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?                      (10 Marks)

Answer 1

Form a Lagragian equation

$Q=4$$\left(L^{\smash{0.5}}\right)\left(K^{\smash{0.5}}\right)$ subject to $wL+rK=C$

$L=4$ $\left(L^{\smash{0.5}}\right)\left(K^{\smash{0.5}}\right)$ $-λ(wL+rK-C)$

$δL/δL=2$$\left(L^{\smash{-0.5}}\right)\left(K^{\smash{0.5}}\right)$$-λw=0..........(i)$

$δL/δK=2$ $\left(L^{\smash{0.5}}\right)\left(K^{\smash{-0.5}}\right)$ $5-λr=0...........(ii)$

$δL/δλ=wL+rK-C=0.................(iii)$

Divide equation (i) and (ii)

$K/L=w/r$ and thus  $K=wL/r$ and $L=rK/w$

Replacing the two equation on equation $(iii)$

$w(kr/w)+rK=C$ thus $K*=C/2r$

$\left(K^{\smash{*}}\right)=200/\left( 2^{\smash{*}}\right) 5=20$

$wL+r(wL/r)=C$

$wL+wL=C$

$\left(L^{\smash{*}}\right)=C/2w$

$\left(L^{\smash{*}}\right)=200/\left(2^{\smash{*}}\right)3 = 33.33$

Hence the optimum output is;

$Q=4$ $\left(33.33^{\smash{0.5}}\right)\left(20^{\smash{0.5}}\right)$

$Q=4(4.58)(4.5)$

$Q=104.40$ units - Maximum Output.