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QUESTION 1 Your entire wealth consists of the balance of your bank account, which is $ 1, . 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 R)iej. Assume that all of these functions are continuous, strictly quasiconcave and strictly monotone. Unlike in class, we are going to treat individual endowments as variables. For individual i, her endowment is denoted by WERE. Define the functions x' : R4 + x R! -> R. by x' (p, w') = argmaxxER! {u'(x) |p - x

RL by Z(p, w) = [[x' (p. wi) - wij (2) and the set M = {(p, W) ER! x RY [ Z(p, w) = 0} . (3) Here, x' is individual i's demand function and Z is the aggregate excess demand function. Ob- viously, p is a vector of competitive equilibrium prices of exchange economy {], (u', wilies} if, and only if, (p, w) ; (wi - wi ).3. Argue that for any I: and any w, there exists W E R}: such that {a} [p,W} is in M; and {h} for all i, \"[3 ~ W1 = 11 ~ WE. (Him: Third: ofa prole of endowments such that at prices p each individual i. demands xifp,wi} and 1which guarantees that markets clear.) 4. Use the previous steps to explain haw an analyst who only observes M can construct function Z in a unique manner. 5. Argue that there artists a sub-prole of individual endowments for all agents other than i=1, say [W2 ..... WILsuehthatforallpandalel, xl[p,w1] = Z[p,wl,;2,...,w ] +w. (Hint: Think of a way of kicking all agents but i = 1 out of the market) 6. Explain how, once the analyst of part 4 has constructed Function Z, she can construct mction :1 in a unique manner