What are the monopoly price quantity and profits in this

Question

Assignment

1. 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:

Q_w = 80 - 2p_W + p_R         Q_R = 80 - 2p_R + p_W

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 p_w before the manufacturer of Rah could set p_R. 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(q_I) = 0.025 q²_I and MC(q_I) = 0.05q_I. The market demand for hand sanitizer is given by P = 50 - 0.1Q. Under monopoly, Q = Q_I.

(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(q_E) = 10q_E + 0.025 q²_E and
MC(q_E) = 10 + 0.05q_E. 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) = 100 - Q/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 q_i = q*
(ii) If any firm produces q>q*, then each firm believes that both will revert to the one-shot Cournot quantity q^c, forever.

What is the critical value of the firms' discount factor δ such that q* = 0.5Q^m (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 p_i = p*
(ii) If any firm produces pB, forever.

What is the critical value of the firms' discount factor δ such that p* = p^m 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 environmentally-hazardous substitutes over the next five years. At that point, production and sales of this product will be banned.