Ranger Construction Is Preparing to Repair Potholes,under Contract to the Local

Question

Examlex · Accepted Answer

The Answer is Ranger's objective is: Maximize Q = K^0.⁵L^0.⁵ subject to 8K + 12L = 96,000.
The Lagrange multiplier is: G = K^0.⁵L^0.⁵ +

\lambda

(8K + 12L - 96,000).The three partial derivatives need to be computed and set equal to zero:

\partial \mathrm{G}

/

\partial \mathrm{K}

= 0.5K^-^0.⁵L^0.⁵+ 8

\lambda

= 0 ---- (1)

\partial \mathrm{G}

/

\partial \mathrm{L}

= 0.5K^0.⁵L^-0^.5 + 12

\lambda

= 0 ---- (2)

\partial \mathrm{G}

/

\partial\lambda

= 8K + 12L - 96,000 = 0 ---- (3)
Multiply (1)by 1.5:
0.75K^-⁰^.5L^0.⁵ + 12

\lambda

= 0 and equate this expression with (2)to get:
0.75K^-⁰^.5L^0.⁵ + 12

\lambda

= 0.5K^0.⁵L^-⁰^.5 + 12

\lambda

Cancel 12

\lambda

,and then combine terms and simplify:
0.75L = 0.5K,or K = 1.5L
Substitute this last expression into (3):
8(1.5L)+ 12L - 96,000 = 0,or 24L = 96,000
Thus,L = 4,000 units and K = (1.5)(4,000)= 6,000 units.
The firm's level of output is: Q = (6,000)^0.⁵(4,000)^0.⁵ = 4,899 potholes.Ranger should hire 4,000 units of labor,6,000 units of capital,and agree to fill 4,899 potholes.

Ranger Construction Is Preparing to Repair Potholes,under Contract to the Local

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Ranger Construction Is Preparing to Repair Potholes,under Contract to the Local

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Correct Answer:VerifiedShow Answer

Correct Answer:
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