Necessary and Sufficient Conditions for Existence and Uniqueness of Recursive Utilities

• Date: 01 June 2020
• Author: JAROSLAV BOROVIČKA,JOHN STACHURSKI
• DOI: http://doi.org/10.1111/jofi.12877
• Published date: 01 June 2020

THE JOURNAL OF FINANCE •VOL. LXXV, NO. 3 •JUNE 2020

Necessary and Sufficient Conditions for

Existence and Uniqueness of Recursive Utilities

JAROSLAV BOROVI ˇ

CKA and JOHN STACHURSKI∗

ABSTRACT

We obtain exact necessary and sufficient conditions for existence and uniqueness of

solutions of a class of homothetic recursive utility models postulated by Epstein and

Zin. The conditions center on a single test value with a natural economic interpreta-

tion. The test sheds light on the relationship between valuation of cash flows, impa-

tience, risk adjustment, and intertemporal substitution of consumption. We propose

two methods to compute the test value when an analytical solution is not available.

We further provide several applications.

RECURSIVE PREFERENCE MODELS SUCH AS those discussed in Koopmans (1960),

Epstein and Zin (1989), and Weil (1990) play an important role in macroeco-

nomic and financial modeling. For example, the long-run risk models analyzed

in Bansal and Yaron (2004), Hansen, Heaton, and Li (2008), Bansal, Kiku, and

Yaron (2012), and Schorfheide, Song, and Yaron (2018) employ such prefer-

ences in discrete-time infinite-horizon settings with a variety of consumption

path specifications to help resolve long-standing empirical puzzles identified in

the literature.

In recursive utility models, the lifetime value of a consumption stream from

a given point in time is expressed as the solution to a nonlinear forward-looking

equation. While this representation is convenient and intuitive, it can also be

vacuous, in the sense that no finite solution to the forward-looking recursion

exists. Moreover, even when a solution is found, this solution lacks predictive

content unless some form of uniqueness can also be established. In general,

∗Jaroslav Boroviˇ

cka is with New York University, Federal Reserve Bank of Minneapolis, and

NBER. John Stachurski is with the Research School of Economics, Australian National University.

The authors thank Anmol Bhandari, Tim Christensen, Ippei Fujiwara, Jinill Kim, Stefan Nagel,

Daisuke Oyama, Guanlong Ren, and the anonymous referees for useful comments and sugges-

tions. Special thanks are due to Mirosława Zima for her valuable input on local spectral radius

conditions. Boroviˇ

cka gratefully acknowledges financial support from New York University and

Federal Reserve Bank of Minneapolis. Stachurski gratefully acknowledges financial support from

ARC grant FT160100423. The views expressed herein are those of the authors and not necessarily

those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. We have read

The Journal of Finance’s disclosure policy and have no conflicts of interest to disclose.

Correspondence: Jaroslav Borovicka, Department of Economics, New YorkUniversity, 19 W 4th

Street, 6th floor, New York, NY 10012; e-mail: jaroslav.borovicka@nyu.edu.

DOI: 10.1111/jofi.12877

C2020 the American Finance Association

1457

1458 The Journal of Finance R

identifying restrictions that imply both existence and uniqueness of a solution

for an empirically relevant class of consumption streams is challenging.

The aim of this paper is to obtain existence and uniqueness results that are

as tight as possible in a range of empirically plausible settings, while restricting

attention to practical conditions that can be tested in applied work. To this end,

we provide conditions for existence and uniqueness of solutions to the class of

homothetic preferences studied in Epstein and Zin (1989). These conditions ad-

mit both stationary and nonstationary consumption paths. Moreover, they are

necessary as well as sufficient, and hence as tight as possible in the setting we

consider. In particular, if the conditions hold, then a unique, globally attracting

solution exists, while if they do not, then no finite solution exists. Existence of

a finite solution is equivalent to the existence of a finite wealth-consumption

ratio, a central object of interest in asset pricing.

To describe the setting in more detail, let preferences be defined recursively

by

Vt=(1 −β)C1−1/ψ

t+β{Rt(Vt+1)}1−1/ψ1/(1−1/ψ )

,(1)

where {Ct}is a consumption path, Vtis the utility value of the path extending

from time t,andRtis the Kreps-Porteus certainty-equivalent operator

Rt(Vt+1)=(EtV1−γ

t+1)1/(1−γ).(2)

The parameter β∈(0,1) is a time discount factor, while γ= 1 governs risk

aversion and ψ= 1 is the elasticity of intertemporal substitution. We take the

consumption stream as given and seek a solution for normalized utility Vt/Ct.

The first step in our approach is to associate with each consumption process

the risk-adjusted long-run mean consumption growth rate

MC:=lim

n→∞RCn

C01/n

,(3)

where Ris the unconditional version of the Kreps-Porteus certainty equivalent

operator. Beginning with the case in which the state vector driving the condi-

tional distribution of consumption growth takes values in a compact set—which

is where the sharpest results obtain—we show that a unique solution exists if

andonlyif<1, where

:=βM1−1/ψ

C.(4)

Under the same compactness restriction, we also show that the condition <1

is both necessary and sufficient for global convergence of successive approxi-

mations associated with a natural fixed-point mapping. Indeed, our results

establish that convergence of successive approximations itself implies that a

unique solution exists, and that the limit produced through this process is

equal to the solution. Furthermore, we prove that when the condition <1

Necessary and Sufficient Conditions for Existence and Uniqueness 1459

fails, not only does existence and uniqueness of a solution fail, but existence

itself fails more specifically.

The value represents the risk-adjusted long-term consumption growth

rate, modified by both the degree of impatience and the intertemporal substi-

tutability of consumption. Thus, despite the fact that the preference recursion

(1) intertwines the contributions of impatience, intratemporal risk aversion,

and intertemporal elasticity of substitution to value, the condition <1 effec-

tively separates these forces. Aspects of the consumption growth process, such

as its persistence or higher moments of its innovations, matter only through

the long-run distribution of consumption growth encoded in MC.

From an asset pricing perspective, the condition <1 imposes a bound on

the average growth rate in the value of long-dated consumption strips as their

maturity increases. In particular, in an IIDgrowth setting, log is exactly equal

to this growth rate, which must be negative for a finite wealth-consumption

ratio to exist. We provide additional discussion on the intuition behind condition

(4) in Section II and the applications.

In addition to the preceding results, we use a local spectral theorem to show

that

MC=r(K)1/(1−γ),(5)

where r(K) is the spectral radius of a valuation operator Kdetermined by

the primitives and clarified below. This result is useful on two levels. First,

spectral radii and dominant eigenfunctions associated with valuation operators

have been increasingly used to understand long-run risks and long-run values

in macroeconomic and financial applications by inducing a decomposition of

the stochastic discount factor (see, e.g., Alvarez and Jermann (2005), Hansen

and Scheinkman (2009), Qin and Linetsky (2017), or Christensen (2017)). The

identification in (5) allows us to connect and draw insights from this literature.

Second, computationally, when the state space for the state process is finite,

the valuation operator Kis just a matrix and the spectral radius is easy to

compute, which allows us to compute the test statistic via (5). When the

state space is not finite, one can still implement this idea after discretization.

When the state space is high dimensional, accurate discretization is nontriv-

ial and calculation of the spectral radius becomes computationally expensive.

For these scenarios, we instead propose a Monte Carlo method to calculate

the test value that is based on the idea of simulating consumption paths

from a given specification and calculating the risk-adjusted expectation on the

right-hand side of (3) by averaging over these paths. This approach is straight-

forward to implement and relatively insensitive to the dimension of the state

space. The routine is easy to parallelize by simulating independent consump-

tion paths along multiple execution threads.

All of the theoretical results on existence, uniqueness, and convergence

of successive approximations discussed above are stated in the context of a

compact-valued state process, which drives the persistent component of con-

sumption growth. In this setting, we apply a fixed-point theorem due to Du