Price Leadership in a Homogeneous Product Market Assignment

Assignment Task

1. Consider a homogeneous product industry with inverse demand given by p = 100 − 2Q. Variable cost is given by C = 10q. There is currently one incumbent firm and one potential competitor. Entry into the industry implies a sunk cost of F.

a) Determine the incumbent’s optimal output in the absence of potential competition.

b) Suppose the entrant takes the incumbent’s output as given. Show that the entrant’s equilibrium profit is decreasing in the incumbent’s output.

c) What output should the incumbent firm set to deter entry?

d) Determine the lowest value of F such that the incumbent firm prefers to deter entry.

2. Suppose that two products, i= 1, 2, are located at the extreme ends of the [0,1] interval. Let li denote the location of the product and x the consumer locations. Consumers buy one unit and are uniformly distributed on the unit interval; their indirect utility is given by r − pi − t |li − x|, where r is the willingness to pay, pi the price set by firm i, and t a parameter that measures the disutility from not consuming their ideal product. Marginal cost of production is constant and equal to c. Solve for the Stackelberg equilibrium in prices where the firm located at zero is the leader firm. Is there an advantage for being a price leader, that is, is there a first-mover advantage? Why? Explain. Compare the results with simultaneous choice of prices.

3. Firm 1 is the incumbent in a market lasting two periods with inverse demand curve p = 74 − 9Q. Its first-period costs are C(q) = 15 + 20q and it faces entry in the second period by Firm 2, which has identical costs. There is an asymmetry between the firms, however, in that only Firm 1 has the option of investing $63.5 in R&D in the first period in order to reduce its second-period marginal costs to $2 per unit. Show that it is optimal for Firm 1 to make this investment even though Firm 2 enters regardless. Also show that it would not be optimal for Firm 1 to make the investment if there were no threat of entry.

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