chiappori (1992) on twitter

Collective Labor Supply and Welfare


Written by mixingale@twitter for private study

June 24, 2010

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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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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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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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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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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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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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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

�

◮ no distinct pair of (L2,C 2) attains the same cost?

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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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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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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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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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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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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

�

◮ 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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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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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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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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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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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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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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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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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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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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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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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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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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Extensions

Possible Extensions

◮ Possible extensions:

◮ multiple consumption goods◮ public goods (e.g. expenditures for their children)◮ multiple sources of income

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