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105 306 Chapter 6 INCOMPLETE I .HIWISTMENT FIGURE 6.10 The distribution of  pf{t) in the CaplinLeahy model (c) (j) Do anticipated changes in m (that is, changes that are expected a< of when rigidprice firms set their prices) affect y? Why or why not (ii) Do unanticipated changes in m affect y? Why or why not? 6.14. Consider an economy consisting of many imperfectly competitive, prictsetting firms. The profits of the representative firm, firm i, depend oaggregate output, y, and the firms real price, r,: , = 7 ( , ,), where < (subscripts denote partial derivatives). Let r*(y) denote the profltmav: mizing price as a function of y; note that r*(y) is characterized b. 7 2( , *( )) = 0. Assume that output is at some level , and that firm zs real price is *( ). Now suppose there is a change in the money supply, and suppose that other firms do not change their prices and that aggregate output therefore changes to some new level, yj. (a) Explain why firm zs incentive to adjust its price is given by G = ( 1, *( 1)) ( *( )). (Jb) Use a secondorder Taylor approximation of this expression in yi around 1 = to show that G  7 22( , *( ))[ *( )]( 1  )/2. ( ) What component of this expression corresponds to the degree of real rigidity? What component corresponds to the degree of insensitivity of the profit function?
6.15. Multiple equilibria with menu costs. (This follows Ball and D. Romer, 1991.) Consider an economy consisting of many imperfectly competitive firms. The profits that a firm loses relative to what it obtains with p, = p* are K{p,  p*),K > 0. As usual, p* = p + and = m  p. Each firm faces a fixed cost Z of changing its nominal price. Initially m is zero and the economy is at its flexibleprice equilibrium, which is = 0 and p = m = 0. Now suppose m changes to m. (a) Suppose that fraction f of firms change their prices. Since the firms that change their prices charge p* and the firms that do not charge zero, this implies p = fp*. Use this fact to find p, y, and p* as functions of m and f. ib) Plot a firms incentive to adjust its price, K(0p*) = Kp*, as a function of f. Be sure to distinguish the cases < 1 and > 1. ( ) A firm adjusts its price if the benefit exceeds Z, does not adjust if the benefit is less than Z, and is indifferent if the benefit is exactly Z. Given this, can there be a simatlon where both adjustment by all firms and adjustment by no firms are equilibria? Can there be a situation where neither adjustment by all firms nor adjustment by no firms are equilibria? 6.16. (This follows Diamond, 1982.)* Consider an island consisting of N people and many palm trees. Each person is in one of two states, not carrying a coconut and looking for palm trees (state P) or carrying a coconut and looking for other people with coconuts (state C). If a person without a coconut finds a palm tree, he or she can climb the tree and pick a coconut; this has a cost (in utility units) of c. If a person with a coconut meets another person with a coconut, they trade and eat each others coconuts; this yields IT units of utility for each of them. (People cannot eat coconuts that they have picked themselves.) A person looking for coconuts finds palm trees at rate b per unit time. A person carrying a coconut finds trading partners at rate aL per unit time, where L is the total number of people carrying coconuts, a and b are exogenous. Individuals discount rate is r. Focus on steady states; that is, assume that L is constant. [a) Explain why, if everyone In state P climbs a palm tree whenever he or she finds one, then rVp = b(Vc  Vp  c), where Vp and Vc are the values of being in the two states. (Jb) Find the analogous expression for Vc (c) Solve for Vc  Vp, Vc, and Vp in terms of r,b,c,u,a, and L. (d) What is L, still assuming that anyone in state P climbs a palm tree whenever he or she finds one? Assume for simplicity that aN = 2b. "The solution to this problem requires dynamic programming (see Section 10.4).
(e) For what values of is it a steadystate equilibrium for anyone in state P to climb a palm tree whenever he or she finds one? (Continue to assume oN = 2b.) if) For what values of is it a steadystate equilibrium for no one who finds a tree to climb it? Are there values of for which there is more than one steadystate equilibrium? If there are multiple equilibria, does one involve higher welfare than the other? Explain intuitively.
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