Journal of Economic Theory 145 (2010) 1776–1804
www.elsevier.com/locate/jet
Unique solutions for stochastic recursive utilities ✩
Massimo Marinacci a,∗ , Luigi Montrucchio b
a Department of Decision Sciences, Dondena, and Igier, Università Bocconi, Italy
b Department of Statistics and Applied Mathematics and Collegio Carlo Alberto, Università di Torino, Italy
Received 18 June 2007; final version received 27 September 2009; accepted 18 December 2009
Available online 18 February 2010
Abstract
We study unique and globally attracting solutions of a general nonlinear stochastic equation, widely used
in Finance and Macroeconomics and closely related to stochastic Koopmans equations. The equation is
specified by a temporal aggregator W and a certainty equivalent operator M. The main contribution of the
paper is the introduction of the new class of Thompson aggregators. Other contributions of the paper are:
(i) a detailed analysis of quasi-arithmetic operators M that generalize those of Kreps and Porteus (1978)
[18]; (ii) a clarification of the nature and properties of the stochastic recursive preferences that underlie
Koopmans equations.
© 2010 Elsevier Inc. All rights reserved.
JEL classification: D81; D91
Keywords: Recursive utilities; Koopmans equations; Stochastic recursive utilities
1. Introduction
Consider a standard intertemporal stochastic setting, where uncertainty is modeled by a probability space (Ω, Σ, P) and information by an increasing filtration {Σt }t defined on Ω.
In such settings, most economic applications are based on a ranking of some relevant stochastic payoff streams, like for example a consumption stream c = {ct }t , adapted to the given
✩
We thank Donato Cifarelli, Peter Klibanoff, Jianjun Miao, Sujoy Mukerji, as well as an anonymous referee and an
associate editor, for insightful discussions and suggestions. The financial support of the European Research Council
(advanced grant, BRSCDP-TEA) is gratefully acknowledged.
* Corresponding author.
E-mail address: [email protected] (M. Marinacci).
0022-0531/$ – see front matter © 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.jet.2010.02.005
M. Marinacci, L. Montrucchio / Journal of Economic Theory 145 (2010) 1776–1804
1777
filtration. The ranking is made through a stream of preference functionals V = {Vt (c)}t , also
adapted to the filtration, and their basic form is
Vt (c) = Et
β τ −t u(ct ) ,
(1)
τ t
where Et is the expectation conditional on Σt , β ∈ (0, 1) is a discount factor, and u is an instantaneous utility function, constant over time.
This basic representation is analytically very tractable and has the following main behavioral
features: it is dynamically consistent and independent of both unrealized alternatives and past
consumption levels. It has, however, an important drawback: it is unable to disentangle risk
attitudes from the intertemporal elasticity of substitution.
In order to overcome this difficulty, starting from Kreps and Porteus [18] and Epstein and
Zin [11], the more general preference functional
Vt (c) = W ct , Mt Vt+1 (c)
(2)
has been widely used in Finance and Macroeconomics, where W : R2 → R is a (temporal) aggregator and Mt is a time t certainty equivalent of future utility Vt+1 . Clearly, (2) generalizes
the additive case (1), which is the special case:
(3)
Vt (c) = u(ct ) + βEt Vt+1 (c) .
Standard specifications of (2) are, for example, the CES aggregator W (x, y) = (x ρ + βy ρ )1/ρ
and the quasi-arithmetic certainty equivalent operator
(4)
Mt (Vt+1 ) = φ −1 Et (φ ◦ Vt+1 ) ,
where φ is a strictly increasing function.
The recursive preference functional (2) allows a separation between risk attitudes and the degree of intertemporal substitution. It is still analytically tractable and retains the main behavioral
features of the basic preference functional (1), that is, dynamic consistency and independence of
both unrealized alternatives and past consumption levels.
Because of its tractability and theoretical soundness, the recursive preference functional (2)
has become very popular in applications, especially in Finance and Macroeconomics. These
works typically specify an aggregator W and a certainty equivalent M, and the stochastic payoff
streams c = {ct }t are then evaluated by using the stochastic process V that solves the recursive
equation (2).
For this reason a key feature of the stochastic equation (2) is the existence of a solution V for
a given specification of W , M, and c. Even more important, a further key property is that the
solution be unique and globally attractive. In fact, uniqueness is crucial both to properly interpret
the solution as an evaluation of the stream c, and to carry out comparative statics exercises that
describe how the solution/evaluation varies as M, W and c vary.1 Global attractivity is, on the
other hand, a very useful property that allows to find the solution iteratively by starting from any
possible initial point.
1 Uniqueness is also important because it allows to specify a well defined utility function over consumption streams.
This delicate point will be discussed in Section 7.
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Together, uniqueness and global attractivity make the solution economically meaningful and
computable. For this reason in this paper we investigate in depth the uniqueness and global
attractivity of the solution of the recursive equation (2). In particular, our aim is to find conditions
directly on W and M, so that a direct check of the properties of W and M is enough to determine
whether there is a unique and globally attracting solution. This direct verifiability is important in
applications. For example, for the specification (4) this means to have conditions directly on the
function φ.
Despite the importance of (2) in applications, surprisingly little is known about the existence
of unique and globally attracting solutions of these recursive stochastic equations. Our results
therefore fill a significant gap in the literature that uses them, as we will discuss in Section 1.1.
In particular, in our analysis we consider two types of aggregators W , which we call Thompson and Blackwell aggregators. Thompson aggregators are based on condition (W-iii) below, that
is,
W (x, αy) αW (x, y) + (1 − α)W (x, 0),
∀α ∈ [0, 1], ∀x, y ∈ R+ ,
a simple “concavity” condition on W , while Blackwell aggregators use condition (W-v) below,
that is,
W (x, y) − W x, y β y − y , ∀x, y, y ∈ R+ ,
for some β ∈ (0, 1), a standard Lipschitzian condition often imposed on aggregators.2 As we
show in Section 3.1, for some common specifications of the aggregators the conditions that make
them either Thompson or Blackwell nicely complement each other, and together cover a wide
range of values of the parameters of these specifications. For example, the classic CES aggregator
W (x, y) = (x ρ + βy ρ )1/ρ , with β < 1, is Thompson for ρ 1 and Blackwell for ρ 1. The
Thompson and Blackwell conditions are thus fully complementary.
For the standard specification (4), the recursive equation (2) takes the form
Vt = W ct , φ −1 Et (φ ◦ Vt+1 )
(5)
and our Theorem 3 establishes the existence of a unique and globally attracting solution to this
version of Eq. (2) provided either W is Thompson and φ exhibits increasing relative risk aversion
(IRRA) or W is Blackwell and φ exhibits increasing absolute risk aversion (IARA). These well
known conditions on φ include some of the most popular specifications of φ, such as CRRA,
CARA, and the quadratic. In particular, for the CES aggregator Eq. (5) becomes
ρ
ρ 1/ρ
(6)
Vt = ct + β φ −1 Et (φ ◦ Vt+1 )
with β < 1. Theorem 3 shows that there is a unique and globally attracting solution to Eq. (6)
provided either ρ > 1 and φ is IARA or ρ 1 and φ is IRRA. The latter set of conditions include
the classic CRRA specification of φ, which gives the most popular version of (6), that is,
ρ
1−γ ρ 1/ρ
.
(7)
Vt = ct + β Et Vt+1 1−γ
These results are special cases of the general results established in Theorems 1 and 2. They
solve the general nonlinear stochastic equation (13) below, which includes (2) as a very special
case. Besides its generality, the advantage of studying the general equation (13) is that it makes it
2 See, e.g., Stokey and Lucas [29, condition (W3), p. 115].
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1779
possible to focus on the essential features that lead to uniqueness and global attractivity, without
being distracted by the peculiar features that some more special cases might have.
On a technical level, our main novel contribution is the introduction of Thompson aggregators.
For these concave aggregators the nonlinear equation (2) cannot be solved via standard contraction arguments à la Blackwell. Instead, we need to use different contraction techniques based on
Thompson [31]. The use of these techniques to study the recursive equation (2) is a secondary
contribution of our paper.
It should be emphasized that the Thompson contraction technique has nothing to do with
Boyd’s Weighted Contraction Theorem [4], which is an extension of the classic Blackwell Contraction Theorem to spaces of unbounded functions. In our analysis of unbounded aggregators
we will make use of γ -subhomogeneity, a property satisfied by most of the relevant aggregators
used in Economics.
The paper is organized as follows. After some preliminaries in Section 2, we present in
Section 3 the general nonlinear stochastic equation (13), which we solve in the bounded case
in Section 4 and in the unbounded case in Section 5. In that section we provide the main
theorem, Theorem 2, on existence and uniqueness of the solution to our functional equation.
It will be then used throughout the paper. Section 6 analyzes the specification of the model
for the class of quasi-arithmetic certainty equivalent operators. Section 7 studies Koopmans
equations in our setting. In particular, Section 7.1 shows that even in the deterministic case
Thompson aggregators are a novel class of aggregators. Section 7.2 studies the stochastic preferences generated by our equation. In Section 7.3 we study the existence of an “atemporal”
utility that generates the recursive utilities. The appendices contain all proofs and related material.
1.1. Related analysis
We now discuss some related works. In so doing, we discuss recursive preferences as solutions
to Eq. (2), something that in the literature is not always well clarified and that might give rise to
misinterpretations.
The structure of Koopmans [14] recursive utilities over deterministic consumption streams
(8)
V (c0 , c1 , . . .) = W c0 , V (c1 , c2 , . . .)
has been extensively investigated since the work of Lucas and Stokey [20], who initiated
the approach of taking as a primitive the aggregator W . They studied bounded aggregators,
while later Boyd [4] extended their analysis to unbounded aggregators. Since then, Koopmans equations and their solutions have been widely analyzed (see, e.g., Duran [8], Le
Van and Vailakis [19], Rincon-Zapatero and Rodriguez-Palmero [28], Martins-da-Rocha and
Vailakis [24], and the references therein contained). For instance, in [19] the recursion operator may be unbounded from below and above, and the recoverability of a utility function
does not rely on contraction arguments: it is derived, instead, as a pointwise limit of “partial
sums.”
The uncertainty case, fundamental for applications, is quite complicated and there are very
few results on it (see [11,21–23,27] and [9]). Moreover, different formalizations coexist and
their comparison is not always easy. Our approach is perhaps the simplest and our results on the
existence and uniqueness of fixed points of (2) are easy to formulate. Like Lucas and Stokey [20],
we take the time aggregator W and the stochastic aggregator M as primitives. But, this does not
mean that a Koopmans equation is a primitive in our theory. Though (2) could be interpreted
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as a stochastic Koopmans equation, it differs from (8) in that the solution to (2) is not a utility
function, but rather a sequence of values of utilities. It is actually unclear what is the stochastic
version of Koopmans equations (8), and the literature does not help much in clarifying this aspect.
Sections 7.2 and 7.3 try to shed some light on this point and show how to derive and study
Koopmans equations (8) through the formulation (2).
Turning to the few papers that deal with the uncertainty case, they rely on different formalizations of the stochastic environment that in turn lead to different specifications of recursive
utilities. Consider first Epstein and Zin [11]. Though the applied papers Epstein and Zin [12]
and Epstein and Melino [10] consider economic applications of Eq. (2), the theoretical paper
[11] uses a quite different framework where, roughly speaking, random variables are replaced
by their distributions. Specifically, they start with the Polish space D of infinite horizon temporal lotteries, a generalization of the finite horizon temporal lotteries of Kreps and Porteus [18].
A consumption plan can be modeled as a temporal lottery d ∈ D, viewed as a pair d = (c0 , m)
where c0 is current consumption and m belongs to M(D). Preference functionals V : D → R are
defined on D and, given a certainty equivalent operator μ : M(R+ ) → R+ , Epstein and Zin [11]
say that V is recursive provided
V (c0 , m) = W c0 , μ(Vm ) ,
(9)
where Vm denotes the image measure on R induced by the mapping V : D → R.
The advantage of [11]’s approach is a sharp formulation of a stochastic version of Koopmans
equation (9). Its drawback, however, is that the analysis of existence and uniqueness of solutions
to (9) is far more difficult than in our setting since the operator associated with (9) presents
several analytical difficulties. In fact, [11, Theorem 3.1] establishes few results for Eq. (9), and
existence of solutions is only proved for the CES aggregator W = (x ρ + βy ρ )1/ρ . Ma [23] studies
(9) and proves the existence of regular (upper or lower semicontinuous) recursive preferences
when the utility generator W exhibits suitable attitudes toward the resolution of uncertainty.
Though elegant, the existence and uniqueness results in [23] require nontrivial restrictions (e.g.,
the distribution of the consumption plans has to lie in a compact and convex subset of Rn+ ) and
are not analytically tractable.
A further different approach is proposed by Ozaki and Streufert [27], who provide a comprehensive study of the existence of optima through dynamic programming techniques. In place
of (3), they start from the additive intertemporal relation
(10)
Vt zt = u c zt + β Vt+1 zt , zt+1 π(dzt+1 | zt ),
where zt = (z0 , z1 , . . . , zt ) denotes time-t history of the exogenous states and π(dzt+1 | zt ) is a
time homogeneous Markov stochastic kernel. The recursive version of (10) that they study is
Vt zt = W c zt , Mzt Vt+1 zt , · .
This setting is particularly well suited for Dynamic Programming, where standard Markov operators are replaced by nonadditive operators f (z ) → Mz [f (z )]. The part of [27] that is more
closely related with our paper is their Section 3, devoted to nonnegative objectives. There they
establish sufficient conditions to recover utility functions from W and M. Due to the generality
of their setting, their conditions are sometimes difficult to check (see, for instance, some of their
conditions N.1–N.12). Moreover, our Thompson aggregators generally fail to be bi-convergent,
and so they are beyond the scope of [27]’s analysis. In Section 7.3 we will study Koopmans
equations in this topological framework, though with a narrower scope than theirs.
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2. Preliminaries
2.1. Uncertainty and information
We model uncertainty and information in a standard way.3 Specifically, uncertainty is modeled by a probability space (Ω, Σ, P), where Σ is an event σ -algebra of a state space Ω
and P : Σ → [0, 1] is a countably additive probability measure. Given a discrete time horizon
N = {0, . . . , t, . . .}, information is modeled through an increasing filtration {Σt }t0 of σ -algebras
contained in Σ.
A real-valued adapted stochastic process X = {Xt }t0 is a collection of Σt -measurable functions Xt : Ω → R. We denote by L the space of all adapted processes. Henceforth any process
will be adapted to the filtration {Σt }t0 , even if not explicitly mentioned.
A process X ∈ L can be regarded as a suitably measurable4 function X : Ω × N → R. As
such, we can consider on L the pointwise order , i.e., given any X , X ∈ L, we write X X if Xt Xt P-a.e. for all t 0. In particular, we write [X , X ] = {X ∈ L: X X X }. We
also set L+ = {X ∈ L: X 0}. A process will be said to be integrable if E(Xt ) < ∞ for all
t 0.
We denote by L∞ the set of all essentially bounded processes, that is, X ∈ L∞ if
X∞ ≡ ess
|Xt (ω)| < +∞.
sup
(ω,t)∈Ω×N
In particular, D denotes the subset of L∞ consisting of all (almost) deterministic processes; that
is, X ∈ D if there exists d : N → R such that Xt (ω) = d(t) for all t 0 and P-a.e. ω ∈ Ω.
Endowed with the essential supnorm · ∞ , L∞ becomes a Banach space (under the usual
identification of its P-a.e. equal elements).
Notation. (i) With a slight abuse of notation, k denotes the constant process Xt such that
Xt (ω) = k ∈ R for all t 0 and P -a.e. ω ∈ Ω. (ii) Et (Xt+1 ) denotes the conditional expectation EP (Xt+1 | Σt ), provided X is an integrable process; moreover, equalities and inequalities
between Σ-measurable functions are understood to hold P -a.e. even where not stated explicitly.
(iii) Given X ∈ L+ , we set
[X]∞ = ess
inf
(ω,t)∈Ω×N
Xt (ω).
(11)
2.2. Weighted norms
In order to deal with unbounded processes, it is useful to consider weighted supnorms, a standard generalization of the supnorm (see, e.g., Wessels [32] and Boyd [4]). A weight function
is a deterministic process w ∈ L+ with w 1, and it induces an (essential) weighted supnorm
· w : L → [0, ∞] by
Xw ≡ ess
|Xt (ω)|
.
wt
(ω,t)∈Ω×N
sup
3 See, e.g., Stokey and Lucas [29] for canonical interpretations of this setting in terms of shocks/observations.
4 More details are given in Appendix A.
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Set L(w) = {X ∈ L: Xw < +∞}. The pair (L(w), · w ) is easily seen to be a Banach space.
Note that for any element X ∈ L(w), Xt is essentially bounded for every t. Consequently, all
X ∈ L(w) are integrable.
Since Xw = w −1 X∞ , the norms · w and · ∞ are equivalent, and so L(w) = L∞
when w is bounded. Weighted supnorms therefore become relevant when w is unbounded. There
is no loss of generality by assuming wt 1 for all t and w0 = 1, and in the sequel we always
maintain this condition. Like in the bounded case, we use the notation
[X]w = ess
Xt (ω)
.
(ω,t)∈Ω×N wt
inf
(12)
A deterministic function w : N → R is said to be an admissible weight function if there is a
positive constant aw 1 such that wt+1 aw wt for all t 0.
Example 1. Given a > 1, let wt = a t be the exponential weight, for all t 0. This is the standard
case considered, for instance, by Boyd [4]. Exponential weights a t are clearly admissible, with
a = aw .
2.3. Risk attitudes
In Decision Theory there are two classic indices associated to a given continuous function φ : R+ → R, twice differentiable on (0, ∞), that is, Aφ (t) = −φ (t)/φ (t) and Rφ (t) =
−tφ (t)/φ (t) for all t > 0. The index Aφ is the coefficient of absolute risk aversion and Rφ is
the coefficient of relative risk aversion. The interpretation of these classic indices is well known
(see, e.g., Kreps [17]).
We can classify the functions φ using these indices. A function φ : R+ → R is decreasing
absolute risk averse (DARA) if its index Aφ : (0, ∞) → R is nonincreasing, it is increasing
absolute risk averse (IARA) if its index Aφ is nondecreasing, and it is constant absolute risk
averse (CARA) if it is both DARA and IARA.
Decreasing (DRRA), increasing (IRRA), and constant (CRRA) relative risk averse functions
are similarly defined by replacing the index Aφ with Rφ . In applications the most important
classes of functions are the DARA and IRRA (see, e.g., [1, p. 96]), which in turn include the
CARA and CRRA.
3. A general nonlinear stochastic operator equation
The nonlinear stochastic equations used in Economic Dynamics that we discussed in the Introduction are special cases of the following general nonlinear stochastic operator equation defined
on the space of positive adapted processes L+ :
X = W Y, M(X) ,
(13)
that is, for all t 0,
Xt (ω) = W Yt (ω), Mt (X)(ω) ,
P-a.e.,
M. Marinacci, L. Montrucchio / Journal of Economic Theory 145 (2010) 1776–1804
1783
where Y ∈ L+ is a given stochastic process, M : L+ → L+ is an operator, and W : R2+ → R+ is
a function.5
For example, the stochastic equation (2) is a special case of (13), where Yt = ct , Xt = Vt , and
Mt (X) only depends on Xt+1 , that is,
Xt+1 = Xt+1
⇒ Mt (X) = Mt X , ∀X, X ∈ L+ .
Eq. (13) can be written in the form X = T (X), where T : L+ → L+ is the operator given by6
T (X) = W Y, M(X) ,
(14)
that is, for all t 0, Tt (X)(ω) = W (Yt (ω), Mt (X)(ω)) P-a.e.
A solution of Eq. (13) is thus a fixed point of the operator T , and so the resolution of this
stochastic equation can be reduced to finding the fixed points of the operator T .
3.1. Aggregator functions
We will solve the stochastic equation (13) for suitable M and W . Specifically, in the spirit
of Lucas and Stokey [20], we say that W : R2+ → R is an aggregator function if it satisfies the
following two properties:
(W-i) W is nonnegative and monotone, i.e., 0 W (x, y) W (x , y ) if x x and y y ,
(W-ii) equation W (x, y) = y has at least a nonnegative solution for each x 0.
Though standard, these conditions are not innocuous. For example, condition (W-i) will rule
out utilities that, like the logarithmic, are unbounded from below.
In some results we will need some of the following further properties:
(W-iii) W (x, ·) is concave at 0 for each x 0, i.e.,
W (x, αy) αW (x, y) + (1 − α)W (x, 0)
for each α ∈ [0, 1] and each x, y ∈ R+ ,
(W-iv) W (x, 0) > 0 for each x > 0,
(W-v) W is a contraction in y, i.e., there is some β ∈ (0, 1) such that
W (x, y) − W x, y β y − y , ∀x, y, y ∈ R+ .
(15)
By standard contraction arguments, any W : R2+ → R that obeys the Lipschitz condition (W-v)
has a unique fixed points in y for each x 0. Hence, (W-v) implies (W-ii).
In our main results the aggregators are required to satisfy either both (W-iii) and (W-iv) or
(W-v). In the former case, our results rely on a fixed point for suitably concave functions based
on Thompson [31], while in the latter case we use a standard contraction argument à la Blackwell [3], Wessels [32], and Boyd [4]. This motivates the following terminology.
5 We are using the notation M (X)(ω) ≡ (MX)(ω, t). Moreover, with a slight abuse of notation, we write
t
W (Y, M(X)) in place of W ◦ (Y, M(X)), where we regard each element of the pair (Y, M(X)) as a function from
Ω × T into R2 .
6 We implicitly assume that W is Borel measurable, something that will be guaranteed by the monotonicity assumption
(W-i). Moreover, the operator T should depend on Y . To ease notation we write T in place of more precise notation TY .
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Definition 1. An aggregator that satisfies both properties (W-iii) and (W-iv) is called a Thompson
aggregator, while an aggregator that satisfies (W-v) is called a Blackwell aggregator.
The last property we consider is a generalization of subhomogeneity:
(W-vi) W is γ -subhomogeneous, i.e., there is some γ > 0 such that:
W α γ x, αy αW (x, y),
for each α ∈ (0, 1] and each x, y ∈ R+ .
Standard subhomogeneity corresponds to γ = 1. Some aggregators enjoy the stronger property W (α γ x, αy) = αW (x, y) for every α 0, which may be called γ -homogeneity. For example, the “asymmetric” CES function W (x, y) = (x + βy ρ )1/ρ is clearly ρ-homogeneous.
Next we illustrate with few examples the conditions on W that we have just introduced.
Example 2. The family of functions
1/ρ
W (x, y) = x η + βy σ
,
(16)
with η, σ, ρ, β > 0, includes many interesting cases. Conditions (W-i) and (W-iv) always hold.
Condition (W-ii) holds if either σ < ρ or σ = ρ and β < 1.
The concavity condition (W-iii) holds iff σ 1 and σ ρ. It is easy to check that the contraction property (W-v) is satisfied iff σ = ρ 1 and β < 1. Summing up:
Thompson
Blackwell
if σ 1 and either σ < ρ or σ = ρ and β < 1
if β < 1 σ = ρ
Hence, Thompson and Blackwell cases share only the very special case σ = ρ = 1 > β. They
are otherwise different and nicely complement each other. Finally, as to condition (W-vi), W is
clearly γ -subhomogeneous, with γ = σ/η, provided σ ρ.
When the aggregator (16) is Thompson, it does not obey any Lipschitz condition. This is an
important observation since most of the existing results on recursive utilities in deterministic
settings rest on a Lipschitz assumption (see [4,19,28]). Hence, also in the deterministic case
Thompson aggregators are not covered by the literature, as further discussed in Section 7.
Example 3. Koopmans, Diamond, and Williamson [15, p. 97] study the aggregator
W (x, y) = θ −1 log 1 + ηx δ + βy
with θ, β, δ, η > 0. Conditions (W-i) and (W-ii) are true, and so W is an aggregator. Moreover,
W always satisfies (W-iii), (W-iv), while it satisfies (W-v) iff β < θ . Hence:
Thompson
Blackwell
always satisfied
if β < θ
Note that (W-vi) holds since W is δ −1 -subhomogeneous.
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3.2. Certainty equivalent operators
As to the operator M, we say that M : L+ → L+ is a certainty equivalent operator if it is
adapted (i.e., it maps adapted processes into adapted processes) and:
(M-i) M(k) = k for all k 0,
(M-ii) M is monotone, i.e., M(X ) M(X ) if X X .
Besides the basic properties (M-i) and (M-ii), other properties of the operator M will play a
key role. In particular, M : L+ → L+ is:
(M-iii)
(M-iv)
(M-v)
(M-vi)
constant subadditive if M(X + k) M(X) + k for all k 0 and all X ∈ L+ ,
subhomogeneous if M(αX) αM(X) for all α ∈ [0, 1] and all X ∈ L+ ,
a shift operator if M(d) = S(d) for all d ∈ D+ , where S : D → D is the shift operator,7
shift subadditive if M(X + d) M(X) + S(d) for all d ∈ D+ and all X ∈ L+ .
Since S(k) = k for all k 0, (M-v) implies (M-i) and (M-vi) implies (M-iii).
Example 4. Given a strictly increasing function φ : [0, ∞) → R, consider the operator M :
∞
L∞
+ → L+ given by:
Mt (X) = φ −1 Et (φ ◦ Xt+1 ) , P-a.e.,
for each t 0. This operator satisfies (M-i), (M-ii), and (M-v) and so it is a certainty equivalent
operator. By Proposition 7 of Appendix C, M satisfies (M-vi) whenever φ is IARA, and M
satisfies (M-iv) whenever φ is IRRA.
4. Bounded case
In this section we study the stochastic equation (13) when the process Y is bounded. In the
next section the analysis will be extended to the case in which Y may be unbounded.
We begin with a simple existence result, which shows that fixed points exist under very general
conditions.
Proposition 1. Consider the operator T given by (14), where W is an aggregator function and
∞
M is a certainty equivalent operator. If Y ∈ L∞
+ , then T has a fixed point in L+ .
This result is a simple consequence of the classic Tarski Fixed Point Theorem [30]. By this
theorem, the set of fixed points is a nonempty complete lattice. Therefore, there exist both a
M fixed point. As X
m 0 and T is monotone, by iterating T
m and a maximal X
minimal X
n
it follows that 0 T 0↑ Xm . A large literature studies the limit T n 0↑, which is viewed as
the natural candidate to be the fixed point of T . A thorough analysis along these lines can be
found for example in [19] for deterministic models. Assume, for instance, that Xn ↑X implies
m . A simlimn→∞ MXn = MX. Under this assumption, it easily follows that limn→∞ T n 0 = X
ilar argument applies to the unbounded case of the next section.
7 That is, S(d)(t) = d(t + 1) for all t 1 and all deterministic processes d ∈ D.
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Turn now to uniqueness, the main property which we are interested in. It is easy to check (see
Lemma 1 of Appendix D) that there exist unique y∗ , y ∗ ∈ R+ such that:
W [Y ]∞ , y∗ = y∗ and W Y ∞ , y ∗ = y ∗ ,
provided W is either Blackwell or Thompson (see (11) for the definition of [Y ]∞ ).
Theorem 1. Consider the operator T given by (14), where W is an aggregator function and M
∞
is a certainty equivalent operator. Given any Y ∈ L∞
+ , then T has a unique fixed point X in L+
provided at least one of the following conditions holds:
(i) M is constant subadditive and W is Blackwell;
(ii) M is subhomogeneous, W is Thompson, and [Y ]∞ > 0.
belongs to [y∗ , y ∗ ] and is globally attracting on L∞
Moreover, X
+ , that is,
n
T (X) − X
→ 0, ∀X ∈ L∞
+.
∞
Example 5. Given Y ∈ L∞
+ with [Y ]∞ > 0, consider the CES operator
ρ
ρ 1/ρ
Tt (X) = Yt + β Mt (X)
, P -a.e.,
where β, ρ ∈ (0, 1) and M is a subhomogeneous certainty equivalent operator. Simple algebra
shows that y ∗ = (1 − β)−1/ρ Y ∞ and y∗ = (1 − β)−1/ρ [Y ]∞ . By Theorem 1, there is a unique
of T , with X
∈ [(1 − β)−1/ρ [Y ]∞ , (1 − β)−1/ρ Y ∞ ].
and globally attracting fixed point X
Remark. The positivity condition (W-iv) may not be satisfied in some important cases (e.g., for
the Cobb–Douglas aggregator W (x, y) = x α y β , with α, β > 0). The failure of (W-iv) can generate multiple fixed points. For example, suppose that W (x, 0) = 0 for each x > 0. By (W-i),
W (0, 0) = 0. Hence, T (0) = 0 and so there always exists a trivial fixed point. Fortunately, a variation of Theorem 2(ii) can be proved (for brevity we omit the proof) where (W-iv) is replaced by
the following condition: there exists k > 0 such that, given any ε > 0 small enough, W (kε, ε) > ε
and, for all x > 0, there is y ε with W (kε + x, y) = y. Under this modified condition (ii), satisfied by the Cobb–Douglas aggregator when α + β 1, there is a unique and globally attracting
fixed point in int L∞
+ (that is, provided we consider processes that are uniformly bounded away
from 0).
5. Unbounded case
Theorem 1 covers the case in which the process Y is bounded. Interpreting Y as a consumption
stream, this means that utility functions are well defined as long as their domains are bounded
consumption streams. In this section we extend our results to unbounded Y , that is, to unbounded
consumption streams.
In the unbounded case the following condition plays an important role
−1/γ
,
lim W (1, t)/t < aw
t→∞
(17)
where γ > 0 and aw is the maximal growth rate associated with a admissible weight w (see Section 2.2). By Lemma 1 in Appendix D, the scalar function ρ(t) = W (1, t)/t is strictly decreasing
M. Marinacci, L. Montrucchio / Journal of Economic Theory 145 (2010) 1776–1804
1787
when W is Thompson. Therefore, the limit in (17) exists. Under (17) a unique scalar yw > 0 ex−1/γ
ists for which W (1, yw ) = aw yw . Notice that, by Lemma 2 in Appendix D, the existence and
uniqueness of yw is assured also in the Blackwell case provided aw β γ < 1.
We can now state our uniqueness theorem for unbounded processes Y . In reading the statement, we recall that the assumption wt 1 = w0 will be always maintained in the sequel and
that M is a shift operator if it satisfies (M-v). Moreover, see (12) for the notation [W (Y, 0)]w1/γ
in (18).
Theorem 2. Consider the operator T given by (14), where W is a γ -subhomogeneous aggregator
function and M is a shift certainty equivalent operator. Let Y ∈ L+ (w) for some admissible
weight function w with maximal growth rate aw .
(i) If M is shift subadditive and W is Blackwell for some β ∈ (0, 1) such that aw β γ < 1, then
in L+ (w 1/γ ).
T has a unique globally attracting fixed point X
in
(ii) If M is subhomogeneous and W is Thompson, then T has at least a fixed point X
1/γ
1/γ
L+ (w ) under (17). Moreover, X is unique and globally attracting in L+ (w ) provided
W (Y, 0) w1/γ > 0.
(18)
−1/γ
belongs to [0, aw
In both cases, X
yw (1 + Y w )1/γ w 1/γ ].
The proof of point (i) of this theorem is essentially based on weighted contraction techniques
([32] and [4]). Point (ii), instead, requires our Thompson’s contraction technique.
Example 6. Given Y ∈ L+ (w) for some admissible w, consider the CES aggregator Tt (X) =
ρ
(Yt + βMt (X)ρ )1/ρ , with β < 1. We treat separately two cases.
Case 1. ρ 1. The aggregator is Blackwell with contraction constant β 1/ρ . Since W is 1 in L+ (w) under the growth
homogeneous, there is a unique globally attracting fixed point X
1/ρ
and with M subadditive.
condition aw < 1/β
Case 2. ρ 1. The aggregator W is Thompson and condition (17) holds under the same condition
aw < 1/β 1/ρ . Therefore, if M is subhomogeneous, there is at least a fixed point in L+ (w).
Condition (18) is equivalent to Yt (ω) εwt for some ε > 0, that is, Y ∈ int L+ (w). Under this
is unique and globally attracting in L+ (w).
condition, the solution X
∈ [0, (1 + Y w )w 1/γ (1 − βaw )−1/ρ ].
Since yw = aw (1 − βaw )−1/ρ , in both cases X
1/γ
ρ/γ
ρ/γ
We close with few remarks:
(i) Setting wt ≡ 1 in Theorem 2, we get back to the bounded case considered by Theorem 1.
However, Theorem 1 holds without assuming that W is γ -subhomogeneous.
(ii) By Lemma 1, we have limt→∞ W (1, t)/t < 1. Therefore, condition (17) is fulfilled for some
aw > 1. For some aggregators (e.g., those of Example 2 when σ < ρ) limt→∞ W (1, t)/t
= 0. In this case, condition (17) holds for any admissible weight function w.
(iii) Theorem 2(ii) requires condition (18) in order to have uniqueness of the fixed point of T .
Namely, setting
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1/γ
U w,γ = Y ∈ L+ (w): ∃ε > 0 such that W (Yt , 0) εwt for all t 0 ,
Y ∈ U w,γ implies uniqueness of the fixed point. For instance, U w,γ = int L+ (w 1/γ ) in Example 6. In general, U w,γ might well be either larger or smaller than int L+ (w 1/γ ).
6. Quasi-arithmetic certainty equivalent operators
Theorems 1 and 2 contain crucial assumptions on the certainty equivalent operator M. We
study these assumptions for the recursive stochastic equations of the form (2) discussed in the
Introduction, with the specification (4). More precisely, given a strictly monotone function φ :
[0, ∞) → R, define M : L+ → L+ by:
Mt (X) = φ −1 Et (φ ◦ Xt+1 ) , P-a.e.,
(19)
for all t 1. Under this specification of M, the nonlinear operator equation (13) takes the recursive form:
(20)
Xt = W Yt , φ −1 Et (φ ◦ Xt+1 ) , P-a.e.,
for all t 0. The associated operator T : L+ (w 1/γ ) → L+ (w 1/γ ) is given by, for all t 0,
(21)
Tt (X) = W Yt , φ −1 Et (φ ◦ Xt+1 ) , P-a.e.,
where Y ∈ L+ (w). The nonlinear stochastic equation (20) is the quasi-arithmetic specification
(4) of the classic recursive equation (2) discussed in the Introduction, which is widely used in
Macroeconomics and Finance.
For brevity we only consider the unbounded case. This is done in the next uniqueness result, which is based on Theorem 2 and on the results on quasi-arithmetic means derived in
Appendix C. Mutatis mutandis, an analogous result, based on Theorem 1, holds in the bounded
case.
2 (R ) denotes the set of all functions φ : R → R that are twice
A piece of notation: Cm
+
+
differentiable on (0, ∞), with either φ (t) < 0 for all t > 0 or φ (t) > 0 for all t > 0.
Theorem 3. Consider the operator T given by (21), where W is a γ -subhomogeneous aggregator
2 (R ). Suppose that Y ∈ L (w) for some admissible weight function.
function and φ ∈ Cm
+
+
(i) If φ is IARA and W is Blackwell for some β ∈ (0, 1) such that aw β γ < 1, then T has a
in L+ (w 1/γ ).
unique globally attracting fixed point X
(ii) If φ is IRRA, W is Thompson, and (17) holds, then T has at least a fixed point X
1/γ
1/γ
in L+ (w ). Moreover, X is unique and globally attracting in L+ (w ) provided
[W (Y, 0)]w1/γ > 0.
For instance, the stochastic equation
ρ
1−γ ρ/(1−γ ) 1/ρ
,
Xt = Yt + β Et Xt+1
(22)
with β ∈ (0, 1), ρ > 0, and γ = 1, features a CES aggregator and a CRRA certainty equivalent,
and is a special case of the recursive equation (20) that is widely used in applications (see, e.g.,
[6] and [7]). In view of Theorem 3, (22) has a unique and globally attracting solution provided
at least one of the following conditions holds:
M. Marinacci, L. Montrucchio / Journal of Economic Theory 145 (2010) 1776–1804
1789
(i) φ is IARA, ρ 1, and aw β < 1;
(ii) φ is IRRA, ρ 1, aw β 1/ρ < 1, and [w 1/γ Y ]∞ > 0.
7. Koopmans equations
As mentioned in the Introduction, the stochastic equation (13) is not a Koopmans equation
like (8) since it only implicitly defines an underlying utility function. In this section we study
Koopmans equations and we show what are the implications for these equations of our results
and techniques. It is convenient to consider separately the deterministic case (the one mostly
studied in the literature) and the stochastic case, a topic still not well settled in the literature. To
fix ideas, throughout this section we will use a classic consumption setting.
7.1. Deterministic recursive utilities
In this case L+ reduces to the space l+ of nonnegative sequences of real numbers, which
model deterministic outcome streams. More specifically, the elements of l+ are consumption
streams c = (c0 , c1 , . . .).
Under the one step ahead certainty equivalent operator, the stochastic equation (13) becomes:
Vt = W (ct , Vt+1 ),
∀t 0,
(23)
where c ∈ l+ and V = {Vt }t0 . Here Vt is the time-t value of utility evaluated along c.
As customary, at least in the deterministic setting, we can associate with (23) a functional
equation in which the unknown is a utility function U : l+ → R. Specifically, consider the Koopmans equation
U (c0 , c1 , . . .) = W c0 , U (c1 , c2 , . . .) .
(24)
A function U : l+ → R that satisfies (24) is called a (stationary) recursive utility. Clearly, if one
fixes a consumption stream c ∈ l+ and a recursive utility U , the sequence Vt = U (ct , ct+1 , . . .)
solves (23). The converse does not hold, however, as it will be seen later.
Eq. (24) can be viewed as a fixed point problem U = K(U ), where K is the Koopmans
operator acting over suitable spaces of preferences functionals (which will be specified below).
For simplicity, we will use the exponential specification; i.e., wt = a t with t 0 (see Example 1). The space l+ (a) denotes the space of nonnegative sequences c = {ct }t0 such that
ca = supt0 ct /a t < ∞. We thus adopt the same framework as Boyd [4], though the special
structure of our aggregators suggests to take the parametrized family ϕγ (c) = (1 + ca )1/γ as
the “ϕ-function.” Accordingly, a function U : l+ (a) → R is ϕγ -bounded provided
|U (c)|
< ∞.
1/γ
c∈l+ (a) (1 + ca )
U γ = sup
The vector space Fγa of all ϕγ -bounded functions is a Banach space. Its closed subspace Cγa ⊆ Fγa
of continuous functions will be particularly relevant.
The proof of point (ii) of the next theorem will rely on Theorem 2. For this reason, set U γ ,a =
{c ∈ l+ (a): [W (c, 0)]w1/γ > 0} and enlarge the domain U γ ,a by setting
Aγ ,a = c ∈ l+ (a): ∃1 b a such that lim sup ct /bt < ∞ and
t→∞
lim inf W (ct , 0)/bt/γ > 0 .
t→∞
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Clearly, U γ ,a ⊆ Aγ ,a and Aγ ,a is shift invariant. Point (iii) of the next theorem relates recursive
utilities to the utility U 0 defined below in (25) as a limit of “partial sums,” which play a central
role in this setting (see [4] and [19]).
Theorem 4. Let W be a continuous and γ -subhomogeneous aggregator, and let a > 1.
(i) If W is Blackwell with β ∈ (0, 1) such that aβ γ < 1, then there is a unique recursive utility
is globally attracting in Fγa and U
∈ Cγa .
∈ Fγa . Moreover, U
U
(ii) If W is Thompson and (17) holds for aw = a, then
∈ [0, a −1/γ ya (1 +
∈ Fγa , with U
(a) there is an upper semicontinuous recursive utility U
1/γ
ca ) ];
is unique and continuous on the domain Aγ ,a ;
(b) U
pointwise on Aγ ,a for all U ∈ Fγa .
(c) K n U → U
(iii) The function
(25)
U 0 (c) = lim W c0 , W c1 , W c2 , . . . , W (cn , 0), . . .
n→∞
. It is continuous in the
is, in the Blackwell case, the unique recursive utility; i.e., U 0 = U
relative product topology on the norm bounded sets of l+ (a). In the Thompson case, U 0 (c)
is recursive and continuous in the product topology on norm bounded sets of Aγ ,a , and
on Aγ ,a .
U0 = U
Interestingly, Theorem 2 implies that under the assumptions of Theorem 4 aggregators rule
out nonstationary recursive utilities as well. A sequence of utilities {Ut (c)}t0 is called recursive
provided Ut (ct , ct+1 , . . .) = W (ct , Ut+1 (ct+1 , . . .)) for all t 0.
Proposition 2. Under the conditions of Theorem 4, if {Ut (c)}t0 is a sequence of recursive
utilities and Ut γ m for all t 0 and some m 0, then Ut = Ut for all t and t (in the
Thompson case this is true on the domain Aγ ,a ).
The assumption Ut γ m is needed since “explosive” nonstationary preferences always
Consider the additive aggregator W (c, V ) = u(c) + βV . The utilities Ut (c) =
coexist.t−τ
u(c
)β
+ β −t A are recursive and nonstationary preferences for every scalar A > 0.
t
τ t
7.2. Stochastic Koopmans equations
Consider the equation Vt (ω) = W (ct (ω), Mt (Vt+1 (ω))). For simplicity, consider again the
exponential specification wt = a t , with a 1. We thus write L+ (a) in place of L+ (w). We need
to consider spaces of consumptions not starting at t = 0. For a given t 0, t L denotes the space
of the sequences (ct , ct+1 , . . .) adapted to the filtration {Στ }τ t . The spaces t L(a) have a similar
meaning.
Two linear operators will play an important role: the usual shift operator S : t L(a) → t+1 L(a)
and the projection operators πs : t L(a) → L∞ (Ω, Σt+s , P), with s 0, that associate the random variable ct+s ∈ L∞ (Ω, Σt+s , P) to every element c = (ct , ct+1 , . . .) ∈ t L(a).
We now define utility sequences {Ut }t0 in this setting, where Ut : t L+ (a) → L∞ (Ω, Σt , P)
for each t 0. Though tempting, note that in general it is not correct to write Ut (c, ω) ∈ R in
place of Ut (c)(ω) because we are using equivalence classes of random variables.
M. Marinacci, L. Montrucchio / Journal of Economic Theory 145 (2010) 1776–1804
The stochastic Koopmans equation in this setting is:
Ut (c) = W ct , Mt Ut+1 S(c) , P-a.e.,
1791
(26)
for all t 0 and c ∈ t L+ (a). A sequence of utilities {Ut }t0 that satisfies (26) is called recursive.
Observe that (26) is an equality between two random variables in L∞ (Ω, Σt , P ), which says
that Ut (c)(ω) = W (ct (ω), Mt [Ut+1 (Sc)](ω)) holds P-a.e.
To establish general results on existence and uniqueness of stochastic recursive utilities, it is
convenient to treat Blackwell and Thompson aggregators separately.
Theorem 5. Assume that the aggregator W is continuous, γ -subhomogeneous, and Blackwell for
some β ∈ (0, 1) such that aβ γ < 1. Let M be constant subadditive. Then, there exists a sequence
{Ut }t0 of recursive utilities that, for some m 0, satisfy
Ut (c) ∞ m a t + ca 1/γ , ∀t 0, ∀c ∈ t L(a).
(27)
L
Moreover, if {Ut }t0 are recursive utilities that satisfy (27) for some m 0, then Ut = Ut for
all t 0. Finally, each Ut (c) is pointwise continuous in the product topology on norm bounded
sets.8
An analogous theorem holds for Thompson aggregators. Set:
γ ,a
= c ∈ t L+ (a): ∃ε > 0 such that W (ct+τ , 0) εa τ/γ for all τ 0 .
tU
Theorem 6. Suppose that W is continuous, Thompson, and γ -subhomogeneous, and that (17)
holds for aw = a. Let M satisfy the following conditions:
(i) M is subhomogeneous;
(ii) Xn ↑X implies MXn ↑MX;
(iii) for all t 0, Zn → Z in L∞ (Ω, Σt+1 , P) implies Mt Zn → Mt Z in L∞ (Ω, Σt , P).
Then, there is a sequence {Ut }t0 of recursive utilities that satisfies (27). If Ut are recursive
preferences that satisfies the condition (27) for some m, then Ut (c) = Ut (c) for every c ∈ t U γ ,a .
γ
Finally, each Ut (c) is pointwise continuous in the product topology on norm bounded sets of t Uε .
7.3. Atemporal recursive utilities
The recursive utilities {Ut } described in the previous subsection are necessarily nonstationary,
given that their domains differ for each t. The framework that we used so far is, in fact, too general
to check whether there exists a deeper relation among the utilities Ut (c). To this end, we here
specialize the stochastic model that describes uncertainty. This will enable us to show that the
recursive utilities Ut (c) are actually generated by a unique “atemporal” utility.
We keep using the classic consumption setting. Let Z be a compact metric space9 and set
Ω = Z ∞ = Z × Z × · · · . The elements of Z ∞ are denoted by ω = {zt }t0 . Denote by πt (ω) = zt
8 That is, if cn ∈ L (a) is a sequence such that π (cn ) → π (c) in L∞ (Ω, Σ
n
t +
s
s
t+s , P ) for every s 0 and c a A,
then limn→∞ Ut (cn )(ω) = Ut (c)(ω), P -a.e.
9 Compactness of Z simplifies the analysis.
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the canonical projections for t 0 and endow Z ∞ with the canonical filtration {Σt }t0 , where
Σt is the least σ -algebra that makes measurable the projections {πs }0st . The time-t history is
denoted by zt = (z0 , z1 , . . . , zt ) ∈ Z t .
Given a 1, the space L(a) is given by the processes Y for which
Y a = sup a −t Yt zt = sup a −t supYt zt = sup a −t Yt ,
zt ∈Z t , t0
t0
zt
t0
where Yt = sup{|Yt
Moreover, C(a) ⊆ L(a) is the Banach subspace of continuous processes, that is, Y ∈ C(a) if Yt : Z t → R are continuous for every t 0. We also consider
the space of lower semicontinuous processes LC(a). With this notation we can often drop the
time index and write Yt (zt ) = Y (zt ), with no ambiguity.
In this setting, the certainty equivalent operator is no longer the primitive object, which is instead a nonadditive Markov operator M : C(Z) → C(Z). It is convenient to denote by Mz (f (·))
the continuous function z → (Mf )(z), with f ∈ C(Z) and z ∈ Z.
The following assumptions that we make on the operator M are similar to those we made
on M. Specifically:
(zt )|:
(A-i)
(A-ii)
(A-iii)
(A-iv)
zt
∈ Z t }.
M(k) = k for all k 0.
Mf Mg if f g.
M(f + k) Mf + k for all k 0.
M(αf ) αMf for α ∈ [0, 1].
A property stronger than the Feller property M : C(Z) → C(Z) may be needed. Let f (p, z)
be a continuous function on P × Z, where P is any metric space.
(A-v) The function (p, z) → Mz [f (p, ·)] is continuous on P × Z for all continuous functions
f (p, z) over P × Z.
In this topological setting, the general stochastic equation (13) becomes:
V zt = W c zt , Mzt V zt , · , ∀t 0, ∀zt ∈ Z t ,
(28)
where Mt (Vt+1
is the certainty equivalent operator.
zt [Vt+1
Clearly, (A-i) implies (M-i), (A-ii) implies (M-ii), (A-iii) implies (M-iii), and (A-iv) implies
(M-iv). Moreover, by construction, M is a shift operator. It is also shift subadditive provided
(A-iii) holds.
)(zt ) = M
(zt , ·)]
Proposition 3. Under (A-i), (A-ii) and (A-iii), M satisfies (A-v).
Another useful property of the operator M is the following.
(A-vi) M(f ) is defined for all f ∈ LC(Z) and if fn ↑f for fn ∈ C(Z), then Mfn ↑Mf .
Proposition 4. Under conditions (A-v) and (A-vi), the map (p, z) → Mz (f (p, ·)) is lower semicontinuous if f (p, z ) is lower semicontinuous.
Example 7. The standard example of a nonadditive Markov operator is
M. Marinacci, L. Montrucchio / Journal of Economic Theory 145 (2010) 1776–1804
(Mf )(z) = φ −1
1793
φ f z π dz | z ,
where π(dz | z) is a weakly continuous stochastic kernel. It is the counterpart of (19) and the
properties (A-iii) and (A-iv) are related to φ like those stated in Example 4. Note that (A-v) and
(A-vi) always hold for these operators.
Theorem 2 has a straightforward counterpart in this setting. The operator T associated with
(28) is
(T V ) zt = W c zt , Mzt V zt , · .
(29)
Theorem 7. Consider the operator T given by (29), where W is continuous and γ -subhomogeneous aggregator. Let c ∈ C+ (a). Then,
(i) If W is Blackwell, with aβ γ < 1, and M is constant subadditive, then T has a unique globally
attracting fixed point in C+ (a 1/γ ).
(ii) If W is Thompson, M satisfies (A-v) and (A-vi), and (21) holds, then T has at least a fixed
point in LC+ (a 1/γ ). The fixed point is unique and globally attracting in LC+ (a 1/γ ) provided
inf W c zt , 0 /a t/γ > 0, ∀zt , ∀t 0.
Moreover, the fixed point lies in C+ (a 1/γ ).
This theorem implies an analogous of Theorem 5, that is, the existence of a sequence of
recursive utilities. We treat for simplicity the Blackwell case only. Note that we can here write
Ut (c; zt ) in place of Ut (c)(zt ). The functions Ut (c; zt ) are actually quite regular, as the next
result shows.
Proposition 5. In case (i) of Theorem 7, the utilities Ut (c; zt ) are continuous on t C+ (a) × Z t .
Fix z ∈ Z. For any function Y : Z t → R, t > 0, define the map σ (z)Y : Z t−1 → R by
(σ (z)Y )(·) = Y (z, ·). If t = 0, σ (z)Y denotes the constant function Y (z).
This operation can be extended to the elements in c ∈ C+ (a), where σ (z)c = (σ (z)c0 , σ (z)c1 ,
. . .). For an element zt ∈ Z t , we can similarly define σ (zt )Y for Y : Z s → R if s t.
Theorem 8. Let W be continuous, γ -subhomogeneous, and Blackwell for some β ∈ (0, 1) such
that aβ γ < 1. Let M be constant subadditive. Then, there exists a sequence of recursive utilities
Ut (c; zt ), with t 0, c ∈ t C+ (a), and zt ∈ Z t , which are generated by
Ut c; zt = U0 σ zt−1 c; zt ,
(30)
where zt = πt (zt ). More generally,
Ut (c; z0 , z1 , . . . , zt ) = Ut−1 σ (z0 )c; z1 , z2 , . . . , zt ,
∀t 1.
(31)
This theorem implies, inter alia, that (26) and (30) together yield the Koopmans equation
U0 (c; z) = W c0 (z), Mz U0 σ (z)Sc; · ,
(32)
for all c ∈ C+ (a) and z ∈ Z, which generalizes (24).
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Appendix A. The space of adapted processes
We can regard the space L of all adapted processes as the space of measurable functions wrt
a suitable σ -algebra. It is a rather standard construction in probability theory and we just outline it. Denote by λ the counting measure on (N, 2N ). Consider the measure space (Ω × T ,
Σ ⊗ 2T , P ⊗ λ). Define the σ -algebra F ⊆ Σ ⊗ 2T as A ∈ F iff At ∈ Σt for all t 0
and where At denotes the t-section of the set A ⊆ Ω × T . It is easy to check that the process X(ω, t) is adapted iff X is F -measurable. Therefore, we may identify L with the space
L(Ω × T , F , P ⊗ λ), where two elements X, X ∈ L(Ω × T , F , P ⊗ λ) are identified iff Xt = Xt
P-a.e. for all t.
The spaces L∞ and L+ (w) can then be regarded as subspaces of L(Ω × T , F , P ⊗ λ). An
important property is that L+ (w) turns out to be a Banach lattice. Moreover, the positive cone
L+ (w) is normal (see [16] and [31]). Actually, X1 w X2 w if X1 X2 in L+ (w).
Appendix B. A contraction theorem
In this appendix we present a contraction theorem based on the Thompson metric, which
adapts to the setting of this paper some results due to Montrucchio [25]. The results below hold in
any ordered normed space with a complete and normal positive cone. For concreteness, however,
we will use the space L+ (w), which suffices for our purposes.
Following [31] (see also [26]), two adapted processes X and Y of L+ (w) are said to be
comparable if there exist scalars α > 0 and β > 0 such that αX Y βX. This is an equivalence
relation on L+ (w) and CX = {Y : Y is comparable to X} denotes the component containing X.
Given two comparable elements X and Y , set M(Y | X) = inf{α > 0; Y αX}. Observe that
the inf is a minimum, so that M(Y | X) > 0.
If X and Y are two comparable elements in L+ (w), their Thompson distance dτ is defined as
dτ (X, Y ) = max ln M(X | Y ), ln M(Y | X) .
If C is a component of L+ (w), one can easily prove that dτ gives a metric on C. Moreover,
Thompson [31] proves the following result.
Theorem 9. Each component C of L+ (w) is a complete metric space wrt the metric dτ .
This theorem relies on the fact that the cone L+ (w) is complete and normal. Next we show
when an operator is a contraction wrt the Thompson metric.
Theorem 10. Suppose an operator T : [0, X1 ] → [0, X1 ], with [0, X1 ] ⊆ L+ (w), has the following properties:
(i) T is monotone, i.e., X Y implies T (X) T (Y ) for all X, Y ∈ [0, X1 ];
(ii) T (0) = X0 is comparable to X1 , i.e., X0 ∈ CX1 ;
(iii) T (αX) αT (X) + (1 − α)T (0) for all α ∈ [0, 1] and X ∈ [X0 , X1 ].
Then, T is a contraction over [X0 , X1 ] wrt the Thompson distance dτ ; that is,
dτ T (X), T (Y ) γ dτ (X, Y ), ∀X, Y ∈ [X0 , X1 ],
where γ = 1 − μ−1 < 1 and μ = M(X1 | X0 ).
M. Marinacci, L. Montrucchio / Journal of Economic Theory 145 (2010) 1776–1804
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Proof. We begin with the following easy claim, a variant of Bernoulli’s inequality.
Claim. It holds t γ 1 − γ + γ t for all t 0 and all 0 γ 1.
Proof of the claim. The function t γ is concave over t 0. From the superdifferentiability property at t = 1, it follows the desired inequality. 2
The result is trivial if X0 = X1 . Suppose that X0 = X1 . Set μ = M(X1 | X0 ). Clearly, μ > 1.
Otherwise, X1 μX0 X0 . Hence, X0 X1 . As X0 X1 , it would be X0 = X1 , a contradiction.
Consider two nonidentical elements X, Y ∈ [X0 , X1 ]. They are clearly comparable. Set β1 =
M(X | Y ), β2 = M(Y | X) and β = max{β1 , β2 }. As X and Y are distinct, β > 1. Suppose that
β = β1 = M(X | Y ). This means βY X, i.e., Y β −1 X with β −1 < 1. By (ii) and (iii),
T (Y ) T β −1 X β −1 T (X) + 1 − β −1 X0 .
(33)
From μ = M(X1 | X0 ) it follows X0 μ−1 X1 . Plugging this into (33),
T (Y ) β −1 T (X) + 1 − β −1 μ−1 X1 β −1 T (X) + 1 − β −1 μ−1 T (X)
= β −1 + μ−1 1 − β −1 T (X),
where in the second inequality we are using the fact that T (X) X1 .
By setting t = β −1 and γ = 1 − μ−1 in the claim, we obtain
β −1 + μ−1 1 − β −1 = μ−1 + β −1 1 − μ−1 β −γ .
Therefore, β γ T (Y ) T (X). Hence, M(T (X) | T (Y )) β γ and in turn,
ln M T (X) | T (Y ) γ ln M(X | Y ) = γ dτ (X, Y ).
If also β2 > 1, the same argument leads to ln M(T (Y ) | T (X)) γ ln M(Y | X) γ dτ (X, Y ).
This implies d(T X, T Y ) γ d(X, Y ), which ends the proof. Suppose, on the contrary, that
β2 1. We then have X Y . Thus T X T Y , and so M(T Y | T X) 1. Its logarithm is nonpositive and therefore the same result holds. 2
Theorem 10 implies the following contraction result.
∈ [0, X1 ] under the conditions of
Theorem 11. The operator T has a unique fixed point X
w → 0 for each
is globally attracting on [0, X1 ]; i.e., T n (X) − X
Theorem 10. Moreover, X
X ∈ [0, X1 ].
Proof. We begin with a claim, which is essentially [31, Lemma 1].
Claim. If X and Y are two comparable elements of L+ (w) and Xw , Y w A, then
X − Y w A exp dτ (X, Y ) − 1 .
Proof of the claim. Set λ = M(X | Y ), ν = M(Y | X) and μ = max{λ, ν}. If X = Y , then μ > 1.
It follows μY X and μX Y . Therefore, |X − Y | (μ − 1)(X ∨ Y ), and so
w −1 |X − Y | (μ − 1)w −1 (X ∨ Y ) (μ − 1) Xw ∨ Y w (μ − 1)A.
In turn, this implies X −Y w (μ−1)A = A(exp dτ (X, Y )−1), because μ = exp dτ (f, g). 2
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As T (0) 0 and T is monotone, X ∈ [0, X1 ] implies T (X) ∈ [X0 , X1 ]. Consequently, the
fixed points lie in [X0 , X1 ]. On the other hand, T : [X0 , X1 ] → [X0 , X1 ] is a contraction mapping
wrt the Thompson distance. By hypothesis, [X0 , X1 ] is contained in a component C of L+ (w).
The interval [X0 , X1 ] is closed and bounded wrt the weighted supnorm · w . By the claim,
[X0 , X1 ] is then closed in C wrt the Thompson metric, and so ([X0 , X1 ], dτ ) is a complete
such that
metric space. By the Banach Contraction Mapping Theorem, there exists a unique X
w → 0 for any X ∈ [0, X1 ]. 2
= X.
The claim guarantees that T n (X) − X
T (X)
Appendix C. Quasi-arithmetic means
n
n
Let n−1 = {q ∈ Rn+ :
i=1 qi = 1} be the unit simplex of R . Given a strictly monotone
n
function φ : R+ → R and a vector q ∈ n−1 , define Mφ (·; q) : R+ → R by
n
−1
Mφ (x; q) = φ
φ(xi )qi , ∀x ∈ Rn+ .
i=1
The function Mφ is often called a quasi-arithmetic mean (see, e.g., [13, Ch. III] and [5, Ch. IV]).
In the paper two properties of these means play a key role:
• Mφ (·; q) : Rn+ → R is constant subadditive if
Mφ (x + k; q) Mφ (x; q) + k,
∀x, k ∈ Rn+ ,
for any q ∈ n−1 and any n 1, where k denotes both a scalar and the corresponding
constant vector (k, . . . , k).
• Mφ (·; q) : Rn+ → R is subhomogeneous if
αMφ (x; q) Mφ (αx; q),
∀x ∈ Rn+ , ∀α ∈ [0, 1],
for any q ∈ n−1 and any n 1.
These two properties are closely connected, and next we provide a duality between them
(we omit the simple proof). Here ex for x ∈ Rn stands for the vector (ex1 , . . . , exn ), with the
convention e−∞ = 0.
Proposition 6. The quasi-arithmetic mean Mφ is subhomogeneous iff Mφ(x; q) = log Mφ (ex ; q)
(t) = φ(et ) for each t 0. If, in addition, φ is twice differenis constant subadditive, where φ
tiable on (0, ∞), then Aφ(t) = Rφ (et ) − 1.
is a one-to-one correspondence between IRRA and IARA functions, as
The duality φ → φ
well as between DRRA and DARA functions, and CRRA and CARA ones. In other words, this
duality preserves the classification of functions according to absolute and relative risk aversion
mentioned in Section 2. Observe that this duality also preserves monotonicity, that is, φ is in is.
creasing iff φ
We begin by characterizing constant subadditive quasi-arithmetic means.
Theorem 12. Suppose φ : R+ → R is twice differentiable on (0, ∞), with either φ (t) < 0 for all
t > 0 or φ (t) > 0 for all t > 0. Then, Mφ (·; q) : Rn+ → R is constant subadditive iff φ is IARA.
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Proof. Wlog, assume φ(0) = 0, so that φ(0) = φ −1 (0) = 0. Following Beck [2] (see also [5,
p. 251]), for all k 0, define fk : φ(R+ ) → R by fk (s) = φ(φ −1 (s) + k), ∀s ∈ φ(R+ ).
Claim. Suppose φ is strictly increasing. Then, Mφ (·; q) is constant subadditive iff fk is concave
on φ(R++ ) for all k 0.
Proof of the claim. Let fk be concave on φ(R++ ). On the other hand, fk (s) fk (0) because φ
is strictly increasing. Hence, fk is concave on the entire φ(R+ ).
Let {si }ni=1 ⊆ φ(R+ ) and let xi = φ −1 (si ). Then,
n
n
n
n
−1
φ(xi + k)qi =
φ φ (si ) + k qi =
fk (si )qi fk
si qi
i=1
i=1
=φ φ
−1
n
i=1
i=1
si qi + k = φ φ
−1
i=1
n
φ(xi )qi + k ,
i=1
and so, being φ −1 increasing, φ −1 ( ni=1 φ(xi + k)qi ) φ −1 ( ni=1 φ(xi )qi ) + k.
This shows that Mφ is constant subadditive. As to the converse, assume that all Mφ , given any
q ∈ n−1 and any n 1, are constant subadditive. Let s1 , s2 , ∈ φ(R++ ) with xi = φ −1 (si ), and
let α ∈ [0, 1]. Then,
fk ts1 + (1 − t)s2 = φ φ −1 ts1 + (1 − t)s2 + k = φ φ −1 tφ(x1 ) + (1 − t)φ(x2 ) + k
= φ Mφ x; (t, 1 − t) + k φ Mφ (x + k); (t, 1 − t)
= tφ(x1 + k) + (1 − t)φ(x2 + k) = tfk (s1 ) + (1 − t)fk (s2 ),
and so fk is concave on Rn++ .
2
In view of the claim, we need to show that fk is concave on φ(R++ ). We first assume that
φ > 0 on R++ . Set ψ = φ −1 , so that fk (s) = φ(ψ(s) + k), for all s ∈ φ(R++ ). For all x > 0,
we have
−1
−3
and φ (x) = −ψ φ(x) ψ φ(x)
,
φ (x) = ψ φ(x)
so that (ψ )2 /ψ = −(φ /φ ) ◦ ψ . The function fk is concave iff fk 0. We have: fk (s) =
φ (ψ(s) + k)(ψ (s))2 + φ (ψ(s) + k)ψ (s), and so fk 0 iff
2
φ ψ(s) + k ψ (s) −φ ψ(s) + k ψ (s).
(34)
Hence, inequality (34) holds iff φ (ψ(s) + k)/φ (ψ(s) + k) −ψ (s)/(ψ (s))2 . That is,
−φ ψ(s) + k /φ ψ(s) + k −φ ψ(s) /φ ψ(s) .
This holds iff φ is IARA, as R+ is the domain of φ. We conclude that fk 0 iff φ is IARA, as
desired. Finally, if φ < 0, we get easily the same result by setting ϕ = −φ. 2
By Theorem 12, a simple application of the duality established in Proposition 6 gives the next
characterization of subhomogeneous quasi-arithmetic means.
Corollary 1. Assume φ : R+ → R is twice differentiable on (0, ∞), with either φ (t) < 0 for all
t > 0 or φ (t) > 0 for all t > 0. Then, Mφ (·; q) : Rn+ → R is subhomogeneous iff φ is IRRA.
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established in Proposition 6, the subhomogeneous counterpart of
Thanks to the duality φ → φ
the claim in Theorem 12 is easily established: Mφ (·; q) is subhomogeneous iff the scalar function
s → φ(αφ −1 (s)) are convex for all α ∈ [0, 1].
We close by showing what form Theorem 12 and Corollary 1 take in our more general probability setting (we omit the proof, which is similar to that of Theorem 12 using the Jensen
inequalities for conditional expectations).
Proposition 7. Let X ∈ L+ and let Σ be a σ -algebra contained in Σ . Suppose φ : R+ → R is
twice differentiable on (0, ∞), with either φ (t) < 0 for all t > 0 or φ (t) > 0 for all t > 0. Then,
φ is IARA iff, for all k 0,
φ −1 ◦ E φ ◦ (X + k) | Σ φ −1 ◦ E φ ◦ X | Σ + k, P -a.e.,
while φ is IRRA iff, for all α ∈ [0, 1],
αφ −1 ◦ E φ ◦ X | Σ φ −1 ◦ E φ ◦ (αX) | Σ ,
P -a.e.
Appendix D. Proofs in the main text
We give first a few basic properties of aggregators that are needed in proving our results.
Lemma 1. Let W (x, y) be a Thompson aggregator. Then:
(i) W (x, ·) is continuous over (0, ∞) for all x 10 ;
(ii) W (x, y) = y has a unique solution for all x > 0;
(iii) the function W (x, y)/y is strictly decreasing in y ∈ (0, ∞) for all x > 0.
Proof. (i) Suppose yn ↓y > 0. There exists {αn }n ⊆ [0, 1], with αn ↑1, such that y = αn yn for
each n. By (W-iii),
W (x, y) = W (x, αn yn ) αn W (x, yn ) + (1 − αn )W (x, 0) αn W (x, yn )
and so limn W (x, yn ) W (x, y). By (W-i), limn W (x, yn ) W (x, y). The proof for the left limit
yn ↑y is similar.
(iii) Let y > y > 0 and y = ty with t ∈ (0, 1). Hence W (x, y) tW (x, y ) + (1 − t) ×
W (x, 0) > tW (x, y ). It follows W (x, y)/y > tW (x, y )/y = W (x, y )/y which is the desired
property.
(ii) It is an obvious consequence of (iii). 2
By Lemma 1, limy→+∞ W (x, y)/y exists when W is a Thompson aggregator. Note further
that from (iii) of Lemma 1, from W (x,
y) = y it follows
W (x, y) − y (y − y ) < 0, ∀y = y.
(35)
The proof of the following property is immediate by the change of variable W (x, y1 /λ) = y1 .
Lemma 2. If W (x, y) is Blackwell with Lipschitz constant β, then equation W (x, y) = λy has
one and only one solution for all x, provided λ > β.
10 We thank an anonymous referee for having pointed out to us a mistake in an earlier draft of this paper.
M. Marinacci, L. Montrucchio / Journal of Economic Theory 145 (2010) 1776–1804
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Proof of Proposition 1. By (W-ii), there is a point y ∗ such that W (Y ∞ , y ∗ ) = y ∗ . Let X ∈
[0, y ∗ ]. (W-i) and (M-ii) imply that T (X) = W (Y, M(X)) W (Y ∞ , y ∗ ) = y ∗ . Therefore,
T : [0, y ∗ ] → [0, y ∗ ]. Observe that [0, y ∗ ] is a complete lattice wrt the pointwise order . Since
the operator T is monotone, by the Tarski Fixed Point Theorem [30] we conclude that T has a
fixed point. 2
Proof of Theorem 1. (i) We omit the proof because it is a straightforward application of the
Banach Contraction Mapping Theorem.
(ii) If X ∈ [0, y ∗ ], W (Yt , Mt (X)) W (Y ∞ , Mt (y ∗ )) = y ∗ , and so T : [0, y ∗ ] → [0, y ∗ ].
Let us show that the conditions of Theorem 10 in Appendix B are satisfied (by setting w ≡ 1).
Condition (i) clearly holds since T is monotonic. Moreover, for all α ∈ [0, 1] and all X ∈ L∞
+,
we have:
T (αX) = W Y, M(αX) W Y, αM(X) = W Y, αM(X) + (1 − α)M(0)
αW Y, M(X) + (1 − α)W Y, M(0) = αT (X) + (1 − α)T (0),
and so also condition (iii) of Theorem 10 holds. As W (Y, 0) W ([Y ]∞ , 0), T 0 is compara ∈ [0, y ∗ ], with
ble with y ∗ and (ii) holds. By Theorem 11, there exists a unique fixed point X
n
∗
T (X) − X∞ → 0, ∀X ∈ [0, y ].
∗
∗
∈ L∞
Let X
+ and set ỹ = X∞ ∨ y . Clearly X ∈ [0, ỹ]. Let X ∈ [0, ỹ]. As ỹ y , (35)
implies:
T (X) T (ỹ) = W (Y, ỹ) W Y ∞ , ỹ ỹ
and so T (X) ∈ [0, ỹ]. Hence, T : [0, ỹ] → [0, ỹ]. Once again, Theorem 11 implies T n (X) −
∞ → 0 for each X ∈ [0, ỹ]. Hence, T n (X)
− X
∞ → 0, as desired. Finally, we obviously
X
∗
have X ∈ [y∗ , y ]. 2
1/γ
Proof of Theorem 2. Case (i). As W is γ -subhomogeneous and wt 1, W (Yt , 0)/wt
W (Yt /wt , 0) W (Y w , 0). Therefore, T 0 = W (Y, 0) ∈ L+ (w 1/γ ).
Let X1 , X2 ∈ L+ (w 1/γ ). From X1 X2 + X1 − X2 w1/γ w 1/γ , it follows
M(X1 ) M X2 + X1 − X2 w1/γ w 1/γ M(X2 ) + X1 − X2 w1/γ S w 1/γ
1/γ
M(X2 ) + aw
X1 − X2 w1/γ w 1/γ .
1/γ
Interchanging X1 and X2 , we get |M(X1 ) − M(X2 )| aw X1 − X2 w1/γ w 1/γ . Hence, for all
t 0 and P-a.e. ω ∈ Ω,
Tt (X1 )(ω) − Tt (X2 )(ω) = W Yt (ω), Mt (X1 )(ω) − W Yt (ω), Mt (X2 )(ω) β Mt (X1 )(ω) − Mt (X2 )(ω) a 1/γ βX1 − X2 1/γ w 1/γ ,
w
1/γ
aw βX1 − X2 w1/γ . This implies T (X) − T 0w1/γ
w
1/γ
so that T (X1 ) − T (X2 )w1/γ
aw ×
βXw1/γ . Hence, T (X)w1/γ T (X) − T 0w1/γ + T 0w1/γ < ∞ for all X ∈ L+ (w 1/γ ).
1/γ
1/γ
To conclude, T maps L+ (w 1/γ ) into itself and is a aw β-contraction provided aw β < 1, i.e.,
γ
1/γ
∈ L+ (w ).
aw β < 1. This shows the existence and uniqueness of the fixed point X
−1/γ
−1/γ
for all t yw . Setting taw
= λ, we get
Case (ii). In view of (17), W (1, t) taw
1/γ
−1/γ
W (1, aw λ) λ for all λ aw yw .
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Fix Y ∈ L+ (w), we prove that T : [0, λ(1 + Y w )1/γ w 1/γ ] → [0, λ(1 + Y w )1/γ w 1/γ ] for
−1/γ
all λ aw yw . Let X ∈ [0, λ(1 + Y w )1/γ w 1/γ ]. We have
Tt (X)(ω)
1/γ
(1 + Y w )1/γ wt
=
1
1/γ
(1 + Y w )1/γ wt
W Yt (ω), Mt (X)(ω)
Yt (ω)
Mt (X)(ω)
,
(1 + Y w )wt (1 + Y w )1/γ wt1/γ
1/γ Y w
−1/γ
(ω)
, λwt
Mt w
W
1 + Y w
−1/γ 1/γ −1/γ 1/γ (ω) = W 1, λwt
S w
wt+1
W 1, λwt
1/γ
W 1, λaw λ.
W
1/γ
Hence, Tt (X)(ω) λ(1 + Y w )1/γ wt , the desired result. By the Tarski Fixed Point Theorem,
T has a fixed point because T is monotone and [0, λ(1 + Y w )1/γ w 1/γ ] is a complete lattice in
L(w 1/γ ).
To prove the last part of the theorem, it is now enough to show that the conditions of Theorems
10 and 11 are satisfied by setting L(w 1/γ ) and taking an interval [0, λ(1 + Y w )1/γ w 1/γ ],
−1/γ
with λ aw yw . Conditions (i) and (iii) of Theorem 10 clearly hold. From (18) it follows
that W (Y, 0) εw 1/γ for some ε > 0. Hence, there is a scalar σ such that σ W (Y, 0) λ(1 +
Y w )1/γ w 1/γ . Thus, W (Y, 0) and λ(1 + Y w )1/γ w 1/γ are comparable, and so (ii) of Theorem
10 holds. It follows the uniqueness and attractivity of the fixed point in [0, λ(1 + Y w )1/γ w 1/γ ].
As λ can be taken arbitrarily large, we have the desired result. 2
Proof of Theorem 3. In view of Theorem 2, it is enough to observe that M given by (19)
satisfies (M-v), and by Theorem 12 it is constant subadditive if φ is IARA, while by Corollary 1
it is subhomogeneous if φ is IRRA. 2
Proof of Theorem 4. (i) We omit the proof, a well-known application of the Weighted Contraction Theorem (see Boyd [4]).
(ii-a) Fix λ 0. By γ -subhomogeneity,
K(λϕγ ) W (c0 , λ(1 + Sca )1/γ )
c0
1 + Sca 1/γ
=
W
,λ
ϕγ
1 + ca
1 + ca
(1 + ca )1/γ
1/γ
W 1, λa
.
Like in the proof of Theorem 2, under our assumption we have W (1, λa 1/γ ) λ provided λ a −1/γ ya . Hence, K(λϕγ ) λϕγ if λ a −1/γ ya . As K is monotone, K : [0, a −1/γ ya ϕγ ] →
[0, a −1/γ ya ϕγ ] and the existence of a fixed point is a consequence of Tarski’s Theorem.
Note that by iterating the operator K, with initial condition U = a −1/γ ya ϕγ , we have that
n
= limn→∞ K n U is an upper semicontinuous fixed point.
K U ↓. Consequently, U
In the rest of the proof we first consider the domain U γ ,a and then the asymptotic domain Aγ ,a .
(ii-b) Let U 1 and U 2 be any two fixed points of K in the space Fγw . Fix a consumption
sequence c ∈ U γ ,a . Given c, consider the problem according to the approach that led to Theorem 2. Associated with utility U 1 (c), we have the stream of utilities Vt1 = U 1 (ct , ct+1 , . . .)
for t 0. Likewise, Vt2 = U 2 (ct , ct+1 , . . .) according to utility U 2 . Let us check that the
M. Marinacci, L. Montrucchio / Journal of Economic Theory 145 (2010) 1776–1804
1801
sequences (V01 , V11 , V21 , . . .) and (V02 , V12 , V22 , . . .) are in l(a 1/γ ). Since U 1 ∈ Fγw , it follows
U 1 (c) L(1 + ca )1/γ for some scalar L 0. Hence,
1/γ
Vt1
U 1 (S t c)
1 + S t ca 1/γ
1 + a t ca 1/γ
=
L
L
L 1 + ca
,
t/γ
t/γ
t
t
a
a
a
a
and thus (V01 , V11 , V21 , . . .) ∈ l(a 1/γ ).
1 ) for t 0, and so it is the fixed
The sequence (V01 , V11 , V21 , . . .) satisfies Vt1 = W (ct , Vt+1
point of the operator T . By (ii) of Theorem 2, the fixed point is unique. It follows that V01 = V02
implies U 1 (c) = U 2 (c).
Now observe that among the fixed points of K in the interval [0, λ0 ϕγ ], with λ0 = a −1/γ ya ,
there is both a lower semicontinuous function and an upper semicontinuous function. Actually,
consider the two fixed points U 0 = limn→∞ K n 0 and U1 = limn→∞ K n (λ0 ϕγ ). They coincide
on U γ ,a and this proves the l(a)-continuity of the unique recursive function on U γ ,a .
(ii-c) For any function U in Fγw , 0 U λϕγ with λ large enough. This implies K n 0 K n U K n (λϕγ ). This proves the convergence of K n U on U γ ,a .
(iii) Both in the Blackwell and Thompson cases, the conditions aβ γ < 1 and (17) are still
satisfied for a > a sufficiently close to a. Therefore, the conclusions of this theorem hold by
replacing l+ (a) with l+ (a ). It is well known (see [4, Lemma 2]) that on l+ (a)-bounded sets the
l+ (a )-topology agrees with the relative product topology. Therefore the claims follow.
To complete the proof it remains to extend to Aγ ,a what established for the domain U γ ,a . Let
c ∈ Aγ ,a . By definition, c ∈ l+ (b), with b a, and W (ct , 0)/bt/γ ε > 0 for each t N . That
is, W (ct , 0) εbN/γ b(t−N )/γ . Observe that l+ (b) ⊆ l+ (a) and that, if U is any recursive utility
on l+ (a), its restriction on l+ (b) is a recursive utility as well. Moreover, the condition (17) holds
by replacing a with b.
Let U 1 and U 2 be two recursive utilities on l+ (a). Restrict them on l+ (b) and apply the
theorem with a replaced by b. Since U 1 and U 2 are recursive, we have, for i = 1, 2,
U i (c) = W c0 , W c1 , . . . , W cN −1 , U i S N c
.
On the other hand, S N c ∈ U γ ,b . By point (ii-b), U 1 (S N c) = U 2 (S N c). Consequently, U 1 (c) =
U 2 (c) for all c ∈ Aγ ,a . Clearly, all other properties easily follow. 2
Proof of Proposition 2. It is similar to point (ii-b) of the previous proof. Define Vt1 = Ut (S t c)
for a fixed element c ∈ l+ (a). Under the assumption Ut γ m, it follows that (Vt1 )t0 ∈
0 (S t c), where U
0 is the function
l+ (a 1/γ ). By the uniqueness result of Theorem 2, Ut (S t c) = U
(25). 2
Proof of Theorem 5. To emphasize the dependence of operator T on the process c ∈
L+ (a), in the following we write V = Tc (V ). Fix c = (ct , ct+1 , . . .) ∈ t L(a). Set c =
(0, . . . , 0, ct , ct+1 , . . .) ∈ L(a), where S t c = c. Define Ut (c) = Vtc , where Vtc is the unique
solution to V = Tc (V ). By Theorem 2, |Vtc (ω)| m(1 + c a )1/γ a t/γ . On the other hand,
c a = a −t ca . It follows that Ut (c) = Vtc m(a t + ca )1/γ . The converse is proved
similarly.
The last part of the proof uses a technique similar to that adopted by [19]. Set β = βa 1/γ < 1.
Fix t 0 and c ∈ t L(a). Once again set c = (0, . . . , 0, ct , ct+1 , . . .) ∈ L(a). By Theorem 2,
V c − TcN 0a 1/γ β N (1 − β)−1 Tc 0a 1/γ . By the property of β-contraction of T , we have
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M. Marinacci, L. Montrucchio / Journal of Economic Theory 145 (2010) 1776–1804
a −t/γ Vtc − TcN 0 t L∞ β N (1 − β)−1 Tc 0a 1/γ a −t/γ β N (1 − β)−1 W ca , 0 ,
where the last inequality follows from the γ -subhomogeneity and monotonicity of W .
If ca A and recalling that Vtc = Ut (c), then Ut (c) − (TcN 0)t L∞ Mβ N . Hence,
Ut (c)(ω) − T N 0 (ω) Mβ N ,
c
t
Ut (c)(ω) TcN 0 t (ω) + Mβ N ,
which holds a.e. for any ca A. As in the proof of Theorem 4, if cn → c for the product
topology and cn a A, we get lim supn→∞ Ut (cn )(ω) (TcN 0)t (ω) + Mβ N , provided c →
(TcN 0)t is continuous. In this case, as N → ∞, lim supn→∞ Ut (cn )(ω) Ut (c)(ω).
Note that Ut (cn )(ω) (TcN 0)t (ω), and so
n
lim inf Ut (cn )(ω) TcN 0 t (ω)
n
⇒
lim inf Ut (cn )(ω) Ut (c)(ω),
n
and in turn, Ut (cn ) → Ut (c) a.e., as desired.
It remains to check the continuity of the map c → (TcN 0)t . More precisely, we must prove
the following fact. If cn ∈ L+ (a) and πs (cn ) → πs (c) in L∞ (Σs ) for all s 0, and cn a A,
then, (TcNn 0)t → (TcN 0)t in L∞ (Σt ), for all N 1 and t 0.
Observe first that, for all t and N , (TcN 0)t L∞ m(a t + ca )1/γ . It follows that
(TcN 0)t L∞ Kt , where the scalar Kt is independent of N , provided ca A.
The continuity claim will be proved by induction.
(i) It is true for N = 1. Actually, (Tc 0)t = W (ct , 0). On the other hand, ct L∞ a t ca a t A. As W is uniformly continuous over bounded sets, it follows that (Tcn 0)t → (Tc 0)t in
L∞ (Σt ).
(ii) Suppose that the property holds for N = m. We prove it for N = m + 1. We have (Tcm+1 0)t =
W (ct , Mt ((Tcm 0)t+1 )). As the operator M is constant subadditive,
Mt (Xt+1 ) − Mt (Zt+1 ) ∞ Xt+1 − Zt+1 L∞ .
(36)
L
From (36), Mt ((Tcm 0)t+1 )L∞ (Tcm 0)t+1 L∞ Kt+1 . Moreover, the uniform continuity of W and (36) imply easily that (Tcm+1 0)t has the desired continuity property and this
completes the proof. 2
Proof of Theorem 6. We omit the details of the proof. Note that (ii) implies the existence of a
minimal solution given by limN →∞ TcN 0, while (iii) is used to prove the continuity property. 2
Proof of Proposition 3. Let zn → z and pn → p. We have
Mz f (p, ·) − Mz f (pn , ·) Mz f (p, ·) − Mz f (p, ·) n
n
+ Mz f (p, ·) − Mz f (pn , ·) .
n
n
Clearly, |Mz (f (p, ·)) − Mzn (f (p, ·))| → 0. Thanks to compactness of Z, given ε > 0, we have
|f (p, z ) − f (pn , z )| ε for all n n0 (ε) and all z ∈ Z. In other words, f (p, z ) f (pn , z )
+ ε. By (A-iii), M[f (p, ·)] M[f (pn , ·)] + ε. Hence, |Mzn (f (p, ·)) − Mzn (f (pn , ·))| ε.
Consequently, also |Mzn (f (p, ·)) − Mzn (f (pn , ·))| → 0 and the claim is proved. 2
M. Marinacci, L. Montrucchio / Journal of Economic Theory 145 (2010) 1776–1804
1803
Proof of Proposition 4. Since f (p, z) is lower semicontinuous, there exists a sequence fn (p, z)
of continuous functions such that fn (x, z)↑f (x, z). By (A-vi), Mz (fn (p, ·))↑Mz (f (p, ·)) over
P × Z. By (A-v), Mz (f (p, ·)) is lower semicontinuous on P × Z. 2
Proof of Theorem 7. (i) By Proposition 3, (A-v) holds. Consequently, by continuity of W , the
operator T maps continuous processes to continuous processes. The proof then proceeds as in (i)
of Theorem 2, where the cone L+ (a) is replaced by the cone C+ (a).
(ii) Also here the proof is similar to part (ii) of Theorem 2. Note, however, that we cannot
invoke here Tarski’s Theorem. On the other hand, thanks to conditions (A-v) and (A-vi) and
Proposition 4, there exits the lower semicontinuous process limN →0 T N 0 that is the minimal
fixed point of the operator. The rest of the proof easily follows by using the closed cone LC+ (a)
of L(a). 2
Proof of Proposition 5. We prove only the statement for the function U0 (c; z). The general case Ut (c; zt ) will be a consequence of (30) in Theorem 8. The operator T turns out to
be a β-contraction on C+ (a 1/γ ) for all c ∈ C+ (a), where β = a 1/γ β < 1 (see the proof of
(i) of Theorem 2). Consider an element V (c, zt ) such that (c, zt ) → V (c, zt ) is continuous
for all t and such that {V (c, zt )} ∈ C+ (a 1/γ ) for all fixed c. The image of {V (c, zt )} by T
(c, zt ) is continuous. Therefore,
(c, zt ) = W (ct (zt ), Mzt (V (c, zt , ·))). Hence, (c, zt ) → V
is V
N
U0 (c; z) = limN →0 (Tc 0)0 is a limit of continuous functions. If the limit is uniform, U0 (c; z) is
continuous. By the usual inequality, we have
N
N
U0 (c, z) − T N 0 (z) U (c) − T N 0 1/γ β Tc 0 1/γ β W ca , 0 .
a
c
c
0
a
1−β
1−β
Hence (TcN 0)(z) → U0 (c, z) uniformly over any norm bounded set. The desired property is thus
proved. 2
Proof of Theorem 8. By induction on N it is easy to check that
N Tc 0 t (z0 , z1 , . . . , zt ) = Tc̃N 0 t−1 (z1 , z2 , . . . , zt )
holds for all N 1 and t 1, where c̃ = σ (z0 )c. Letting N → ∞, we get (31). By iterating the relation (31) we obtain Ut (c; z0 , z1 , . . . , zt ) = Ut−2 (σ (z0 , z1 )c; z2 , . . . , zt ) and, at last,
Ut (c; zt ) = U0 (σ (zt−1 )c; zt ). 2
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