Outline
Introduction
Frisch Elasticity of Labor Supply
Daolu Cai
April 8, 2010
Daolu Cai
Frisch Elasticity of Labor Supply
Outline
Introduction
Introduction
Daolu Cai
Frisch Elasticity of Labor Supply
Outline
Introduction
Type of Supply Function of Labor
I
Marshallian labor supply. Hold Income constant.
Daolu Cai
Frisch Elasticity of Labor Supply
Outline
Introduction
Type of Supply Function of Labor
I
Marshallian labor supply. Hold Income constant.
I
Hicksian labor supply. Hold utility level constant.
Daolu Cai
Frisch Elasticity of Labor Supply
Outline
Introduction
Type of Supply Function of Labor
I
Marshallian labor supply. Hold Income constant.
I
Hicksian labor supply. Hold utility level constant.
I
Frisch labor Supply. Hold the marginal utility of wealth
constant.
Daolu Cai
Frisch Elasticity of Labor Supply
Outline
Introduction
Frisch Elasticity
The Frisch labor supply Elasticity ( constant marginal utility of
wealth) is defined as
∂n w
kλ
(1)
ηλ =
∂w n
Daolu Cai
Frisch Elasticity of Labor Supply
Outline
Introduction
Simple Standard Consumer Problem
max
ct ,at+1 ,n
∞
X
β t U(ct , nt )
(2)
t=0
subject to:
ct + at+1 = (1 + r )at + wt nt
I
multiplier λt
β is the discount factor, at is the household asset in the
beginning of period t, and nt , units of labor.
Daolu Cai
Frisch Elasticity of Labor Supply
(3)
Outline
Introduction
FOC’s
The FOC’s are:
Uct = λt
(4)
−Unt = λt wt
(5)
λt = β(1 + r )λt+1
(6)
Daolu Cai
Frisch Elasticity of Labor Supply
Outline
Introduction
FOC’s
Notice (4) and (5) are in function of λ and w:
∂U(c(λ, w ), n(λ, w ))
=λ
∂c
(7)
∂U(c(λ, w ), n(λ, w ))
= −λw
∂n
(8)
Daolu Cai
Frisch Elasticity of Labor Supply
Outline
Introduction
Computing Frisch Elasticity
Taking derivative of (4) and (5) with respect to w, we have
Ucc
∂c
∂n
+ Ucn
=0
∂w
∂w
∂n
∂c
+ Unn
= −λ
∂w
∂w
Notice this is a system of 2 equations and two unknowns.
Unc
Daolu Cai
Frisch Elasticity of Labor Supply
(9)
(10)
Outline
Introduction
Computing Frisch Elasticity
Solving for
∂n
∂w
we have:
λUcc
∂n
= 2
∂w
[Ucn − Unn Ucc ]
(11)
Replacing λ from (5) we have
− Un Ucc
∂n
= 2 w
∂w
[Ucn − Unn Ucc ]
Daolu Cai
Frisch Elasticity of Labor Supply
(12)
Outline
Introduction
Computing Frisch Elasticity
Using the last expression we have that the Frisch labor supply
elasticity is
Un
ηλ =
n[Unn −
Daolu Cai
2
Ucn
Ucc ]
Frisch Elasticity of Labor Supply
(13)
Outline
Introduction
Example: Separable Utility
Consider the following utility function
1
U(c, n) = ln(c) − α
FOC’s:
n1+ ν
1 + ν1
1
=λ
c
1
αn ν = λw
Daolu Cai
Frisch Elasticity of Labor Supply
(14)
(15)
(16)
Outline
Introduction
Example: Separable Utility
The labor supply is given by
n=(
w ν
)
αc
(17)
The Fischer elasticity is given by
ηλ = ν
Daolu Cai
Frisch Elasticity of Labor Supply
(18)
Outline
Introduction
Example: Non Separable Utility
Consider the another utility function
U(c, n) =
(c γ (1 − n)1−γ )(1−σ)
1−σ
(19)
The Frisch Elasticity is given by
ηλ =
1 − n 1 − γ(1 − σ)
[
]
n
σ
Daolu Cai
Frisch Elasticity of Labor Supply
(20)