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1- Consider a couple consisting of a husband {H} and wife {H ]. Each is endowed with one unit of time which has three possible uses: market production {m}; home production {31:}. and leisure {if}. Labor devoted to market production. If\"; is paid the wage wi- i = H} H'. where w\": s". 10H. The market price of m is normalized to 1:_ all of the couples earnings are devoted to purchasing m- The household good is produced according to the technology h 2 LI\": + til-Elm: where L\": denotes the amount of time i devotes to home production- H and 11' both consume the entire anlou-nt of we purchased and the entire alum-Int of h produced. Howeverj they consume only their own leisure- Agent is utility is given by aim. h? [i]. which is increasingj strictly concave and twice continuously di'erentiable- a \"hat variables must be determined by the couple? ii- Assuming the couples objective is to maximize joint welfare; uH +u" 3 characterize an interior optimum. c. Argue that if the agents have identical preferences3 i.e.3 if 11:3 = Law. then as long as w\"- s: wH. it is optimal for H to consume less leisure and to supply more market labor than H' but for H' to devote more time to home production than H. d For the general case in which the agents" preferences might differ; could there be an interior optimum in which they devote equal amounts of time to home production even though w": s: 13H? Explain. For parts eg. belowj assume H and II' decide separately how to allocate their time rather than jointlyj as above. They each maximize their own utility; taking the otheris choices as given. e. 1FJ-Irite the decision problem facing each of the two agents and describe the principle di'erence between this and the previous joint welfare maximization. f. Argue that in this case too little time will be devoted to both home production and market production, and too much will be devoted to leisure- g. Agaim suppose the agents have identical preferences and that w\"- {I wH . Can you infer in this case that H will sunnlv more market labor than H"? leain Whenever possible. justify your answers and prove that they are correct. A rm manager has already offered to hire a worker. IThe worker knows that working for the rm is either pleasant or unpleasant and initially believes that it is pleasant with probability p E (El. 1}. The manager knows if thejob is pleasant or unpleasant and has no way to affect the pleasant ness of the job for the worker. Before the worker decides whether to accept or reject the job oer. the manager can give the worker a gift. Suppose rst that the manager does not give the gift. If the worlmr does not work for the rm._ then both the manager and worker get payon I} from the interaction. If the worker works for the rm. then both get payoff l.li|I if working for the rm is pleasant; if it is not._ then the manager gets payoff 5 and the worker gets payoff -1. If._ instead. the manager gives the gift. then it raises the worker's payo" by -1 units no matter what the worker decides. [The worker keeps the gift whether she accepts or rejects the job offer.) Giving the gift reduces the manager's payon by c : 1] no matter what. In other words. no matter what the worker decides. c is subtracted from the manager's payon specied above. Before deciding whether to accept the job. the worker knows if a gift has been given. The manager and worker are both rational expected payo' maximizers with common knowledge of the entire interaction described above. {In particular. the manager knows the value of P-l a. The above interaction can be represented as a sequential {extensive form] game. Draw a complete game tree representing this game. The tree should include all the possible choices of the manager and worker given their respective information. Dene all the notation you introduce in labeling the tree. b. How many pure strategies does the manager have'ir Give an example of one. c. How many pure strategies does the worker have\"? Give an example of one. d. Suppose that. for some values of the parameters 3:: and c. there is a pure strategy sequential equilibrium {SE} in which the manager never gives a gift under any circum stances. 'I'b formulate the SE. it is still necessary to specify the worker's beliefs in case the manager did give a gift. Show that sequential equilibrium places no restriction on those beliefs. c. Find all parameter 1values of the game in part a such that there is a pure strategy SE in which the manager never gives a gift under any circumstances. Describe the SE completely when it exists and show that it is SE. f. Find all parameter values of the game in part a such that there is a pure strategy SE in which the manager gives a gift if and only if working for the rm is pleasant. Describe the SE completely when it exists and show that it is SE. Explain why the equilibrium strategies of manager and worker are equilibrium strategies. Consider a pure exchange private ownership economy with two consumers and two goods. Consumer 1 has utility function eaten. $11} = 11311 and endowment e] = {3. 1}. and consumer '2 has utility function rie\". $13} = when and endowment e: = {3.3}. where 3:\" 3: {I is the quantity of good I consumed by consumer i. for 11 = 1. 2. a. Find all the Pareto efficient allocations in this economy. b. Find a competitive {\\E'alrasian] equilibrium for this economy. Is there any other competitive equilibriLu'n allocation? c. Is the allocation in part b Pareto efficient? Justify your answer. d. Could the welfare of consumer '2 increase if the consLuuer could change one of the prices by a small amount from its equilibrium level in part b'.' For this assume that consumer 1 would act as a competitive price taker at the new prices and consumer '2 would get whatever goods are available after satisfying consumer l's demand. If your answer is yes. show that there is a small price change that would make consumer '2 better o'. If you answer no. explain why the welfare of consumer 2 could not increase with a small change in a price. 'Would the allocation resulting from a small change in a single price be Pareto eicient'? Explain. e. Suppose that the original economy is modied so that consumer I now owns a competitive rm. [The consumers" utility functions and endowment vectors remain as before.) The rm has constant returns to scale production. producing 2 units of good '2 for each unit of good 1 it uses as input. Find competitive equilibrium prices for this economy and compare them to the prices in the equilibrium of part b. Find the cor responding consumption vector of consumer 2 and compare that consumer's utility in the new equilibrium to the utility in the equilibrium of part b. Interpret and explain the comparison. Two researchers with identical characteristics try to complete scientic projects in a single period. They can work separately andy'or together. Each researcher can complete at most one project working alone and also at most one project jointly during the period. The endowment of time is l for each researcher. Researcher 1' working alone for e.- units of time has probability pl[e.-} of completing his work {time invested represents the effort expended}. Here. pH is a continuous function satisfying pm} = .p'l:e.-:I 3.:- IEI and p"{e.-]I e: {I for 1 .3 e.- l3 4]. If researcher 1' completes his work alone. he consumes the credit [monetary or not] 41:5, = 1. If he fails at the project working alone. he consumes the credit 93,-. = 41 If the researchers work together and researcher 11 spends 1".- units of time on the joint work. then the probability of the joint work being successful is pl[-r1 + r1}. but each 1' = 1. 2 gets the credit mg = c. I} a: c :: 1 if they succeed. If they fail. $2} = 4]. 13 = 1.2. Each researcher i has a utility function n{l..m.-:I = LEE. where l.- = 1 e.- r.- is the amount of leisure time i consumes and x.- = 3;. + 335* is the urn-usual 1.11.1\"! I- .- I u.\" \"11.. Pluwullll-J anus-ulna\": J- 1"! In run-u. u. c. Now. suppose that efforts of the researchers are not directly observable and not contractible. The researchers choose their labor inputs independently. but they can only do joint work and this can be enforced Derive the necessary conditions for strictly positive Nash equilibrium eort levels. Compare these effort levels with the effort level derived in b for general probability function p[-}. Compute eort levels r.- = 1: 2 for the probability function e}. cl. 'Ir'ir'hat is the eccpected number of successfully completed projects for the two re- searchers together in cases a_. b and c_. respectively when the probability function is given by f{-}'? Interpret the results. e. Suppose in part c. the researchers can also work on the side on their own if they so choose. Researcher 11 works r.- units of time for the partnership and e,- on his own private work. The effort inputs are not directly observable or contractible and the scholars choose their levels independently. i. \"That are the rst order conditions for a symmetric {i.e.: el = ea: r1 = 1'1} and strictly positive Nash equilibrium under the general probability function pH when [I e: s c l? . c v: s: ii. Suppose. instead: that the probability function is given by: 9H} = { l :2: D E? I 1 - Compute a syn'unetric Nash equilibrium in this case