Please help me to solve this problem Suppose that a firm has a production functi
ID: 1181224 • Letter: P
Question
Please help me to solve this problem
Suppose that a firm has a production function of the form: q = K1/2L1/2. Find the firm's marginal product of labor, marginal product of capital, and the firm's MRTS. Does this production function exhibit diminishing marginal returns to labor? Explain. Suppose that the rental rate of capital is r = $5 per unit and that labor costs w = $20per unit. What is the cost of producing q=100 using 100 units of K and 100 units of L? What is the minimum cost of producing q=100? (Again, suppose that is r = $5 and w = $20.)Explanation / Answer
1. MP of capital=MP1= dq/dk=0.5*sqrt(L/K)
MP of labor = MP2=dq/dl=0.5*sqrt(K/L)
MRTS=MP1/MP2=L/K
d(dq/dl)/dl=-0.25*sqrt(k/L^3)
Since d(MP2)/dl is negative MP labour will decrease and hence it exhibits Diminishing Marginal rerurns to labor.
2. Cost=$5*100 +$20*100=$2500 for 100 units
3. cost=5K+20L
q=100=sqrt(K*L)
k*L=10000
k=10000/l
cost=50000/L+20L
d(cost)/dl=0
=-50000/L^2+20=0
L=50
k=200
cost=5*200+20*50=$2000