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