辅导 MFIN 7007 HW3讲解 Prolog

MFIN 7007 HW3

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Question 1 (optimal reserve). Assume there is a single buyer. The seller knows that the buyer's valuation for the object is a random number (real number) that is uniformly distributed over [20,100], whereas the buyer knows the realization. The buyer will purchase if the price does not exceed his valuation. The seller has no use for the object (values it at zero). She is risk neutral (maximizes expected profit). She sets a reserve price of r. What is the optimal reserve? What is the seller's expected profit at this reserve?

Question 2 (Cournot competition). Assume demand is 100 - p. Two firms have constant but different marginal costs: cost function for firm A is 10q and cost function for firm B is 15q. Suppose in equilibrium the two firms produce qA and qB respectively. What are qA and qB? What is the profit for each firm?

Question 3 (Collusion). Consider the setting in Question 2. Suppose the two firms collude to maximize the sum of the two firms'profits. What will be the total profit of the two firms?

1 Question: Switching Costs

Consider two airlines, A and B, flying the same route and engaging in a single-period Cournot competition. Industry demand (the demand from the entire population of customers) is q = 100 - p, and each firm has a constant unit cost of 10 (total cost is 10 times the quantity produced). Suppose in the past 50% (randomly determined) of the consumers had flown A and the other 50% flown B. Now each firm announces that it will offer a "frequent-flyer" discount of 10 to anyone who flew on its flight before.

Furthermore, suppose that firm B suffers an unexpected capacity constraint so that its quantity is fixed at a low level of qB = 15. Firm A does not have such a constraint and is free to choose any qA.

For all qA ≥ 15, find the market prices (before discounts) pA and pB for the two firms'products as a function of qA.

Question: When target has superior information

Suppose the target has private information about VT (target's standalone value). The target knows the realization of VT , but the acquirer only knows VT is uniformly distributed over [0,120]. The acquirer has standalone value of VA = 200, which is publicly known (no information asymmetry about it). Once the two firms merge, a synergy of s = 30 (publicly known) emerges.

Suppose the acquirer makes a take it or leave it cash offer of c ≤ 120 to the target.

— Find the acquirer's expected profit πA as a function of c. What value of c maximizes πA? What is the value of πA at this c?

Suppose the acquirer makes a take it or leave it offer of paying the target with a fraction α of the joint firm's equities.

— Find the acquirer's expected profit πA as a function of α. What is the value of πA when α = 0:14, is this value higher than the maximum value of πA (which you find in the previous question) when the acquirer offers cash?

Question: Target screening when acquirer has superior information

A single acquirer has standalone value VA. The acquirer knows the realization of VA, whereas the target (or financial market) only knows that VA is distributed uniformly over [1; 2]. The target's standalone value is publicly known (without information asymmetry), which we set to be zero to simplify calculation. The synergy the acquirer can create in the merger is an increasing function of VA (a higher type acquirer can create more synergy): (this functional form. is publicly known). The target makes a take it or leave it proposal to the acquirer.

Consider the following screening mechanism in which the target presents a menu with infinite choices of payment methods, asking the acquirer to choose one of them. Choices are indexed by a continuous variable z.

— Ask the acquirer to report a type (a real number) z 2 [1; 2].

— Given the reported z, the acquirer pays the target with a fraction of equity in the joint firm (which decreases in z) plus a cash payment of 1+0:05z2 (which increases in z).

— The reporting of type is voluntary: the acquirer is free to report any z 2 [1; 2], or to refuse to report (hence decline the takeover) if reporting leads to a negative profit.