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g _ Each worker's utility is given by the dierence between ie wage she is paid and le cost of education. (a) [Note: for this part do not assume that each worker must be paid a wage equal to her productivity] Is there an incentivecompatible situation where {1) the employer o'ers two wages, depending on the education level: wage wt to those whose education level is e: and wage \"in i w: to those whose education level is 3\" at 3* and reises to hire anybody with education a e {a'= en} , (2) both types 3L and EH choose education level a', while type 5'\" choose education EM ? [Note that you should make no assumptions about whether EM =1: 2* or 2* :15 2M and similarly for w\" and w* _] It'iere is such an incentivecompatible situation, please describe it in detail. It'your claim is that it does not exist, please prove it. For parts (b) and (c) assume that the employer pays each worker a wage equal to the worker's expected productivity (as computed by the employer, who is risk neutral). 0)} Dene and describe in detail a pooling equilibrium, that is, a signaling equilibrium where all three types make the same choice of education level, call it E- [Assume that the employer believes that anybody who shows up with education level e as E must be of type L-] {c} Find allthe pooling equilibria when 19L =1, HM = 2, EH = 5. Now let us change the situation as follows- There are only two types of potential workers: those with productivity EL and those with productivity EH , with EH 3- EL l} U . The 'action of type 3L in the population is equal to the 'action of type 3H. Asslnne that the cost of education is the somefor both types and is given by C(eJ = e- Suppose that the utility of worker of type 5' E {HD4513} who is paid wage w and chooses education level as is U(w,e,) =Ewe_ Assume also that e E [19,5] withtl 6:. a {E}, that is, there is a minimum level of education a that every worker must have (it is mandated by the government) and a maximum level of education I) (e_g_ corresponding to a \"x Consider an economy with two goods, a private good and a public good subject to congestion. One unit of private good can be transformed into a unit of public good : if 2 units of private good are used then 1: = z, where y is the quantity of public good produced. There are I agents with initial resources {wilizl in private good who can use the public good with varying intensity. Let qf denote the intensity of use of agent 1'. Think of y as a number which sunmlarises the characteristics of a freeway (width, length, road quality) and of if as the miles that agent 1' drives on the freeway. 1When agent 1' uses the public good with intensity of, it costs the agent cm") of private good {think of ch'i) as the cost of gas). The utility of an agent for the public good depends not only on 3,: but also on the congestion Q = ELI if of the public good. That is, the agents' preferench depend on Y = y, Q}, which we can call the quality of public good, where f is increasing in y and decreasing in Q. The preferences of agent 1' are thus represented by a function uTmq'HY} where I": is the consumption of private good, q': the intensityr of use of the public good, Y is the quality of the public good and of is an increasing function. All functions considered in this problem are continuously differentiable. {a} Write a maximum problem whose solutions are the Pareto optimal allocations of this economy. Be careful to write correctly the feasibility constraints. {b} Write the rstorder conditions for an interior solution {If :a U, y r:- I], q'i 1: D for all i J of this maximum problem. {c} By eliminating the multipliers in {b}, show that {i} the optimality of the provision 3-,." of the public good requires that a condition, akin to the Samuelson condition, must be satised, involving all the agents' valuations gig: %:. \f{E} amount 3; produced, and the agents' marginal pollingoeas to pa},r or need to be compensated for a marginal change in the intensity:r if of agent is use of the public good. Consider an equilibrium ({i", q'izl, ,a,?, T) such that: {i} the government taxes the agents' endowments at the rate f to nance the provision 1,? = 1"" 2:21 mi of the public good; {ii} agents are small and talce the level of congestion a as independent of their own intensityr of use. Shovnr that, no matter how T is chosen, the equilibrium is not Pareto optimal. Explain why, and suggest a via},r of improving on the equilibrium. Question 4 We were all pretty relieved that Ali from our first prelim got such a simple consumption problem. Yet, we also realized that putting him into the CobbDouglas straightjacket missed some features of the story. Since the overall theme of the second prelim should be \"We can do betterl\