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Differentiated Bertrand competition versus price leadership. The demand for two brands of laundry detergent, Wave (W) and Rah (R), are given by the following demands:

Differentiated Bertrand competition versus price leadership. The demand for two brands of
laundry detergent, Wave (W) and Rah (R), are given by the following demands:
Qw = 80 – 2pW + pR QR = 80 – 2pR + pW
The firms have identical cost functions, with a constant marginal cost of 10.
The firms compete in prices.
(a) What is the best response function for each firm? (that is, what is firm W’s optimal price
as a function of firm R’s price, and vice-versa?) What is the equilibrium to the one-shot
pricing game? What are the profits of each firm?
(b) Suppose the manufacturer of Wave could commit to setting pw before the manufacturer of
Rah could set pR. How would this change the equilibrium? What are the profits of each
firm in this case? Should Wave take advantage of this commitment possibility? Why or
why not?
(c) Is there a first or second-mover advantage in this game? First-mover advantage is like the
conventional Stackelberg quantity-leadership story, while second-mover advantage is
reversed. Explain the intuition for your answer, and compare / contrast with the Stackelberg
quantity-setting story.
2. Entry deterrence via quantity pre-commitment. The U.S. market for hand sanitizer is controlled
by a monopoly (firm I, for incumbent) that has a total cost given by TC(qI) = 0.025 2
I q and
MC(qI) = 0.05qI. The market demand for hand sanitizer is given by P = 50 – 0.1Q. Under
monopoly, Q = QI.
(a) What is the monopolist’s optimal price and output?
(b) Now let there be a foreign firm (firm E, for entrant) that is considering entry into the
market. Because the entrant must ship hand sanitizer all the way across the ocean, its costs
are higher. Specifically, the entrant’s costs are given by TC(qE) = 10qE + 0.025 2
E q and
MC(qE) = 10 + 0.05qE. Suppose that the incumbent monopolist has committed to the
monopoly output level. What is the residual demand faced by the entrant? How much
output will the entrant export to the U.S.? What will be the U.S. price of hand sanitizer?
(c) Show that the monopolist would need to commit to produce 400 units in order to deter
entry of the foreign firm. (Hint: figure out the monopolist’s output level q
*
such that the
entrant loses money if it exports anything other than zero.) What are the incumbent’s profits
if it commits to this output level and deters entry?
(d) If the incumbent decides to accommodate entry, what quantity will it commit to?
(e) Will the incumbent deter or accommodate entry in this market?
3. Collusion and punishment. Suppose the market demand for lumber is given by:
P Q Q ( ) 100 / 2  
There are two symmetric producers in the market, each with a constant marginal cost of 10.
(a) What are the monopoly price, quantity, and profits in this market?
(b) What are the Cournot price, quantities, and profits in this market?
(c) Suppose the two firms compete in the following infinitely repeated game:
(i) Each firm produces qi = q*
(ii) If any firm produces q>q*
, then each firm believes that both will revert to the oneshot
Cournot quantity q
c
, forever.
What is the critical value of the firms’ discount factor δ such that q
* = 0.5Qm
(where Q
m
is
the monopoly output) is the equilibrium outcome to this game?
(d) Suppose the firms instead set price, given the cost functions above and no capacity
constraints. What is the equilibrium price and quantity to this one-shot stage game?
(e) Let the firms in part (d) compete repeatedly in the following infinitely repeated Bertrand
game:
(i) Each firm sets pi = p*
(ii) If any firm produces p<p*
, then each firm believes that both will revert to the oneshot
Bertrand price p
B
, forever.
What is the critical value of the firms’ discount factor δ such that p
* = pm
is the equilibrium
outcome to this game? Which type of competition, price or quantities, is more likely to
sustain the monopoly outcome? Why?
4. Factors affecting the sustainability of collusion. Consider an infinitely repeated Bertrand trigger
pricing game (for example, question 3(e) above). Describe how each of the following
conditions would affect the sustainability of a collusive outcome, if at all.
(a) The government’s Competition Commission announces plans to publish a monthly list of
all transactions prices and volumes in this market, in an effort to improve “market
transparency” for consumers.
(b) Recent regulations require users of the product to convert to less environmentallyhazardous
substitutes over the next five years. At that point, production and sales of this
product will be banned

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