EXPONENTIAL UTILITY WITH NON-NEGATIVEwueth/Papers/2012_muraviev...Exponential Utility with Non-Negative Consumption 3 setting is violated for incomplete markets. Then, we turn to analyzing

EXPONENTIAL UTILITY WITH NON-NEGATIVE

CONSUMPTION

ROMAN MURAVIEV AND MARIO V. WUTHRICH

DEPARTMENT OF MATHEMATICS AND RISKLAB

ETH ZURICH

Abstract. This paper investigates various aspects of the discrete-time ex-

ponential utility maximization problem, where feasible consumption policies

are not permitted to be negative. By using the Kuhn-Tucker theorem, some

ideas from convex analysis and the notion of aggregate state price density, we

provide a solution to this problem, in the setting of both complete and incom-

plete markets (with random endowments). Then, we exploit this result and use

certain fixed-point techniques to provide an explicit characterization of a het-

erogeneous equilibrium in a complete market setting. Moreover, we construct

concrete examples of models admitting multiple (including infinitely many)

equilibria. By using Cramer’s large deviation theorem, we study the asymp-

totic behavior of endogenously determined in equilibrium prices of zero coupon

bonds. Lastly, we show that for incomplete markets, un-insurable future in-

come reduces the current consumption level, thus confirming the presence of

the precautionary savings motive in our model.

1. Introduction

Utility maximization in the setting of decision making under uncertainty is a cen-

tral topic both in mathematical finance and financial economics. The most preva-

lent class of preferences which has been extensively studied by numerous researchers

over many decades is the so-called class of HARA (hyperbolic absolute risk-aversion)

utility functions. HARA preferences are commonly split into two primary sub-

classes: CRRA (constant relative risk-aversion) preferences, which are also referred

to as power utility functions; and CARA (constant absolute risk-aversion) prefer-

ences, which are known as exponential utilities. This paper concentrates on the

latter type of preferences. Exponential preferences are widely used when dealing

with indifference pricing (see e.g. Frei and Schweizer (2008), Henderson (2009),

and Frei et. al. (2011)); incomplete markets with portfolio constraints (e.g. Svens-

son and Werner (1993)); and asset pricing in equilibrium (see Christensen et. al.

Date: January 9, 2012.

2000 Mathematics Subject Classification. Primary: 91B50 Secondary: 91B16 .

Key words and phrases. Exponential utility, Incomplete markets, Equilibrium, Heterogeneous

economies, Zero coupon bonds, Precautionary savings.1

2 R. Muraviev and M. V. Wuthrich

(2011a, b)). We investigate various aspects of the discrete-time exponential util-

ity maximization problem from consumption, where a significant and quite novel

dimension introduced here (when comparing to other papers dealing with CARA

preferences and consumption) is the constraint that feasible consumption policies

are not permitted to be negative.

An essential drawback challenging the classical paradigm of exponential utility

is the fact that negative consumption policies are included in the set of feasible

controls (see Caballero (1990), Christensen et. al. (2011a, b), and Svensson and

Werner (1993)). Economically, and intuitively, this is a vague assumption which

cannot be reasonably justified, but is rather made to allow a simplified frame-

work when solving analytically the corresponding problem. Namely, consumption

of goods over a life-cycle cannot be apparently measured in negative units, since it

models unrealistic situations when for instance one is capable of consuming a nega-

tive amount of bread. However, mathematically, exponential preferences often tend

to extinguish some complicated affects (e.g., the indifference price is independent

of the size of the wealth) caused by random shocks, and thus substantially ease the

associated problems. Even though some seminal works rely on this negativity con-

vention, many authors are aware of this problematic issue, for example, Caballero

(1990) writes:

”This paper specializes to this type of preferences (exponential) in spite of some

of its unpleasant features like the possibility of negative consumption. (Unfor-

tunately, explicitly imposing non-negativity constraints impedes finding a tractable

solution.)”

Nonetheless, it is demonstrated in the current paper that it makes a practical

sense to impose these non-negativity constraints: not only that it genuinely refines

the model, but it allows us finding the related solution in a closed form and employ

it then for a variety of economic applications.

We explain now in more detail the contributions of this work and the links it

has to some other related papers. First, by using some ideas from convex analysis,

we solve the constrained exponential utility maximization problem in a complete

market setting, with an arbitrary probability space. We show that the solution is

equal to the non-negative part of the solution associated with the unconstrained

problem (ass also Cox and Huang (1989), for a related problem in a continuous-time

framework). Next, we use the Kuhn-Tucker theorem to solve the preceding maxi-

mization problem in a setting of incomplete markets, with a finite probability space,

and express the solution in terms of the aggregate state price density introduced

by Malamud and Trubowitz (2007). In particular, we show that the transparent

relation between the constrained and unconstrained solution in complete markets

Exponential Utility with Non-Negative Consumption 3

setting is violated for incomplete markets. Then, we turn to analyzing heteroge-

neous equilibria in the framework of complete markets. There is a vast body of

literature studying equilibrium asset prices with heterogeneous investors (see e.g.

Mas-Colell (1986), Karatzas et. al. (1990, 1991), Dana (1993a, 1993b), Constan-

tinides and Duffie (1996), Malamud (2008)). Buhlmann (1980) was apparently the

first one to write down the formulas in a one period risk-exchange economy with

exponential preferences, and negative consumption. We express the equilibrium

state price density as a ’non-smooth’ sum over indicators depending on the agents’

characteristics and the total endowment of the economy. In affect, our finding is

one of very few examples when one can characterize in a closed-form manner the

associated state price density in heterogeneous economies. A further issue that we

sort out here is the existence of equilibrium (this is a quite standard chronological

order in equilibrium: first a characterization result, and then existence), by em-

ploying a fixed-point argument involving the excess-demand function in the spirit

of Mas-Collel et. al. (1995). Some other authors have studied various equilibrium

existence problems in a bit different settings: Dana’s papers (1993a, b) do not cover

multi-period models and Karatzas et. al. (1990, 1991) work in continuous-time.

Hence, we provide a self-contained proof that takes advantage of our underlying

exponential preferences. Next, we turn treating the issue of non-uniqueness of

equilibria. Non-uniqueness is usually anticipated in this type of models: it is re-

sulted by the fact that the Inada condition is not fulfilled, which in turn leads to the

associated excess-demand function not to satisfy the ’gross-substitution’ property

of Dana (1993a), that would guarantee uniqueness. However, we are not aware of

any papers (apart from Malamud and Trubowitz (2006)) that construct an explicit

example as ours of multiple equilibrium state price densities in a risk-exchange

economy. We concentrate then on studying the important question of long-run

behavior (see e.g. Wang (1996), Lengwiler (2005), and Malamud (2008)) of zero

coupon bonds, whose price is determined endogenously in equilibrium. Despite of

the non-uniqueness of the state price densities, modulo some multipliers, we show

that asymptotically these multipliers carry no impact. We conduct our analysis in

two settings: first we consider economies where agents differ with respect to their

risk-aversion and then economies where agents differ only with respect to their im-

patience rate. The main tool we use for the preceding problems is Cramer’s large

deviation theorem. Finally, we are concerned with the phenomenon of precaution-

ary savings (see Kimball (1990) for a comprehensive introduction) in incomplete

markets and exponential preferences. That is, we verify that un-insurable future in-

come forces an investor to save more (or equivalently, consume less) in the present.

We examine this phenomenon in rather enriched stochastic frameworks of incom-

plete markets (markets of type C, see Malamud and Trubowitz (2007)), whereas


most classical papers on this subject (see e.g. Dreze and Modigliani (1972) and

Miller (1976)) consider mainly riskless bonds.

The paper is organized as follows. In Section 2 we introduce the model. In

Section 3 we establish a solution to the exponential utility maximization problem

with a non-negativity constraint in both complete and incomplete markets. Section

4 analyzes the associated heterogeneous equilibrium for complete markets, and

Section 5 deals with the non-uniqueness of this equilibrium. In Section 6 we study

the long-run behavior of the equilibrium zero coupon bonds. Finally, in Section 7

we study the precautionary savings motive in incomplete markets.

2. Preliminaries

We consider a discrete-time market with a maturity date T . In Sections 2-5 and

Section 7, we set T ∈ N, and in Section 6, T = ∞. The uncertainty in our model is

captured by a probability space (Ω,F , P ) and a filtration F0 = ϕ,Ω ⊆ F1 ⊆ ... ⊆FT = F . Adaptedness and predictability of stochastic processes is always meant

with respect to the filtration (Fk)k=0,...,T , unless otherwise stated. We use the

notation R+ = [0,∞) and R++ = (0,∞). There is no-arbitrage in the market which

consists of n risky stocks: (Sjk)k=0,...,T , j = 1, ..., n and one riskless bond paying

an interest-rate rk, at each period k = 1, ..., T. Each price process (Sjk)k=0,...,T is

non-negative and adapted, and the interest rate process (rk)k=1,...,T is non-negative

and predictable. The economy is inhabited by N (types of) individuals, labeled by

i = 1, ..., N. Each agent i receives a random income (ϵik)k=0,...,T , which is assumed to

be non-negative and adapted. The preferences of each agent i are characterized by

an impatience rate ρi ≥ 0 and an exponential utility function ui(x) = − exp(−γix),defined on R+, for a given level of risk aversion γi > 0. The underlying utility

maximization problem from consumption of each agent i is formulated as

(2.1) sup(c0,...,cT )∈Bi

T∑k=0

−e−ρikE [exp (−γick)] ,

where (ck)k=0,...,T is a consumption stream lying in a certain set of budget con-

straints Bi. As described below, we will tackle problem (2.1) in both complete and

incomplete markets, under different assumptions on the model and the budget set

Bi. However, we will preserve the same notations as above for both settings without

mentioning it explicitly.

2.1. Complete markets. When dealing with complete markets, the probability

space is allowed to be infinite1. The budget set Bi in this setting consists of all

non-negative adapted processes ck that satisfy the equation

(2.2)T∑

k=0

E [ξkck] =T∑

k=0

E[ξkϵ

ik

].

1Therefore, the amount of securities completing the market might be infinite as well.


Here, (ξk)k=0,...,T is the unique positive and normalized (ξ0 = 1) state price density

(SPD) of the market, i.e.,

Sjkξk = E

[Sjk+1ξk+1|Fk

], j = 1, ..., n,

and

ξk = E [ξk+1(1 + rk+1)|Fk] ,

for k = 0, ..., T − 1. Let us stress that the completeness of the market implies

that stock prices are not directly involved in the corresponding utility maximiza-

tion problem (2.1). We introduce now the standard notion of equilibrium in the

framework of complete markets and risk exchange economies.

Definition 2.1. An equilibrium is a pair of processes (cik)k=0,...,T ;i=1,...,N and

(ξk)k=1,...,T such that:

(a) The process (ξk)k=1,...,T is a SPD and (cik)k=0,...,T is the optimal consumption

stream of each agent i (i.e., solving (2.1)).

(b) The market clearing condition holds:

(2.3)

N∑i=1

cik = ϵk :=

N∑i=1

ϵik,

for all k = 0, ..., T .

2.2. Incomplete markets. We assume that the probability space is finite. Thus

incompleteness of markets is modeled standardly by allowing infinitely many SPDs.

For each k = 0, ..., T , we denote by L2(Fk) the Hilbert space of all Fk−measurable

random variables, endowed with the inner product ⟨X,Y ⟩ = E[XY ], X,Y ∈L2(Fk). Each agent i selects a portfolio strategy (πj

k)k=0,...,T−1, j = 1, ..., n, and

(ϕk)k=0,...,T−1. Here, πjk and ϕk are Fk−measurable and denote the shares invested

in asset j and the riskless bond at period k, respectively. We set further πj−1 = 0,

j = 1, ..., n, ϕ−1 = 0, πjT = 0, j = 1, ..., n and ϕT = 0. The last two assumptions

formalize the convention that no trading is executed in the last period T . For each

k = 1, ..., T , we denote by

Lk =

n∑

j=1

πjk−1S

jk + πk−1(1 + rk)

∣∣∣∣ϕk−1, πjk−1 ∈ L2 (Fk−1) , j = 1, ..., n

the wealth space at time k, and remark that L2 (Fk−1) ⊆ Lk ⊆ L2 (Fk). By

Lemma 2.5 in Malamud and Trubowitz (2007) there exists a unique normalized

SPD (Mk)k=0,...,T such thatMk ∈ Lk, for all k = 1, ..., T, called the aggregate SPD.

For incomplete markets, the budget constraints are more sophisticated than the

single constraint (2.2) arising in a complete market setting. Namely, the budget set

Bi is composed of all non-negative adapted processes (ck)k=0,...,T of the form

(2.4) ck = ϵik +

n∑j=1

πjk−1S

jk + πk−1(1 + rk)−

n∑j=1

πjkS

jk − πk,


for all k = 0, ..., T, where (πjk)k=0,...,T−1, j = 1, ..., n, and (ϕk)k=0,...,T−1 is a port-

folio strategy. Notice that (2.4) can be rewritten as

(2.5) ck = ϵik +Wk − E

[Mk+1

MkWk+1

∣∣Gk

],

where Wk ∈ Lk, k = 1, ..., T, and W0 = WT+1 = 0, see Section 2 in Malamud and

Trubowitz (2007).

3. Optimal Consumption

The current section is devoted to the derivation of the optimal consumption pol-

icy of an individual solving the utility maximization problem (2.1) in both frame-

works of complete and incomplete markets.

3.1. Complete Markets. We investigate the utility maximization problem (2.1)

in the framework of complete markets, as specified in Subsection 2.1. We make

use of convex conjugates and other related ideas from convex analysis to derive

an explicit formula for the optimal consumption stream in this setup. As will be

shown below, there is a link between the utility maximization problem (2.1) and

the corresponding unconstrained (allowing negative consumption) version of this

problem. Hence, we first solve the latter problem.

Theorem 3.1. Assume that 0 <∑T

k=0E[ξkϵik] <∞ and −∞ < −

∑Tk=0E[ξk log ξk],

for all i = 1, ..., N. Consider the utility maximization problem

(3.1) sup(c0,...,cT )

T∑k=0

−e−ρikE [exp (−γick)] ,

where (ck)k=0,...,T is an adapted process that satisfies the equation (2.2). Then,

there exists a unique solution given by

(3.2) cik = cik(λ) :=1

γi

(log

(γi

λ

)− ρik − log (ξk)

),

for all k = 0, ..., T, where λ is a positive real number specified uniquely by the

equation

(3.3)1

γi

T∑k=0

E

[ξk

(log

(γi

λ


)]=

T∑k=0

E[ξkϵ

ik

].

Proof of Theorem 3.1. First, observe that the function fi : R++ → R, fi(λ) =∑Tk=0E

[ξk(log(γi

λ


)]is strictly monotone decreasing with limλ→0+

fi(λ) = ∞ and limλ→∞ fi(λ) = −∞. Therefore, there exists a unique solution λ

for equation (3.3). Next, consider the Legendre transform vi(y) : R++ → R of the

function −e−γix with the domain of definition R :

vi(y) = supx∈R

(−e−γix − xy

).


Denote Ii(y) :=1γi

log γi

y , and note that vi(y) = −e−γiIi(y) − Ii(y)y. Therefore, we

get

vi

(λeρikξk

)≥ −e−γick − λeρikξkck,

or equivalently

−e−ρike−γiIi(λeρikξk) ≥ −e−ρike−γick + λξk

(Ii

(λeρikξk

)− ck

),

for all Fk−measurable random variables ck. In particular, we have

−T∑

k=0

e−ρikE[e−γiIi(λe

ρikξk)]

≥ −T∑

k=0

e−ρikE[e−γick

]+ λ

T∑k=0

E[ξk

(Ii(λe

ρikξk)− ck

)],

for all adapted processes (ck)k=0,...,T such that∑T

k=0E [ξkck] < ∞. Thus, we

conclude the existence of an optimal consumption stream of the form (3.2), be-

cause the last term disappears due to (2.2). Uniqueness follows from the inequality

vi(y) > −e−γix − xy, which holds for all x = Ii(y). Next, we return to the treatment of the utility maximization problem with non-

negative consumption policies.

Theorem 3.2. Assume that∑T

k=0E[ξkϵ

ik

]> 0. Then, the constrained utility max-

imization problem (2.1) in the setting of a complete market admits a unique solution

given by

(3.4) cik =(cik(λ

∗))+

=1

γi

(log( γiλ∗


)+,

where the constant λ∗ is determined as the unique positive solution of the equation

(3.5)1

γi

T∑k=0

E

[ξk

(log( γiλ∗


)+]=

T∑k=0

E[ξkϵ

ik

].

We first prove the following auxiliary lemma.

Lemma 3.3. Denote Ii(y) = 1γi

(log(

γi

y

))+, and consider the function ψi :

R++ → R+ defined by

ψi(λ) =

T∑k=0

E[ξkIi(λe

ρikξk)].

Then, ψi(λ) is a decreasing continuous function of the following form: if ψi(b) >

0 for some b ∈ R++, then ψi(a) > ψi(b) for all 0 < a < b. Furthermore,

limλ→0+ ψi(λ) = ∞ and limλ→∞ ψi(λ) = 0.


Proof of Lemma 3.3. First observe that E [ξkIi(cξk)] <∞ for all c > 0 since

E [ξkIi(cξk)] =1

γiE

[ξk log

(γicξk

)11≤ γi

cξk

]< 1/c,

which follows from the fact that log t < t, for all t ≥ 1. This shows that ψi(λ)

is well defined for all λ ∈ R++. Next, we prove that ψi is continuous. Let λn

be a sequence such that λn ↑ λ, hence, limn→∞ ξkIi(λn eρikξk) = ξkIi(λeρikξk),

P−a.s, and thus the dominated convergence theorem implies that limn→∞ ψi(λn) =

ψi(λ). The same argument holds for a sequence λn ↓ λ. Now, we treat the limits.

Consider an arbitrary increasing sequence λn → ∞, then, ξkIi(λneρikξk) → 0 and

ξkIi(λ1eρikξk) ≥ ξkIi(λne

ρikξk), for all n. Therefore, the dominated convergence

theorem implies that limn→∞ ψi(λn) = 0. Next, pick an arbitrary sequence λn ↓ 0

and note that limn→∞ ξkIi(λneρikξk) = ∞, P−a.s. Therefore, Fatou’s lemma

implies that

lim infn→∞

E[ξkI(λne

ρkξk)]≥ E

[lim infn→∞

ξkI(λneρkξk)

]= ∞.

This shows that limλ→0+ ψi(λ) = ∞. Next, note that λ 7→ E

[ξkIi

(λeρikξk

)]is a

decreasing function for each k = 1, ..., T since Ii is decreasing. Therefore, ψi(λ)

is decreasing. Finally, assume in a contrary that ψi(b) > 0 and ψi(a) = ψi(b),

for some a < b. By the previous observations, it follows that E[ξkIi

(aeρikξk

)]=

E[ξkIi

(beρikξk

)], for each k = 0, ..., T . By definition, Ii is a strictly decreas-

ing function on the interval (0, γi], thus Ii(beρikξk

)< Ii

(aeρikξk

), on the set

beρikξk < γi. Since ψi(b) > 0, it follows that there exists some k ∈ 0, ..., Tsuch that P

[beρikξk < γi

]> 0. This is a contradiction, since beρikξk < γi ⊆

aeρikξk < γi, and thus

E[ξkIi

(aeρikξk

)1beρikξk<γi

]< E

[ξkIi

(beρikξk

)1beρikξk<γi

],

finishing the proof.

Proof of Theorem 3.2. First, observe that Lemma 3.3 yields the existence of a

unique solution to equation (3.5) denoted by λ∗ > 0. Next, consider the Legendre

transform of the function −e−γix with the domain of definition R+:

vi(y) = supx∈R+

(−e−γix − xy

),

for all y > 0. Note that vi(0) = 0 and recall that Ii(y) = 1γi

(log γi

y

)+. Ob-

serve that ∂∂x (ui(x) − xy) = γie

−γix − y. Therefore, if y < γi, then vi(y) =

ui

(1γi

log(

γi

y

))− y

γilog(

γi

y

). If y ≥ γi, then vi(y) = ui(0). Thus, we can rewrite

vi(y) = ui(Ii(y))− Ii(y)y. The rest of the proof is identical to the one of Theorem

3.1 and thus omitted.


For a sufficiently large level of endowments, and under some boundness assump-

tions on the SPD, the optimal consumption stream is strictly positive, and hence,

in particular, coincides with the optimal consumption streams corresponding to the

setting of unconstrained exponential utility.

Corollary 3.4. Assume that ξk < C and ϵik >1γi

(log(

Cξk

)+ ρi (T − k)

), P−a.s.,

for all k = 0, ..., T, and some constant C > 0. Then, the optimal consumption

stream of the i−th agent (see (3.4)) is strictly positive and given by

(3.6) cik =1

γi

(∑Tl=0E

[ξl(γiϵ

il + log ξl + ρil

)]∑Tl=0E [ξl]

− ρik − log ξk

)> 0,

for all k = 0, ..., T .

Proof of Corollary 3.4. Fix z = γi

CeρiTand observe that γi

zeρikξk= C

ξkeρi(T−k) > 1,

for all k = 0, ..., T . Therefore, by applying the same argument as in the proof of

Lemma 3.3, one concludes that ψi(λ) is a strictly decreasing function on the interval

(0, z). Moreover, note that

ψi(z) =1

γi

T∑k=0

E

[ξk

(log

(C

ξk

)+ ρi (T − k)

)]<

T∑k=0

E[ϵikξk

],

by assumption. Therefore, a solution λ∗ to equation (3.5) is attained for some

λ∗ ∈ (0, z), and thus γi

λ∗eρikξk> γi

zeρikξk> 1. In view of (3.4), we obtain that the

optimal consumption stream admits the form

cik =1

γilog

(γi

λ∗eρikξk

)> 0.

By plugging this back into the budget constraints equation (3.5), one solves this

equation explicitly and verifies the validity of (3.6).

3.2. Incomplete Markets. In the current section we deal with incomplete mar-

kets (see Subsection 2.2). In this setting, we provide an explicit construction of

the optimal consumption for the utility maximization problem (2.1). The methods

employed here rely on the Kuhn-Tucker theorem and the notion of aggregate SPD

(defined in Subsection 2.2) introduced by Malamud and Trubowitz (2007).

Theorem 3.5. For an investor solving the utility maximization problem (2.1) in

an incomplete market, the optimal consumption stream (ck)k=0,...,T is uniquely de-

termined through the following scheme:

(3.7) P kL

[e−ρikγi exp(−γick) + λk

e−ρi(k−1)γi exp(−γick−1) + λk−1

]=

Mk

Mk−1,

for all k = 1, ..., T, where (λk)k=0,...,T is a non-negative adapted process satisfying

λk ck = 0,


for all k = 0, ..., T. Here, (Mk)k=0,...,T and P kL stand for the aggregate SPD and the

orthogonal projection on the space Lk, respectively (see Subsection 2.2).

Proof of Theorem 3.5. Consider the function

li(π, ϕ, λ0, ..., λT ) =

T∑k=0

e−ρikE

− exp

−γi

ϵik +n∑

j=1

πjk−1S

jk + ϕk−1(1 + rk)−

n∑j=1

πjkS

jk − ϕk

−

T∑k=0

E

λkϵik +

n∑j=1

πjk−1S

jk + ϕk−1(1 + rk)−

n∑j=1

πjkS

jk − ϕk

,where ϕ = (ϕk)k=0,...,T−1, π = (πj

k)k=0,...T−1, j = 1, ..., n, is a portfolio strategy

and λk ∈ L2 (Fk), k = 0, ..., T . Since our probability space is finite, we can employ

the Kuhn-Tucker theorem. Namely, by differentiating the function li with respect

to the partial derivatives ϕk, πjk, k = 0, ..., T , j = 1, ..., n, and equalizing the re-

sulting expression to 0, we get that the optimal consumption stream (ck)k=0,...,T is

determined by requiring that the process(e−ρikγi exp (−γick) + λk

)k=0,...,T

is a SPD and λk ck = 0, k = 0, ..., T , for some non-negative adapted process

(λk)k=0,...,T . Finally, Lemma 2.5 in Malamud and Trubowitz (2007) yields the

validity of (3.7). Uniqueness follows from strict concavity.

Remark 3.1. The Kuhn-Tucker theorem could not be applied directly for complete

markets, since we allowed for arbitrary probability spaces.

3.2.1. One-period incomplete markets. We fix T = 1 and consider a market of

type C, introduced by Malamud and Trubowitz (2007). That is, we assume that

L1 = L2 (H1) and thus PL1 [·] = E

[·∣∣H1

], where H1 is a sigma-algebra satisfying

F0 ⊆ H1 ⊆ F1. Let c0 and c1 denote the optimal consumption stream (we drop the

index i). As above, λ0 and λ1 stand for the multipliers. Recall that by (2.5), we

have c1 = ϵi1 + W1, where W1 ∈ L1. Next, by Theorem 3.5 we get

exp(−γiW1

)=M1 (γi exp(−γic0) + λ0)− E

[λ1∣∣H1

]e−ρiγiE

[exp

(−γiϵi1

) ∣∣H1

] ,

hence

(3.8) c1 = ϵi1 + W1 =

1

γilog

eγiϵi1E[e−γiϵ

i1

∣∣H1

](exp (ρi − γic0) + λ0

eρiγi

)M1 − eρi

γiE[λ1∣∣H1

] .


Now, denote λ = exp (ρi − γic0) + λ0eρi

γi, or equivalently c0 = 1

γilog(

γi

λγie−ρi−λ0

).

Recall that λ0c0 = 0, λ0 ≥ 0 and c0 ≥ 0. Therefore, we get

(3.9) c0 =1

γi

(log

(1

λ

)+ ρi

)+

.

Now, we claim that

(3.10) c1 =1

γilog

eγiϵi1E[e−γiϵ

i1

∣∣H1

]λM1

1essinf [eγiϵ

i1 |H1]E[e−γiϵ

i1 |H1]>λM1

+

1

γilog

(eγiϵ

i1

essinf [eγiϵi1 |H1]

)1essinf [eγiϵ

i1 |H1]E[e−γiϵ

i1 |H1]≤λM1

,

where essinf [eγiϵi1 |H1] is the essential infimum of the random variable eγiϵ

i1 condi-

tioned on the sigma-algebraH1. To this end, assume first that eγiϵi1E[e−γiϵ

i1

∣∣H1

]>

λM1. Then, identity (3.8) yields c1 > 0 and thus λ1 = 0, since λ1c1 = 0. Thereby,

we have

λ1 = Λ1eγiϵ

i1E

[e−γiϵ

i1

∣∣H1

]≤λM1

,for some F1−measurable non-negative random variable Λ. Now, the condition

λ1c1 = 0 can be rewritten as

(3.11) Λ1eγiϵ

i1E

[e−γiϵ

i1

∣∣H1

]≤λM1

×

log

eγiϵi1E[e−γiϵ

i1

∣∣H1

]λM1 − eρi

γiE

[Λ1

eγiϵi1E

[e−γiϵ

i1

∣∣H1

]≤λM1

∣∣H1

] = 0.

Hence, if

essinf [eγiϵi1 |H1]E

[e−γiϵ

i1

∣∣H1

]> λM1,

then c1 = 1γi

log

(eγiϵ

i1E

[e−γiϵ

i1

∣∣H1

]λM1

). On the other hand, we claim that if


[e−γiϵ

i1

∣∣H1

]≤ λM1,

then

log

essinf[eγiϵ

i1

∣∣H1

]E[e−γiϵ

i1

∣∣H1

]λM1 − eρi

γ E

[Λ1

eγiϵi1E

[e−γiϵ

i1

∣∣H1

]≤λM1

∣∣H1

] = 0.

Assume that it is not the case. It follows that

log

eγiϵi1E[e−γiϵ

i1

∣∣H1

]λM1 − eρi

γiE

[Λ1

eγiϵi1E

[e−γiϵ

i1

∣∣H1

]≤λM1

∣∣H1

] > 0,


and thus by (3.11) we get

Λ1eγiϵ

i1E

[e−γiϵ

i1

∣∣H1

]≤λM1

= 0.

By substituting it back, we get


[e−γiϵ

i1

∣∣H1

]> λM1,

and this is a contradiction, proving the identity (3.10). Finally, let us remark that

λ is derived from the equation c0 + E [M1c1] = ϵi0 + E[M1ϵ

i1

].

Remark 3.2. For complete markets, the optimal consumption stream in the con-

strained case (non-negative consumption) is given in the form of a call-option on

the unconstrained (possibly negative) optimal consumption stream with a strike be-

ing equal to zero (see (3.4)). In contrast to this, for incomplete markets this relation

is no longer valid. Indeed, if we allow for negative consumption, one can show that

the associated unconstrained optimal consumption stream is given by

c′1 =1

γilog

eγiϵi1E[e−γiϵ

i1

∣∣H1

]λ′M1

,

and

c′0 =1

γi

(ρi + log

1

λ′

),

where the constant λ′ is determined by the equation c′0+E [M1c′1] = ϵi0+E

[M1ϵ

i1

].

Thus by comparing it to (3.9) and (3.10), we conclude that the structure of the

solution presented in the complete setting disappears here.

4. Equilibrium

We adapt the notion of a complete market equilibrium as introduced in Subsec-

tion 2.1 (see Definition 2.1). In the current section we show that there exists an

equilibrium under the assumption that all endowments are positive. Moreover, we

describe the set of all feasible equilibrium SPDs and the associated optimal con-

sumption policies. For this purpose we introduce the following quantities. For each

vector (λ1, ..., λN ) ∈ RN++, we define

(4.1) βi(k) = β(λi)i (k) =

γiλieρik

,

for all i = 1, ..., N and all k = 0, ..., T . For a fixed k = 0, ..., T, let i1(k), ..., iN (k)

denote the order statistics of β1(k), ..., βN (k), that is, i1(k), ..., iN (k) = 1, ..., Nand βi1(k)(k) ≤ ... ≤ βiN (k)(k). We set βi0(k)(k) = 0, for all k = 0, ..., T. With the

preceding notations, we denote

(4.2) ηj(k) = η(λ1,...,λN )j (k) =

N∑l=j+1

log(βil(k)(k)

)− log

(βij(k)(k)

)γil(k)

≥ 0.


Note that η0(k) = +∞ and ηN (k) = 0, for all k = 0, ..., T. At last, we introduce a

candidate for the equilibrium SPD

(4.3) ξk (λ1, ..., λN ) =

N∑j=1

N∏l=j

β(∑N

m=j(γil(k)/γim(k)))−1

il(k)exp

(− ϵk∑N

l=j 1/γil(k)

)1ηj(k)≤ϵk<ηj−1(k),

for all k = 0, ..., T. Recall that Ii(y) =(log γi

y

)+.

Theorem 4.1. Assume that ϵik > 0 for each period k and each agent i, P−a.s.

Then, every equilibrium SPD is given by (ξk (λ∗1, ..., λ

∗N ))k=0,...,T , where λ

∗1, ..., λ

∗N ∈

R++ are constants that solve the following system of equations

(4.4)T∑

k=0

E[ξk(λ1, ..., λN )Ii(λie

ρikξk(λ1, ..., λN ))]=

T∑k=0

E[ξk (λ1, λ2, ..., λN ) ϵik

],

for i = 1, ..., N . The optimal consumption stream of agent i is given by

(4.5) cik = Ii(λ∗i e

ρikξk(λ∗1, ..., λ

∗N )),

for all k = 0, ..., T.

Remark. Theorem 4.1 constitutes a preliminary tool for proving the existence

of an equilibrium (see Theorem 4.3) and yields a closed-form characterization of all

feasible equilibrium state price densities.

Proof of Theorem 4.1. Let (cik)k=0,...,T denote the optimal consumption stream

of agent i. Recall that, by (3.4), we have that cik = Ii(λ∗i e

ρikξk)for some λ∗i > 0.

Plugging this into the market clearing condition (2.3), we obtain that the following

holds in equilibrium:

(4.6)N∑i=1

Ii(λ∗i e

ρikξk)= ϵk,

for all k = 0, ..., T . Here, λ∗1, ..., λ∗N are constants that will be derived from the

budget constraints in the sequel. Using the explicit form of Ii, this is equivalent to

N∑i=1

log

( γiγi1λ∗

i eρikξk>γi + λ∗i e

ρikξk1λ∗i e

ρikξk≤γi

)1/γi = ϵk,

a further transformation yields,

N∏i=1

(γi1ξk>βi(k) + λ∗i e

ρikξk1ξk≤βi(k))1/γi

=N∏i=1

γ1/γi

i exp(−ϵk),


where βi(k) = β(λ∗

i )i (k) was defined in (4.1). This is equivalent to

(4.7)N∑j=1

Y(k)j 1

βij−1(k)(k)<ξk≤βij(k)(k) +

N∏i=1

γ1/γi

i 1ξk>βiN (k)(k) =

N∏i=1

γ1/γi

i exp (−ϵk) ,

where

Y(k)j =

j−1∏l=1

γ1/γil(k)

il(k)

N∏l=j

(λ∗il(k))1/γil(k) exp

k N∑l=j

ρil(k)

γil(k)

ξk∑Nl=j

1γil(k) .

The strict-positivity assumption on the endowments implies that ϵk > 0, P -a.s, and

thus ξk ≤ βiN (k) holds P−a.s. Next, for each k = 0, ..., T , we have that

Y(k)i 1

βij−1(k)(k)<ξk≤βij(k)(k) =

N∏i=1

γ1/γi

i exp (−ϵk)1βij−1(k)(k)<ξk≤βij(k)(k)

,which implies that the following holds on each set

βij−1(k)(k) < ξk ≤ βij(k)(k)

:

ξk =

N∏l=j

(βil(k)(k))(∑N

m=j γil(k)/γim(k))−1

exp

(− ϵk∑N

l=j 1/γil(k)

).

In particular, one checks that ξk ≤ βij(k)(k) is equivalent to ϵk ≥ ηj(k) and

βij−1(k)(k) < ξk is equivalent to ϵk < ηj−1(k), where, ηj(k) = ηλ∗1 ,...,λ

∗N

j (k) is given

in (4.2). Now, one revises the above identity in terms of λ∗1, ..., λ∗N and concludes

that every equilibrium SPD is of the form (4.3) for some λ∗1, ..., λ∗N . At last, observe

that due to the budget constraints (2.2), in equilibrium, the constants λ∗1, ..., λ∗N

solve the system of equations (4.4). This concludes the proof.

The following result is essential for proving an existence of equilibrium.

Lemma 4.2. For each i = 1, ..., N , consider the excess demand function gi :

RN++ → R defined by

(4.8) gi(λ1, ..., λN ) =

λ−1i

T∑k=0

(E[ξk(λ

−11 , ..., λ−1

N )Ii(λ−1i eρikξk(λ

−11 , ..., λ−1

N ))]−E

[ξk(λ

−11 , ..., λ−1

N )ϵik])

,

where ξk(λ1, ..., λN ) is defined in (4.3). Then, the following properties are satisfied:

(1) Each function gi(λ1, ..., λN ) is a homogeneous function of degree 0.

(2) We have

N∑i=1

λigi(λ1, ..., λN ) = 0,


for all (λ1, ..., λN ) ∈ RN++.

(3) Each function gi(λ1, ..., λN ) is continuous.

(4) (i) The following limit holds

limλi→0

gi(λ1, ..., λN ) = −∞.

(ii) Each function gi(λ1, ..., λN ) is bounded from above.

Proof of Lemma 4.2. (1) The assertion follows from the identity

ξk((cλ1)−1, ..., (cλN )−1) = cξk(λ

−11 , ..., λ−1

N ),

for all c > 0, which holds due to (4.3) and the observations that η(λ1,...,λN )j =

η(cλ1,...,cλN )j .

(2) By the construction of the SPDs in the proof of Theorem 4.1 (see (4.6)), it follows

that cik = Ii(λ−1i eρikξk

(λ−11 , ..., λ−1

N

))satisfies the market clearing condition (2.3),

for all (λ−11 , ..., λ−1

N ) ∈ RN++. By changing the order of summation, the claim

becomes equivalent to

T∑k=0

E

[ξk(λ

−11 , ..., λ−1

N )

(N∑i=1

Ii(λ−1i eρikξk

(λ−11 , ..., λ−1

N

))−

N∑i=1

ϵik

)]= 0,

which follows from the latter observation concerning the market clearing condition.

(3) Consider some x = (x1, ..., xN ) ∈ RN++ and a sequence xm = (xm1 , ..., x

mN ) ∈ RN

++

such that xm → x. One checks that the random function ξk(λ1, ..., λN ) : RN++ →

R++ is P−a.s continuous, and hence limm→∞ ξk(xm) = ξk(x), and consequently

limm→∞ ξk(xm) Ii(xmi e

ρikξk(xm)) = ξk(x)Ii(xieρikξk(x)). Therefore, it suffices to

show uniform integrability in order to obtain L1−convergence. Thus, it is enough

to check that

supm≥1

E[(ξk(xm)Ii(x

mi e

ρikξk(xm)))p]

<∞ , supm≥1

E [(ξk(xm)ϵk)p] <∞,

for some p > 1. Since xi > 0, for all i = 1, ..., N , we can assume that there

exists ε > 0 such that xmi > ε, for all i = 1, ..., N and all m. Therefore, we

can estimate ξk(xm) ≤ Ke−Mϵk , for all m and some constants K,M > 0. Hence,

supm≥1E [(ξk(xm)ϵk)p] ≤ KpE

[e−Mpϵkϵpk

]<(

KMe

)p. Next, one checks that

supm≥1

E[(ξk(xm)Ii(x

mi e

ρikξk(xm)))p]

≤ 1

γpisupm≥1

E

[(γi

xmi eρik

− ξk(xm)

)p

1 γi

xmi

eρik−ξk(xm)≥0

]≤ 1

γpisupm≥1

E

[(γi

xmi eρik

+Ke−Mϵk

)p]<

1

γpi

( γiεeρik

+K)p,

where the first inequality follows from the fact that 1+ log t ≤ t, for all t ≥ 1. This

shows that g(λ1, ..., λN ) is continuous.


(4) (i) For each i = 1, ..., N and k = 0, ..., T , consider the random function (de-

pending on ω ∈ Ω) f i(k, λ1, ..., λN ) : RN++ → R given by

f i(k, λ1, ..., λN ) = λiξk(λ1, ..., λN )(Ii(λie

ρikξk(λ1, ..., λN ))− ϵik

).

The claim is equivalent to showing that limλi→∞E[f i(k, λ1, ..., λN )

]= −∞. First,

we show that limλi→∞ ξk(λ1, ..., λN )(Ii(λie


)< 0, P−a.s.

Assume that λi is sufficiently large so that i = i1(k) and λi = λi1(k) > maxλ1, ...,λi−1, λi+1, ..., λN. One checks that limλi→∞ η1(k) = ∞. This implies that

(4.9) limλi→∞

ξk (λ1, ..., λN )

=

N∏l=j

β(∑N


il(k)exp

(− ϵk∑N

l=j 1/γil(k)

)1η2(k)≤ϵk<+∞+

N∑j=3

N∏l=j

β(∑N


il(k)exp

(− ϵk∑N

l=j 1/γil(k)

)1ηj(k)≤ϵk<ηj−1(k) > 0,

P−a.s. Next, we claim that limλi→∞ Ii(λie

ρikξk(λ1, ..., λN ))= 0. Indeed, recall

that

Ii(λie

ρikξk(λ1, ..., λN ))=

1

γilog

(γi

λieρikξk(λ1, ..., λN )

)1

γi

eρikξk(λ1,...,λN )≥λi

,and note that, by (4.9),

1γi

eρikξk(λ1,...,λN )≥λi

= 0

is satisfied identically P -a.s, for sufficiently large λi. This yields that

limλi→∞

ξk(λ1, ..., λN )(Ii(λie


)< 0.

Now, consider an arbitrary sequence (an)∞n=1 such that an → ∞, and denote

bn = (λ1, ..., λi−1, an, λi+1, ..., λN ) ∈ RN++. The same arguments as in (3) can be

applied to show that the sequenceξk(bn)

(Ii(ane

ρikξk(bn))− ϵik

)∞n=1

is uniformly

integrable. Therefore,

limλi→∞

E[f i(k, λ1, ..., λN )

]= lim

λi→∞λiE

[lim

λi→∞ξk(λ1, ..., λN )

(Ii(λie


)]= −∞.

This accomplishes the proof of part (i).

(ii) This follows by the fact that the function h : R++ → R given by h(x) =

x((log 1/x)+ −a

)is bounded from above, for any fixed a > 0.

The next statement establishes the existence of an equilibrium.


Theorem 4.3. Assume that ϵik > 0 for each period k and each agent i, P−a.s.

Then, there exists an equilibrium. Furthermore, a process (ξ′k)k=0,...,T is an equilib-

rium SPD if and only if ξ′k = ξk(λ∗1, ..., λ

∗N ) for some (λ∗1, ..., λ

∗N ) ∈ RN

++ that solves

the system of equations (4.4). The optimal consumption stream is given by (4.5).

Proof of Theorem 4.3. By Theorem 4.1 it is sufficient to prove that ξk(λ∗1, ...,

λ∗N ) and Ii(λ∗i e

ρikξk(λ∗1, ..., λ

∗N )) is an equilibrium. Therefore, it is left to check that

the system of equations (4.4) has a solution, or equivalently, the system of equa-

tions gi(λ1, ..., λN ) = 0, for i = 1, ..., N (see (4.8)) has a solution. The existence

of a solution follows by properties (1)-(4) in Lemma 4.2 and fixed point arguments

similar to those appearing in Theorem 17.C.1 in Mas-Colell et al. (1995).

As will be shown in the next section, the equilibrium SPD is, in general, not unique.

Nevertheless, when the economy is homogeneous, the utility maximization problem

is similar to the one in the non-constrained setting for consumption, and in partic-

ular assures uniqueness.

Example: Homogeneous Economy. In an economy populated only by agents

of type i that hold strictly positive endowment streams (ϵik)k=0,...,T , there exists a

unique (normalized) equilibrium and the corresponding homogeneous SPD process

ξikk=0,...,T is given by

ξik = eγi(ϵi0−ϵik)−ρik,

for all k = 1, ..., T. The optimal consumptions obviously coincide with the endow-

ments: cik = ϵik, for all k = 0, ..., T .

5. Non-uniqueness of equilibrium

In the present section we show that the equilibrium established in Section 4 is

generally non-unique. First, we construct a specific example when endowments are

positive, and then explore the case when endowments may vanish with a positive

probability and provide some examples.

5.1. Non-uniqueness with positive endowments. It is evident that the sys-

tem of equations (4.4) is related to the uniqueness of the equilibrium. The sys-

tem of equations (4.4) is somewhat cumbersome in certain aspects: the functions

ξk(λ1, ..., λN ) are not differentiable with respect to each variable λi; it might happen

that (4.4) has infinitely many different solutions but the equilibrium is still unique;

the property of “gross substitution” (see Definition 3.1 in Dana (1993b)), which

would be a sufficient condition for the uniqueness of the equilibrium, is not neces-

sarily satisfied. The next example demonstrates the existence of multiple equilibria.


Example: Non-Uniqueness of Equilibrium. We assume a one period mar-

ket with F0 = F1 = Ω, ∅. Consider two agents i = 1, 2 represented by u1(x) =

u2(x) = −e−x and ρ1 = ρ2 = 0. The agents hold different endowments ϵ10, ϵ11 and

ϵ20, ϵ21 respectively. Let ϵ0 and ϵ1 denote the aggregate endowments. By Theorem

4.1, every equilibrium state price density is of the form

ξ1(x, y) =1

minx, y1ϵ1<log

maxx,yminx,y e

−ϵ1 +1

√xy

1log maxx,yminx,y ≤ϵ1

e−ϵ1/2,

where x and y are to be determined by the budget constraints. One can check that

it is possible to rewrite it as

ξ1(x, y) =

1

eϵ11x if 0 < x < y

eϵ1 ,

1√yeϵ1/2

1√x

if ye−ϵ1 ≤ x ≤ yeϵ1 ,

1yeϵ1 if x > yeϵ1 .

The positive arguments x, y solve equations (4.4) which take the form:

(1) log(1/y)1y≤1 + ξ1(x, y) log

(1

yξ1(x, y)

)1xξ1(x,y)≤1 = ϵ10 + ϵ11ξ1(x, y),

(2) log(1/x)1x≤1 + ξ1(x, y) log

(1

xξ1(x, y)

)1xξ1(x,y)≤1 = ϵ20 + ϵ21ξ1(x, y).

Let us note that we work with a normalized state price density, i.e., ξ0 = 1. We

denote

h(x, y) = ξ1(x, y)

(log

(1

yξ1(x, y)

)1xξ1(x,y)≤1 − ϵ11

),

and

g(x, y) = ξ1(x, y)

(log

(1

xξ1(x, y)

)1xξ1(x,y)≤1 − ϵ21

).

Observe that

h(x, y) =

−ϵ11 1

eϵ11x if 0 < x < y

eϵ1 ,

1√yeϵ1/2

1√x

(log(√

xeϵ1/2

√y

)− ϵ11

)if ye−ϵ1 ≤ x ≤ yeϵ1 ,

1yeϵ1

(ϵ1 − ϵ11

)if x > yeϵ1 .

and that

g(x, y) =

−ϵ21 1

eϵ11x if 0 < x < y

eϵ1 ,

1√ye−1/2ϵ1

1√x

(log(√

ye1/2ϵ1√x

)− ϵ21

)if ye−ϵ1 ≤ x ≤ yeϵ1 ,

1yeϵ1

(ϵ1 − ϵ21

)if x > yeϵ1 .

Hence, equations (1) and (2) can be rewritten as

(1′) log(1/y)1y≤1 + h(x, y) = ϵ10,

(2′) log(1/x)1x≤1 + g(x, y) = ϵ20.

We are going to define the endowments in a particular way that will yield two

distinct solutions to (1′) and (2′). More precisely, we are going to construct two


solutions (x1, y1) and (x2, y2) such that yl < 1 and 0 < xl < yle−ϵ1 , for l = 1, 2. Set

ϵ11 = ϵ21, e2ϵ11−1 > ϵ11 and ϵ11 < e−1. We start by treating equation (2′). Consider

the function h(x) = log(1/x)− ϵ11eϵ1

1x on the interval [0, ye−ϵ1 ], for arbitrary y > 0.

Note that h′(x) = 1x2

ϵ11eϵ1 − 1

x , and hence h′(x) > 0 if x <ϵ11eϵ1 , and h′(x) < 0 if

x >ϵ11eϵ1 , which implies that xmax =

ϵ11eϵ1 is a maximum of h. Furthermore, yl (to

be determined explicitly in the sequel) will satisfy ϵ11 < yl, and this will guarantee

that the maximum is indeed in the domain of definition of h, i.e.,ϵ11eϵ1 ∈ [0, yle

−ϵ1 ].

Next, note that h(ϵ11eϵ1 ) = log

(eϵ1

ϵ11

)− 1 > 0 due to the assumption e2ϵ

11−1 > ϵ11.

Now let δ > 0 be some small quantity to be determined below. One can pick ϵ20such that the equation h(x) = ϵ20 has exactly two solutions x1 and x2 (see Figure

1) in the interval [ϵ11eϵ1 − δ, ϵ11

eϵ1 + δ]. Now, equation (1′) has two solutions denoted by

y1 and y2 (depending on ϵ10) corresponding to x1 and x2 that are given by

yl = exp

(−ϵ10 −

ϵ11eϵ1

1

xl

),

for l = 1, 2. Obviously, y1, y2 < 1. It is left to check that maxxl, ϵ21eϵ1 < yle

−ϵ1 ,

for l = 1, 2, that is,

max

xl,

ϵ11eϵ1

< exp

(−ϵ1 − ϵ10 −

ϵ11eϵ1

1

xl

).

Since xl ∈ [ϵ21eϵ1 − δ,

ϵ21eϵ1 + δ], it suffices to verify that

eϵ1(ϵ11eϵ1

+ δ

)< exp

(−ϵ10 −

ϵ11eϵ1

1ϵ11eϵ1 − δ

),

for an appropriate choice of δ > 0 and ϵ10. This inequality is equivalent to

ϵ11 + δeϵ1 < exp

(−ϵ10 −

ϵ11ϵ11 − δeϵ1

).

By continuity, it suffices to prove this inequality for δ = 0 and ϵ10 = 0, which

becomes ϵ11e < 1, and follows from the assumptions imposed on ϵ11.

5.2. Non-uniqueness with vanishing endowments. In Theorems 4.1 and 4.3

we have assumed that P (ϵik > 0) = 1, for all k = 1, ..., T and all i = 1, ..., N . This

assumption was crucial for proving that every equilibrium SPD is of the form (4.3).

It turns out that once this assumption is relaxed, there exist necessarily infinitely

many equilibria all of the same canonical form.

Theorem 5.1. Assume that P (ϵk = 0) > 0, for all k = 0, ..., T and P (∪Tk=0ϵik >

0) > 0, for all i = 1, ..., N . Then, in the setting of Section 4, there exist infinitely

many equilibria. Every equilibrium SPD (ξk)k=0,...,T is of the form

(5.1) ξk(λ1, ..., λN ) = ξk(λ1, ..., λN )1ϵk =0 +Xk1ϵk=0,


Figure 1. Multiple equilibria

for all k = 1, ..., T , where ξk(λ1, ..., λN ) is given by (4.3) and Xk is some non-

negative Fk-measurable random variable. The constants λ1, ..., λN are determined

by the budget constraints

T∑k=0

(E[ξk(λ1, ..., λN ))Ii

(λie

ρik ξk(λ1, ..., λN ))]

− E[ξk(λ1, ..., λN )ϵik

])= 0

for i = 1, ..., N .

Proof of Theorem 5.1. The proof is identical to the proofs of Theorem 4.1

and Theorem 4.3 apart from a slight modification as follows. Consider equa-

tion (4.7) and note that in the current context this equation admits the form

1ξk>βiN (k)(k)= 1 on the set ϵk = 0, which implies that ξk is of the form

(5.1). The rest follows by similar arguments to those found in Section 4.

We illustrate the above phenomenon in the following elementary example.

Example: Infinitely Many Equilibria. Let (Ω, F1, P ) be a probability space

where Ω = ω1, ω2, P (ω1), P (ω2) > 0, F0 = Ω, ∅ and F1 = 2Ω. Consider

a one period homogeneous economy with an individual represented by the utility

function u(x) = −e−x and ρ = 0. The endowments of the agent are denoted by

ϵ0 and ϵ1. For the sake of transparency, we analyze the following two simple cases

directly by using the definition of equilibrium rather than by using Theorem 5.1.

(i) Let ϵ0 = 0 and ϵ1 be an arbitrary F1−measurable positive random variable.

Theorem 3.2 implies that the optimal consumption policies c0 and c1 are given by


c0 = − log(1λ>1 + λ1λ≤1

)and c1 = − log

(1λξ1>1 + λξ11λξ1≤1

). The mar-

ket clearing condition c0 = 0 and c1 = ϵ1 implies that ξ1 = e−ϵ1

λ is an equilibrium

SPD, for all λ ≥ 1. Note that the budget constraints of the type (2.2) are automat-

ically satisfied due to the fact that the market clears and due to the homogeneity

of the economy. We stress out that for the corresponding unconstrained problem

sup(c0,c1)

−e−c0 − E[e−c1 ],

where c0 ∈ R and c1 ∈ L2 (F1) are such that

c0 + E[ξ1c1] = ϵ0 + E[ξ1ϵ1],

there exists a unique equilibrium corresponding to λ = 1, that is, ξ1 = e−ϵ1 .

(ii) Let ϵ0 > 0 be arbitrary, ϵ1(ω1) > 0 and ϵ1(ω2) = 0. Then, by Theorem 3.2 we

obtain that c0 = log(1/λ) = ϵ0 and c1 = maxlog( 1λξ1

), 0 = ϵ1. It follows that

λ = e−ϵ0 , and that there are infinitely many equilibrium state price densities of

the form ξ1(y) = eϵ0−ϵ11ϵ1 =0 + y1ϵ1=0, for every y > eϵ0 . As in (i), the market

clearing condition is redundant due to the homogeneity of the economy.

6. Long-run yields of zero coupon bonds

In the current section we work with an infinite time horizon T = ∞. We em-

phasize that all our results in the context of equilibrium hold in this setting due

to a specific choice of the aggregate endowment process (see (6.3)). Denote by

ξk = ξk(λ1, ..., λN ), k ∈ N the equilibrium SPD (see (4.3)). The equilibrium price

at time 0 of a zero coupon bond maturing at a period t ∈ N is defined by

(6.1) Bt = E [ξt] .

Based mainly on Cramer’s large deviation theorem, we study the asymptotic be-

havior (as t→ ∞) of the yield of this bond, defined by

(6.2) − logBt

t.

Let us stress that we omit the weights λ1, ..., λN (in most cases) since they will

not be crucial when dealing with the corresponding limits. As will be shown in the

sequel, the long-run behavior of this bond will depend on the agents’ characteris-

tics. Throughout this section we assume that the total endowment process in the

economy is a random walk with a drift, i.e.,

(6.3) ϵk =k∑

j=1

Xj ,

for all k ∈ N, where X0 = 0 and X1, X2, ... are non-negative independent identically

distributed random variables with a finite mean E[X1] > 0. This specification of

endowments is quite natural in this setting, due to the exponential preferences of


the investors. We consider two categories of economies: agents with heterogeneous

risk-aversion and agents with heterogeneous impatience rates.

6.1. Heterogeneous risk-aversion. Consider an economy where agents differ

only with respect to the risk-aversion, that is, γ1 < ... < γN , ϵik = ϵk/N, i = 1, ..., N ,

k ∈ N; ρ1 = ... = ρN = ρ. Recall (4.1) and note that in the current setting, equa-

tions (4.4) can be rewritten as

(6.4)∞∑k=0

E[ξk (log (βi(k))− log ξk)

+]=γiN

∞∑k=0

E [ξkϵk] .

Now, let i, j ∈ 1, ..., N be arbitrary. Notice that the homogeneity of the impa-

tience rate among agents implies that either βi(k) ≤ βj(k), or βi(k) ≥ βj(k), for

all k ∈ N. Therefore, by (6.4), we conclude that β1(k) < ... < βN (k), for all k ∈ Nand thus il(k) = l, for all l = 1, ..., N. Hence, we get (see (4.2))

ηj = log

((λjγj

)1/γj+1+...+1/γN(γj+1

λj+1

)1/γj+1

...

(γNλN

)1/γN),

for all j = 1, ..., N , and (see (4.3))

ξk =N∑j=1

N∏l=j

(γl

λleρk

)(∑Nm=j

γlγm

)−1

exp

(− ϵk∑N

l=j 1/γl

)1ηj≤ϵk<ηj−1,

for all k ∈ N.

Theorem 6.1. In heterogeneous risk-aversion economies, we have

limt→∞

− logBt

t= ρ− logE

[exp

(− X1∑N

l=1 1/γl

)].

Proof of Theorem 6.1. First, we have evidently

− logBt

t≤ −

logE

[∏Nl=1

(γl

λleρt

)(∑Nm=1

γlγm

)−1

exp(− ϵt∑N

l=1 1/γl

)1η1≤ϵt<η0

]t

.

Next, note that the law of large numbers implies that limt→∞ P (a ≤ ϵt ≤ b) = 0,

for all 0 ≤ a ≤ b, and limt→∞ P (c ≤ ϵt) = 1, for any 0 ≤ c. Therefore, since

η0 = ∞, we get

−logE

[N∏N

l=1

(γl

λleρt

)(∑Nm=1

γlγm

)−1

exp(− ϵt∑N

l=1 1/γl

)1η1≤ϵt<η0

]t

≤ − logBt

t,

for sufficiently large t. It is left to prove that

limt→∞

−logE

[exp

(− ϵt∑N

l=1 1/γl

)1η1≤ϵt<η0

]t

= − logE

[exp

(− X1∑N

l=1 1/γl

)].


First, observe that

limt→∞

−logE

[exp

(− ϵt∑N

l=1 1/γl

)1η1≤ϵt<η0

]t

≥ limt→∞

−logE

[exp

(− ϵt∑N

l=1 1/γl

)]t

= − logE

[exp

(− X1∑N

l=1 1/γl

)].

On the other hand, we have

−logE

[exp

(− ϵt∑N

l=1 1/γl

)1η1≤ϵt<η0

]t

≤

−logE

[exp

(− ϵt∑N

l=1 1/γl

)1 η1

t ≤X1...1 η1t ≤Xt

]t

=

− logE

[exp

(− X1∑N

l=1 1/γl

)1 η1

t ≤X1

],

lastly, the dominated convergence theorem yields

limt→∞

− logE

[exp

(− X1∑N

l=1 1/γl

)1 η1

t ≤X1

]= − logE

[exp

(− X1∑N

l=1 1/γl

)],

accomplishing the proof.

6.2. Heterogeneous impatience rates. Consider an economy which is composed

of agents who differ only with respect to the impatience rates, that is, γ1 = ... =

γN = γ; ϵik = ϵk/N, i = 1, ..., N , k = 1, 2, ...; ρN < ... < ρ1. Let λ1, ..., λN be the

weights corresponding to the equilibrium SPD ξk = ξk(λ1, ..., λN ), k ∈ N. Note

that there exists t′ ∈ N such that λ1eρ1t > .... > λNe

ρN t, for all t > t′. Therefore,

by recalling (4.1), we get il(t) = l, for all t > t′, and consequently (see (4.2) and

(4.3)), we have

(6.5) ηj(t) =1

γ

N∑l=j+1

log(βl/βj) =1

γ

N∑l=j+1

log(λj/λl) +1

γ

N∑l=j+1

(ρj − ρl)t,

for all j = 1, ..., N, and

ξt = γN∑j=1

(λj ...λN )−(N−j+1)−1

exp

(−ρj + ...+ ρN

N − j + 1t− γ

N − j + 1ϵt

)×1ηj(t)≤ϵt<ηj−1(t),

for all t > t′. Consider the logarithmic moment generating function of X1

Λ(x) = logE[eλX1

],

and denote by

Λ∗(y) = supx∈R

(xy − Λ(y)) ,


the corresponding Legendre transform of Λ. We set

aj = ρj + infx∈[ 1γ

∑Nl=j+1(ρj−ρl),

1γ

∑Nl=j(ρj−1−ρl)]

Λ∗(x),

bj = ρj−1 + infx∈( 1

γ

∑Nl=j+1(ρj−ρl),

1γ

∑Nl=j(ρj−1−ρl))

Λ∗(x),

for j = 2, ..., N, and

a1 = ρ1 + infx∈[ 1γ

∑Nl=2(ρ1−ρl),∞)

Λ∗(x).

We are ready to state now the main result of this subsection.

Theorem 6.2. In economies with heterogeneous impatience rates, we have

(6.6) lim supt→∞

− logBt

t≤ min b2, ..., bN ,

and

(6.7) lim inft→∞

− logBt

t≥ min a1, ..., aN .

Proof of Theorem 6.2. Observe that the following inequality is satisfied

ξt ≤ γN∑j=1

(λj ...λN )−(N−j+1)−1

exp

(− 1

N − j + 1((ρj + ...+ ρN ) t+ γηj(t))

)

×1ηj(t)≤ϵt<ηj−1(t) ≤ γN∑j=1

(λj)−1 exp (−ρjt)1ηj(t)≤ϵt<ηj−1(t),

for all t > t′. Denote aj(t) := (λj)−1 exp (−ρjt)1ηj(t)≤ϵt<ηj−1(t). We have

lim inft→∞

− logBt

t≥ lim inf

t→∞−logE[

∑Nj=1 aj(t)]

t

≥ lim inft→∞

log

(1

(N maxE [a1(t)] , ..., E [aN (t)])1/t

)

= lim inft→∞

log

(1

(maxE [a1(t)] , ..., E [aN (t)])1/t

)

= lim inft→∞

min

log

(1

(E [a1(t)])1/t

), ..., log

(1

(E [aN (t)])1/t

)

= min

lim inft→∞

log

(1

(E [a1(t)])1/t

), ..., lim inf

t→∞log

(1

(E [aN (t)])1/t

).

Fix an arbitrary ε > 0. Next, recall (6.5) and observe that the following inequality

holds true for each j ∈ 2, ..., N.

(6.8) lim inft→∞

− logE [aj(t)]

t= ρj + lim inf

t→∞− logP (ηj(t) ≤ ϵt < ηj−1(t))

t


≥ ρj + lim inft→∞

−logP

(1γ

∑Nl=j+1(ρj − ρl)− ε ≤ ϵt ≤ 1

γ

∑Nl=j(ρj−1 − ρl) + ε

)t

≥ ρj + infx∈[ 1γ

∑Nl=j+1(ρj−ρl)−ε, 1γ

∑Nl=j(ρj−1−ρl)+ε]

Λ∗(x),

where in the last inequality we have employed Cramer’s large deviation theorem

(see e.g. Theorem 2.2.3 in Dembo and Zeitouni (1998)). Since ε > 0 was arbitrary,

we conclude that

lim inft→∞

− log aj(t)

t≥ ρj + inf

x∈[ 1γ∑N

l=j+1(ρj−ρl),1γ

∑Nl=j(ρj−1−ρl)]

Λ∗(x).

Similarly, we have

lim inft→∞

− logE [a1(t)]

t≥ ρ1 + inf

x∈[ 1γ∑N

l=2(ρ1−ρl),∞)Λ∗(x).

The preceding inequalities combined with (6.8) prove the validity of (6.7). On the

other hand, we have

ξt(λ1, ..., λN ) ≥

γN∑j=2

(λj ...λN )−(N−j+1)−1

exp

(− 1

N − j + 1((ρj + ...+ ρN ) t+ γηj−1(t))

)

×1ηj(t)≤ϵt<ηj−1(t) ≥ γ

N∑j=2

(λj−1)−1 exp (−ρj−1t)1ηj(t)≤ϵt<ηj−1(t).

Finally, by exploiting the preceding inequality, one can use the same arguments as

in the lower-limit case to obtain inequality (6.6).

7. Precautionary savings

The current section is concerned with the phenomenon of precautionary savings,

namely, savings resulting from future uncertainty. When markets are complete, one

would not anticipate this phenomenon to occur, since in principle, all risks can be

hedged. Thus we concentrate on incomplete markets. Here, as commonly referred to

in the literature (see e.g. Carroll and Kimball (2008)), savings should be understood

literally as less consumption. We consider a one-period incomplete market of type

C, as in Subsection 3.2.1, and stick to the same notation. The only distinction is the

particular specification of the random endowment. Let X ∈ L2 (F1) be arbitrary

non-negative random variable. Denote by

(7.1) ϵ1 = ϵ1(ε) :=eεX

E[eεX

∣∣H1

] ,the endowment at time T = 1 of the agent, for some ε ∈ [0, 1].Note that E

[ϵ1(ε)

∣∣H1

]= 1, and ϵ1 = 1 in case that X ∈ L2 (H1) or in case that the market is complete

(i.e., if F1 = H1). As shown in the next statement, the variance of ϵ1(ε) is an

increasing function of ε.


Lemma 7.1. The conditional variance on H1 of the random variable ϵ1(ε) is an

increasing function of ε, namely, the function

V ar(ϵ1(ε)) = E[(ϵ1(ε)− E

[ϵ1(ε)

∣∣H1

])2 ∣∣H1

],

satisfies the inequality

V ar(ϵ1(ε1)) ≤ V ar(ϵ1(ε2)),

P − a.s., for all ε1 ≤ ε2.

Proof of Lemma 7.1. First note that

V ar(ϵ1(ε)) =E[e2εX

∣∣H1

](E[eεX

∣∣H1

])2 − 1.

We can view V ar(ϵ1(ε)) : [0, 1] → L2 (H1) as a curve. Since the probability space

Ω is finite, the function V ar(ϵ1(ε)) is differentiable. Furthermore, to complete the

proof, it suffices to show that the corresponding differential is non-negative. One

checks that the differentiable is non-negative if and only if

E[Xe2εX |H1

]E[eεX |H1

]− E

[e2εX |H1

]E[XeεX |H1

]≥ 0.

To see this, consider the measure Q defined by the Radon-Nykodym derivative

dQ

dP=

e2εX

E [e2εX |H1],

and note that the above inequality is equivalent to

EQ[X∣∣H1

]EQ

[e−εX

∣∣H1

]≥ EQ

[Xe−εX

∣∣H1

].

The latter inequality follows from the Fortuin-Kasteleyn-Ginibre (see Fortuin et.

al. (1971); FGK, for short) inequality, since it can be rephrased as

EQ[f(X)

∣∣H1

]EQ

[g(X)

∣∣H1

]≥ EQ

[f(X)g(X)

∣∣H1

],

where f(x) = x is increasing and g(x) = e−εx is decreasing.

According to Lemma 7.1, the parameter ε ∈ [0, 1] quantifies the degree of income

uncertainty. More precisely, an increase in ε yields an increase in the variance of

the income but preserves the mean. Therefore, we learn that the un-insurability of

the income is an increasing function of the parameter ε. Now, recall the explicit

formulas (see (3.9) and (3.10)) for the optimal consumption stream in the current

setting. One can show (similarly to Corollary 3.4) that by choosing a sufficiently

large ϵ0 (the initial endowment), the corresponding optimal consumption stream

coincides with unconstrained one and is given by

(7.2) c0(ε) =1

γ

(log

(1

λ(ε)

)+ ρ

),


and

(7.3) c1(ε) =1

γlog

(eγϵ1(ε)E

[e−γϵ1(ε)

∣∣H1

]λ(ε)M1

)

=1

γlog

(eγϵ1(ε)E

[e−γϵ1(ε)

∣∣H1

]M1

)+ c0(ε)−

1

γρ

where λ(ε) (or equivalently, c0(ε)) solves uniquely the equation

(7.4) c0(ε) + E [M1c1(ε)] = ϵ0 + E [M1ϵ1(ε)] .

Now, we are ready to state the main result of the section.

Theorem 7.2. In the setting of the current section, the precautionary savings

motive holds true. Namely, c0(ε) is a decreasing function of ε. Hence, in particular,

investors consume less in the present, as the variance of future-income increases.

Proof of Theorem 7.2. First, it is possible to rewrite the constraint (7.4) as

c0(ε) (1 + E [M1]) +1

γE[M1 logE

[e−γϵ1(ε)

∣∣H1

]]= K,

where K := ργE [M1] + ϵ0 + 1

γE [M1 logM1] is a constant not depending on ε.

Obviously, c0(ε) is a differentiable function of ε. By differentiating the above identity

with respect to ε, we get

∂c0∂ε

=1

1 + E [M1]E

[M1

E[e−γϵ1(ε) ∂ϵ1

∂ε

∣∣H1

]E[e−γϵ1(ε)

∣∣H1

] ],

where the differential of ϵ1(ε) is given by (see (7.1))

∂ϵ1∂ε

=XeεX

E[eεX

∣∣H1

] − eεXE[XeεX

∣∣H1

](E[eεX

∣∣H1

])2 .

Therefore, to accomplish the proof, it suffices to check that

E

[e−γϵ1(ε)

∂ϵ1∂ε

∣∣H1

]≤ 0.

This boils down to proving that

E[XeεXe−γϵ1(ε)

∣∣H1

]E[eεX

∣∣H1

]≤ E

[e−γϵ1(ε)eεX

∣∣H1

]E[XeεX

∣∣H1

].

Set a new measure Q given by the Radon-Nykodym derivative

dQ

dP:=

eεX

E[eεX

∣∣H1

] ,and note that by dividing the preceding inequality by

(E[eεX

∣∣H1

])2, we arrive at

EQ[Xe−γϵ1(ε)

∣∣H1

]≤ EQ

[e−γϵ1(ε)

∣∣H1

]EQ

[X∣∣H1

].


Assume first that ε > 0. By (7.1), we have

X =(log ϵ1(ε) + logE

[eεX

∣∣H1

]) 1ε.

Therefore, the required inequality admits the form

EQ[e−γϵ1(ε) log ϵ1(ε)

∣∣H1

]≤ EQ

[e−γϵ1(ε)

∣∣H1

]EQ

[log ϵ1(ε)

∣∣H1

],

which follows by the FGK inequality, as in Lemma 7.1. Finally, if ε = 0, the proof

follows from the fact that ϵ1(0) = 1.

Acknowledgments. We would like to thank Jeffrey Collamore and Yan Dolinsky

for useful discussions and Paul Embrechts for a careful reading of the preliminary

version of the manuscript. The constructive remarks of two anonymous referees are

highly appreciated. Roman Muraviev gratefully acknowledges financial support by

the Swiss National Science Foundation via the SNF Grant PDFM2-120424/1.

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Department of Mathematics and RiskLab, ETH, Zurich 8092, Switzerland, E-mails:

[email protected], [email protected]

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EXPONENTIAL UTILITY WITH NON-NEGATIVEwueth/Papers/2012_muraviev...Exponential Utility with Non-Negative Consumption 3 setting is violated for incomplete markets. Then, we turn to analyzing