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assignment

1. To model the problem of deciding whether or not to attend college, suppose an individual, Ann, consumes in each of two periods. She is endowed with income is; in period 1, and her second period income depends on college attendance. If she does not attend, she receives an additional amount of mg in period 2. If she does attend, she incurs the cost a in period 1, but she receives an, :5 mg in period 2. In addition, she can save any amount 5 of her rst period income and earn lnl'rBIIIt at rate 11. Suppose Ann's interteinporal preferences are additively separable of the form U(c1,cg) = vial} + shes), where 5 E (0,1), and the price of c is 1 in both periods. For concreteness, assume u(c) = inc. Ann must decide whether or not to attendfinvest in college and how much to save. Assume initially that she can not borrow and must nance the cost e entirely from rst period income as. a. First, for the casein which Ann does not attend college, set up and solve the problem of determining her optimal savings, am. Verify that this constitutes a maximum. Determine conditions under which the optimum 3:\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 a. Suppose the bank kiiows whether the rm built the new facility or not in the rst period when it offers a contract. Draw the game tree of this game. b. Find all sequential equilibria of the game in [a]. c. Suppose from now on that the rm's possible investment in the rst period is in software and the bank does not know whether or not the rm 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 rm invested in the software with probability p. e. Find all sequential equilibria of this game. 4. Consider a rm with the technology {3 = {311:2}? that purchases inputs 3:1 and an in competitive factor markets at the prices url = mg 2 1. Initially? the rm faces the demand curve qu} = 12D p for its output. a. Determine the rm's optimal supply decision and prots. b. Next} suppose the rm's prots were taxed at the constant rate of 20%. How would this affect the rms optimal decisions? How much would the rm pay in taxes and what would be its post tax prots? c. Suppose that? in addition to producing :13, the rm could also contribute money? y? to charity and thereby reduce its overall tax rate. The rate would then become y) 2 mas-{\"02 DlyL D}. 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 rm. What are the rms prots? lCompare the amount of taxes paid in part {b} to the amount of taxes and charity paid here. Also compare the rm's post tax prots. d. Suppose again that the tax rate is xed at t = [1.2. Now. while charitable contributions do not affect the tax rate? they garner publicity for the rm and increase demand for its product. Thus? the rm faces the demand {y} = 12'] p + Z. J'l'lgainJ assume the eon trihution is from pretax earnings. Determine the rms optimal suppl}r decision in this case. Also determine the amounts of taxes and charityr paid and the post tax prots. e. Assuming that tax revenues and charitable contributions are devoted to precisel}r the same uses? rank the cases {h}? [c] and {d} in terms of the resources {tax revenues and charity} generated and post tax prots