Answer to Question #258318 in Microeconomics for Zack

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=4L^0.5K^0.5 Subject to wL+rK=C

L=4L^0.5K^0.5 -$\lambda$(wL+rK-C)

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

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

$\delta$L/$\delta$$\lambda$=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

K^*=200/2*5 =20

wL+r(wL/r)=C

wL+wL=C

L^*=C/2w

L^*=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.