ANSWER all questions they are micro

QUESTION 1 Your entire wealth consists of the balance of your bank account, which is $ W. . You also own a painting which is worthless to you. You have a chance to sell the painting to a potential buyer. All you know about the buyer is that she has a reservation price (i.e. a maximum price she is willing to pay) re(L, M, H) with L 0 for k e {1. ..., L}. As usual we denote r' = (r;, ...,;). Consider price vectors (p1, .... pz) and wealth levels (w], ...,w') 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, wi)? c.) Find a condition as general as possible on parameters ox, By, i e {1,..., / }, 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 Derive the wealth-expansion path for a given price vector (P1, p2)- e.) In problem d.), when does the wealth-expansion path intersect the r,-axis and when does it intersect the ry-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', wl ), (u', w?)). Define its replica as the four-person exchange economy 82 = ((u', w'), (u', wa), (u', w'), (u', wa)). where (u', w3) = (u', w') and (u', wi ) = (uz, w?). 1. Argue that if (p, x', x?) is a competitive equilibrium for &, then (p, x', x], x3, x' ) with x3 = x' and x* = x', is an equilibrium for &?. 2. Argue that if both utility functions are strictly quasi-concave, and (p, x', x', x', x* ) is a competitive equilibrium for &', then, x' = x3 and x? = x*. 3. Argue that if both utility functions are strictly quasi-concave, and (x], x?, x3, x* ) is in the core of &', then, x' = x3 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], x3, x], x?) is in the core of &?. 5. Suppose that u' (x) = u'(x) =x'x?, wi = (1, 0) and w2 = (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." 1 We are using the term "shrink" loosely, since the presence of more agents changes the dimension of the allocation space, so comparing the sizes of the cores will require some refinement of the argument. In the limit, one can show that replication ad infinitum reduces the core to just the set of equilibrium allocations