c. Characterize the set of interior Pareto efficient (PE) allocations (in which both con- sumers consume positive amounts of both goods). In what way does this set depend on #7 Starting from a PE allocation, does a small transfer of good 1 from consumer 1 to consumer 2 leave the allocation Pareto efficient? d. Let # =0. Characterize a competitive (Walrasian) equilibrium (CE) for this economy. Is the allocation Pareto efficient? e. Let 0 = 1/2. Characterize a CE when consumer 2 acts as if the equilibrium consump- tion of good 2 by consumer 1 is fixed, unaffected by actions of consumer 2. f. Under what, if any, condition is the CE allocation in part e Pareto efficient? g. Is it possible that in the CE allocation in part e consumer 1 consumes inefficiently little of good 2, given consumer I's consumption of good 1? Consider the case in which (t) = Int. Explain and interpret your answer. 3. A government devises an optimal tax schedule for a population, half of whom are of type H, half of type L. A type H agent produces qu = 2ex units of output (income) when it provides ey E [0, 1] units of effort. A type L agent produces qz = (3/2)er, units of output when it provides er E [0, 1] units of effort. Outputs produced by different agents are physically homogeneous, hence indistinguishable, and (except in part a, below) the government cannot directly observe the type of any particular agent or its effort level. The government observes agents' incomes (outputs) and assigns a tax to each income level. The government knows that if a type i agent produces q units of output, providing e; units of effort and paying t; $ 9 - e, in tax, the agent gets utility (q, - 4, - e;)1/2. (If to 1 for = > 1. a. Find the researchers' optimal efforts and expected utilities if they work alone. b. Now, suppose that efforts of the researchers are not directly observable and not con- tractible. The researchers choose their labor inputs independently, but they can only do joint work and this can be enforced. Derive strictly positive Nash equilibrium levels of effort of the two researchers and the corresponding expected utilities. Show why there cannot be a Nash equilibrium involving a researcher devoting either 0 or 1 of his time. c. Now consider the following game: Reseacher 1 either decides to work alone ($) or proposes to Reseacher 2 to work together (7). If S is chosen, the researchers work alone. If T is chosen, then researcher 2 either decides to work alone (s) or to work together (t). If researcher 2 chooses s, both researchers work alone. If t is chosen, the workers work together but their efforts are chosen simultaneously and the efforts are neither observable nor contractible. Draw an extensive form game tree describing this game and find pure strategy subgame perfect Nash equilibria for each possible value of c. (When a researcher faces the same expected utility working alone (s or $) and working jointly (t or T), assume that he chooses to work alone). d. Suppose the researchers' efforts are observable and contractible. What are the ex- pected utility maximizing cooperative efforts for joint work in which the researchers contribute the same effort levels