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AN ANALYTICAL CONSTRUCTION OF

CONSTANTINIDES SOCIAL UTILITY FUNCTION*

Lilia Maliar and Serguei Maliar**

WP-AD 2005-25

Corresponding author: L. Maliar. Departamento de Fundamentos del Anlisis Econmico, Universidad deAlicante, Campus San Vicente del Raspeig, Ap. Correos 99, 03080 Alicante, Spain. E-mail:maliarl@merlin.fae.ua.es.

Editor: Instituto Valenciano de Investigaciones Econmicas, S.A.Primera Edicin Septiembre 2005

Depsito Legal: V-3792-2005

IVIE working papers offer in advance the results of economic research under way in order to

encourage a discussion process before sending them to scientific journals for their final

publication.

* This paper is a revised version of Chapter 1 of Serguei Maliar's Ph.D. thesis. We are grateful to Peter

Hammond, Carmen Herrero, Iigo Iturbe, Andreu Mas-Colell, Morten Ravn, William Schworm andAntonio Villar for helpful comments. This research was partially supported by the Instituto Valenciano de

Investigaciones Econmicas, the Ministerio de Educacin, Cultura y Deporte, SEJ2004-08011ECON and

the Ramn y Cajal program.

** Universidad de Alicante.

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1

AN ANALYTICAL CONSTRUCTION OF

CONSTANTINIDES SOCIAL UTILITY FUNCTION

Lilia Maliar and Serguei Maliar

ABSTRACT

This paper studies the properties of the social utility function defined by

the planner's problem of Constantinides (1982). We show one set of restrictions

on the optimal planner's policy rule, which is sufficient for constructing the

social utility function analytically. For such well-known classes of utility

functions as the HARA and the CES, our construction is equivalent to Gorman's

(1953) aggregation. However, we can also construct the social utility function

analytically in some cases when Gorman's (1953) representative consumer does

not exist; in such cases, the social utility function depends on "heterogeneity"

parameters. Our results can be used for simplifying the analysis of equilibrium

in dynamic heterogeneous-agent models.

Keywords: Aggregation of preferences, Planner's problem, Social utility function,

Social welfare, Gorman aggregation

Classification numbers: D11, D58, D60, D70

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1 Introduction

In general, a characterization of equilibrium in dynamic heterogeneous-agentmodels is complicated and relies heavily on numerical methods, see, e.g., themodel of Krusell and Smith (1998). In the presence of Gormans (1953) ag-gregation, this task is simplified considerably, as, in essence, a heterogeneous-agent model is reduced to the familiar one-consumer setup, see, e.g., Chat-terjee (1994), Caselli and Ventura (2000). However, by construction, suchmodels do not provide a framework for analyzing the role of heterogeneityin the aggregate behavior of actual economies, which, in fact, is the issue ofgreatest interest for the current literature on heterogeneous agents.

There are several papers that show examples of the so-called imperfectaggregation, where the aggregate dynamics depends on distributions but ina manner which is relatively easy to characterize and understand. Here, theaggregate dynamics are still described by a one consumer model but sucha model has parameters (shocks) whose values (properties) depend on dis-tributions. This class of models proved to be very convenient for empiricalwork, see Atkeson and Ogaki (1996), Maliar and Maliar (2001, 2003a, 2003b).In this paper, we therefore attempt to establish general results concerningthe possibility of describing the aggregate behavior of heterogeneous-agenteconomies by one-consumer models without assuming Gormans representa-tive consumer.

A starting point for our analysis is the result of Negishi (1960) who showsthat a competitive equilibrium in a multiconsumer economy can be restoredby solving the problem of a social planner whose objective is to maximize aweighted sum of individual utilities subject to the economys feasibility con-straint. Constantinides (1982) reformulates a dynamic version of Negishis(1960) problem in the form of two sub-problems: First, a social (intra-period)utility function is constructed by solving a one-period multi-consumer model,and then, an aggregate allocation is computed by solving a multi-period one-consumer model. In general, the social utility function defined in Constan-tinides (1982) is a complicated object, which depends on a distribution of

welfare weights in an unknown way. There are certain cases, however, inwhich the social utility function (i.e., the mapping between the distributionof welfare weights and the social preferences) can be constructed analytically.The well-known case is Gormans (1953) aggregation, in which the preferenceordering on the aggregate commodity space is the same for all distributionsof welfare weights. However, there are also examples of the analytical con-

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struction of the social utility function in the absence of Gormans (1953)

aggregation, see Atkeson and Ogaki (1996), Maliar and Maliar (2001, 2003a,2003b), and Ogaki (2003).

We start by generalizing the property that is common to all known ex-amples of analytical construction of the social utility function. This propertyhappens to be a particular kind of the optimal planners sharing rule: First,the amount of a commodity that each agent gets from the planner dependson the total endowment of this given commodity, but not on the endowmentsof any other commodities. Secondly, a change in the total endowment of eachcommodity is distributed among agents in fixed proportions that are deter-mined by the distribution of the welfare weights. We refer to the planners

policy rule that satisfi

es the above two properties as a linear sharing rule.We show that under the assumption of the linear sharing rule, the so-cial utility function is additive in some partition of commodities, with eachsub-function being composed of two multiplicatively separable terms, onedepending on the aggregate commodity endowment and another dependingon the distribution of the welfare weights. The terms that depend on wel-fare weights capture all of the effects that the distribution of the welfareweights (wealth) have on the social preference relation. The actual numberof such heterogeneity parameters does not exceed the number of additivesub-functions in the social utility function. If either the social utility functionconsists of just one additive component, or if the values of all the heterogene-ity parameters are equal, we have Gormans (1953) representative consumer.Otherwise, the preference relation on the commodity space is not invariantto redistributions of the welfare weights, and Gormans (1953) representativeconsumer does not exist.

We illustrate the construction of social utility functions under the plan-ners linear sharing rule for three classes of utility functions. Our first twoclasses are the Hyperbolic Absolute Risk Aversion (HARA) and the ConstantElasticity of Substitution (CES); here, the agents preferences are similarlyquasi-homothetic and our construction is equivalent to Gormans (1953) ag-gregation. Our third class is defined by assuming that the individual util-

ity functions are given by identical-for-all-agents (up to possibly differenttranslated origins) linear combinations of distinct members from the HARAand the CES classes. The considered preferences are not similarly quasi-homothetic and hence, they are not consistent with Gormans (1953) ag-gregation. Still, the planners sharing rule is linear, and the social utilityfunction can be constructed analytically.

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The income endowment of agent i in period t is denoted by yit. The dis-

tribution of income endowments in period tis Yt {yit}iI = RI++. We

assume thatYtfollows a stationary first-order Markov process. To be precise,let< be the Borel-algebra on=, and let us define a transition function forthe distribution of income endowments :=

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the agents the interest rate and wages in exchange for their capital and labor.

Finally, we can consider other sources of uncertainty, for example, aggregateuncertainty in the form of shocks to production technology as in Maliar andMaliar (2001) or idiosyncratic uncertainty in the form of shocks to individuallabor productivities and discount factors as in Maliar and Maliar (2003a).

Negishi (1960) demonstrates that a competitive equilibrium allocationin a deterministic one-period market economy can be restored by solvingthe problem of a social planner who maximizes the weighted sum of theindividual utilities subject to the economys feasibility constraint. Under theassumption of complete markets, this result also holds for dynamic stochasticeconomies like ours. Indeed, the First Order Condition (FOC) of the agents

utility maximization problem(1),(2)with respect to Arrow securities is

itqt(Z) =it+1Pr {Yt+1 Z |Yt=z} , (3)

where it is the Lagrange multiplier associated with the budget constraint(2). Note that equation(3)implies that for any two agentsi0, i00 I, we have

i0

t

i00

t

=i

0

t+1

i00

t+1

for all Z < it=t/i. (4)

That is, we can rewrite each agents Lagrange multiplier as a ratio of a

common-for-all-agents time-dependent variable tand an agent-specific time-invariant parameteri. This result is the standard consequence of the com-plete markets assumption that the ratio of marginal utilities of any two agentsremains constant in all periods and states of nature. Let us assume that theplanner weighs the utility of each agent i by i and solve the following prob-lem:

max{Xit}

T

t=0

(E0

TXt=0

tZ

iI

iUi

Xit

di

|

ZiI

Xitdi= Xt, PtXt= yt

), (5)

where yt RiIy

iidi. The constraintPtXt = yt follows by aggregating (2)

across agents and by imposing market clearing conditions for Arrow securitiesRiI

mit(Yt) di= 0 andRiI

mit+1(Z) di= 0 for all Z

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Constantinides (1982) shows that the social planners problem (5) has

an equivalent representation in the form of two sub-problems. The first oneis to distribute the economys endowment of commodities X among agentsto maximize the weighted sum of the individual utilities. This sub-problemdefines the (momentary) social utility function:

V (X, ) max{Xi}iI

ZiI

iUi

Xi

di |

ZiI

Xidi= X

, (6)

where

iiI

. We assume that the solution to the problem (6), Xi :H RK+ for all i I, is unique interior and continuously differentiable

in the region H

.The second sub-problem is to compute the aggregate optimal allocationgiven the social utility function:

max{Xt}

Tt=0

(E0

TXt=0

tV (Xt, ) | PtXt= yt

). (7)

We must draw attention to an important difference between the notionof the social utility function of Constantinides (1982) and the one of Bergson(1938) and Samuelson (1947) that is standard in the social choice litera-ture.2 In the Bergson-Samuelson case, the planner owns all commodities

and distributes them across agents to maximize social welfare; the plan-ners choice defines the socially-optimal distribution of wealth across agents.In the Constantinides case, the planners solution should replicate a com-petitive equilibrium in the underlying market economy (1), (2) and hence,should be consistent with a given wealth distribution or equivalently, witha given set of welfare weights; the distribution of wealth / welfare weightsneed not be socially optimal. Consequently, the planner in the sense of Con-stantinides (1982) faces an additional set of restrictions (i.e., a fixed wealth /welfare weights distribution) compared to the planner in the sense of Bergson-Samuelson.

Constantinides (1982) argues that one can interpret V(X, )with a fixedset of the welfare weights as the utility function of a representative consumer.Indeed, under our assumptions, the function V (X, ) is single-valued and

2See Mas-Colell et al. (1995) for a formal definition of the Bergson-Samuelson welfarefunction. The literature studying the Bergson-Samuelson planners problem includes, e.g.,Eisenberg (1961), Chipman (1974), Chipman and Moore (1979).

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twice continuously differentiable onH. Moreover, for any fixed and

allX H,V(X, )is strictly increasing and strictly concave. Therefore, fora fixed , the functionV(X, )induces a binary (transitive and convex)preference relation on the aggregate commodity space H. This aggregationconcept is often referred to in the literature as aggregation in equilibriumpoint because the constructed composite consumer represents the economyonly for just one fixed set of welfare weights.

The construction of the planners problem in Constantinides (1982) hasan advantage over the one in Negishi (1960), since it allows us to explicitlyseparate the intra-temporal and the inter-temporal aspects of the plannerschoice. In other words, instead of solving the original multi-consumer multi-

period problem (1), (2), we can fi

rst construct the social utility functionby solving a multi-consumer but one-period problem(6), and then computethe aggregate quantities from a multi-period but one-consumer problem (7).This result is particularly useful for empirical applications if the social utilityfunction can be constructed analytically. The well-known case is Gormans(1953) aggregation, where the preference relation induced by V(X, ) onH is the same for all sets of weights (see Blackorby and Schworm, 1993,for a detailed discussion). However, there are also examples of economiesthat are not consistent with aggregation in the sense of Gorman (1953),but for which the social utility function can be constructed analytically, seeAtkeson and Ogaki (1996), Maliar and Maliar (2001, 2003a, 2003b), andOgaki (2003). Our subsequent objective, therefore, is to distinguish theproperty that is common to all of the above examples and to provide generalresults concerning the possibility of constructing the social utility functionanalytically.

3 Constructing the social utility function

Let us first illustrate the construction of the social utility function, V(X, ),on the example of Atkeson and Ogaki (1996) where the agents momentary

utility functions are of the addilog type.3

Example 1 AssumeUi (Xi) =PK

k=1(xik bk)

ck , where0 < ck

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bk < xik fork = 1,...,K. The FOC of(6)with respect to x

ik is

i

xik bkck1 =k, (8)

where k is the Lagrange multiplier associated with the constraint on thekth commodity,

RiI

xikdi = xk. By expressingxik from (8) and integrating

across agents, we obtain that the individual and the aggregate quantities arerelated by

xik =bk+

i1/(1ck)R

iI

i1/(1ck) di (xk bk) . (9)

Substituting xik for k = 1,...,K in the objective function in (6) yields the

social utility function

V(X, ) =KX

k=1

k() (xk bk)ck , k() =

ZiI

i1/(1ck) di1ck .

(10)Note that ifck = c for all k = 1,...,K, the individual preferences are iden-tical quasi-homothetic. In this case, we have k() = () andV (X, ) =()

PKk=1(xk bk)

c, i.e., the social utility function is identical to the in-dividual utility functions (up to a multiplicative constant ()), which isthe case of Gormans (1953) aggregation. However, ifcks differ across com-

modities, the individual preferences are not quasi-homothetic, and Gormans(1953) representative consumer does not exist. Still, we have an analyticalexpression for the social utility function although the parameters of such afunction, k(), depend on a specific distribution of welfare weights.

The introspection of all known examples of the analytical construction ofthe social utility function in Atkeson and Ogaki (1996), Maliar and Maliar(2001, 2003a, 2003b), and Ogaki (2003) reveals that they all have the plan-ners sharing rule of the type(9), one which is linear in aggregate commodities(for a fixed set of welfare weights). We therefore proceed in two steps: Wefirst postulate a general form of the planners linear sharing rule and we then

construct the corresponding social utility function.

4

4In fact, a sharing rule Xi (X,) in our planners problem is an analogue of the indi-vidual demand functions in a market economy. Our approach is therefore similar to thatof Gorman (1953), which imposes a restriction on the individual demand functions byassuming linear Engel curves, i.e., Xi

P, yi

= i (P) + (P) yi, where i (P) and (P)

are the agent-specific and the common-for-all-agents functions of prices, respectively, andthen identifies the preference classes that are consistent with such demand functions.

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Definition The linear sharing rule is a planners policy rule such that theoptimal allocation of each consumer i Iis given by

Xi (X, ) = i () + i () X, (11)

for all X H, , where i () and i () are defined as

i ()

i1()...

iK()

, i ()

i1() ... 0... ... ...0 ... iK()

,

withRiI

ik() di= 0 and

RiI

ik() di= 1 for k= 1,...,K.

The linear sharing rule (11) has two characteristic features. First, theplanners optimal policy for distributing akth commodity across the popula-tion does not depend on the economys endowment of the other commodities.Secondly, an increase in the endowment of a kth commodity is always dis-

tributed among agents in fixed proportions,

ik()iI

. The linear sharingrule(11)is sufficiently general: in particular, in Section 4, we will show thatsuch well-known classes of utility functions as the HARA and the CES leadto linear sharing rules.5

For the purpose of our analysis, we shall express the individual preferencesin the (strongly) additive form. Let us consider a partition of the commodity

vector Xi into Nsub-vectorsXi =n

(Xi)[1]

,..., (Xi)[n]

, ..., (Xi)[N]o

such that

the utility function of each individual can be represented as a direct sum ofNsub-functions in the same partition, thus:

Ui

Xi

=NX

n=1

Ui[n]

Xi[n]

, (12)

where each (Xi)[n]

is a vector of the dimension #n 1. The partition we

consider is maximal in the sense that no partition with more than Nadditivecomponents exists. We shall also notice that the above representation doesnot impose the additivity restriction on the individual preferences but merely

5Restrictions on sharing rules have been used previously in the context of the (surplus-)cost-sharing problem, see, e.g., a survey on cooperative decision making theory in Moulin(1988).

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makes the additive structure of the individual preferences explicit if such a

structure is in fact present.The following theorem states the main result of the paper.

Theorem 1 Assume that Xi (X, ) is a linear sharing rule given by (11).The social preferences corresponding to (12) can therefore be represented by:

V(X, ) =[0] () +NX

n=1

[n] () W[n]

X[n]

, (13)

whereW[n] :R#n++ R, [0] : R and[n] : R++, n= 1,...,N.

Proof. See Appendix.

The constructed social utility function is additive in the same partition asthe individual utility functions, and it can depend on N+ 1"heterogeneity"parameters, [n] (), n = 0,...,N. The values of the heterogeneity parame-ters are determined by a specific distribution of the welfare weights. Sincean increasing linear transformation of preferences has no effect on the op-timal allocation, the social preferences can be equivalently represented byV(X,)[0]()

[1]() . Under the latter representation, the number of the heterogene-

ity parameters is reduced from N+ 1 to N 1.We finally discuss the implications of our results for the existence of Gor-

mans (1953) representative consumer, which is the case when the preferencerelation induced by V (X, ) on H is the same one for all sets of welfareweights.

Corollary 1 Gormans (1953) aggregation.Assume (13) and let W(X)

PNn=1 W

[n]

X[n]

.If N= 1, thenV (X, ) W(X).6

If N >1, thenV (X, ) W(X) [n] () () for all n= 1,...,N.

Indeed, if the individual utility functions have only one additive com-ponent, N = 1, we therefore obtain V(X, ) = [0] () +[1] () W(X) W(X), i.e., the linearity of the sharing rule is sufficient for Gormans (1953)

6Notation and mean identical and not identical, respectively, up to anincreasing linear transformation.

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aggregation. However, if there is more than one additive component, N >1,

then the linear sharing rule does not necessarily imply the existence of Gor-mans (1953) representative consumer, since there is the possibility that notall of the heterogeneity parameters are equal and thus, V(X, ) W(X).

4 Three classes of utility functions

In this section, we illustrate the construction of the social utility functionfor three different classes of utility functions that lead to linear sharing rulesof type (11). The first two classes are the HARA and the CES; here, theagents preferences are similarly quasi-homothetic, and we have Gormans

(1953) aggregation. Our third class is composed of identical-for-all-agents(up to possibly different translated origins) linear combinations of HARAand CES members. Such preferences are not similarly quasi-homothetic, andGormans (1953) representative consumer does not exist. Still, the plannerssharing rule is linear, so that the social utility function takes the form of(13)and can be constructed analytically.

4.1 The HARA Class

Pollack (1971) shows that all additive utility functions leading to linear Engel

curves are members of the generalized Bergson family, which is also referredto in the literature as the HARA class:

Ui() (Xi) =NP

n=1

an(n(xin b

in))

c,

n= 1, c b

in;

n= 1, 0< c 0, xin> b

in;

n= 1, c >1, an< 0, xin< b

in;

Ui(ln) (Xi) =NP

n=1

anln (xin b

in) , an> 0, x

in> b

in;

Ui(exp)

(Xi

) =

NPn=1b

i

nexp (anxi

n) , an< 0, bi

n

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pressing the individual optimal allocation,xin, from the first-order conditions,

by computing xn =RiIxindi and combining the formulas for xin and xn to

eliminate the Lagrange multiplier, for n = 1,...,N, we obtain

(xin)()

=bin+ (i)

1/(1c)

RiI(

i)1/(1c)

di(xn bn) , bn

RiI

bindi;

(xin)(ln)

=bin+ iRiI

idi(xn bn) , bn

RiI

bindi;

(xin)(exp)

= ln(bn/bin)ln(i)+

RiIln(

i)dian

+ xn, ln (bn)

RiIln (bin) di.

(15)Given that the planners sharing rule is linear, by Theorem 1, we have thatthe social utility function takes the form (13). SubstitutingXi into(6)yields

V() (X, ) =()NP

n=1

an(n(xn bn))c , () =

RiI

i1/(1c)

di1c

;

V(ln) (X, ) =(ln) +NP

n=1

anln (xn bn) , (ln) =

RiI

i ln

iRiI

idi

di;

V(exp) (X, ) =(exp)NP

n=1

bn

exp (an

xn

) , (exp) = exp RiI

ln idi .(16)

In none of the above cases does the heterogeneity parameter influence the so-cial preference relationship induced byV(X, )on the aggregate commodityspace H. Hence, we have Gormans (1953) representative consumer.

4.2 The CES Class

In a one-period market economy, an increasing non-linear transformation ofthe individual utility function does not affect the solution, i.e., the maximiza-

tion ofU

i

(X

i

)leads to the same optimal allocation as does the maximizationofF[Ui (Xi)], where F :R R with F0 >0. However, such a transforma-tion does affect the individual optimal allocation in the planners economysince the maximization of

RiI

iUi (Xi) di andRiI

iF[Ui (Xi)] di leads todifferent solutions. As a result, the linearity of the planners sharing ruledoes not, in general, survive a non-linear transformation of the individual

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utility functions, although in certain cases, it does. An example of such a

case is discussed below.Consider a planners economy, in which all agents possess preferences

given by a power transformation of the CES utility function:

Ui(CES)

Xi

= 1

KXk=1

ak

xik bik

!/, (17)

where 1, 6= 0, < 1, 6= 0 and ak > 0,PK

k=1 ak = 1, xik > b

ik,

k = 1,...,K. The limiting case of the transformed CES utility functionunder 0 is the Cobb-Douglas utility function, thus:

Ui(CD)

Xi

= 1

KYk=1

xik b

ik

ak! . (18)The utility functions (17) and (18) are transformations of members of theHARA class.7

In the case of the CES class, by following the same procedure we employedin Example 1, we shall now show that the individual optimal allocations isgiven by a linear sharing rule:

xik(CES) =bik+

i1/(1)RiI

i1/(1)

di(xk bk) , bk

ZiI

bikdi, (19)

wherek = 1,...,K. Theorem 1 implies that the social utility function is givenby(13). After substituting (19) into (6), we obtain:

V(CES) (X, ) =(CES)

KXk=1

ak(xk bk)

!/, (CES)

ZiI

i1/(1)

di

1.

(20)Regarding the Cobb-Douglas case, the results are similar. The individual

optimal allocations are also given by (19), i.e., (xik)(CD) = (xik)(CES). The

7Although the CES and the Cobb-Douglas utility functions under = 1 are strictlyquasi-concave, they do not satisfy our assumption of strict concavity. As a result, theindividual optimal allocations in the planners economy are either indeterminate or non-interior. In the market economy, the property of strict quasi-concavity is sufficient, how-ever, for unique interior optimal allocations, see Maliar and Maliar (2003b) for a discussion.

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social utility function is

V(CD) (X, ) = (CD)

KYk=1

(xk bk)ak

!, (CD) =(CES),

i.e., the formula for (CD) is the same as the one for (CES).In all of the above cases, we again have Gormans (1953) aggregation.

4.3 Linear combinations of HARA and CES members

We shall now construct a class of the utility function that is consistent withthe linear sharing rule but not with Gormans (1953) representative con-sumer. We shall assume that the individual utility functions have the form

(12)withN >1, where each sub-function(Ui)[n]

(Xi)[n]

is given by a CES-

or HARA-class member that is identical for all agents (up to the value of theparameters bik). Therefore, by Theorem 1, we have that the social utility

function takes the form(13), wheren

[n] ()oN

n=1are the heterogeneity para-

meters from(20)and(15)corresponding to the given CES and HARA mem-

bers, and that each sub-function W[n]

X[n]

is identical to (Ui)[n]

(Xi)[n]

(again, up to the value of the parameters bk). Below, we elaborate another

related example.Example 2 Let the agents have the preferences given by

U

Xi

= a1

xi1 bi1

c1 + bi2exp a2xi2 (21)+

1

a3

xi3 bi3

+ a4

xi4 b

i4

/,

where the parameters satisfy the corresponding restrictions outlined in Sec-tions 4.1 and 4.2. From the individual optimality conditions, we obtain thatthe individual optimal allocations xi1 andx

i2 are given, respectively, by the

formulas for(xin)()

and(xin)(exp)

in(15), andxi3 andxi4 are given by the for-

mulas for (x

i

n)

(CES)

in (19). The planners sharing rule is, therefore, linear.By Theorem 1, we have that the social utility function is of the form (13).By substituting Xi into(6), we obtain

V(X, ) = ()a1(x1 b1)c1 + (exp)b2exp (a2x2) (22)

+(CES)

(a3(x3 b3)

+ a4(x4 b4))

/,

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where (), (exp) are given in (16), and (CES) is given in (20). Since the

heterogeneity parameters (

), (exp) and(CES) are given by different func-tions of the welfare weights, the social preference relation induced by V (X, )on Hwill depend on the distribution of the welfare weights assumed. Wetherefore do not have Gormans (1953) aggregation.

5 Recovering the competitive equilibrium

In this section, we discuss how to recover the competitive equilibrium in thedecentralized heterogeneous-agent economy (1), (2) by using the associatedplanners problem (6), (7). We argue that aggregation results considerably

simplify the task of recovering the competitive equilibrium.To characterize the relation between the distribution of initial endow-

ments in the decentralized heterogeneous-agent economy and the distribu-tion of welfare weights in the planners economy, we use the agents ex-pected life-time budget constraints. To derive such constraints, we proceed

as follows. First, we use the FOC (3) to show that Et1

h

it

it1mit(Yt)

i =R

Z