Collection of utility functions and corresponding
indirect utility, expenditure and demand functions
corrected version
Thomas Herzfeld
Advanced Microeconomics ECH-32306
September 23, 2011
Introduction
This list serves as a tool to check your own work. Starting from the various
utility functions try to find the corresponding demand, indirect utility and
expenditure functions. Prove their respective properties.
CES utility function u(x) = (xρ1 + xρ2 )1/ρ where 0 6= ρ < 1
Marshallian demand functions:
x1 (p, y) =
pr−1
pr−1
1 y
2 y
and
x
(p,
y)
=
with r = ρ/(ρ − 1)
2
pr1 + pr2
pr1 + pr2
Indirect utility function:
v(p, y) = y(pr1 + pr2 )−1/r
Expenditure function:
e(p, u) = u(pr1 + pr2 )1/r
Hicksian demand functions:
xh1 (p, u) =
pr−1
p1r−1 u
h
2 u
and
x
(p,
u)
=
2
r
r
r
1−1/r
(p1 + p2 )
(p1 + pr2 )1−1/r
Cobb-Douglas utility function u(x) = xα1 x21−α where 0 < α < 1
Marshallian demand functions:
x1 (p, y) =
αy
(1 − α)y
and x2 (p, y) =
p1
p2
Indirect utility function:
α α
1 − α 1−α
v(p, y) = y
p1
p2
1
Expenditure function:
e(p, u) = u
p α p 1−α
1
2
α
1−α
Hicksian demand functions:
xh1 (p, u) = u
αp2 1−α
(1 − α)p1 α
and xh2 (p, u) = u
(1 − α)p1
αp2
Quasi-linear utility function u(x) = v(x1 ) + x2
1/2
Let’s look at a concrete example of this function u(x) = x1 + x2 .
Marshallian demand functions:
( y
p1 if y ≤ p2
x1 (p, y) =
p22
if y > p2
4p2
1
x2 (p, y) =
y−p1 x1
p2
=
y
p2
−
0 if y ≤ p2
if y > p2
p2
4p1
Indirect utility function:
 1/2

y
if y ≤ p2
p
1
v(p, y) =
 y + p2 if y > p2
p2
4p1
Expenditure function:
(
e(p, u) =
p1 u2 if y ≤ p2
up2 −
p22
2p1
+
p22
4p1
= up2 −
p22
4p1
if y > p2
Hicksian demand functions:
( 2
u if y ≤ p2
h
x1 (p, u) =
p22
if y > p2
4p2
1
xh2 (p, u) =
u−
0 if y ≤ p2
if y > p2
p2
2p1
Linear utility function u(x) = x1 + x2
Marshallian demand functions:

y/p1

any number between 0 and y/p1
x1 (p, y) =

0

0

any number between 0 and y/p1
x2 (p, y) =

y/p2
2
if p1 < p2
if p1 = p2
if p1 > p2
if p1 < p2
if p1 = p2
if p1 > p2
Indirect utility function:
y/p1 if p1 < p2
v(p, y) =
y/p2 if p1 > p2
Expenditure function:
up1 if p1 < p2
e(p, u) =
up2 if p1 > p2
Hicksian demand functions: Apply Shephard’s lemma to the expenditure function yields straight vertical Hicksian demand functions.
xh1 (p, u) = uif p1 < p2
xh2 (p, u) = uif p2 > p1
Stone-Geary utility function u = (x1 − a1 )b1 (x2 − a2 )b2 where b1 , b2 ≥ 0
and b1 + b2 = 1
This is the utility function underlying the Linear Expenditure System.
Marshallian demand functions:
x1 (p, y) = a1 +
b1
b2
(y − p2 a2 ) and x2 (p, y) = a2 + (y − p1 a1 )
p1
p2
Indirect utility function:
b1 b2
b2
b1
(y − p2 a2 )
(y − p1 a1 )
v(p, y) =
p1
p2
Expenditure function:
e(p, u) = p1 a1 + p2 a2 + e
u
p1
b1
b1 p2
b2
b2
Hicksian demand functions:
b2 u p1 b1 p2 b2
b1 u p1 b1 p2 b2
h
h
and x2 (p, u) = a2 + e
x1 (p, u) = a1 + e
p1
b1
b2
p2
b1
b2
Leontief utility function u = min {x1 , x2 }
y
Marshallian demand functions: x1 = x2 = x(p, y) = p1 +p
2
y
Indirect utility function: v(p, y) = p1 +p2
Expenditure function: e(p, u) = u(p1 + p2 )
Substitution between commodities in the angle would not change utility. Therefore, the Hicksian demand functions are constant xh1 = u
and xh2 = u
3