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Consider a simple utility function for two goods, x1 and x2: u = (x1 1) 1 (x2 2) 2 , with 1 > 0, 2 > 0. The parameters 1 and 2 are constants, but they could be positive or negative. (a) Someone suggests that "it would be okay" to assume that 1 + 2 = 1. Yet someone else claims that this restriction may be unrealistic. Which person is correct? Briefly explain your answer. (b) Suppose that the person with the above utility function is struck by lightning. He survives, except now his utility function is u = 1 log(x1 1) + 2 log(x2 2). Will his consumption decisions change as a result of being struck by lightning? Briefly explain your answer. (c) Returning to the original utility function, assume that the consumer faces the budget constraint p1x1 + p2x2 = w, where p1 > 0, p2 > 0 are prices and w > 0 is wealth. Set up the Lagrangean and solve for the two first-order conditions. Then use the budget constraint to solve for the Walrasian demands of both goods. These should be functions of p1, p2, w, 1, 2, 1, and 2. Finally, use your answer to part (a) to simplify your answer. (d) Finally, turn to a more general utility function with L goods: u = Y L k=1 (xk k) k , where k > 0 for all k, which is maximized subject to the budget constraint PL k=1 pkxk = w. Using the same approach as in part (c), derive the Walrasian demands for the L goods. (Hint: first substitute out the Lagrangean multiplier using the first-order conditions for two goods, i and j, then find an expression for pixi and sum that expression over all i, using a normalization similar to the one used in part (c)).

Consider an expenditure function that has the form log[e(p, u)] = X L k=1 k log(pk) + u Y L k=1 p k k . Don't worry that this is expressed as log[e(p, u)] instead of as e(p, u). After all, if we apply the exp() function to both sides, the left-hand side will become e(p, u). (a) Apply Shephard's lemma to this expenditure function to obtain the Hicksian demand for good i. (Hint: Differentiate with respect to log(pi) to obtain an elasticity that includes the Hicksian demand. Note also that p k k equals (e log(pk) ) k . The Hicksian demand should be a function of the 's, the 's, prices, and u.) (b) Derive the indirect utility function that is associated with this cost function. This is easier than part (a). (c) Finally, use your answer to (b) to obtain the Walrasian demands. What is the name of the derivation that is used to obtain the Walrasian demands?

Consider a firm that produces a single output q 0 using inputs z1 0 and z2 0, where the input-requirement set is nonempty, strictly convex, closed, and satisfies weak free disposal. Assume the firm operates in competitive markets. The firm's profit function is (r1, r2, p) = p 4(r1 + r1r2 + r2) , where p > 0 is the price of output, r1 > 0 and r2 > 0 are the input prices, and is a constant parameter. (a) What condition on (if any) is required for (r1, r2, p) to satisfy the price homogeneity property of a valid profit function? Justify your answer. (b) Use (r1, r2, p) to derive the firm's unconditional supply and factor demands. (c) Derive the firm's conditional factor demands and cost function. (d) It is easy to verify that the conditional input demands in (c) are non-increasing in their own price. Show that this result holds in general for a cost function derived from a production possibility set with N inputs and M outputs.

Consider a firm that produces output q 0 at a cost of c(q), where c 0 (q) > 0 and c 00(q) > 0. Also assume that there is a probability 1 > > 0 that the firm experiences an equipment failure and incurs additional repair costs equal to cR > 0 per unit of output or cRq in total. The competitive price of output is p > 0. (a) Derive the firm's first-order condition for an interior solution assuming its objective is to maximize expected profit. What is the economic intuition of this condition? (b) Derive the firm's first-order condition for an interior solution assuming its objective is to maximize its expected utility of profit, where the strictly increasing and strictly concave function u() characterizes its risk preferences. (c) Will the firm produce more if its objective is to maximize expected profit or if its objective is to maximize the expected utility of profit? Justify your answer and provide the economic intuition for your result.

Consider an expenditure function that has the form log[e(p, u)] = X L k=1 k log(pk) + u Y L k=1 p k k . Don't worry that this is expressed as log[e(p, u)] instead of as e(p, u). After all, if we apply the exp() function to both sides, the left-hand side will become e(p, u). (a) Apply Shephard's lemma to this expenditure function to obtain the Hicksian demand for good i. (Hint: Differentiate with respect to log(pi) to obtain an elasticity that includes the Hicksian demand. Note also that p k k equals (e log(pk) ) k . The Hicksian demand should be a function of the 's, the 's, prices, and u.) (b) Derive the indirect utility function that is associated with this cost function. This is easier than part (a). (c) Finally, use your answer to (b) to obtain the Walrasian demands. What is the name of the derivation that is used to obtain the Walrasian demands?

Consider a firm that produces a single output q 0 using inputs z1 0 and z2 0, where the input-requirement set is nonempty, strictly convex, closed, and satisfies weak free disposal. Assume the firm operates in competitive markets. The firm's profit function is (r1, r2, p) = p 4(r1 + r1r2 + r2) , where p > 0 is the price of output, r1 > 0 and r2 > 0 are the input prices, and is a constant parameter. (a) What condition on (if any) is required for (r1, r2, p) to satisfy the price homogeneity property of a valid profit function? Justify your answer. (b) Use (r1, r2, p) to derive the firm's unconditional supply and factor demands. (c) Derive the firm's conditional factor demands and cost function. (d) It is easy to verify that the conditional input demands in (c) are non-increasing in their own price. Show that this result holds in general for a cost function derived from a production possibility set with N inputs and M outputs.