How would you characterize the posterior distribution of

Question

Assignment -

Question 1 - Consider the following offers by Microsoft for Yahoo. Microsoft feels that it would benefit from acquiring Yahoo as long as the price does not exceed $52 million. [At a price above 52 million it expects negative profits]. It does not know Yahoo's thinking on the price it would be willing to be acquired at. Its prior is that Yahoo could be one of three types. A type i Yahoo would be willing to sell at $v_i with v₁ = 40 million, v₂ = 50 million and v₃ = 55 million. [Type i makes zero profits at v_i] Microsoft also believes the probability for each type i is q_i: q₁ = 0.1, q₂ = 0.6 and q₃ = 0.3. All this is common knowledge. Also common knowledge that Yahoo employs a per period discount factor of 0.4, and Microsoft a per period discount factor d.

We know that Microsoft made an initial offer of $44 million. This has been rejected by Yahoo. Next period Microsoft can choose to make another offer or withdraw. If the offer the next period is again rejected Microsoft can make another offer the following period and so on.

What value must d take for the play so far to be part of an equilibrium outcome?

To answer follow the following steps:

a. What would make a type I Yahoo indifferent between accepting a price p, in period 1 and a price p2 in period 2? How does this condition apply to the three different types?

b. How would you characterize the posterior distribution of types in period 2, given what you know happened thus far? (Break indifference in Microsoft's favor).

c. If Microsoft had only one chance to bid, what would its bid have been? Given what you know has happened, what would you expect Microsoft to bid next (assuming it had only one more chance to bid now)?

d. Do you think Yahoo's rejection was a best response?

e. What value must d take for the initial bid to be part of an equilibrium outcome?

Question 2 - Consider the following example: A seller of an appliance, say a refrigerator, knows that there is a probability p that the refrigerator has a defect that cannot be observed. Suppose that buyers also know this rate, and the defect rate is common knowledge. The seller can be either of two types, which produce models with different quality levels (defect rates). Type i has a defect rate of p_i, i = 1, 2 and p₁ < p₂ and a cost of production c_i. Assume the probability of type 1 is 0. Moreover, assume that the value to a buyer of a good refrigerator is $1000 and a defective one is 0, and that buyers are risk neutral.

a. Suppose p₁ = 0.1, p₂ = 0.2, θ = 0.5, c₁ = 845 and c₂ = 750. How would the sellers price the refrigerator under a PBNE?

b. Now suppose the seller can offer a money back guaranty that costs $25 to administer. How would the sellers price the refrigerator under a PBNE in this case?