A firm manufactures widgets according to the production function Q = 8L3/4 K1/4,

Question

A firm manufactures widgets according to the production function Q = 8L3/4 K1/4, where Q is the number of skis made, K is the amount of capital, and L is the amount of labor employed. The firm is currently producing Q = 120 using L = 15 and K = 15. The factor price of Labor (w) is $10 and the factor price of Capital (r) is $2. The firm suspects that it is not minimizing the cost of producing Q = 120 and has hired you, a student taking a course with the most famous economist in campus, to help it understand the nature of its inefficiency. a) At the current level of production (Q = 120) what are the marginal productivities of Labor and Capital respectively? MPL = Answer MPK = Answer b) What are the current cost of production for the company? TC = $Answer c) In order to minimize the costs of production the company should use more Answer d) In order to miminize costs of production the company should use the following input ration: K = Answer L e) Say the company wanted to continue use the same total costs as to when K and L = 15, and instead produce more output. How much output can the company produce? Q2 = ?

Explanation / Answer

Answer:

The production function (Q) = 8L3/4 K1/4

The firm is currently producing Q = 120; L = 15 and K = 15.

The factor price of Labour (w) is $10.

The factor price of Capital (r) is $2.

a) At the current level of production (Q = 120) what are the marginal productivities of Labor and Capital respectively? MPL = Answer MPK = Answer

                                Q = 8L3/4 K1/4

Marginal productivities of Labour is:        MPL = dQ/dL = 6-1/4K1/4

Marginal productivities of Capital is:        MPK = dQ/dK = 8L3/4 1-3/4

b) What are the current costs of production for the company?

   The Total Cost (TC) = wL + rK

                                TC = 15 (10) + 15 (2)

Therefore,          TC = $180

c) In order to minimize the costs of production the company:

The cost-minimizing combination of capital and labour is the one where

                                MRTS = MPL/MPK = w/r

Therefore, the Marginal Rate of Technical Substitution is:            

                                = (6-1/4K1/4)/( 8L3/4 1-3/4) = K/L

      The cost minimizing of production with at K = 15 and L = 15

d) In order to minimize costs of production the company should use the following input ration: K = L:

   To determine the optimal capital-labour ratio set the marginal rate of technical substitution equal to the ratio of the wage rate to the rental rate of capital:

                                K/L = 15/15, (or ) L = K

   Therefore, the ratio of L and K is: 1: 1

e) Say the company wanted to continue use the same total costs as to when K and L = 15, and instead produce more output. How much outputs can the company produce?

                We can substitute for L in the production function and solve where K yields an output of 120 units for calculate Q2

                                180 = 8K3/4 K1/4

                                180 =8(K)1

                                K = 22.5

Thus L = 22.5, because L = K.

     Therefore, the company wanted to continue use total costs as to when K and L = 15, and 60 units of output require instead produce more output. That is: Q2 = 180

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