**1.** Sustaining Tacit Collusion. Consider the infinitely repeated Bertrand-Nash duopoly interaction where, in each period, the market will either be in a “low demand” state in which case the demand function is given by DL(p) = 1 for p < v or in a “high demand” state in which case the demand curve is given by DH(p) = z >1 for p ≤ v. The low and high demand states occur with probabilities x and (1–x), respectively. Production costs are zero. Consider Nash reversion trigger strategies of the following form: charge price pH in a high-demand state if no previous deviation has occurred, charge pL in a low-demand state if no previous deviation has occurred, and set a price equal to 0 if a deviation has previously occurred.

**a)** Find the threshold discount rate < 1 required to sustain the monopoly price in both in demand states as a SPNE: i.e., pH = pL = v.

**b)** Suppose (½, ). Find the highest price pH that the firms can sustain in the high demand period. [HINT: This price will depend upon the specified value of . Assume that pL = v.]

**c)**Verify that the firms can still sustain pL = v in the low demand state while setting pH equal to the value found in part b).

**d) **Show that when < ½, the only possible SPNE involve pH = pL = 0.

**2.** Merger Policy: Consider a homogeneous product Cournot industry in which there are, initially, 3 firms. Their (constant) marginal costs are given by c1 = 0 < c = c2 = c3. All firms have fixed costs equal to f. The market (inverse) demand curve is given by P = 1 – Q, where Q = Q1 + Q2 + Q3.

**a)**Under what conditions (on c and f) will all three firms profitably produce positive quantities in Cournot – Nash equilibrium? [Assume that these conditions are satisfied throughout the remainder of the problem.]

**b)**Derive formulae for the amounts of Consumers’ Surplus, Total Surplus, and individual firm profits for this benchmark case.

**c)** Suppose the two high cost firms merge. Redo parts a) and b) for the newly formed Cournot duopoly. Determine the effects of the merger on equilibrium levels of Consumers’ Surplus, Total Surplus, and the equilibrium profits of each of the three firms.

**d)**Suppose the low cost firm merges with one of the high cost firms. Redo parts a) and b) for the newly formed Cournot duopoly. Determine the effects of the merger on equilibrium levels of Consumers’ Surplus, Total Surplus, and the equilibrium profits of each of the three firms.

**e)**Discuss what you have learned from your analysis.

**3.** Vertical Merger Policy. Two manufactures produce widgets and sell to two retailers at a wholesale price w. These retail duopolists sell the widgets to final consumers whose (inverse) demand curve is given by P = 1 – Q. For simplicity, assume that the fixed and marginal costs of the manufacturers and retailers are zero and that Cournot rivalry determines the market outcome in both the wholesale and retail markets.

**a)**Derive the formula for the retail industry’s derived demand curve for widgets made by the manufacturers.

**b)** What are the equilibrium firm wholesale and retail prices in this “double duopoly” situation?

**c)** What are the equilibrium levels of firm profits and consumers’ surplus?

Now suppose that one of the manufacturers merges with one of the retailers. They integrate their operations so thoroughly that the merged firm is unable to participate in the wholesale market. Thus the non-merging retailer buys only from the non-merging manufacturer, which, in turn, sells only to the non-merging retailer. The non-merging retailer engages in (asymmetric) Cournot rivalry with the vertically integrated firm when making sales to final consumers.

**d)** Derive the formula for the non-merging retailer’s derived demand curve for widgets made by the non-merging manufacturer.

**e)**What are the equilibrium wholesale and retail prices in this partially integrated situation?

**f)**What are the equilibrium levels of firm profits and consumers’ surplu

**g)** Suppose the other manufacturer and the other retailer merge. What would be the Cournot equilibrium retail price?

**h)**What are the equilibrium levels of firm profits and consumers’ surplus?

**i)** Finally, turn your attention to possible horizontal mergers in this industry. Econ 782 has (hopefully) convinced you that, in contrast to the vertical mergers analyzed above, these would bad for consumers and total surplus. But, which would be worse (in terms of total surplus): a retailer – retailer merger or a manufacturer – manufacturer merger? Explain your answer.

**4.** Exclusion. An Incumbent (I) with cost c faces a Buyer (B) who values one unit of its product at v. An entrant (E), whose product B values at ve can enter with unit cost ce. Consider the following two stage game: (i) I makes a take-it-or-leave-it offer to B of an exclusive contract (p,d), where p is the selling price and d is the penalty for breach of contract: i.e., for switching to E. (ii) E decides whether or not to enter and the firms compete in prices ala Bertrand. (E’s fixed cost of entry is f.) Assume that it is socially efficient for the Entrant to provide the product: i.e., ve – ce – f > v – c.

**a)** What is the profit-maximizing choice of (p,d)? Describe the resulting market equilibrium.

**b)**Next, suppose 1 > ve = v > c, f = ε ≈ 0 and ce is uniformly distributed over the interval [0,1]. The timing of the game is as above, except the Incumbent and the Buyer do not know the Entrant’s unit cost at the time they contract. (The Entrant knows its cost before making its entry decision and its cost becomes common knowledge when it enters.) What is the equilibrium contract? Is it efficient? Explain.

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