2. (20 points) Consider the following investment decision. The rm's future output is given by Yr : ZI(K:)O.4(N1)0.6, where: Y' is future output; K ' is future capital; N ' = 19779 is future labor, measured in hours per year; and z' = 0.08 is future total factor productivity. Capital evolves according to r K = (1 d) K + I, where: K = 296 is current capital; d = 0.07 is the depreciation rate and I denotes investment. The rm chooses future capital to maximize V = z (K)'4 (N)'6 wN [K' (1 00K] 1 [2' (K004 (MO-6 w'N' + (1 aye] . + 1+7" where 'r = 0.03 is the real interest rate. (a) (4 points) Compute the optimal value of K '. After you compute the desired future capital, compute the desired investment. (b) (4 points) Suppose 2 (K )0'4 (N )0'6 wN w' N ' : 0. Show that the value of K ' computed in part (a) is indeed the optimal value of K' by completing the following table: | 291000 | value from part (a) | 300000 (c) (2 points) Suppose that current capital, K , increases from 296 to 308. What is desired investment now? (d) (4 points) Suppose that 2' increases to 0082, While K is back at its original value of 296. Compute the new optimal value of K' and the new desired investment. (e) (2 points) Suppose that 2' increases to 0.082, while K increases from 296 to 308. What is desired investment now? (f) (4 points) On a single graph, show how the changes you considered in parts (c), (d) and (e) shift the desired investment schedule. You should clearly label the four curves: one for part (a), one for part (c), one for part ((1), and one for part (e), and include the values of desired investment at 7' = 0.03 under each case