Question

This question is very important and I need solution for this issue with all the details a.b.c , and help me with all the details? BR/Ha

a) The firm ACME has the production function f ( K , L)=K^2/3 L^2/3 . Calculate an expression for the marginal product of labour, L , and establish if it is increasing, constant or decreasing. Verify if ACME’s production technology exhibits diminishing, constant or increasing returns to scale.

b) Set up ACME’s long run profit maximization problem and derive the factor demands for optimal choice of y.

c)Try to derive a long run supply function for ACME using your calculations in b). Did your effort result in a viable supply function? Explain why.

Answer 1

Solution:

a) Production function: f(K, L) = K^2/3*L^2/3

Marginal product of labor, MPL =

MPL = (2/3)*K^2/3*L^{2/3 - 1} = (2/3)*(K²/L)^1/3

Clearly, L and MPL are inversely related, so as the value of L increases, MPL decreases. So, we've established that it is decreasing

Finding returns to scale: Increasing K and L by same factor of t:

f(tK, tL) = (tK)^2/3*(tL)^2/3

f(tK, tL) = t^2/3+2/3 K^2/3*L^2/3

f(tK, tL) = t^4/3(K^2/3L^2/3) = t^4/3*f(tK, tL)

Since, 4/3 > 1, it means increasing the inputs by factor t, increases output by factor greater than t, thus exhibiting increasing returns to scale.

b) Denoting the wage rate by w and rental rate by r, the total cost = w*L + r*K

Denoting price be p, total revenue = p*Y = p*f(K, L) = p*(K^2/3*L^2/3)

Profit = total revenue - total cost

Profit, W = p*(K^2/3*L^2/3) - w*L - r*K

Finding optimal levels of K and L:

That occurs where MPL/MPK = w/r

So, [(2/3)*(K²/L)^1/3]/[(2/3)*(L²/K)^1/3] = w/r

K/L = w/r

So, optimal y is: Y* = (w/r)^2/3