Recursive Utility with Narrow Framing: Existence and
Uniqueness
Jing Guo∗
Xue Dong He†
February 16, 2017
Abstract
We study the total utility of an agent in a model of narrow framing with constant
elasticity of intertemporal substitution and relative risk aversion degree and with
infinite time horizon. In a finite-state Markovian setting, we prove that the total
utility uniquely exists when the agent derives nonnegative utility of gains and losses
incurred by holding risky assets and that the total utility can be non-existent or
non-unique otherwise. Moreover, we prove that the utility, when uniquely exists, can
be computed by a recursive algorithm with any starting point. We then consider
a portfolio selection problem with narrow framing and solve it by proving that the
corresponding dynamic programming equation has a unique solution. Finally, we
propose a new model of narrow framing in which the agent’s total utility uniquely
exists in general.
Key words: recursive utility, narrow framing, existence, uniqueness, Markov processes, portfolio selection, dynamic programming
JEL Codes: G02, G11
∗
Department of Industrial Engineering and Operations Research, Columbia University in the City of
New York, New York, NY, 10027, US. Email: [email protected].
†
Room 609, William M.W. Mong Engineering Building, Department of Systems Engineering and
Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. Email:
[email protected].
1
1
Introduction
Barberis and Huang (2008, 2009) and Barberis et al. (2006) propose a model of narrow
framing in a discrete-time multiple-period setting in which an agent derives utility not
only from her consumption stream but also from the investment gain and loss incurred by
holding certain risky assets; the former is referred to as consumption utility and the latter
is referred to as investment utility. The total utility of the agent is computed based on
the classical recursive utility model (Epstein and Zin, 1989, Kreps and Porteus, 1978): the
total utility of the agent’s consumption and investment after time t is the aggregation of
1) her consumption at time t, 2) her investment utility in period from t to t + 1, and 3)
the time-t certainty equivalent of the agent’s total utility of consumption and investment
after time t + 1. In particular, when the investment utility is set to be zero, the model of
narrow framing degenerates into the classical recursive utility model.
Just as in the classical recursive utility model, the aggregation of different components
of utility in the model of narrow framing is achieved by a function named aggregator and the
certainty equivalent is computed under the expected utility theory. The aggregator thus
measures the elasticity of intertemporal substitution (EIS) and the certainty equivalent
measures the relative risk aversion degree (RRAD) of the agent. As in many applications
of the classical recursive utility model to portfolio selection and asset pricing, in their model
of narrow framing, Barberis and Huang (2008, 2009) and Barberis et al. (2006) select a
specific aggregator in which the EIS is constant and a specific certainty equivalent in which
the RRAD is constant; see the exact forms in (2.2) and (2.3). Furthermore, the authors
adopt an infinite-horizon setting. Both the specific choice of the aggregator and certainty
equivalent and the infinite-horizon setting are known to be simple and helpful in obtaining
close-form solutions to a variety of problems.
The model of narrow framing is successful in explaining some empirical findings, such as
why people are averse to a small, independent gamble, even when the gamble is actuarially
favorable; see for instance Barberis et al. (2006). The model of narrow framing is further
extended by De Giorgi and Legg (2012) and He and Zhou (2014) with various applications,
and these authors also assume constant EIS and RRAD and adopt the infinite-horizon
setting. Even with many successful applications, however, the existence and uniqueness of
the agent’s total utility in the model of narrow framing have not been established. Indeed,
in the infinite-horizon setting, the agent’s total utility is defined recursively without an
end date, so its existence and uniqueness cannot be taken for granted. Surprisingly, even
2
for the classical recursive utility, its existence and uniqueness have not been completely
established; see Section 2.1.
In the present paper, assuming constant EIS and RRAD, we study the existence and
uniqueness of the agent’s total utility in the model of narrow framing in a Markovian
setting. More precisely, we assume a Markovian process {Xt } and a process {Yt } that is
i.i.d. conditioning on {Xt }. Thus, {Xt } can be regarded as market states and {Yt } can be
interpreted as random noise. The asset returns in period from t to t + 1 are assumed to be
functions of Xt , Xt+1 , and Yt+1 , so the agent’s consumption propensity and investment in
the assets in that period are functions of Xt . We further assume that {Xt } is irreducible
and its state space is finite.
The Markovain setting here is the same as in Hansen and Scheinkman (2012), who
show the existence of the classical recursive utility with non-unitary EIS and RRAD and
the uniqueness of the utility with further conditions on the EIS and RRAD, except that we
assume the state space to be finite; see Section 2.1. This setting is not restrictive in terms
of applying the model of narrow framing because (i) finite-state Markovian processes can
be sufficiently flexible to describe financial data and (ii) we do not impose any assumption
on {Yt }.
We prove that with the same condition as used by Hansen and Scheinkman (2012)
to obtain the existence of the classical recursive utility, the agent’s total utility in the
model of narrow framing uniquely exists when her investment utility in each period is
nonnegative, regardless of the values of the EIS and RRAD. When the investment utility
becomes negative in some states, however, the total utility in the model of narrow framing
can be non-existent or non-unique even in some simple settings, such as in the setting in
which the EIS is less than or equal to one and the state space of {Xt } is singleton. In this
case, we propose a sufficient condition under which the total utility in the model of narrow
framing uniquely exists, and this condition is nearly necessary.
We also prove that if the total utility in the model of narrow framing uniquely exists, it
can be obtained by applying the recursive equation defining the utility repeatedly with any
positive utility value as the starting point. This result is not only computationally useful
but also economically important: the infinite-horizon setting can be regarded as a limit of
the finite-horizon setting when the number of periods in the latter is sent to infinity, and
the utility in the latter setting converges to that in the former setting regardless of the
terminal utility in the latter setting.
We then consider a portfolio selection model with narrow framing. We prove that a
3
consumption-investment plan is optimal if and only if it, together with the value function
of the portfolio selection problem, satisfies the dynamic programming equation. Moreover,
we prove that the solution to the dynamic programming equation uniquely exists and can
be computed by solving the equation recursively with any starting point. As a result, the
portfolio selection problem in a finite-horizon setting approaches that in the infinite horizon
setting when the number of periods in the former goes to infinity.
Finally, to solve the issue that the model of narrow framing may fail to define the agent’s
total utility when the agent’s investment utility is negative and the EIS is less than or equal
to one, we propose a new model of narrow framing, in which the agent’s investment utility
is scaled by her total utility, and prove that in this model the agent’s total utility uniquely
exists in general. Moreover, the total utility in this new model can also be obtained by
applying the recursive equation defining the utility repeatedly and portfolio selection with
this new model can also be solved by dynamic programming.
The remainder of the paper is organized as follows: In Section 2 we review the literature
and in Section 3 we introduce the model of narrow framing that is used in the literature. In
Section 4 we prove the existence and uniqueness of the total utility in the model of narrow
framing in a finite-state Markovian setting and apply the results to a portfolio selection
problem. In Section 5, we propose the new model of narrow framing. Section 6 concludes.
Proofs are placed in Appendix A and some extensions of the existence and uniqueness
results to the non-finite-state Markovian setting are provided in Appendix B.
2
2.1
Literature
Existence and Uniqueness of Recursive Utility
Recursive utility is a classical model for individual’s preferences for discrete-time consumption streams; see Kreps and Porteus (1978) and Epstein and Zin (1989). In an infinitehorizon setting, the recursive utility of consumption stream Ct , t = 0, 1, . . . derived by an
agent is represented by Vt , t = 0, 1, . . . , where Vt stands for the utility of the consumption
stream starting from time t, i.e., Cs , s ≥ t. The recursive utility process {Vt } is defined
recursively:
Vt = H(Ct , Mt (Vt+1 )),
4
t = 0, 1, . . . ,
(2.1)
where Mt (X) stands for the certainty equivalent of random quantity X conditioning on the
information at time t and H(c, z) is an aggregator. There are many choices for the certainty
equivalent and aggregator, but the following one, which was first proposed by Kreps and
Porteus (1978), is popular due to its tractability in deriving asset pricing results (see e.g.,
Epstein and Zin, 1990, 1991):

1
[(1 − β)c1−ρ + βz 1−ρ ] 1−ρ
, 0 < ρ 6= 1,
H(c, z) : =
e(1−β) ln c+β ln z ,
ρ = 1,

x1−γ , 0 < γ 6= 1,
Mt (X) : = u−1 Et [u(X)] , u(x) :=
ln(x), γ = 1,
(2.2)
(2.3)
where Et stands for the expectation operator conditioning on the information at time t.
In addition, β ∈ (0, 1) is a discount rate, γ stands for the relative risk aversion degree
(RRAD), and 1/ρ is the elasticity of intertemporal substitution (EIS); see for instance
Kreps and Porteus (1978) and Epstein and Zin (1989).
In the following, when ρ ≥ 1, we set H(c, 0) = limz↓0 H(c, z) = 0 and H(0, z) =
limc↓0 H(c, z) = 0. As a result, H(c, z) is well defined, takes real values, and continuous in
(c, z) ∈ [0, ∞)2 . Similarly, when γ = 1, we define u(0) := −∞ and u−1 (−∞) := 0; when
γ > 1, we define u(0) := +∞ and u−1 (+∞) := 0. As a result, Mt (X) is well defined
for any nonnegative random variable X. Moreover, when γ ≥ 1 and X = 0 with positive
probability, Mt (X) = 0.
Note that in the infinite-horizon setting the recursive utility process is defined recursively without a terminal condition, so the existence and uniqueness of this process is not
automatically guaranteed. Epstein and Zin (1989) prove the existence when the aggregator is given by (2.2). Ma (1993, 1996, 1998) prove the existence and uniqueness of the
recursive utility process by assuming that Hz (c, z), the derivative of the aggregator H(c, z)
with respect to z, is bounded uniformly in c and z by a number strictly less than one.1
However, this assumption does not hold for H as defined in (2.2) for any ρ > 0. Ozaki
and Streufert (1996) prove the existence and uniqueness of the recursive utility process by
assuming Hz (c, z) to be uniformly bounded in c and z and a set of conditions.2 However,
1
See Assumption W4 in Ma (1993, p. 246) and Ma (1996, p. 568). In Ma (1998), the author assumes
the recursive utility for deterministic consumption flows is well defined, but this requires Hz (c, z) to be
bounded by a number strictly less than one as well; see Footnote 5 of Ma (1998) and Assumption W5 in
Lucas and Stokey (1984).
2
In Ozaki and Streufert (1996, Theorem D), the authors assume that δ̄ and δ therein are finite, which
5
these conditions are difficult to verify; see conditions N1–N12 in Ozaki and Streufert (1996,
pp. 404–405); in addition, for H as defined in (2.2), Hz (c, z) is not bounded when ρ ≤ 1.
Marinacci and Montrucchio (2010) consider Thompson and Blackwell aggregators and
study the existence and uniqueness of the recursive utility with these two types of aggregators. One can check that H as defined in (2.2) satisfies properties (W-i), (W-ii), and
(W-iii) in Marinacci and Montrucchio (2010, p. 1783), satisfies property (W-iv) therein if
and only if ρ < 1, and does not satisfy property (W-v) therein for any ρ > 0. Thus, H
as defined in (2.2) with ρ < 1 is a Thompson aggregator, but the case in which ρ ≥ 1 is
neither Thompson nor Blackwell. Moreover, u as defined in (2.3) is constant relative risk
averse (CRRA), i.e., −xu00 (x)/u0 (x) is constant in x, so Theorem 3-(ii) of Marinacci and
Montrucchio (2010) applies, showing that the recursive utility process uniquely exists if
ρ < 1 and the growth rate of the consumption process {Ct } is properly constrained. The
case in which ρ ≥ 1, however, is not studied by Marinacci and Montrucchio (2010).3
Hansen and Scheinkman (2012) assume that the consumption growth rate Ct+1 /Ct =
exp[κ(Xt , Xt+1 , Yt+1 )] for some function κ, where {Xt } is a Markov process and the joint
distribution of (Xt+1 , Yt+1 ) conditioned on (Xt , Yt ) depends only on Xt . They show that for
H and Mt as defined, respectively, in (2.2) and (2.3) with ρ 6= 1 and γ 6= 1, the recursive
utility process exists. They also show the uniqueness when (1 − γ)/(1 − ρ) ≥ 1.
2.2
A Recursive Utility Model with Narrow Framing
Barberis and Huang (2008, 2009) and Barberis et al. (2006) consider a model of narrow
framing: at time t, an agent consumes Ct and invests dollar amount Θi,t in asset i, i =
0, 1, . . . , n, where asset 0 is a risk-free asset and the other assets are risky. For simplicity,
assume that short-selling of the risky assets is not allowed, i.e., Θi,t ≥ 0, i = 1, . . . , n. The
agent’s utility process {Ut } is defined recursively as follows:
Ut = H
Ct , Mt (Ut+1 ) +
n
X
!
bi Gi,t ,
t = 0, 1, . . . ,
(2.4)
i=1
is equivalent to assuming that Hz (c, z) is bounded; see pages 403–406 therein.
3
Alternatively, one can consider the following transformation: Ṽt := f (Vt ), where f (x) := x1−ρ when
ρ 6= 1 and f (x) := ln(x) when ρ = 1. Then, we have Ṽt = H̃(Ct , M̃t (Ṽt+1 )) for a new aggregator H̃.
However, H̃(0, z) is finite only if ρ < 1, and the aggregators considered in Marinacci and Montrucchio
(2010) are assumed to take real values for any c, z ≥ 0, so the results in Marinacci and Montrucchio (2010)
do not apply to the case ρ ≥ 1 either even if we do the transformation.
6
where H is an aggregator, Mt (X) is the certainty equivalent of X given information at
time t, bi ≥ 0 is a constant, and Gi,t stands for the utility of the gain and loss experienced
by the agent for her investment in asset i. More precisely, in Barberis and Huang (2008,
2009) and Barberis et al. (2006),
Gi,t = Et Θi,t (Ri,t+1 − Rf,t+1 )1Ri,t+1 >Rf,t+1 + kΘi,t (Ri,t+1 − Rf,t+1 )1Ri,t+1 <Rf,t+1
for some k ≥ 1, where Ri,t+1 and Rf,t+1 are the total returns of asset i and the risk-free
asset, respectively, in period t to t + 1. Indeed, Gi,t represents the preference value of the
agent’s position in asset i under prospect theory (Kahneman and Tversky, 1979, Tversky
and Kahneman, 1992) with the reference point to be the risk-free return, utility function
to be a piece-wise linear function in which parameter k measures the loss aversion degree
of the agent, and no probability weighting. Thus, Gi,t captures the agent’s utility of the
gain and loss for her investment in each individual risky asset due to narrow framing.
There are several variants and extensions of the model of narrow framing in the literature. De Giorgi and Legg (2012) generalize it by considering a piece-wise power utility
function and nonlinear probability weighting functions. He and Zhou (2014) consider the
case in which there is only one risky asset, but the reference point therein can be different
from the risk-free return. In Barberis and Huang (2001), Barberis et al. (2001), and Li and
Yang (2013), bi in (2.4) is set to be a constant proportion of a power transformation of the
aggregate consumption in the market.
Although the model of narrow framing and its variants have been proven useful to
explain individuals’ choice under risk and the equity premium puzzle (Barberis and Huang,
2008, 2009, Barberis et al., 2006, De Giorgi and Legg, 2012, He and Zhou, 2014), the
existence and uniqueness of the utility process in these models have not been studied. The
main objective of the present paper is to establish the existence and uniqueness.
3
3.1
Model and Motivation
Model
Consider the following equation
Vt = H (ct , Mt (At+1 Vt+1 ) + Bt ) ,
7
t = 0, 1, . . . ,
(3.1)
where the aggregator H and certainty equivalent Mt are given by (2.2) and (2.3), respectively. Here, {ct } stands for a consumption process, {At } is a process that is used to model
portfolio returns in portfolio selection problems, and {Bt } is used to model the utility of
investment gains and losses in the model of narrow framing. Our goal is to establish the
existence and uniqueness of the solution {Vt } to this equation. Compared to (2.1), which is
the equation considered in the literature to study the existence and uniqueness of recursive
utility, our formulation (3.1) has two additional terms At+1 and Bt . The presence of Bt is
motivated by the model of narrow framing, and the motivation for considering At+1 will be
clear momentarily.
Following Hansen and Scheinkman (2012), we consider equation (3.1) in a Markov
environment. More precisely, we consider a Markov process {(Xt , Yt )} and suppose the
following:
Assumption 1 (i) {(Xt , Yt )} is a Markov process and the joint distribution of (Xt+1 , Yt+1 )
conditioned on (Xt , Yt ) depends only on Xt .
(ii) Consumption dynamics evolve as
log(ct+1 ) − log(ct ) + log At+1 = κ(Xt , Xt+1 , Yt+1 ),
t = 0, 1, . . .
for some real-valued measurable function κ.
(iii) Bt /ct = $(Xt ), t = 0, 1, . . . for some real-valued measurable function $.
(iv) For any state x, Et u eκ(Xt ,Xt+1 ,Yt+1 ) |Xt = x exists.
Assumption 1-(i) is the same as Assumption 1-a) in Hansen and Scheinkman (2012):
it posits that given {Xt }, {Yt } is an independent sequence. This assumption also implies
that {Xt } is a Markov process, and we denote its state space as X. Assumption 1-(ii) and
-(iii) are parallel to Assumption 1-b) in Hansen and Scheinkman (2012), which ensure a
Markovian structure in equation (3.1). We assume the state space X to be a metric space,
so the measurability in Assumption 1 is with respect to Borel σ-algebra of X.
Dividing (3.1) by ct on both sides and using the homogeneity of H, we obtain
Vt /ct = H 1, Mt At+1 (ct+1 /ct )(Vt+1 /ct+1 ) + Bt /ct ,
t = 0, 1, . . .
(3.2)
Thus, to solve equation (3.1), we only need to solve {Vt /ct } from (3.2). Moreover, because of
8
Assumption 1, we restrict ourselves to Markovian solutions to (3.2), i.e., Vt /ct = f (Xt ), t =
0, 1, . . . for some function f . Then, the solution to equation (3.2) becomes the fixed point
of operator T defined as
Tf (x) := H 1, u−1 Et u eκ(Xt ,Xt+1 ,Yt+1 ) f (Xt+1 ) |Xt = x + $(x) ,
x ∈ X.
(3.3)
Denote X as the space of measurable functions on X, X+ as the space of nonnegative
measurable functions on X, i.e., X+ := {f ∈ X |f (x) ≥ 0, x ∈ X}, X+o as the space of
nonnegative functions on X that are not zero, i.e., X+o := {f ∈ X+ |f 6= 0}, and X++ as
the space of positive functions on X, i.e., X++ := {f ∈ X |f (x) > 0, x ∈ X}. Recalling the
definitions of H, u, and T, we can see that the domain of T is contained in X+ .
In the following, for any f ∈ X , we denote f + as its positive part, i.e., f + (x) :=
max(f (x), 0). For any f1 , f2 ∈ X , f1 ≥ f2 means f1 (x) ≥ f2 (x), x ∈ X and f1 > f2 means
f1 (x) > f2 (x), x ∈ X. Denote R as the set of real numbers. For any a ∈ R, it also denotes
the function on X that takes value a in all states.
3.2
Motivation
We justify Assumption 1 by considering the model of narrow framing proposed by Barberis
and Huang (2008, 2009) and Barberis et al. (2006), i.e., model (2.4). Denote {Wt } as the
agent’s wealth process corresponding to a consumption strategy, i.e., a consumption process
{Ct }, and an investment strategy, i.e., the process of the dollar amount invested in asset i,
{Θi,t }, i = 1, . . . , n. Then, the wealth dynamics evolve as
Wt+1 =
W t − Ct −
n
X
!
Θi,t Rf,t+1 +
i=1
n
X
Θi,t Ri,t+1 ,
t = 0, 1, . . .
i=1
Because of the homogeneity of H, Mt , and Gi,t , the agent’s utility per unit wealth, Ut /Wt ,
satisfies
!
n
X
Ut /Wt = H ct , Mt (1 − ct )Rp,t+1 (Ut+1 /Wt+1 ) + (1 − ct )
bi θi,t gi,t
(3.4)
i=1
9
where ct := Ct /Wt is the consumption propensity at time t, θi,t := Θi,t /(Wt − Ct ) ≥ 0 is
the percentage allocation to risky asset i at time t, i = 1, . . . , n,
Rp,t+1 := Rf,t+1 +
n
X
θi,t (Ri,t+1 − Rf,t+1 )
(3.5)
i=1
is the portfolio return in period t to t + 1, and
gi,t = Et (Ri,t+1 − Rf,t+1 )1Ri,t+1 >Rf,t+1 + k(Ri,t+1 − Rf,t+1 )1Ri,t+1 <Rf,t+1 .
(3.6)
Suppose the total return rate of risky asset i in period t to t+1 is Ri,t+1 = ri (Xt , Xt+1 , Yt+1 ),
t = 0, 1, . . . , for some function ri and the total return rate of the risk-free asset in period
t to t + 1 is Rf,t+1 = r0 (Xt ), t = 0, 1, . . . , for some function r0 . Because conditioning on
(Xt , Yt ), the joint distribution of (Xt+1 , Yt+1 ) depends only on Xt , it is natural for the agent
to consider Markovian strategies only, i.e., consider ct = c(Xt ), θi,t = θi (Xt ), t = 0, 1, . . . ,
i = 1, . . . , n, for some functions c and θi ’s. Then, gi,t depends on Xt only. Moreover, {Ut }
is a solution to (2.4) if and only if {Ut /Wt } is a solution to (3.1) with
At+1 = (1 − ct )Rp,t+1 ,
Bt = (1 − ct )
n
X
bi θi,t gi,t .
i=1
Furthermore, Assumption 1 holds in this example.
4
Existence and Uniqueness When the State Space is
Finite
In this section, we study the existence and uniqueness of the solution to (3.1), i.e., of the
fixed point of (3.3), when the state space of {Xt } is finite. Thus, we impose
Assumption 2 The state space for {Xt } is finite and {Xt } is irreducible.
Hansen and Scheinkman (2012) consider a general Markov process when studying the
solution to (2.1), but they assume the existence of Perron-Frobenius eigenvalue and eigenvector of a linear operator and the stochastic stability of {Xt } after a change of measure.
These assumptions are not easy to verify in general. Moreover, when applying model (3.1)
to a decision problem, such as the portfolio selection example demonstrated in Section 3.2,
10
it becomes even more difficult to find a feasible set of decisions so that these assumptions
hold. Thus, we choose to focus mainly on the setting of finite-state Markov processes at
the moment, and the setting of general Markov processes is discussed in Appendix B.
Note that we assume {Xt } to be irreducible. This assumption is necessary for the
existence of the stationary distribution of {Xt }, which will be used in the following. Note
also that we do not impose any assumptions on {Yt }.
We also note that when X is finite, T is continuous. However, we cannot apply the
classical Brouwer fixed point theorem to prove the existence and uniqueness of the fixed
point of T. First, the domain of T under consideration in the following, i.e., X++ , is not
compact. Second, the Brouwer theorem does not imply uniqueness of the fixed point.
Third, the Brouwer theorem does not show how to compute the fixed point; we, however,
will provide an easy algorithm to compute the fixed point.
4.1
Changing the Probability Measure
To prove the existence and uniqueness of the fixed point of T, we follow Hansen and
Scheinkman (2012) to do a change of probability measure based on the classical PerronFrobenius theory. To this end, consider the following operator
Uh(x) := Et u eκ(Xt ,Xt+1 ,Yt+1 ) h(Xt+1 )|Xt = x ,
x ∈ X.
With Assumptions 1 and 2, this operator is well defined. Denote P as the transition
matrix of {Xt }, i.e., Px,y = P(Xt+1 = y|Xt = x), x, y ∈ X. Define matrix P̃ by P̃x,y :=
Px,y Et u eκ(Xt ,Xt+1 ,Yt+1 ) |Xt = x, Xt+1 = y , x, y ∈ X.
Proposition 1 Suppose Assumptions 1 and 2 hold.
(i) Suppose γ 6= 1. Then, there exist η > 0 and v ∈ X++ such that
Uv(x) = ηv(x),
x ∈ X.
(4.1)
Moreover, η and v are the Perron-Frobenius eigenvalue and eigenvector of P̃, respectively.
(ii) Suppose γ = 1. Then, there exist η ∈ R and v ∈ X such that
Et [κ(Xt , Xt+1 , Yt+1 )|Xt = x] = −Et [v(Xt+1 )|Xt = x] + v(x) + η,
11
x ∈ X.
(4.2)
In addition,
η=
X
πx Et [κ(Xt , Xt+1 , Yt+1 )|Xt = x],
x∈X
where vector (πx )x∈X is the stationary distribution of {Xt }.
(iii) Define δ := u−1 (η). Then,
δ = max min
u−1 Et u eκ(Xt ,Xt+1 ,Yt+1 ) f (Xt+1 ) |Xt = x
f (x)
f ∈X++ x∈X
.
Proposition 1-(i) is the same as in Hansen and Scheinkman (2012, p. 11968), but
Proposition 1-(ii) is new.4 Proposition 1-(iii) transforms η obtained in Proposition 1-(i)
and -(ii) into δ that is easy to use in the following. Moreover, it provides a representation
of δ, which shows that δ is related to the certainty equivalent of exp[κ(Xt , Xt+1 , Yt+1 )]. One
can see that δ is decreasing with respect to the RRAD γ.
As we will see, δ is critical in proving the existence and uniqueness of the fixed point
of T. Thus, it is important to compute δ, i.e., to compute η. When γ 6= 1, η is the
Perron-Frobenius eigenvalue of P̃, so its computation has been studied extensively in the
literature; see for instance Chanchana (2007). When γ = 1, η is actually the expectation of
κ(Xt , Xt+1 , Yt+1 ) under the stationary distribution of {Xt }, which is also easy to compute.
4.2
The Case in Which $ is Nonnegative
Theorem 1 Suppose Assumptions 1 and 2 hold. Assume $(x) ≥ 0, x ∈ X. Recall δ as
defined in Proposition 1 and assume βδ 1−ρ < 1. Then, the fixed point of T in X++ uniquely
exists. Moreover, for any f ∈ X++ , {Tn f }n≥0 converges to the fixed point.
Theorem 1 shows that when the state space of {Xt } is finite and $ is nonnegative,
the fixed point of T in X++ and, thus, the recursive utility defined by (3.1) uniquely exist
provided that βδ 1−ρ < 1. Condition βδ 1−ρ < 1 is the same as the one in Hansen and
Scheinkman (2012, Proposition 6) where the authors study the existence and uniqueness
of the classical recursive utility (without narrow framing).
4
Note that the notations in Hansen and Scheinkman (2012) are different from ours: η therein corresponds
ln η in the present paper.
12
Note that we restrict the domain of T to X++ although T is well defined on X+ . This
is because T can have nonpositive fixed points. For example, when ρ ≥ 1 and $ ≡ 0,
0 is a fixed point of T. When ρ ≥ 1, γ ≥ 1, the transition matrix of {Xt } is positive,
and $(x) = 0 for some x ∈ X, we can verify that H(1, $(x)), x ∈ X is a fixed point of T
but is not in X++ . The fixed points in these two examples, however, are not economically
meaningful to represent the agent’s total utility of her consumption and investment: given
a positive consumption stream and nonnegative investment utility, we expect the agent’s
total utility to be positive. Thus, we need to exclude such fixed points by restricting the
domain of T to X++ and, by doing so, we obtain the uniqueness of the fixed point.
Theorem 1 also provides a simple algorithm to compute the fixed point: one can start
from any positive function, e.g., a positive constant function, to do iteration, and one
can obtain a sequence that eventually converges to the fixed point. This result provides
another reason why nonpositive fixed points of T, if exist, are not desirable: these fixed
points cannot be obtained by a recursive algorithm with any positive starting point.
In the above algorithm, one can also choose a nonnegative function, i.e., f ∈ X+ , as the
starting point, provided that Tm f ∈ X++ for some m. Such m exists (i) for any f ∈ X+
if ρ < 1 because H(1, 0) = (1 − β)1/(1−ρ) > 0 and (ii) for any f ∈ X+o if γ < 1 because
{Xt } is irreducible and u−1 (E[u(Z)]) > 0 for any nonnegative and nonzero random variable
Z when γ < 1. If ρ ≥ 1 and γ ≥ 1, however, {Tn f }n≥0 may not converge to the fixed
point of T in X++ . For instance, suppose X contains two elements, e.g., x1 and x2 , the
transition matrix of {Xt } is positive, and $ ≡ 0. Consider f ∈ X+o such that f (x1 ) = 0
and f (x2 ) > 0. Note that H(1, 0) = 0 because ρ ≥ 1 and that u−1 (E[u(Z)]) = 0 for any
nonnegative random variable taking zero with positive probability because γ ≥ 1. We then
immediately obtain that Tf = 0 and thus the limit {Tn f }n≥0 is 0; i.e., this sequence does
not converge to the fixed point of T in X++ .
The convergence of {Tn f }n≥0 to the fixed point of T for any positive f is economically
important: it shows that a finite-horizon model of narrow framing, in which the utility at
the terminal time is positive, converges to the infinite-horizon model when the number of
periods in the former model goes to infinity. Moreover, the utility at the terminal time in
the former model is irrelevant, provided that it is positive.
13
4.3
The Case in Which $ is Not Nonnegative
We first illustrate that when $(x) < 0 for some x ∈ X, T can have zero, one, or multiple
fixed points, depending on the parameter values.
Example 1 Suppose X is singleton. Then, operator T becomes a function on [0, +∞), and
we denote this function as T (f ). In this case, δ defined in Proposition 1 becomes
u−1 E u(eκ(Xt ,Xt+1 ,Yt+1 ) ) .
Then, function T (f ) can be written as
T (f ) = H(1, δf + $).
We assume βδ 1−ρ < 1, and Theorem 1 shows that the fixed point of T in (0, +∞) uniquely
exists when $ ≥ 0. Next, we consider the case in which $ < 0.
It is obvious that the domain of T is [−$/δ, +∞). Straightforward computation yields

+∞,
ρ ≤ 1,
ρ < 1,
lim T (f ) =
lim T (f ) =
(1 − β)1/(1−ρ) , ρ > 1,
0,
f ↓−$/δ
ρ ≥ 1, f ↑+∞


+∞,
(βδ 1−ρ )1/(1−ρ) , ρ < 1,
ρ ≤ 1,
0
0
lim T (f ) =
lim T (f ) =
(βδ 1−ρ )1/(1−ρ) , ρ > 1, f ↑+∞
0,
f ↓−$/δ
ρ ≥ 1.

(1 − β)1/(1−ρ) ,
Moreover, T is strictly increasing and concave.
We first consider the case in which ρ ≥ 1. Note that in this case T (−$/δ) = 0 < −$/δ.
Because T 0 (−$/δ) > 1 and T 0 (+∞) < 1, we conclude that except in a very special case in
which the identity line is tangent to T , it is either the case in which T has no fixed point
or the case in which T has two fixed points; see Figure 4.1.
Next, consider the case in which ρ < 1. If T (−$/δ) = (1 − β)1/(1−ρ) ≤ −$/δ, we
conclude, as in the case in which ρ ≥ 1, that except in a very special case in which the
identity line is tangent to T , it is either the case in which T has no fixed point or the case
in which T has two fixed points. If (1 − β)1/(1−ρ) > −$/δ, then the fixed point exists and
is unique; see Figure 4.2.
Note that exp[κ(Xt , Xt+1 , Yt+1 )] stands for the portfolio and consumption growth rate in
the model of narrow framing in Section 3.2, so δ stands for the certainty equivalent of the
14
No fixed point, ρ>1
Two fixed points, ρ>1
β = 0.5
1.5
6
β = 0.5
ρ = 1.5
ϖ = −0.1
0.5
3
T(f)
1.0
δ=1
ϖ = −0.4
1
2
T(f)
4
5
ρ = 1.5
δ=1
0
1
2
3
4
5
T(f)
identity line
x= − ϖ/δ
0.0
0
T(f)
identity line
x= − ϖ/δ
6
0.0
0.5
f
1.0
1.5
f
Figure 4.1: T (f ) in Example 1 without a fixed point (left panel) and with two fixed points
(right panel) when ρ > 1. The solid lines in the two panels stand for T (f ) and the dashed
lines stand for the identity function. Note that the domain of T is [−$/δ, +∞), as indicated
by the dash-dotted lines.
portfolio and consumption growth rate and thus is decreasing with respect to the RRAD. On
the other hand, −$ stands for the disutility of loss. We can see that with ρ < 1, inequality
(1 − β)1/(1−ρ) > −$/δ holds if β is small, δ is large, and −$ is small. Thus, we can
conclude that the agent’s total utility with narrow framing is well defined when her EIS is
strictly larger than one, her time discounting is large, her portfolio and consumption growth
rate is high, her RRAD is low, and her disutility of loss is small.
Example 1 shows that we need some conditions on model parameters to establish the
existence and uniqueness of the fixed point of T when $ is negative in some states.
Assumption 3 Denote
f0 (x) := H(1, $+ (x)),
x ∈ X.
(4.3)
Assume Tf0 is well defined, i.e.,
u−1 Et u eκ(Xt ,Xt+1 ,Yt+1 ) f0 (Xt+1 ) |Xt = x + $(x) ≥ 0,
and Tm f0 > f0 for some m ≥ 1.
15
x ∈ X,
No fixed point, ρ<1
Two fixed points, ρ<1
1.5
0.8
β = 0.5
ρ = 0.5
δ=1
δ=1
δ=1
ϖ = −0.4
ϖ = −0.1
ϖ
(1 − β)(1 (1−ρ))< −
(1 − β)(1 (1−ρ))> −
δ
ϖ
δ
0.2
2
0.5
0.4
T(f)
ϖ
T(f)
δ
4
(1 − β)(1 (1−ρ))< −
ρ = 0.5
1.0
ρ = 0.5
ϖ = −0.5
T(f)
One fixed point, ρ<1
β = 0.4
0.6
6
β = 0.5
0
2
4
f
6
0.0
0.2
0.4
f
0.6
0.8
T(f)
identity line
x= − ϖ/δ
0.0
T(f)
identity line
x= − ϖ/δ
0.0
0
T(f)
identity line
x= − ϖ/δ
0.0
0.5
1.0
1.5
f
Figure 4.2: T (f ) in Example 1 without a fixed point (left panel), with two fixed points
(middle panel), and with one fixed point (right panel) when ρ < 1. The solid lines in the
two panels stand for T (f ) and the dashed lines stand for the identity function. Note that
the domain of T is [−$/δ, +∞), as indicated by the dash-dotted lines. In the left and
middle panels, (1 − β)1/(1−ρ) < −$/δ, and in the right panel, (1 − β)1/(1−ρ) > −$/δ.
Theorem 2 Suppose Assumptions 1–3 hold. Assume βδ 1−ρ < 1, where δ is defined as in
Proposition 1. Then, the fixed point of T in its domain uniquely exists and is strictly larger
than f0 point-wisely. Moreover, for any f such that Tf is well defined, sequence {Tn f }n≥0
converges to the fixed point of T.
Theorem 2 shows the existence and uniqueness of the fixed point of T when $ can go
negative. Moreover, the calculation of the fixed point is easy: start from any f such that Tf
is well defined to do iteration. Then, the resulting sequence converges to the fixed point. As
discussed in the case of nonnegative $, this algorithm implies that a finite-horizon model
of narrow framing converges to the infinite-horizon model when the number of periods in
the former goes to infinity.
Assumption 3 is critical to obtain the existence and uniqueness of the fixed point of T,
so we discuss it in full details in the following:
(i) Note that if Tf0 is well defined, we must have Tf0 ≥ f0 . However, this is insufficient
to guarantee the uniqueness of the fixed point of T. Indeed, in the setting of Example 1,
if (1 − β)1/(1−ρ) = −$/δ, Tf0 is well defined, and actually Tf0 = f0 . We already showed
in that example that T has two fixed points and one of them is f0 , and both can represent
the utility process. Thus, to guarantee the uniqueness, we need further conditions and
Assumption 3 serves the purpose.
(ii) Assumption 3 implies that Tf0 (x) > f0 (x) for some x ∈ X. The reverse is also true
when γ < 1 or f0 ∈ X++ . Indeed, for any y ∈ X such that the transition probability from
16
y to x is positive, we have
u−1 Et u eκ(Xt ,Xt+1 ,Yt+1 ) Tf0 (Xt+1 ) |Xt = y > u−1 Et u eκ(Xt ,Xt+1 ,Yt+1 ) f0 (Xt+1 ) |Xt = y .
As a result, T2 f0 (y) = T(Tf0 )(y) > Tf0 (y), and because of the irreducibility of {Xt }, we
conclude that Tm f0 > f0 for some m ≥ 1.
(iii) When γ ≥ 1 and f0 (x) = 0 for some x ∈ X, which is the case if and only if $(x) ≤ 0
for the same x and ρ ≥ 1, it is possible that Tf0 is well defined, Tf0 (x) > f0 (x) for some
x ∈ X, and the fixed point of T is not unique. For instance, consider {Xt } with state space
X = {x1 , x2 , x3 } such that P(Xt+1 = x3 |Xt = x1 ) = 1, P(Xt+1 = x3 |Xt = x2 ) = 1, p1 :=
P(Xt+1 = x1 |Xt = x3 ) > 0, and p2 := P(Xt+1 = x2 |Xt = x3 ) > 0. Then, {Xt } is irreducible.
Suppose κ ≡ 0. Suppose $(x2 ) > 0, $(x3 ) > 0, and $(x1 ) := −H(1, $(x3 )) < 0. Then,
one can verify that Tf0 (x1 ) = H(1, 0) = f0 (x1 ) = 0, Tf0 (x2 ) = H(1, f0 (x3 ) + $(x2 )) >
f0 (x2 ) > 0, and Tf0 (x3 ) = H(1, $(x3 )) = f0 (x3 ) > 0. Moreover, it is straightforward to
see that Tf0 is a fixed point of T. On the other hand, suppose ρ > 1 and γ > 1. Then,
straightforward calculation shows that
dT(f0 + 1)(x1 ) = Hz (1, 0) = β 1/(1−ρ) ,
d
=0
dT(f0 + 1)(x2 ) = Hz (1, f0 (x3 ) + $(x2 )),
d
=0
dT(f0 + 1)(x3 ) 1/(1−γ)
= Hz (1, $(x3 ))p1
,
d
=0
where 1 stands for the constant function taking value 1 and Hz is the partial derivative of
H(c, z) with respect to z. Because ρ > 1, γ > 1, and β < 1, with sufficiently small (but
i) > 1, i = 1, 2, 3. As a result, there
positive) p1 , $(x2 ), and $(x3 ), we have dT(f0 +1)(x
d
=0
exists > 0 such that T(f0 + 1) ≥ f0 + 1. Consequently, {Tn (f0 + 1)}n≥0 is increasing
and converges because Tf ≤ (1 − β)1/(1−ρ) for any f . It is obvious that the convergent
point is a fixed point of T and is different from Tf0 because f0 (x1 ) + > 0 = Tf0 (x1 ).
(iv) When ρ ≥ 1, γ ≥ 1, and the transition matrix of {Xt } is positive, Assumption 3
does not hold and thus Theorem 2 cannot apply if $(x) < 0 for some x ∈ X. Indeed, in
this case, we have
u−1 Et u eκ(Xt ,Xt+1 ,Yt+1 ) f0 (Xt+1 ) |Xt = x = 0
17
because f0 (x) = 0 and P(Xt+1 = x|Xt = x) > 0, and γ ≥ 1. Consequently, Tf0 is not well
defined because $(x) < 0.
(v) Theorem 2 cannot cover Theorem 1. Indeed, suppose $ ≡ 0 and ρ ≥ 1. Then, Assumption 3 does not hold, so Theorem 2 cannot apply. Theorem 1, however, can still apply.
Therefore, Theorem 1 is more comprehensive than Theorem 2 when $ is nonnegative, and
Theorem 2 is useful when $ goes negative.
4.4
4.4.1
Portfolio Selection with Narrow Framing
Model
Consider the problem of portfolio selection with narrow framing discussed in Sections 2.2
and 3.2. The agent’s total utility Ut is given by (2.4) and thus her total utility per unit
wealth Ut /Wt satisfies (3.4), where Rp,t+1 and gi,t are given as in (3.5) and (3.6), respectively.
Suppose the total return rate of risky asset i in period t to t+1 is Ri,t+1 = ri (Xt , Xt+1 , Yt+1 )
for some function ri and the total return rate of the risk-free asset in period t to t + 1 is
Rf,t+1 = r0 (Xt ) for some function r0 . Suppose the agent chooses consumption propensity
ct = c(Xt ) and portfolio θt = (θ1 (Xt ), . . . , θn (Xt ))0 at time t for some functions c and θi ’s.
Then, the agent’s total utility per unit wealth Ut /Wt = Fc,θ (Xt ), where Fc,θ is a fixed point
of
Vc,θ F (x) := H c(x), u−1 Et u (1 − c(x))Rp (x, Xt+1 , Yt+1 )F (Xt+1 ) |Xt = x
+(1 − c(x))
n
X
bi θi (x)gi (x) ,
x∈X
i=1
with Rp (Xt , Xt+1 , Yt+1 ) := r0 (Xt ) +
Pn
i=1 θi (Xt )(ri (Xt , Xt+1 , Yt+1 )
− r0 (Xt )) and
gi (x) := Et (Ri,t+1 − Rf,t+1 )1Ri,t+1 >Rf,t+1 + k(Ri,t+1 − Rf,t+1 )1Ri,t+1 <Rf,t+1 |Xt = x , x ∈ X.
For any c and θ such that 0 < c(Xt ) < 1 and Rp (Xt , Xt+1 , Yt+1 ) > 0, F is a fixed point
of Vc,θ if and only if f (x) := F (x)/c(x) is a fixed point of Tc,θ , where
Tc,θ f (x) := H 1, u−1 Et u eκc,θ (Xt ,Xt+1 ,Yt+1 ) f (Xt+1 ) |Xt = x + $c,θ (x) ,
18
x∈X
(4.4)
and
κc,θ (Xt , Xt+1 , Yt+1 ) : = ln c(Xt+1 ) − ln c(Xt ) + ln(1 − c(Xt )) + ln Rp (Xt , Xt+1 , Yt+1 ), (4.5)
n
1 − c(Xt ) X
$c,θ (Xt ) : =
bi θi (Xt )gi (Xt ).
(4.6)
c(Xt ) i=1
Denote δ in Proposition 1 as δc,θ when κ and $ therein are set to be κc,θ and $c,θ , respectively.
For each x ∈ X, consider a set Ix ⊂ (0, 1) and a set Jx ⊂ Rn . Define
A := {(c, θ)|c(x) ∈ Ix , θ(x) ∈ Jx , x ∈ X}.
1−ρ
Assumption 4 For each (c, θ) ∈ A, Rp (Xt , Xt+1 , Yt+1 ) > 0 and βδc,θ
< 1. Moreover,
for each (c, θ) ∈ A, it is either the case in which $c,θ (x) ≥ 0, x ∈ X or the case in which
+
Tc,θ f0,c,θ with f0,c,θ (x) := H(1, $c,θ
(x)), x ∈ X is well defined and Tm
c,θ f0,c,θ > f0,c,θ for some
m ≥ 1.
With Assumptions 2 and 4 in place, Theorems 1 and 2 show that the fixed point of Tc,θ
in X++ uniquely exists for any (c, θ) ∈ A. Thus, if the agent consumes Cs = c(Xs )Ws and
invests Θi,s = θi (Xs )(Ws − Cs ) dollars in risky asset i, i = 1, . . . , n at time s ≥ t, her utility
Ut is well defined. As a result, the following portfolio selection problem
max
({Cs }s≥t ,{Θs }s≥t )∈Bt
Ut
(4.7)
is well defined, where
Bt := {({Cs }s≥t , {Θs }s≥t )|Cs = c(Xs )Ws , Θs = θ(Xs )(Ws − Cs ) for some (c, θ) ∈ A}.
It is obvious that problem (4.7) is equivalent to
max Fc,θ (x),
(c,θ)∈A
19
∀x ∈ X.
(4.8)
4.4.2
Dynamic Programming
The dynamic programming equation associated with the portfolio selectin problem (4.8)
can be heuristically derived as
Φ(x) = WΦ(x),
x ∈ X,
(4.9)
WΦ(x) := max H c̄, (1 − c̄) max DΦ (x, θ̄) ,
x ∈ X,
(4.10)
where
c̄∈Ix
DΦ (x, θ̄) :=
n
X
θ̄∈Jx
θ̄i bi gi (x)
i=1
h n
i
X
+u
θ̄i (ri (x, Xt+1 , Yt+1 ) − r0 (x)) Φ(Xt+1 ) |Xt = x .
Et u r0 (x) +
−1
(4.11)
i=1
Proposition 2 Suppose Assumptions 2 and 4 hold. Suppose Φ ∈ X++ is a solution to
(4.9). Then, Φ(x) ≥ Fc,θ (x), x ∈ X for any (c, θ) ∈ A. Moreover, if there exists (c∗ , θ∗ ) ∈ A
such that (c∗ (x), θ∗ (x)) is a maximizer of (4.10) for each x ∈ X, then (c∗ , θ∗ ) and Φ are a
maximizer and the optimal value, respectively, of (4.8).
Proposition 2 shows that the solution to the dynamic programming equation, if exists,
is the solution to (4.8).
Proposition 3 Suppose Assumptions 2 and 4 hold, and for each x ∈ X, Ix and Jx are
compact. Then, the fixed point of W in X++ uniquely exists. Moreover, the following are
true:
(i) If $c,θ ≥ 0 for any (c, θ) ∈ A, then {Wn Φ}n≥0 converges to the fixed point of W in
X++ for any Φ ∈ X++ .
(ii) If $c,θ (x) < 0 for some x ∈ X and some (c, θ) ∈ A, then {Wn Φ}n≥0 converges to the
fixed point of W in X++ for any Φ such that WΦ is well defined and in particular for
Φ(x) := max H
c̄∈Ix
c̄, (1 − c̄) max
θ̄∈Jx
n
X
!
+
θ̄i bi gi (x)
.
(4.12)
i=1
Proposition 3 shows the existence and uniqueness of the solution to the dynamic programming equation when the control sets Ix and Jx are compact. Moreover, it shows that
20
starting from any Φ that is positive when $c,θ ≥ 0 for any (c, θ) ∈ A or that makes WΦ
well defined in other cases, by applying the dynamic programming equation repeatedly, one
eventually obtains the solution to the equation. This result shows that the optimal consumption and portfolio in a finite-horizon model converges to those in the infinite-horizon
model when the number of periods in the former goes to infinity.
Note that DΦ (x, θ̄) is strictly concave in θ̄ for each x and Φ and H(c̄, (1 − c̄)z) is strictly
concave in c̄ for any given z ≥ 0. Thus, for each x and Φ, the maximization problem in the
right-hand side of the dynamic programming equation (4.9), i.e., in (4.10), can be easily
solved. As a result, WΦ can be easily computed, and once we find the fixed point of W,
the optimal control (c∗ , θ∗ ) can also be solved easily.
Finally, when $c,θ (x) < 0 for some x ∈ X and some (c, θ) ∈ A, equation (4.12) provides
a simple choice of Φ such that WΦ is well defined and thus {Wn Φ}n≥0 converges to the
+
Pn
is convex
fixed point of W in X++ . This Φ is easy to compute: note that
i=1 θ̄i bi gi (x)
in θi and H (c̄, (1 − c̄)z) is concave in c̄ for any z > 0, so the maximization in θ̄i and c̄ can
be easily computed.
4.4.3
Example
We consider a market with a risky stock and a risk-free asset. We can regard the stock as
the market portfolio and we set the length of each period to be one year. To construct the
return of the stock, we assume that the stock pays a dividend every year and the dividend
growth rates are i.i.d. following the distribution given as in Table 4.1.
Table 4.1: Distribution of the dividend growth rate. The distribution is assumed to be the
same as in Table I of Chapman and Polkovnichenko (2009), which is obtained using the
historical gross consumption growth from 1949 to 2006.
State
1
Outcome 0.976
Probability 0.03
2
0.993
0.03
3
1.002
0.10
4
1.011
0.16
5
1.019
0.24
6
1.028
0.19
7
1.037
0.13
8
1.045
0.09
9
1.054
0.03
We assume the market is governed by a two-state Markovian process {Xt } that takes
values in X = {0, 1}. We assume the price-dividend ratio at time t to be ϕ(Xt ) and the
risk-free total return rate in period t to t + 1 to be r0 (Xt ); i.e., both are functions of Xt .
As a result, the total return rate of the stock in period t to t + 1 is
r(Xt , Xt+1 , Yt+1 ) = Yt+1 (ϕ(Xt+1 ) + 1)/ϕ(Xt ),
21
where Yt+1 refers to the dividend growth rate in period t to t + 1.
We assume the transition matrix of {Xt } to be
0.6 0.4
0.2 0.8
!
.
We also set the risk-free total return rate and the price-dividend ratio to be
r0 (Xt ) =
1.03
1.03
!
,
ϕ(Xt ) =
30.25
39.75
!
,
respectively, so that the mean and variance of the stock return under the stationary distribution of {Xt } are 6% and 15%, respectively, and consequently the equity premium is
3%.
We set the loss aversion degree k = 1.5, so g(Xt ), which measures the investment utility,
is
!
0.1532
g(Xt ) = Et ν (r(Xt , Xt+1 , Yt+1 ) − r0 (Xt )) |Xt =
,
−0.0551
where ν(x) := x1x>0 + kx1x≤0 . Finally, we set β = 0.95, b = 0.001, ρ = 0.5, and γ = 8.
Consider the feasible set Ix = [0.1%, 10%] and Jx = [0, 100%], x = 0, 1. We can show
that Assumption 4 holds for any (c, θ) ∈ A.5 Then, we apply Propositions 2 and 3 to
calculate the optimal consumption and portfolio and the value function, and the results
are shown as follows:
!
!
!
3.65%
100%
0.0686
c∗ (Xt ) =
, θ∗ (Xt ) =
, Φ(Xt ) =
.
4.62%
19.0%
0.0541
5
A New Model of Narrow Framing
Example 1 and the discussions after Theorem 2 show that the classical model of recursive
utility with narrow framing (2.4) may fail to uniquely define the utility process, especially
when ρ ≥ 1. In this section, we propose a revision for this model so that the utility process
is uniquely defined.
5
In general, given A, Assumption 4 is not straightforward to verify, but in the two-state case, the
verification can be done. The verification result is available from the authors upon request.
22
Consider the same portfolio selection problem with narrow framing as in Section 4.4.1
except that the agent’s utility process Ūt is now defined by
Ūt = H
n X
Mt (Ūt+1 )
Ct , Mt (Ūt+1 ) +
b̄i
Gi,t
Mt (Wt+1 )
i=1
!
,
(5.1)
Mt (Ut+1 )
,
where b̄i ’s are nonnegative constants. In other words, bi in (2.4) is replaced by b̄i M
t (Wt+1 )
meaning that the weight of the investment utility in period t to t + 1 is scaled by the total
utility per unit capital from the next period.
Following the same derivation as in Section 3.2, we conclude that {Ūt } is a solution to
(5.1) if and only if {Ūt /Wt } is the solution to
Ūt /Wt = H
n
X
b̄i θi,t gi,t
ct , Mt (1 − ct )Rp,t+1 (Ūt+1 /Wt+1 ) 1 + Mt (Rp,t+1 )−1
!!
,
i=1
i.e., if and only if {Ūt /Wt } is the solution to (3.1) with
At+1 = (1 − ct )Rp,t+1 1 + Mt (Rp,t+1 )−1
n
X
!
b̄i θi,t gi,t
,
Bt = 0.
(5.2)
i=1
In consequence, we can apply Theorem 1 to obtain the existence and uniqueness of the
solution to (5.1). More precisely, we have the following corollary:
Corollary 1 Given a consumption-investment strategy (c, θ), assume {At } defined as in
(5.2) satisfy Assumptions 1 and 2. Then, {Ūt } is a solution to (5.1) if and only if f (Xt ) :=
(Ūt /Wt )/ct is a fixed point of
T̄c,θ f (x) := H 1, u−1 Et u eκ̄c,θ (Xt ,Xt+1 ,Yt+1 ) f (Xt+1 ) |Xt = x
,
x ∈ X,
(5.3)
where
κ̄c,θ (Xt , Xt+1 , Yt+1 ) := log(ct+1 ) − log(ct ) + log(1 − ct ) + log Rp,t+1
!
n
X
− log(Mt (Rp,t+1 )) + log Mt (Rp,t+1 ) +
b̄i θi,t gi,t .
(5.4)
i=1
1−ρ
Moreover, with β δ̄c,θ
< 1, where δ̄c,θ is the value of δ in Proposition 1 with κ set to be κ̄c,θ ,
the fixed point of T̄ in X++ uniquely exists and for any f ∈ X++ , {T̄nc,θ f } converges the
23
fixed point.
Corollary 1 shows that in a finite-state Markovian setting, the utility process defined
by (5.1) uniquely exists if
−
n
X
b̄i θi,t gi,t < Mt (Rp,t+1 ),
(5.5)
i=1
which is necessary for At defined by (5.2) to satisfy Assumption 1, and a growth condition
1−ρ
holds (i.e., β δ̄c,θ
< 1). The condition specified by inequality (5.5) stipulates that the
disutility of the loss incurred by the investment in the risky assets cannot exceed the
average return of the investment, which is measured by the certainty equivalent of the
return on the investment portfolio.
The dynamic programming analysis in Section 3.2 can still go through with the new
model of narrow framing. Formally, for each x ∈ X, consider a set Ix ⊂ (0, 1) and a set
Jx ⊂ Rn . Define A := {(c, θ)|c(x) ∈ Ix , θ(x) ∈ Jx , x ∈ X}. Assume for each (c, θ) ∈ A, the
assumptions in Corollary 1 holds. Denote
h n
i
X
D̄Φ (x, θ̄) := u
Et u r0 (x) +
θ̄i (ri (x, Xt+1 , Yt+1 ) − r0 (x)) Φ(Xt+1 ) |Xt = x
−1
i=1

× 1 +

Pn
i=1 b̄i θ̄i gi (x)
 .
P
u−1 Et u r0 (x) + ni=1 θ̄i (ri (x, Xt+1 , Yt+1 ) − r0 (x)) |Xt = x
(5.6)
Then, Propositions 2 and 3 still hold with DΦ replaced by D̄Φ and thus W is replaced by
W̄, where
W̄Φ(x) := max H c̄, (1 − c̄) max D̄Φ (x, θ̄) ,
c̄∈Ix
θ̄∈Jx
x ∈ X.
In solving the dynamic programming equation Φ = W̄Φ, we need to solve maxθ̄∈Jx D̄Φ (x, θ̄).
The maximization of D̄Φ (x, θ̄) in θ̄ is more difficult than that of DΦ (x, θ̄), but it is easy to
solve when the state space of {Xt } is singleton or when γ = 1. Indeed, in these two cases,
24
we have
n
hX
D̄Φ (x, θ̄) = Mt (Φ(Xt+1 ))
b̄i θ̄i gi (x)
i=1
n
X
i
−1
+u
Et u r0 (x) +
θ̄i (ri (x, Xt+1 , Yt+1 ) − r0 (x)) |Xt = x .
i=1
In consequence, the agent is myopic in portfolio selection in the sense that the optimal
portfolio in the current period does not depend on the optimal utility in the future; see
applications in Guo and He (2017).
6
Conclusion
We examined the model of narrow framing in the literature in which an agent derives not
only consumption utility but also investment utility of gains and losses; in particular, this
model degenerates into the classical recursive utility model when the investment utility is
zero. Assuming constant EIS and RRAD, we studied the existence and uniqueness of the
agent’s total utility in the model of narrow framing.
We assumed a Markovian setting: The asset returns in period from t to t + 1 are
assumed to be functions of Xt , Xt+1 , and Yt+1 , so the agent’s consumption propensity and
investment in the assets in that period are functions of Xt , where {Xt } is a Markov process
that represents market states and {Yt } is i.i.d. conditioning on {Xt } and thus represents
random noise. We further assumed that {Xt } is irreducible and its state space is finite.
We proved that the agent’s total utility in the model of narrow framing uniquely exists
when her investment utility is nonnegative, regardless of the values of the EIS and RRAD.
We then illustrated by an example that when the state space of {Xt } is singleton and the
EIS is less than or equal to one, the agent’s total utility is either non-existent or non-unique
if her investment utility is negative. We then proposed a sufficient condition under which
the total utility in the model of narrow framing with negative investment utility uniquely
exists, and this condition is nearly necessary. We also proved that if the total utility in
the model of narrow framing uniquely exists, it can be obtained by applying the recursive
equation defining the utility repeatedly with any positive utility value as the starting point.
We then considered portfolio selection with narrow framing and proved that a consumption and portfolio plan is optimal if and only if it, together with the value function of the
portfolio selection problem, satisfies the dynamic programming equation. Moreover, we
25
proved that the solution to the dynamic programming equation uniquely exists and can be
computed by solving equation recursively with any starting point.
Finally, we proposed a new model of narrow framing in which the agent’s investment
utility is scaled by her total utility. We proved that in this model the agent’s total utility
uniquely exists in general. Moreover, the total utility in this new model can also be obtained
by applying the recursive equation defining the utility repeatedly and portfolio selection
with this new model can also be solved by dynamic programming.
A
Proofs
Proof of Proposition 1 We first consider (i), i.e., the case in which γ 6= 1. Because of
Assumption 1-(i), we have
X
P̃x,y h(y).
Uh(x) := Et h(Xt+1 )Et u eκ(Xt ,Xt+1 ,Yt+1 ) |Xt , Xt+1 |Xt = x =
y∈X
Because P is irreducible and eκ is positive, we conclude that P̃ is also irreducible. Thus,
we have (4.1), where η and v are the Perron-Frobenius eigenvalue and eigenvector of P̃,
respectively, and η > 0, v ∈ X++ ; see for instance Meyer (2000, p. 673).
Next, we consider (ii), i.e., the case in which γ = 1. It is straightforward to see that
(4.2) is equivalent to
Pv = v + η1 − w,
(A.1)
where w denotes the vector of Et [κ(Xt , Xt+1 , Yt+1 )|Xt = x], x ∈ X and 1 denotes the vector
of all ones. Because P is an irreducible stochastic matrix, the kernel of I − P> , where I is
the identity mapping, is the linear space spanned by the left-Perron-Frobenius eigenvector
of P, i.e., by the stationary distribution π of {Xt }. As a result, the range of I − P is the
space of all vectors that are orthogonal to π. By the definition of η, η1 − w is orthogonal
to π and thus is in the range of I − P. As a result, there exists v such that (4.2) holds.
Moreover, by multiplying the stationary distribution π on both sides of (A.1), we can see
that η is uniquely determined.
Finally, we prove (iii). We first consider the case in which γ 6= 1. Because η is the
Perron-Frobenius eigenvalue of P̃, according to max-min version of the Collatz-Wielandt
26
formula (Meyer, 2000, p. 673), we have
P
P̃x,y g(y)
g∈X+ x∈X,g(x)6=0
g(x)
P
κ(Xt ,Xt+1 ,Yt+1 )
g(Xt+1 )|Xt = x, Xt+1 = y
y∈X Px,y Et u e
= maxo min
g∈X+ x∈X,g(x)6=0
g(x)
κ(Xt ,Xt+1 ,Yt+1 )
Et u e
g(Xt+1 )|Xt = x
= maxo min
.
g∈X+ x∈X,g(x)6=0
g(x)
η = maxo
y∈X
min
Moreover, it is straightforward to see that the maximum in the above formula is attained
when g is chosen to be the Perron-Frobenius eigenvector of P̃. Because the eigenvector
lies in X++ , we conclude that X+o can be replaced with X++ in the above formula. Now,
recalling δ = η 1/(1−γ) and setting f = g 1/(1−γ) , we conclude that when γ < 1,
u
δ = max min
−1
κ(Xt ,Xt+1 ,Yt+1 )
Et u e
f (Xt+1 ) |Xt = x
f (x)
f ∈X++ x∈X
.
(A.2)
Similarly, according to the min-max version of the Collatz-Wielandt formula (Meyer, 2000,
p. 669),6 we have
Et u eκ(Xt ,Xt+1 ,Yt+1 ) g(Xt+1 )|Xt = x
η = min max
.
g∈X++ x∈X
g(x)
Recalling δ = η 1/(1−γ) and setting f = g 1/(1−γ) , we conclude that (A.2) also holds when
γ > 1.
Finally, we show that (A.2) also holds when γ = 1. For each f ∈ X++ , denote
ξf : = min
u−1 Et u eκ(Xt ,Xt+1 ,Yt+1 ) f (Xt+1 ) |Xt = x
f (x)
x∈X
= exp min Et [κ(Xt , Xt+1 , Yt+1 ) + ln f (Xt+1 )|Xt = x] − ln f (x)
x∈X
.
As a result, ln(ξf ) + ln f (Xt ) ≤ Et [κ(Xt , Xt+1 , Yt+1 ) + ln f (Xt+1 )|Xt ]. Taking expectation
on both sides under the stationary distribution of {Xt } and recalling η that is derived in
6
The formula therein is presented for postive matrices, but it also holds for irreducible nonnegative
matrices because the Perron-Frobenius eigenvectors for these matrices are positive; see for instance Meyer
(2000, p. 673).
27
part (ii) of the proof, we conclude that ln(ξf ) + E[ln f (Xt )] ≤ η + E[ln f (Xt+1 )], which
implies ln(ξf ) ≤ η. Therefore, we conclude
η ≥ max min Et [κ(Xt , Xt+1 , Yt+1 ) + ln f (Xt+1 )|Xt = x] − ln f (x) .
f ∈X++ x∈X
(A.3)
On the other hand, recall v defined in part (ii) of the proof. Then, ev ∈ X++ and (A.1) can
be written as
Et ln ev(Xt+1 ) |Xt = x = ln ev(x) + η − Et [κ(Xt , Xt+1 , Yt+1 )|Xt = x] ,
x ∈ X.
Combining with (A.3), we immediately conclude that
η = max min Et [κ(Xt , Xt+1 , Yt+1 ) + ln f (Xt+1 )|Xt = x] − ln f (x) .
f ∈X++ x∈X
Therefore, (A.2) holds.
Proof of Theorem 1 For ease of exposition, the proof is divided into two parts.
Part One: existence and uniqueness of the fixed point
In the first part of the proof, we show the existence and uniqueness of the fixed point
of T in X++ . Observe that Tf is well-defined for any f ∈ X+ and T is increasing.
We first note from Proposition 1-(i) that when γ 6= 1, with η, δ, and v as defined in
Proposition 1, we can define Mt+1 := η −1 e(1−γ)κ(Xt+1 ,Yt+1 ,Xt ) v(Xt+1 )/v(Xt ) and show that
Mt+1 > 0 and Et [Mt+1 ] = 1. As a result, we can define a new measure P̃ by using Mt+1 as
the Radon-Nikodym density. Note that {Xt } is still an irreducible Markov chain under P̃.
Denote the corresponding expectation as Ẽ. Then,
Et u eκ(Xt ,Xt+1 ,Yt+1 ) f (Xt+1 ) |Xt = x
=Et Mt+1 v(Xt+1 )−1 v(Xt )ηf (Xt+1 )1−γ |Xt = x
=Ẽt v(Xt+1 )−1 v(Xt )ηf (Xt+1 )1−γ |Xt = x .
As a result, we obtain
u
Ẽt u e
f (Xt+1 ) |Xt = x
= δu−1 v(x) u−1 Ẽt u f (Xt+1 )/u−1 v(Xt+1 ) |Xt = x .
−1
κ(Xt ,Xt+1 ,Yt+1 )
28
(A.4)
After a careful calculation, one can conclude from Proposition 1-(ii) that (A.4) holds for
the case γ = 1 as well with Ẽ replaced by E. Therefore, in the following, we will use
(A.4) regardless of the value of γ, and Ẽ stands for E when γ = 1. Using (A.4) and the
homogeneity of H(c, z), we obtain
Tf (x)
=H
−1
u v(x)
"
1
, δ u−1
u−1 (v(x))
"
f (Xt+1 )
−1
u v(Xt+1 )
Ẽt u
!
#!
|Xt = x
δ −1 $(x)
+ −1
u v(x)
#!
.
(A.5)
1−ρ
We first consider the case in which ρ 6= 1. In this case, denoting g(x) := f (x)/u−1 (v(x))
,
we conclude that f is a fixed point of T in X++ if and only if g is a fixed point of S in X++ ,
where S is an operator on X+ defined as
1−β
+ βδ 1−ρ
Sg(x) := −1
ũ (v(x))
(
h
1
i 1−ρ
δ −1 $(x)
−1
ũ
+ −1
Ẽt ũ g(Xt+1 ) |Xt = x
u v(x)
)1−ρ
(A.6)
with ũ(x) := u(x1/(1−ρ) ), x ≥ 0.
It is easy to see that S is an increasing mapping from X+ into X++ . Consider function
1
1−ρ
ϕ(z) := z 1−ρ + a
, z ≥ 0 for some a ≥ 0. It is straightforward to see that ϕ0 (z) =
ρ
1
1
z 1−ρ /(z 1−ρ + a) ≤ 1. Consequently, |ϕ(z1 ) − ϕ(z2 )| ≤ |z1 − z2 | for any z1 , z2 ≥ 0. As a
result, for any g1 and g2 , we have
|Sg1 (x) − Sg2 (x)| ≤ βδ
1−ρ
−1
−1
Ẽt ũ g1 (Xt+1 ) |Xt = x − ũ
Ẽt ũ g2 (Xt+1 ) |Xt = x .
ũ
When α := (1 − γ)/(1 − ρ) ≥ 1, ũ(x) = xα is a convex, power function, so we conclude
that
i α1
h
|Sg1 (x) − Sg2 (x)| ≤ βδ 1−ρ Ẽt (|g1 (Xt+1 ) − g2 (Xt+1 )|α |Xt = x) ,
x ∈ X,
which implies
|Sg1 (Xt ) − Sg2 (Xt )|α ≤ βδ 1−ρ
α
Ẽt (|g1 (Xt+1 ) − g2 (Xt+1 )|α |Xt ) .
Recall that {Xt } is an irreducible Markov chain under measure P̃, so it has a unique
29
stationary distribution. Taking expectation on both sides of the inequality under this
stationary distribution, and noting that the marginal distributions of Xt and Xt+1 are the
same, we conclude
h
i1/α
h
i1/α
Ẽ (|Sg1 (Xt ) − Sg2 (Xt )|α )
≤ βδ 1−ρ Ẽ (|g1 (Xt ) − g2 (Xt )|α )
.
(A.7)
h
i1/α
Because βδ 1−ρ < 1, S is a contraction mapping on X+ with norm Ẽ|g(Xt )|α
. Consequently, for any g ∈ X+ , the limit of {Sn g}n≥0 exists and is the unique fixed point of S in
X+ . Moreover, because Sg(x) ≥ (1 − β)/ũ−1 (v(x)) > 0, x ∈ X, the fixed point must lie in
X++ . As a result, T has a unique fixed point in X++ .
When α < 1, we consider the following operator:
S̃g(x) :=
1−β
ũ−1 (v(x))
+ βδ 1−ρ
(
1
1−ρ
δ −1 $(x)
+ −1
Ẽt [g(Xt+1 )|Xt = x]
u v(x)
)1−ρ
.
(A.8)
Because ũ is either a concave power or a logarithmic function when α < 1, we have
ũ−1 (Et [ũ(Z)]) ≤ Et [Z] for any nonnegative random variable Z. As a result, Sg(x) ≤
S̃g(x), x ∈ X for any g and in particular for g0 (x) := (1 − β)/ũ−1 (v(x)) > 0, x ∈ X. One
can see that both {Sn g0 }n≥0 and {S̃n g0 }n≥0 are increasing sequences and the former is
dominated by the latter. On the other hand, following the same proof as in the case in
which α ≥ 1, we can show that S̃ is a contraction mapping from X+ into X++ . As a result,
{S̃n g0 }n≥0 converges and so does {Sn g0 }n≥0 . Consequently, the limit of {Sn g0 }n≥0 is a fixed
point of S and lies in X++ , and thus the fixed point of T in X++ exists. We then show
the uniqueness of the fixed point of T in X++ when α < 1. For the sake of contradiction,
suppose there are two distinct fixed points f1 and f2 in X++ . Without loss of generality,
we assume f1 (x) < f2 (x) for some x ∈ X. Define x∗ := argmin f1 (x)/f2 (x) and denote the
x∈X
corresponding minimum value as r∗ . Because X is finite and fi ’s are positive, x∗ is well
defined and r∗ ∈ (0, 1). Define f (x) := r∗ f2 (x), x ∈ X. Then, f (x) ≤ f1 (x), x ∈ X and
f (x∗ ) = f1 (x∗ ). Denote I as the identity mapping. Then, for each x ∈ X, we have
(T − I)f (x) = H 1, u−1 Et u eκ(Xt ,Xt+1 ,Yt+1 ) r∗ f2 (Xt+1 ) |Xt = x + $(x) − r∗ f2 (x)
> r∗ H 1, u−1 Et u eκ(Xt ,Xt+1 ,Yt+1 ) f2 (Xt+1 ) |Xt = x + $(x) − r∗ f2 (x)
= r∗ (T − I)f2 (x) = 0,
30
where the inequality is the case because β < 1 and $(x) ≥ 0 and the last equality is the
case because f2 is a fixed point of T. In particular, we have Tf (x∗ ) > f (x∗ ) = f1 (x∗ ). On
the other hand, because T is increasing and f ≤ f1 , we have Tf ≤ Tf1 = f1 , where the
equality is the case because f1 is a fixed point of T. In particular, Tf (x∗ ) ≤ f1 (x∗ ). Thus,
we have a contradiction, so the fixed point of T must be unique.
Next, we consider the case in which ρ = 1. In this case,
h
i
−1
κ(Xt ,Xt+1 ,Yt+1 )
Tf (x) = exp β ln u
Et u(e
f (Xt+1 ))|Xt = x + $(x) ,
x ∈ X.
Because X is finite, there exists > 0 such that
β
≤ min u−1 Et u(eκ(Xt ,Xt+1 ,Yt+1 ) )|Xt = x 1−β .
x∈X
(A.9)
It is straightforward to verify that for such > 0, T ≥ > 0. Because T is increasing,
{Tn }n≥0 is an increasing sequence. On the other hand, because β < 1 and X is finite,
there exists N > such that TN ≤ N . Consequently, Tn ≤ Tn N ≤ N, n ≥ 0, so the limit
of {Tn }n≥0 exists and is a fixed point of T in X++ . Using the same proof as in the case in
which ρ 6= 1 and α < 1, we can show that the fixed point of T in X++ is unique.
Part Two: computation of the fixed point.
In the second part of the proof, we show that {Tn f }n≥0 converges to the fixed point of
T for any f ∈ X++ . Denote the fixed point as f ∗ .
Because f ∈ X++ , f ∗ ∈ X++ , and X is finite, there exists r ≥ 1 such that f ≤ rf ∗ and
f ≥ (1/r)f ∗ . Then,
T(rf ∗ ) ≤ rT(f ∗ ) = rf ∗ ,
where the inequality is the case because β ≤ 1 and $ ≥ 0 and the equality is the case
because f ∗ is the fixed point of T. Consequently, {Tn (rf ∗ )}n≥0 is a decreasing sequence.
Similarly, {Tn ((1/r)f ∗ )}n≥0 is an increasing sequence. Moreover, Tn ((1/r)f ∗ ) ≤ Tn f ≤
Tn (rf ∗ ) because T is increasing. As a result, both {Tn (rf ∗ )}n≥0 and Tn ((1/r)f ∗ ) converge
in X++ and the convergent points are fixed points of T. Because the fixed point of T is
unique, both {Tn (rf ∗ )}n≥0 and Tn ((1/r)f ∗ ) converge to this fixed point, i.e,. to f ∗ . By
the squeeze theorem, {Tn f }n≥0 also converges to f ∗ . 31
Proof of Theorem 2.
Define operator T+ on X+ by
T+ f (x) := H 1, u−1 Et u eκ(Xt ,Xt+1 ,Yt+1 ) f (Xt+1 ) |Xt = x + $+ (x) ,
x ∈ X.
It is obvious that Tf ≤ T+ f for any f . According to Assumption 3, sequence {Tn f0 }n≥0
is increasing. Consequently, T+ f0 ≥ Tf0 ≥ f0 and thus {Tn+ f0 }n≥0 is also an increasing
sequence and dominates {Tn f0 }n≥0 . By Assumption 3, Tm f0 (x) > f0 (x) ≥ 0, x ∈ X
for some m ≥ 0. As a result, Tn+ f0 ∈ X++ for sufficiently large n and thus {Tn+ f0 }n≥0
converges to the fixed point of T+ in X++ according to Theorem 1. Consequently, the limit
of {Tn f0 }n≥0 exists, is a fixed point of T, and is strictly larger than f0 point-wisely.
Next, we show the uniqueness of the fixed point of T. We first note that for any fixed
point f ∗ of T, we have f ∗ = Tf ∗ ≥ f0 . Because Tm f0 > f0 for some m ≥ 0, f ∗ must be
strictly larger than f0 point-wisely. Now, for the sake of contradiction, suppose we have two
distinctive fixed points f1 and f2 . We already showed that fi (x) > f0 (x), x ∈ X, i = 1, 2.
Without loss of generality, we assume f1 (x) < f2 (x) for some x ∈ X. Define
x∗ := argminx∈X
f1 (x) − f0 (x)
f2 (x) − f0 (x)
and denote the corresponding minimum value as r∗ . Because X is finite, x∗ must exist
and r∗ ∈ (0, 1). Define f (x) := r∗ f2 (x) + (1 − r∗ )f0 (x), x ∈ X. Then, one can verify that
f0 (x) < f (x) ≤ f1 (x), x ∈ X and f (x∗ ) = f1 (x∗ ). Because Tf0 is well defined, so is Tf .
Recall that Tm f0 (x) > f0 (x), x ∈ X for some m ≥ 1. Because T is increasing and concave,
so is Tm . Denote I as the identity mapping. Then, for any x ∈ X,
(Tm − I)f (x) ≥ r∗ (Tm − I)f2 (x) + (1 − r∗ )(Tm − I)f0 (x) = (1 − r∗ )(Tm − I)f0 (x) > 0,
where the first inequality is the case due to the concavity of Tm and the equality is the
case because f2 is a fixed point of T. Thus, Tm f (x∗ ) > f (x∗ ) = f1 (x∗ ). On the other hand,
Tm f (x) ≤ Tm f1 (x) = f1 (x), x ∈ X because T is increasing and f1 is a fixed point of T. In
particular, Tm f (x∗ ) ≤ f1 (x∗ ), which is a contradiction.
Finally, we show that for any f such that Tf is well defined, {Tn f }n≥0 converges to the
fixed point of T. We first note that Tf ≥ f0 and Tf0 is well defined according to Assumption
3. As a result, Tn f is well defined for any n ≥ 0. Recall that Tm f0 (x) > f0 (x), x ∈ X for
some m ≥ 1, so Tm+1 f (x) ≥ Tm f0 (x) > f0 (x), x ∈ X. Thus, in the following, we assume
f (x) > f0 (x), x ∈ X without loss of generality.
32
Denote f ∗ as the unique fixed point of T, and we already showed that f ∗ (x) > f0 (x), x ∈
X. Because X is finite, there must exist r ∈ (0, 1] such that f − f0 ≤ (f ∗ − f0 )/r, i.e.,
rf + (1 − r)f0 ≤ f ∗ . Then,
f ∗ = Tf ∗ ≥ T(rf + (1 − r)f0 ) ≥ rTf + (1 − r)Tf0 ,
where the equality is the case because f ∗ is the fixed point, the first inequality is the
case because T is increasing, and the second inequality is the case because T is concave.
Applying T on both sides of the above inequality, we conclude that f ∗ ≥ rTn f +(1−r)Tn f0
for any n ≥ 0, which implies
Tn f ≤ [f ∗ − (1 − r)Tn f0 ] /r.
On the other hand, we have Tn f ≥ Tn f0 . Because {Tn f0 }n≥0 converges to f ∗ , the squeeze
theorem shows that {Tn f }n≥0 converges to f ∗ as well. Proof of Proposition 2. One can observe from (4.9) that Φ ≥ Vc,θ Φ for any (c, θ) ∈ A.
Consequently, {Vnc,θ Φ}n≥0 is a decreasing sequence and so is {Tnc,θ (Φ/c)}. By Theorem 1
its limit is the fixed point of Vc,θ , i.e., is Fc,θ . Thus, Φ(x) ≥ Fc,θ (x), x ∈ X.
If there exists (c∗ , θ∗ ) ∈ A such that (c∗ (x), θ∗ (x)) is a maximizer of (4.10) for each
x ∈ X, then Φ = Vc∗ ,θ∗ Φ. By the uniqueness of the fixed point of Vc∗ ,θ∗ , we conclude that
Φ = Fc∗ ,θ∗ . As a result, (c∗ , θ∗ ) and Φ are a maximizer and the optimal value, respectively,
of (4.8). Proof of Proposition 3. It is straightforward to see from (4.10) that W is continuous and
increasing, and
WΦ(x) = max H (c(x), (1 − c(x))DΦ (x, θ(x))) = max c(x)Tc,θ (Φ/c)(x),
(c,θ)∈A
(c,θ)∈A
x ∈ X.
(A.10)
Moreover, because Ix and Jx are compact for each x ∈ X, we conclude from (4.10) and
(A.10) that for each Φ such that WΦ is well defined, there exists (c, θ) ∈ A such that
WΦ = cTc,θ (Φ/c), i.e., the maximum in (A.10) is attained by (c, θ).
In the following, we divided the proof the theorem into three parts.
Part One: existence of the fixed point
33
We prove the existence of the fixed point of W in this part. We first consider the case
in which there exists (c, θ) ∈ A such that $c,θ (x) < 0 for some x ∈ X.
+
Recall f0,c,θ as defined in Assumption 3, i.e., f0,c,θ = H(1, $c,θ
) for each (c, θ). Define
Φ0 (x) := max c(x)f0,c,θ (x) = max H
(c,θ)∈A
c̄∈Ix
c̄, (1 − c̄) max
θ̄∈Jx
n
X
!
+
θ̄i bi gi (x)
,
x ∈ X.
i=1
By Assumption 4, for each (c, θ) ∈ A, Tc,θ f0,c,θ is well defined. Because Φ0 ≥ cf0,c,θ ,
i.e., Φ0 /c ≥ f0,c,θ , for each (c, θ) ∈ A, we conclude that Tc,θ (Φ0 /c) is well defined for any
(c, θ) ∈ A, so WΦ0 is well defined. Moreover, because Tc,θ (f ) ≥ f0,c,θ for any (c, θ) ∈ A and
f such that Tc,θ (f ) is well defined, we conclude that Tc,θ (Φ0 /c) ≥ f0,c,θ for any (c, θ) ∈ A.
As a result, WΦ0 ≥ Φ0 . Now, define Φn := Wn Φ0 , n ≥ 1. Becuase W is increasing,
{Φn }n≥0 is an increasing sequence.
Because $c0 ,θ0 (x) < 0 for some x ∈ X and some (c0 , θ0 ) ∈ A, Assumption 4 yields that
m
Tc0 ,θ0 fc0 ,θ0 > fc0 ,θ0 for some m ≥ 1. On the other hand,
WΦ0 ≥ c0 Tc0 ,θ0 (Φ0 /c0 ) ≥ c0 Tc0 ,θ0 (fc0 ,θ0 ),
which implies Φ1 /c0 = (WΦ0 )/c0 ≥ Tc0 ,θ0 (fc0 ,θ0 ). Repeating the the above calculation,
we conclude that Φm /c0 ≥ Tm
c0 ,θ0 (fc0 ,θ0 ) > fc0 ,θ0 . As a result, Φm > c0 fc0 ,θ0 , showing that
Φm ∈ X++ . Therefore, in the following we assume the sequence {Φn }n≥0 is in X++ without
loss of generality.
Now, we show that {Φn }n≥0 is bounded from above and thus its limit exists. Note that
for each n ≥ 1, there exists (cn , θn ) ∈ A such that
Φn ≤ Φn+1 = WΦn = cn Tcn ,θn (Φn /cn ),
which implies Φn /cn ≤ Tcn ,θn (Φn /cn ). Because Φn ∈ X++ , according to Theorems 1 and
2, {Tm
cn ,θn (Φn /cn )}m≥0 converges to fcn ,θn , the fixed point of Tcn ,θn , as m goes to infinity. Consequently, Φn /cn ≤ fcn ,θn , i.e., Φn ≤ cn fcn ,θn . Thus, we only need to show that
{cn fcn ,θn }n≥1 is bounded for any sequence (cn , θn ) ∈ A. Because cn ≤ 1, we only need to
show that {fcn ,θn }n≥1 is bounded. Moreover, when ρ > 1, we have H(1, z) ≤ (1 − β)1/(1−ρ) ,
so fcn ,θn = Tcn ,θn fcn ,θn ≤ (1 − β)1/(1−ρ) . Thus, in the following, we only need to consider
the case in which ρ ≤ 1.
We prove the boundedness of {fcn ,θn }n≥1 for the case in which ρ < 1 and γ 6= 1 as other
34
cases can be dealt with similarly. When γ 6= 1, recall the Perron-Frobenius eigenvalue
ηcn ,θn and eigenvector vcn ,θn as defined in Proposition 1 corresponding to (cn , θn ). Here and
hereafter, we always choose the Perron-Frobenius eigenvector such that its L1 norm is one.
Recalling that fcn ,θn is the fixed point of Tcn ,θn , we conclude from (A.5) that
1
− 1−γ
fcn ,θn (x)vcn ,θn (x)
1
1
= H vcn ,θn (x)− 1−γ , $cn ,θn (x)vcn ,θn (x)− 1−γ
i
+ δcn ,θn u
Ẽt u fcn ,θn (Xt+1 )vcn ,θn (Xt+1 )
|Xt = x
1
1
1
− 1−γ
− 1−γ
− 1−γ
+
≤ H kvcn ,θn k∞ , k$cn ,θn vcn ,θn k∞ + δcn ,θn kfcn ,θn vcn ,θn k∞ ,
−1
h
1
− 1−γ
where k · k∞ stands for the L∞ norms of functions on X. Consequently,
− 1
kfcn ,θn vcn1−γ
,θn k∞
1
1
1
− 1−γ
− 1−γ
− 1−γ
+
≤ H kvcn ,θn k∞ , k$cn ,θn vcn ,θn k∞ + δcn ,θn kfcn ,θn vcn ,θn k∞ .
With ρ < 1, we have (y1 + y2 )1−ρ ≤ y11−ρ + y21−ρ for any y1 , y2 ≥ 0, so
− 1
1−ρ
kfcn ,θn vcn1−γ
,θn k∞
≤ (1 −
− 1
1−ρ
β)kvcn1−γ
,θn k∞
−
+β
− 1
k$c+n ,θn vcn1−γ
,θn k∞
1
−
+
− 1
δcn ,θn kfcn ,θn vcn1−γ
,θn k∞
1
−
1−ρ
1
1−ρ
1−γ 1−ρ
1−γ 1−ρ
+
1−ρ
≤ (1 − β)kvcn1−γ
,θn k∞ + βk$cn ,θn vcn ,θn k∞ + βδcn ,θn kfcn ,θn vcn ,θn k∞ .
As a result,
−
− 1
1−ρ
kfcn ,θn vcn1−γ
,θn k∞
≤
1
−
1
1−γ 1−ρ
+
1−ρ
(1 − β)kvcn1−γ
,θn k∞ + βk$cn ,θn vcn ,θn k∞
1 − βδc1−ρ
n ,θn
.
(A.11)
Note that the Perron-Frobenius eigenvalue and eigenvector of an irreducible matrix A is
continuous in A.7 As a result, δc,θ and vc,θ are continuous in (c, θ). Because Ix and Jx are
7
To prove the continuity, we only need to show that for any sequence of irreducible matrices An with
Perron-Frobenius eigenvalue rn and eigenvector pn (normalized under L1 norm) such that it converges to
an irreducible matrix A, there exists a subsequence, indexed by nk , such that {rnk } and {pnk } converge to
the Perron-Frobenius eigenvalue and eigenvector of A, respectively. Because An converges to A and A is an
irreducible matrix, by the min-max and max-min version of the Collatz-Wielandt formula, we conclude that
rn ’s are uniformly bounded from above and uniformly bounded away from zero. On the other hand, pn ’s
are uniformly bounded because they are normalized. As a result, there exists a subsequence, indexed by nk ,
such that {rnk } and {pnk } converge. Denote the convergent points as r and p, respectively. Then, r > 0,
p ≥ 0, and p is normalized, i.e., the L1 norm of p is one. Moreover, because Ank pnk = rnk pnk , we conclude
Ap = rp, i.e., p is an eigenvector of A with eigenvalue r. Because there is no nonnegative eigenvector for
35
1−ρ
compact for each x ∈ X, A is also compact. Because βδc,θ
< 1 and vc,θ ∈ X++ for each
1−ρ
(c, θ) ∈ A, we conclude that supn βδcn ,θn < 1 and vcn ,θn ’s are uniformly bounded from above
and bounded away from zero. Moreover, because Ix and Jx are compact, $n+ ’s are also
uniformly bounded. Consequently, because ρ < 1, we conclude from (A.11) that fcn ,θn ’s
are uniformly bounded.
We have proved that the limit of {Φn } exists and must be in X++ . Then, by the
continuity of W, the limit must be a fixed point of W in X++ .
Next, we consider the case in which $c,θ ≥ 0 for any (c, θ) ∈ A. In this case, for
any (c0 , θ0 ) ∈ A, WFc0 ,θ0 is well defined. Moreover, we have WFc0 ,θ0 ≥ c0 Tc0 ,θ0 (Fc0 ,θ0 /c0 ) =
Fc0 ,θ0 , so {Wn Fc0 ,θ0 }n≥0 is an increasing sequence. On the one hand, the sequence {Wn Fc0 ,θ0 }n≥0
is in X++ because Fc0 ,θ0 ∈ X++ . On the other hand, following the same proof as in the
previous case, we can show that this sequence is bounded from above. As a result, this
sequence converges and the convergent point is a fixed point of W in X++ .
Part Two: uniqueness of the fixed point
Because Ix and Jx are compact for each x ∈ X, according to Proposition 2, any fixed
point of W in X++ is the optimal value of (4.8), so the fixed point of W in X++ is unique.
Part Three: computing the fixed point
Denote the unique fixed point of W as Φ∗ . We first consider the case in which $c,θ ≥ 0
for any (c, θ) ∈ A. Note that for any Φ ∈ X++ , because X is finite, there exists r > 1 such
that (1/r)Φ∗ (x) ≤ Φ(x) ≤ rΦ∗ (x), x ∈ X. Because $c,θ ≥ 0 for all (c, θ) ∈ A and β < 1,
W(rΦ∗ )(x) = max c(x)Tc,θ (rΦ∗ /c)(x) ≤ max c(x)rTc,θ (Φ∗ /c)(x) = rWΦ∗ (x) = rΦ∗ (x).
(c,θ)∈A
(c,θ)∈A
Therefore, {Wn (rΦ∗ )}n≥0 is a decreasing sequence. Similarly, {Wn ((1/r)Φ∗ )}n≥0 is an
increasing sequence. Moreover, Wn ((1/r)Φ∗ ) ≤ Wn Φ ≤ Wn (rΦ∗ ) because W is increasing.
As a result, both {Wn (rΦ∗ )}n≥0 and {Wn ((1/r)Φ∗ )}n≥0 converge in X++ and the convergent
points are fixed points of W in X++ . Because the fixed point of W in X++ is unique, both
{Wn (rΦ∗ )}n≥0 and {Wn ((1/r)Φ∗ )}n≥0 converge to this fixed point, i.e., to Φ∗ . By the
squeeze theorem, {Wn Φ}n≥0 converges to Φ∗ as well.
Next, we consider the case in which $c,θ (x) < 0 for some x ∈ X and some (c, θ) ∈ A.
A except for positive multiples of the Perron-Frobenius eigenvector regardless of the eigenvalue (Meyer,
2000, p. 673), we immediately conclude that r and p are the Perron-Frobenius eigenvalue and eigenvector
of A, respectively.
36
In this case, for any Φ such that WΦ is well defined, we have
WΦ = max c(x)Tc,θ (Φ/c) ≥ max c(x)f0,c,θ = Φ0 .
(c,θ)∈A
(c,θ)∈A
As a result, Wn Φ ≥ Wn−1 Φ0 , n ≥ 1. On the other hand, consider the following operator:
W+ Φ := max c(x)T+,c,θ (Φ/c),
(c,θ)∈A
+
where T+,c,θ is defined by replacing $c,θ in Tc,θ with $c,θ
. We already showed that W+ Φ
has a unique fixed point in X++ , and denote this fixed point as Φ∗+ . Because X is finite,
there exists r ≥ 1 such that Φ ≤ rΦ∗+ . Then,
W(Φ) ≤ W(rΦ∗+ ) ≤ W+ (rΦ∗+ ) ≤ rW+ Φ∗+ = rΦ∗+ ,
where the first inequality is the case because W is increasing, the second inequality is the
+
case because W+ dominates W, the third inequality is the case because β < 1 and $c,θ
≥0
for all (c, θ) ∈ A, and the equality is the case because Φ∗+ is the fixed point of W+ . As
a result, {Wn (rΦ∗+ )}n≥1 is a decreasing sequence and dominates {Wn (Φ)}n≥1 and thus
dominates {Wn−1 (Φ0 )}n≥1 . We already showed that {Wn−1 (Φ0 )}n≥1 converges to Φ∗ , so
{Wn (rΦ∗+ )}n≥1 must converge in X++ and the convergent point is a fixed point of W in
X++ . Because the fixed point of W in X++ is unique, we conclude that {Wn (rΦ∗+ )}n≥1
converges to Φ∗ as well. By the squeeze theorem, we conclude that {Wn Φ}n≥0 converges
to Φ∗ . B
When the State Space is Not Finite
In this section, we study the existence and uniqueness of the solution to (3.1), i.e., the fixed
point of T, when the state space of {Xt } is not finite. We consider only the case in which $
is nonnegative for two reasons. First, by imposing a similar condition to Assumption 3, we
can prove that the fixed point of T exists, but we do not have uniqueness, so we chose not
to present the results here. Second, our new model of narrow framing proposed in Section
5 is a special case of (3.1) with $ ≡ 0.
Proposition 4 Suppose Assumption 1 holds, ρ 6= 1, and $(x) ≥ 0, x ∈ X. Suppose the
results in Proposition 1-(i) and -(ii) hold.
37
(i) When γ 6= 1, f is a fixed point of T in X++ if and only if g := f 1−ρ v −1/α is a fixed
point of
1
Sg(x) := (1 − β)v(x)− α
h
1−ρ
1
i α(1−ρ)
1−ρ
α
−1
−1/(1−γ)
,
+ βδ
Ẽt (g(Xt+1 ) |Xt = x)
+ δ $(x)v(x)
x∈X
in the same space, where α := (1 − γ)/(1 − ρ) and Ẽ is the expectation operator
corresponding to probability measure P̃ that is obtained by a change of measure using
the Radon-Nikodym density Mt+1 := η −1 e(1−γ)κ(Xt ,Xt+1 ,Yt+1 ) v(Xt+1 )/v(Xt ). Assume
βδ 1−ρ < 1 and the stationary distribution of {Xt } under P̃ exists. For α ≥ 1, denote
space X+ when equipped with the Lα norm under the stationary distribution of {Xt }
under P̃ as L̃α+ (X). Then, if α ≥ 1 and S0 ∈ L̃α+ (X), S is a contraction mapping
0
in L̃α+ (X) and its unique fixed point is positive. If α < 1 and S0 ∈ L̃α+ (X) for some
0
α0 ≥ 1, then the limit of {Sn 0} exists, belongs to L̃α+ (X), and is the minimum fixed
point of S in X++ .
(ii) When γ = 1, f is a fixed point of T in X++ if and only if g := f 1−ρ e−(1−ρ)v is a fixed
point of
Sg(x) := (1 − β)e−(1−ρ)v(x)
n
o1−ρ
1
1−ρ
Et (ln g(Xt+1 )|Xt =x) 1−ρ
−1
−v(x)
+ βδ
e
+ δ $(x)e
,
x∈X
in the same space. Assume βδ 1−ρ < 1 and the stationary distribution of {Xt } exists.
0
0
Suppose for some α0 ≥ 1, S0 ∈ Lα+ (X), the space X equipped with the Lα norm
under the stationary distribution of {Xt }. Then, the limit of {Sn 0} exists, belongs to
0
Lα+ (X), and is the minimum fixed point of S in X++ .
Proof of Proposition 4. We first prove (i). Following the proof of Theorem 1, we can
show that f is a fixed point of T in X++ if and only if g := f 1−ρ v −1/α is a fixed point of
S in X++ . In addition, when α ≥ 1, inequality (A.7) holds for any g1 , g2 ∈ L̃α+ (X). As a
result, because S0 ∈ L̃α+ (X), this inequality implies that Sg ∈ L̃α+ (X) for any g ∈ L̃α+ (X).
Moreover, because βδ 1−ρ < 1, S is a contraction mapping on L̃α+ (X), so it admits a unique
fixed point. In particular, {Sn 0}n≥0 is an increasing sequence converging to the fixed point.
1
Because S0(x) ≥ (1 − β)v(x)− α > 0, x ∈ X, we conclude that the fixed point must be
38
positive.
When α < 1, define
1
−α
S̃g(x) := (1 − β)v(x)
+ βδ
1−ρ
1−ρ
h
i 0 1
1
α (1−ρ)
− 1−γ
α0
−1
Ẽt g(Xt+1 ) |Xt = x
+ δ $(x)v(x)
.
0
Because S̃0 = S0 ∈ L̃α+ (X) and βδ 1−ρ < 1, we can show that S̃ is a contraction mapping on
0
Lα+ (X). In particular, {S̃n 0}n≥0 is an increasing sequence and converges to the unique fixed
point of S̃. Noting that {Sn 0}n≥0 is an increasing sequence and dominated by {S̃n 0}n≥0
0
because α < 1, we conclude that the limit of {Sn 0}n≥0 exists and belongs to Lα+ (X).
Moreover, by the monotone convergence theorem and the observation that S0(x) > 0, x ∈
X, the limit, denoted as S∞ 0, must be a fixed point of S in X++ . Finally, for any other
fixed point g of S, we have g ≥ 0, so g = Sn g ≥ Sn 0 for any n ≥ 0. As a result, g ≥ S∞ 0.
The case in which γ = 1 can be treated in the same way as for the case in which γ 6= 1
and α < 1. Proposition 4 is completely parallel to Hansen and Scheinkman (2012, Proposition 6):
when ρ 6= 1 and $ is nonnegative, if (i) η, v, and δ in Proposition 1 are well defined
and βδ 1−ρ < 1 and (ii) the stationary distribution of {Xt } exists after a specific change of
measure, then the fixed point of T exists. Moreover, if (1 − γ)/(1 − ρ) ≥ 1, then the fixed
point is unique and {Tn f }n≥0 converges to the fixed point for any positive f . Note that just
as in Hansen and Scheinkman (2012), we do not have uniqueness when (1 − γ)/(1 − ρ) < 1.
Proposition 5 Suppose Assumption 1 holds and the stationary distribution of {Xt } exists.
Suppose ρ = γ = 1 and $(x) ≥ 0, x ∈ X. Then, f is a fixed point of T in X++ if and only
if g := ln f is a fixed point of S in X , where
Sg(x) :=β ln eEt (κ(Xt ,Xt+1 ,Yt+1 )|Xt =x) eEt (g(Xt+1 )|Xt =x) + $(x) ,
x ∈ X.
(B.1)
Denote L1 (X) as X equipped with the L1 norm under the stationary distribution of {Xt }
and assume there exists g ∈ L1 (X) such that Sg ∈ L1 (X). Then, S is a contraction mapping
on L1 (X) and thus the fixed point of S uniquely exists in L1 (X).
Proof of Proposition 5 Consider function ϕ(z) := ln(aez + b) for some a > 0 and b ≥ 0.
Straightforward computation yields ϕ0 (z) = aez /(aez + b) ≤ 1, so |ϕ(z1 ) − ϕ(z2 )| ≤ 1.
Consequently, recalling the definition of S, we conclude that for any functions g1 and g2 on
39
X,
|Sg1 (x) − Sg2 (x)| ≤ βEt [|g1 (Xt+1 ) − g2 (Xt+1 )| | Xt = x] ,
x ∈ X.
Replacing x with Xt in the above inequality and take expectation on both sides under the
stationary distribution of {Xt }, we immediately conclude that
E [|Sg1 (Xt ) − Sg2 (Xt )|] ≤ βE [|g1 (Xt+1 ) − g2 (Xt+1 )|] = βE [|g1 (Xt ) − g2 (Xt )|] .
This inequality implies (i) S is a mapping from L1 (X) into L1 (X) because Sg ∈ L1 (X) for
some g ∈ L1 (X); and (ii) S is a contraction mapping. Proposition 5 shows the existence and uniqueness of the fixed point of T when ρ = γ = 1
provided that {Xt } has a stationary distribution and $ is nonnegative. Moreover, because
S as defined in (B.1) is a contraction mapping, {Sn g} converges to the fixed point of S for
any g and as a result {Tn f } converges to the fixed point of T for any positive f .
Finally, we show that the solution to (3.1) uniquely exists when γ = ρ = 1 even in a
non-Markovian setting.
Proposition 6 Suppose ρ = γ = 1, At > 0, and Bt ≥ 0. Then, {Vt } is a positive solution
to (3.1) if and only if {ln(Vt /ct )} is a fixed point of
(SZ)t := β ln eEt (ln ct+1 −ln ct +ln At+1 ) eEt (Zt+1 ) + (Bt /ct ) ,
t = 0, 1, . . .
(B.2)
Moreover, if there exist α ∈ (β, 1) and {Zt } ∈ L1,α , the space of {Ft }-adapted processes
P
t
1,α
, then S is a contraction mapping
with norm ||Z|| := ∞
t=0 α E(|Zt |), such that SZ ∈ L
on L1,α and thus the fixed point of S on this space uniquely exists.
Proof of Proposition 6 Using the same proof as of Proposition 5, we can show that for any
two adapted processes {Zi,t }, i = 1, 2,
|(SZ1 )t − (SZ2 )t | ≤ βEt (|Z1,t+1 − Z2,t+1 |) = (β/α) · αEt (|Z1,t+1 − Z2,t+1 |) ,
In consequence,
E[|(SZ1 )t − (SZ2 )t |] ≤ (β/α) · αE [|Z1,t+1 − Z2,t+1 |] ,
40
t = 0, 1, . . .
which yields ||SZ1 − SZ2 || ≤ (β/α)||Z1 − Z2 ||. Because SZ ∈ L1,α for some Z ∈ L1,α and
L1,α is a complete normed space, S is a contraction mapping and thus admits a unique
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