Risk sharing and the household collective model

F UNO A

ç

Ã

O

GETULIO VARGAS

EPGE

Escola de Pós-Graduação

em Economia

Ensaios Econômicos

Escola de

Pós-Graduação

em Economia

da Fundação

Getulio Vargas

N° 497

ISSN 0104-8910

Risk Sharing and the Household Collective

Model

Carlos Eugênio da Costa

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ESCOLA DE PÓS-GRADUAÇÃO EM ECONOMIA

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Diretor de Ensino: Luis Henrique Bertolino Braido

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Eugênio da Costa, Carlos

Risk Sharing and the Household Collective Model/

Carlos Eugênio da Costa - Rio de Janeiro

FGV,EPGE, 2010

(Ensaios Econômicos; 497)

Inclui bibliografia.

N"497

ISSN 0104-8910

Risk sharing and the household collective model

Risk Sharing and the Household Collective

Model*

Carlos

E. da Costa

t

Getulio Vargas School of Economics

October 3, 2003

Abstract

When the joint assumption of optimal risk sharing and coincidence of beliefs is added to the collective model of Browning and Chiappori (1998) income pooling and symmetry of the pseudo- Hicksian matrix are shown to be restored. Because these are also the features of the unitary model usually rejected in empirical studies one may argue that these assumptions are at odds with evidence. We argue that this needs not be the case. The use of cross-section data to generate price and income variation is based Oil a definition of income pooling or symmetry suitable for testing the unitary

model, but not the collective model with risk sharing. AIso, by relaxing assumptions on beliefs, we show that symmetry and income pooling is lost. However, with usual assumptions on existence of assignable goods, we show that beliefs are identifiable. More importantly, if di:fferences in beliefs are not too extreme, the risk sharing hypothesis is still testable. Keywords: Collective Model, risk sharing. JEL Classification: Dll, D13.

1 Introduction

Neoclassical theory defines rationality at the individual leveI, however, consump-tion data is typically available only at the household leveI. It is, then, common practice in empirical work to assume that households behave as single agents, in what carne to be know as the unitary model.

Though appealing for its simplicity the unitary model lacks strong theo-retical support. Arguably, in its early versions, when it was but an 'ad hoc' extension of the rationality assumption to the household leveI, it would even be in contradiction with traditional theory, in that it did not take into account self-interested behavior. Later, some attempts were made to justify, from first

*1 thank Luis Braido for his valuable comments. The usual disclaimer applies.

principIes, this extension; most notably, Becker (1974) and his rotten kid the-orem. Unfortunately, as shown by Bergstrom (1989), this result is valid only under strong restrictions on preferences and on the decision process within the household 1 .

No less important is the fact that the hypothesis of rationality is almost always rejected in empirical studies where household instead of individual data is used. This is not, however, a rejection of rationality, which most studies are aimed at testing, but a rejection of the unitary model, i.e., the joint hypothesis of rationality and everything else that is needed to justify its extension to the

household.

As an alternative to this way of modeling household behavior, Chiappori

(1988 and 1992) and Browning and Chiappori (1998) developed lhe collective model. While alIowing for rationality only at the individual leveI, further re-strictions on household behavior are imposed by the assumption of efficiency of alIocations between members of the household.

From a theoretical perspective, the model uses weaker assumptions - effi-ciency on the bargaining process is all that is needed, while, from an empirical perspective, strong testable implications are retained.

As long as the allocations are efficient, the optimization problem of a house-hold can be written as a Pareto problem, but with the possibility that the Pareto weights depend on prices and income2_{- to account for effects of these variables}

on relative power within the household. It is exactly this effect of price changes on Pareto weights that breaks symmetry of compensated demands.

Whal Browning and Chiappori (1998) show is lhal lhe devialion from lhe

unitary model is restricted by the dimensionality of the Pareto frontier. Hence, a new testable property for the households 'pseudo-Hicksian' matrix - symmetry plus rank 1 (henceforth SR1) - replaces symmetry. That is, if one treats the household as a single agent and constructs its compensated demand matrix, instead of symmetric, it will be the sum of a symmetric matrix and another of rank at most one .

Another nice feature of their model appears when individual preferences are of the egotistic type, or of the caring type3 - that is, when there are no externalities in consumption. The problem is, in this case, mapped into one where, first income is split between members of a household according to a pre-specified rule - a sharing rule, then, agents are alIowed to individualIy chose their preferred bundle.

The colIective model, does not impose any restrictions on this sharing rule. AlI testable implication hinge on its being a function taking prices and incomes (and, possibly, other distribution factors) into the realline. What we do in this 1 M ore recently, Comes and Silva (1999) showed that the theorem is valid for a much broader class of preferences - even with non-transferrable utility - whenev er the solution to the problem is interior. However, Chiappori and Werning (2002) proved that, generically, the solution is not interior, which sends us back to Bergstrom's negative resulto

2 Samuelson (1956) also proposed the use of a household welfare function. However, he

paper is to evaluate the consequences for the sharing rule of extending the idea of efficiency to the allocation of risk within the household4 .

More importantly, assuming that only the aggregate consumption, i.e., the bundle chosen by the household but not the consumption of each agent is ob-served, we ask whether it is possible to test the hypothesis of optimal risk sharing.

With additional identifying assumption on beliefs, we give a positive answer to this question and show that very strong restrictions are imposed in house-hold behavior by this hypothesis. Not only income pooling is restored but also symmetry. In fact, though the SR1 property is preserved, the rank 1 matrix is itself symmetric - thus, so is the pseudo-hicksian matrix.

Does it mean that the model is observationally equivalent to the unitary model? Yes and no. Yes, we do have symmetry and income pooling. However, we argue that the way these properties are defined in most studies though suitable for testing the unitary model, is of no use in testing the collective model with optimal risk sharing.

The problem here is that the variations in prices and incomes explored in cross-sections may capture differences that were present when risk sharing agree-ments were made. In this case, though, sharing rules for each household may be defined according to optimal risk sharing, data across households may be measuring these initial differences. In fact, if we maintain the definitions used in these studies then we are bound to reject the model even if it is a reasonable description of reality.

In the last part of the paper, we relax the assumptions on beliefs and show that optimal risk sharing no longer guarantees symmetry or income pooling. Nonetheless, when divergence in opinions is not too severe, some testable re-strictions remain. We construct these tests, under the assumption that there exist two exclusive goods.

The remainder of the paper is organized as follows. The general setting is presented in 2. The implications of risk sharing with identical beliefs are shown in section 3, with the empirical evidence discussed in 3.1. Beliefs are generalized in 4, with testable implications described in 4.2. Section 5 concludes.

2 General Setup

The model we consider here is essentially the one described by Browning and Chiappori (1998). A household is composed of two members with preferences that satisfy usual properties. Each agent acts in his or her own interest, as prescribed by classic consumer theory. It is then assumed that whatever the bargaining process within the household it leads to an efficient outcome.

This means that bundles chosen by a household - and observed by an econo-metrician - are allocations at the Pareto frontier of this household.

In their paper, Browning and Chiappori (1998), allow for externalities in consumption and the presence of public goods. We, however, restrict ourselves to the case where there are no externalities in consumption and no public goods and assume that preferences are of the egotistic type.

When this is the case, a simple application of the second welfare theorem shows that the problem can be mapped into a two stages problem as follows. First, income is split between the two agents according to a given sharing rule and, then, consumption decisions are made individualIy.

A sharing rule is a function p : 1FtP _{---t 1Ft, that maps a p-dimensional vector of}

relevant variables - like prices, q E 1Ftm; full income brought into the household

by each agent, fi

(i

= 1,2); and other distribution factors - into a certain amount of disposable income5 _{yl for agent one. For the moment we shall not}

be explicit about distribution factors, writing the function p : (fI, f2, q) f----t yl.

Later on, we bring them back into the discussion in a very natural way. We start the description of the problem solved by the household with the second stage: the individual maximization of utility after resources having been split according to the sharing rule.

Thus, we define agent one's indirect utility function conditional on his or her share of total income p as:

V'

(q,p(I'f,q))

'= {

max

U'

(x')

x'

Analogously for agent two:

V

2

_(q,I

_-

_p(I'f,q))

'= {

where f

==

fI

+

f2. Observe that an increase in agent i' s wage can be decomposed into an increase in his fulI income, fi, and an increase in the price of leisure for this agent. In this sense we notice that not alI prices need to affect both indirect utility functions6 .

Let the total demand by the couple be denoted x, then:

(1)

RecalIing that yi is the income available for each agent - i.e., yl

==

f - p and

y2

==

P -, we have that xi

(q,

yi) , i = 1,2, are usual marshallian demands, with

Up to this point we are just repeating one of the specifications used by Browning and Chiappori (1998) . The assumption that will specialize even fur-ther this model is the one of optimal risk sharing.

Though we have not explicitly introduced risk in our model, we should think of prices and incomes as being randomly determined. Before the realization of uncertainty the two members of the household agree on an 'ex-post' rule p that

allows for optimally sharing the risks 7 .

Under the assumption that inter-temporal discount factors {3h

(h

=

1,2)

are identical across agents, optimal risk sharing implies

(2)

where li is as a measure of relative power within the household, w, a state of

nalure, and

1"

C)

(h

ｾ＠ 1,2.), lhe densily funelion assoeialed lo beliefs of agenl

h.

In section 4, when we investigate in detail the role of beliefs in generating testable implications for the model, we shall discuss the meaning of these den-sities. For now, one should only notice that we have abused notation somewhat by wriling V_y

h (w)

lo denole V_y

h

(q

(w)

,

yh (w))

,

lhe marginal ulilily of ineome in state

w.

By the same token, fh

(w)

should be seen as a short hand for

1"

(I'

(w),12 (w),

q

(w)) .

Either this notation, or fh (11,12 ,

q)

-

with the omission of w - shall be used,

whenever convenient.

3 Main Model

In this section we adopt a very important identifying assumption: the coinci-denee of beliefs. Thal is, we assume

f' (w)

ｾ＠

f2 (w), \fw.

The idea here is, firsl to use this assumption to characterize the sharing rule and, then to check what are the implications of such rule for observable behavior.

With coincidence of beliefs, the so called Borch's rule obtains.

(3)

for some li

>

O. The relation holds for all values, (11,12 ,

q)

,

at which the

equation is evaluated.

The first consequence of this assumption is income pooling, as shown in proposition 1.

Proposition 1 Optimal risk sharing 'ex-ante' implies income pooling 'ex-post', i.e., p (11, 12

_,q)

₌₌

P

_(l,q)

_,

_{with 1}

₌₌

₁₁

₊

₁2_{. Mo reo ver,}

(4)

where (Ih

==

yh /1 for h = 1,2, is the share of income household income agent h

gets and R h

== -

ｖｹｾ＠ yh /Vyh is his or her coefficient of relative risk aversion.

The last part of the proposition is a straightforward consequence of the sharing rule found in Wilson (1968) for the case where there are only two agents. In fact, in this classic paper it is shown that the share of aggregate risk carried by an agent j is given by

l/Ai (Li l/Ai)-1

,which in the case of two agents simplifies lo (4).

Nolice lhal

(8)

implies dyhjd1

>

O,

h ｾ＠

1,2,

if agenls are risk averse. In words, the amount of income each agent is entitled to, according to the sharing rule is increasing in the household's full income.

From now on, we write

p

(11,12 ,

q)

==

P

(1,

q),

but abuse notation a little bit, by dropping lhe hal over p.

We shall now consider how changes in prices affect income sharing. That is, we shall investigate the derivative of p with respect to prices.

Proposition 2 Let the sharing rvie be defined by p (1, q), with 1 then

1 dlogp

(T

-dlogqj

(T} (R'

-

'7}) -

(TJ (R2

-

'73)

R'

j

(TI

+

R2

j

(T2

( 5)

where (I7

==

X7%/yh is the share of agent h's income spent on good j and

"77

= dlogx7/dlogyh is agent h 's income elasticity of demand for good j. What matters for risk sharing is the relative effect prices have on each mem-ber's marginal utility of income. The agent whose marginal utility of income increases most (decreases least) when the price increases receives a transfer that increases his or her share in total income. Or, put in better terms, it is the sen-sitivity of an agent 's marginal utility of income to price fluctuation compared to the other agent that determines whether he bears more or less of the price fluctuation risk.

The total effect of an increase in the price of commodity j on the marginal utility of income can be decomposed in two parts. The first one is simply due to the decline in real income. Up to a first order approximation this decline is equal to the share of the good in total expenditure, and changes the marginal utility of income according to the concavity of the indirect utility function as measured by the agent 's risk aversion. The other component, the direct effect of changing the price on the marginal cost of increasing welfare, is captured by the income elasticity of demand for this goods .

Corollary 3 Let

Zh

denote agent h' s consumption of leisure, then, the effect of

changes in wages in the sharing rule is given by:

and

dlogp dlogw2

â-l 2R2

+

O"l21]r

R' j

(TI

+

R2 j (T2 '

w

' _ _dlog dlogp w '

â-l 1RI +O"l11]1

R'j(T' +R2j(T2

where â-[lt

==

(l-Zh) /yh,

is the proportion of agent h' s labor income to his or

her disposable income and wI

==

wI

/y

is the participation of agent 2's earnmgs

potential on the h01lSehold's full income.

From proposition 2 it is not hard to see that, among other things, p is

homogeneous of degree I in q and l. Other properties of p might be explored.

However, we do not observe p. It is then important to derive restrictions on the observable household demands.

What we show next is that the assumption of optimal risk sharing makes the "Hieksian Matrix" for the household, symmetrie and negative semi-definite. However, before spelling out the result we must introduee some notation.

Let

h

i denote the household eompensated demand functions for good i. It is obtained from the observed household uneompensated demands by diseounting the ineome effeets - aeeording to the Slutsky equation - as if the data were generated by a single agent . We shall eall these functions "pseudo-Hieksian demands". We use h_i to denote the true - but unobservable - eompensated (Hieksian) individual demand for good i.

Proposition 4 The household pseudo- Hicksian demands

h

are symmetric, and

negative semi-definite.

Let S be the pseudo-Hieksian matrix. Chiappori and Browning showed that the eollective model implies:

S = L; - uv',

where L; is the sum of the two individual Hieksian matriees and uv' is a rank

one matrix - we are borrowing their notation.

What we show in proposition 4 is that, under the optimal risk sharing as-sumption, uv' beeomes vv' whieh is still one dimensional but now, also symmet-rie and positive semi-definite.

It is worth emphasizing here that, even though the pseudo Hieksian matrix is symmetrie it is not the sum of the agents' Hieksian demands. It is the Hieksian demand of the fictitious agent that has preferenees represented by UI

+

liU2 .

Under our assumptions, li may depend on many different variables. These

available at the time the risk sharing agreement was made - e.g., when marriage took place.

Qur findings relate to Constantinides' (1982) representation of a complete markets equilibrium by the choices made by a representative agent. AIso in that paper, beliefs and inter-temporal discount factors are assumed to coincide.

Notice that we have imposed no restrictions on the Bernoulli functions. This is important because we are not assuming aggregation of preferences of any sort. As is well known, aggregation would require linearity - and equality of slopes - of Engle curves. In a context similar to ours, Mazzocco (2002a) has shown that, for these properties to come up, individual utility functions must be of the harmonic absolute risk aversion - HARA - class with identical curvature parameters.

Here, however, the only assumption on preferences is that they satisfy usual regularity properties. It is not the case, then, that the distribution of income within the household is of no importance, as would be the case with aggregation. Thus, the main consequences of adding the assumption of optimal risk shar-ing and coincidence of beliefs to the collective model are: income poolshar-ing and symmetry of pseudo-Hicksian demands. The problem here is that both seem to be at odds with evidence. But, are they?

In section 3.1 we discuss the evidence regarding both these consequences of optimal risk sharing

3.1 Empirical Evidence

The model we present here has, for obvious reasons, never been intentionally tested. However, because many of the empirical consequences are akin to the ones derived from the unitary model, it is worthwhile discussing in some detail these studies.

What we argue here is that most studies that reject these two consequences of optimal risk sharing use a definition that is suitable for testing the unitary model but not the collective with risk sharing. When the proper definition of variables and data sets compatible with testing those variables are used, the evidence against optimal risk sharing is, at best, inconclusive.

3.1.1 Incorne Pooling

Income pooling means that, for a given household, any change in individual incomes such that dI1

₊

_dI2 _{= O should cause no changes in the bundle chosen}

by lhe household.

It does not mean that two different households with the same I

==

11

+

12 and facing the same set of prices, p, should choose the same bundle independently

(1996), and Pezzin and Sehone (1997). This may be lhe eorreel Ihing lo do when testing the unitary model, but does not work when testing risk sharing.

The problem with the evidence cited above is that if current differences in the distribution of income are correlated with what the distributions were at the moment the insurance agreements were made, then the former are proxying for the latter, which will most likely enter K, according to the collective model.

To understand the issue, suppose that we observe two households identical in all respects but the distribution of income received, i.e., 11

+

12 _{= I =}

_Íl

₊

_Í2,

but

Ih

#-

Íh

1 h = 1,2. If the unitary model were an accurate description of

the functioning of households we should expect both households to have the same pattern of consumption. We should also expect this to be the case in the collective model with risk sharing if we knew that the Pareto weights were the same in the two households.

Assume, however, that this latter model is valid, but incomes today are identical to what they were at the time when the couple was formed. In this case, there is no reason to expect that the Pare to weights will coincide for the two households. Neither should we expect the consumption bundles to be the same.

It is clear that such a test is bound to reject optimal risk sharing, even though we have assumed it to existo

For the definition of income pooling more appropriate to our identifying assumptions the evidence is mixed. The two main contributions to the area are: Attanasio and Lechene (2002), which rejects income pooling and Braido et aI. (2003) whieh fails lo rejeel.

It is also worth mentioning the work of Thomas et aI. (2002) who use data from the Indonesia Family Survey to estimate the effects of assets brought into marriage by each partner on child illnesses. This is a clear example of a study that tests the unitary model but not the collective model with risk sharing. In fact, we would not expect this type of study to favor income pooling, since assets brought into marriage should affect li and should produce departure from

income pooling also in our modeI. The evidence is, however, mixed. They fail to reject the unitary model in Indonesia, but reject it in Java and Sumatra10 .

3.1. 2 Syrnrnetry

Given that the matrix of compensated demands has the symmetry and negative semi-definiteness properties, one might wonder if the rejection of the unitary model as characterized by the same properties, wouldn't imply the rejection of the collective model with optimal risk sharing. Were it the case, the evidence would be very disturbing for the model, as the discussion in the introduction to this article has pointed out.

There are some reasons why this need not be the case. The rejection of one model need not imply the rejection of the other.

The point here is similar to the one we made for income pooling in section 3.1.1. Price variations do not affect Pareto weights for a couple according to our model. N onetheless, they may affect Pareto weights across households if prices today are correlated to prices at the time risk sharing arrangements were made. Of course this would not happen if we were following a household through time, in a panel study. However, alI studies produced thus far in the area use the price variation that is present in a cross section. Compensated demand func-tions are estimated and tested for symmetry and negative definiteness, under the assumption that household heterogeneity can be controlIed for, using some demographics.

The problem is that the same price diversity that alIows for estimating the demand functions should probably be relevant in determining the initial distrib-ution of power within each household. If prices today are correlated with prices at the moment risk sharing agreements were made, then the diversity explored in the data, may be measuring diversity at that moment in time as well. Once again we should not expect the li to coincide. Changes in li that justify the

rejection of the unitary model, but not the colIective model, could be causing what is perceived as non-optimal insurance.

The general message is that using a cross section we would accept the plain collective model and reject the colIective with risk sharing. A panel study would correctly fail to reject both, while still rejecting the unitary model.

80 far we have been discussing evidence related to the joint hypothesis of optimal risk sharing and coincidence of beliefs. In the next section we relax the assumption on beliefs, first to show that if beliefs are too general, no additional testable restrictions arise from the optimal risk sharing assumption, and then to show that even with some disagreement - in a sense to be made precise, the optimal risk sharing assumption is testable.

4 Generalizing Beliefs

We could never overemphasize the need for coincidence of beliefs in the deriva-tion of our results in secderiva-tion 3. What we discuss in this secderiva-tion is whether, after relaxing this assumption, there still remain testable implications from optimal risk sharing.

In section 4.1 we present some negative results. First, for very general di-vergence of beliefs, no additional restrictions over 8R1 are generated by optimal risk sharing. 8econd, even when we allow for divergence only to take place in what regards individual incomes (and possibly wages), income pooling and symmetry are no longer valido

4.1 Negative Results

We start with very general divergence on beliefs. In this case, all additional structure attained with risk sharing is lost . To introduce the discussion, however, we need to invest in some new notation.

First let L = 11 / l. Then, we rede fine the sharing rule

p(I,L,q)

'=

p(IL,I(l- L) ,q)

Notice that

PI

is the deriva tive of the sharing rule with respect to total income holding L fixed . That is, dlog l' ｾ＠ dlog 12 ｾ＠ dlog I.

N ext, let the beliefs of agent h with respect to the behavior of the rele-vant variables be given by the distribution function

ph

(1,

L, q)

,

with associated density

fh (I, L, q)

.

Finally assume that

f2 (I, L, q)

cf

O \j

(I, L, q),

then define

""(I

ＩｾｦＧＨｉＬｌＬｱＩ＠

'P , L, q -

j2

(I,

L, q)

With this notation, expression (2) becomes:

It is easy to see that income pooling is equivalent to ｰｾ＠ = O. In fact,

p (I, L, q)

'=

p (fL,I

(1 -

L)

,

q)

'=

P

(fL

+ I

(1 -

L)

,

q)

ｾ＠

P

(I, q) .

To understand how this very general structure on beliefs breaks the ad-ditional testable implications of the model, just notice that the model is iso-morphic, in what comes to its empirical content, to one with Pareto weights

Fi

(I, L, q)

.

Define Fi

(I, L, q)

'= ,,<jJ

(I, L, q)

,

and we are back to the collective model in what regards the testable implications.

We are not arguing that risk sharing has no implications in this case. In fact, condition (6) need not hold in the collective model. The point here is that observationalIy they are equivalent. Therefore, if we retain this very general assumption about beliefs, not much structure is expected to be found.

To give testability a chance we shalI now impose some restrictions on the way beliefs are alIowed to differ. We shalI make the following assumption about beliefs.

Assumption A Agents beliefs about behamor of prices and incomes are given

by

1"

(I,

L,

q)

'= gh

(I,

L) 'P

(q).

Though this may seem restrictive it allows for an important class of problems in which agents have different opinions about their own prospects, but agree about the behavior of the economy as a whole.

where beliefs are also different with respect to w1 _{and w2 . We come back to} this issue later.

With assumption A,

Hence,

rP

q = O.

Unfortunately, what the next proposition shows is that income pooling fails to hold even in this case.

Proposition 5 lf agents have different beliefs about the behavior of individual

incomes, then there is no income pooling.

Finally, notice that, even with coincidence of beliefs regarding the behavior of prices - assumption A, symmetry is lost.

Proposition 6 lf agents have different beliefs about the behavior of income,

then the pseudo-Hicksian matrix is still 3R1, but no longer symmetric.

Thus, all additional structure we bought with our assumption on risk sharing is apparently lost. What we do in the next section is to show that under the

same assumplions used by Chiappori (1992), Browning el a!. (1994) or Fong

and Zhang (1994), for example, we are able to identify some parameters that will allow for testing the model.

4.2 Identification and Testab!e Implications

What one shows when proving proposition 6 is that, with optimal risk sharing and assumption A,

where

What we shall do in this section is to show that, under some usual assump-tions about observability, we can identify (jCi - (iCj up to a multiplicative

constant . This being the case the model can be tested by taking two pairs of goods i and j and i' and j', and comparing the two ratios

h

ij -

h

ji

hi'j' - _hj'i'

(jCi - (iCj

(j'ci' - (itCi' .

First we are assuming that: i) household consumption decisions are Pareto efficient; ii) preferences are of the egotistic type; iii) there are no public goods; and iv) there is optimal risk sharing among members of the household.

When compared to the assumptions adopted in other studies like Browning

el a!. (1994), Chiappori (1992) and Fong and Zhang (2001). The obvious novelly

is assumption iv, which is the essence of this paper. We are, however, adding also assuming iii, for simplicityl l and ii, instead of alIowing preferences to be of the caring type. This latter assumption is made because the idea of risk sharing with preferences of the caring type is not so clear.

We now borrow from the same studies the term exclusive to denote a good that is consumed by only one of the members. We also folIow them in adopting assumption B, below.

Assurnption B There exist two exc1usive goods, one for each member.

This along with assumption A will be crucial in deriving testable implications of our model. To do so, we first need the folIowing.

Lernrna 7 lf there are two exclusive goods then, -rjJjrjJ (AI

+

A2)-1 can be

identified.

The variable L IS a distribution factor in the sense we mentioned earlier.

Distribution factors are variables that affect neither individual preferences nor a household resource constraint, but that to affect choices. They are very useful in identifying parameters in the collective model.

Nexl, we use

-q"jqJ

(A'

+

A2r'

lo idenlify

(xJy

-

xJy).

Lernrna 8 Under the conditions of lemma

7,

xJy - xJy is identi.fied for all j.

The important thing is that we are now able to construct the term that, ac-cording to our model, as the difference between the cross derivatives of hicksian demands.

Proposition 9 Under assumptions A and B, ((jCi - (iCj) is identified up to

a multiplicative constant.

The test of risk sharing works as follows. Consider any pair of goods j and i.From proposition 9 we may identify (jCi - (iCj. Now we construct the number,

h

ji -

h

ji

(jCi - (iCj

Use the same procedure for another pair j' and i' ,and construct )fi' The testable implication is that '{ji = '{fil for alI i, i', j and j'.

Before moving on to the conclusion of this paper, a word about labor supply

IS needed. The main motivation for the specific assumption we made about

beliefs was that the principal source of divergence in opinions should be about each one's of his or her own prospects. Most of one's assessment about future income streams is related to labor income. More so ifwe assume that investment decisions are made in agreement. In this case, it is natural to include among the variables for which there is disagreement in beliefs each one's wage.

That is, we replace assumption A with assumption A', below.

Assumption A ｾ＠ Agents beliefs about behavior of prices and incomes are given by

1" (1,

L,

w" w

2,

q)

=' gh

(1,

L,

w" w

2) 'P

(q) .

Abusing notation somewhat, the vector q should, now, be regarded as a

(n

-

2)-dimensional vector, with the exclusion of the two prices w1 _{and w2 .}

What we can show, in this case, is that, we may drop assumption B and get the following alterna tive resulto

Proposition 10 Under assumption A /, ((jCi - (iCj) is identified up to a

mv1-tiplicative constant.

With this result, testable implications on all other goods are derived, along the same lines of what was done in the previous case.

5 Conclusion

In this paper we show that the pseudo-Hicksian matrix for a household who shares risk efficiently is negative semi-definite and symmetric, when agents be-liefs about the behavior of prices and income coincide.

This additional restrictions - optimal risk sharing and coincidence of beliefs - on behavior of a household makes the empirical content of a collective model very close to that of a unitary model. In fact, we argue that they only can be distinguished with panel data (in the case of symmetry tests) or in random experiments studies (in income pooling tests).

Most studies use cross section data to create enough price variation to test the models. The problem here is that the same price variation should make the Pareto weights differ across households even if they are fixed across states of nature for each household, as predicted by the risk sharing version of the collective model. In that case, we show that while these studies would correctly fail to reject the collective model they would likely reject optimal risk sharing, even if it is present.

References

[1] ATTANASIO , O . AND LECHENE, V . (2002) "Tests of Income Pooling in

Household Decisions" Review of Economic Dynamics, 5: 720-748 .

[2] BECKER, GARY (1991) "A Treatise on the Family," Enlarged Edition,

Har-vard University Press.

[3] BERGSTROM, THEODORE C. (1989) "A Fresh Look at the Rotten Kid

Theorem - and Other Household Mysteries." Journal of Political Economy,

97: 1138-1159

[4] BROWNING, M., BOURGUIGNON, F., CHIAPPORI, P.-A. AND LECHENE, V . (1994) "Incomes and Outcomes: a Structural Model of Intra-household Allocations" Journal of Political Economy, 102: 1067-1096

[5] BOURGUIGNON, F ., CHIAPPORI, P .-A . (1992) "Collective Models of

Household Behavior: An Introduction" European Economic Review, 36 :

355-64

[6] BRAIDO, L. H., OLINTO, P ., PERRONE. H . (2003) "Intra-household

Allo-cation in a Random Encouragement Design" Mimeo., Getulio Vargas Grad-uate School of Economics.

[7] BROWNING, M. ; CHIAPPORI, P .-A. (1998) "Efficient Intra-household

Al-locations: A General Characterization and Empirical Tests" Econometrica

66: 1241-78

[8] CHIAPPORI, PIERRE-ANDRE (1992) "Collective Labor Supply and

Wel-fare" Journal of Political Economy, 100: 437-67.

[9] CHIAPPORI, P .-A . AND WERNING, I. (2002) "Comment on 'Rotten Kids,

Purity and Perfection'" Journal of Political Economy ,110: 475-480.

[lO]

CONSTANTINIDES, GEORGE M . (1982) "Inter-temporal Asset Pricing with

Heterogeneous Consumers and without Demand Aggregation" Journal of BWiiness 55: 253-67.

[11] CORNES, R . AND SILVA, E. (1999) "Rotten Kids, Purity and Perfection"

Journal of Political Economy 107: 1034-1040.

[12] HANDA, S . (1996) "Expenditure Behavior and Children's Welfare: An

Analysis of Female Headed Households in Jamaica." Journal of Develop-ment Economics, 50: 165-187.

[13] MAZZOCCO, M . (2002a) "Savings, Risk Sharing and Preferences for Risk"

Manuscript, University of Wisconsin .

[14] MAZZOCCO, M. (2002b) "Household Inter-temporal Behavior: a Collective

Characterization and Empirical Tests." Manuscript, University of

[15] PEZZIN, L. AND SCRONE (1997) "The Allocation of Resources in

Intragen-erational Households: Adult Children and Their Elderly Parents."

Ameri-can Economic Review Papers and Proceedings, 87: 460-464.

[16] SAMUELSON, P. (1956) "Community Indifference Curves" Quarterly Jour-nal of Economics, 70:1-22.

[H] TROMAS, D. (1990) "Intra-household Resource Allocation: An Inferential Approach" Journal of Human Resources, 25: 635-664.

[18] TROMAS, D., CONTRERAS, D. AND FRANKENBERG, E. (2002)

"Distribu-tion ofPower Within the Household and Child Health" Manuscript, DeLA.

[19] WILSON, ROBERT (1968) "The Theory of Syndicates" Econometrica 36:

119-131.

A Proofs

Praof af Proposition 1. Because (3) is an identity under optimal risk sharing we may differentiate it with respect to 11 to obtain

(7)

2 · 2 ( 1 2 ) - 1

The same procedure for I glves PI2 = li V_yy V_yy

+

li V_yy . Hence PI2 = Pll = PI' Which says that if there is optimal risk sharing ex-ante there must be income pooling ex-posto

In this case, dividing both the numerator and denominator of (7) by V_y1_{, and}

noticing that V_y1 = fi V_y2 it is easy to see that

(8)

where

A

h

== -

ｖｹｾｪｖｹｨ＠ is the coefficient of absolute risk aversion for agent

h.

Finally, define 0"1,= yl/I '= (I-p)/I, and 0"2,= y2/I '= p/I, and Rh

- ｖｹｾｹｨ＠ jVyh the coefficient of relative risk aversion. Then,

where

•

Ih

(Jh = _

Praaf af Prapasitian 2. Next, the same procedure is used to find an expression for Pj' First differentiate (3) with respect to qj to obtain

Vy'j

+

ｖｹｾｐｪ＠ - ｉ＼ｖｹｾ＠

+

I<Vy2y Pj ｾ＠

O.

(9)

Then differentiate Roy's identity with respect to y to write ｖｹｾ＠

VyhxJy, h ｾ＠ 1,2. Subsliluling ｖｹｾ＠ inlo (9)

(-ｖｹｬｹｘｾ＠ - ｖｹｬｘｾｹＩ＠

+

_Vy1_{y Pj} - K (-ｖｹＲｹｘｾ＠ - ｖｹＲｘｾｹＩ＠

+

KVy2_{y Pj} = O.

and rearranging yields:

(Vy1y xJ

+

Vy1 xJy) - K (Vy2y xJ

+

Vy2xJy) Pj ｾ＠ V' _yy

₊

V2

K yy

Using, again the fact that V_y1 = K V_y2,

ａｬｸｾ＠ - ｸｾ＠ - ＨａＲｸｾ＠ - ｸｾ＠ )

p.= J JY J JY

J

A'

+

A2

To gel expression (5) we define

h

h _ xjqj d _ dlogxj

(T.

- - -

an 1]' - -

--J Ih J dlogy ,

and rewrite (10) using elasticities

for h = 1,2,

1 dlogp

(T3 (R'

-

'I}) -

(T; (R2

-'I;)

(T

- -

ｾ＠ ＭＭＧＭＭＭＧＭＭＭＭＭＭ［ ＢＧＧＧＧＭＧ［Ｍ Ｍｾ Ｍ［Ｍ［［Ｍ Ｍ ＧＭＧＭ

dlogqj

R'

/

(TI

+

R2

/

(T2

'

where

(TI

'=

p/I,

and

(T2

'= (1-

p) /1 . •

(10)

Praaf af Carallary 3. First note that, as stated before full income li is

defined by li

==

i/ +

wi , where

i/

is non-Iabor income of agent i and wi is his or her labor market wage. We then write,

P (1,

w1_{, w}2_,

_q)

_,

_{just to make explicit}

the fact that two of the prices we are considering are the prices of leisure, i.e., wages. Then, it is immediate to see that

dp

-d _w' . ｾｐｉＫｐｷ Ｂ＠

Using (8), and (10), in (11), we gel

dp

A 2

+

X, (A'I'

-

ｉｾＩ＠

-

X, (A212

-

I;)

dw

i

_{A' +A2}

(11)

where, li denotes leisure of agent i, and Xl and X₂ are indicator functions. The presence of this indicator stems from our assumption on the absence of externalities in consumption. Hence,

dp A2

(1-

12)

₊

₈₁2_/8y

and

1 _ _ d_p

ｾ＠

_A_'

-'--(

ｬ｟ＭＭ［Ｍ［ｚｾＧ＠

)'---+-;o8Z---,' /_8-,,-Y

dw ' A'

+

A2 .

Finally,

and

•

dp w2 P

-dw2

_{P I}

R

2 (1 - Z2) w 2 / y2

+

o-r (8log Z2 / 8log y2)

R' +R2

w ' dp w ' p

- -

-I dw' p I

R' (1 - ZI) _{w '}

/y'

+

0'1

(8IogZ ' /8log

yl)

R' +R2

Praof af Proposition 4. We define each entry of a pseudo-hicksian matrix

by

(12)

where Xi

==

x}+xJ

denote the uncompensated household demand for commodity

,. From

(1)

we have:

Notice that the income effect on the consumption of commodity i by the

house-hold is defined by:

XiI =

X7

yPI

+

x7y

(1 -

PI)

Using these expressions, equation (12) may be re-written as

h

ij

xi

j

+

X7j

+

(x}y

-

x7y)

Pj

+ [X;yPI

+ X;y

(1-

PI)]

Ｈｸｾ＠

+ x;J,

where

h

_i is the pseudo-hicksian demand for commodity i. We may rearrange terms to get:

X7j

+

x7y

(Pi

+

PI (X]

+

xn)

+x;j

-

X;y (Pj

-

(1-

PI)

Ｈｸｾ＠

+ xm·

(13)

Notice that the expression for PI can be used to re-write (10) as

(14)

;- - ( 1 2 ) (A' A2) -1 .

Now, define '-,j = X_jy - X_jy

+

1 then.

Pj

-

(1 -

PI)

Ｈｘｾ＠

+

x;J

(1 -

PI)

ｸｾ＠

-

PIX;

-

(j

-

(1 -

PI)

Ｈｸｾ＠

+

x;J

Also

Pj

+

PI

Ｈｸｾ＠

+

x])

(1 -

PI)

ｘｾ＠

- PIX] - (j

+

PI

Ｈｘｾ＠

+ x])

ｘｾ＠ - (j

Now, by substituting for these expressions in (13) we come up with the following expreSSlOn:

Slutsky equation and the definition of Qur variables imply that

ｨｾｪ＠ = ｘｾｪ＠

+

ｘｾｹｘ［＠ k = 1,2.

Hence, expression (15) becomes

hij h7j

+

h7j -

(x7y

-

X;y)

(j

h7j +h7j

-

(i(j (A' +A2).

Observe that this pseudo-compensated demand is still symmetric since

A A 1 2 1 2

hij - hji = hij

+

hij - hji - hji = O.

Finally, define the v

==

((AI

+

A2/12 ,where,

with ｸｾ＠

(h

= 1,2) defined as the n-dimensional vectors of income effects.

(15)

The pseudo-Hicksian matrix is the sum of a symmetric negative semi-definite matrix - representing the sum of the two agents Hicksian matrix - and -vv', which is a symmetric, rank one negative semi-definite matrix. •

Proof of Proposition 5. Sinee (6) holds ai every

(q.

l. ,) . lhen we ean proceed as in the proof of proposition 1 and differentiate (6), first with respect

to 11,

ｖｹｬｹｰｾ＠ = ｋｲｪｊｾ＠ Vy2

+

ｋｲｪｊｖｹＲｹｰｾＬ＠ or

"<p,

Vy2 p, ｾ＠ -;-;;---'---"-;=

Vy'y

+

"<P

_{V y2y}

Dividing both the numerator and the denominator by V_y1 _{and noticing that}

V' ｾ＠ ".!.V2 _{we gel'}

y ｾ＠ y .

Ineome pooling means lhal

p,

ｾ＠ O li

(q.I.

,).

This will only oeeur if

<p,!<p

ｾ＠ O.

Proof of Proposition

6.

When beliefs differ.

(8)

becomes

But now,

Lei

_ <PI

(x}

+

xJ)

Cj =

if;

A' +A2

Then. (14) becomes:

A'

+

<Pljq,

I

A

2

-

<Pljq,

2

A' +A2

xj -

A' +A2

xj

A'x}

-

A

2

_xJ

_<Pljq,

_(x}

₊

_xJ)

A'+A2

+

A'+A2

.

Substituting this into (13) and following the same procedure used in the proof

of proposition 4 we get:

hij h}j

+

h7j -

(x7y

-

x7y)

(j -

(x7y

-

x7y)

Cj.

h;j

+

h7j - (j(i

(A'

+

A2)

-

(iCj

(A'

+

A2)

Symmetry is lost, since

and

•

Praof af lemrna 7. Let goods l and k be exclusive. That 18 one can

observe

xl

and

x%.

Therefore,

Then,

and

. I

-XII = X1yPI

So, one can identify both ｐｾ＠ and PIo Now, from (16) we have that ｐｾ＠

-<Pc/<P

(A'

+

A2r1 •

Praaf af lemma 8.

(x}y

-

xJy)

ｰｾＮ＠ Hence,

Consider any good j. First notice that ｘｪｾ＠

But lemma 7 guarantees that

ＭｲｪｊｾＯ＠

rjJ

(A

1

+

A2)

-1 is identified. So,

(X}y

-

XJy)

is also identified. •

Praaf af Prapasitian 9. One just have to notice that

x}

+

xJ

=

Xj

is

oh-servable, to realize that ((jCi - (iCj) may be constructed up to a multiplicative constant . •

Praaf af Prapasitian 10. Let LI and L2 denote labor supply of agents

1 and 2, respectively. In this case,

Then,

L'

ｾ＠ =

L'-

ｹｐｾ＠

L'

_I =

L'-

yPI and

L2 ｾ＠ = - L2-ｹｐｾ＠

ｌｊｾ＠ ｌｾ＠

(1-

PI)

ENSAIOS ECONÔMICOS DA EPGE

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