MICROECONOMIC THEORY I. ECONOMICS 665. FALL 2014.
PROFESSOR: Alex Anas. TA: Yiqian Lu. MIDTERM EXAM 2.
TIME: Thursday, Nov. 6; 3:30-5:30 PM. PLACE: 424 FRONCZAK.
Books, notes or calculators are not allowed in the examination room.
Please answer each question. TOTAL POINTS = 100.
Solution
PROBLEM 1 (40 points)
There are two goods and a consumer's preference are given by the utility function
𝑈(𝑥1 , 𝑥2 ) = 𝑎√𝑥1 + 𝑥2 , where 𝑥1 is the quantity of good 1 and 𝑥2 is the quantity of
good 2. a > 0 is a parameter.
(i)
Assume that the income m is fixed. Find the Marshallian demand functions
for both goods. (10 points)
(ii)
Derive the indirect utility function. (10 points)
(iii)
Suppose that both commodities are consumed in positive amounts. There
is a change in the price of good 1 from 𝑝1 to 𝑝1′ (𝑝1′ > 𝑝1). Define the
compensating variation (CV) as V(𝑝1, 𝑝2 , m)=V(𝑝1′, 𝑝2 , m-CV). Solve for
the compensating variation (CV) when price increases from 𝑝1 to 𝑝1′ (15
points)
(iv)
Suppose that, as in (iii), both goods are consumed in positive
quantities. Explain the signs of the income effects. (5 points)
Answer:
The utility function is quasi-linear which helps to determine the solution.
(i) The consumer wants to choose 𝑥1 to max
m − x1 p1
a√x1 +
p2
1
FOC is
a −2
x
2 1
−
p1
p2
=0
So the demand function is:
1
x1 (p1 , p2 , m) = (
ap2 2
)
2 p1
and x2 (p1 , p2 , m) =
𝑚
𝑝2
−
𝑎 2 p2
4 p1
But the above solution is for an interior solution ( x1 , x2  0) and ignores
that we may be at a corner.
Since 𝑥2 > 0 holds strictly when we have an interior solution, we must
have p1 > p* =
a2 p22
4m
Finally the Marshallian demand function is
ap2 2
) , 𝑤ℎ𝑒𝑛 𝑝1 > 𝑝 ∗
2
p
1
x1 (p1 , p2 , m) =
𝑚
, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
{
𝑝1
(
𝑚 𝑎2 p2
−
, 𝑤ℎ𝑒𝑛 𝑝1 > 𝑝 ∗
x2 (p1 , p2 , m) = {𝑝2 4 p1
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝑎 2 p2
(ii) V(p1 , p2 , m) = {
4p1
+
𝑚
𝑝2
, 𝑤ℎ𝑒𝑛 𝑝1 > 𝑝 ∗
𝑚
𝑎√ , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
p1
(iii) We have V(𝑝1 , 𝑝2 , m) =
V(𝑝1′ , 𝑝2 , m
𝑎 2 p2
4p1
+
𝑚
𝑝2
𝑎2 p2 𝑚 − 𝐶𝑉
− CV) =
+
4𝑝1′
𝑝2
Let V(𝑝1 , 𝑝2 , m) = V(𝑝1′ , 𝑝2 , m − CV) we have
𝑎2 𝑝22 1
1
CV =
( ′− )
4 𝑝1 p1
(iv) In this case the income effect on good 1 is 0 and the sign of the
income effect on good 2 is positive.
We can get this conclusion from Marshallian demand function.
2
PROBLEM 2 (25 Points)
Suppose that the consumer’s utility function is U  x1a x21 a , 0  a  1. The
consumer’s income is i 1,2 pii where   1, 2  is the endowment vector. There
is no cash income.
What are the Marshallian demand functions for 𝑥1 and 𝑥2 ? You should not do
(i)
any calculations. Just explain in words how you know the answer. (5 points)
(ii)
Derive the Hicksian demand function for good 1. (10 points)
(iii)
Write two Slutsky equations: 1) That of the demand for good 1 with respect to
the price of good 1, and 2) that of the demand for good 1 with respect to the
price of good 2. In each Slutsky equation correctly identify and sign the
income and substitution effects. (10 points).
Answer:
(i)
Since the utility function is in C-D form. So the Marshallian
Demand should be a fraction of each good.
𝑥1 =
𝑥2 =
(ii)
𝑎(𝑝1 𝜔1 +𝑝2 𝜔2 )
𝑝1
(1−𝑎)(𝑝1 𝜔1 +𝑝2 𝜔2 )
𝑝2
Let m=𝑝1 𝜔1 + 𝑝2 𝜔2
Then the indirect utility function is V(p,m)=(
𝑎(𝑚) 𝑎 (1−𝑎)𝑚 1−𝑎
) (
)
𝑝1
𝑝2
So the Expenditure function should be
𝑝1
𝑝2 1−𝑎
𝐸(𝑝1 , 𝑝2 , 𝑢) = ( )𝑎 (
) 𝑢
𝑎
1−𝑎
ℎ1 (𝑝1 , 𝑝2 , 𝑢) =
𝜕𝐸(𝑝1 , 𝑝2 , 𝑢)
𝑝1
𝑝2 1−𝑎
= ( )𝑎−1 (
) 𝑢
𝜕 𝑝1
𝑎
1−𝑎
(iii)
𝜕𝑥1 (𝑝, 𝜔) 𝜕ℎ1 (𝑝, 𝑢) 𝜕𝑥1 (𝑝, 𝜔)
=
−
(𝑥1 − 𝜔1 )
𝜕𝑝1
𝜕𝑝1
𝜕𝑚
𝜕ℎ1 (𝑝,𝑢)
𝜕𝑝1
< 0 is the Substitution Effect
3
𝜕𝑥1 (𝑝,𝜔)
−
𝜕𝑚
(𝑥1 − 𝜔1 ) is the Income Effect and good 1 is normal. So if the
consumer is a net buyer of good 1 then IE<0. If he is a net seller of
good 1, IE>0.
𝜕𝑥1 (𝑝, 𝜔) 𝜕ℎ1 (𝑝, 𝑢) 𝜕𝑥1 (𝑝, 𝜔)
=
−
(𝑥2 − 𝜔2 )
𝜕𝑝2
𝜕𝑝2
𝜕𝑚
𝜕ℎ1 (𝑝,𝑢)
𝜕𝑝2
−
> 0 is the Substitution Effect
𝜕𝑥1 (𝑝,𝜔)
𝜕𝑚
(𝑥2 − 𝜔2 ) is the Income Effect and good 2 is also normal. If
the consumer is a net buyer of good 2 then IE<0. If he is a net seller of
good 2, IE>0.
PROBLEM 3 (35 points)
In a competitive market, there are 100 identical consumers. Each consumer has
the utility 𝑈(𝑥, 𝑦) = 𝑥 −
𝑥2
2
+ 𝑦, where x and y are the quantities of the two
goods. The price of x is p and the price of y is 1. The income of each consumer is
m.
(i)
Write down the aggregate Marshallian demand function for x. (10 points)
(ii)
There are n producers of the x good with the same technology. Each
producer has a fixed cost of 1 and a marginal cost of q/8. Derive the total
and average cost functions. What kind of returns to scale does the
technology possess? Explain. (10 points)
(iii)
Assuming the market is in long run equilibrium, calculate how many
producers n will exist in the market (ignoring the integer nature of n ).
What is the market price level p and the total quantity of good x ? (15
points)
4
Answer:
(i)
Each consumer wants to max 𝑥 −
St
𝑥2
2
+𝑦
px+y=m
We construct Lagrangian
𝐿=𝑥−
𝑥2
2
+ 𝑦 − 𝜆(𝑝𝑥 + 𝑦 − 𝑚)
FOC:
1 − 𝑥 − 𝜆𝑝 = 0
𝜆=1
𝑝𝑥 + 𝑦 − 𝑚 = 0
We can get x = 1-p. Since there are 100 consumers in the market, so
D(p)=100-100p
(ii)
𝑞2
𝑞
𝑇𝐶(𝑞) = 1 + ∫0 𝑙𝑑𝑙/8 = 1 + 16
AC(q)= (1 +
Since
𝐴𝐶(𝑞)
𝑞2
16
=
𝑀𝐶(𝑞)
)/𝑞
MC(q)=q/8
𝑞 2⁄
2
𝑞2
8+
(*) when 0<q<4 (*)>1 IRTS
when q>4 (*)<1 DRTS
(iii)
Suppose there are N producers and price level is p. Because market is
competitive, then p=MC(q)=q/8. So the total supply is 8Np. When price
level is p, demand=100-100p.We solve 100-100p=8Np
100
We get 𝑝 = 100+8𝑁
At the equilibrium the producer should gain nonnegative profit:
𝜋 = 𝑝𝑞 − 𝑐(𝑞) = 8𝑝2 − 4𝑝2 − 1 ≥ 0
1
Then 𝑝 ≥ 2
100
1
Then we solve 𝑝 = 100+8𝑁 ≥ 2 to get 𝑁 ≤ 12.5.
There are at most 12 producers in the market. At this time p=1/2 and
q=100-100p=50
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