Economics 401
Sample questions 1
1. Consider a price-taking consumer in a two-good world. Let w denote money wealth,
(p1 , p2 ) prices, xj (p1 , p2 , w) Marshallian demands, hj (p1 , p2 , u) Hicksian demands, V (p1 , p2 , w)
the indirect utility function and e (p1 , p2 , u) the expenditure function. Fill in the following
tables, and verify the Slutsky equation for the effect of changing the price of good 1 on the
demand for good 2, for someone whose preferences are represented by:
u = x1 x 2 .
x1 (p1 , p2 , w) x2 (p1 , p2 , w)
h1 (p1 , p2 , u)
h2 (p1 , p2 , u)
V (p1 , p2 , w)
e (p1 , p2 , u)
Range
Range
ANSWER
x1 (p1 , p2 , w) x2 (p1 , p2 , w)
1 w
2 p1
1 w
2 p2
h1 (p1 , p2 , u)
−1/2 1/2
p1 p2 u1/2
h2 (p1 , p2 , u)
1/2 −1/2
p1 p2 u1/2
V (p1 , p2 , w)
1 w2
4 p1 p2
e (p1 , p2 , u)
1/2 1/2
2p1 p2 u1/2
Range
p1 , p2 > 0, w ≥ 0
Range
p1 , p2 > 0, u ≥ 0
Slutsky equation: from the duality between the UMP and the EMP we know
x2 (p1 , p2 , e (p1 , p2 , u)) = h2 (p1 , p2 , u) .
Differentiate this with respect to p1 and use the envelope theorem to obtain
∂x2
∂x2
∂h2
+ x1
=
∂p1
∂w
∂p1
LHS = 0 +
1w1 1
2 p1 2 p2
1/2 1/2
But w = e (p1 , p2 , u) = 2p1 p2 u1/2 , so
1 −1/2 −1/2 1/2
LHS =
p
p2 u = RHS
2 1
1
2. Consider a price-taking consumer in a two-good world. Let w denote money wealth,
(p1 , p2 ) prices, xj (p1 , p2 , w) Marshallian demands, hj (p1 , p2 , u) Hicksian demands, V (p1 , p2 , w)
the indirect utility function and e (p1 , p2 , u) the expenditure function. Fill in the following
tables for someone whose preferences are represented by:
u = min (x1 , 2x2 ) .
x1 (p1 , p2 , w) x2 (p1 , p2 , w)
h1 (p1 , p2 , u)
V (p1 , p2 , w)
h2 (p1 , p2 , u)
e (p1 , p2 , u)
Range
Range
ANSWER
x1 (p1 , p2 , w) x2 (p1 , p2 , w) V (p1 , p2 , w)
2w
2p1 +p2
w
2p1 +p2
2w
2p1 +p2
Range
p1 , p2 > 0, w ≥ 0
h1 (p1 , p2 , u) h2 (p1 , p2 , u) e (p1 , p2 , u)
Range
u
u/2
u (p1 + p2 /2) p1 , p2 > 0, u ≥ 0
The Slutsky substitution matrix is
"
∂h1
∂p1
∂h2
∂p1
∂h1
∂p2
∂h2
∂p2
0 0
0 0
S =
=
#
.
3. Susan is a price taker and lives in a three-good world. Denote her money wealth by
w, prices by (p1 , p2 , p3 ) and her consumption bundle by (x1 , x2 , x3 ) . Find her Marshallian
demands if her preferences can be represented by
u = ln x1 + 2 ln x2 + x3 .
2
ANSWER
Solution type
Interior
Corner
x1 (p1 , p2 , p3 , w)
x2 (p1 , p2 , p3 , w)
p3
p1
1 w
3 p1
2p3
p2
2 w
3 p2
x3 (p1 , p2 , p3 , w)
w
−3
p3
0
w
p3
w
p3
Range
−3≥0
−3<0
4. Suppose a consumer’s preferences can be represented by
1/2
1/2
u = 2x1 + 2x2 + x3 .
Find Marshallian demands and the indirect utility function. Find Hicksian demands and
the expenditure function. Be sure to specify the range over which each function is valid.
Verify the Slutsky equation for the effects of a change in the price of good 2 on the demand
for good 3.
ANSWER
The Marshallian demands are:
Solution type
Interior
Corner
x1
x2
p23
p21
p23
p22
x3
p2 w
p1 (p1 +p2 )
p1 w
p2 (p1 +p2 )
w
p3
−
p3
p1
−
p3
p2
0
2
V (p1 , p2 , w)
w
+ pp31 + pp32
p3
1/2
w
p3
w(p1 +p2 )
p1 p2
w
p3
Range
− pp31 − pp32 ≥ 0
−
p3
p1
−
p3
p2
<0
The Hicksian demands are:
Solution type
h1
h2
Interior
p23
p21
p23
p22
Corner
p2 u
2(p1 +p2 )
2
u − 2p3
p1 u
2(p1 +p2 )
2
h
3
1
p1
e(p1 , p2 , u)
+
1
p2
p3 u −
p23
p1
p1 p2 u2
4(p1 +p2 )
0
The Slutsky equation we want is
∂h3
∂x3
∂x3
=
+ x2
∂p2
∂p2
∂w
Verifying we have
Solution type
Interior
Corner
LHS
2p3
p22
0
p3
p22
RHS
p2
+ p23 p13
2
0
3
−
w
p3
w
p3
Range
− pp31 − pp32 ≥ 0
− pp31 − pp32 < 0
p23
p2
Range u − 2p3 p11 + p12 ≥ 0
1
1
u − 2p3 p1 + p2 < 0
5. Consider the utility function
u = xa1 xb2 , a > 0, b > 0.
Assume x1 , x2 > 0.
(a) Under what conditions is this function strictly concave?
(b) Under what conditions is this function strictly quasi-concave?
(c) Use your answers to (a) and (b) to discuss the relationship between concavity and
quasi-concavity.
ANSWER
The function f (x1 , x2 ) = xa1 xb2 , a > 0, b > 0 is strictly concave only if f11 < 0 and
2
> 0. And the function is strictly quasi-concave only if f12 f22 −2f1 f2 f12 +f22 f11 < 0.
f11 f22 −f12
We have
u1 =
u2 =
u11 =
u12 =
u22 =
au
x1
bu
x2
a(a − 1)u
x21
abu
x1 x2
b(b − 1)u
x22
Thus
u11 u22 − u212 =
ab(a − 1)(b − 1) − a2 b2
= ab(1 − a − b)
u2
x21 x22
u2
x21 x22
and
u21 u22 − 2u1 u2 u12 + u22 u11
2
2
au b(b − 1)u
au
bu abu
bu a(a − 1)u
=
−2
+
2
x1
x2
x1
x2 x1 x2
x2
x21
abu3
= 2 2 (a(b − 1) − 2ab + b(a − 1))
x1 x2
abu3
= − 2 2 (a + b)
x1 x2
4
So strict concavity holds only if a+b < 1 but strict quasi-concavity holds for any a, b > 0.
Thus strict quasiconcavity is a weaker condition than strict concavity.
6. Suppose a consumer’s preferences can be represented by
1/2
u = x1 + 2x2 .
Find Marshallian demands and the indirect utility function. Find Hicksian demands and
the expenditure function. Be sure to specify the range over which each function is valid.
Verify the Slutsky equation for the effects of a change in the price of good 2 on the demand
for good 1.
ANSWER
The Marshallian demands are:
Solution type
Interior
Corner
x1
w/p1 − p1 /p2
0
x2
p21 /p22
w/p2
V (p1 , p2 , w)
w/p1 + p1 /p2
2(w/p2 )1/2
Range
w/p1 − p1 /p2 ≥ 0
w/p1 − p1 /p2 < 0
The Hicksian demands are:
Solution type
Interior
Corner
h1
u − 2p1 /p2
0
h2
2
p1 /p22
2
u /4
e(p1 , p2 , u)
p1 u − p21 /p2
p2 u2 /4
Range
u − 2p1 /p2 ≥ 0
u − 2p1 /p2 < 0
The Slutsky equation we want is
∂h1
∂x1
∂x1
=
+ x2
∂p2
∂p2
∂w
Verifying we have
Solution type
Interior
Corner
Range
u ≥ 2p1 /p2
u < 2p1 /p2
LHS
2p1 /p22
0
p1 /p22
5
RHS
Range
2
2
+ (p1 /p2 )(1/p1 ) w/p1 − p1 /p2 ≥ 0
0
w/p1 − p1 /p2 < 0
7. Prove that the expenditure function, e(p1 , p2 , u) is concave in prices (p1 , p2 ).
ANSWER
Consider two sets of prices (p11 , p12 ) and (p21 , p22 ) and the convex combination of them
= t(p11 , p12 ) + (1 − t)(p21 , p22 ) for 0 ≤ t ≤ 1. Then e(p1 , p2 , u) is concave in prices (p1 , p2 )
(pt1 , pt2 )
if
e(pt1 , pt2 , u) ≥ te(p11 , p12 , u) + (1 − t)e(p21 , p22 , u).
Let (h∗1 , h∗2 ) be the expenditure-minimizing bundle for prices (pt1 , pt2 ) and utility level u.
Then
e(pt1 , pt2 , u) =
=
=
≥
pt1 h∗1 + pt2 h∗2
(tp11 + (1 − t)p21 )h∗1 + (tp12 + (1 − t)p22 )h∗2
t(p11 h∗1 + p12 h∗2 ) + (1 − t)(p21 h∗1 + p22 h∗2 )
te(p11 , p12 , u) + (1 − t)e(p21 , p22 , u).
6