Question

A firm has a production function of Q = 10L0.3K 0.6 . The price of L is w = 9 and the price of K is r = 18

. a. What is its short-run marginal cost curve?

b. What is its average variable cost curve?

Answer 1

a)

Q = 10L^0.3K^0.6

In the short-run capital is fixed say at the level K₁ so the short-run production function is

Q = 10L^0.3K₁^0.6

Q/10 = L^0.3K₁^0.6

(Q/10)K₁^{- 0.6} = L^0.3

(Q/10)^1/0.3(K₁^{- 0.6} )^1/0.3 = L

L = (Q/10)^10/3(K₁^{- 2})

L = (Q/10)^10/3/K₁²

Therefore short run total cost is

STC = wL + rK₁  

STC = 9L + 18K₁  

STC = [9(Q/10)^10/3]/K₁² +  18K₁

STC = [9(Q^10/3/10^10/3)]/K₁² + 18K₁²

dSTC/dQ = [9(10/3)Q^{10/3 -1} /10^10/3]/K₁²

dSTC/dQ = [30(Q^7/3/10^10/3)]/K₁²  

SMC =  [30(Q^7/3/10^10/3)]/K₁²  

b)

AVC = TVC/Q  

TVC = wL  

TVC = 9(Q/10)^10/3/K₁²

AVC = TVC/Q

AVC = [9(Q/10)^10/3/K₁²​​​​​​​]/Q

AVC = [9(Q^10/3/10^10/3)/K₁²]/Q

AVC = 9(Q^{10/3 -1} /10^10/3)/K₁²

AVC = 9(Q^7/3/10^10/3)/K₁²