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Question 3 This question consists of two independent parts, each of which carries one half of the ques- tion's grade. 1. Let J1 = {1,..., I,} and 32 = (1, + 1,..., I, + 12] be two (disjoint) local societies, and denote by ] = J, UJz the global society resulting from their integration. For each individual in these societies, let u be continuous, strongly quasiconcave and strictly monotone, and let wi E R- . In this exercise you are going to argue that each society can protect itself against any damage brought about by globalization. For each local society, k = 1,2, let (pk, (x' )ien ) be a competitive equilibrium of ex- change economy (3, (u', wiley,). Show that there exist (wilies such that: (a) for each k = 1,2, ELEJK Wi = Lien wt; and (b) for any competitive equilibrium (p. (x' hes) of the (global) economy (J. (ut. wiles) one has that u' (x ) > u' (x' ), for all i e J. (Hint: The claim is that each society can implement a local fiscal policy that guarantees that any global equilibrium is Pareto superior, for itself, to its own local equilibrium. Now, how would you make sure that nobody can be made worse off, through voluntary trade, than at some given allocation?) 2. In a standard exchange economy {], (u', w')teo} the distribution of wealth is biased in favour only of agent 1 if for any competitive equilibrium (p, x) of the economy the following three properties are true: (a) for each i 2 2, u'(x' ) > u' (x) for every x such that p . x u'(x ); and (c) for every i, j > 2, u'(x ) > u' (x) for every x such that p . x w. for all individuals, where w, is some minimal bundle. Determine assumptions under which fiscal policy can ensure that the distribution of wealth is biased in favour only of agent 1 and permits subsistence. State your claim formally and prove it.QUEETIDN ti. Players 1 and 2 have agreed to play tennis tomorrow. Hoytrever1 this 1will be possible only if at least one of them shows up early to claim the court. Each player is either a morning person or a night person. Each player knows his own type1 and, regardless of his own type, assigns probability % to the other player being a morning person. Each player chooses between showing up early (E) to claim the court or showing up late {L}; the choices are made independently and simultaneously. The basic payoffs are as follows: (1} if either player shows up early. then playing is pomible. and each player receives a payoff of 4. [2} if both show up late1 then they will not be able to play and each player receives a payoff of [it in addition to those payoffs, a morning person who shows up early also receives an extra payoff of l. whereas a night person 1who shows up early suifers a payoff penalty of 3. Thus. for example1 if they are both night persons and Player l shows up early. while Player 2 shows up late then Player l's payoff is 43 = l and Player 2's payoff is 4. All these payoffs are von Nemmm-Morgenstem parsf'fs {a} Describe this situation using states and information partitions. With every state associate the relevant game. {h} Find the common prior. {c} Use the Harsanyi transformation to convert the representation of part (a) into an extensive- fonn game. {:1} (ii) How many pure strategies does Player 2 have?I {til} Write down one possible mixed sn'ategy for Player l1 which is not a pure snategy. {:13} Write down one possible behavior strategy for Player 1. {c} Find a weak sequential equilibrium of the game of Part {c} and prove that there are no other (pure- or mixed-strategy} weak sequentiai equilibria. {f} Prove that the weak sequential equilibrium found in Part (e) is also a sequential equilibrium