1. (a)Suppose the wealth of one consumer is. The problem is that
By first order condition and , we can get ,. Thus, .
(b) The equilibrium value of each asset is.
(c) When , by first order condition and, we can get ,. Thus,.
(d) We can find that the claims price of state 2 is higher in than in .Because they have two different levels of risk aversion.
(e) We can get ’. Solving, .
(f)The same for (e),suppose there exist a portfolio ()
we get . The same to , we can get
(g)Only in state 2 and 3 they have the chance to trade. And only by trading the consumer can arrival one point that everyone will not improve themselves welfare but not hurt others.
2.
(a) The degree of absolute risk aversion for this individual is ., thus it is an increasing function of .
(b) The problem for individual
We can get , then F.O.C..
Solving, ,.
(c) The net demand for state is.
(d) The same to (c). .
(e) Suppose that and .
In the equilibrium market,. Thus,.
In a word,,. Also, we can get.
(f) An increase in will not change the equilibrium market odds. The net demand always equal to 0 in the equilibrium so that the equilibrium will not change. The intuition is that will not change the relative price.
3.
(a)The profit for firm 1:
The profit for firm 2:
The first order condition for firm 1 and firm 2, we can get:
Thus, we can solve .
(b) If firm 1 move first, then is given for firm 2. The profit function for firm 2 is
By first order condition, we can get
Thus, the response curve for firm 2 is . Then, the profit function for firm 1 is . By first order condition, we can get. We can find. This because firm 1 has first move advantage that will make firm 1 can consider what firm 2 will do when firm1 makes decision. Thus, firm 1 will produce more products.
(c) If the two firms collude and maximize total profit, then the problem is like
F.O.C. for ,
Solving and we can get. But , then joint profit is maximized at , and the profits are .
(d) The two firms will not reach a collusive agreement. Since the firm 2 will get the profit 200 under no collusive agreement instead of the profit 0 in a collusive agreement.
4.
Solution：
(a) ,,, are the best response. Let us take an example for buyer 1, the true value of buyer 1 is. We should prove that bidding price of buyer 1 will be better than and. Suppose that the highest price of all buyers except buyer 1 is . First, we consider one situation. When, he will lose if he bids. He will win when he bids, but the utility of him is. He will not improve his utility when he bids. When, he will win when he bids, not which will not change the result. When, his utility is whatever he takes. Thus, buyer will choose which will be better than in any case. The same method will be used to prove will be better than. And we can prove other buyers will bid their value in this way.
(b) The equilibrium bidding strategies are,,,. The main idea is the same in (a).
(c) Yes. The dominant strategy for each buyer to bid in either of these auctions is .
(d) Because if they bid the price more than theirs value, they will get negative utility; if they bid the price less than theirs value, it is no profit to get. Thus the best response is bid their really value. The buyer 1 should bid 120 for the first unit, but only bid 4o for the second unit.
(e) The buyer 2 will always keep her light green until the one unit price upon 80. The buyer 2 will always keep her light green until the one unit price equal 81. The buyer 2 also get the another unit at price 81.