Question

Problem 1 Your firm produces output of some generic manufactured good Q according to the following production function: Q(L,K) = 2L1/451/2 Assume the wage paid to labor is $10, the rental rate of capital is $40, and the price of your product is $200. (a) What is the technical rate of substitution for this production function? (b) Suppose you have 80 units of capital in the short run. Find the profit maximizing amount of labor you should employ. (c) Find your short run profits using the amount of labor and capital from the previous part. (d) Now find the profit maximizing amount of labor and capital you should employ in the long run. (e) Find your long run profits from your solution above.

Answer 1

Q (L,K) = 2 1/4K/2 (a) marginal rete of Technical substitution (MRTS) Labor to It is the retio of marginal product of marginal product of lepifde. mpa do = 2x1 (2/4-1) kih - OL - Į Kh z mpka da Эk 2x1 (kty L4 . kthl/ 2 I MRTS = mh MRTS z + 18/4 kl 2 mpk ] K / LY 30, MRTS 2 I K

. Protit maximizira Londitione MRTS - w We wage rate = ra rental rate of $10 elital = $40 when kz 80 units I will be I - Le 160 units TC and L2 160 units at the K= 80 units output will be @z_2 13.55) (8.94) a = 63.56 or 64 Approx, frice of output per unit is $200 Topal Revenue = Pxa $2004 64 $ 12,800 Total lost = - WXL & r XK. 10x160 + 40 x 80 1600+ 3200 = 4800

Profit Revenue =12,800 - = $8,000 - lost 4,800 (d) Profit Labor maximizing in the londo combination of ron. celital and MRTS = w 2kzL The brott Laber in meximizing tombination the long run is of elital and Units of unif of labor a 2K lapit of a ko