Assignment how to do it

1. Consider an individual who lives for two periods. In each period she is endowed with one unit of time which can he devoted to leisure or labor. In period 1 she earns $1.9 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 [(1,1) 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 to] and pear the unemployment tea: on her period 2 earnings as well. Assume the individual has stationary von Neumann - Morgenstern preferences represented by HEW, ft] = Till-g HELP, where my and E1: denote her net earnings and leisure in period t, respectively. Her intertemporal preferences are given by U = 1t[m1,-1] | u[m2,fg], where ,8 E {l}, 1]. Suppose the probability of injury or layoff is r c: 1 and there are no private savings or accumulated interest on tax payments. a. Set up and solve the agent's decision problem assuming her objective is to maximise expected lifetime utility. b. Does it matter whether the agent chooses 6: ex ante or ex postT that is, whether she chooses g in period 1 (before knowing if she will be employed in period 2) or in period 2 [knowing that she will be employed)? o. Discuss how the solution in part a depends on each of the parameters r1 w, {3 and s'. d. What level of T E [I], 1} maximises the agent's expected lifetiJne utility? s. Now suppose it is pomihle to have (nonnegativej private savings in addition to the public unemployment insurance described above. Assruning the interest rate on private savings is t 2 i}, modify the above model to include private savings and explain what effect this would have on the individual. Are private savings and public unemployment insurance substitutes? i. What change in the agent's utility function might improve the realism of the model with saving and make the unemployment tax and benet payment more desirable for the agent? Explain. 2. Consider an economy with one rm, two consumers and two goods: food and time for either labor or leisure. IConsumer 1 initially owns 4 units of time and gets utility 1.1031, c1) when E, 3 {I is its leisure consumption and c, 23 is its food consumption. The function u is twice continuously diferentiable, homogeneous of degree 1, and strictly quasiconcave, with strictly positive partial derivatives. Consumer 2 owns the rm but no goods initially; it gets utility equal to its food consumption as I: U. The rm producm t} 2 10sr s2 9 units of food when it uses a E [1, 5] units of labor input and He} = {i if i] E s s". 1. The feasible allocation with (8,, c1] = {2, 7) is the only i'areto eicient (PE) allocation with c; = ID. 3.. Compare f'{s) and sh's, the marginal and average products of the firm at c = 3,}? and :2: = 4 and use the comparisons to describe the local returns to scale of the rm at input levels near 2 and 4. b. Graph the set of feasible allocations, represented by the feasible consumption vectors for consumer 1. Show the PE consumption vector (2, T} for consumer I with a. possible indier ence curve through it. Prove that there is no price quasiequilibrimn with transfers with this PE allocation. {For this economy, in a price quasi-equilibrium with transfers, the rm max\" imises its prot and neither consumer prefers a cheaper consumption vector.) Find prices and transfers such that, in this allocation, both consumers maximise their utilities subject to budget constraints and the rm satises the rst order condition for prot maximisation. c. In the graph of part b, Show how the whole set of PE allocations with El 2::- U can look. Be sure your graph is compatible with all the assumptions above. Prove that for each level of input for the rm there is at most one PE allocation. Show that Cl ,' 1 *4: W2 in each PE allocation with 1 2':- l}. d. Find all the PE allocations with El :> 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 [Walrusian} 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 331 in period 1 and if necessary posts price p; in period 2. There is no {additional} cost in providing the units so the monopolist attempts to maximise the total [undiscounted] revenue from the two periods. If H buys the good in the rst period at p1 {and so gets to use it in both periods}, his payoff is $9013 _. 101. If he buys it in the second period at 352, his payoff is $3 - pg. If he does not buy in either period, his payoff is 0. Consumer L gets payo $5M p1 if he buys the good in period 1, $205 P2 if he buys it in period 2, and $3 if he does not buy in either period. Additional continuiption of the durable good adds nothing to the commonersT payois. in period 1, the monopolist posts price 391. Consumers H and L make simultaneous decisions whether to buy the durable good (B) or not buy {N}. If both oonsumere 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 502. Any consumer(s} who did not buy in period 1 mice {simultaneous} buy-or-nothuy decision{s) in period 2 after learning 39;. Assume that a consumer buys the good if he is indifferent between buying and not buying. Denote a period 1 history as pIBN if M chooses p1, H buys, and L does not buy in period 1 and similarly for other histories.