MICROECONOMIC THEORY I. ECONOMICS 665. FALL 2014.
PROF. Alex Anas. TA: Yiqian Lu. MIDTERM EXAMINATION 3.
December 4, 3:30-5:30 PM, 2014, 424 FRONCZAK HALL.
Books, notes or calculators are not allowed in the examination room. Please answer
each question. TOTAL POINTS = 100.
Solutions:
PROBLEM 1 (35 points)
This is a general equilibrium problem. Assume there are 200 consumers and 3 goods
in the market. Initially we have 100 consumers each of whom owns 10 units of good 1;
50 consumers each of whom owns 5 units of good 2; 50 consumers each of whom
owns 20 units of good 3. All consumers have the same utility function which is:
𝑈=
1
1
1
𝑙𝑜𝑔 𝑥1 + 𝑙𝑜𝑔 𝑥2 + 𝑙𝑜𝑔 𝑥3
2
4
4
Let 𝑥12 be the amount of good 1 consumer 2 has after trading. Similarly you can
define 𝑥11 , 𝑥13 , etc.
(i) Write down the excess demand function for good 1 using 𝑥11 , 𝑥12 , 𝑥13 etc. (5
points)
(ii) What are the equilibrium prices prices 𝑝2 , 𝑝3 ? You may assume that 𝑝1 = 1.
(20 points)
(iii) Which group of consumers are best off? (10 points)
Answer:
(i) Excess demand for good 1=100𝑥11 +50 𝑥12 + 50𝑥13 − 1000
(ii) 𝑚1 = 10𝑝1 , 𝑚2 = 5𝑝2 , 𝑚3 = 20𝑝3 , Each group want to max
1
1
1
𝑈 𝑖 = log 𝑥1𝑖 + log 𝑥2𝑖 + log 𝑥3𝑖
2
4
4
3
𝑠𝑡 ∑ 𝑝𝑗 𝑥𝑗𝑖 ≤ 𝑚𝑗
𝑗=1
Using Lagrangian to solve this problem, we have
1
𝑗
𝑥1
𝑚𝑗
=
, 𝑓𝑜𝑟 𝑗 = 1,2,3
2𝑝1
𝑗
𝑥2
𝑚𝑗
=
, 𝑓𝑜𝑟 𝑗 = 1,2,3
4𝑝2
𝑚𝑗
, 𝑓𝑜𝑟 𝑗 = 1,2,3
4𝑝3
Actualy the production function is in Cobb-Douglas form, you can get
𝑗
𝑥3 =
above three functions immediately according to the characteristics of
Cobb-Douglas function.
Excess demand for good 1=100𝑥11 +50 𝑥12 + 50𝑥13 − 1000
Excess demand for good 2=100𝑥21 +50 𝑥22 + 50𝑥23 − 250
So we have
-500+125p2/p1+500p3/p1=0
250p1/p2-750/4+250p3/p2=0
Under 𝑝1 = 1 we have 𝑝2 = 2 and 𝑝3 =
1
2
(iii) Since 𝑚1 = 𝑚2 = 𝑚3 = 10, all groups of traders are equally well
off.
PROBLEM 2 (40 points)
There are two firms in the market. Their products are perfect substitutes for the
1
consumer . Their cost functions are 𝐶1 (𝑥1 ) = 5𝑥1 and 𝐶2 (𝑥2 ) = 2 𝑥22 . The market
demand function is P = 100 - (𝑥1 + 𝑥2 )/2.
(i) Suppose that the firms compete as a Cournot duopoly, what is the market
equilibrium? Calculate the quantity produced, and the price charged by each firm. (15
points)
(ii) What is the profit of each firm? Calculate the consumer surplus? And then
calculate the social welfare. (5 points)
(iii) How does price and outputs change if the marginal cost of firm 1 increase by
20%? (5 points)
2
(iv) Assume that firm 1 is a Stackelberg quantity leader and firm 2 is the follower.
Calculate the equilibrium price charged and the quantity produced by each firm. (15
points)
Answer:
(i)
𝜋1 (𝑥1 , 𝑥2 ) = (100 −
𝑥1 +𝑥2
2
) 𝑥1 − 5𝑥1
𝑆𝑜𝑙𝑣𝑒 𝐹𝑂𝐶 𝑤𝑒 𝑐𝑎𝑛 ℎ𝑎𝑣𝑒 𝑥1 = 95 −
𝜋2 (𝑥1 , 𝑥2 ) = (100 −
𝑥2
2
𝑥1 + 𝑥2
) 𝑥2 − 𝑥22 /2
2
𝑆𝑜𝑙𝑣𝑒 𝐹𝑂𝐶 𝑤𝑒 𝑐𝑎𝑛 ℎ𝑎𝑣𝑒 𝑥2 = 50 −
𝑥1
4
Solve these two equations we can have 𝑥1 = 80, 𝑥2 = 30,
p=100-(80+30)/2=45
(ii)
Profit of company 1= 45*80-5*80=3200;
Profit of company 2=45*30-30*30/2=900
Consumer Surplus = (100-45)/2 *110=3025
(iii)
Social Welfare= 7125
Now we have 𝐶1 (𝑥1 ) = 6𝑥1 We have 𝑥1 deceases, 𝑥2 increases,
total supply decreases, p increases.
(iv)
From (i) we have 𝑥2 = 50 −
So 𝜋1 (𝑥1 , 𝑥2 ) = (100 −
𝑥1
4
𝑥
𝑥1 +50− 1
4
2
) 𝑥1 − 5𝑥1
3
We have FOC: 70 − 𝑥1 = 0
4
We can get 𝑥1 =
280
3
and 𝑥2 = 80/3. Total Supply is 120 and
price level is 40.
3
PROBLEM 3 (25 points)
(i) This part of this problem is from the textbook and the lecture. Two companies
producing oil want to form a Cartel. They agree to reduce output to increase the price.
Is the agreement stable? Prove the economic reason. Devise and explain your own
clear notation. (5 points)
(ii) Suppose that the time horizon is infinite (years are t=0,1,2,…,  ). The present
value profit function to be maximized by each firm is i  t 0  t it , where 0 < 𝛽 ≤

1 is a discount rate per year, 𝛽 𝑡 means 𝛽 raised to the power t and  t is the profit of
firm i in year t. Keep in mind that


t 0
t 
1
. The aggregate demand for oil is
1 
Q  1  p each year, Q  q1  q2 , and qi i  1, 2 is what each firm produces. The
marginal cost is constant and it is c1  c2  0.1 for each firm.
(ii-a) If there is no Cartel, the firms compete as in Bertrand competition.
Explain what will be the Bertrand-Nash equilibrium price and the quantities produced
by each firm? Write an expression for the profit 1,BN of firm 1. (5 points)
(ii-b) Find an expression for the profit of firm 1,  1,C , if the Cartel agreement
is assumed to remain forever unbroken. (10 points)
(ii-c) Suppose that in the beginning of year 1, firm 1 is considering to secretly
defect from the Cartel in year 1, thus breaking the agreement. It knows that if it does
so, its rival will find out not immediately, but in the beginning of year 2. Once the
rival finds out it will also defect from year 2 to infinity, thus “punishing” firm 1.
Calculate for what value of  , firm 1 will prefer to defect from the Cartel agreement.
(5 points)
Answer:
(i)
A cartel is unstable.
𝑚𝑎𝑥𝑦1,𝑦2 𝑝(𝑦1 + 𝑦2 )[𝑦1 + 𝑦2 ] − 𝑐1 (𝑦1 ) − 𝑐2 (𝑦2 )
𝐹𝑂𝐶: 𝑝(𝑦1 + 𝑦2 ) + 𝑝′(𝑦1 + 𝑦2 )[𝑦1 + 𝑦2 ] = 𝑐1′ (𝑦1 )
4
𝜕𝜋
= 𝑝(𝑦1 + 𝑦2 ) + 𝑝′ (𝑦1 + 𝑦2 )[𝑦1 ] − 𝑐1′ (𝑦1 )
𝜕𝑦1
= −𝑝′(𝑦1 + 𝑦2 ) 𝑦2 > 0
So every firm has the incentive to deviate from the agreement to
max its profit.
(ii)
(a) Since it is Bertrand competition. The firms will choose the
price at their marginal cost level. So p=0.1. q1=q2=0.45 In this
case Π1,𝐵𝑁 = 0
(b) If they agree to form a Cartel. According to the symmetry,
q1=q2=q
Each firm wants to max (1-2q)q-0.1q
FOC: 1-4q-0.1=0
We have q=0.225 Thus q1=q2=0.225, p=0.55
∞
Π1,𝐶 = ∑ 𝛽 𝑡 0.225 ∗ (0.55 − 0.1) =
𝑡=0
0.10125
1−𝛽
(c) Firm 1 (the firm that decides to cheat or not) believes that firm
2 will continue to produce at the quantity level 0.225.
When firm 1 chooses to defect, it will
max (1-0.225-q)q-0.1q in year 1
FOC
we have q=0.3375
Total Profit of firm 1 if it defects is 0.3375*(0.4375-0.1)=0.1139
When
0.10125
1−𝛽
< 0.1139 i.e. when 𝛽 < 0.11 Firm 1 will prefer
to defect.
5