Business Application: Price Leadership in Differentiated Product Markets: We have considered how oligopolistic firms in a differentiated product market price output when the firms simultaneously choose price. Suppose now that two firms have maximally differentiated products on the Hotel ling line [0, 1] and that the choice of product characteristics is no longer a strategic variable. But let’s suppose now that your firm gets to move first — announcing a price that your opponent then observes before setting her own price. This is similar to the Stackelberg quantity-leadership model we discussed in Chapter 25 except that firms now set price rather than quantity.
A: Suppose you are firm 1 and your opponent is firm 2, with both firms facing constant marginal cost (and no fixed costs).
(a) Begin by reviewing the logic behind sequential pricing in the pure Bertrand setting where the two firms produce undifferentiated products. Why does the sequential (sub game perfect) equilibrium price not differ from the simultaneous price setting equilibrium?
(b) Now suppose that you are producing maximally differentiated products on the Hotel ling line. When firm 2 sees your price p1, illustrate its best response in a graph with p2 on the horizontal and p1 on the vertical axis.
(c) Include in your graph the 45-degree line and indicate where the price equilibrium falls if you and your competitor set prices simultaneously.
(d) Let p be the price that results in zero demand for your goods assuming that your competitor observes p before setting her own price. Indicate p in a plausible place on your graph. Then, on a graph next to it, put p1 on the vertical axis and x1 — the good produced by your firm —on the horizontal. Where does your demand curve start on the vertical axis given that you take into account your competitor’s response?
(e) Draw a demand curve for x1 and let this be the demands for x1 given you anticipate your competitor’s response to any price you set. Include MC and MR in your graph and indicate
p∗1 —the prices you will choose given that you anticipate your competitor’s price response once she observes your price.
(f) Finally, find your competitor’s price p∗2 on your initial graph. Does it look like p∗1 is greater or less than p∗2?
(g) Who will have greater market share on the Hotel ling line — you as the price leader, or your competitor?
B: Suppose that the costs (other than price) that consumers incur is quadratic as in the text—i.e. a consumer n whose ideal point is n ∈ [0, 1] incurs a cost α (n − y) 2 from consuming a product with characteristic y ∈ [0, 1]. Continue to assume that firm 1 has located its product at 0 and firm 2 has located its product at 1—i.e. y1 = 0 and y2 = 1. Firms incur constant marginal cost c (and no fixed costs).
(a) For what value of α is this the Bertrand model of Chapter 25? In this case, does the equilibrium price differ depending on whether one firm announces a price first or whether they announce price simultaneously? (Assume sub game perfection in the sequential case.)
(b) Now suppose α > 0. If the firms set price simultaneously, what is the equilibrium price?
(c) Next, suppose firm 1 announces its price first, with firm2 then observing firm 1’s price before setting its own price. Using the same logic we used in the Stackelberg model of quantity competition, derive the price firm 1 will charge (as a function of c and α.) (Hint: You can use the best response function for firm 2 derived in the text—substituting y1 = 0 and y2 = 1—to set up firm1’s optimization problem.)
(d) What price does this imply firm 2 will set after it observes p1? Which price is higher?
(e) Derive the market shares for firms 1 and 2. In the Stackelberg quantity setting game, the firm that moved first had greater market share. Why is that not the case here?
(f) Derive profit for the two firms. Which firm does better — the leader or the follower? True or False: The quantity leader in the Stackelberg model has a first mover advantage while the price leader in the Hotel ling model has a first mover disadvantage.
(g) True or False: Both firms prefer sequential pricing in the Hotel ling model over simultaneous pricing (given maximal product differentiation).