Undergraduate Industrial Organization
Homework 1
Due: 5/13/2022
Spring 2022
Total is 120 points, you can get 100 points at most
Guidance
1. You can form a study group (at most 4 in one group) to finish the homework, but you
must clearly state your group members in your individual submission.
2. You must answer the questions in English.
3. Write down necessary steps if you are unable to fully solve a question.
Question 1 (10 points each)
Suppose there are two producers of silver in the world: Mexico and Peru. In 2014, total
production was 280 (millions of ounces), whereas price was $18 per ounce. Marginal cost is
constant at the level of $10 per ounce (Mexico) and $12 per ounce (Peru). Finally, suppose that
firms compete according to the Cournot model (output levels are determined simultaneously
and the equilibrium price is such that total demand equals total supply.
(a) Assuming that demand is linear, determine the coefficients of the demand curve that
are consistent with the observed data.
(b) Suppose that a third country, China, discovers silver mines with a marginal cost of
production equal to Peru. What impact do you expect this will have on the price of silver?
Question 2 (10 points each)
In 2010, global sales of Windows-based desktop computers were 351 million units, whereas
average price was $605. The leading market shares were as follows: HP, 17.9%; Acer, 13.9%;
Dell, 12%; Lenovo, 10.9%; Asus, 5.4%; others, 39.9%. Suppose that the Cournot model provides
a good approximation of the industry’s behavior. Suppose moreover that inverse demand is
given by p = a − b Q; each firm has constant marginal cost; and that, from previous studies, the
price elasticity of market demand is estimated to be -0.5 (at the market equilibrium level).
(a) Based on the observable data, estimate each firm’s marginal cost and margin.
(b) Suppose that Dell’s marginal cost decreases by 5%. Based on your answer to the
previous question, estimate the impact of this cost decrease on equilibrium price and on Dell’s
market share.
Question 3 (10 points each)
WeRCell offers wireless phone service. Market research has established that there are three
types of customers, in equal number: light users (Type 1), medium users (Type 2) heavy users
(Type 3). Type 1 users are willing to pay calls for at most $1 per minute, and if price per call
were zero they would make 100 minutes of calls per month. Type 2 are willing to pay calls
for at most $1.4, and if price per call were zero they would make 140 minutes of calls. Type 3
users are willing to pay up to $2 per minute, and if price per call were zero they would make
200 minutes of calls. Suppose moreover that demands are linear and that the marginal cost of
placing calls is zero.
(a) Determine the optimal menu of calling plans whereby each calling plan has a fixed fee
and a price per minute of call.
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(b) Show that there exists an alternative menu where each plan consists of a fixed fee and
a monthly minute allowance (and possibly a high price per minute for minutes exceeding the
allowance).
Question 4(5 points each)
There are two firms i = 1, 2. Without investment, they have constant and identical marginal
cost c = 5. In period 1, Firm 1 can make an investment I to reduce its marginal cost. In
particular, to reduce its marginal cost by x, Firm 1 must invest I = x2 . (That is, if Firm
1 invests I = x2 , its marginal cost becomes 5–x. Firm 2 cannot make any cost-reducing
investment and its marginal cost is 5). In period 2, the firms observe I and simultaneously
choose their outputs q1 , q2 . The market demand curve is given by P = 10–Q. The payoff to
Firm 1 is (its profit in the second period) – (its investment in the first period (I)). The payoff
to Firm 2 is its profit in the second period.
(a) Compute the investment level I ∗ and the realized outputs q1∗ , q2∗ at the subgame perfect
equilibrium.
(b) Is it right to say that the subgame perfect equilibrium strategy of Firm 2 is q2∗ ? Discuss.
(c) Let I 0 be the optimal investment for Firm 1, assuming that outputs are fixed at q1∗ ,
∗
q2 . I 0 can be regarded as the technologically optimal investment, or ex-post optimal investment. How does it compare to the subgame perfect equilibrium investment I ∗ ? Do we have
overinvestment (I 0 < I ∗ ) or underinvestment (I 0 > I ∗ )?
(d) Can you provide an intuitive explanation why we have over or under-investment? In
your explanation, use the following keywords: strategic substitutes or complements, commitment.
Question 5
Consider a three-person bargaining game with alternating offers defined as follows. The
three players have to divide a “pie” of size 1: In period 1, player 1 makes an offer (x1 , x2 , x3 )
where x1 +x2 +x3 = 1 and all xi ≥ 0. Then players 2 and 3 respond simultaneously with “yeses”
and “nos”. If both say “yes” the game ends with the division (x1 , x2 , x3 ). If either player 2 or
player 3 says “no”, then in the next period player 2 makes an offer (y1 , y2 , y3 ) to which players
1 and 3 respond simultaneously with “yeses” and “nos”. If both say “yes” the game ends with
the division (y1 , y2 , y3 ). If either player 1 or player 3 says “no”, then in the next period player
3 makes an offer (z1 , z2 , z3 ) and so on. All players have linear utilities u(w) = w and discount
future payoffs with a common discount factor δ ∈ (0, 1).
(a) Find a stationary subgame perfect equilibrium of this game. (10 points)
(b) (Hard) Find another subgame perfect equilibrium of this game. (30 points)
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