Question

2. Suppose a production function is given by  F ( K , L ) = K 1 2 L 1 2, the price of capital “r” is $16, and the price of labor “w” is $16.
a. (5) What combination of labor and capital minimizes the cost of producing 100 units of output in the long run?
b. (5) When r falls to $1, what is the minimum cost of producing 100 pounds of pretzels in the short run? In the long run?
c. (5) When r falls to $1, will the cost of producing 100 units of output increase or decrease in the long-run? Explain.

Answer 1

a)

Production cost is given as

First we find marginal product of labor and marginal product of capital.

Marginal Product of labor is determined as

Marginal Product of capital is determined as

Cost minimization requires that

MPL/MPK=w/r

K/L=1

K=L (optimal ratio of inputs)

Now set K=L and Q=100 in production function

So,

Cost minimizing number of labor units=L*=100

Cost minimizing number of capital units=K*=L*=100

b)

If r falls to $1, it is not possible to alter the quantities of the inputs. So, input combination will remain the same as calculated in part a

Hence, short run cost=wL+rK=16*100+1*100=$1700

In long run input combination can be changed. Cost minimization requires that

MPL/MPK=w/r

K=16L

Set K=16L and Q=100 in production function

100=4L

L*=25

Cost minimizing amount of labor=L*=25

Cost minimizing amount of capital=K*=16L*=400

Hence

Minimum cost of producing 100 units in long run=wL+rK=16*25+1*400=$800

c)

Price change for any of the inputs calls for the need for change in optimal ratio for inputs. In short run it is not possible to change input combination but it can be done in long run to minimize cost. So, total cost of production is reduced in long run as a result of reduction in r.

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