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Federal Reserve Bank of Chicago
Survival and long-run dynamics with
heterogeneous beliefs under recursive
preferences
Jaroslav Borovička
WP 2011-06
Survival and long-run dynamics with heterogeneous
beliefs under recursive preferences∗
Jaroslav Borovička
University of Chicago
Federal Reserve Bank of Chicago
borovicka@uchicago.edu
May 23, 2011
Abstract
I study the long-run behavior of a two-agent economy where agents differ in their
beliefs and are endowed with homothetic recursive preferences of the Duffie-EpsteinZin type. When preferences are separable, the economy is dominated in the long
run by the agent whose beliefs are relatively more precise, a result consistent with
the market selection hypothesis. However, recursive preference specifications lead to
equilibria in which both agents survive, or to ones where either agent can dominate
the economy with a strictly positive probability. In this respect, the market selection
hypothesis is not robust to deviations from separability. I derive analytical conditions
for the existence of nondegenerate long-run equilibria, and show that these equilibria
exist for plausible parameterizations when risk aversion is larger than the inverse of
the intertemporal elasticity of substitution, providing a justification for models that
combine belief heterogeneity and recursive preferences.
∗
I am indebted to Lars Peter Hansen for his advice and continuous support. I appreciate helpful comments from Fernando Alvarez, Gadi Barlevy, Katarı́na Borovičková, Hui Chen, Valentin Haddad, Narayana
Kocherlakota, Alan Moreira, Stavros Panageas, Christopher Phelan, Thomas Sargent, Harald Uhlig, Yuichiro
Waki, Mark Westerfield, and the participants of the Economic Dynamics working group at the University of
Chicago. The views expressed herein are those of the author and do not necessarily represent those of the
Federal Reserve Bank of Chicago or the Federal Reserve System. All typos and errors are mine.
1
Introduction
The market selection hypothesis first articulated by Alchian (1950) and Friedman (1953)
is one of the supporting arguments for the plausibility of the rational expectations theory.
The hypothesis states that agents who systematically evaluate the distributions of future
quantities incorrectly (and are therefore called ‘irrational’) lose wealth on average, and will
ultimately be driven out of the market. Thus, in a long-run equilibrium, the dynamics of
the economy are only determined by the behavior of the rational agents whose beliefs about
the future are in line with the true probability distributions.
However, rationality does not guarantee survival, nor do deviations from rational preferences imply extinction. Ultimately, survival in a market is driven by the consumption-saving
decision and willingness to take risky positions with high expected return vis-à-vis market
prices. In this sense, rationality may facilitate survival if it prevents overconsumption and/or
leads the agent to take appropriate bets. On the other hand, specific forms of irrationality
may, at least in theory, provide even stronger incentives for survival in the long run, despite
not being optimal in the rational sense. Since individual decisions depend on equilibrium
prices, survival analysis only makes sense in the context of a fully specified model, including
preferences and belief formation of the market participants and their trading opportunities.
Survival of agents with incorrect beliefs has been studied extensively in complete market
models populated by agents endowed with separable preferences. The existing literature
on market survival has demonstrated how differences in beliefs can be counteracted by differences in preferences. Yan (2008) analyzes a model with constant relative risk aversion
(CRRA) preferences and constructs a quantity called the survival index that aggregates the
role of intertemporal elasticity of substitution (IES), time preference, and belief distortion
into a single number that determines survival. When preferences are identical across agents,
then only agents whose beliefs are closest to the truth will survive in the long run.
This insight is central for the understanding of the survival mechanism and can be
rephrased as follows. If rich and poor agents are alike, in the sense that rich agents behave
as scaled versions of poor ones, then agents with relatively more incorrect beliefs cannot
survive in the long run. This is precisely correct for the homothetic CRRA preferences. In
their lucid analysis, Kogan, Ross, Wang, and Westerfield (2009) show that this statement is
also true for a class of preferences with bounded relative risk aversion, i.e., preferences that
are in some norm uniformly ‘close’ to the homothetic CRRA case.
This paper shows that support in favor of the market survival hypothesis weakens considerably once the assumption of separability in preferences is relaxed. In order to focus solely
1
on the impact of belief heterogeneity, I endow agents with identical homothetic recursive
preferences axiomatized by Kreps and Porteus (1978), and developed by Epstein and Zin
(1989) and Weil (1990) in discrete time, and by Duffie and Epstein (1992b) in continuous
time. These preferences allow one to disentagle the risk aversion with respect to intratemporal gambles from the intertemporal elasticity of substitution, and include the CRRA utility
as a special case. Thanks to the additional degree of flexibility, this class of preferences is
widely used in the asset pricing literature to provide a better fit of the constructed models to
empirically observed patterns in asset returns. Homotheticity assures that survival results
are not driven by exogenous differences in the local properties of the utility functions.
The decoupling of risk aversion and IES proves to have a crucial impact on survival.
Consider an endowment economy with agents who are optimistic or pessimistic about the
growth rate of the stochastic aggregate endowment. A higher risk aversion in the economy
increases risk premia associated with risky assets and improves survival chances of relatively
more optimistic agents who are willing to invest in these assets. On the other hand, a
higher IES facilitates survival of agents who are relatively more optimistic about the return
on their own portfolio because of a stronger willingness to postpone consumption vis-à-vis
the higher expected returns. This mechanism can also facilitate survival of the relatively
pessimistic agents if they are willing to take a sufficiently large short position in the claim
to the aggregate endowment.
With CRRA preferences, risk aversion and IES are inversely related, so the two effects
offset each other. Increasing the risk aversion increases relatively more the expected returns
on portfolios held by optimistic agents, but the associated decrease in IES makes them, at
the same time, consume more out of their wealth, relative to agents with correct beliefs.
Pessimistic agents, on the other hand, are paying more to insure against the states with
low consumption growth as risk aversion increases; and the relatively lower willingness to
consume out of wealth linked to the associated decrease in IES in conjunction with lower
perceived expected returns is not strong enough to compensate for the cost.
I analyze these mechanisms in a two-agent, continuous-time endowment economy with
complete markets and an aggregate endowment process modeled as a geometric Brownian
motion. I find that both agents survive and a nondegenerate equilibrium exists in the long
run for wide regions of the parameter space when risk aversion is larger than the inverse of
IES. An optimistic agent will dominate the economy in the long run when risk aversion is
sufficiently large. A pessimistic agent survives in the long run when IES is sufficiently high
and risk aversion is not excessive. In the opposite case, when risk aversion is sufficiently
2
lower than the inverse of IES, only one agent survives in the long run, but the surviving
agent can be either of the two agents with a strictly positive probability.
The market selection hypothesis is thus not robust to departures from separable preferences. Survival crucially depends on the interaction between risk attitudes that drive the
portfolio selection decision and the IES that influences the consumption-saving decision. Recursive preferences provide an additional degree of freedom compared to the separable case
that allows one to separate these two effects.
Given the homotheticity of preferences and complete markets, the survival results are
driven purely by prices endogenously determined in general equilibrium. Crucially, nondegenerate long-run equilibria arise for preference parameterizations that are considered
plausible in the asset pricing literature, which lends support to asset pricing models that
combine heterogeneous beliefs and recursive preferences.
1.1
Methodology and literature overview
The modern approach in the market survival literature1 originates from the work of De Long,
Shleifer, Summers, and Waldmann (1991), who study wealth accumulation in a partial equilibrium setup with exogenously specified returns and find that irrational noise traders can
outgrow their rational counterparts and dominate the market. Similarly, Blume and Easley
(1992) look at the survival problem from the vantage point of exogenously specified saving
rules, albeit in a general equilibrium setting.
Subsequent research has shown that taking into account general equilibrium effects and
intertemporal optimization eliminates much of the support for survival of agents with incorrect beliefs that models with ad hoc price dynamics produce. Sandroni (2000) and Blume
and Easley (2006) base their survival results on the evolution of relative entropy as a measure
of disparity between subjective beliefs and the true probability distribution. In their models,
aggregate endowment is bounded from above and away from zero. As a result, changes in the
1
Modeling of economies populated by agents endowed with heterogeneous beliefs constitutes a quickly
growing branch of literature, and a thorough overview of the literature is beyond the scope of this paper.
Here, I primarily focus on the intersection of this literature with the analysis of recursive nonseparable
preferences. Bhamra and Uppal (2009) provide a more general survey that also focuses on asset pricing
implications of belief and preference heterogeneity.
I also omit the discussion of evolutionary literature which predominantly focuses on the analysis of the
interaction between agents with exogenously specified portfolio rules and price dynamics. The survival
mechanism in this paper critically hinges on the interaction the of endogenous consumption-saving decision
and portfolio allocation vis-à-vis general equilibrium prices driven by the dynamics of the wealth shares, and
is thus only loosely related. See Hommes (2006) for a survey of the evolutionary literature, and Evstigneev,
Hens, and Schenk-Hoppé (2006) for an analysis of portfolio rule selection.
3
curvature of the utility function are immaterial for survival when mild regularity conditions
are satisfied. Controlling for pure time preference, the long-run fate of economic agents is
determined solely by belief characteristics, and only agents whose beliefs are in a specific
sense asymptotically ‘closest’ to the truth can survive.
With unbounded aggregate endowment, local properties of the utility function become
an additional survival factor. Even if preferences are identical across agents, the local curvature of the utility function at low and high levels of consumption can be sufficiently different
to outweigh the divergence in beliefs, and lead to survival of agents with relatively more
incorrect beliefs. Kogan, Ross, Wang, and Westerfield (2009) show that a sufficient condition to prevent this outcome is the boundedness of the relative risk aversion coefficient.
This condition can be interpreted as a bound on deviations of the utility function from
homotheticity.
Importantly, survival analysis under separable preferences corresponds to analyzing a sequence of time- and state-indexed static problems that are only interlinked through the initial
marginal utility of wealth, which is largely innocuous for the long-run characterization of the
economy. The survival literature frequently exploits martingale methods to characterize the
long-run divergence of subjective beliefs and marginal utilities of consumption.
Nonseparability of preferences breaks this straightforward link, and I therefore develop a
different method that is more suitable for this environment. I utilize the planner’s problem
derived in Dumas, Uppal, and Wang (2000) and extend it to include heterogeneity in beliefs.
The solution of the planner’s problem involves endogenously determined processes that can
be interpreted as stochastic Pareto weights.
The analysis of market survival then corresponds to investigating the long-run behavior
of scaled Pareto weights. I present tight sufficient conditions for the existence of nondegenerate long-run equilibria and for dominance and extinction. While the full model requires a
numerical solution, I show that the behavior at the boundaries, which is essential for survival
analysis, can be established analytically. I thus provide closed-form solutions for the regions
of the parameter space in which the survival conditions are satisfied.
The applicability of the derived solution method is not limited to fixed distortions. I
discuss how to extend the procedure to include learning and robust preferences of Anderson,
Hansen, and Sargent (2003). Explicit solutions of these problems are left for future work.
The approach based on the characterization of the behavior of the endogenously determined Pareto weights is closely linked to the literature on endogenous discounting, initiated
by Koopmans (1960) and Uzawa (1968), and to models of heterogeneous agent economies
4
under recursive preferences, studied by Lucas and Stokey (1984) and Epstein (1987) under
certainty and by Kan (1995) under uncertainty. The survival conditions derived in this paper
resemble a sufficient condition for the existence of a stable interior steady state in Lucas and
Stokey (1984), called increasing marginal impatience. This condition postulates that agents
discount future less as they become poorer. I show that my analysis crucially depends on
a similar quantity that I call relative patience. The key difference lies in the determination
of the two quantities. While Lucas and Stokey require that the time preference exogenously
encoded in the utility specification changes with the level of consumption, in this paper the
variation in relative patience arises endogenously as an equilibrium outcome driven by belief
differences.
Anderson (2005) studies Pareto optimal allocations under heterogeneous recursive preferences in a discrete-time setup using similar methods but he does not consider survival
under belief heterogeneity. Mazoy (2005) discusses long-run consumption dynamics when
agents differ in their IES. Colacito and Croce (2010) prove the existence of nondegenerate
long-run equilibria in a two-good economy when agents are endowed with risk-sensitive preferences and differ in the preferences over the two goods. However, none of these papers
treats systematically the case of belief heterogeneity. This work aims at filling this gap.
The paper is organized as follows. Section 2 outlines the economic environment, provides
a theoretical exposition to recursive preferences, and derives the planner’s problem that is
central to the analysis. Section 3 presents the survival results. I provide in analytical form
tight sufficient conditions for survival and extinction and discuss the economic interpretation of the results. This analytical part is followed by numerical analysis of consumption
and price dynamics for economies with nondegenerate long-run equilibria in Section 4. Section 5 summarizes the findings and outlines extensions of the developed framework involving
learning and endogenously determined belief distortions derived, for instance, from robust
preferences. The Appendix contains proofs omitted from the main text. Further material
that provides more details and extends the analysis is available in the online appendix.2
2
Optimal allocations under heterogeneous beliefs
I analyze the dynamics of equilibrium allocations in a continuous-time endowment economy
populated by two types of infinitely-lived agents endowed with identical recursive preferences.
I call an economy where both agents have strictly positive wealth shares a heterogeneous
2
Available at http://home.uchicago.edu/∼borovicka/files/research/heterogeneous beliefs online appendix.pdf.
5
economy. A homogeneous economy is populated by a single agent only. The term ‘agent’
refers to an infinitesimal competitive representative of the particular type.
Agents differ in their subjective beliefs about the distribution of future quantities but
are firm believers in their probability models and ‘agree to disagree’ about their beliefs as in
Morris (1995). Since they do not interpret their belief differences as a result of information
asymmetries, there is no strategic trading behavior.
Without introducing any specific market structure, I assume that markets are dynamically complete in the sense of Harrison and Kreps (1979). This allows me to sidestep the
problem of directly calculating the equilibrium by considering a planner’s problem. The discussion of market survival then amounts to the analysis of the dynamics of Pareto weights
associated with this planner’s problem. Optimal allocations and continuation values generate a valid stochastic discount factor and a replicating trading strategy for the decentralized
equilibrium.
In this section, I specify agents’ preferences and belief distortions, and lay out the planner’s problem. I utilize the framework introduced by Dumas, Uppal, and Wang (2000),
and exploit the observation that belief heterogeneity can be analyzed in their framework
without increasing the degree of complexity of the problem. The method then leads to a
Hamilton-Jacobi-Bellman equation for the planner’s value function.
2.1
Information structure and beliefs
The stochastic structure of the economy is given by a filtered probability space (Ω, F , {Ft } , P )
with an augmented filtration defined by a family of σ-algebras {Ft } , t ≥ 0 generated by a
univariate Brownian motion W . Given the continuous-time nature of the problem, equalities
are meant in the appropriate almost-sure sense. I also assume that all processes, in particular
belief distortions and permissible trading strategies, satisfy regularity conditions like square
integrability over finite horizons, so that stochastic integrals are well defined and pathological cases are avoided. Under the parameter restrictions below, constructed equilibria satisfy
these assumptions.
The scalar aggregate endowment process Y satisfies
dYt
= μy dt + σy dWt ,
Yt
Y0 > 0
(1)
with constant parameters μy and σy .
Agents of type n ∈ {1, 2} are endowed with identical preferences but differ in their
6
subjective probability measures Qn that they use to assign probabilities to future events. I
assume that measures Qn and P are equivalent for all finite-horizon events, so that there
exists the Radon-Nikodým derivative
dQn
dP
t
=M
˙ tn
t
1 t n2
n
= exp −
|u | ds +
us dWs ,
2 0 s
0
(2)
where un is an adapted process. The martingale M n measures the disparity between the
subjective and true probability measures and is commonly called the belief ratio. While
a likelihood evaluation of the past observed data reveals that the view of an agent with
distorted beliefs becomes less and less likely to be correct as time passes, absolute continuity
of the measure Qn with respect to P implies that he can never refute his view of the world
as impossible. The main results of the paper are developed using a constant un , but the
computational strategy allows me to incorporate more general distortion processes, which I
discuss in the concluding remarks.
The Girsanov theorem implies that agent n, whose deviation from rational beliefs is
described by M n , views the evolution of the Brownian motion W as distorted by a drift
component un , i.e., dWt = unt dt + dWtn , where W n is a Brownian motion under Qn . Consequently, the aggregate endowment is perceived to contain an additional drift component
un σy , and un can be interpreted as a degree of optimism or pessimism about Y . When
σy = 0, this distinction loses its meaning but the the survival problem is still nondegenerate,
as long as the agents can contract upon the realizations of the process W .
2.2
Recursive utility
Agents endowed with separable preferences reduce intertemporal compound lotteries (different payoff streams allocated over time) to atemporal simple lotteries that resolve uncertainty
at a single point in time. In the Arrow-Debreu world with separable preferences, once trading
of state-contingent securities for all future periods is completed at time 0, uncertainty about
the realized path of the economy can be resolved immediately without any consequences for
the ex-ante preference ranking of the outcomes by the agents.
Kreps and Porteus (1978) relaxed the separability assumption by axiomatizing discretetime preferences where temporal resolution of uncertainty matters and preferences are not
separable over time. While intratemporal lotteries in the Kreps-Porteus axiomatization
still satisfy the von Neumann-Morgenstern expected utility axioms, intertemporal lotteries
cannot in general be reduced to atemporal ones. The work by Epstein and Zin (1989,
7
1991) extended the results of Kreps and Porteus (1978), and initiated the widespread use of
recursive preferences in the asset pricing literature. Duffie and Epstein (1992a,b) formulated
the continuous-time counterpart of the recursion.3
I utilize a characterization based on the more general variational utility approach studied
by Geoffard (1996) in the deterministic case and El Karoui, Peng, and Quenez (1997) in
a stochastic environment.4 They show that recursive preferences can be represented as a
solution to the maximization problem
λnt Vtn
subject to
=
n
sup EtQ
νn
∞
t
λns F
(Csn , νsn ) ds
dλnt
= −νtn dt, t ≥ 0; λn0 = 1,
n
λt
(3)
(4)
where ν n is called the discount rate process, and λn the discount factor process. The felicity
function F (C, ν) encodes the contribution of the consumption stream C to present utility. This representation closely links recursive preferences to the literature on endogenous
discounting, initiated by Koopmans (1960) and Uzawa (1968).
For the case of the Duffie-Epstein-Zin preferences, the felicity function is given by
Cγ
F (C, ν) = β
γ
γ − ρ βν
γ −ρ
1− γρ
,
with parameters satisfying γ, ρ < 1, and β > 0. Preferences specified by this felicity function5
are homothetic and exhibit a constant relative risk aversion with respect to intratemporal
wealth gambles α = 1 − γ and (under intratemporal certainty) a constant intertemporal
elasticity of substitution η =
1
.
1−ρ
Parameter β is the time preference coefficient. Assump-
tion 2 below restricts parameters to assure sufficient discounting for the continuation values
to be finite in both homogeneous and heterogeneous economies. In the case when γ = ρ, the
3
Duffie and Epstein (1992b) provide sufficient conditions for the existence of the recursive utility process
for the infinite-horizon case but these are too strict for the preference specification considered in this paper.
However, the Markov structure of the problem allows me to utilize existence results derived Duffie and Lions
(1992). Schroder and Skiadas (1999) establish conditions under which the continuation value is concave,
and provide further technical details. Skiadas (1997) shows a representation theorem for the discrete time
version of recursive preferences with subjective beliefs.
4
Hansen (2004) offers a tractable summary of the link between the recursive and variational utility.
Interested readers may refer to the online appendix for a more detailed discussion.
5
The cases of ρ → 0 and γ → 0 can be obtained as appropriate limits. The maximization problem (3)
assumes that the felicity function is concave in its second argument. When it is convex, the formulation
becomes a minimization problem.
8
utility reduces to the separable CRRA utility with the coefficient of relative risk aversion α.
Formula (3), together with an application of the Girsanov theorem, suggests that it is advantageous to combine the contribution of the discount factor process λn and the martingale
M n that specifies the belief distortion in (2):
Definition 1 A modified discount factor process λ̄n is a discount factor process that incor˙ λn M n .
porates the martingale M n arising from the belief distortion, λ̄n =
Applying Itô’s lemma to λ̄n leads to a maximization problem under the true probability
measure
λ̄nt Vtn
subject to
= sup Et
νn
∞
t
λ̄ns F
(Csn , νsn ) ds
dλ̄nt
= −νtn dt + unt dWt , t ≥ 0; λ̄n0 = 1.
λ̄nt
(5)
(6)
The problem (5-6) indicates that F (C, ν) can be viewed as a generalization of the period
utility function with a potentially stochastic rate of time preference ν that depends on the
properties of the consumption process and thus arises endogenously in a market equilibrium.
Moreover, belief distortions are now fully incorporated in the framework of Dumas, Uppal,
and Wang (2000) — the only difference is that the modified discount factor process is not
locally predictable.
The diffusion term uns dWs has an intuitive interpretation. Consider an optimistic agent
with un > 0. This agent’s beliefs are distorted in that the mass of the distribution of dWs
is shifted to the right — the agent effectively overweighs good realizations of dWs . Formula
(6) indicates that under the true probability measure, positive realizations of dWs increase
the term dλ̄ns /λ̄ns , which implies that the optimistic agent discounts positive realizations of
dWs less than negative ones.
From the perspective of the utility-maximizing agent, assigning a higher probability to an
event and a lower discounting of the utility contribution of this event have the same effect.
In fact, equation (3) suggests that we can understand the belief distortion as a preference
shock and view λ̄n F (C n , ν n ) as a state-dependent felicity function. However, interpreting
the martingale M n as a belief distortion is more appealing since it bears a clearer economic
meaning, separating the structure of beliefs and preferences.
9
2.3
Planner’s problem and optimal allocations
The problem of an individual agent (3–4) is homogeneous degree one in the modified discount
factors and homogeneous degree γ in consumption. In the homogeneous economy, there exists
a closed-form solution for the continuation value Vtn (Y ) = γ −1 Ytγ Ṽ n where
n
Ṽ =
β
−1
− γρ
1
n
2
β − ρ μy + u σy − (1 − γ) σy
2
with the associated discount rate
− γρ
β
1
n
n
n
2
ν =
= β + (γ − ρ) μy + u σy − (1 − γ) σy .
γ + (ρ − γ) Ṽ
ρ
2
(7)
(8)
Assumption 2 The parameters in the model satisfy the restrictions
1
n
2
β > max ρ μy + u σy − (1 − γ) σy ,
(9)
n
2
1
1 (un − u∼n )2
ρ
∼n
2
n
∼n
(u − u ) σy +
(10)
β > max ρ μc + u σy − (1 − γ) σy +
n
2
1−ρ
2
1−γ
where ∼ n is the index of the agent other than n.
The first restriction is sufficient for the continuation values in the homogeneous economies
to be well-defined. The second restriction, which may be, depending on the parameterization,
somewhat tighter, is a sufficient condition assuring that the wealth-consumption ratio is
asymptotically well-behaved in the survival proofs when the agent becomes infinitesimally
small. Observe that both conditions are restrictions on the time-preference parameter of
the agents and can always be jointly satisfied by making the agents sufficiently impatient.
Since survival results will not depend on β, Assumption 2 does not introduce substantial
restrictions for the analysis of the problem.
In the heterogeneous economy, I can follow Dumas, Uppal, and Wang (2000) and introduce a fictitious planner who maximizes a weighted average of the continuation values of the
two agents. Given a pair of strictly positive initial Pareto weights α = (α1 , α2 ), the planner’s
time-0 objective function J0 (α) is the solution to the problem
2
J0 (α) =
sup
(C 1 ,C 2 ,ν 1 ,ν 2 ) n=1
E0
10
0
∞
λ̄nt F
(Ctn , νtn ) dt
(11)
subject to the law of motion for the modified discount factors,
dλ̄nt
= −νtn dt + unt dWt , t ≥ 0; λ̄n0 = αn
n
λ̄t
(12)
for n ∈ {1, 2}, and the feasibility constraint C 1 + C 2 ≤ Y .
The validity of this approach for a finite-horizon economy is discussed in Dumas, Uppal,
and Wang (2000) and Schroder and Skiadas (1999). The infinite-horizon problem in (11-12)
is a straightforward extension when individual continuation values are well-defined. The
planner’s objective function is bounded from above by the weighted average of continuation
values from the homogeneous economies, J0 (α) ≤ α1 V0n (Y ) + α2 V0n (Y ), and the supremum
in (11) thus exists. Since the continuation values are concave, first-order conditions are
sufficient for the supremum problem. The following Lemma describes the behavior of the
objective function at the boundaries.
Lemma 3 The objective function J0 (α) can be continuously extended at the boundaries as
α1 0 or α2 0 by the continuation values calculated for the homogeneous economies, i.e.,
for α2 > 0
˙ lim
J0 α1 , α2 = α2 V02 (Y )
J0 0, α2 =
1
α 0
(13)
and limα1 0 C 2 (α1 , α2 ) = Y . The case α2 0 is symmetric.
The planner’s problem (11-12) suggests that we can interpret the modified discount factor
processes λ̄n as stochastic Pareto weights. Indeed, if λ̄n0 = αn are the initial weights, then
λ̄nt are the consistent state-dependent weights for the continuation problem of the planner
at time t.6,7
The evolution of the weights involves the drift component ν n and thus can only be
determined in equilibrium unless agent n ’s preferences are separable, in which case ν n = β.
6
Similar techniques, which extend the formulation of the representative agent provided by Negishi (1960)
to representations with nonconstant Pareto weights, can be used to study models with incomplete markets
where changes in the Pareto weights reflect the tightness of the binding constraints. See Cuoco and He
(2001) for a general approach in discrete time and Basak and Cuoco (1998) for a model with restricted stock
market participation in continuous time.
7
Jouini and Napp (2007) approach the problem from a different angle to show that a planner’s problem
formulation with constant Pareto weights is in general not feasible under heterogeneous beliefs. Given
an equilibrium with heterogeneous beliefs, they define a hypothetical representative agent with a utility
function constructed as a weighted average of individual utility functions, with weights given by the inverses
of marginal utilities of wealth. The implied consensus belief of the representative agent that would replicate
the equilibrium allocation is not a proper belief but can be decomposed into the product of a proper belief
and a discount factor. This discount factor would mimic the dynamics of the Pareto shares in problem
(11–12).
11
The variation in Pareto weights arises from the interaction of two components in the model
— the nonseparable preference structure and the belief distortion that drives the diffusion
component in (12).8
Observe that the introduction of belief heterogeneity kept the structure of the problem
unchanged. For instance, Dumas, Uppal, and Wang (2000) show that in a Markov environment, the discount factor processes λn serve as new state variables that allow a recursive
formulation of the problem using the Hamilton-Jacobi-Bellman (HJB) equation. The same
conclusion is true for the modified discount factor processes λ̄n , once belief heterogeneity is
incorporated. Belief distortions thus do not introduce any additional state variables into the
problem, as long as the distorting processes un are functions of the existing state variables.
2.4
Hamilton-Jacobi-Bellman equation
From now on, I assume that both agents have constant belief distortions un , a frequently
considered case in the survival literature. Extensions involving endogenously determined
distortion processes including learning dynamics are considered in Section 5.
The planner’s problem has an appealing Markov structure. Homogeneity of the planner’s
problem (11-12) in λ̄1 , λ̄2 suggests a transformation of variables
θ1 = λ̄1 λ̄1 + λ̄2
−1
θ2 = λ̄1 + λ̄2 .
(14)
The single state variable θ1 represents the Pareto share of agent 1. The dynamics of θ1
are central to the study of survival in this paper. Obviously, θ1 is bounded between zero
and one. It will become clear that for strictly positive initial weights, the boundaries are
unattainable, so that θ1 evolves on the open interval (0, 1). Since the objective function of
the planner is also homogeneous degree γ in Y , the planner’s problem can be characterized
as a solution to an ordinary differential equation with a single state variable θ1 .
Proposition 4 The objective function for the planner’s problem (11-12) is
J0 (α) = α1 + α2 γ −1 Y0γ J˜ α1 / α1 + α2
8
,
The belief heterogeneity introduces an additional risk component arising through the stochastic reweighing of wealth shares. The diffusion component in the weight dynamics will have a direct impact on local risk
prices. Notice that other sources of heterogeneity, including differences in risk aversion and IES parameters,
do not lead to a diffusion component in the dynamics of the Pareto weights (12), and reweighting therefore
has no impact on local risk prices in these cases. Different types of market participation constraints as in
Basak and Cuoco (1998) may also introduce a diffusion term into the Pareto weight dynamics.
12
where J˜ (θ1 ) is the solution to the nonlinear ordinary differential equation
1β
1 ρ
1− γρ
β
J˜1
1 − ζ1
+ 1 − θ1
ρ
ρ
J˜2
1− γρ
ζ
+
ρ
β
1
1 1
1
2
2
+ − + μy + θ u + 1 − θ u σy + (γ − 1) σy J˜ +
ρ
2
1
1
2
2
2
+θ1 1 − θ1 u1 − u2 σy J˜θ1 +
1 − θ1
θ1
u1 − u2 J˜θ1 θ1
2γ
0 = θ
(15)
with boundary conditions J˜ (0) = Ṽ 2 and J˜ (1) = Ṽ 1 , where Ṽ n are defined in (7). The
functions J˜n (θ1 ) are the continuation values of the two agents scaled by γ −1 Y γ ,
J˜1 θ1
=
˙ J˜ θ1 + 1 − θ1 J˜θ1 θ1
J˜2 θ1
=
˙ J˜ θ1 − θ1 J˜θ1 θ1 .
(16)
and the consumption share ζ 1 is given by
1
ζ 1 θ1 =
1
(θ1 ) 1−ρ
1
1−ρ
1−ρ/γ
1−ρ
1
1
˜
J (θ )
(θ )
.
1−ρ/γ
1−ρ/γ
1
1−ρ
1−ρ
1
1
1
2
1
˜
˜
+ (1 − θ ) 1−ρ J (θ )
J (θ )
(17)
Unfortunately, equation (15) does not have a general closed-form solution. However,
the Pareto share θ1 of agent 1 remains the only state variable. This considerably simplifies
numerical solutions, and, more importantly, allows one to formulate the survival problem
in terms of the boundary behavior of a scalar Itô process. Despite the nonexistence of a
closed-form solution for J˜ (θ1 ), this boundary behavior can be characterized analytically by
studying the limiting behavior of the objective function.
Equation (15) is not specific to the planner’s problem (11-12). For instance, Gârleanu
and Panageas (2010) use the martingale approach to directly analyze the equilibrium in
an economy with agents endowed with heterogeneous recursive preferences, and show that
they can derive their asset pricing formulas in closed form up to the solution of a nonlinear
ODE that has the same structure as (15), which they have to solve for numerically. The
analytical characterization of the boundary behavior of the ODE derived in this paper is thus
applicable to a wider class of recursive utility models, and can aid numerical calculations
which are often unstable in the neighborhood of the boundaries in this type of problems.
13
3
Survival
Survival chances of agents with distorted beliefs have been studied extensively under separable utility. Kogan, Ross, Wang, and Westerfield (2009) show a tight link between the
behavior of the belief ratio, consumption shares, and the risk aversion coefficient as a measure
of curvature of the utility function. To provide a simple illustration, consider a period utility
function U (C) and the corresponding Euler equation that prices a payoff Zt+s at time t
Ptz = EtQn e−βs
n
n
n
U Ct+s
−βs U Ct+s Mt+s
=
E
e
Zt+s .
Z
t+s
t
U (Ctn )
U (Ctn ) Mtn
Since prices are observed in equilibrium, agents have to agree on them. When markets
are complete, the objects
n
n
U Ct+s
Mt+s
U (Ctn ) Mtn
have to be equalized across agents n, and deviations in beliefs have to be offset by reciprocal
deviations in marginal utilities. Survival analysis thus corresponds to analyzing a sequence
of state- and time-indexed static problems that are interlinked only by the initial relative
marginal utilities of wealth of the two agents, whose choice is largely innocuous for the
long-run results. If agent 1 has a constant belief distortion u1 = 0 and agent 2 is rational,
1
= 0 ( P -a.s.), and thus
then M 1 is a strictly positive supermartingale and lims→∞ Mt+s
1
2
/U Ct+s
= +∞ ( P -a.s.). For a class of utility functions that includes the
lims→∞ U Ct+s
1
2
CRRA utility (the special case when γ = ρ in this paper), this implies lims→∞ ζt+s
/ζt+s
=0
( P -a.s.).
When preferences are not separable, this straightforward link breaks down because marginal utilities also depend on continuation values and the stochastic discount factor involves
the evolution of the endogenously determined discount rate process ν n between t and t + s.
Since these continuation values and discount rate processes are not available in closed form,
they have to in general be solved for numerically.
I show in this section that in order to evaluate the survival chances of individual agents,
a complete solution for the consumption allocation, continuation values, and the implied
discount rate processes is not necessary. In fact, it is sufficient to characterize the wealth
dynamics in the limiting cases when the wealth share of one of the agents becomes negligible,
and this limiting behavior can be solved for in closed form. This characterization of survival
requires taking an approach that is different from the majority of the literature, which
typically analyzes the global properties of relative entropy as a measure of disparity between
14
subjective beliefs and the true probability distribution, and its convergence as t
∞.
1
Instead, I derive the local dynamics of the Pareto share θ and rely on its ergodic properties, which allow me to investigate the existence of a unique stationary distribution for θ1
that is closely related to survival. The derived sufficient conditions are tightly linked to the
behavior of the difference of endogenous discount rates of the two agents. In a decentralized
economy, these relative patience conditions can be reinterpreted in terms of the difference in
expected logarithmic growth rates of individual wealth.
Since the analyzed model includes growing and decaying economies, I am interested in
a measure of relative survival. The following definition distinguishes between survival along
individual paths and almost-sure survival.
Definition 5 Agent 1 becomes extinct along the path ω ∈ Ω if limt→∞ θt1 (ω) = 0. Otherwise,
agent 1 survives along the path ω. Agent 1 dominates in the long run along the path ω if
limt→∞ θt1 (ω) = 1.
Agent 1 becomes extinct (under measure P ) if limt→∞ θt1 = 0, P -a.s. Agent 1 survives if
lim supt→∞ θt1 > 0, P -a.s. Agent 1 dominates in the long run if limt→∞ θt1 = 1, P -a.s.
Kogan, Ross, Wang, and Westerfield (2009) or Yan (2008) use the consumption share
ζ 1 as a measure of survival. Since the consumption share (17) is continuous and strictly
increasing in θ1 and the limits are limθ1 0 ζ 1 (θ1 ) = 0 and limθ1 1 ζ 1 (θ1 ) = 1, the two
measures are equivalent in this setting.
3.1
Dynamics of the Pareto share and long-run distributions
Recall the dynamics of the modified discount factor processes λ̄n in (12). An application of
Itô’s lemma to θ1 = λ̄1 / λ̄1 + λ̄2 yields
dθt1
=
θt1
1 − θt1
+ 1 − θt1
νt2 − νt1 + θt1 u1 + 1 − θt1 u2
u2 − u1
dt +
(18)
u1 − u2 dWt .
Both heterogeneous beliefs and heterogeneous recursive preferences lead to nonconstant
dynamics of the Pareto share, although with different implications. Under nonseparability, preference heterogeneity induces a smooth evolution of the Pareto weights, while belief
heterogeneity leads to dynamics with a nonzero volatility term. Identical belief distortions
(u1 = u2 ) under separable preferences with identical time preference coefficients or under
15
identical recursive preferences imply a constant Pareto share θt1 ≡ α1 / (α1 + α2 ). In what
follows, I abstract from this situation, and assume u1 = u2 .
Under nonseparable preferences, the discount rates are determined endogenously in the
model as a solution to problem (11–12) and are given by
ν n θ1
β
=
ρ
γ + (ρ − γ)
ζ n (θ1 )
J˜n (θ1 )1/γ
ρ
.
(19)
The discount rates ν n are twice continuously differentiable functions of the state variable
θ1 , and thus θ1 is an Itô process on the open interval (0, 1) with continuous drift and volatility
coefficients.9 Intuitively, one would expect a stationary distribution for θ1 to exist if the
process exhibits sufficient pull toward the center of the interval when close to the boundaries.
This is formalized in the following Proposition:
Proposition 6 Define the following ‘repealing’ conditions (i) and (ii), and their ‘attracting’
counterparts (i’) and (ii’).
1 2
2 2
(i) limθ1 0 [ν (θ ) − ν (θ )] > (u ) − (u )
2
2
(ii) limθ1 1 [ν 2 (θ1 ) − ν 1 (θ1 )] < 12 (u1 ) − (u2 )
2
1
1
1
1
2
(i’)
<
(ii’) >
Then the following statements are true:
(a) If conditions (i) and (ii) hold, then both agents survive under P .
(b) If conditions (i) and (ii’) hold, then agent 1 dominates in the long run under P
(c) If conditions (i’) and (ii) hold, then agent 2 dominates in the long run under P .
(d) If conditions (i’) and (ii’) hold, then there exist sets S 1 , S 2 ⊂ Ω which satisfy
S 1 ∩ S 2 = ∅,
P S 1 = 0 = P S 2 , and P S 1 ∪ S 2 = 1
such that agent 1 dominates in the long run along each path ω ∈ S 1 and agent 2
dominates in the long run along each path ω ∈ S 2 .
The conditions are also the least tight bounds of this type.
9
The unattainability of the boundaries follows from the proof of Proposition 6.
16
Given the dynamics of the Pareto share (18), conditions (i) and (ii) are jointly sufficient
for the existence of a unique stationary density q (θ1 ). The proof of Proposition 6 is based
on the classification of boundary behavior of diffusion processes, discussed in Karlin and
Taylor (1981). The four ‘attracting’ and ‘repealing’ conditions are only sufficient and their
combinations stated in Proposition 6 are not exhaustive. There are delicate cases involving
equalities, which are however only of limited importance in the analysis below.
I call the difference in the discount rates ν 2 (θ1 ) − ν 1 (θ1 ) relative patience because it
captures the difference in discounting of future felicity in the variational utility specification
(3) between the two agents. Conditions in Proposition 6 have an intuitive interpretation.
Survival condition (i) states that agent 1 survives under the true probability measure even in
cases when his beliefs are more distorted, |u1 | > |u2 |, as long as his relative patience becomes
sufficiently high to overcome the distortion when his Pareto share vanishes.
Lucas and Stokey (1984) impose a similar condition called increasing marginal impatience
that is sufficient to guarantee the existence of a nondegenerate steady state as an exogenous
restriction on the preference specification. This condition requires the preferences in their
framework to be nonhomothetic, and rich agents must discount future more than poor ones.
In this model, preferences are homothetic, and variation in relative patience arises purely as
a response to the market interaction of the two agents endowed with heterogeneous beliefs.
The discount rate ν n encodes not only a pure time preference but also an interaction through
the dynamics of the optimal consumption stream.
3.2
CRRA preferences
The framework introduced in this paper includes as a special case the separable constant
relative risk aversion preferences when γ = ρ. Yan (2008) and Kogan, Ross, Wang, and
Westerfield (2009) show that in the economy presented in this paper, the agent whose beliefs
are less distorted dominates in the long run under measure P . The conditions in Proposition 6
confirm these results as follows:
Corollary 7 Under separable CRRA preferences (γ = ρ), agent n dominates in the long
run under measure P if and only if |un | < |u∼n |. Agent n survives under P if and only if
the inequality is non-strict. Further, agent n survives under measure Qn and dominates in
the long run under Qn if and only if un = u∼n .
Under separable CRRA preferences, the dynamics of the Pareto share (18) do not depend on the characteristics of the endowment process. The survival result in Corollary 7
17
thus extends to an arbitrary adapted aggregate endowment process Y that satisfies elementary integrability conditions, including a constant one, as long as the two agents can write
contracts on the realizations of the Brownian motion W . It is a special case of the analysis in
Kogan, Ross, Wang, and Westerfield (2009), who show that this survival result holds, under
mild conditions, for any separable preferences with bounded relative risk aversion. In this
sense, the separable environment generates a robust result about the extinction of agents
whose beliefs are relatively imprecise.
A specific situation in Corollary 7 arises when un = −u∼n . The proof of the corollary
shows that although none of the agents becomes extinct, a nondegenerate long-run distribution for θ1 does not exist.
3.3
The nonseparable case
When preferences are not separable, consumption choices across periods are interlinked
through the endogenously determined discount rate processes ν n , which opens another channel for intertemporal tradeoff and thus potential survival. This endogenous discounting is
reflected in the evolution of the Pareto share θ1 . In this section, I derive closed-form formulas
for the boundary behavior of ν n , and evaluate analytically the region in the parameter space
in which the conditions of Proposition 6 hold.
The proof strategy in this section relies on a decentralization argument and utilizes the
asymptotic properties of the differential equation (15) for the planner’s continuation value.
The economy is driven by a single Brownian shock, and two suitably chosen assets that
can be continuously traded are therefore sufficient to complete the markets in the sense of
Harrison and Kreps (1979). Let the two traded assets be an infinitesimal risk-free bond
in zero net supply that yields a risk-free rate rt = r (θt1 ) and a claim on the aggregate
endowment with price Ξt = Yt ξ (θt1 ), where ξ (θ1 ) is the aggregate wealth-consumption ratio.
Individual wealth levels are denoted Ξnt = Yt ζ n (θt1 ) ξ n (θt1 ), where ξ n (θ1 ) are the individual
wealth-consumption ratios.
The results reveal that as the Pareto share of one of the agents converges to zero, the
infinitesimal returns associated with the two assets converge to those which prevail in a
homogeneous economy populated by the agent with the large Pareto share. Solving the
individual optimization problems yields the required limits for the discount rates ν n . The
limiting problems correspond to the analysis of homogeneous economies for which analytical
solutions exist, and thus the limits for ν n are also available in closed form.
The proof also shows that the conditions on the limiting behavior of the discount rates
18
in Proposition 6 that assure the existence of a nondegenerate long-run equilibrium can be
directly restated as conditions on the limiting expected growth rates of the logarithm of
individual wealth levels in a decentralized economy.
3.3.1
Equilibrium prices
Homotheticity of preferences implies that individual wealth-consumption ratios are given by
ξ n θ1
1
=
β
1/γ
J˜n (θ1 )
ζ n (θ1 )
ρ
.
(20)
I start by assuming that ξ n (θ1 ) are functions that are bounded and bounded away from
zero. This, among other things, implies that the discount rate functions ν n (θ1 ) in (19) are
bounded and that the drift and volatility coefficients in the stochastic differential equation
for θ1 , (18), are bounded as well. The assumption will ultimately be verified by a direct
calculation of the limits of ξ n (θ1 ) as θ1 0 or θ1
1.
Without loss of generality, it is sufficient to focus on the case θ1 0. First notice some
asymptotic results for the planner’s continuation value J˜ (θ1 ).
Lemma 8 The solution of the planner’s problem implies that
lim θ1 J˜θ1 θ1 = lim
θ1
1
θ 1 0
θ 0
2
J˜θ1 θ1 θ1 = lim
θ1
1
θ 0
3
J˜θ1 θ1 θ1 θ1 = 0.
The Markov structure of the problem implies that the evolution of the continuation values
and consumption shares can be written as
dJ˜n (θt1 )
= μJ˜n θt1 dt + σJ˜n θt1 dWt
1
n
˜
J (θt )
dζ n (θt1 )
= μζ n θt1 dt + σζ n θt1 dWt ,
ζ n (θt1 )
(21)
(22)
where the drift and volatility coefficients are functions of θ1 , and the results from Lemma 8
allow the characterization of their limiting behavior.
Lemma 9 The coefficients in equations (21–22) for agent 2 satisfy
μJ˜2 θ1 = lim
σJ˜2 θ1 = lim
μζ 2 θ1 = lim
σζ 2 θ1 = 0.
lim
1
1
1
1
θ 0
θ 0
θ 0
19
θ 0
(23)
The result follows from an application of Itô’s lemma to J˜2 and ζ 2 . Utilizing formulas
(16) and (17), the coefficients will contain expressions for the value function J˜ (θ1 ) and its
partial derivatives up to the third order, and all the expressions can be shown to converge
to zero using Lemma 8.
Using the construction from Duffie and Epstein (1992a), the stochastic discount factor
process for agent 2 under the subjective probability measure Q2 is given by
γ−1 2 1 ρ−1 ˜2 1 1− γρ
t
Yt
J (θt )
ζ (θt )
St2 = exp −
ν 2 θs1 ds
.
1
2
Y0
ζ (θ0 )
J˜2 (θ01 )
0
(24)
Since limθ1 0 ν 2 (θ1 ) = ν 2 , which is given in (8), and Lemma 9 states that the local drift and
volatility of the last two terms decline to zero as θ1 0, the infinitesimal risk-free rate and
the local price of risk converge to their homogeneous economy counterparts. Moreover, the
price of aggregate endowment Ξ converges as well, and so does the local return on aggregate
wealth. The following Proposition summarizes the limiting pricing implications.
Proposition 10 As θ1 0, the infinitesimal risk-free rate r (θ1 ), the aggregate wealthconsumption ratio ξ (θ1 ), and the drift and volatility coefficients of the aggregate wealth process dΞt /Ξt = μΞ (θt1 ) dt + σΞ (θt1 ) dt converge to their homogeneous economy counterparts:
lim
r θ1
1
θ 0
lim ξ θ1
θ 1 0
lim μΞ θ1
θ 1 0
1
= r (0) = β + (1 − ρ) μy + u2 σy − (2 − ρ) (1 − γ) σy2 ,
2
−1
1
2
2
= ξ (0) = β − ρ μy + u σy − (1 − γ) σy
,
2
= μy , and lim
σΞ θ1 = σy .
1
θ 0
Consequently, the infinitesimal return on the claim on aggregate wealth,
ξ θt1
−1
+ μΞ θt1
dt + σΞ θt1 dWt ,
(25)
has coefficients that converge as well.
Notice that the convergence of the coefficients of the wealth process is not an immediate
consequence of the convergence of the aggregate wealth-consumption ratio. It may be that
the wealth-consumption ratio ξ (θ1 ) converges as θ1 0, yet its price dynamics are such that
μΞ (θ1 ) and σΞ (θ1 ) do not converge to μy and σy , respectively. The fact that this does not
happen is closely linked to the dynamics of log θ1 . The bounded drift and volatility coefficient
of log θ1 assure that the local variation in ξ (θ1 ) becomes irrelevant as log θ1 −∞.
20
The results in Proposition 10 are sufficient to proceed with the construction of the main
result. As a side note, prices of finite-horizon risk-free claims and individual cash flows from
the aggregate endowment converge as well:
Corollary 11 For every fixed maturity t, prices of a zero-coupon bond and a claim to a
payout from the aggregate endowment stream converge to their homogeneous economy counterparts as θ1 0.
3.3.2
Decision problem of an agent with negligible wealth
I have now established that the actual general equilibrium price dynamics that agent 1 with
infinitesimal wealth takes as given when solving his portfolio problem are locally the same
as those in an economy populated only by agent 2. However, the construction of the main
result is not completed yet. The marginal utility of agent 1 is forward looking due to the
nonseparable nature of the preferences, and will thus depend on agent’s 1 continuation value.
It remains to show that the continuation value of agent 1 converges as well.
Agent 1, whose wealth Ξ1 is close to zero, solves
λ̄1t Vt1
= max
Et
1 1 1
C ,π ,ν
∞
t
λ̄1s F
Cs1 , νs1
ds
subject to (6) and the budget constraint,
C1
dΞ1t
1
1
1 −1
1
1
= r θt + πt ξ θt
+ μΞ θt − r θt − t1 dt + πt1 σΞ θt1 dWt ,
Ξ1t
Ξt
(26)
where π 1 is the portfolio share invested in the stock. The homogeneity of the problem
motivates the guess
γVt1 = Ξ1t
γ
V̂ 1 θt1 .
(27)
The drift and volatility coefficients depend explicitly on θ1 because Ξ1 and θ1 are linked
through
Ξ1t
= Yt ζ
1
θt1
β
1
− 1−ρ
1
γρ 1−ρ
1
V̂ (θ)
.
(28)
Recall that we are interested in the characterization of the limiting solution as θ1 0.
21
The associated HJB equation leads to a second-order ODE (omitting dependence on θ1 )
1
1− γρ 1−ρ
1
1 1−ρ
1
β
2
1
1
1
V̂
0 = max
+ V̂
− + μΞ1 + u σΞ1 − (1 − γ) (σΞ1 ) + (29)
β
C 1 ,π 1 ,ν 1 ρ
ρ
2
1
2 11
μθ1 + u1 σθ1 + σθ1 σΞ1 + V̂θ11 θ1 θ1
(σθ1 )2 ,
+V̂θ11 θ1
γ
2γ
which yields the first-order conditions on Ct1 and πt1 :
1
− γρ 1−ρ
1
Ct1
1
1
1−ρ
= β
V̂ θt
Ξ1t
πt1
=
[ξ (θt1 )]
−1
(30)
+ μΞ (θt1 ) + u1 σΞ (θt1 ) − r (θt1 ) +
(1 − γ) (σΞ (θt1 ))
θ V̂ 11 (θ 1 )
θ
V̂ 1 (θ 1 )
σθ1 (θt1 ) σΞ1 (θt1 )
2
,
where μΞ1 and σΞ1 are the drift and volatility coefficients on the right-hand side of (26),
and μθ1 and σθ1 are the coefficients associated with the evolution of dθt1 /θt1 . Notice that the
portfolio choice π 1 almost corresponds to the standard Merton (1971) result, except the last
term in the numerator which explicitly takes into account the covariance between agent’s 1
wealth and the evolution in the state variable θ1 imposed by (28).
The solution of this equation determines the consumption-wealth ratio of agent 1 and,
consequently, the evolution of his wealth. While a closed-form solution of this equation is
not available, it is again possible to characterize the asymptotic behavior as θ1 0.
Lemma 12 The following results hold:
lim θ1 V̂θ11 θ1 = lim
θ1
1
θ 1 0
θ 0
2
V̂θ11 θ1 θ1 = 0.
These results are similar to those in Lemma 8. They imply that the derivative terms in
the ODE (29) vanish as θ1 0, and we obtain the limit for V̂ 1 (θ1 ) and the evolution of Ξ1
in closed form.
Proposition 13 The consumption-wealth ratio of agent 1 converges to
lim β
θ 1 0
1
1−ρ
V̂
1
θ
1
1
− γρ 1−ρ
1
2
2
= β − ρ μy + u σy − (1 − γ) (σy ) −
2
ρ
−
1−ρ
22
2
1
u −u
2
1 (u1 − u2 )
σy +
2 1−γ
(31)
and the wealth share invested into the claim on aggregate consumption to
π 1 θ1 = 1 +
lim
1
θ 0
u1 − u2
.
(1 − γ) σy
(32)
It follows that the asymptotic coefficients for the evolution of agent’s 1 wealth are
2
u1 (u1 − u2 )
1
1 2 − ρ (u1 − u2 )
(u1 − u2 ) σy +
−
1−ρ
21−ρ 1−γ
1−γ
1
2
u −u
.
= σy +
(1 − γ)
μΞ1 θ1
lim
1
= μy +
θ 0
lim σΞ1 θ1
θ 1 0
Naturally, the wealth evolution must track the evolution of the aggregate endowment
when u1 = u2 .
3.3.3
Limiting relative patience and relationship to wealth growth
Importantly, these results allow one to calculate the limiting discount rate ν 1 (θ1 ) and state
the main result of this section.
Proposition 14 The expressions for the limiting behavior of the relative patience in Proposition 6 are
lim ν
2
θ 1 0
lim ν
θ 1 1
2
θ
θ
1
1
−ν
1
−ν
1
θ
θ
1
1
ρ−γ
=
1−ρ
ρ−γ
=
1−ρ
2
1
2
1
2
u −u
u −u
1 (u1 − u2 )
σy +
2 1−γ
1 (u1 − u2 )
σy −
2 1−γ
,
(33)
.
(34)
2
Section 3.4 discusses which regions of the parameter space satisfy the individual survival
and extinction conditions from Proposition 6. It remains for me to verify that the assumption
about the boundedness of wealth consumption ratios indeed holds.
Corollary 15 Under parameter restrictions in Assumption 2, the wealth-consumption ratios
are bounded and bounded away from zero.
Notice that while Assumption 2 imposes a restriction on the time preference parameter
β of the agents, the survival conditions do not explicitly depend on β. The survival results
thus always hold with the implicit assumption that time discounting is sufficiently large.
The construction of the main survival result utilized the link between the planner’s problem and the competitive equilibrium. It turns out that relative patience conditions that
assure survival can be restated as conditions on the relative growth rates of individual wealth.
23
Corollary 16 The survival conditions in part a) of Proposition 6 are equivalent to:
2
2
2
2
(i) limθ1 0 μΞ1 (θ1 ) − 12 [σΞ1 (θ1 )] > limθ1 0 μΞ2 (θ1 ) − 12 [σΞ2 (θ1 )] ,
(ii) limθ1 1 μΞ1 (θ1 ) − 12 [σΞ1 (θ1 )] < limθ1 1 μΞ2 (θ1 ) − 12 [σΞ2 (θ1 )] .
Verifying the conditions in Proposition 6 therefore amounts to checking that the expected
growth rate of the logarithm of wealth is higher for the agent who is at the brink of extinction.
This is of course a natural condition for a competitive equilibrium, but the planner’s problem
was still required to determine these limiting growth rates.
3.4
Limiting behavior of asset prices
The results in the previous section establish that, as the Pareto share of one of the agents
becomes negligible, agents make their portfolio and consumption-saving decisions as if they
observed prices in a homogeneous economy populated only by the large agent. If there exists
a nondegenerate stationary distribution of the Pareto share θ1 , then an agent will always
have a nontrivial price impact in the future, even if his current Pareto share is negligible.
The forward looking nature of the optimization problem then implies that this price impact
should be taken into account when making current decisions. The results show that as the
Pareto share of the agent vanishes, the time when his price impact becomes relevant is so
distant that it is immaterial for current decisions.
The logic manifests itself in the behavior of the last term in the numerator of the portfolio
share π 1 in equation (30). This term explicitly takes into account agent 1’s knowledge about
the impact of his portfolio decision on equilibrium prices. Since this term vanishes as θ1 0,
the agent understands that asymptotically the portfolio decisions made by agents of his type
will not have any impact on local equilibrium price dynamics, and thus behaves as if he
resided in an economy populated only by agent 2.
This implies that the survival question, whose answer only depends on the behavior at
the boundaries, can be resolved by studying homogeneous economies with an infinitesimal
price-taking agent. Even if the agent survives with probability one and has an impact on
equilibrium prices in the long run, he does not influence current prices and returns.
The dynamics of the Pareto share (18) that has bounded drift and volatility coefficients
is also informative about experiments that ‘inject’ infinitesimal heterogeneous agents into
an initially homogeneous economy. By making the initial θ01 arbitrarily close to zero, one
can extend the time before the presence of the new agent becomes noticeable (measured,
24
e.g., by sufficiently large deviations in prices or return distributions from their homogeneous
economy counterparts) arbitrarily far into the future.
3.5
Survival regions
This section analyzes the regions of the parameter space in which agents with distorted
beliefs survive or dominate the economy. It turns out that all four combinations outlined
in Proposition 6 can occur, and Figure 1 visualizes the survival regions. Each panel fixes
the belief distortions (u1 , u2 ) and the volatility of aggregate endowment σy , and plots the
regions in the risk aversion / inverse of IES (1 − γ, 1 − ρ) plane. The results do not reveal
what happens at the boundaries of the regions where conditions from Proposition 6 hold
with equalities, but the survival characteristics inside the individual regions are well-defined.
3.5.1
Asymptotic results
It is useful to describe the asymptotic results as either risk aversion or intertemporal elasticity
of substitution moves toward extreme values, holding the other parameters fixed.
Corollary 17 Holding other parameters fixed, the survival restrictions imply the following
asymptotic results.
(a) As risk aversion increases (γ −∞), the agent who is relatively more optimistic about
the growth rate of aggregate endowment always dominates in the long run.
(b) As agents become risk neutral (γ
1), each agent dominates in the long run with a
strictly positive probability.
(c) As intertemporal elasticity of substitution increases (ρ
1), the relatively more opti-
mistic agent always survives. The relatively more pessimistic agent survives (and thus
a nondegenerate long-run equilibrium exists) when risk aversion is sufficiently small.
(d) As the intertemporal elasticity of substitution decreases to zero (ρ −∞), a nondegenerate long-run equilibrium cannot exist.
In order to shed more light on the results, it is useful to consider the discrete-time version
of the recursive preferences, featured by Epstein and Zin (1989):
Ṽt =
−β
1−e
ρ
−β
(Ct ) + e
25
EtQ
γ γρ ρ1
Ṽt+1
.
1
2
1
Optimistic agent 1, rational agent 2 (u = 0.25, u = 0)
either agent dominates
1
agent 2
dominates
1.5
(IES)−1
(IES)
−1
1.5
nondegenerate
long−run equilibrium
0.5
2
Mildly optimistic agent 1, rational agent 2 (u = 0.1, u = 0)
2
either agent dominates
2
1
agent 2
dominates
nondegenerate
long−run equilibrium
agent 1
dominates
0.5
agent 1
dominates
0
0
1
2
3
4
5
risk aversion
6
7
1
8
9
0
10
0
1
2
2
either agent dominates
1.5
agent 2
dominates
(IES)−1
1
5
risk aversion
6
7
8
9
10
8
9
10
9
10
2
Mildly pessimistic agent 1, rational agent 2 (u = −0.1, u = 0)
2
either agent dominates
(IES)
−1
1.5
4
1
Pessimistic agent 1, rational agent 2 (u = −0.25, u = 0)
2
3
0.5
1
agent 2
dominates
nondegenerate
long−run equilibrium
0.5
nondegenerate
long−run equilibrium
0
0
1
2
3
4
5
risk aversion
6
7
8
9
0
10
0
1
4
5
risk aversion
6
7
2
Pessimistic agent 1, optimistic agent 2 (u = −0.08, u = 0.0775)
10
either agent dominates
1
agent 2
dominates
0.5
nondegenerate
long−run equilibrium
0
agent 1
dominates
8
(IES)−1
−1
1.5
(IES)
3
1
Incorrect agent 1, rational agent 2 (u1 = 0.25, u2 = 0), deterministic aggregate endowment
2
0
2
1
2
3
4
5
risk aversion
6
7
either agent
dominates
6
agent 2
dominates
4
2
nondegenerate
long−run equilibrium
8
9
10
0
0
1
2
3
4
5
risk aversion
6
7
8
Figure 1: Survival regions for different parameterizations. All panels except the bottom left panel
assume σy = 0.02. In the bottom left panel, aggregate endowment is deterministic, σy = 0. Belief
distortion parameters un are shown in the titles of individual panels.
The risk aversion parameter γ drives the risk adjustment of the next-period continuation
value Ṽt+1 ; and as risk aversion increases, the lower tail of the distribution of Ṽt+1 will
contribute with an increasingly larger penalty to the expected value. When two agents differ
in their beliefs, the more pessimistic agent assigns a higher probability to the tail events.
Since the penalty increases as γ −∞, he is willing to sacrifice an increasingly large amount
of wealth in the remaining states in order to insure against the low-probability tail event
that will ultimately drive him to extinction.
In the other extreme, when γ
1 and agents become risk neutral with respect to
intratemporal gambles, all that matters in the next period’s contribution is the expected
continuation value. Depending on the current distribution of wealth, there will be a beliefratio threshold that divides the realizations of the shock tomorrow into two sets, where in
each set one of the agents appropriates all wealth while the other becomes immediately
26
extinct.
A higher IES provides more incentives to substitute consumption across time and accept
steeper consumption profiles. In an economy with ρ > 0 (IES > 1), a relatively more
optimistic agent with negligible wealth who faces prices essentially determined by the more
pessimistic agent is willing to postpone consumption into the future vis-à-vis higher expected
returns on his portfolio, as indicated by the last term in the consumption-wealth ratio (31).
As ρ
1, this saving motive dominates, and the relatively more optimistic agent outsaves
the other agent whenever the Pareto share of the relatively more optimistic agent becomes
sufficiently small, thus guaranteeing his survival.
A similar mechanism operates when the Pareto share of the relatively more pessimistic
agent becomes negligible. Observe that the last term in brackets in the consumption-wealth
ratio (31), which dominates the saving decision when ρ
1, is equal to
1 1
u − u2 σy 1 + π 1 (0) .
2
If agent 1 is relatively more pessimistic, then u1 − u2 < 0, and thus π 1 (0) < −1 is needed
for the saving motive of agent 1 to increase as ρ
1. In that case, despite being relatively
more pessimistic about the growth rate of aggregate endowment, agent 1 becomes sufficiently
optimistic about the return on his own portfolio, which contains a short position in the claim
on aggregate endowment, so that he outsaves the relatively more optimistic agent 2 when
the Pareto share of agent 1 declines to zero. This will happen when γ is not too large — a
high risk aversion prevents the agents from taking sufficiently disparate portfolio positions
that would imply a short stock position in the portfolio of the pessimistic agent.
Finally, when preferences of the agents become inelastic (ρ −∞), formulas in Proposition 14 imply that the survival conditions cannot hold simultaneously. Inelastic preferences
imply that the agents are unwilling to substantially change the slope of their consumption
profiles; and the mechanism based on differences in saving rates, which operated for high
IES, is largely absent. The consumption-wealth ratio when one agent has a negligible Pareto
share is dominated by the second term in expression (31), which is common for both agents.
An increased willingness to save when the Pareto share decreases is thus not strong enough
to compensate for mistakes in portfolio allocation for at least one of the agents. When IES
is sufficiently low, then a pessimistic agent 1 with relatively more distorted beliefs can dominate the economy in situations when the sum of the belief distortions of the two agents is
not too large (the exact condition requires u1 + u2 + 2σy > 0).
27
3.5.2
Economic interpretation of the equilibrium price mechanism
Figure 1 documents the survival regions results for different belief parameterizations. Previous literature on survival under separable preferences already confirmed that along the
dotted diagonal, the agent with the smaller belief distortion dominates. This paper establishes that although this conclusion holds in a neighborhood of the diagonal, there are wide
regions of the parameter space where nondegenerate long-run equilibria exist. Moreover,
such equilibria generically arise for plausible parameterizations when risk aversion is larger
than the inverse of IES that are typically used in the asset pricing literature.
The two boundaries in the top left panel which delimit the region with a nondegenerate
stationary distribution of the Pareto share are asymptotically parallel as γ −∞ with
slope 2σy / (u1 + u2 + 2σy ). The graphs confirm the asymptotic results from Section 3.5.1,
where I explained why the relatively more optimistic agent dominates the economy if risk
aversion is sufficiently high (γ −∞). But why do we obtain an intermediate region in
the parameter space where nondegenerate long-run equilibria exist? The existence of these
equilibria critically depends on general equilibrium price effects.
Consider a parameterization that falls into the region where a nondegenerate long-run
equilibrium exists in an economy populated by an optimistic and a rational agent, shown in
the top two panels of Figure 1. Proposition 10 shows that as the Pareto share of one of the
agents becomes negligible, prices are determined by the large agent. The optimistic agent
holds a leveraged position in the stock, which is relatively overpriced due to his presence,
but the extent of overpricing varies with the Pareto share. When the Pareto share of the
optimistic agent is large, overpricing is large as well; investment in the overpriced asset slows
down the growth rate of the optimistic agent’s wealth, and allows the survival of the rational
agent. On the other hand, when the Pareto share of the rational agent is large, overpricing
disappears, and the leveraged high expected return strategy of the optimistic agent prevents
his extinction.
The equilibrium price mechanism also explains why survival is not sufficient for dominance. Consider an economy populated by a pessimistic and a rational agent (the middle
two panels of Figure 1) and the survival region when risk aversion is larger that the inverse
of IES. Section 3.5.1 shows that a pessimistic agent is optimistic about the growth rate of his
own wealth if his portfolio involves a short position in the stock. When the short position
is sufficiently large and IES is larger than one, his consumption-wealth ratio can decline
enough so that he is able to outsave the rational agent. However, when the Pareto share of
the pessimistic agent is large, he determines equilibrium prices, and his wealth share invested
28
in the stock approaches one. This makes the agent pessimistic again about the growth rate
of his own portfolio, and the saving motive disappears.
The general equilibrium price effects thus play a central role in the construction of nondegenerate long-run equilibria. Partial equilibrium models with exogenous price dynamics
that do not depend on wealth shares of individual agents cannot replicate this survival
mechanism.
3.5.3
Comparative statics
Figure 1 also shows the sensitivity of the survival regions to changes in parameter values.
Survival regions do not depend on the time preference parameter β and the growth rate of
the economy μy , because these parameters influence the decision rules of the two agents in a
symmetric way and offset each other in the difference in growth rates of individual wealth.
The only remaining parameters to analyze are the belief distortions un of the two agents,
and the volatility of aggregate endowment σy .
The belief distortions used in Figure 1 may be considered large — the incorrect agent
misperceives the growth rate of the economy by |u1 σy | = 0.005, which is a quarter of a
plausible value for the growth rate of μy = 0.02. The top two panels illustrate how survival
regions change when the magnitude of the belief distortion u1 decreases in a situation when
agent 2 is rational.
Decreasing the magnitude of the belief distortion in general improves the survival chances
of the relatively more optimistic agent. The top right panel shows that decreasing the belief
distortion of an optimistic agent shrinks the regions of the parameter space in which the
rational agent dominates and where nondegenerate long-run equilibria exist. A decrease in
u1 > 0 leads to a lower leverage of agent 1 (see the asymptotic formula (32) for π 1 (θ1 )) and
less overpricing of the risky asset in which the optimistic agent overinvests. Yet the stronger
insurance motive of the relatively more pessimistic agent is still present. As u1 declines to
zero, the relative patience ν 2 (θ1 ) − ν 1 (θ1 ) close to the boundaries in expressions (33) and
(34) is dominated by the linear term u1 σy , but the threshold in the survival condition in
Proposition 6 decreases quadratically. This makes it easier for agent 1 to survive, despite
the fact that the speed of convergence to a steady state distribution of θ1 , influenced by the
relative patience in the drift term of (18), may be slower.
Interestingly, the middle right panel of Figure 1 shows that decreasing the distortion of
a pessimistic agent in general diminishes his survival chances when risk aversion is larger
than the inverse of IES. In the previous sections, I argued that a pessimistic agent can
29
coexist with a rational agent in the long run if he is able to outsave the rational agent when
his Pareto share becomes small. This will occur in economies with IES > 1 in situations
when the pessimistic agent is sufficiently optimistic about the growth rate of his own wealth,
i.e., when he chooses a sufficiently large short position in the stock. But decreasing the
magnitude of u1 also decreases the agent’s willingness to short the stock (see equation (32)),
which subsequently diminishes the relative incentives of the pessimistic agent to save.
The region in which nondegenerate long-run equilibria exist can be expanded if we consider economies populated by an optimistic and a pessimistic agent (the bottom right panel
of Figure 1, with a different scale on the vertical axis). In these situations, motives to misallocate assets arising from belief distortions work in opposite directions for the two agents.
For instance, a pessimistic agent can coexist with a rational agent (u1 < 0, u2 = 0)
in the long run only when IES is larger than one (middle panels of Figure 1), although
a nondegenerate long-run equilibrium can exist for IES smaller than one if we consider
parameterizations where agent 2 is optimistic, albeit with a smaller belief distortion (u1 <
−u2 < 0). With an at least somewhat optimistic agent 2, survival chances of the pessimistic
agent no longer depend solely on his ability to outsave the other agent when preferences are
elastic, but also on the willingness of the optimistic agent to overpay for the claim on the
aggregate endowment.
The case of ‘symmetric’ optimism and pessimism, 0 < u1 = −u2 , which under CRRA
preferences generates a rather delicate economy without a stationary distribution for the
Pareto share yet with both agents surviving in the long run, is dissected in a straightforward
way when ρ = γ. The parameter space is divided into four regions by the 45◦ line and a
vertical boundary for γ that satisfies (1 − γ) σy = u1 , and one of the four survival possibilities
from Proposition 6 arises in each of the four regions. The online appendix analyzes this case
in more detail.
Finally, the bottom left panel in Figure 1 illustrates that the analysis is still plausible
when aggregate endowment is deterministic, σy = 0, as long as agents can write contracts
on the realizations of W . Volatility of the aggregate consumption stream has an impact on
the shape of the survival regions but is not central for the existence of parameterizations
under which a nondegenerate long-run equilibrium exists. The essential component of the
model is the existence of a betting mechanism with a probability distribution of outcomes
about which the agents disagree. Not surprisingly, the bottom left panel in Figure 1 also
corresponds to an economy where aggregate endowment is driven by a shock uncorrelated
with W over which there is no disagreement.
30
3.6
Comparison to economies with only terminal consumption
In this paper, I analyze economies with intermediate consumption. Kogan, Ross, Wang, and
Westerfield (2006) deal with a different framework with two agents endowed with CRRA
preferences. In their economy, there is no intermediate consumption and the agents split and
consume an aggregate dividend payoff at a terminal date T . The dividend evolves according
to a geometric Brownian motion (1) as in this paper, and agents can continuously retrade
claims on the terminal payoff during the lifetime of the economy. The notion of survival in
that framework is captured by analyzing the limit of the consumption share distribution in
a sequence of economies as T
∞.
Without intermediate consumption, the agent’s intertemporal decision is reduced to the
maximization of the (risk-adjusted) expected growth rate of the portfolio. In this respect, the
framework is similar to a model in this paper under unitary IES when agents’ consumptionwealth ratio is constant and equal to β, and intermediate consumption has no impact on the
difference of the wealth growth rates.
The difference that prevents a direct comparison of the results lies in the valuation of
wealth. In the absence of intermediate consumption, Kogan, Ross, Wang, and Westerfield
(2006) use the price of a bond maturing at time T as numeraire and define the initial wealth
in the economy with horizon T as the time 0 price of the terminal payoff. (In this paper,
this quantity corresponds to the price of a single cash flow from the aggregate endowment
paid out at time T , scaled by the price of a bond with corresponding maturity.) Then they
consider two approaches to survival analysis.
In the ‘general equilibrium’ approach, they study the limiting properties of the terminal
consumption allocation obtained as a solution of a sequence of planner’s problems as T
∞.
A critical assumption in this approach is the choice of the initial Pareto shares. These are
chosen so that the initial wealth shares of the two agents are identical, which requires the
initial Pareto share of the irrational agent to approach one as T
∞. This mechanism
reweighs the behavior of the tail and allows an optimistic agent to ‘survive’ in the sequence
of planner’s problems. In economies with intermediate consumption, consumption at distant
dates contributes only little to the wealth levels, and thus the reweighting of initial Pareto
shares in order to achieve equal initial wealth levels would have no effect on the survival
results. Under the ‘general equilibrium’ notion of survival in Kogan, Ross, Wang, and
Westerfield (2006), optimistic agents can survive when risk aversion is larger than one but
the survival regions differ from the results in this paper under unitary IES (ρ = 0).
Kogan, Ross, Wang, and Westerfield (2006) contrast their ‘general equilibrium’ to a
31
5
RA = 3
RA = 4
RA = 6
RA = 8
q(ζ1)
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ζ1
Figure 2: Stationary distributions for the consumption share of the agent with distorted beliefs
ζ 1 θ 1 . All models are parameterized by u1 = 0.25, u2 = 0, IES = 1.5, β = 0.05, μy = 0.02,
σy = 0.02, and differ in levels of risk aversion.
simplified approach that is analogous to the boundary analysis in this paper and that they
call the ‘partial equilibrium’ method. This method constructs a homogeneous economy
injected with an infinitesimal agent with different beliefs under the assumption that he does
not affect local price dynamics. It turns out that this method delivers exactly the same
survival regions as those derived in this paper under unitary IES.
Propositions 10 and 11 show that in the model with intermediate consumption considered
in this paper, the return on aggregate wealth and prices of individual finite-horizon cash flows
from the aggregate endowment converge to their homogeneous economy counterparts and
thus the ‘partial equilibrium’ approach is actually the correct method for this paper under
general equilibrium. However, these results do not translate to the setup considered in
Kogan, Ross, Wang, and Westerfield (2006). Although prices of individual cash flows from
the aggregate endowment converge for every fixed T ≥ 0, this convergence is not uniform on
T ∈ [0, ∞), which in general invalidates the result on converging returns and prices for the
limit as T
4
∞.
Dynamics of long-run equilibria
In Section 3, I derived parametric restrictions on the survival regions. However, even if a
nondegenerate long-run equilibrium exists, the question remains whether this equilibrium
delivers quantitatively interesting dynamics under which each of the agents can gain a significant wealth share. This section investigates numerically the equilibrium allocations and
prices and their dynamics by solving the ODE (15) and decentralizing the allocations.
32
−3
4
0.06
0.02
2
1
0
−0.02
RA = 0.1
RA = 0.75
RA = 4
−0.04
−0.06
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2
drift of the θ1 process
survival threshold
1
1
ν (θ ) − ν (θ )
0.04
x 10
0
−2
−4
−6
RA = 0.1
RA = 0.75
RA = 4
−8
1
−10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
θ1
θ
Figure 3: Relative patience
ν 2 θ 1 − ν 1 θ 1 / u1
2
(left panel) and the drift component of the
Pareto share evolution E dθt1 | Ft /dt (right panel) as functions of the Pareto share θ 1 . All models
are parameterized by u1 = 0.25, u2 = 0, IES = 1.5, β = 0.05, μy = 0.02, σy = 0.02, and differ in
levels
The dotted horizontal line in the left panel represents the survival threshold
of risk aversion.
2
2
1
1
2
− u
from Proposition 6.
u
2
4.1
Consumption allocation
Figure 2 plots the densities q (ζ 1 ) for the stationary distribution of the consumption share in
economies with an optimistic agent. The parameterizations10 are chosen along a horizontal
line in the top left panel of Figure 1. As risk aversion increases, the distribution of consumption shifts toward the optimistic agent, but the equilibria in general permit substantial
variation over time in the consumption shares of the two agents.
The existence of nondegenerate long-run equilibria depends on the behavior of the relative
patience ν 2 (θ1 ) − ν 1 (θ1 ) in the neighborhood of the boundaries. Figure 3 displays three
different cases. The dashed line represents the low risk aversion case in which both attracting
conditions from Proposition 6 hold and each of the agents dominates with a strictly positive
probability. The solid line corresponds to a parameterization that is close to the CRRA
case when only the survival condition for the rational agent 2 is satisfied (with CRRA
preferences, the relative patience would be identically zero). Finally, a case for which both
survival conditions hold is shown by the dot-dashed line.
Figure 3 also plots the impact of relative patience on the drift component of the Pareto
share process. The drift vanishes at the boundaries and the boundaries are unattainable
(a reflection of the Inada conditions), but sufficiently large positive (negative) slopes at the
left (right) boundaries assure the existence of a nondegenerate stationary equilibrium of the
Pareto share.
The essential components of the survival mechanism are the propensity to save and the
10
A full solution of the consumption dynamics requires setting additional parameters that do not influence
the survival regions. I set β = 0.05 and μy = 0.02. The high value for the time preference coefficient is
chosen merely to assure that restrictions in Assumption 2 hold for all compared models.
33
0.02
0.02
IES = 0.75
IES = 1.50
IES = 2.00
IES = 3.00
0.01
0.005
0
−0.005
−0.01
IES = 0.75
IES = 1.50
IES = 2.00
IES = 3.00
0.015
(ξ2)−1 − (ξ1)−1
(ξ2)−1 − (ξ1)−1
0.015
0.01
0.005
0
−0.005
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−0.01
1
0
0.1
0.2
0.3
0.4
1
0.5
0.6
0.7
0.8
0.9
1
ζ1
ζ
Figure 4: Difference in consumption-wealth ratios ξ n (θ 1 ) as a function of the consumption share
ζ 1 (θ 1 ). The left panel considers an optimistic agent 1 (u1 = 0.25) while the right panel a pessimistic
agent 1 (u1 = −0.25). The remaining parameters are u2 = 0, RA = 2, β = 0.05, μy = 0.02, σy =
0.02, and individual curves correspond to different levels of intertemporal elasticity of substitution.
6
6
RA = 3
RA = 4
RA = 6
4
RA = 3
RA = 4
RA = 6
4
πn
2
πn
2
0
0
−2
−2
−4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ζ1
−4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ζ1
Figure 5: Wealth shares π n (θ 1 ) of the two agents invested in the claim to aggregate endowment as
functions of the consumption share ζ 1 (θ 1 ). The left panel considers an optimistic agent 1 (u1 = 0.25)
while the right panel a pessimistic agent 1 (u1 = −0.25). The remaining parameters are u2 = 0,
IES = 1.5, β = 0.05, μy = 0.02, σy = 0.02, and individual curves correspond to different levels of
risk aversion. Wealth share curves originating at 1 for ζ 1 θ 1 = 1 (ζ 1 θ 1 = 0) belong to agent 1
(2).
portfolio allocation of the two agents. Figure 4 displays the differences in the consumptionwealth ratios [ξ n (θ1 )]
−1
of the two agents, which are primarily driven by the intertemporal
elasticity of substitution. For the case of IES = 1, the difference is exactly zero since each
agent consumes a fraction β of his wealth per unit of time. A higher IES improves the survival
chances of the agent who is relatively more optimistic about the return on his own wealth,
as he is willing to tilt his consumption profile more toward the future. Figure 4 captures this
effect for both an optimistic agent 1 (u1 = 0.25, left panel), as well as a pessimistic agent 1
(u1 = −0.25, right panel).
The portfolio allocation mechanism is depicted in Figure 5 and is closely related to the
behavior of the consumption-wealth ratios. The share of wealth invested in the risky asset is
primarily driven by the risk aversion parameter γ. A higher risk aversion limits the amount
of leverage. For the pessimistic agent, this implies that if risk aversion is high, he does not
form a large enough short stock position that would make him sufficiently optimistic about
34
the return on his own wealth and outsave the rational agent when IES > 1.
4.2
Evolution over time
In empirical applications, it may be advantageous if θ1 converges to its stationary distribution
from any initial condition fast enough, so that data observed over finite horizons are a
representative sample of the stationary distribution. Proposition 6 gives sufficient conditions
for the existence of a unique stationary distribution for θ1 but it does not say anything about
the rate of convergence.
It turns out that under the conditions in Proposition 6, convergence occurs at an exponential rate, so that the process θ1 does not exhibit strong dependence properties, although
the exponent may be small. Yan (2008) conducts numerical experiments under separable
utility when one of the agents always vanishes, and shows that the rate of extinction can be
very slow. The same quantitative result can hold under recursive preferences.
Having at hand numerical solutions for the evolution of θ1 and the function ζ 1 (θ1 ), one
can investigate conditional distributions of ζ 1 (θt1 ) conditional on θ01 by solving the corresponding Kolmogorov forward equation
1 ∂2 1
∂ 1
∂qt (θ1 )
1
1
θ σθ1 θ1
+ 1 θ μθ1 θ qt θ
−
2
1
∂t
∂θ
2 ∂ (θ )
2
qt (θ) = 0
for the conditional density qt (θ1 ) of θt1 with the initial condition q0 (θ1 ) = δθ01 (θ1 ), where δ
is the Dirac delta function, and then transforming to obtain the conditional density for ζ 1
pt ζ
1
θ
1
= qt θ
1
∂ζ 1 1
θ
∂θ1
−1
.
Figure 6 considers the evolution of conditional densities for the consumption share in
different economies. The speed of convergence depends on relative patience ν 2 (θ1 ) − ν 1 (θ1 )
that governs the magnitude of the drift term of θ1 and the shape of the function ζ 1 (θ1 )
depicted in bottom right panel.
For high levels of risk aversion, convergence of the conditional distribution pt is slow, due
to the low slope of ζ 1 (θ1 ). With a high level of risk aversion, agents are not willing to engage
in large bets on the realizations of the Brownian motions W , and wealth and consumption
shares evolve only slowly. In the example in Figure 6, it takes 2,500 periods until the density
pt is indistinguishable from the stationary density.
As risk aversion decreases, and agents are willing to bet larger portions of their wealth,
35
5
10
t = 25
t = 100
t = 250
t = 1000
t = 2500
t=∞
t
t
3
4
2
2
1
0
0
0.1
0.2
t=1
t=5
t = 10
t = 50
t = 100
4
p (ζ1)
6
1
p (ζ )
8
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0
0.1
0.2
0.3
0.4
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
1
3
t=1
t=5
t = 10
t = 50
t = 100
2
RA = 0.25
RA = 0.75
RA = 2
RA = 8
0.8
ζ1(θ1)
2.5
t
0.6
ζ
ζ
p (ζ1)
0.5
1
1
1.5
0.6
0.4
1
0.2
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
1
1
θ
ζ
Figure 6: Distributions of ζ 1 θt1 conditional on ζ 1 θ01 = 0.5, and the consumption share of agent
1, ζ 1 θ 1 , as a function of the Pareto share θ 1 (bottom right panel). In the top left panel, the
economy has a nondegenerate long-run distribution. In the top right panel, agent 2 dominates, and
in the bottom left panel, each agent dominates with a strictly positive probability. The parameters
are u1 = 0.25, u2 = 0, IES = 1.5, β = 0.05, μy = 0.02, σy = 0.02. Risk aversion is equal to 8 in the
top left panel, 0.75 in the top right panel, and 0.25 in the bottom left panel.
the evolution of the conditional density pt speeds up. In the middle panel of Figure 6,
consumption is substantially skewed toward the dominating agent 2 already after 50 periods.
In the bottom panel, when the risk aversion coefficient drops to 0.25 and each of the agents
dominates with a strictly positive probability, the mass of the density quickly shifts toward
both boundaries.
5
Extensions and concluding remarks
Before concluding, I consider two extensions of the analyzed model that involve Bayesian
learning about the underlying model and representation of other preference structures as
belief distortions. The online appendix outlines in more detail how to set up these problems
within the framework of this paper.
5.1
The role of learning
The analysis in this paper focuses on the case of fixed belief distortions. Agents are firm
believers in their probability models, and do not use new data to update their beliefs. A
36
natural question is to ask what happens when agents are allowed to learn.
Blume and Easley (2006) provide a detailed analysis of the impact of Bayesian learning
on survival under separable utility, and they are able to characterize the relationship between
survival chances and the complexity of the learning problem. The central message arising
from the analysis is that learning, which reduces belief distortions over time, in general aids
survival of agents with incorrect beliefs.
It seems to be reasonable to expect that this insight should hold also under nonseparable
preferences. Unfortunately, results presented in the previous analysis indicate that this
logic is not generally correct. For instance, the middle panels of Figure 1 show that the
survival region of a pessimistic agent can shrink if his belief distortion diminishes, and the
pessimistic agent moves from a region of the parameter space where a nondegenerate longrun equilibrium exists to one where only the rational agent survives. Whether the pessimist
can then learn quickly enough so that his beliefs converge to rational expectations at a rate
that allows survival depends on the complexity of the learning problem, as shown by Blume
and Easley (2006). As the beliefs converge, the evolution of the Pareto share process θ1
settles. The limiting distribution of θ1 as t
∞ from which we can deduce the wealth and
consumption distribution remains an open question.
5.2
Robust utility
The economic interpretation of the distortionary processes un is not limited to ‘irrationality’,
and other preference specifications lead to representations which are observationally equivalent to belief distortions. Consider, for instance, an agent who believes that the model for
the aggregate endowment dynamics is misspecified and views (1) only as a reference model
that approximates the true dynamics, as in the robust utility models of Anderson, Hansen,
and Sargent (2003) and Skiadas (2003). This class of models leads to a representation where
agent n views as relevant the realization of the worst case scenario, characterized by the least
favorable dynamics
dYt
= μy dt + σy (unt dt + dWtn ) ,
Yt
where W n is a Brownian motion under Qnu associated with an endogenously determined
distortionary process un . Epstein and Miao (2003) and Uppal and Wang (2003) construct
models with ambiguity aversion where the optimal solution to the minimization problem
involves a constant un , and thus exactly corresponds to the framework in this paper.
37
Under separable preferences, agents who fear misspecification more (and therefore assign
a lower penalty θ to deviations from the reference model) choose a more distorted worst
case scenario, which tends to worsen their survival chances. However, the results for constant belief distortions un indicate that survival chances of the more fearful agents may well
look much better for appropriate nonseparable parameterizations of preferences. A detailed
analysis of the dynamics of these models is left for future research.
5.3
Summary
Survival of agents with heterogeneous beliefs has been studied extensively under separable
preferences. The main conclusion arising from the literature is a relatively robust argument
in favor of the market selection hypothesis. Under complete markets and identical utility
functions, a two-agent economy is dominated in the long run by the agent whose beliefs are
closest to the true probability measure for a wide class of preferences and endowments. In
particular, Kogan, Ross, Wang, and Westerfield (2009) show elegantly that this result holds,
irrespective of the specification of the aggregate endowment process,11 as long as relative
risk aversion is bounded.
This paper shows that the robust survival result is specific to the class of separable
preferences. Under nonseparable recursive preferences of the Duffie-Epstein-Zin type, nondegenerate long-run equilibria exist for a broad set of plausible parameterizations when risk
aversion is larger than the inverse of the intertemporal elasticity of substitution. It is equally
easy to construct economies dominated by agents with relatively more incorrect beliefs.
The analysis reveals the important role played by the interaction of risk aversion with
respect to intratemporal gambles that determines risk taking, and intertemporal elasticity of
substitution that drives the consumption-saving decision. Critical for obtaining the survival
results, and in particular the nondegenerate long-run equilibria, are the general equilibrium
price effects generated by the wealth dynamics.
The survival results are obtained by extending the planner’s problem formulation of
Dumas, Uppal, and Wang (2000) to a setting with heterogeneous beliefs. Long-run survival
of the agents is determined by the dynamics of a stochastic process that models the Pareto
share of one of the agents as the share becomes negligible. This dynamics can be characterized
in closed form by studying the boundary behavior of a nonlinear ODE resulting from the
planner’s problem. This type of ODE arises in a wider class of recursive utility problems, so
11
The survival results under separable utility thus also hold for ‘exotic’ endowment processes like the rare
disaster framework in Chen, Joslin, and Tran (2010).
38
these results can be utilized in a broader variety of economic applications.
I provide in analytical form tight sufficient conditions that guarantee survival or extinction. These conditions can be interpreted as relative patience conditions analogous to
those in Lucas and Stokey (1984). An agent survives in the long run if his relative patience
becomes sufficiently large as his wealth share vanishes. However, in this framework, the
dynamics of relative patience arises endogenously as an equilibrium outcome, and is not a
direct property of agents’ preferences. I also show that the survival conditions are equivalent
to conditions on the limiting expected growth rates of the logarithm of individual wealth
levels in a decentralized economy
These results are obtained for a two-agent economy with an aggregate endowment process
that is specified as a geometric Brownian motion, but the theoretical framework can also be
utilized to derive an analog HJB equation for multi-agent economies with more sophisticated
Markov dynamics. In principle, the qualitative survival results should extend to a wider
class of models with stable consumption growth dynamics, although the analysis of the
existence of a stationary distribution for the Pareto share becomes more complicated in a
multidimensional state space.
Importantly, the developed solution method is not limited to constant distortions and
applies to a much wider class of preferences that are interpretable as deviations in beliefs.
I outline how to use the method in a framework with model uncertainty and learning and
in a model where agents are endowed with robust preferences. Solutions of these problems
are left as open questions for future research. Similarly, formulas for survival regions can
be extended by incorporating heterogeneity in preferences, as in Dumas, Uppal, and Wang
(2000), in a straightforward way.
The bad news for the market selection hypothesis is in some sense good news for models
with heterogeneous agents. Models with agents who differ in preferences or beliefs often
have degenerate long-run limits in which only one class of agents survives. This paper shows
that coupling belief heterogeneity (including preferences that can be interpreted as belief
distortions) and recursive preferences with empirically plausible parameters leads to models
in which the heterogeneity does not vanish over time.
39
A
Proofs
Schroder and Skiadas (1999) prove that V n (C n ) is concave. Consider
Proof of Lemma 3.
the case α1 0. Given optimal consumption streams C n (α), we have
J0 (α) = α1 V01 C 1 (α) + α2 V02 C 2 (α)
(35)
and since V01 C 1 (α) is bounded from above as a function of α, it follows that
α1 0
α1 V01 C 1 (α) −→ v 1 ≤ 0
and thus J0 0, α2 ≤ limα1 0 α2 V02 C 2 (α) ≤ α2 V02 (Y ).
1
2|γ|
Assume suboptimal policies Ĉ 1 α1 , α2 = α1
α1 V01 Ĉ 1 α1 , α2 = α1
and
Y and Ĉ 2 α1 , α2 = 1 − α1
γ
1+ 12 |γ|
α2 V02 Ĉ 2 α1 , α2 = α2 1 − α1
γ
1
2|γ|
1
2|γ|
Y . Then
α1 0
γ −1 Y0γ Ṽ n −→ 0
α1 0
γ −1 Y0γ Ṽ n −→ α2 V02 (Y )
which implies J0 0, α2 ≥ α2 V02 (Y ). Therefore (13) holds, and the convergence of C 2 α1 , α2 is a
direct consequence.
Proof of Proposition 4. The planner’s problem has an appealing Markov structure. Denoting
λ̄ = λ̄1 , λ̄2
and u = u1 , u2 , the state vector is Z = λ̄ , Y , and the planner’s problem (11-12)
leads to the Hamilton-Jacobi-Bellman equation for J (Z),
2
0≡
sup
(C 1 ,C 2 ,ν 1 ,ν 2 ) n=1
where
Σ=
1
λ̄n [F (C n , ν n ) − Jλ̄n ν n ] + Jy μy Y + tr (Jzz Σ) ,
2
diag λ̄ u
diag λ̄ u
σy Y diag λ̄ u
diag λ̄ u σy Y
(36)
σy2 Y 2
and diag λ̄ is a 2 × 2 diagonal matrix with elements of λ̄ on the main diagonal.
The maximization over ν 1 , ν 2 in the HJB equation (36) can be solved separately. Under the
optimal discount rate process ν n for agent n,
˙ sup F (C n , ν n ) − Jλ̄n ν n =
f (C n , Jλ̄n ) ≡
νn
β n ρ
1− ρ
(C ) (γJλ̄n ) γ − γJλ̄n .
ρ
(37)
The function f is the aggregator in the stochastic differential utility representation of recursive
preferences postulated by Duffie and Epstein (1992b). The online appendix gives more detail on
40
this relationship. Optimal consumption shares ζ n are given by the first-order conditions in the
consumption allocation
1−ρ/γ
1
(γJλ̄n ) 1−ρ λ̄n 1−ρ
Cn
ζ =
=
˙
,
1−ρ/γ
1
Y
2
k
1−ρ
1−ρ
(γJ
)
λ̄
λ̄k
k=1
n
where Jλ̄n are agents’ continuation values under the optimal consumption allocation.
The HJB equation (36) further implies that J is homogeneous degree one in λ̄ and homogeneous
degree γ in Y . The transformation of variables (14) leads to the guess
J (Z) = γ −1 Y γ θ 2 J˜ θ 1 = γ −1 Y γ θ 2 θ 1 J˜1 θ 1 + 1 − θ 1 J˜2 θ 1 ,
where J˜n θ 1 are continuation values of the two agents scaled by γ −1 Y γ , defined in (16). The ODE
for J˜n θ 1 then immediately follows. The continuity at the boundaries follows from Lemma 3.
In addition, the same logic and derivation of the HJB equation applies to multi-agent economies
and more sophisticated Markov dynamics of the aggregate endowment process. In an N -agent
economy, the state vector includes N − 1 Pareto shares as state variables. The boundary conditions
for θ n = 0, n ∈ {1, . . . , N } associated with the N -agent version of the ODE (15) are given by the
solutions of (N − 1)-agent economies that exclude agent n. In this way, solutions to multi-agent
economies can be calculated by iteratively adding individual agents.
Proof of Proposition 6.
Given an initial condition θ01 ∈ (0, 1), the process (18) lives on
the open interval (0, 1) with unattainable boundaries (the preferences satisfy an Inada condition
at zero). For any numbers 0 < a < b < 1, the process θ 1 has bounded and continuous drift and
volatility coefficients on (a, b), and the volatility coefficient is bounded away from zero. It is thus
sufficient to establish the appropriate boundary behavior of θ 1 in order to make the process positive
Harris recurrent (see Meyn and Tweedie (1993)). Since the process will also be ϕ -irreducible for the
Lebesgue measure under these boundary conditions, there exists a unique stationary distribution.
Denote μθ (θ) and σθ (θ) the drift and volatility coefficients in (18). The boundary behavior of
the process θ 1 is captured by the scale measure S : (0, 1)2 → R defined as
s (θ) = exp −
θ
θ0
2μθ (τ )
dτ
σθ2 (τ )
S [θl , θh ] =
θl
θh
s (θ) dθ
for an arbitrary choice of θ0 ∈ (0, 1), and the speed measure M : (0, 1)2 → R
1
m (θ) = 2
σθ (θ) s (θ)
M [θl , θh ] =
θl
θh
m (θ) dθ.
Karlin and Taylor (1981, Chapter 15) provide an extensive treatment of the boundaries.
41
The boundaries are nonattracting if and only if
lim S [θl , θh ] = ∞
and
θl 0
lim S [θl , θh ] = ∞
(38)
θh 1
and this result is independent of the fixed argument that is not under the limit. With nonattracting
boundaries, the stationary density will exist if the speed measure satisfies
lim M [θl , θh ] < ∞
and
θl 0
lim M [θl , θh ] < ∞,
(39)
θh 1
again independently of the argument that is not under the limit.
In our case,
s (θ) = exp −
where
ssep (θ) =
τ (1 − τ ) (u1 − u2 )2
θ0
2 ν 2 (τ ) − ν 1 (τ )
θ
1−θ
1 − θ0
−
2u1
u1 −u2
θ
θ0
dτ
ssep (θ) ,
2u2
u1 −u2
(40)
is the integrand of the scale function in the separable case, when ν 2 (θ) − ν 1 (θ) ≡ 0.
For the left boundary, assume that in line with condition (i), there exist θ ∈ (0, 1) and ν ∈ R
such that ν 2 (θ) − ν 1 (θ) ≥ ν for all θ ∈ (0, θ). Taking θ0 = θ, the scale measure can be bounded as
S [θl , θ] ≥
θ
exp −
θ
2ν
1−θ
1−θ
−
dτ
τ (1 − τ ) (u1 − u2 )2
2u1
2ν
θ 12u2 2 − 2ν 2
θ u −u (u1 −u2 )
1 − θ (u1 −u2 )2 − u1 −u2
dθ
=
θ
1−θ
θl
θl
θ
2u1
u1 −u2
12u2 2
θ u −u
dθ =
θ
The left limit in (38) thus diverges to infinity if
2ν
2u2
−
≤ −1,
− u2 (u1 − u2 )2
u1
which is satisfied when ν ≥
1
2
u1
2
− u2
2
.
The argument for the right boundary is symmetric. Taking θ̄ ∈ (0, 1) and ν̄ ∈ R such that
2
2
ν 2 (θ) − ν 1 (θ) ≤ ν̄ for all θ ∈ θ̄, 1 , the calculation reveals that we require ν̄ ≤ 12 u1 − u2 .
It turns out that the bounds implied by conditions (39) are marginally tighter. Following the
same bounding argument as above, sufficient conditions for (39) to hold are
ν>
1 1
u
2
2
− u2
2
and ν̄ <
1 1
u
2
2
− u2
2
.
(41)
The construction reveals that these bounds are also the least tight bounds of this type under which
42
the proposition holds.
It is also useful to note that the unique stationary density q (θ) is proportional to the speed
density m (θ). Finally, if the limits in Proposition 6 do not exist, they can be replaced with
appropriate limits inferior and superior.
This discussion has sorted out case (a). Conditions (i’) and (ii’) are sufficient conditions for
the boundaries to be attracting. Lemma 6.1 in Karlin and Taylor (1981) then shows that if the
‘attracting’ condition is satisfied for a boundary, then θ 1 converges to this boundary on a set of
paths that has a strictly positive probability. This probability is equal to one if the other boundary
is non-attracting. Combining these results, we obtain statements (b), (c), and (d).
Proof of Corollary 7. Assume without loss of generality that u2 ≤ u1 . The sufficient part
is an immediate consequence of Proposition 6. Under separable preferences, ν 2 − ν 1 ≡ 0, and thus
if u2 < u1 then conditions (i’) and (ii) hold, and agent 2 dominates in the long run under P .
For the necessary part, when u2 = u1 , then θ 1 is constant and both agents survive under P .
When −u2 = u1 = u, then it follows from inspection of formula (40) in the proof of Proposition 6
that conditions (38) are satisfied and the boundaries are non-attracting. Lemma 6.1 in Karlin and
Taylor (1981) then implies that both agents survive under P .
Note that even though both agents survive when −u2 = u1 , the speed density m (θ) ∝
θ −1 (1 − θ)−1 is not integrable on (0, 1) and thus there does not exist a finite stationary measure.
The result on survival under measure Qn follows from the fact that the evolution of Brownian
motion W under the beliefs of agent n is dWt = un dt + dWtn . Since the evolution of θ 1 completely
describes the dynamics of the economy, substituting this expression into (18) and reorganizing
yields the desired result.
Proof of Lemma 8.
ously extended at
θ1
Lemma 3 implies that the planner’s objective function can be continu-
= 0 by the continuation value for agent 2 living in a homogeneous economy.
Expression (35) scaled by α1 + α2 γ −1 Y γ leads to an equation in scaled continuation values
J˜ θ 1 = θ 1 J˜1 θ 1 + 1 − θ 1 J˜2 θ 1
and the proof of Lemma 3 yields
˜ 1 ) = lim J˜2 (θ 1 ) = Ṽ 2 ,
lim J(θ
θ 1 0
θ 1 0
where Ṽ 2 is defined in (7). Since J˜2 θ 1 = J˜ θ 1 − θ 1 J˜θ1 θ 1 , then
lim θ 1 J˜θ1 θ 1 = 0.
θ 1 0
43
(42)
Further, consider the behavior of individual terms in ODE (15) as θ 1 0. Using expression
(17), the first term is proportional to
θ1 ζ 1 θ1
ρ
J˜1 θ 1
1− ρ
γ
θ1
=
1−ρ/γ
−ρ
1−ρ
=
K θ1
J˜1 θ 1
1−ρ
,
K θ1
1
1−ρ
= ζ 1 θ1
where K θ 1 is the denominator in the formula for the consumption share (17), and limθ1 0 K θ 1 =
1−ρ/γ
1−ρ
Ṽ 2
, which is a finite value. Since limθ1 0 ζ 1 θ 1 = 0, the first term in (15) vanishes as
θ 1 0. The sum of the second and third term converges to
ρ
β
1
β 2 1− γ
2
2
+ − + μy + u σy + (γ − 1) σy Ṽ
Ṽ
ρ
ρ
2
and formula (7) implies that this expression is zero. Since the fourth term in (15) also converges
to zero due to result (42), the last term in (15) must also converge to zero, or
2
lim θ 1
θ 1 0
J˜θ1 θ1 θ 1 = 0.
(43)
Finally, differentiate the PDE (15) by θ 1 and multiply the equation by θ 1 . Using comparisons
1/γ
are bounded and bounded away from
with results (42–43), the assumption that ζ n θ 1 /J˜n θ 1
zero and limθ1 0 ζ 1 θ 1 = 0, it is possible to determine that all terms in the new equation containing derivatives of J˜ θ 1 up to second order vanish as θ 1 0. The single remaining term that
3
contains a third derivative of J˜ θ 1 is multiplied by θ 1 and must necessarily converge to zero
as well, and thus
lim θ 1
3
θ 1 0
J˜θ1 θ1 θ1 θ 1 = 0.
Proof of Lemma 9. Itô’s lemma implies
dJ˜2 θt1
= d J˜ θt1 − θt1 J˜θ1 θt1 =
= −
2
θt1 J˜θ1 θ1
θt1
dθt1 1 1
θt
−
2
θt1
2
J˜θ1 θ1 θt1 + θt1
3
J˜θ1 θ1 θ1 θt1
dθ 1 2
t
θt1
and since the drift and volatility coefficients in the dynamics of θ 1 given by equation (18) are
bounded by assumption, applying results from Lemma 8 proves the claim about the drift and
volatility coefficients of J˜2 θ 1 (J˜2 itself converges to a nonzero limit so the scaling is innocuous).
44
Further notice that
= d J˜ θt1 + 1 − θt1 J˜θ1 θt1 = − θt1
dJ˜1 θt1
+
and that
1 1
θt
2
2
ζ 1 θ1
J˜1 (θ 1 )
1
γ
J˜θ1 θ1 θt1 + 1 − θt1
= θ
1
1−ρ
1
dθt1
+
J˜θ1 θ1 θt1
θt1
dθ 1 2
t
1 2 ˜
1
θt Jθ1 θ1 θ1 θt
θt1
1−1/γ
1−ρ
K θ1
J˜1
2
(44)
−1
(45)
is bounded and bounded away from zero by assumption. Denote the numerators of ζ 1 and ζ 2
Z
1
θ
1
= θ
1
1
1−ρ
1−ρ/γ
1−ρ
1
1
˜
J θ
Z
2
θ
1
= 1−θ
1
1
1−ρ
J˜2 θ 1
1−ρ/γ
1−ρ
.
Then ζ 2 = Z 2 / Z 1 + Z 2 and, omitting arguments,
dZ
1
1 2
1
1 − γρ 1 dJ˜1 1
ρ
1
1 dθ
1 dθ
Z 1 +
Z
=
+
Z
+
1−ρ
θ
1−ρ
2 (1 − ρ)2
θ1
J˜1
2
ρ
ρ
ρ
−
1 ˜1
1
−
˜1
1 − γρ
γ
γ
d
J
1
1
1 dθ dJ
Z
+
Z
+
2
θ 1 J˜1
J˜1
(1 − ρ)2
(1 − ρ)2
1 2 1 2
ρ
ρ
1
θ 1 dθ 1 1 − γ 2 dJ˜2 1
θ
dθ
2
Z2
Z
+
+
Z
+
2
1
2
˜
1 − ρ 1 − θ1 θ1
1−ρ
2
1
−
θ
θ1
J
(1 − ρ)
2
ρ
ρ
ρ
−
1
1
−
1 − γρ
γ
γ
dθ 1 dJ˜2
dJ˜2
1
2
2 θ
Z
−
Z
.
+
2
J˜2
(1 − ρ)2
(1 − ρ)2 1 − θ 1 θ 1 J˜2
dZ 2 = −
Since the drift and volatility coefficients of dJ˜2 /J˜2 vanish as θ 1 0, and limθ1 0 Z 2 θ 1 =
1−ρ/γ
1−ρ
, the drift and volatility coefficients in the equation for dZ 2 vanish. In the equation for
Ṽ 2
dZ 1 , it remains to determine the behavior of terms containing dJ˜1 (the remaining contributions to
drift and volatility terms converge to zero because limθ1 0 Z 1 θ 1 = 0):
Z1
= θ1
J˜1
θ
1
1
1−ρ
1−1/γ
1−ρ
J˜1
ρ
,
where the term in brackets is bounded and bounded away from zero by utilizing (45). Using the
first θ 1 to multiply the coefficients in dJ˜1 in formula (44), we conclude that the coefficients in
45
2
Z 1 dJ˜1 /J˜1 vanish as θ 1 0. Finally, the drift term arising from dJ˜1 vanishes, and
Z
1
dJ˜1
2
=
J˜1
θ1
5
2
J˜θ1 θ1
J˜ + (1 − θ 1 ) J˜θ1
θ
1
1
1−ρ
˜1
J
ρ
1−1/γ
1−ρ
dθt1
θt1
2
.
Here, the last term has bounded drift, the second last term is bounded, and the first term converges
to zero as θ 1 0, which can be shown by using the l’Hôpital’s rule (the numerator converges to
zero and the denominator to zero or +∞, depending on the sign of γ):
lim
θ 1 0
θ1
5
2
J˜θ1 θ1
J˜ + (1 − θ 1 ) J˜θ1
= lim
5 θ1
4
θ 1 0
J˜θ1 θ1 + 2 θ 1
1 − θ1
5
J˜θ1 θ1 θ1
= 0.
Thus all terms in the drift and volatility coefficients of dZ 1 vanish.
Applying Itô’s lemma to ζ 2 yields
dζ 2 =
1
Z2
2
dZ
−
dZ 1 + dZ 2 +
Z1 + Z2
(Z 1 + Z 2 )2
1
Z2
1
2 2
−
dZ 2 dZ 1 + dZ 2
+
3 dZ + dZ
1
2
1
(Z + Z )
(Z + Z 2 )2
and the results on the behavior of dZ 1 and dZ 2 as θ 1 0 lead to the desired conclusion about the
convergence of drift and volatility coefficients of dζ 2 .
Proof of Proposition 10.
calculation of
Convergence of the risk-free interest rate follows from the direct
1
r (0) = lim − log E Mt2 St2 (0) | F0
t0
t
where St2 (0) is the limiting stochastic discount factor corresponding to the one prevailing in a
homogeneous economy populated only by agent 2. Lemma 9 shows that the local behavior of St2
converges to St2 (0) as θ01 0. Similarly, convergence of the wealth-consumption ratio follows from
ξ θ1 = ξ1 θ1 ζ 1 θ1 + ξ2 θ1 ζ 2 θ1 .
Since ξ n θ 1 are bounded and ζ 1 θ 1 converges to zero, we have
lim ξ θ 1 = lim ξ 2 θ 1 =
θ 1 0
θ 1 0
where Ṽ 2 is given by (7).
46
1 2 ρ
Ṽ
,
β
In order to obtain the convergence of the infinitesimal return, observe that
ξ 1 θ 1 ζ 1 θ 1 = β −1 θ 1 J˜1 θ 1
and
Z 1 θ1 + Z 2 θ1
ρ−1
1
dθ 1
1 ˜1
1
1 ˜1
1 dθ
+
θ
d
J
d
J
.
θ
+
θ
θ
d θ 1 J˜1 θ 1 = θ 1 J˜1 θ 1
θ1
θ1
The drift and volatility coefficients of the first term on the right-hand side vanish as θ 1 0 by the
proof of Lemma 8, and the coefficients of the other two terms vanish by combining the results in
that Lemma with equation (44). Further,
d
ρ−1
ρ−2
dZ 1 + dZ 2 +
= (ρ − 1) Z 1 θ 1 + Z 2 θ 1
Z1 + Z2
ρ−3
1
dZ 1 + dZ 2
+ (ρ − 2) (ρ − 1) Z 1 θ 1 + Z 2 θ 1
2
2
and since dZ 1 and dZ 2 have vanishing coefficients by the proof of Lemma 9 and the remaining
terms are bounded, we obtain that dξ 1 θ 1 ζ 1 θ 1 has vanishing drift and volatility coefficients as
θ 1 0. The same argument holds for dξ 2 θ 1 ζ 2 θ 1 , and thus dξ θ 1 has vanishing coefficients
as well. Therefore all but the first term in
dYt
+ Yt dξ θt1 + dξ θt1 dYt
dΞt = d ξ θt1 Yt = Ξt
Yt
have coefficients that decline to zero as θt1 0, which proves the result.
Proof of Proposition 11.
The evolution of θ 1 given by equation (18) implies that for every
fixed t ≥ 0
θ01 0
θt1 → 0, P -a.s.
=⇒
and thus also ζ 2 θt1 → 1 and J˜2 θt1 → Ṽ 2 , P -a.s.12 The last two terms in the expression for the
stochastic discount factor, St2 , equation (24), converge to one, P -a.s., and since ν 2 θs1 , 0 ≤ s ≤ t
P
also converges to ν 2 (0) and is bounded, we have St2 −→ St2 (0). Consider a family of random
variables Mt2 St2 θ01 indexed by the initial Pareto share θ01 . Since this family is uniformly integrable,
12
This result becomes more transparent if we consider ζ 2 and J˜2 as functions of log θ1 . The dynamics of
log θ1
1
1 2 2
1
1
2
1
1
1
1 2
1
1
2 2
− θt u − u
u
d log θt = 1 − θt νt θt − ν θt +
− u
dt +
2
2
+ 1 − θt1
u1 − u2 dWt
has bounded drift and volatility coefficients and thus for ∀ε > 0, ∀k > 0, it is possible to achieve
P θt1 < k = P log θt1 < log k > 1 − ε
by setting log θ01 sufficiently low.
47
then convergence in probability implies convergence in mean, and we obtain the convergence result
for bond prices
θ01 0 2 2
E Mt2 St2 θ01 | F0 −→
E Mt St (0) | F0 .
The same argument holds for Mt2 St2 θ01 Yt , which yields the result for the price of individual cash
flows from the aggregate endowment.
For the proof of the next lemma, the following result will be useful:
Lemma 18 Let f : R → R be differentiable with a monotone first derivative in a neighborhood of
−∞ and have a finite limit limx→−∞ f (x). Then limx→−∞ f (x) = 0.
Proof of Lemma 12. Transformation (27) together with the previously used γVt1 = Y γ J˜1 θt1
imply that
V̂ 1 θ 1 = β γ
1/γ
J˜1 θ 1
ζ 1 (θ 1 )
γ(1−ρ)
.
(46)
Think for a moment of V̂ 1 as a function of log θ 1 , where we are interested in the limiting behavior
as log θ 1 → −∞. We have
1
1
θ 1 V̂θ11 = V̂log
θ 1 and θ
2
1
1
− V̂log
V̂θ11 θ1 = V̂(log
θ1 .
θ 1 )2
(47)
Differentiating repeatedly expression (46) and exploiting the local behavior of J˜ θ 1 as θ 1 0, we
conclude that the assumptions of Lemma 18 hold, and thus both expressions in (47) converge to
zero as θ 1 0.
Proof of Lemma 13.
Utilizing Lemma 12 to deduce which terms in ODE (29) vanish and
Proposition 10 to determine the limiting values of the remaining coefficients, we obtain
lim β
θ 1 0
1
1−ρ
− ρ 1
γ 1−ρ
V̂ 1 θ 1
1
2
= β − ρ μy + u σy − (1 − γ) (σy ) −
2
ρ
−
1−ρ
2
1
2
u −u
1 u1 − u2
σy +
2 1−γ
2
,
which is the limiting consumption-wealth ratio for agent 1. The formulas for the wealth share
invested in the claim on aggregate consumption and the coefficients of the wealth process are
obtained by plugging in the previous results into expressions (26) and (30).
Proof of Proposition 14.
the expression for limθ1 0
ν2
θ1
Given convergence to the homogeneous economy counterpart,
is given by equation (8). Utilizing the formula for the wealth-
48
consumption ratio (20) and the result from Lemma 13 then yields
lim ν
θ 1 0
1
θ
1
1 1 −1
γ
1
2
2
= β + (γ − ρ) μy + u σy − (1 − γ) σy +
= lim β + (ρ − γ) ξ θ
2
θ 1 0 ρ
+
γ−ρ
1−ρ
u1 − u2 σy +
1 u1 − u2
2 1−γ
2
.
The first two terms in the last expression are equal to the limit for ν 2 θ 1 , which yields the result
for the difference of the discount rates. The expression for part (ii) is obtained by symmetry.
Proof of Corollary 15.
The critical point is the limits for the consumption-wealth ratios as
the Pareto share of one of the agents becomes small. Since the large agent’s consumption-wealth
ratio converges to that in a homogeneous economy, the relevant parameter restriction is the same
as restriction (9) in Assumption 2. The consumption-wealth ratio of the small agent is given in
expression (20), and restriction (10) in Assumption 2 assures that this quantity is strictly positive,
and the wealth-consumption ratio finite.
Proof of Corollary 16. Utilize results in Proposition 13 and the fact that limθ1 0 μΞ2 θ 1 = μy
and limθ1 0 σΞ2 θ 1 = σy , then form the differences in the limiting expected logarithmic growth
rates, and compare them to inequalities in Proposition 6.
Proof of Corollary 17.
The results are obtained by taking limits of the expressions in
Proposition 14.
49
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53
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