GR5215 Ron Miller Assignment 5. Due December 8th 1) Consider an economy with many imperfectly competitive, price-setting firms. The profits of a representative firm, firm i, depend on aggregate output, y, and the firm’s real price, ri, that is: πi = π(y,ri ), where π22 < 0 (that is, the second partial with respect to own-price is negative) Let r∗(y) denote the profit-maximizing price as a function of y. This function is, of course, implicitly defined by the first-order condition for profit maximization: π2(y, r∗(y)) = 0. Assume that output is initially at some level y0, and that firm i’s real price is r*(y0). 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, y1. a) Explain why firm i’s incentive to adjust its price is given by: G = π(y1,r*(y1)) − π(y1,r∗(y0)). b) Use a second-order Taylor approximation of this expression for G in y1 expanding around the point y1 = y0 to show: c) What component of this expression corresponds to the degree of real rigidity? What component corresponds to the degree of insensitivity of the profit function? 2) Suppose the economy is as in the Fischer model as done in class and in the text, but instead of half of firms setting their prices each period, a fraction f set their prices in odd-numbered periods and a fraction (1 – f) set their prices in even-numbered periods. Thus the price level is if t is even and if t is odd. Derive expressions analogous to those for the regular Fischer model for p and y in even and odd periods. 3) Suppose the structure of supply and demand is as given in the simplest Lucas model: (1) yt = mt - pt + vt (quantity theory) (2) yt = a(pt - E[pt | t-1] ) + et (Lucas supply function, a>0) with the usual symbols. (vt and et are mean zero, i.i.d. disturbances) However, the agents are unaware that this is the true model of the economy and believe (1); but instead of (2) believe: (3) yt = b(pt - E[pt | t-1] ) + et (wrong supply function, b¹a) G ! −π22( y0 ,r *( y0))[r *′( y0)]2( y1 − y0)2 /2 fpt1 +(1− f )pt2 (1− f )pt1 + fpt2 GR5215 Ron Miller a) Will this error lead to any real effects of anticipated monetary policy under any of the following monetary policy rules? (Assume that the central bank understands the true structure of the economy where relevant.) (4.1) mt = m0+ ut (constant money supply plus white noise) (4.2) mt = m0+ r(mt - 1 - m0) + ut (gradual adjustment: 0
0) b) Now suppose the public makes a different mistake and gets the lag structure wrong rather than the coefficient. Now they believe that: (5) yt = a(pt - E[pt|t-2] ) +et Repeat the analysis of part (a) for this mistake. c) Give some intuition for why the answers to parts (a) and (b) differ. d) Imagine that the second policy rule (4.2) had a different error term, εt, with stochastic characteristics different from those of ut. Suppose we repeated the analysis in parts (a) and (b). (Don't actually do it, just suppose!) In terms of the underlying model of output behavior as described by Lucas, what would be the problem with this analysis? 4) a) Some authors have proposed that a version of the New Keynesian Phillips curve, of the following form: " = ""&' + (1 − )"-' + " where xt is some measure of “inflationary pressure.” How do you interpret this equation? Provide a one-sentence rationale that might justify modelling price-setting in this way. b) Assume that the inflation forcing process, xt, is exogenous with respect to price-setters, and further assume that it follows a random walk. Use the method of undetermined coefficients to find a solution for inflation in terms of lagged inflation and the level of xt. 欢迎咨询51作业君
