Question

Given the following long run production and cost functions: q=LPK1/4 C = 12L +4K (A) What input has diminishing marginal returns? (B) Does this production function display increasing, decreasing or constant returns to scale? (C) What is this firm's expansion path assuming input prices do not change?

Answer 1

(A) MPL =  $\frac{\partial q}{\partial L} = 3L^{2}K^{1/4}$

So, as L increases, MPL will also increase which means that L does not have diminshing marginal returns.

MPK = $\frac{\partial q}{\partial K} = L^{3}(1/4)K^{(1/4)-1} = (1/4)L^{3}K^{-3/4}$

So, as K increases, MPK will decrease which means that K have diminshing marginal returns.

(B) Let L = tL and K = tk where t > 0.
So, q' = (tL)³(tK)^1/4 = t³L³t^1/4L^1/4 = t^3+(1/4)L³K^1/4 = t^3.25q
As power of t is greater than 1 so there are increasing returns to scale.

(C) From the cost function, w = 12 and r = 4

Long run expansion path is determined at the optimal combination which is where MRTS = w/r = 12/4 = 3
MRTS = MPL/MPK = 3L²K^1/4/(1/4)L³K^-3/4 = 3*4K^(1/4)+(3/4)/L^3-2 = 12K/L

So, 12K/L = 3
So, K = 3L/12
So, K = 0.25L is the firm's expansion path.