Consider the following bond market. It is common knowledge that the bond pays off either $20 or $100 at t=1. Both states are equally likely.

Agent S owns the bond. Since he needs cash at t=0 he is willing to sell the bond at a $8 discount. It means if the true value of the bond is $20 agent S is willing to sell for (at least) $12. If the true value of the bond is $100 agent S is willing to sell for (at least) $92.

Agent B is interested in buying the bond. If the true value of the bond is $20 (or $100) agent B is willing to buy for (at most) $20 (or $100).

The seller (agent S) proposes a price to sell the bond. Both agents are risk neutral and maximize expected utility.

a) What price does the seller offer in equilibrium? Is there trade? [3 Points]

Now suppose agent B is a sophisticated buyer. After seeing the price offer of the seller, the buyer can try to learn something about the bond. If agent B spends $10, he can find out the true value of the bond.

b) Suppose the seller offers the expected payoff of the bond and thus the price p=$60. What is the best response of the buyer? Should the seller offer this price? [4 Points

c) Suppose the seller offers a price p=$50. What is the best response of the buyer? Should the seller offer this price? [4 Points]

d) Suppose the seller offers a price p=$90. What is the best response of the buyer? [3 Points]

e) Is there any price at which agent S is willing to sell and agent B is willing to buy? Please provide a complete proof. [6 Points]