Question

A production function is given by q = L1/5K1/5 and a total cost function is given by TC = q2/5. Under what conditions could these belong to the same firm?

Answer 1

Consider the given problem here the production function is given by, “q=L^1/5*K^1/5”, => the MRTS is given by.

=> MRTS = MPL/MPK = [1/5*L^(-4/5)*K^1/5]/[ 1/5*L^1/5*K^(-4/5)] = K/L = MRTS.

=> at the optimum “MRTS” must be equal to “W/R”.

=> K/L = W/R, => K = (W/R)*L.

=> the production function is given by.

=> q = L^1/5*K^1/5, => q^5 = L*K, => q^5 = L*(W/R)*L, => L = (R/W)^1/2*q^5/2.

Now, K = (W/R)*L = (W/R)* (R/W)^1/2*q^5/2 = (W/R)^1/2*q^5/2 = K.

=> the cost function is given by, “C = W*L + R*K”.

=> C = W*(R/W)^1/2*q^5/2 + R*(W/R)^1/2*q^5/2= (W*R)^1/2*q^5/2 + (W*R)^1/2*q^5/2.

=> C = 2*(W*R)^1/2*q^5/2, be the cost function.

Now, if we compare the cost function corresponding to the production function with the given cost function, we can get that the coefficient of “q” term can be “1” for some values of “W” and “R”, but the exponent of “q” is different, => the given cost function and the production function belongs to different industry.