Question

From now on, let the production function of the firm be Y = 2K*N1-a, where a is a parameter between 0 and 1. 4. Verify that this production function has constant return to scale. 5. Derive the marginal product of labor MPx and marginal product of capital MPK. How does MPN change with N and K? 6. Solve the firm's optimization problem analytically. That is, to derive the firm's opti- mal choice as a function of exogenous variables (i.e., in terms of parameters). 7. Based on your answer to the last question, how does the firm's choice respond to (a) an increase in the real wage w; (b) an increase in the capital stock K; (c) an increase in the total factor productivity z.

Answer 1

Production from Chion for chion on of the firm bet Let production the y = K & Nice 0 and 1. where x is a parametu between 4. Now we raise both Kand N by a scalar factor E.) Hace 24 Y(AK, XN) = z Aka (angla - 1 Z Kenie =XY Hace is creasing both k and N by 1 in the ass out but by the same pro portion t o thence this production function has constant restums to scale. N. 52 Now y=zkania Now, marginal product of N. . nie. MPN = 24 = (1-4) z kaw-a Now, effect of change in n on mp n will be aw d (MPN) = -22-2) AN --(-) ZkQE NE -1) 20 Now, 02221 =) all => (2-1) LO => (1-2))0 Hace -x(1-27 LO Huce, as n inueass d (MPN) Lo, ie MPN declines Scanned with CamScanner

Now the effect of Change in K on mpn will be demon), 211-) 2k&WW-X Do Hace as k increasis, mpN in creass. is ak 62 NOW, MPK = Oy - 20 KQ-'N 1- Now, suppose price of N is to and cost of k 7 Now, equation of total cost cauce - Te = watok Now optimal input mix condition will be t- (12) zkena 4x Za K-IN (1-2) K 2 N r => => (1-2)*8= wan WAN . K = T- ar in the production function supstituting this value of k We get y = Z Ka Nina =) y = z[WIN 74 NI-a Tw272 csomos Y = 2^ [lees ne CamScanner

and, as ka WON (1-2) substituting value of w* weget ka ed an Ž [wana - a tor] the firm will be Hence the optimal choius of 1-2 - N* = ² [ we are ] and KX = 2 [ ] 7 Now the effect of innease in real wage on wat jie. But = (*)() Lien = -x26 ) [posts] 26 the increase in real wage w will redace optimal N* Now the effect of increase in real wage on k*re. own = (-2) / WX_7- ł 21272/x x = 2(1-2)} , want n yo ak* increase optimal w real wage w wil will in - in crease Hanned with KmScanner