Question

3. Consider the production function, Q = [L^0.5 + K^0.5] ² . The marginal products are given as follows: MPL = [L^0.5 + K^0.5] L^-0.5 and MPK = [L^0.5 + K^0.5] K^-0.5 and w = 2, r = 1.

A). what is the value of lambda

B). Does this production function exhibit increasing, decreasing or constant returns to scale?

C).Determine the cost minimizing value of L

D).Determine the cost minimizing value of K

E).Determine the total cost function

F).Determine the average cost function

G).Indicate whether the firm is experiencing economies or diseconomies of scale or neither.

H).Suppose in the short run capital is fixed: K = 9. Determine the value of L.

I).Determine the short run total cost.

J).what is the Fixed cost?

Answer 1

Q = [L^0.5 + K^0.5] ²

MPL = [L^0.5 + K^0.5] L^-0.5

MPK = [L^0.5 + K^0.5] K^-0.5

w = 2, r = 1

A). what is the value of lambda

lambda = MPL/MPK = [L^0.5 + K^0.5] L^-0.5/[L^0.5 + K^0.5] K^-0.5 = K^0.5/L^0.5

B). Does this production function exhibit increasing, decreasing or constant returns to scale?

increase k and L by t. Q(t) = [(tL)^0.5 + (tK)^0.5] ² = t^2*0.5[L^0.5 + K^0.5] ² = t[L^0.5 + K^0.5] ² = t.Q

therefore constant returns to scale.

C).Determine the cost minimizing value of L

cost minimising combination of K and L would be such that RTS(=MPL/MPK) = w/r

from part A, ration of MPL/MPK = K^0.5/L^0.5

K^0.5/L^0.5 = 2/1 = 2

K/L = [2] ² = 4

L = K/4

in Q = [L^0.5 + (4L)^0.5] ² = [L^0.5 + 2(L)^0.5] ² = 3(L)^0.5] ² = 3L

L = Q/3

D).Determine the cost minimizing value of K

from part C, K = 4L and Q = 3L = 3K/4

so K = 4Q/3

E).Determine the total cost function

TC = wL+rK = 2*K/4 + 1*K = K/2 + K = 3K/2 = (3/2)(4/3)Q = 2Q

F).Determine the average cost function

AC = TC/Q = 2

G).Indicate whether the firm is experiencing economies or diseconomies of scale or neither.

answer is neither as the AC of firm is constant.