Question

A cob Douglas production function for a firm is given as Q=4L ½K½. The firm has also established that wage rate and interest paid on capital are $3 and $5 respectively for a production period. The firm intents to spend $200 million for the period on production cost. Compute the levels of capital and labor that will maximize output. What is the maximum output?

Answer 1

Form a Lagragian equation

$Q=4L^{0.5}K^{0.5}$ Subject to $wL+rK=C$

$L=4L^{0.5}K^{0.5}-λ(wL+rK-C)$

$\frac{\delta{L}}{\delta{L}}=2L^{-0.5}K^{0.5}-\lambda{w}=0........(i)$

$\frac{\delta{L}}{\delta{K}}=2L^{0.5}K^{-0.5}-\lambda{r}=0........(ii)$

$\frac{\delta{L}}{\delta{\lambda}}=wL+rK-C=0........(iii)$

Divide equation (i) and (ii)

$\frac{K}{L}=\frac{w}{r}$ and thus $K=\frac{wL}{r}$ and $L=\frac{rK}{w}$

Replacing the two equation on equation (iii)

$w(\frac{kr}{w})+rK=C$ thus $K^*=\frac{C}{2r}$

$K^*= \frac{200}{2}×5 =20$

$wL+r(\frac{wL}{r})=C$

$L^*=\frac{C}{2w}$

$L^*=\frac{200}{2}×3 = 33.33$

Hence the optimum output is;

$Q=4(33.33)^{0.5}(20)^{0.5}$

$Q=4(4.58)(4.5)$

Q=104.40 units - Maximum Output.