Collective Labor Supply and Welfare Collective Labor Supply and Welfare Written by mixingale@twitter for private study June 24, 2010 1/1
Collective Labor Supply and Welfare
Collective Labor Supply and Welfare
Written by mixingale@twitter for private study
June 24, 2010
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Collective Labor Supply and Welfare
Introduction
Summary
◮ Construct a collective model of household consumption and laborsupply
◮ impose Pareto efficiency assumption
◮ old models such as neoclassical labor supply model and bargainingmodel also imply Pareto efficiency
◮ trim unnecessary assumptions
◮ show that the program is equivalent to a representation with“sharing rule” (Proposition 1)
◮ derive restrictions on labor supply functions (Proposition 2)◮ derive identification conditions for model primitives (Proposition 3)
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Collective Labor Supply and Welfare
Settings
Model Component
◮ Setting:
◮ members: i = 1, 2◮ leisures: Li → Labor supplies: T − Li , observed◮ wages: wi , observed◮ Hicksian composite consumption: C i , unobserved◮ aggregate consumption: C ≡ C 1 + C 2, observed◮ price: 1◮ ← observer has a cross-sectional data◮ utilities: U i(Li ,C i )
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Collective Labor Supply and Welfare
Settings
Pareto Efficiency Approach
◮ Definition: (L1(w1,w2, y), L2(w1,w2, y)), together with anconsumption function defined by the budget constraint, is said to becollectively rational if there exists (C 1(w1,w2, y),C 2(w1,w2, y))and some function u2(w1,w2, y) s.t. ∀(w1,w2, y), the followingshold:
◮ C 1(w1,w2, y) + C 2(w1,w2, y) = C(w1,w2, y)◮ (L1, L2,C 1,C 2) is a solution to:
(P)
maxU1(L1,C 1)
s.t. µ : U2(L2,C 2) ≥ µ2
λ : w1L1 + w2L
2 + C1 + C
2≤ (w1 + w2)T + y
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Collective Labor Supply and Welfare
Settings
Sharing Rule Approach
◮ First, nonlabor income y is shared between the members withψ(w1,w2, y) for 1 and y − ψ(w1,w2, y) for 2, where ψ can benegative or greater than y
◮ Then, each member independently solve the problem:
(Pi )maxU i (Li ,C i)
s.t. wiLi + C i ≤ wiT + ψi (w1,w2, y)
where ψ1 = ψ and ψ2 = y − ψ
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Collective Labor Supply and Welfare
Settings
Proposition 1: Equivalence
◮ Proposition 1: Le L1(w1,w2, y) and L2(w1,w2, y) be arbitraryfunctions. Then, there exists a function u2(w1,w2) s.t. L1 and L2
are solutions of (??) if and only if there exists a functionψ(w1,w2, y) s.t. Li is a solution of (??)
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Collective Labor Supply and Welfare
Settings
Proof
◮ Necessity:
◮ let L1∗(w1,w2, y ; u2) and L2∗(w1,w2, y ; u2) be a solution to (??)given a function u2 together with C 1∗(w1,w2, y ; u2) andC 2∗(w1,w2, y ; u2)
◮ ψ(w1,w2, y ; u2) ≡ w1(T − L1∗(w1,w2, y ; u2))− C 1∗(w1,w2, y ; u2)◮ 1’s budget constraint is now:
w1L1 +C
1−w1T ≤ w1L
1∗(w1,w2, y ; u2)+C1∗(w1,w2, y ; u2)−w1T
◮ If L1∗(w1,w2, y ; u2) and C 1∗(w1,w2, y ; u2) were not solutions to (P1)with this constraint, it implies that 1 could raise his utility withoutreducing 2’s expenditure, contradiction
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Collective Labor Supply and Welfare
Settings
Proof
◮ Sufficiency:
◮ let L1∗(w1,w2, y ;ψ) and L2∗(w1,w2, y ;ψ) be a solution to (??) givena function ψ together with C 1∗(w1,w2, y ;ψ) and C 2∗(w1,w2, y ;ψ)
◮ u2(w1,w2, y ;ψ) ≡ U2(L2∗(w1,w2, y ;ψ),C 2∗(w1,w2, y ;ψ))◮ w2L
2∗(w1,w2, y ;ψ) + C 2∗(w1,w2, y ;ψ) = e2(w2, u2(w1,w2, y ;ψ))where e2 is a expenditure function associated with U2
◮ hence, any pair (L2′ ,C 2′) providing the utility level at leastu2(w1,w2, y ;ψ) is no less than e2(w2, u2(w1,w2, y ;ψ))
◮ if (L2′ ,C 2′) 6= (L2∗(w1,w2, y ;ψ),C 2∗(w1,w2, y ;ψ) attains the
optimum of (??), then, 1 can adjust (L2′ ,C 2′) so that the costreduces down to e2(w2, u2(w1,w2, y ;ψ)), contradiction
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◮ no distinct pair of (L2,C 2) attains the same cost?
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Collective Labor Supply and Welfare
Characterization
Three Questions
◮ Implication:
◮ what kind of restrictions does the model impose on L1 and L2?
◮ Integrability:
◮ it is possible to recover a sharing rule and a pair of individualpreferences from any pair of labor supply functions satisfying theabove restrictions?
◮ Uniqueness:
◮ it is uniquely determined from observations?
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Collective Labor Supply and Welfare
Characterization
Regularity Conditions
◮ Continuous differentiability:
◮ Li is three-times continuously differentiable◮ ψ is twice continuously differentiable
◮ Notation:
◮ Xz ≡ ∂X/∂z , A ≡ L1w2/L1
y , B ≡ L2w1/L1
y whenever L1yL
2y 6= 0
◮ Assumption R: for almost all (w1,w2, y) ∈ R2+ × R
◮ L1y 6= 0, L2
y 6= 0◮ ABy − Bw2 6= BAy − Aw1
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Collective Labor Supply and Welfare
Characterization
Proposition 2: Implication
◮ Proposition 2: If (L1, L2) is a solution to (P1) and (P2) for somecontinuously differentiable sharing rule ψ, respectively, then, itgenerically satisfies:
◮ αyA + αAy − αw2 = 0◮ βyB + βBy − βw1 = 0◮ L1
w1− L1
y [(T − L1 − βB)/α] ≤ 0◮ Lw2 − L2
y [(T − L2 − αA)/β] ≤ 0◮ where
α =
8
<
:
“
1−BAy − Aw1
ABy − Bw2
”
−1
if ABy − Bw2 6= 0
0 otherwise
β = 1− α =“
1−ABy − Bw2
BAy − Aw1
”
−1
�
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Collective Labor Supply and Welfare
Characterization
Proposition 3: Integrability
◮ Proposition 3: If three-times continuously differentiable functionsL1 and L2 satisfy Assumption R and the necessary conditions ofProposition 2, then, for w ≡ (w 1,w 2, y), an arbitrary point inR2
++ × R , there exists a neighborhood of w , V s.t:
◮ there exists a sharing rule ψ defined over V◮ there exists a pair of utility functions (U1,U2) with the property that
the solution of (Pi ), at any point of V, is the couple (Li ,C i ) forsome C i ≥ 0
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Collective Labor Supply and Welfare
Characterization
Proposition 4: Uniqueness
◮ Proposition 4: Under the same hypothesis as Proposition 3:
◮ the sharing rule is defined up to an additive constant k ; specifically,its partials are given by:
ψy = α
ψw2 = Aα
ψw1 = B(α− 1) = −βB
◮ for each k , the preferences represented by U1 and U2 are uniquelydefined
◮ the indifference curve corresponding to different values of k can bededuced from one another by translation
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Collective Labor Supply and Welfare
Welfare Analysis
Welfare Analysis using Collective Utility Model
◮ We can use U1 and U2 deduced using proposition 4 for welfareanalysis:
◮ Corollary 1: Let ψ,U1 and U2 be associated with a pair of givenlabor suppy functions in the sense of Proposition 3 and 4. If U1 andU2 is increased by the reform, then, for any k , U1
k and U2k are also
increased by the reform
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◮ ψk (w1,w2, y) = ψ(w1,w2, y) + k◮ U1
k (L,C) = U1(L,C − k)◮ U2
k (L,C) = U2(L,C + k)
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Collective Labor Supply and Welfare
Welfare Analysis
Indirect Collective Utility
◮ Now we can define collective indirect utility functions v1(w1,w2, y)
◮ note that this function implicitly indexed by ψ
◮ Let V i (wi ,Y ) be the traditional indirect utility functions, then:
v1(w1,w2, y) = V 1(w1,w1T + ψ(w1,w2, y))
v2(w1,w2, y) = V 2(w2,w2T + y − ψ(w1,w2, y))
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Collective Labor Supply and Welfare
Welfare Analysis
Proposition 5: Comparative Statistics
◮ Let v1 and v2 be indirect utilities associated with L1 an L2, then:
v1w1
= λ(T − L1 − βB)
v2w1
=λ
µβB
v1w2
= λαA
v2w2
=λ
µ(T − L2 − αA)
v1y = λα
v2y =
λ
µβ
�
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Collective Labor Supply and Welfare
Welfare Analysis
Indifference Curve
◮ Let define ΨK by: v1(w1,w2, y) = K ⇔ y = ΨK (w1,w2)
◮ This defines the generic indifference curve:
ΨKw1
= −v1w1
v1y
ΨKw2
= −v1w2
v1y
⇔ ΨKw1
= −1
α(T − L1
− βB)
ΨKw2
= −A
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Collective Labor Supply and Welfare
Example
Start from a Particular Functional Form
◮ Suppose that our estimates of the labor supply functions have thefollowing forms:
L1 = a1 + b1y + c1y log y + d11w1 + d2
1w2
L2 = a2 + b2y + c2y log y + d12w1 + d2
2w2
◮ Starting from these estimates, we try to recover ψ and v i by:
◮ calculate A,B, α, and β◮ check the identification conditions in Proposition 3◮ derive sharing rule ψ using Proposition 4◮ derive collective indirect utility v i using Proposition 5
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Collective Labor Supply and Welfare
Example
Calculate A,B , α, and β
◮ Straightforward calculation gives us:
A = d21 (b1 + c1 + c1 log y)−1
B = d12 (b2 + c2 + c2 log y)−1
α =c2
c2b1 − b2c1(b1 + c1 + c1 log y)
β =c1
c1b2 − b1c2(b2 + c2 + c2 log y)
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Collective Labor Supply and Welfare
Example
Check the Identification Conditions
◮ Straightforward calculation shows that (a) and (b) hold:
αyA + αAy − αw2 = 0
βyB + βBy − βw1 = 0
◮ (c) and (d) are now:
d11 +
c1
c2d1
2 −D
c2(T − L1) ≤ 0
d22 +
c2
c1d2
1 −D
c1(T − L2) ≤ 0
◮ assume that the parameters satisfy these conditions
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Collective Labor Supply and Welfare
Example
Derive ψ using Proposition 4
◮ By Proposition 4, the sharing rule ψ satisfy the following partialequations:
ψy = α =c2
D(b1 + c2 + c1 log y)
ψw2 = Aα =d2
1 c2
D
ψw1 = −βB = −d1
2 c1
D
◮ ψ is identified up to the location parameter k :
ψ =c2
D(b1y + c1y log y) +
d21 c2
Dw2 +
d12 c1
Dw1 + k
where D = c1b2 − b1c2
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Collective Labor Supply and Welfare
Example
Derive vi using Proposition 5 (1/2)
◮ Let Ψ be the indifference curve defined in Slide p.17◮ By Proposition 5, we have:
Ψw1 = −1
α(T − L1 − βB)
Ψw2 = −A
◮ Defining θ ≡ b1Ψ + c1Ψ log Ψ, we have:
θw2 = −d21
θw1 =D
c2(−T + a1 + θ1 + d1
1 w1 + d21 w2 −
c1d12
D)
◮ Solving these differential equations, we obtain:
θ = KeDw1/c2 − d11 w1 − d2
1 w2 +d1
1 c2 + d12 c1
D+ T − a1
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Collective Labor Supply and Welfare
Example
Derive vi using Proposition 5 (2/2)
◮ Since v1(w1,w2,ΨK (w1,w2)) = K , we have:
v1(w1,w2, y)
= e−Dw1/c2(−γ1 + b1y + c1y log y + d11w1 + d2
1w2)
where γ1 =d1
1 c2+d12 c1
D+ T − a1
◮ By symmetry,
v2(w1,w2, y)
= e−Dw2/c1(−γ2 + b2y + c2y log y + d12w1 + d2
2w2)
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Collective Labor Supply and Welfare
Discussion
Bergsonian Index
◮ Suppose the household behavior can be represented by themaximization of Bergsonian Index W (U1,U2)
◮ with this index, the result is always Pareto efficient◮ it requires stronger conditions for identification
◮ Proposition 6: Let Π denote the following program:
(Π)maxW [U1(L1,C 1),U2(L2,C 2)]
s.t. w1L1 + w2L
2 + C 1 + C 2 ≤ (w1 + w2)T + y
then, there exists C 1,C 2,U1,U2 and W s.t. L1 and L2 are solutionsof (??) if and only if
◮ necessary conditions of Proposition 2 hold◮ the Slutsky conditions hold:
L1w2− (T − L
2)L1y = L
2w1− (T − L
1)L2y
L1w1− (T − L
1)L1y ≤ 0, L2
w2− (T − L
2)L2y ≤ 0
�
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Collective Labor Supply and Welfare
Discussion
Bargaining Model
◮ Suppose the household behavior can be represented by Nashbargaining program:
(NB)max[U1(L1,C 1) − U
1][U2(L2,C 2) − U
2]
s.t. w1L1 + w2L
2 + C 1 + C 2≤ (w1 + w2)T + y
◮ any solution of (??) is Pareto efficient◮ entails several degrees of freedom◮ collective model provide a useful framework for testing more specific
approaches such as bargaining
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Collective Labor Supply and Welfare
Extensions
Caring
◮ Assume that i maximizes some altruistic indexW i [U1(L1,C 1),U2(L2,C 2)] instead of egoistic index U1(L1,C 1)
◮ this does not fundamentally alter the conclusions of the model
◮ Relax the separability assumption and assume that i maximizesU i (L1,C 1, L2,C 2)
◮ no uniqueness conclusion can be expected to hold
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Collective Labor Supply and Welfare
Extensions
Possible Extensions
◮ Possible extensions:
◮ multiple consumption goods◮ public goods (e.g. expenditures for their children)◮ multiple sources of income
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