Econ 561bSpring 2004Yale UniversityProf. Tony SmithHOMEWORK #61. In this problem, you will compute the stationary equilibrium of a competitive industrypopulated by a continuum of firms, each producing an identical product. Let the totalnumber (measure) of firms be one. In period t, each firm produces output accordingto yt= f(nt, zt), where ntis the amount of labor hired by the firm (at exogenous andtime-invariant wage rate w) in period t and ztis the firm’s idiosyncratic productivityshock in period t. The industry (inverse ) demand curve is given by pt= D(Yt), whereptis the price of the industry’s good in period t and Ytis total industry output inperiod t. Each firm faces costs of adjusting its labor demand: in period t this costis given by c(nt− nt−1), where c is twice continuously differentiable, strictly conve x,and satisfies c(x) > 0 for x 6= 0, c(0) = 0, and c0(0) = 0. The firm seeks to maximizeexpected discounted profitsE0∞Xt=0βt(ptf(nt, zt) − wnt− c(nt− nt−1) ,given n−1.(a) Let f (nt, zt) = ztnαt, where 0 < α < 1, and c(x) = dx2, where d > 0. Let the shockztfollow a two-state Markov chain. Pick values for the structural parameters andthen use your favorite computational method to find the decis ion rule of a typicalfirm, assuming that the industry price is constant and equal to p. (Note that thisdecision rule takes the form nt= g(nt−1, zt).)(b) Use the decision rule from part (a) to compute (a numerical approximation to)the invariant dis tribution Γ of firms over pairs (n−1, z).(c) Use the invariant distribution from part (b) to compute total industry outputY =Rf(g(n−1, z), z) Γ(dn−1, dz). Does the implied industry price equal the pricep that you took as given in part (a)? If not, continue iterating on steps (a)–(c) tofind the equilibrium industry price.2. Suppose that the industry demand curve in problem 1 is perturbed by an aggregateshock at: pt= D(Yt, at), where at∈ {aH, aL} follows a two-state Markov chain and isstatistically independent of zt. Your task in this problem is to compute an approxima-tion to the dynamic behavior of the industry, assuming that a typical firm does not1keep track of the distribution of firms when forecasting future prices but instead keepstrack of total industry output. In particular, firms assume that Yt+1= bH0+ bH1Ytifat= aHand Yt+1= bL0+ bL1Ytif at=
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