Question

A firm has a Cobb Douglas production function given as

q=AL

α K β

a.

Solve for the factor demand functions

b.

If the firms’ competitive output price is p find the wage rate

c.

What is the share of the firms revenue paid to labour and

capital?

d.

If α =0.6, β =0.2 and A=1 find the LR labour and capital

demand curve equations

Answer 1

$q=AL^αK^β$

a. Let the price of the output «q» be p.

Total cost of the firm $TC=wL+rK$

Profit $Π = pq -wL -rK$

$Π = pAL^αK^β -wL -rK$

Profit is maximized when $\frac{∂Π}{∂L} = 0 \; and \; \frac{∂Π}{∂K} =0$

$αpAL^{α-1}K^β -w = 0 \;\;\;(1) \\ βpAL^αK^{β-1} -r =0 \;\;\;(2)$

Dividing two equations, we get:

$\frac{α}{β} \frac{K}{L} = \frac{w}{r} \\ wL = \frac{α}{β}Kr$

Substituting this in C = wL +rK, we get:

$\frac{α}{β}rK +rK=C \\ rK[\frac{α+β}{β}] =C \\ K^* = [\frac{β}{α +β} \times \frac{C}{r}] \\ L^* = [\frac{α}{α +β} \times \frac{C}{w}]$

b. From (1), we get:

$αpAL^{α-1}K^β =w$

So, the wage rate w is

$w=αpAL^{α-1}K^β$

c. Total wage paid to labour $= wL = αpAL^{α}K^β$

Total sum paid to capital $= rK = βpAL^αK^{β}$

Share of firms revenue paid to labour $= \frac{wL}{pq}$

$= \frac{αpAL^{α}K^β}{pAL^{α}K^β} = α$

Share of firms revenue paid to capital $= \frac{rK}{pq}$

$= \frac{βpAL^αK^{β}}{pAL^{α}K^β}=β$

d. α=0.6, β=0.2, A=1

$K^* = \frac{0.2}{0.2+0.6} \times \frac{C}{r} = 0.25 \frac{C}{r} \\ L^* = \frac{0.6}{0.2+0.6} \times \frac{C}{w} = 0.75 \frac{C}{w}$