Question 1 Consider an economy with / consumers and & goods. For each consumer i e {1, ..., /}. the consumption set is R4. Her utility function is given by "(x') = - where ax, B; > 0 for k E {1. ..., L). As usual we denote r' = (r;, ..., r)). Consider price vectors (P1, ...,pz) and wealth levels (w), ...,ur ) for which the solution to the utility maximization problem is interior for every consumer i e {1. .... /}. a.) Derive the Walrasian demand function for good j by consumer i. (Be careful with your calculations. Double-check! It is easy to make mistakes.) b.) What is the slope of consumer i's Engel curve for good j at (p, w')? c.) Find a condition as general as possible on parameters or, A i e (1,.. .I), k E {1,.., L} guaranteeing the existence of a positive representative consumer. Do we need restrictions on parameters ox, ke {1, .... L}? d.) Consider now the special case with just a single consumer and two goods. The consumer's utility function is given by "(1, 12) = -ce Bin - age Mara Derive the wealth-expansion path for a given price vector (p1, pz). e.) In problem d.), when does the wealth-expansion path intersect the ri-axis and when does it intersect the x2-axis?Question 2 It is intuitive to think that the presence of more agents in the economy "shrinks" its core, since there are more coalitions that can object a given allocation. You will understand in this question why this is indeed the case.' Fix a standard, two-person exchange economy & = {(u', w' ), (u', w )). Define its replica as the four-person exchange economy 82 = ((u', w'), (uz, wi), (w', w'). (u', w.)). where (u', w) = (u', w ) and (u', wi) = (uz, w/ ). 1. Argue that if (p, x', x') is a competitive equilibrium for &, then (p, x', x3, x], x' ) with x' = x' and x* = x', is an equilibrium for 82. 2. Argue that if both utility functions are strictly quasi-concave, and (p, x', x', x', x' ) is a competitive equilibrium for & , then, x' = x' and x? = x*. 3. Argue that if both utility functions are strictly quasi-concave, and (x', x', x', x* ) is in the core of &', then, x' = x and x =x'. 4. Argue that if both utility functions are monotone and strictly quasi-concave, and (p, x', x?) is a competitive equilibrium for &, then (x], x', x', x") is in the core of E?. 5. Suppose that u' (x ) = u' (x ) = x'x, w' = (1,0) and w = (0, 1). Argue that allocation ((0, 0), (1, 1)) is in the core of &, yet allocation ( (0, 0), (1, 1), (0,0), (1, 1)) is not in the core of &?. 6. Use these results to argue, informally, that the replication of agents does not affect the set of equilibrium allocations of the economy but shrinks its core.QUESTION 3 You won $200M ($200,000,000) playing the lottery. Two charities have approached you for a donation. You summoned the directors of the two charities and informed them that they will play the following sequential game: First, the director of Charity 1 (call her D, ) will make a public request, which can be any integer dollar amount, starting from $1 and up to $200M. Then the director of Charity 2 (call him D, ) - having heard D, 's request - will announce his own request, which, again, can be any integer dollar amount starting from $1 and up to $200M. You inform them that one of your objectives is to punish greed and thus you will proceed as follows: Let d, denote the amount requested by D, and d, the amount requested by D, . If d, > d, then D, - being the greedy one - will get nothing, while D, will be rewarded with two times what he asked for, up to $200M; that is, D, will get min (2d,, 200M) . Symmetrically, if d,