++}$. Errors in Euler
equations provide a good method to analyze solutions to dynamic optimization
problems (see Santos (2000) or Judd (1992)). However, in models with
heterogeneous agents, it is more difficult to interpret them. Agents’
incorrect choices need to be coordinated to ensure that the definition of an
approximate equilibrium holds. Agents’ mistakes are determined by the
requirement that at any time, agents’ rational expectations from the previous
period need to be correct in the current period, and that markets clear.
Errors in the market clearing condition are similarly problematic.

These shortcomings seem to make a rigorous numerical analysis of rational
expectations equilibria impossible and, in consequence, call for an
alternative to the rational expectations assumption that is more
tractable.111This refers to tractability in the sense of the first paragraph.
We do not formally discuss computational tractability in this paper. One way
to solve this is to assume that agents are boundedly rational—as Sargent
(1993) argues—a sensible approach to relax rational expectations is “expelling
rational agents from our model environment and replacing them with
artificially intelligent agents who behave like econometricians.” In this
paper, we develop an alternative to rational expectations equilibria and
consider temporary equilibria (see, e.g., Grandmont (1977)) with forecasting
functions that are simple in the sense that they are easy to compute
numerically. These forecasting functions approximate (a selection of) the
temporary equilibrium correspondence, but the agents might make significant
mistakes in their forecasts. These stem from the definition of an equilibrium
that we are proposing below: The agents make no mistakes in choices, given
their forecasts, and markets clear. Agents’ forecasts are prevented from being
arbitrarily accurate only because of the computational cost associated with
more accurate forecasts.222We do not formalize the concept of “computational
costs” in this paper. However, it is obvious that, for example, the use of a
low degree polynomial for forecasting purposes comes at a lower computation
cost than, for example, using a large number of radial basis functions.

To deal with the aforementioned issues, this paper makes four contributions.
First, we introduce a novel equilibrium concept—that of a “uniformly self-
justified equilibrium” (USJE)—for stochastic general equilibrium models with
heterogeneous agents. Second, we prove existence. Third, we develop an
algorithm to approximate equilibria numerically. Fourth, we demonstrate the
applicability of our new equilibrium concept and the corresponding numerical
method in the context of a stochastic overlapping generations (OLG) model with
many generations.

The basic idea of our proposed USJE approach is as follows: In a temporary
equilibrium, agents use the current endogenous variables such as current
prices and/or asset choices across all agents, and the exogenous shock, to
forecast future marginal utilities for assets; prices for commodities and
assets in the current period ensure that markets clear. Forecasting functions
are assumed to lie in a pre-specified, finite-dimensional class of simple
functions (such as polynomials of a fixed degree), and an agent chooses a
function that minimizes the maximum error of his forecast (i.e., that is the
uniformly best approximation to equilibrium marginal utilities). In a USJE,
the agents make optimal choices, given their forecasts, and the forecasts are
(constrained) optimal, given equilibrium prices and choices.

We introduce the concept of USJE in an abstract framework that encompasses
both economies with heterogeneous, infinitely lived agents, as well as
economies with overlapping generations. This general modeling framework allows
us to investigate the properties of a USJE with as little notation as
possible. We prove existence by relying on relatively weak assumptions on
fundamentals and using well-known tools from mathematical economics. In
addition, we introduce a simple algorithm to compute a USJE numerically. We
then consider an exchange economy with overlapping generations to illustrate
the abstract framework and to show that the assumptions necessary for
existence translate to standard assumptions on fundamentals in a concrete
economic model. Finally, we provide an explicit numerical example to
illustrate the concept and our proposed algorithm in a simple framework.

In an earlier working paper— Kübler and Scheidegger (2019)—we introduced a
related concept of “self-justified equilibria”, where agents choose forecasts
that minimize the average squared error along the equilibrium path.
Unfortunately, this concept is theoretically much harder to analyze, and there
are no existence proofs available. However, as we argue in that paper, it
lends itself better to existing computational methods. The main reason for
that is that least-squares approximation methods are standard in numerical
analysis, and there is a large variety of methods that can be employed
efficiently. Moreover, the least-squares approximation is not as delicate with
respect to outliers as a uniform approximation.

There is a large and diverse body of work exploring deviations from rational
expectation (see, e.g., Jordan (1976), Blume and Easley (1982), Sargent
(1993), Kurz (1994), Woodford (2013), Gabaix (2014), Adam et al. (2016)). Much
of this literature is motivated by insights from behavioral economics about
agents’ behavior or the search for simple economic mechanisms that enrich
standard models’ observable implications. Much of this research strand also
focuses on single-agent models, where many of the technical difficulties
discussed in this paper disappear. Our work’s motivation is very different in
that we develop a simple alternative to rational expectations that allows
researchers to rigorously analyze stochastic dynamic models with heterogeneous
agents. The existence of an equilibrium and the tractability of the
equilibrium functions seem to be a prerequisite for this. Our concept of a
USJE is somewhat similar to the idea of Berk-Nash equilibria discussed in
Esponda and Pouzo (2016). However, the focus of their paper and much of the
literature that follows is on learning, whereas ours is on the existence and
simplicity of the equilibrium functions. Moreover, to the best of our
knowledge, there is no other concept in the literature that ensures the
general existence and tractability of an equilibrium for models with
incomplete markets and heterogeneous agents.

The remainder of the paper is organized as follows. In Section 2, the general
economy is introduced. Furthermore, we define uniformly self-justified
equilibria and show that they exist under general conditions. Finally, we
develop a simple algorithm to approximate USJE numerically. In Section 3, we
demonstrate how the concept of USJE and the related algorithm can be applied
in practical applications. To this end, we consider an exchange economy with
overlapping generations. Section 4 concludes.

##  2 A general dynamic Markovian economy

Time is discrete and indexed by $t\in\mathbb{N}_{0}$. Exogenous shocks $z_{t}$
realize in a finite set ${\mathbf{Z}}=\\{1,\ldots,Z\\}$, and follow a first-
order Markov process with transition probability
$(\pi(z^{\prime}|z))_{z,z^{\prime}\in{\mathbf{Z}}}$. A history of shocks up to
some date $t$ is denoted by $z^{t}=(z_{0},z_{1},\ldots,z_{t})$ and called a
“date event”. For $\tau>t$, we write $z^{\tau}\succ z^{t}$ if a date event
$z^{\tau}$ is a successor of $z^{t}$.

###  2.1 An abstract economy

The following abstract setup is borrowed from Duffie et al. (1994) and Kubler
and Schmedders (2003). It is useful to assume that each agent that is active
in markets in a given period makes a current decision, given expectations over
the next period’s endogenous and exogenous variables such as the next period’s
choices and prices. These expectations are formed based on the current
endogenous variables and the exogenous shock. While these expectations are
always correct in a rational expectations equilibrium, we allow them here to
be imprecise (in a way which we will formalize below). The endogenous
variables at a given date event,
$\left(x(z^{t}),y(z^{t})\right)\in{\mathbf{X}}\times{\mathbf{Y}},$ consist of
variables, $x(z^{t})\in{\mathbf{X}}$, that are predetermined (possibly
stochastically) by the previous period’s choices, as well as the current
period’s choices and prices $y(z^{t})\in{\mathbf{Y}}$, where in our setting
both ${\mathbf{X}}$ and ${\mathbf{Y}}$ are subsets of the Euclidean space. As
in Duffie et al. (1994), the expectations correspondence,

${\mathbf{H}}:{\mathbf{X}}\times{\mathbf{Y}}\times{\mathbf{Z}}\rightrightarrows({\mathbf{X}}\times{\mathbf{Y}})^{Z},$

embodies all short-run equilibrium conditions, that is, inter- and intra-
temporal first order conditions as well as market clearing conditions. If
agents expect the next period’s prices and choices to lie in this
correspondence, they have no incentives to deviate from the current choices.
We focus on environments where these conditions are sufficient for optimality,
that is, if at all nodes $z^{t}$,

$(x(z^{t+1}),y(z^{t+1}))_{z_{t+1}\in\mathbf{Z}}\in{\mathbf{H}}(x(z^{t}),y(z^{t}),z_{t}),$

then the process $(x(z^{t}),y(z^{t}))_{z^{t}}$ constitutes a competitive
equilibrium. For a given economy, there can be several different ways to write
the expectations and to fix the set of endogenous variables, ${\mathbf{X}}$,
for example, one can include continuation utilities. This turns out to make a
difference for our equilibrium concept defined below.

We define $s=(x,z)\in{\mathbf{S}}={\mathbf{X}}\times{\mathbf{Z}}$ to be the
state of the economy, and refer to $x\in{\mathbf{X}}$ as the endogenous state,
and to $z\in{\mathbf{Z}}$ as the exogenous shock. At $t=0$, the state $s_{0}$
is given, and we refer to it as the initial conditions of the economy. A
transition function $T$ maps the state today and actions in the current period
to a probability distribution over states in the subsequent period:

$T:{\mathbf{X}}\times{\mathbf{Y}}\times{\mathbf{Z}}\rightarrow{\mathbf{X}}^{Z}.$

The latter is exogenously given and describes how choices in the current
period lead to predetermined variables in the next period. Note that in
Section 3 below, we will provide concrete examples for the expectations
correspondence and the transition function.

A (forecasted) policy function333In our framework, all policy functions
represent the agents’ expectations about future optimal choices and prices. We
sometimes refer to these as “forecasts”, or “forecasted policies”. maps states
to current endogenous variables, that is,

$P:{\mathbf{X}}\times{\mathbf{Z}}\rightarrow{\mathbf{Y}}.$

This function describes an agent’s expectations about which equilibrium prices
and allocations arise from the predetermined variables and the shock. The
temporary equilibrium correspondence, ${\mathbf{N}}_{P}$, maps the exogenous
shock and the predetermined variables to all endogenous variables that are
consistent with an equilibrium (more precisely, with the expectations
correspondence) in the current period, given that the next period’s choices
and prices are expected to be described by the policy, $P$:

${\mathbf{N}}_{P}:{\mathbf{S}}\rightrightarrows{\mathbf{Y}},$

with

${\mathbf{N}}_{P}(x,z)=\\{y\in{\mathbf{Y}}:\left(T_{z^{\prime}}(x,y,z),P(T_{z^{\prime}}(x,y,z),z^{\prime})\right)_{z^{\prime}\in{\mathbf{Z}}}\in{\mathbf{H}}(x,y,z)\\}.$

In this recursive formulation, we drop the time-index, $t$, and superscript
variables in the subsequent period with primes. While in our environment, the
transition function is exogenously given and therefore known, the policy
function embodies the agents’ expectations about the future. The policy
function is defined as part of our equilibrium concept. In a recursive
(rational expectations) equilibrium, $P(x,z)\in{\mathbf{N}}_{P}(x,z)$ for all
$(x,z)$. In contrast, in a USJE, this strong requirement is relaxed, and
agents approximate the equilibrium correspondence by a “simple” function which
we will assume to be the weighted sum of a given number of basis functions. In
our examples below, we use polynomial basis functions, but there is a large
literature in approximation theory on basis function expansions (Friedman et
al. (2001)) which explores many alternatives. We take
${\mathbf{Y}}\subset\mathbb{R}^{Y}$, and assume that each $P_{k}(\cdot)$,
$k=1,\ldots,Y$, is a “best uniform approximation” of (a selection of) the
equilibrium correspondence within our given class of simple functions.444In
general, we cannot ensure the uniqueness of the best uniform approximation.
Therefore we refer to our approximations as a best. For each $k=1,\ldots,Y$,
the class of these simple functions is denoted by ${\mathbf{P}}_{k}$ and is
exogenously given. Our interpretation of this assumption is that economic
agents cannot evaluate arbitrarily complicated functions and resort to
approximations to decrease computational costs. We define a USJE as follows:

###### Definition 1

A uniformly self-justified equilibrium is given by a policy function,

$P^{*}:{\mathbf{X}}\times{\mathbf{Z}}\rightarrow{\mathbf{Y}},$

a finite set of points $(x_{z}^{1},y_{z}^{1}),\ldots,(x_{z}^{N},y_{z}^{N})$,
for each $z\in{\mathbf{Z}}$, with $x_{z}^{i}\in{\mathbf{X}}$,
$y_{z}^{i}\in{\mathbf{N}}_{P^{*}}(x_{z}^{i},z)$, for all $i=1,\ldots,N$, and a
set of states $\widehat{\mathbf{S}}\subset{\mathbf{S}}$ that satisfy the
following two properties:

  1. 1.

For each $z\in{\mathbf{Z}}$ and all $k=1,\ldots,Y$, the policy function,
$P^{*}_{k}(\cdot,z)$, is a best uniform approximation to the points
$(y_{kz}^{i})_{i=1,\ldots,N}$, that is,

$P^{*}_{k}(\cdot,z)\in\arg\min_{f\in{\mathbf{P}}_{k}}\sup_{i=1,\ldots,N}|y^{i}_{zk}-f(x^{i},z)|.$

  2. 2.

For each $s=(x,z)\in\widehat{\mathbf{S}}$, there is a
$y(s)\in{\mathbf{N}}_{P^{*}}(s)$ such that for all $k=1,\ldots,Y$,

$|P^{*}_{k}(s)-y_{k}(s)|\leq\sup_{i=1,\ldots,N}|y^{i}_{k}-P^{*}_{k}(x^{i}_{z},z)|,$

and

$(T_{z^{\prime}}(x,y(s),z),z^{\prime})\in\widehat{\mathbf{S}}\mbox{ for all
}z^{\prime}\in{\mathbf{Z}}.$

In this definition 1, contrary to what one might expect, the finite set of
points $(x^{i}_{z},y^{i}_{z})$ do not need to satisfy
$(x^{i}_{z},z)\in\widehat{\mathbf{S}}$. Depending on how the set
$\widehat{\mathbf{S}}$ relates to ${\mathbf{S}}$, this fact can lead to
equilibria where for the realized equilibrium variables, the forecasted
policy, $P(\cdot)$, is not a best uniform approximation. However, as we argue
below, it is computationally not feasible to require that
$x^{i}\in\widehat{\mathbf{S}}$. Of interest is then the distance of the points
$x^{i}$ to equilibrium states, that is,

$\max_{i=1,\ldots,N}\inf_{s\in\widehat{\mathbf{S}}}\|x-x^{i}\|.$

In our computational method below, we require that for some small $\eta>0$ and
for each $i$ and $z$, there is a $s=(x,z)\in{\mathbf{S}}$ such that

$\|x-x^{i}_{z}\|\leq\eta.$

As first pointed out by Hellwig (1983) (see also Kubler and Schmedders
(2002)), recursive equilibria do not always exist. The fact that typically the
temporary equilibrium correspondence cannot be shown to be single-valued
implies that one might not be able to always find a selection that is
consistent with the last period’s expectations. While recursive equilibria may
fail to exist, a USJE exists under mild (and standard) assumptions on
fundamentals (similar to the assumptions in Kubler and Schmedders (2003)). We
give a general existence proof here and, in Section 3, provide detailed
assumptions on economic fundamentals that guarantee existence in the concrete
case of a stochastic OLG model.

###  2.2 Assumptions

The following assumption describes the set of admissible approximations to the
next period’s policy. In the framework of a USJE, these are fundamentals of
the economy:

###### Assumption 1

For each $k=1,\ldots,Y$, there are $D$ continuous basis functions

$\psi_{dk}:{\mathbf{X}}\rightarrow\mathbb{R},d=1,\ldots,D,$

such that

${\mathbf{P}}_{k}=\\{f:\
f(\cdot)=\sum_{d=1}^{D}\alpha_{d}\psi_{dk}(\cdot),\quad\alpha\in\mathbb{R}^{D}\\}.$

In assumption 1, the number of basis functions is assumed to be the same for
each ${\mathbf{P}}_{k}$, $k=1,\ldots,Y$. It is conceptually easy to allow for
different approximations for different policies, that is to say, to have $D$
differ across policies $k=1,\ldots,Y$. This extension might be interesting for
applications where part of the agents’ heterogeneity stems from their ability
to forecast. However, since the notation becomes cumbersome, we omit this
extension here. Nevertheless, observe that the basis functions depend on $k$
and can vary with $k$.

Note that, depending on the approximation, it is not guaranteed that every
$f\in{\mathbf{P}}$ maps into the set ${\mathbf{Y}}$ as required by the
definition of a policy function (cf. definition 1). As we will see below, this
turns out not to be problematic in practice. For now, we can think of the
expectations correspondence projecting $P(x,z)$ into ${\mathbf{Y}}$ when
necessary. We will give an example for this construction in Section 3 below.

Assumption 1 allows us to write the temporary equilibrium correspondence,
${\mathbf{N}}_{P}(\cdot)$, as a map from $(x,z)\in{\mathbf{S}}$ and
$\vec{\alpha}\in\mathbb{R}^{DYZ}$, where each $\vec{\alpha}_{k,z}$ denotes the
vector of $D$ coefficients of the policy $P_{k}(\cdot,z)$. We then omit the
subscript $P$, since the policy is determined by the coefficients
$\vec{\alpha}$. As we will show in our OLG example in Section 3 below, the
equilibrium correspondence will be upper-hemi-continuous and non-empty value
in these coefficients. Assumption 2 states this requirement for our abstract
setting:

###### Assumption 2

  1. 1.

Sets ${\mathbf{X}}$ and ${\mathbf{Y}}$ are compact and, the equilibrium
correspondence ${\mathbf{N}}(.)$ has a closed graph as a correspondence in
$s\in{\mathbf{S}}$ and in coefficients $\vec{\alpha}\in\mathbb{R}^{DYZ}$.

  2. 2.

There is a compact ${\mathbf{A}}\subset\mathbb{R}^{DYZ}$ such that, given any
finite $\widehat{\mathbf{X}}\subset{\mathbf{X}}$, there exist policies,
$P_{k}=P_{k}(\cdot,\vec{\alpha})$, $k=1,\ldots Y$, and
$y(x,z)\in{\mathbf{N}}_{P}(x,z)$ for each $x\in\widehat{\mathbf{X}}$ and each
$z\in{\mathbf{Z}}$ such that

$P_{k}(.,z)\in\arg\min_{f\in{\mathbf{P}}_{k}}\sup_{x\in\widehat{\mathbf{X}}}\|y_{k}(x,z)-f(x,z)\|,\quad
k=1,\ldots,Y,$

and such that $\vec{\alpha}\in{\mathbf{A}}$.

This assumption is very abstract—in the OLG model below (cf. Section 3), we
show that it can be derived from fundamentals.

###  2.3 Existence

The main result of this section reads as follows:

###### Theorem 1

Under Assumptions 1 and 2, there exists a uniformly self-justified
equilibrium.

The proof consists of two parts. As a first step, we find a convenient finite-
dimensional representation of self-justified equilibria. In the second step of
the proof, we consider a sequence of points that becomes dense in the true
state space and show that (for a convergent subsequence) the associated
equilibria converge to a USJE.

####  2.3.1 Best uniform approximations

The key to establishing existence lies in the following simple facts about the
best uniform approximation. Let $\\{(x^{1},y^{1}),\ldots,(x^{N},y^{N})\\}$ be
a finite set of $N$ points in $\mathbb{R}^{n}\times\mathbb{R}$. Let
$\psi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{D}$, where the $\psi_{d}(\cdot)$,
$d=1,\ldots,D$, denote basis functions as in Assumption 1 above. For $N>D$,
consider the problem:

$\min_{\alpha\in\mathbb{R}^{D}}\sup_{i=1,\ldots,N}|y^{i}-\sum_{d=1}^{D}\alpha_{d}\psi_{d}(x^{i})|.$
(1)

As Sukhorukova et al. (2018) point out, expression (1) is a convex
optimization problem, and its solution can be characterized by the requirement
that zero lies in the subgradient of a function $g(\cdot)$ that maps
coefficients $\alpha\in\mathbb{R}^{D}$ to positive numbers, that is,

$g(\alpha)=\sup_{i=1,\ldots,N}|y^{i}-\sum_{d=1}^{D}\alpha_{d}\psi_{d}(x^{i})|,$

at the optimal solution of the minimization problem. By Caratheodory’s theorem
(Sukhorukova et al. (2018)), there must be $D+1$ points in the subgradient
that span zero, that is to say, there exist $D+1$ points
$(\hat{x}^{1},\hat{y}^{1}),\ldots,(\hat{x}^{D+1},\hat{y}^{D+1})\in\\{(x^{1},y^{1}),\ldots,(x^{N},y^{N})\\}$
such that for all $j=1,\ldots,D+1$,

$\sup_{i=1,\ldots,N}|y^{i}-\sum_{d}\alpha_{d}\psi_{d}(x^{i})|=|\hat{y}^{j}-\sum_{d}\alpha_{d}\psi_{d}(\hat{x}^{j})|,$
(2)

and such that

$0\in\mbox{ convex
hull}\cup_{j=1}^{D+1}\\{\mbox{sgn}(\hat{y}^{j}-\sum_{d}\alpha_{d}\psi_{d}(\hat{x}^{j}))\cdot\psi(\hat{x}^{j})\\}.$
(3)

Note that this condition is necessary and sufficient for
$\sum_{d}\alpha_{d}\psi_{d}(\cdot)$ to be the best uniform approximation for
$(x^{i},y^{i})_{i=1}^{N}$. This is true even if $x^{i}=x^{j}$ for some $i,j$
with different associated values $y^{i}$, and $y^{j}$. In fact, in this case,
zero might be the combination of only two points in the subgradient, namely if
$y^{i}$ and $y^{j}$ are extreme points on opposite sides of the approximating
function.

Therefore, in the context of our abstract economic setting, we can state the
following lemma:

###### Lemma 1

Given any finite $\widehat{\mathbf{X}}\subset{\mathbf{X}}$, the following two
statements are equivalent:

  1. 1.

There are policies, $P$, and a selection of the equilibrium correspondence,
$y(x,z)\in{\mathbf{N}}_{P}(x,z)$ for each $x\in\widehat{\mathbf{X}}$ and each
$z\in{\mathbf{Z}}$ such that for each $k=1,\ldots,Y$,

$P_{k}(.,z)\in\arg\min_{f\in{\mathbf{P}_{k}}}\sup_{x\in\widehat{\mathbf{X}}}\|y_{k}(x,z)-f(x,z)\|$

  2. 2.

For each $z\in{\mathbf{Z}}$ and each $k=1,\ldots,Y$, there are finite sets
${\mathbf{X}}^{F}_{kz}$ consisting of $D+1$ points
$\hat{x}\in\widehat{\mathbf{X}}$, such that

$\sup_{x\in\widehat{\mathbf{X}}}|P_{k}(x,z)-y_{k}(x,z)|=|P_{k}(\hat{x},z)-y_{k}(\hat{x},z)|,\mbox{
for all }\hat{x}\in{\mathbf{X}}^{F}_{kz}$

and

$0\in\mbox{ convex hull
}\cup_{\hat{x}\in{\mathbf{X}}^{F}_{kz}}\\{\mbox{sgn}(P_{k}(\hat{x},z)-y_{k}(\hat{x},z))\cdot\psi(\hat{x})\\}.$

This result is remarkable for two reasons. First, the characterization of the
equilibrium is independent of the number of points in the finite set
$\widehat{\mathbf{X}}$, and second, because the condition is necessary and
sufficient.

####  2.3.2 Proof of the Theorem 1

We consider a nested sequence of finite sets of points
$(\widehat{\mathbf{X}}_{i})_{i=1}^{\infty}$, where $\widehat{\mathbf{X}}_{i}$
becomes dense in ${\mathbf{X}}$ as $i\rightarrow\infty$. As per Assumption 2
part 2, for every $i$ and $\widehat{\mathbf{X}}_{i}$, there exists a $P^{i}$
and $y(x,z)\in{\mathbf{N}}_{P^{i}}(x,z)$ for every
$x\in\widehat{\mathbf{X}}_{i}$, $z\in{\mathbf{Z}}$ that satisfies for each
$k=1,\ldots,Y,$

$P^{i}_{k}(\cdot,z)\in\arg\min_{f\in{\mathbf{P}}_{k}}\sup_{x\in\widehat{\mathbf{X}}_{i}}|y^{i}_{k}(x,z)-f(x,z)|.$

By Lemma 1, each $P^{i}_{kz}$, $k=1,\ldots,Y,$ $z\in{\mathbf{Z}}$, can be
characterized by $(D+1)$ points, $x^{*i}_{k,z,l}\in\widehat{\mathbf{X}}_{i}$,
$l=1,\ldots,D+1$, coefficients $\alpha_{k,z}\in{\mathbf{A}}$ as well as a
selection of the equilibrium correspondence
$y^{*i}(x^{*i}_{k,z,l},z)\in{\mathbf{Y}}$. Assumption 2 guarantees that
${\mathbf{X}},{\mathbf{Y}}$, and ${\mathbf{A}}$ are all compact sets.
Therefore, as $i\rightarrow\infty$, there must be a convergent sub-sequence
converging to some $(\alpha^{*}_{k,z})_{k=1,\ldots,Y,z\in{\mathbf{Z}}}$, and
some $(x^{*}_{k,z,l})_{k=1,\ldots,Y,z\in{\mathbf{Z}},l=1,\ldots,D+1}$. The
coefficients, $\alpha^{*}$, define forecasting functions $P^{*}$. By upper-
hemi continuity (Assumption 2, part 1), for all $z\in{\mathbf{Z}},\
k=1,\ldots,Y$, if for every $i$,

$0\in\mbox{ convex hull
}\cup_{\hat{x}\in\\{x_{k,z,l}^{*i},l=1,\ldots,D+1\\}}\\{\mbox{sgn}(P^{i}_{k}(\hat{x},z)-y_{k}(\hat{x},z))\cdot\psi(\hat{x})\\},$

then this must also hold in the limit, that is,

$0\in\mbox{ convex hull
}\cup_{\hat{x}\in\\{x_{k,z,l}^{*},l=1,\ldots,D+1\\}}\\{\mbox{sgn}((P^{*}_{k}(\hat{x},z)-y^{*}_{k}(\hat{x},z))\cdot\psi(\hat{x})\\},$

for some $y^{*}(\hat{x},z)\in{\mathbf{N}}_{P^{*}}(\hat{x},z)$. Therefore,
$P^{*}$ satisfies part 1 of the definition of a USJE. For any (other)
$x\in{\mathbf{X}}$, since $\widehat{\mathbf{X}}_{i}$ is becoming dense in
${\mathbf{X}}$, there must be a sequence $x^{i}\rightarrow x$ with some
$y^{i}\in{\mathbf{N}}_{P^{i}}(x^{i},z)$ that satisfies

$|y_{k}^{i}-P_{k}^{i}(x^{i},z)|\leq\sup_{x\in\widehat{\mathbf{X}}_{i}}|y_{k}^{i}(x,z)-P_{k}^{i}(x,z)|.$

By upper-hemi continuity, there must be a $y\in{\mathbf{N}}_{P^{*}}(x,z)$ such
that

$|y_{k}^{i}-P_{k}(x^{i},z)|\leq\sup_{l=1,\ldots,D+1}|y^{i}_{k}(x^{*}_{k,z,l},z)-P^{*}(x^{*}_{k,z,l},z)|.$

This completes the proof of Theorem 1.

###  2.4 A computational method to approximate an USJE

We now introduce two algorithms that will demonstrate how to compute a UJSE
numerically. The first algorithm computes a solution to a discretized problem
(whose existence was assumed in Assumption 2 part 2). The second algorithm
finds extreme points in the sense of the first part of the definition of a
USJE.

To approximate a USJE numerically, an important obstacle seems to lie in the
fact that finding a uniform approximation to a continuous function is a
difficult task. While in the univariate case, the so-called “Remez algorithm”
(Remes, 1934; Pachón and Trefethen, 2009) provides an efficient tool for
computing the best approximation, matters become complicated in higher
dimensions. As Reemtsen (1990) points out, one of the main challenges, in this
case, is to find global optima of the error function.

In our setting, the definition of a USJE implies that this is not necessary
for obtaining a best uniform approximation since we do not require $P$ to be a
best approximation on ${\mathbf{S}}$. We merely need to find points in
${\mathbf{S}}$ together with endogenous variables, for which $P(\cdot)$ is a
best uniform approximation, and we need to ensure that along any simulated
path, the maximum approximation error remains smaller than the error from this
approximation.

A crucial step of the method is to obtain a best uniform approximation for the
discretized problem (see, e.g., Reemtsen (1990)). As Stiefel (1960) points
out, a best uniform approximation to a finite number of points can be obtained
by solving a simple linear programming problem. The problem 1 can be written
as

$\displaystyle\min_{\alpha\in\mathbb{R}^{D},w\geq 0}w\quad\mbox{s.t.}$ (4)
$\displaystyle y_{i}-\sum_{d=1}^{D}\alpha_{d}\psi_{d}(x^{i})\leq w\
i=1,\ldots,N$ $\displaystyle-y_{i}+\sum_{d=1}^{D}\alpha_{d}\psi_{d}(x^{i})\leq
w\ i=1,\ldots,N\quad,$

which can be solved efficiently with standard methods (Schrijver, 1998), even
when the number of points, $N$, becomes large.

Algorithm 1 Time iteration for the discretized problem

  1. 0.

The input consists of a set of states $(s_{t})_{t=1}^{n}$, current policy
function $P:{\mathbf{S}}\rightarrow{\mathbf{Y}}$, and tolerance
$\bar{\epsilon}>0$.

  2. 1.

Solve the equilibrium conditions at $n$ points: For all $t=0,\ldots,n$, solve
the equilibrium conditions at $s_{t}=(x,z)$,

$\left(T_{z^{\prime}}(x,y,z),P(T_{z^{\prime}}(x,y,z),z^{\prime})\right)_{z^{\prime}\in{\mathbf{Z}}}\in{\mathbf{H}}(x,y,z),$
(5)

for $y\in{\mathbf{Y}}$, and set $\hat{x}(z^{t})=x$, $\hat{y}(z^{t})=y$.

  3. 2.

Update the policy function:
Define new policies $\tilde{P}_{k}(\cdot,z)$, $k=1,\ldots Y$,
$z\in{\mathbf{Z}}$ by solving the linear program (4) for
$\left\\{\hat{x}(z^{t}),\hat{y}_{k}(z^{t})\right\\}_{z_{t}=z}$.

  4. 3.

Check stopping criterion:
Compute the error: $\epsilon=\|\tilde{P}-P\|_{\infty}/\|P\|_{\infty}$. Set
$P=\tilde{P}$. If $\epsilon<\bar{\epsilon}$, stop and report $P$; else go to
step 1.

Algorithm 1 solves for a best uniform approximation of the policy in a USJE
that is defined on a finite set of points. It corresponds to a standard time
iteration algorithm as used, for example, in collocation methods (see, for
example, Krueger and Kubler (2004)).

As the next step, we use in Algorithm 2 the ideas from the exchange algorithms
of Remes (1934) and Reemtsen (1990) and, along several simulated paths of
fixed length, determine points where the maximum forecasting error exceeds the
previously computed maximum. If this is the case, the point that yields the
lowest forecasting error is exchanged with the new point.

Algorithm 2 USJE exchange algorithm

  1. 0.

Draw initial points, $(s^{t})_{t=0,\ldots,n}$, $s^{t}\in{\mathbf{S}}$ for all
$t$, and use Algorithm 1 to obtain initial $P(\cdot)$. For all $k=1,\ldots Y$,
$z\in{\mathbf{Z}}$, $t=0,\ldots,n$, define initial $\hat{x}^{t}_{kz}=x^{t}.$
Initialize $\mbox{maxerr}_{kz}=0$ and $\mbox{err}_{kz}^{t}=0$ for all $t$.

  2. 1.

Draw a sequence of $m\gg n$ random shocks $(z_{t})_{t=1}^{m}$.

  3. 2.

Solve equilibrium conditions along a simulated path. For all $t=0,\ldots,m$

    1. (a)

Solve equilibrium equations (5) at $x=x(z^{t}),z=z_{t}$ for $y$ and set
$x(z^{t+1})=T_{z_{t+1}}(x,y,z)$.

    2. (b)

Update the points:
For all $k=1,\ldots,Y$, if $|y_{k}-P_{k}(x,z)|\geq\mbox{maxerr}_{kz}$ search
for a point in $\tilde{x}$ with $\|x-\tilde{x}\|\leq\eta$ and a solution of
the equilibrium equations (5), $\tilde{y}$, that satisfy

$|\tilde{y}_{k}-P_{k}(\tilde{x},z)|>|y_{k}-P_{k}(x,z)|.$

Search for $i$ that solves

$\min_{i=1,\ldots,n}\mbox{err}_{kz}^{i},$

replace $\hat{x}^{i}_{kz}$ by $\tilde{x}$ and set

$\mbox{err}_{kz}^{i}=|\tilde{y}_{k}-P_{k}(\tilde{x},z)|.$

  4. 3.

Find new policy by applying Algorithm 1 on new points $\hat{x}^{t}_{kz}$,
$k=1,\ldots,Y$, $z\in{\mathbf{Z}}$, $t=0,\ldots,n$.

  5. 4.

Check stopping criterion:
If no points where exchanged in step 2.(b) stop and report $P$; else go to
step 1.

Step 2.(b) of the algorithm contains the basic idea. If, along a simulated
path, the previous maximum forecasting error is exceeded, we find a new point
with a new maximum error. In an idealized setup, this point would maximize the
error within some small neighborhood of $x$, but this is obviously
computationally quite expensive. It is not important to find a (local)
maximum, but rather to find a point where the error is larger than at $x$. In
the practical application below, we find this point by taking one gradient
ascent step of pre-described length with possible back-tracking.

While a USJE always exists, there is no general proof that Algorithm 2
converges. However, in our numerical examples below, this is always the case.
Note that in the computational method, we neglect the problems that arise with
the possibility of multiple solutions to expression (5).

We propose a simple error analysis to evaluate the candidate solution of the
above algorithm. Property (1) in definition 1 is easy to verify. Due to finite
precision arithmetics, this property will not hold exactly, but the error is
typically very small, that is, below $10^{-10}$ in the numerical examples that
follow in Section 3. Property (2) in the definition is a condition on an
infinite number of points and thus cannot be verified numerically. We simulate
the economy for a very large number of periods (in the examples below for
$m=100\cdot n$ periods) and check the condition along the simulated path. If
it is violated, one needs to run Algorithm 2 with larger $m$ or possibly
larger $\eta$ and a better method to find points that imply a large error.

##  3 An OLG economy

To illustrate our general results derived in Section 2 in a specific
application, we consider a pure exchange economy with overlapping generations
of agents. We show that in this setting, Assumption 2 follows from standard
assumptions on economic fundamentals. Furthermore, we also illustrate our
computational method with a concrete numerical example.

At each date event, a continuum of identical agents enter the economy and live
for $A$ periods. We denote the set of all date events at time $t$ by
${\mathbf{Z}}^{t}$ and, taking $z_{0}$ as fixed, we write
$z^{t}\in{\mathbf{Z}}^{t}$ for any $t\in\mathbb{N}_{0}$ (including $t=0$). At
each $z^{t}$, there are finitely many different agents actively trading
(distinguishing themselves by the age and the history of shocks), who are
collected in a set ${\mathbf{I}}=\cup_{a=1}^{A}$. A specific agent at a given
node $z^{t}$ is denoted by $a\in{\mathbf{I}}$.

At each date event, there is a single perishable commodity, the individual
endowments are denoted by $e_{a}(z^{t})\in\mathbb{R}_{+}$, and are assumed to
be time-invariant and measurable functions of the current aggregate shock.
Each agent who can be identified by his date event of birth, $z^{t}$, has a
time-separable expected utility function, that is,

$U_{z^{t}}\left((c_{t+a})_{a=0}^{A-1}\right)=\mathbb{E}_{t}\left[\sum_{a=1}^{A}u_{a}\left(c_{z^{t},t+a-1}\right)\right],$

where $c_{z^{t},t+a-1}\in\mathbb{R}_{+}$ denotes the agent’s (stochastic)
consumption at date $t+a-1$.

There are $J$ assets, $j\in{\mathbf{J}}=\\{1,\ldots,J\\}$, traded at each date
event. The assets can be infinitely lived Lucas trees in unit net supply, or
one-period financial assets in zero net supply. The net supply of an asset $j$
is denoted by $\bar{\theta}_{j}\in\\{0,1\\}$. Each asset $j$ is traded at a
price $q_{j}$, and its (non-negative) payoffs depends on the aggregate shock
and possibly on the current prices of the assets, that is,
$f_{j}:\mathbb{R}^{J}_{+}\times{\mathbf{Z}}\rightarrow\mathbb{R}_{+}$. If
asset $j$ is a Lucas tree (i.e., an asset in positive net supply), then
$f_{j}(q,z)=q_{j}+d_{j}(z)$ for some dividends
$d_{j}:{\mathbf{Z}}\rightarrow\mathbb{R}_{+}$. Asset $j$ could also be a
collateralized loan whose payoff depends on the value of the underlying
collateral, or an option, or simply a risk-free asset. The aggregate dividends
of the trees are defined as $d(z)=\bar{\theta}\cdot f(q,z)-\bar{\theta}\cdot
q$. The aggregate consumption is then $e(z)=d(z)+\sum_{a=1}^{A}e_{a}(z)$. An
agent $a$ faces trading constraints
$\theta\in{\mathbf{\Theta}}_{a}\subset\mathbb{R}^{J}$, where
${\mathbf{\Theta}}_{A}=\\{0\\}$. To simplify the notation, we write
$\vec{\theta}=(\theta_{a})_{a\in{\mathbf{I}}}$,
$\vec{\theta}^{-}=(\theta^{-}_{a})_{a\in{\mathbf{I}}}$ and
$\vec{c}=(c_{a})_{a\in{\mathbf{I}}}$.

It is useful to define the set of possible portfolio holdings with market-
clearing built-in as follows:

${\mathbf{\Theta}}=\\{\vec{\theta}:\sum_{a\in{\mathbf{I}}}\theta_{a}=\bar{\theta},\quad\theta_{a}\in{\mathbf{\Theta}}_{a}\mbox{
for all }a-1\in{\mathbf{I}}\\}.$

Similarly let the set of all beginning-of-period portfolio holdings be

${\mathbf{\Theta}}^{-}=\\{\vec{\theta}^{-}:\theta^{-}_{1}=0,\sum_{a-1\in{\mathbf{I}}}\theta^{-}_{a}=\bar{\theta}\mbox{
and }\theta^{-}_{a}\in{\mathbf{\Theta}}_{a-1}\mbox{ for all }a\\}.$

###  3.1 Uniformly self-justified equilibrium

It is straightforward to embed this OLG model into our abstract framework. The
endogenous state is $x=\vec{\theta}^{-}\in{\mathbf{\Theta}^{-}}={\mathbf{X}}$,
current endogenous variables are given by $y=(q,\vec{c},\vec{\theta})$. The
expectations correspondence can be defined as follows:

$(x_{1},y_{1}\ldots x_{Z},y_{Z})\in{\mathbf{H}(\hat{x},\hat{y},\hat{z})}\mbox{
if and only if }$

each $(c_{a},\theta_{a})$, $a=1,\ldots,A$ solves the following problem:

$\displaystyle\max_{\theta\in{\mathbf{\Theta}_{a}},c\geq
0}u_{a}(c)+\beta\sum_{z^{\prime}\in{\mathbf{Z}}}\pi(z^{\prime}|z)u_{a+1}^{\prime}(e_{a+1}(z^{\prime})+\theta\cdot
f(q_{z^{\prime}},z^{\prime})-q_{z^{\prime}}\theta_{z^{\prime}})f(q_{z^{\prime}},z^{\prime})\cdot\theta$
(6) $\displaystyle\mbox{s.t. }c=e_{a}(z)+\theta^{-}_{a}\cdot
f(q,z)-q\cdot\theta\quad.$

Since agents solve finite-dimensional convex optimization problems, the first-
order conditions are necessary and sufficient, and the expectations
correspondence characterizes competitive equilibria.

The transition function is given by
$T_{z^{\prime}}(x,y,z)=(\theta_{1},\ldots\theta_{A-1})$, and a policy function
maps ${\mathbf{\Theta}}^{-}$ to current prices, consumption-levels, and new
asset holdings across agents. A self-justified equilibrium is defined as above
(definition 1). The agents’ expectations about the next period’s asset prices
and optimal choices are not necessarily correct, but the functions chosen are
the best uniform approximation to actual equilibrium values. For the
definition, agents do not need to forecast portfolio choices in the subsequent
period. It suffices to have forecasts for asset payoffs $f(q,z)$ as well as
for individual consumptions, $c$. Since we impose an Inada condition on
utility, we want to ensure that forecasted consumption is bounded away from
zero. Therefore, our definition of the equilibrium correspondence in the
context of the OLG model becomes

$\displaystyle{\mathbf{N}}_{P}(\vec{\theta}^{-},z)=\\{(q,\vec{c},\vec{\theta}):\mbox{For
all }a=1,\ldots,A-1$
$\displaystyle(c_{a},\theta_{a})\in\arg\max_{\theta\in{\mathbf{\Theta}_{a}},c\geq
0}u_{a}(c)+\beta\sum_{z^{\prime}\in{\mathbf{Z}}}\pi(z^{\prime}|z)u_{a+1}^{\prime}(\hat{P}_{c_{a+1}}(T(\vec{\theta}),z^{\prime}))f(P_{q}(T(\vec{\theta}),z^{\prime}),z^{\prime})\cdot\theta$
$\displaystyle\mbox{s.t. }c=e_{a}(z)+\theta^{-}_{a}\cdot
f(q,z)-q\cdot\theta\\},$

where

$\hat{P}_{c_{a}}(\vec{\theta}^{-},z)=\left\\{\begin{array}[]{ll}\epsilon&\mbox{
if }P_{c_{a}}(\vec{\theta}^{-},z)\leq\epsilon\\\
P_{c_{a}}(\vec{\theta}^{-},z)&\mbox{ if
}e(z)>P_{c_{a}}(\vec{\theta}^{-},z)>\epsilon\\\ e(z)&\mbox{ if
}P_{c_{a}}(\vec{\theta}^{-},z)\geq e(z).\end{array}\right.$ (7)

In this setting, agents forecast the next period’s optimal consumption and
prices. This turns out to be computationally advantageous since consumption
functions often do not exhibit strong nonlinearities. Alternatively, we could
assume that agents forecast marginal utilities.

Next, we show that the general existence proof for Theorem 1 readily gives
sufficient conditions for existence in this model. Moreover, we illustrate our
algorithm in the context of a simple numerical example.

###  3.2 Existence in OLG exchange economies

In order to establish existence of a USJE, we need assumptions on fundamentals
that guarantee the high-level Assumption 2 above holds:

###### Assumption 3

  1. 1.

For each $a\in{\mathbf{I}}$, the Bernoulli-utility function $u_{a}(\cdot)$ is
continuously differentiable, strictly increasing, strictly concave, and
satisfies an Inada condition,

$u_{a}^{\prime}(x)\rightarrow\infty\mbox{ as }x\rightarrow 0,$

and individual endowments are positive, that is,

$e_{a}(z)>0\mbox{ for all }z\in{\mathbf{Z}}.$

  2. 2.

The set ${\mathbf{\Theta}}$ is compact, and for each $a\in{\mathbf{I}}$, the
set ${\mathbf{\Theta}}_{a}$ is a closed convex cone containing
$\mathbb{R}^{J}_{+}$.

  3. 3.

The payoff functions,
$f:\mathbb{R}^{J}_{+}\times{\mathbf{Z}}\rightarrow\mathbb{R}^{J}$, are non-
negative valued and continuous. Moreover, for any $i,j=1,\ldots,J$, the payoff
$f_{j}(q)$ only depends on $q_{i}$ if $\bar{\theta}_{i}>0$.

  4. 4.

For all $\theta^{-}\in\Theta_{a-1}$,

$\theta^{-}_{a}\cdot f(q,z)\geq 0\mbox{ for all
}q\in\mathbb{R}^{J}_{+},z\in{\mathbf{Z}}.$

  5. 5.

For all $k=1,\ldots,Y$, the basis functions, $\psi_{k1},\ldots,\psi_{kD}$, are
linearly independent.

Assumptions 3.1-3.3 are standard. Assumption 3.4 is motivated by collateral
and default. Constraints ensure that agents cannot borrow against future
endowments. In our formulation, this is true independently of prices—we
implicitly allow for default (see, e.g., Kubler and Schmedders (2003)).

With these assumptions, the existence of a uniformly self-justified
equilibrium simply reduces to the existence of a finite-dimensional fixed
point, in particular, the main result of this section is as follows:

###### Lemma 2

Given any finite set
$\widehat{\mathbf{\Theta}}^{-}\subset{\mathbf{\Theta}}^{-}$, there exist
$P_{k}\in{\mathbf{P}}_{k}$, $k=1,\ldots,Y$, and $y(\theta^{-},z)$ for all
$z\in{\mathbf{Z}}$, $\theta^{-}\in\widehat{\mathbf{\Theta}}^{-}$ that solve
Equations (6) such that

$P_{k}\in\arg\min_{f\in{\mathbf{P}}_{k}}\sup_{\theta^{-}\in\widehat{\mathbf{\Theta}}^{-}}\|y(\theta^{-},z)-f(x)\|.$

Proof:

Suppose that $\widehat{\mathbf{\Theta}}^{-}$ contains $G$ elements, and let
$\widehat{\mathbf{S}}={\mathbf{Z}}\times\widehat{\mathbf{\Theta}}^{-}$ and
take prices at each $s\in\widehat{\mathbf{S}}$ to lie in the trimmed simplex
$(p,q)\in\Delta_{\epsilon}^{J}=\\{(p,q)\in\mathbb{R}^{J+1}_{+},p+\sum_{j=1}^{J}q_{j}=1,p\geq\epsilon,q_{j}\geq\epsilon,j=1,\ldots,J\\}$.
We decompose the economy into sub-economies for each
$s\in\widehat{\mathbf{S}}$ and construct a map from a compact and convex set
of all agents’ choices, prices, and forecasts, $P$, into itself. We show that
this map is upper hemi-continuous and convex valued, and using Kakutani’s
theorem (see Border (1985)), we can show that this map has a fixed point. As
$\epsilon$ becomes sufficiently small, one can prove market-clearing.

First, we need to find a suitable, convex, and compact domain for the map.
Assumption 1 guarantees the existence of a compact set
${\mathbf{T}}\subset\mathbb{R}_{+}\times\mathbb{R}^{J}$, with

$\\{(c,\theta)\in\mathbb{R}_{+}\times{\mathbf{\Theta}}_{h}:(c-e_{h}(z))+\theta\cdot\frac{1}{p}q-\theta^{-}_{h}\cdot
f(\frac{1}{p}q,z)\leq 0\\}\subset{\mathbf{T}},$

for all $(p,q)\in\Delta^{J}_{\epsilon}$ and all $h\in{\mathbf{H}}.$

We construct an upper hemi-continuous, non-empty and convex-valued
correspondence, ${\mathbf{\Phi}}$, mapping choices, prices and forecasts at
each element in $\widehat{\mathbf{S}}$ to itself, which has a fixed point,
that is,

${\mathbf{\Phi}}:{\mathbf{T}}^{ZG(A-1)}\times(\Delta^{J}_{\epsilon})^{ZG}\times\mathbb{R}^{ZG(A-1)}_{+}\rightrightarrows{\mathbf{T}}^{ZG(A-1)}\times(\Delta^{J}_{\epsilon})^{ZG}\times\mathbb{R}^{ZG(A-1)}.$
(8)

For this construction, for all $a\in{\mathbf{I}}$ and all
$s=(z,\vec{\theta}^{-})\in\widehat{\mathbf{S}}$, agents take prices and
forecasts as given, and their best response correspondence is given by

$\displaystyle{\mathbf{\Phi}}_{h,s}((x_{t},p_{t},q_{t})_{t\in\widehat{\mathbf{S}}})=$
(9)
$\displaystyle\arg\max_{(c,\theta)\in{\mathbf{T}}}u_{h}(x)+\beta_{h}\sum_{z^{\prime}\in{\mathbf{Z}}}\pi(z^{\prime}|z)u_{a+1}^{\prime}(\hat{P}_{c_{a+1}}(T(\vec{\theta}),z^{\prime}))f(P_{q}(T(\vec{\theta}),z^{\prime}),z^{\prime})\cdot\theta$
(10) s.t.
$\displaystyle\quad\quad(c-e_{h}(z))+\theta\cdot\frac{1}{p_{s}}q_{s}-\theta^{-}_{h}\cdot
f(\frac{1}{p_{s}}q_{s},z)\leq 0,$

with $\hat{P}$ being defined as above in Equation (7). The forecasts are
formed by an “artificial” agent who takes choices and prices
$y(z,\vec{\theta})$, $z\in{\mathbf{Z}}$,
$\vec{\theta}\in\widetilde{\mathbf{\Theta}}$ as given, and solves for all
$k=1,\ldots,Y$,

$P_{k}(z,\cdot)=\arg\min_{f\in{\mathbf{P}}}\sup_{\vec{\theta}\in\widehat{\mathbf{\Theta}}}|P(\vec{\theta})-y(z,\vec{\theta})|.$
(11)

By Assumption 3, we can, without loss of generality, take for each agent
$a\in{\mathbf{I}}$, all $z\in{\mathbf{Z}}$ and all $j\in{\mathbf{J}}$
$\alpha\in{\mathbf{A}}$, where ${\mathbf{A}}$ is a compact and convex set.
Therefore, the best uniform approximation always exists, that is, equation
(11) always has a solution. Moreover, since ${\mathbf{A}}$ is convex, and we
minimize a convex function, the set of solutions is convex. Note also that by
the maximum principle, the set of solutions is upper hemi-continuous in the
$ZG$ points $(c,q)(s)$, $s\in\widehat{\mathbf{S}}$. Therefore the map of
forecasts,

${\mathbf{\Phi}}_{H+1}:\mathbb{R}^{ZGHJ}\rightrightarrows\mathbb{R}^{ZGHJ},$
(12)

is upper hemi-continuous, convex valued, and non-empty valued.

For each $s\in\widehat{\mathbf{S}}$, define the price-player’s best response
as follows:

${\mathbf{\Phi}}_{0,s}(\vec{\theta}_{s},\vec{x}_{s})=\arg\max_{(p,q)\in{\mathbf{\Delta}}_{\epsilon}^{J}}p(\sum_{a\in{\mathbf{I}}}(c_{a,s}-e_{a}(z)-d(z)))+q\cdot(\sum_{a\in{\mathbf{I}}}(\theta_{a,s}-\bar{\theta})).$
(13)

Assumptions 1 and 3 therefore, guarantee that the mapping

${\mathbf{\Phi}}=\times_{s\in{\mathbf{S}},a\in{\mathbf{I}}}{\mathbf{\Phi}}_{a,s}\times_{s\in{\mathbf{S}}}{\mathbf{\Phi}}_{0,s}\times{\mathbf{\Phi}}_{H+1}$
(14)

is non-empty, convex valued, and upper hemi-continuous. By Kakutani’s fixed
point theorem, there exists a fixed point with prices
$(\bar{p}_{s},\bar{q}_{s})_{s\in\widehat{\mathbf{S}}}$.

As $\epsilon\rightarrow 0$, Assumption 3 guarantees that there will be a
strictly positive $\epsilon$ such that for all $s\in\widehat{\mathbf{S}}$,
that is,

$(\bar{p}_{s},\bar{q}_{s})\in\arg\max_{(p,q)\in{\mathbf{\Delta}}_{0}^{J}}p(\sum_{a\in{\mathbf{I}}}(c_{a,s}-e_{a}(z)-d(z)))+q\cdot(\sum_{a\in{\mathbf{I}}}(\theta_{a,s}-\bar{\theta})).$
(15)

By a standard argument, markets clear. This finishes our proof. $\Box$

Note that above’s proof implies that there exist convex and compact sets
${\mathbf{A}}$ for each $h\in{\mathbf{H}}$ such that for any finite
equilibrium and any discretization $\widehat{\mathbf{\Theta}}$, all
forecasting coefficients for agent $h$ lie in this set. This follows from the
assumption that ${\mathbf{\Theta}}$ is compact.

As a final lemma, we need to establish the following closed graph
characterization of the equilibrium correspondence. The result follows
directly from the definition of the temporary equilibrium.

###### Lemma 3

Given a sequence of policy functions characterized by coefficients
$\vec{\alpha}$ converging to some $\vec{\alpha}^{*}$ as $i\rightarrow\infty$,
let $P^{i}$ and $P^{*}$ denote the associated policy functions. For any
$z\in{\mathbf{Z}}$, consider any sequence $(\vec{\theta}_{i}^{-},y_{i})$ with
$\vec{\theta}_{i}^{-}\in{\mathbf{\Theta}}^{-}$ and
$y_{i}\in{\mathbf{N}}_{P^{i}}(z,\vec{\theta}_{i}^{-})$ for all $i$, converging
to $(\vec{\theta}^{-*},y^{*})$. Then, the following holds:

$y^{*}\in{\mathbf{N}}_{P^{*}}(z,\vec{\theta}^{*-}).$

Now, our existence theorem (cf. theorem 1) can be applied to ensure the
existence of a USJE for the OLG economy. Note that the key result, Lemma 2,
can be proven with standard methods and that similar results can be
established in models with production, and models with infinitely lived
agents.

###  3.3 Computation in an OLG economy: numerical examples

To illustrate our numerical method proposed in Section 2.4, we consider a
simple example with $A=10$, and a single risk-free asset being traded each
period. Agents face short-sale constraints, that is,

$\theta_{a}\geq-\min_{z\in{\mathbf{Z}}}e_{a+1}(z).$

The purpose of the example is not to claim that the computational method is
competitive compared to other existing methods in terms of performance and
speed, but rather to illustrate that i) USJE can be computed in non-trivial
models and ii) to show that our concept of a USJE allows for straightforward
error analysis of the candidate solution. While the example is almost trivial
with respect to the asset structure, the endogenous state space is
8-dimensional555The old (a=10) do not trade, and in a pure exchange setting,
asset holdings of generation $A-1$ follow from market clearing. which makes it
non-trivial to solve for standard collocation methods (see, e.g., Krueger and
Kubler (2004); Brumm et al. (2021), and references therein). We choose the
stochastic process for individual endowments to ensure that the model exhibits
complex equilibrium dynamics.

Individual endowments have a deterministic part, where endowments initially
increase in age and then decrease,

$\bar{e}_{a}=\left\\{\begin{array}[]{ll}0.6+0.2a&4\geq a\geq 1\\\
1.4-0.2\cdot(a-5)&10\geq a\geq 5,\\\ \end{array}\right.$

and they are subject to three additive random variables,
$\sigma_{1},\sigma_{2},\sigma_{3}$, that are i.i.d., and take values $0.1$ and
$-0.1$ with probability $1/2$, such that

$e_{a}(z)=\bar{e}_{a}+f_{1a}\sigma_{1}(z)+f_{2a}\sigma_{2}(z)+f_{3a}\sigma_{3}(z).$

The presence of three independent shocks implies that ${\mathbf{Z}}$ has
$2^{3}=8$ elements. The values for the coefficients $f_{ja}$ are reported in
Table 1.

Age: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
$f_{1}$ | 1 | $\sqrt{2/3}$ | $\sqrt{1/3}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0
$f_{2}$ | 0 | $\sqrt{1/3}$ | $\sqrt{2/3}$ | 1 | $\sqrt{2/3}$ | $\sqrt{1/3}$ | 0 | 0 | 0 | 0
$f_{3}$ | 0 | 0 | 0 | 0 | $\sqrt{1/3}$ | $\sqrt{2/3}$ | 1 | 0.5 | 0.25 | 0
Table 1: Coefficients on the three shocks for all ages.

Bernoulli utilities are assumed to exhibit constant relative risk aversion,
that is,

$u_{a}(c)=\beta^{a-1}\frac{c^{1-\gamma}}{1-\gamma},$

with $\beta=0.75$ and $\gamma=3$.

In this relatively simple setting, it is sufficient to forecast the next
period’s consumption policy since the prices and new portfolio holdings in the
next period do not enter the equilibrium conditions. We consider three
different specifications for the set of admissible policies, ${\mathbf{P}}$.
First, we consider a very restricted set of basis functions where we postulate
that consumption only depends on the individual’s beginning of period asset
holding, that is,

${\mathbf{P}}^{1}_{c_{a}}=\\{1,\theta^{-}_{a},(\theta^{-}_{a})^{2},(\theta^{-}_{a})^{3}\\}.$

This, of course, neglects the effects of the wealth distribution on asset
prices and optimal choices. As an alternative, we therefore consider

${\mathbf{P}}^{2}_{c_{a}}=\\{1,\theta^{-}_{2},\ldots,\theta^{-}_{A-1},(\theta^{-}_{a})^{2},(\theta^{-}_{a})^{3}\\}.$

In this specification, optimal choices can depend non-linearly on an agent’s
own asset-holdings, but they also depend (linearly) on all other agents
holdings. It does not allow for non-zero cross derivatives. It may be useful
to consider a third specification of ${\mathbf{P}}$, which captures some
interaction terms. We consider therefore the following expression:

${\mathbf{P}}^{3}_{c_{a}}=\\{1,\theta^{-}_{2},\ldots,\theta^{-}_{A-1},\theta^{-}_{a}\theta^{-}_{2},\ldots,\theta^{-}_{a}\theta^{-}_{A-1},(\theta^{-}_{a})^{3}\\}.$

Given the specific economic model, we vary Algorithm 2 slightly. In step $0$
of the method, we solve the model along a simulated path to obtain initial
guesses for policy and points. As explained above, in Step 2.(b), we search
for a point with locally large errors by performing one gradient ascent step
with back-tracking line-search (see, e.g., Nocedal and Wright (2006)).

We choose $n=200$, $\eta=0.1$ and $m=50\cdot n$. In each of the three
specification of admissible forecasts, it took around 50 iterations until
convergence.

Tables 2, 3, and 4 report the maximum errors for our three specification of
admissible forecasting functions (in units of percent). For $a=9$, the
forecasting errors are zero because consumption is a linear function of asset
holdings for $a=10$.

Agent | State 1 | State 2 | State 3 | State 4 | State 5 | State 6 | State 7 | State 8
---|---|---|---|---|---|---|---|---
agent 1 | 8.087 | 7.944 | 8.332 | 7.790 | 8.323 | 7.967 | 8.261 | 8.017
agent 2 | 7.324 | 7.599 | 7.745 | 7.471 | 7.462 | 7.340 | 7.829 | 7.710
agent 3 | 4.710 | 4.445 | 5.000 | 4.2500 | 4.4975 | 4.1751 | 4.3967 | 4.3171
agent 4 | 2.6036 | 2.5321 | 2.5536 | 2.8338 | 2.3538 | 2.3337 | 2.5918 | 2.4610
agent 5 | 2.3683 | 2.3610 | 2.0465 | 2.0528 | 1.9328 | 1.9679 | 1.6598 | 1.4314
agent 6 | 1.3623 | 1.6008 | 0.7356 | 0.8931 | 0.9760 | 1.1620 | 1.3327 | 1.2368
agent 7 | 0.8842 | 1.0090 | 1.2079 | 1.3238 | 1.0501 | 1.1062 | 1.4654 | 1.6153
agent 8 | 0.9123 | 1.1001 | 1.1999 | 1.2297 | 1.0811 | 1.2104 | 1.43666 | 1.4053
Table 2: Maximum error for ${\mathbf{P}}^{1}_{c_{a}}$ ($\times 10^{-2}$). Agent | State 1 | State 2 | State 3 | State 4 | State 5 | State 6 | State 7 | State 8
---|---|---|---|---|---|---|---|---
agent 1 | 0.2494 | 0.2341 | 0.2619 | 0.3009 | 0.2080 | 0.2570 | 0.2764 | 0.3247
agent 2 | 0.3068 | 0.3372 | 0.3579 | 0.3673 | 0.3402 | 0.3732 | 0.4091 | 0.4599
agent 3 | 0.3169 | 0.4021 | 0.3921 | 0.4640 | 0.3646 | 0.4170 | 0.4378 | 0.4677
agent 4 | 0.3361 | 0.3714 | 0.3927 | 0.4650 | 0.3739 | 0.4410 | 0.4058 | 0.4173
agent 5 | 0.3603 | 0.4373 | 0.4426 | 0.4641 | 0.4111 | 0.4615 | 0.4583 | 0.5686
agent 6 | 0.5492 | 0.6472 | 0.5485 | 0.7110 | 0.5752 | 0.6557 | 0.7281 | 0.7092
agent 7 | 0.5844 | 0.6057 | 0.6499 | 0.6416 | 0.6399 | 0.6336 | 0.6497 | 0.7246
agent 8 | 0.5685 | 0.6025 | 0.5725 | 0.6890 | 0.5312 | 0.6511 | 0.6345 | 0.6647
Table 3: Maximum error for ${\mathbf{P}}^{2}_{c_{a}}$ ($\times 10^{-2}$). Agent | State 1 | State 2 | State 3 | State 4 | State 5 | State 6 | State 7 | State 8
---|---|---|---|---|---|---|---|---
agent 1 | 0.0716 | 0.0906 | 0.1069 | 0.0915 | 0.1011 | 0.1285 | 0.1294 | 0.1860
agent 2 | 0.0655 | 0.0873 | 0.0914 | 0.1293 | 0.0854 | 0.1297 | 0.1006 | 0.1400
agent 3 | 0.0643 | 0.0776 | 0.0790 | 0.0999 | 0.0669 | 0.0941 | 0.0849 | 0.1056
agent 4 | 0.0706 | 0.0693 | 0.0728 | 0.0812 | 0.0678 | 0.0753 | 0.1033 | 0.0863
agent 5 | 0.0976 | 0.1167 | 0.1260 | 0.1415 | 0.1079 | 0.1290 | 0.1186 | 0.1330
agent 6 | 0.1246 | 0.1249 | 0.1505 | 0.1563 | 0.1528 | 0.1646 | 0.1629 | 0.1672
agent 7 | 0.1248 | 0.1302 | 0.1402 | 0.1579 | 0.1293 | 0.1452 | 0.1447 | 0.1530
agent 8 | 0.0461 | 0.0384 | 0.0537 | 0.0448 | 0.0382 | 0.0384 | 0.0409 | 0.0420
Table 4: Maximum error for ${\mathbf{P}}^{3}_{c_{a}}$ ($\times 10^{-2}$). Agent 1 | Agent 2 | Agent 3 | Agent 4 | Agent 5 | Agent 6 | Agent 7 | Agent 8
---|---|---|---|---|---|---|---
0.0505 | 0.0406 | 0.0442 | 0.0419 | 0.0509 | 0.0767 | 0.0597 | 0.0128
Table 5: Average error for ${\mathbf{P}}^{3}_{c_{a}}$ ($\times 10^{-2}$).

As expected, adding more terms to the set of admissible functions increases
the accuracy of the approximation. Interestingly the effects are quite
different depending on the agent’s age, $a$. While for
${\mathbf{P}}^{1}_{c_{a}}$ the errors are about 8 times higher for $a=1$ than
$a=8$, for ${\mathbf{P}}_{c_{a}}^{2}$ it is a factor of about 2-3, and for
${\mathbf{P}}_{c_{a}}^{3}$ less than a factor of 3.

In Table 5 we report the average error (along a simulated path) for the set
${\mathbf{P}}_{c_{a}}^{3}$. As in the other two specifications, the average
error is typically about 2-3 times lower than the maximal error. Note that
since our approach minimizes the maximal error, the average error is
relatively high compared to the maximal error (when compared to methods that
minimize the average error).

An obvious shortcoming of our approach lies in the fact that the sets of
admissible forecasts are exogenous. It would be useful to consider a loss
function that chooses forecasts to trade off accuracy and the complexity of
the forecast (see, e.g., Kübler and Scheidegger (2019)). It is worth noting
that forecasting errors for ${\mathbf{P}}^{3}_{c_{a}}$ are generally very
small, and it seems difficult to argue in favor of the need for more accuracy.
On the other hand, errors for the first specification are significant and
imply substantial welfare losses for many agents. For both the examples
considered in Krueger and Kubler (2004) and those in this paper, the wealth
distribution has a substantial impact on prices, and non-trivial aspects of
this wealth distribution need to be used for good forecasts.

##  4 Conclusion

We present a novel equilibrium concept for stochastic dynamic general
equilibrium models with heterogeneous agents. A USJE always exists and can be
approximated numerically with arbitrary precision. The choice of the sup-norm
plays an important technical role in our existence proof. From an economic
point of view, it is reasonable to assume that agents aim to minimize the
maximum forecasting error. In particular, in a framework with overlapping
generations, minimizing the average error (i.e., working with the
computationally more convenient $L^{2}$ norm) allows for a situation where an
agent makes large errors throughout his entire lifetime, hence experiencing
large welfare losses).

Methods that use uniform function approximation, also referred to as Chebyshev
approximation,666in honor of P. L. Chebyshev, who first studied it in
Chebychev (1854). min-max approximation, or $L_{\infty}$ approximation, are no
longer at the heart of computational science, mainly because they are often
thought to be intractable. However, as we show in this paper, for our
purposes, it is not clear whether these methods compare favorably to least-
squares regression.

Our primary motivations for deviating from rational expectations are i) that
recursive rational expectations equilibria do not always exist, and ii) that a
rigorous error analysis of a candidate numerical approximations to a rational
expectations equilibrium is often not possible. Our proposed alternative
equilibrium, the USJE, always exists and allows for a simple numerical
verification procedure.

The fact that USJE always exist and that they can be characterized by a finite
system of non-linear equations and inequalities makes it amenable to formal
investigations of local comparative statics and multiplicity. However, it is
subject to further research to pursue this in more detail.

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