ECO 305 — FALL 2003 — September 25
INDIRECT UTILITY FUNCTION
U ∗ (Px , Py , M ) = max { U (x, y) | Px x + Py y ≤ M }
= U (x∗ , y ∗ )
= U (Dx (Px , Py , M), Dy (Px , Py , M ) )
PROPERTIES OF U ∗ :
(1) No money illusion — Homogeneous degree zero:
U ∗ (k Px , k Py , kM ) = U ∗ (Px , Py , M )
(2) As money income changes:
∗
∂U
∂M
=
¯
∂U ¯¯
¯
∂x ¯∗
"
∗
∂x
+
∂M
∗
∗
∂y ∗
∂M
∗
∂x
∂y
+ Py
∂M
∂M
= λ Px
= λ
¯
∂U ¯¯
¯
∂y ¯
∂M
=λ
∂M
(3) As price changes:
∗
∂U
∂Px
=
¯
∂U ¯¯
¯
∂x ¯
∗
∗
∂x
+
∂Px
1
¯
∂U ¯¯
¯
∂y ¯
∗
∂y ∗
∂Px
#
"
∗
∗
#
∂x
∂y
+ Py
∂Px
∂Px
= −λ x∗
(just like M ↓ by x∗ )
= λ Px
(Last step: differentiate adding-up identity w.r.t. Px :
Px x∗ + Py y∗ = M
∂x∗
∂y ∗
+ Py
=0)
x + Px
∂Px
∂Px
Divide price- and income-change equations :
∗
∂U ∗ /∂Px
Roy’s Identity: x = −
∂U ∗ /∂M
∗
(4) Contours of U ∗ in (Px , Py ) space with M fixed:
(Like theater with stage at NE corner)
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EXPENDITURE FUNCTION
Solve the indirect utility function for income:
u = U ∗ (Px , Py , M)
⇐⇒
M = M ∗ (Px , Py , u)
M ∗ (Px , Py , u) = min { Px x + Py y | U (x, y) ≥ u }
“Dual” or mirror image of utility maximization problem.
Economics — income compensation for price changes
Optimum quantities — Compensated or Hicksian demands
x∗ = DxH (Px , Py , u) ,
y ∗ = DyH (Px , Py , u)
PROPERTIES OF M ∗ :
(1) Homogeneous degree 1 in (Px , Py ) holding u fixed:
M ∗ (k Px , k Py , u) = k M ∗ (Px , Py , u)
(2) Hotelling’s or Shepherd’s Lemma —
Compensated demands partial derivatives w.r.t. prices:
DxH (Px , Py , u) = ∂M ∗ /∂Px , DyH (Px , Py , u) = ∂M ∗ /∂Py
Proof: M ∗ = Px DxH + Py DyH , u = U (DxH , DyH ). So
∂M ∗ /∂Px = DxH + Px ∂DxH /∂Px + Py ∂DyH /∂Px
0 = Ux ∂DxH /∂Px + Uy ∂DyH /∂Px
= λ [ Px ∂DxH /∂Px + Py ∂DyH /∂Px ]
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(3) “Weakly” concave in (Px , Py ) holding u fixed.
Cobb-Douglas example: (Px )1/3 (Py )2/3
PROPERTIES OF HICKSIAN DEMAND FUNCTIONS:
(1) Own substitution effect negative:
¯
∂x ¯¯
¯
∂Px ¯
∂DxH
∂ 2M ∗
=
=
≤0
2
∂P
∂P
x
x
u=const
(2) Symmetry of cross-price effects:
∂DyH
∂DxH
∂ 2M ∗
=
=
∂Py
∂Px ∂Py
∂Px
(Net) substitutes if > 0, complements if < 0
General concept : Comparative statics
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COBB-DOUGLAS EXAMPLE
(Direct) UTILITY FUNCTION:
U (x, y) = α ln(x) + β ln(y),
x∗ = α M/Px ,
α+β =1
y ∗ = β M/Py
INDIRECT UTILITY FUNCTION
U ∗ (Px , Py , M) = α [ln(α) + ln(M) − ln(Px ) ]
+β [ln(β) + ln(M ) − ln(Py ) ]
= junk + ln(M ) − α ln(Px ) − β ln(Py )
Roy’s Identity:
− α/Px
αM
∂U ∗ /∂Px
∗
=
−
=
=
x
−
∂U ∗ /∂M
1/M
Px
EXPENDITURE FUNCTION
M ∗ = M ∗ (Px , Py , u) = eu (Px )α (Py )β
Hicksian demand functions
xH = α eu (Px )α−1 (Py )β , y H = β eu (Px )α (Py )β−1
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SLUTSKY EQUATION
Link between Marshallian and Hicksian demands
Equal if u = U ∗ (Px , Py , M ), M = M ∗ (Px , Py , u).
For good i where i may be either x or y,
DiH (Px , Py , u) = DiM (Px , Py , M ∗ (Px , Py , u) )
Now let Pj change, where j may be x or y
∂DiH
∂Pj
∂DiM
∂DiM ∂M ∗
=
+
∂Pj
∂M ∂Pj
∂DiM
∂DiM H
Dj
=
+
∂Pj
∂M
∂DiM
∂DiM M
=
+
Dj
∂Pj
∂M
For example
¯
∂x ¯¯
¯
∂Py ¯
u=const
=
¯
∂x ¯¯
¯
∂Py ¯
∂x
+y
∂M
M =const
Price derivative of compensated demand =
Price derivative of uncompensated demand
+ Income effect of compensation.
If i = j, LHS is negative. Then Giffen implies Inferior
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