[10 points] Suppose that a firm facing the market interest rate r has the following production function: where &, is capital and a, is the productivity level following an AR(1) process: atti = partErtl, PE (0, 1 ), (1) where Et+1 is an identically and independently normally distributed (iid) in- novation with mean 0 and variance w. The firm's objective is to maximize the expected discounted present value of profits: max E. (2) subject to kit1 = (1 - 8) kit u + nithit 28 (3) given ko, where , is investment, 7+1 is an identically and independently nor- mally distributed (iid) innovation with mean 0 and variance @-, and p, r, 4, and 6 are all positive constants. The Lagrangian for this optimizing problem can be written as Ls = E. E kim - pit - - wi Soif the [(1 - 8 ) k, the - kiti + niti]] (4) (a) [6 points] Derive the first-order conditions for it and kit1. Show that the Euler equation for this problem can be written as a second-order rational expectations stochastic difference equation in k. And then show that one eigenvalue for this difference equation is less than 1 and the other is greater than 1. (b) [4 points] Find the decision rule of ke+1. (Hint: it is a linear combination of its lagged value and a term determined by the exogenous process of.)