Question

Given the Cobb-Douglas production function for Mabel’s factory

Q = (L0.4) * (K0.7)

a) Based on the function above, does Mabel’s factory experiencing economies or diseconomies of scale? Explain.

b) If the manager wished to raise productivity by 50% and planned to increase capital by 25%, how much would she have to increase her labor to reach that desired production level?

Answer 1

Answer

(a)

A function exhibit economies of scale if Average cost is decreasing and A function exhibit Diseconomies of scale if Average cost is increasing

Production function is given by :

Q = (L^0.4) * (K^0.7)

A function exhibit increasing returns to scale if F(zK,zL) > zF(K,L) for all z > 1

Here Q = F(K,L) = (L^0.4) * (K^0.7)

=> F(zK,zL) = (zL^0.4) * (zK^0.7) = z^{0.4 + 0.7}(L^0.4) * (K^0.7) = z^1.1(L^0.4) * (K^0.7) > z((L^0.4) * (K^0.7)) = z*F(K,L)

Hence, F(zK,zL) > zF(K,L) for all z > 1

Thus this function exhibit increasing returns to scale. This means that if we increase all input by some proportion z then Output(Q) will increase by more than z proportion.

Hence In order to produce additional unit of output we need lesser inputs and thus Average Total Cost of producing additional unit of output decreases. Hence, Long Run Average cost is decreasing implies that this function exhibits Economies of scale.

(b)

Formula :

% change in (AB) = % change in A = % change in B

% change in (A^B) = B*% change in A

Q = (L^0.4) * (K^0.7)

=> % change in (Q) = % change in [(L^0.4) * (K^0.7)] = % change in (L^0.4) + % change in (K^0.7)

=> % change in Q = 0.4* % change in L + 0.7*% change in K

It is given that He wishes to increase total product by 50% and he increased capital by 25%

=> % change in Q = 50% and % change in K = 25%

=> % change in Q = 0.4* % change in L + 0.7*% change in K

=> 50% = 0.4*% change in L + 0.7*25%

=> 32.5 = 0.4*% change in L

=> % change in L = 81.25%

Hence, she will have to increase her amount of labor by 81.25% in order to reash that production level.