Micro I Midterm, February 23, 2012
1. There are two goods. Suppose a consumer has the convex utility function u(x1 , x2 ) = x1 + x22 and
consumption set X = R2+ . Let prices be p = (1, p) 0 and income be m > 0. Suppose further that
ū ≥ 0. If possible for this utility function:
a) Find the Marshallian (ordinary) demand x(p, m).
Answer: The indifference curves are parabolas. Because of the convexity of the utility function,
maximum utility will be at a corner. The two possibilities are (m, 0) and (0, m/p), If p2 > m,
the first one is better, if p2 < m, the second is better, and if p2 = m, they are tied. It follows
that the Marshallian demand is:

(m, 0)
if p2 > m




2
m, 0 , 0, m
if p2 = m .
x(p, m) =
p2



 m
if p2 < m
0, p
b) Compute the indirect utility function v(p, m).
Answer: Indirect utility is v(p, m) = u(x(p, m)) = max{m, m2 /p2 }.
c) Find the Hicksian (compensated) demand h(p, ū).
Answer: The convex indifference curves imply that expenditure minization also occurs at a
√
corner. We set u(x) = ū to find the corners are (ū, 0) and (0, ū). The associated expenditures
√
√
√
are ū and p ū. We choose the first if ū < p and the second if ū > p. Thus
√

(ū, 0)
if ū < p


√ √
(ū, 0), (0, ū)
h(p, ū) =
if ū = p .

√
√

(0, ū)
if ū > p
d) Compute the expenditure function e(p, ū).
√
Answer: Expenditure is e(p, ū) = p·h(p, ū) = min{ū, p ū}.
2. There are two consumers and two goods. Consumer 1 has utility u1 (x) = (x1 x2 )1/2 and income m1 > 0.
Consumer 2 has utility u2 (x) = min(x1 , 2x2 ) and income m2 > 0. Prices are p = (p1 , p2 ) 0.
a) Find the Marshallian demand functions for each consumer.
Answer: Equal-weighted Cobb-Douglas utility requires consumer 1 spend half of his income on
each good, so x1 (p, m1 ) = (m1 /2)(1/p1 , 1/p2 ). The second consumer has Leontief preferences
with x1 = 2x2 at the optimum. Total spending is then m2 = p1 x1 + p2 x2 = (2p1 + p2 )x2 so
x2 (p, m2 ) = (2m2 /(2p1 + p2 ), m2 /(2p1 + p2 )).
b) Compute aggregate demand.
Answer: We add the consumer demands to obating aggregate demand. It is
m1
2m2
m1
m2
x(p, m1 , m2 ) =
+
,
+
.
2p1
2p1 + p2 2p2
2p1 + p2
c) Show by example that aggregate demand cannot be written as a function of the price vector p
and aggregate wealth m = m1 + m2 .
Answer: Now x(p, 2, 0) = (1/p1 , 1/p2 ) and x(p, 0, 2) = (2p1 + p2 )−1 (4, 2). Since these only
agree when 2p1 = p2 , the different income distributions yield different demand curves.
MICRO I MIDTERM, FEBRUARY 23, 2012
Page 2
3. Suppose Y is a convex technology set (also is non-empty, closed, obeys inaction, no free lunch, and free
disposal). Suppose y 0. Show there is a price vector with p·y > π(p) (20 points). Show that p ≥ 0
(5 points).
Answer: Now Y is a closed convex set and y is a point outside the set. By the Separation Theorem,
there are p 6= 0 and α ∈ R with p·y > α > p·z for all z ∈ Y . Taking the supremum over z ∈ Y , we find
y·y > α ≥ π(p).
Now consider −ne` ∈ Y by free disposal for n > 0. Then α > p · (−ne` ) = −np` . It follows that
α/n > −p` . Letting n → +∞, we find 0 ≤ p` . Since ` was arbitrary, and p 6= 0, p > 0.
p
4. A consumer with X = R2+ has expenditure function e(p, ū) = p1 + p2 + p1 p2 + (p1 + 2p2 )ū.
a) Find the Hicksian demand function h(p, ū).
Answer: Note that e is concave and homogeneous of degree 1 in prices, as an expenditure
function should be. We apply the Shephard-McKenzie Lemma, which says h = Dp e. Then
h(p, ū) =
r
r
1 p2
1 p1
1+
+ ū, 1 +
+ 2ū
2 p1
2 p2
b) Find the indirect utility function v(p, m).
Answer: We use the duality relation m = e(p, v(p, m)) to find indirect utility. Then m =
p
p1 + p2 + p1 p2 + (p1 + 2p2 )v(p, m), so
m − p1 − p2 −
v(p, m) =
p1 + 2p2
p
p1 p2
c) Find the Marshallian demand function x(p, m).
Answer: According to Roy’s Identity, x` (p, m) = −(∂v/∂p` )/(∂v/∂m). Now ∂v/∂m = 1/(p1 +
2p2 ). After a short calculation, we find
x1 (p, m) = 1 +
x2 (p, m) = 1 +
1
2
r
1
2
r
m − p1 − p2 −
p2
+
p1
p1 + 2p2
p
p1 p2
m + p2 −
=
p
m − p1 − p2 − p1 p2
p1
+2
=
p2
p1 + 2p2
1
2
p
p1 p2 + p2
p1 + 2p2
p
2m − p1 − p1 p2 +
p1 + 2p2
q
p1
2
p2
p1
q
p1
p2