Answer to Question #257716 in Microeconomics for Kahil

Question #257716

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)


1
Expert's answer
2021-10-31T18:31:54-0400

Form a Lagragian equation

Q=4Q=4(L0.5)(K0.5)\left(L^{\smash{0.5}}\right)\left(K^{\smash{0.5}}\right) subject to wL+rK=CwL+rK=C

L=4L=4 (L0.5)(K0.5)\left(L^{\smash{0.5}}\right)\left(K^{\smash{0.5}}\right) λ(wL+rKC)-λ(wL+rK-C)

δL/δL=2δL/δL=2(L0.5)(K0.5)\left(L^{\smash{-0.5}}\right)\left(K^{\smash{0.5}}\right)λw=0..........(i)-λw=0..........(i)

δL/δK=2δL/δK=2 (L0.5)(K0.5)\left(L^{\smash{0.5}}\right)\left(K^{\smash{-0.5}}\right) 5λr=0...........(ii)5-λr=0...........(ii)

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

Divide equation (i) and (ii)

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

Replacing the two equation on equation (iii)(iii)

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

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

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

wL+wL=CwL+wL=C

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

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

Hence the optimum output is;

Q=4Q=4 (33.330.5)(200.5)\left(33.33^{\smash{0.5}}\right)\left(20^{\smash{0.5}}\right)

Q=4(4.58)(4.5)Q=4(4.58)(4.5)

Q=104.40Q=104.40 units - Maximum Output.


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