1 Firm Investment with Uncertainty (10 points) Parts of this problem are based on question 4 from chapter 11 in the book. Suppose we have a rm which seeks to maximize the present value of its prot across two periods: 7r! W+1+r where 71' is the real prot in period 1, ir'is the real prot in period 2, and r is the real interest rate. 0 In the rst period, the rm is given exogenously determined capital K, and chooses how much labor N to hire at wage in, and how much to invest I. Output is given by 2F (K, N) = 23K ClN 1'\" and Investment takes the form of output which is not sold. Thus revenues are 2K \"N 1'\" I and so rst period prots are: 7r = mum's I wN 0 Between periods, some portion of prexisting capital 6 depreciates, and investment is added to the capital stock, so the formula for second period capital is K'=K-(16)+I o In the second period, the rm chooses how much labor N ' to hire at the second period's real wage 10'. And because time ends after the second period, the rm will sell off any remaining capital stock K' - (1 6) and so second period prots are given by: n' = z' (K')' (mlC\" + K' . (1 a) w'N' rblems: (a) Set up the rm's present-value prot maximization problem. (b) Set up the Lagrangian and derive equations describing the rm's optimal labor-hiring and investment decisions. ( These equations should Show the optimal relationship between the rm's costs and the marginal product of labor or capital.) Now suppose that the rm is uncertain about future productivity. The rm believes that there is only a p E [0,1] chance that they will be able to produce anything tomorrow. That is, there is a p chance that z' = 2H > 0 and a (1 p) chance that z' = 0