Suppose that the production function of a representative firm in perfect competi

Question

Suppose that the production function of a representative firm in perfect competition is given as X=K0.25L0.75 and the budget of this firm is defined as PLL+PKK=C for arbitrary C.d. Now drop previous assumption about price of labor, price of capital and budget limit, and just focus on the general case. Find the formula of total cost function, marginal cost function, and average cost function in the long-run. (Hint: TC(X)= PL×L + PK×K. Plug input demand functions into L and K of TC function. Input demand functions are derived by plugging rearranged production function with respect to L or K into the formula of expansion path.)

Explanation / Answer

X=K0.25L0.75

MPL = dX/dL = 0.75(K/L)^0.25

MPK = dX/dK = 0.25(L/K)^0.75

MPL/MPL = PL/PK

0.75(K/L)^0.25/0.25(L/K)^0.75 = PL/PK

K = (1/3)*(PL/PK)*L

Putting it in production function

X = [(1/3)*(PL/PK)*L]^0.25L^0.75

L = X/[(1/3)*(PL/PK)]^0.25

K = X[(1/3)*(PL/PK)]^0.75

TC = PL*L + PK*K

= PL* X/[(1/3)*(PL/PK)]^0.25 + PK*X[(1/3)*(PL/PK)]^0.75

= (1/3)-0.75*X*(PL)^0.75*(PK)^0.25 + X*(1/3)^0.75*(PK)^0.25*(PL)).75

= 2X*(Pk)^0.25*(PL)^0.75*(1/3)^0.75

Marginal Cost MC = dC/dX = 2*(Pk)^0.25*(PL)^0.75*(1/3)^0.75

Average cost AC = 2*(Pk)^0.25*(PL)^0.75*(1/3)^0.75

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