Mcroeconomics questions

Question 1. Throughout this question we assume that all the optimization problems considered have unique solutions which are interior and characterized by the first-order equalities. We also assume that all decision makers take all prices as given, and that the Lagrange multipliers are nonzero. 1.1. This part covers standard price-taking production theory, and we minimally adapt the notation used in class and in Chapter 5 of Mas-Colell et al. We consider a price-taking firm that produces one output by using / inputs, according to a direct production function f: RR: ( 2, , .... = N ) H> / ( =, ,....=N). where (z,,...,ZN) = z denotes the input vector. We denote by q the amount of output, and in order to avoid notational confusion with Parts 1.2 below we use capital letters for prices, i. e., PE R. denotes the price of the output, and W = (W,,...,W)ER, the vector of input prices. We consider the following optimization problems. Problem INPUTPROFITMAX[W, P]. Given (W, P), choose (z,,...,ZN) in order to maximize Pf (z)-W . z. Denote by E(W, P) and by II(W , P) the solution and the value, respectively, of this problem. Problem COSTMIN[ W, q]. Given (W, q), choose (z,,...,Z,) in order to minimize W . z subject to f(z) 2q. Denote by 2(W ,q) and by c(W ,q) the solution and the value, respectively, of this problem, and by B(W ,q) its Lagrange multiplier. Problem OUTPUTPROFITMAX[ W, P]. Given (W, P), choose q order to maximize Pq-c(W,q). Page 1 of 7 1.1(a). Prove that * solves INPUTPROFITMAX[W, P] if and only if, defining q* = A(z*): (i) =* solves COSTMIN[W, q*], and (ii) q* solves OUTPUTPROFITMAX[W, P]. Remark. It is here preferable that you write direct proofs by contradiction rather than manipulating first order conditions 1.1(b). Interpret B(W ,q). 1.1(c). Argue that, if z* solves INPUTPROFITMAX[W, P], and q* =f(z*), then P=B(W,q*).1.2. We now consider a consumer with utility function a : R# -> R :(x,.....Xx) (> u(x,,..-;XN), which is in particular assumed to be locally nonsatiated, where (X,,...,X) = x denotes her consumption vector. We denote by p= (p, .....P,)ER., the price vector for the consumption goods, and by w > 0 the wealth of the consumer. We consider the following optimization problems. Problem UMAX[p, w]. Given (p, w), choose x = (X,,...,Xx ) in order to maximize u(x) subject to p. xs w. Denote by x(p,w) and by v(p,w) the solution and the value, respectively, of this problem, and by A(p, w) its Lagrange multiplier. Problem EMIN[p, u]. Given (p, u), choose x = (X,,...,Xx ) in order to minimize p . x subject to u(x) 24. Denote by h(p,u) and by e( p,u) the solution and the value, respectively, of this problem, and by u(p,u) its Lagrange multiplier. Now a new one: Problem FRISCHIP, r]- Given pe R. and re R. , choose x = (x,,...,X, ) in order to maximize ru(x)- p. x. Denote by q(p,r) and by 2(p,r) the solution and the value, respectively, of this problem. 1.2(a). Interpret Problem FRISCH[p, r] and the parameter r. 1.2(b). What can you say about " (p,r)? Justify your answer. ap, Page 2 of 7 1.2(c). Some of the optimization problems in this part (Part 1.2) are formally (mathematically) identical to some of the ones in Part 1.1 above. Which ones are those? Explain. 1.2(d). Interpret the Lagrange multipliers A(p,w) and u(p,u). 1.2(e). Show that if u* = v(p,w*), then Ap,w* )=- H(p,u*) Interpret. (Hint. Use duality.) 1.2(f). Let x* = Q(p,r) and u* = u(x*). Show that r= u(p,a*) and interpret. 1.2(g). Let x* = Q(p,r), u* = u(x*) and w* = e(p,u*) . Show that r = A(p,w#) - and interpret. 1.3. Comment on the results of parts 1.1-1.2 above.Question 2 2. An economy is composed of two sectors of production and one (representative) consumer. The first sector of production (represented by firm 1) produces good 1 from the two factors of production, capital and labor, with the Leontief production function y = min(3K"], L'). The second sector of production (represented by firm 2) produces good 2 from capital and labor with the Leontief production function y' = min(K2, L'). The representative consumer, who owns the total supply of capital K > 0, the total supply of labor time L > 0 and the firms, has the utility function 2(x1, 12. () = 21 120 where r, denotes the consumption of good 1, 72 the consumption of good 2 and / the consumption of leisure time. (a) Justify the terminology: "the production of good 2 is relatively more capital intensive than the production of good 1". (b) The goal of the exercise is to compute the competitive equilibrium of this economy for different values of the parameters (K, L) and study what happens when the economy becomes relatively richer in capital. Without loss of generality let L = 1. Let pi denote the price of good 1, p2 the price of good 2, r the price of capital and w the price of labor. Argue that, in a competitive equilibrium, w > 0 and r 2 0. Intuitively when do you expect r to be zero? We use the property w > 0 to normalize the wage to 1. (c) Find the relation between the price of each produced good and the prices of the factors that must hold in equilibrium. (d) Show there exists & such that if K 2 k, there is an equilibrium such that r = 0. Describe this equilibrium. (e) Assume K 0 must satisfy to be an equilibrium price. Check that if K 0 to a, the interest rate decreases and the equilibrium ratio y2/y1 increases. Explain these results