bourguignon browning, and chiappori (2009) on twitter

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Efficient Intra-Household Allocations and Distribution Factors: Implications and Identifications

Efficient Intra-Household Allocations andDistribution Factors: Implications and

Identifications

Written by mixingale@twitter for private study

June 24, 2010

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Introduction

Main Results: Testability

Derive testable implication of collective model with Pareto efficiencyassumption when not having price variation

consistent with all possible assumptions on private/public nature ofgoods, all possible consumption externalities between householdmembers, and all types of interdependent individual preferences anddomestic production technology

necessary and sufficient combining with test on unitary model, it allows to check either of

unitary/collective assumption and Pareto efficiency assumption assuming bargaining model gives additional testable implication

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Introduction

Main Results: Identification

Gives a series of identification conditions of individual Engel curves

(= preferences? Or, possible, w.l.o.g?) and decision process whennot having price variation

price variation + labor supply: Chiappori (1992), Blundell, Chiappori,Magnac and Meghir (2000), Chiappori, Fortin and Lacroix (1992)

no price vavriation + exclusive/assignable good: Browning,Bourguignon, Chiappori, and Lechene (1994)

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Model

Notation

two people A and B

n marketable consumption goods, private or public

q

m

R

n

+,m = A,B: a vector of private consumption Q Rn+: a vector of public consumption

q qA + qB,C q+ Q

no price variation normalize all prices to 1: budget constraint:

e

(qA

+ qB

+ Q) = e

C = x

where x is a total income

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Model

Preference, Distribution Factor

individual preferences:

uA(qA, qB,Q; a), uB(qA, qB,Q; a)

where a is a preference factor: a vector of characteristics affects

preferences directly three times differentiable, strictly convex (means convex preference?) private consumption of each member can enter the preferences of

the other

Definition 1: A variale zk is a distribution factor if it does not enter

individual preferences nor the overall household budget constraintbut it does influence the decision process

decision process seems not well-defined and thus distributionfactor either

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Model

z-Conditionl Demands

Cj j(x, a, z): household demand function for good j

Axiom 1: There is at least one good j and one observabledistribution factor zk such that j(x, a, z) is strictly increasing in zk

by strict monotonicity,

z1 = (x, a, z1,C1)

for i= j,

Ci = i(x, a, z1, z1, z1) = i[x, a, (x, a, z1,C1), z1] = i(x, a, z1,C1)

refer to i as z-conditional demand

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Model

Add Unobserved Terms

demand:

Ci = i(x, a, z, i)

z-conditional demand:

Ci = i(x, a, z1, z1, i)

= i[x, a, (x, a, z1,C1, 1), z1, i]

= i(x, a, z1,C1, 1, i)

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Testability

Unitary Rationality

Definition 2: Let (qA, qB,Q) be given demand functions of(x, a, z). These are compatible with unitary rationality if there existsa utility function U(qA, qB,Q; a) such that, for every (x, a, z), thevector (qA, qB,Q) maximizes U() subject to the budget constraint

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Testability

Test on Unitary Rationality

Proposition 1: A given system of demand functions is compatiblewith unitary rationality it satisfies:

i(x, a, z)

zk= 0,i, k

Remark:

consider a collective model in which the household maximized aweighted sum of individual utilities

suppose that the weight is dependent on income but not ondistribution factor

it is not unitary model in a strict sense but is observationallyequivalent to a unitary model under the current setting

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Effi i I H h ld All i d Di ib i F I li i d Id ifi i


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Testability

Collective Ratinality

Definition 3: Let (qA, qB,Q) be given functions of (x, a, z). Theseare compatible with collective rationality if there exist two utilityfunctions uA(qA, qB,Q; a) and uB(qA, qB,Q : a) such that, for every(x, a, z), the vector (qA, qB,Q) is Pareto efficient. That is, for any

other bundle (qA

,qB

,Q) such that

um(qA,qB, Q; a) um(qA, qB,Q; a),m = A,B

e(qA + qB + Q) > e(qA + qB + Q)

it should require that um(eqA,eqB, eQ; a) > um(qA, qB,Q; a) for at leastone m? anyway, strictly convex preferences will imply uniquemaximizer ofu and so imply this

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Efficient Intra Ho sehold Allocations and Distrib tion Factors Implications and Identifications


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Testability

Test on Collective Rationality

Proposition 2: Consider a point P= (x, a, z) at whichi/z1 = 0,i. Without a priori restrictions on individualpreferences um(qA, qB,Q; a),m = A,B, a given system of demand

functions is compatible with collective rationality in some openneighborhood ofP K = 1 or it satisfies any of the followingequivalent conditions:

(i) there exists real value functions 1, ,n and such that:

i(x, a, z) = i[x, a, , (x, a, z)],i

continue to the next page

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Efficient Intra Household Allocations and Distribution Factors: Implications and Identifications


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Testability

Test on Collective Rationality

(ii) household demand functions satisfy:

i/zk

i/zl

=j/zk

j/zl

,i,j, k, l

(iii) there exists at least one good 1 such that:

i(x, a, z1, q1)

zk= 0, i= 1, k= 1

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Testability

Intuition behind Test on Collective Rationality

distribution factors affect consumption but only through their effect

upon the location (the weight ) of the final outcome on the Paretofrontier

this effect is one-dimensional

fixing one value of the distribution factor z1 and hence , the othershave no further effect

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Testability

Bargaining Model

bargaining framework impose additional restrictions

Chiappori and Donni (2006) additional restrictions on the bargaining process and specifically on

the nature of the status quo point any efficient outcome can be constructed as a bargaining solution for

well-chosen status quo values

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p

Testability

Test on Bargaining Model

consider specific assumption as an example:

some distribution factors are known to positively(negatively)correlated with member Bs threat point

then the weight on B, should be increasing (decreasing) in the

factor Proposition 3: Assume that is known to be increasing in z1 and

decreasing in z2. Then, the demand functions consistent with anybargaining model are such that:

i/z1i/z2= j/z1j/z2

0,i= 1, , n,j = 1, n

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Separability

Focus on Caring Preference

impose separability in the preferences of the two household members

refer to the resulting restricted preference as caring:

(16) um(qA, qB,Q; a) = m[A(qA,Q; a), B(qB,Q; a); a]

call m ms felicity function

no externalities for individual felicities altruism works only through their felicity functions

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Identifiability

Sharing Rule Approach

in the followings assume z is a scalar w.l.o.g

Proposition 4 Let (qA, qB) be functions of (x, a, z) compatible withcollective rationality, (16) and (17). Then, there exists a function(x, a, z) such that qm is a solution to

vm(qm; a) s.t. eqm = xm

where

xA = (x, a, z)

xB = x (x, a, z)

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Identifiability

Implication on the Demand (General)

Proposition 5: Assume collective rationality, (16) and (17). Then,

(i) there exists a real-valued function and 2n real-valued functions iand i s.t. for i = 1, , n,

(18)qi(z,x

) = i[(z,x

)] + i[x

(z,x

)

(ii) there exist two real-valued functions F and G s.t. for t, s R+,i = 2, , n,

(19)

i[t+ s,F(t) + G(s)]

= i[t, F(t) + G(0)] + i[s, F(0) + G(S)] i[0, F(0) + G(0)]

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Identifiability

Exclusive and Assignable Good

Assignable goods: goods for which we can observe how much eachperson consumes

Exclusive goods: goods for which consumed by one person only

an exclusive good is assignable an assignable good can be regarded as a pair of exclusive goods

with price variation, they are regarded as two exclusive good with thesame price

without price variation, there is no such restriction

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Id ifi bili


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Identifiability

Identification of Sharing Rule (One Exclusive Good)

Proposition 6: Assume collective rationality, (16) and (17). If theconsumption of exactly one exclusive good (for, say, A) is observed,

and if the demand function of member A for this good is strictlymonotone, then we can recover the sharing rule (z, x) up to astrictly monotone transformation. That is, if(z, x) is one solution,then any solution is of the form F[(z, x)] where F is strictlymonotone.

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Id tifi bilit


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Identifiability

Implication on the Demand (Two Exclusive Goods)

Proposition 7: Assume collective rationality, (16) and (17).Assume that good 1 is exclusive for A and 2 for B. Consider anopen set on which 2/x= 0, 2/q1 = 0. Then, the followingequivalent conditions hold:

(i) there exists a function F s.t. s, t 0,

(20) 2[t+ s,F(t)] = 2[s, F(0)]

(ii) there exist two functions and g s.t.

(21) 2(x, q1) = [x g(q1)]

(iii) 2 satisfies

(22)

x

h2/q12/x

i= 0

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Identifiability


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Identifiability

Identification of the Sharing Rule (Two Exclusive Goods)

Proposition 8: Assume collective rationality, (16) and (17).Assume that good 1 is exclusive for A and 2 for B. Assume that thedirect demand for both goods are observed and that thecorresponding z-conditional demand for good 2 fulfills the necessaryconditions of Proposition 7. Then, the sharing rule is given, up to an

additive constant, by the following equivalent differential equations:(i)

(24) g(q1) = 2/q12/x

, (x, z) = g[q1(x, z)]

(ii)

(25)

x=

q1/xq1/z

q1/xq1/z

q2/xq2/z

,

z=

1q1/xq1/z

q2/xq2/z

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Identifiability


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Identifiability

Implication on the Demand (One Exclusive Good)

Proposition 9: Assume collective rationality, (16) and (17). Assumethat good 1 is exclusive for A and that good 2 is private jointconsumption good. Consider an open set on which 22/

2x= 0and 22/xq= 0. Then, the following, equivalent properties hold:

(i) there exists a function F s.t. s, t> 0,

(26) 2[t+ s,F(t)] = 2[t, F(t)] + 2[s, F(0)] 2[0, F(0)]

(ii) there exist three functions , , and g s.t.

(27) 2(x, q1) = [g(q1)] + [x g(q1)]

(iii)

(28)

x

h22/xq22/x2

i= 0

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Identifiability


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Identifiability

Identification of Individual Engel Curves

differentiate (18) w.r.t. z and x gives:

qiz

= (i

i)

zqi

x

= (i

i)

x

+ i

solve this for i and

i to get:

i =/zqi/x+ (1 /x)qi/z

/z

i =/zqi/x /xqi/z

/z

given the partial derivatives of, i and i (individual Engel curves)are identified up to a constant

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bourguignon browning, and chiappori (2009) on twitter

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