Question

The firm's production function is given as f(L,K) = 4L0.5K0.5 and input prices are w = 4 and r = 16. In the short-run, the level of capital is fixed at K = 2. What is the firm's short-run average cost function? Wählen Sie eine Antwort: O a. 0.125q 32/q b. 0.25q 16/q О с. 0.125q O d, 4 + 32/q

Answer 1

Given, production function  $\large Q = 4L^{0.5}K^{0.5}$

Cost of Labour, w = 4

Cost of capital, r = 16

Short run capital is fixed at K = 2

First of all determine the marginal product of Labour. Differentiate Q wrt L we get

$\large \frac{dQ}{dL} = \frac{d(4L^{0.5}K^{0.5})}{dL}$

$\large MP_L =4[L^{0.5} \frac{dK^{0.5}}{dL} + K^{0.5}\frac{dL^{0.5}}{dL}]$

K is independent of L so the derivative K (wrt L) = 0

$\large MP_L =4*0.5K^{0.5}L^{0.5-1}$

$\large MP_L =0.5*\frac{4K^{0.5}L^{0.5}}{L}$

$\large MP_L =0.5*\frac{Q}{L}$

Similarly differentiating Q wrt K we get the marginal product of capital

$\large MP_K =0.5*\frac{Q}{K}$

Now according to equimarginal criterion satisfy the following formula

$\large \frac{MP_L}{MP_K} =\frac{w}{r}$

$\large \frac{\frac{0.5Q}{L}}{\frac{0.5Q}{K}} =\frac{4}{16}$

$\large \frac{K}{L} =\frac{1}{4}$

Therefore, K = 0.25L

Given, K = 2

$\large Q = 4L^{0.5}K^{0.5}$

$\large Q = 4(L*0.25L)^{0.5}$

$\large Q = 4(0.25L^2)^{0.5}$

$\large Q = 4*0.5*L$

$\large Q = 2L$

Therefore, 4L = 2Q

The cost function can be written as

$\large C = rK + wL$

C = 16K + 4L

Average cost function

$\large AC = \frac{16K + 4L}{Q}$

$\large AC = \frac{16*2 + 4L}{Q}$

$\large AC = \frac{32}{Q} + \frac{2Q}{Q}$

$\large AC = 2 + \frac{32}{Q}$

So it is AC = 2 + 32/Q

I have tried all the methods but I am getting this answer only.