a. What are the pure strategies of the monopolist and the pure strategies of consumer H? b. Consider each history from which the second period starts. Draw the extensive form of the subgame that follows each such history and find a subgame perfect Nash equilibrium for each subgame starting from the histories. c. Draw the extensive form of the original game replacing the subgames starting from period 2 with the payoffs corresponding to the subgame perfect Nash equilibria starting from period 2. Give necessary and sufficient conditions on p, that would induce period 1 histories BB, BN, NB and NN. d. Using the results in c, find all equilibrium paths of subgame perfect Nash equilibria of the original game and the corresponding payoffs for the players. Compare the efficiencies of the equilibria. 4. Several firms in an industry compete to hire from a population of workers. One third of the workers (type 1 workers) can get e 2 0 units of education at a cost of e and then produce 4ve units of output if they are hired. The remaining workers (type 2) can get e units of education at a cost of 5e/6 and then produce 5ve units of output if hired. The workers know their types when they choose their education levels. The firms independently offer wages that can depend on the workers' education levels and the workers accept at most one offer. A worker who accepts an offer gets as a (net) payoff the wage minus the worker's education cost. Workers who reject all offers get (net) payoff 0. For each worker who accepts a firm's offer, the firm gets a profit equal to the worker's output minus the worker's wage. The firms seek to maximize their expected total net profit. All of this information is common knowledge. a. Why could it be reasonable to model the two types of workers as having different education costs even if there is no discrimination in the provision of education and all workers pay the same tuition per unit of education? b. Find competitive equilibrium education levels and wages for both worker types if the firms know each worker's type and the workers know that they will be paid different equilibrium wages depending on their education levels. For the rest of this problem, assume that the firms cannot tell what type any particular worker is. Consider a game played by two of the firms and a randomly selected worker. After the worker learns its type and chooses its education level, firm 1 makes a wage offer, then firm 2 makes an offer without knowing firm 1's offer, then the worker accepts one offer or rejects both offers. Use graphs to illustrate your answers to the following problems. c. Show that the game has a separating perfect Bayesian equilibrium (PBE) in which the worker chooses 16 units of education if it is of type 2. Show that this PBE is constrained inefficient, i.e., that a regulator who could control the firms and interact with the worker the way they do, knowing what the firms know, could obtain a Pareto superior final allocation. On what basis can it be argued that this PBE is more plausible than any PBE in which the type 2 worker gets more than 16 units of education? d. Show that there is a pooling PBE in which both workers get 4 units of education. Compare the expected total surplus in this PBE to the expected total surplus in the PBE in part c. Is this pooling PBE constrained efficient? Justify your answer.c. Suppose now that the firm can offer any (non-linear) contract. What is an optimal con- tract and what is the corresponding expected profit of the firm? Compare your answer with your answer in part a. d. Now, suppose there are two consumers whose types are drawn independently from the identical distribution in the above. The consumers know their types but not the other con- sumer's type. The producer knows only their distributions. Suppose the producer produces q and auctions it off to the two consumers according to the ascending (English) auction. What is the expected revenue of the firm? What is the optimal q and corresponding profit of the firm? 3. The following is a three period model of a financial contract involving a lender (bank) and a borrower (firm). In the first period, the firm either builds a new facility at cost c > 0 or it does not. If the firm builds the new facility, its subsequent production function in value terms becomes 4VL; otherwise it is 2VL. Here, L is the amount of loan used in the production. In the second period, the bank offers a contract (L, () to the firm. A contract consists of L (2 0) the amount of loan and * (2 0), the gross amount the borrower has to repay. Assume that the firm does not default. In the third period, the firm either accepts the contract or rejects the offer. If a contract ( L, #) is offered and accepted, the bank receives the payoff of t - RL, where R > 1, while the payoff to the firm is 4vZ - t - c if it built the facility in the first period and 2v Z -t, if it did not. If the firm rejects the contract, the payoff to the bank is zero and the payoff to the firm is -c if the firm built the facility in the first period and is zero otherwise. Assume that the firm accepts the contract if accepting and rejecting the contract result in the same payoffs. Both the bank and the firm are risk neutral. Assume that { > c. a. Suppose the bank knows whether the firm built the new facility or not in the first period when it offers a contract. Draw the game tree of this game. b. Find all sequential equilibria of the game in (@). c. Suppose from now on that the firm's possible investment in the first period is in software and the bank does not know whether or not the firm made the investment when it offers the contract. Draw the game tree of this game. d. What is the optimal contract offer of the bank as a function of its beliefs? The bank believes that the firm invested in the software with probability p. e. Find all sequential equilibria of this game. 4. Consider a firm with the technology q = (7172)i that purchases inputs z, and r, in competitive factor markets at the prices wj = w2 = 1. Initially, the firm faces the demand curve q(p) = 120 - p for its output. a. Determine the firm's optimal supply decision and profits. b. Next, suppose the firm's profits were taxed at the constant rate of 20%. How would this affect the firm's optimal decisions? How much would the firm pay in taxes and what would be its post tax profits? c Suppose that, in addition to producing q, the firm could also contribute money, y, to charity and thereby reduce its overall tax rate. The rate would then become r(y) = max{(0.2 - 0.00ly), 0}. Assuming the contribution is from pretax earnings, i.e., it can be treated as an additional cost, determine the optimal supply decision and charitable contri- bution of the firm. What are the firm's profits? Compare the amount of taxes paid in part (b) to the amount of taxes and charity paid here. Also compare the firm's post tax profits. d Suppose again that the tax rate is fixed at = 0.2. Now, while charitable contributions do not affect the tax rate, they garner publicity for the firm and increase demand for its 3 product. Thus, the firm faces the demand q(p) = 120 - p + 2\\y. Again, assume the con- tribution is from pretax earnings. Determine the firm's optimal supply decision in this case. Also determine the amounts of taxes and charity paid and the post tax profits. e. Assuming that tax revenues and charitable contributions are devoted to precisely the same uses, rank the cases (b), (c) and (d) in terms of the resources (tax revenues and charity) generated and post tax profits.1. Consider an individual who lives for two periods. In each period she is endowed with one unit of time which can be devoted to leisure or labor. In period 1 she earns $w per hour and is free to choose the number of hours she works. However, her wage earnings are subject to an unemployment tax. She must pay the fraction r e [0, 1) of her wage earnings in tax. Then if she is injured or laid off in period 2, she will collect the tax payment set aside from period 1. Otherwise, she will continue to work (at wage w) and pay the unemployment tax on her period 2 earnings as well. Assume the individual has stationary von Neumann - Morgenstern preferences represented by u(me, () = m -, where m; and & denote her net earnings and leisure in period t, respectively. Her intertemporal preferences are given by U = u(m1, (1) + Bu(m2, (2), where / c [0, 1]. Suppose the probability of injury or layoff is 0 can look. Be sure your graph is compatible with all the assumptions above. Prove that for each level of input for the firm there is at most one PE allocation. Show that c/6 0. d. Find all the PE allocations with 61 > 0 that are part of price equilibria with transfers. What can be said about the transfers? Which consumer gives and which consumer receives a transfer payment? e. Prove that this economy does not have a competitive (Walrasian) equilibrium, Explain why not. [The proof can use the answer to part d.] 3. A monopolist M has two units of a durable good that it wants to sell to two consumers, H and L. There are two periods and the monopolist posts price p in period I and if necessary posts price p2 in period 2. There is no (additional) cost in providing the units so the monopolist attempts to maximize the total (undiscounted) revenue from the two periods. If H buys the good in the first period at pi (and so gets to use it in both periods), his payoff is $900 - pi. If he buys it in the second period at p2, his payoff is $300 - p2. If he does not buy in either period, his payoff is 0. Consumer _ gets payoff $500 - p, if he buys the good in period 1, $200 - p2 if he buys it in period 2, and $0 if he does not buy in either period. Additional consumption of the durable good adds nothing to the consumers' payoffs. In period 1, the monopolist posts price pi. Consumers # and & make simultaneous decisions whether to buy the durable good (B) or not buy (N). If both consumers buy in period 1, the game ends there. If at least one consumer does not buy in period 1, the game moves to the second period and the monopolist posts p2. Any consumer(s) who did not buy in period 1 make (simultaneous) buy-or-not-buy decision(s) in period 2 after learning p2- Assume that a consumer buys the good if he is indifferent between buying and not buying. Denote a period 1 history as p BN if M chooses pi, H buys, and L does not buy in period 1 and similarly for other histories. a. What are the pure strategies of the monopolist and the pure strategies of consumer H? b. Consider each history from which the second period starts. Draw the extensive form of the subgame that follows each such history and find a subgame perfect Nash equilibrium for each subgame starting from the histories. c. Draw the extensive form of the original game replacing the subgames starting from period 2 with the payoffs corresponding to the subgame perfect Nash equilibria starting from period 2. Give necessary and sufficient conditions on p, that would induce period 1 histories BB, BN, NB and NN. d. Using the results in c, find all equilibrium paths of subgame perfect Nash equilibria of the original game and the corresponding payoffs for the players. Compare the efficiencies of the equilibria. 4. Several firms in an industry compete to hire from a population of workers. One third of the workers (type 1 workers) can get e 2 0 units of education at a cost of e and then produce 4ve units of output if they are hired. The remaining workers (type 2) can get e units of education at a cost of 5e/6 and then produce 5ve units of output if ni:ed. The workers know their types when they choose their education levels. The firms independently offer wages that can depend on the workers' education levels and the workers accept at most