Question
Asked Nov 28, 2019
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Let the production function be Q = 4* K1/4 L1/4 assume that both factors are variable.

  1. (a)  Derive the contingent demand functions for K and L

  2. (b)  Substitute the contingent demand functions in the total cost that you minimized in part a) to obtain the total cost function.

  3. (c)  Find the amount of K  and L necessary to produce Q=8 when v=16 and w=1 (with minimal possible cost).

  4. (d)  Find the average and marginal cost functions.

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Expert Answer

Step 1

a.

Given,

Production function

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1 1 Q = 4K4L

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Step 2

Maximize the production function with respect to cost constraint,

And cost constraint C = wL + vK

Here, w is the wage of labor

L = labor, K = capital, and v is the price of capital.

Apply Lagrangian multiplier:

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=Q +2(C wL-vK 1 1 4K L2(C- wL-vK)

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Step 3

Differentiate the Lagrangian function with respect to w, l, ...

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4K L(C - wL-vK K Aw 0 on 11 OL L4 K =-Aw>(1) LA дк L4 =-2v->(2) on =C-wL-vK = 0 -> ( 3)

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