Question

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)

Given

Marginal Product of labor is given as

Marginal Product of capital is given as

Cost minimization requires that

MPL/MPK=w/r

K/L=1

K=L

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

So, Cost minimizing L*=100

Cost minimizing K*=L*=100

Cost of producing 100 units=wL+rK=16*100+16*100=$3200

b)

If r=1

In short run, it is not possible to change all inputs. So, in short run input combination will be the same as derived in part a

Short run cost=wL+rK=16*100+1*100=$1700

In long run input combination will change. It can be determined as under

Cost minimization requires that

MPL/MPK=w/r

K=16L

Now set K=16L and Q=100 in production function

100=L^(1/2)*(16L)^(16L)^1/2=4L

L=100/4=25

So, Cost minimizing L*=25

Cost minimizing K=4L*=16*25=400

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

c)

In short run, it is not possible to change input combination to minimize cost. In long run, all inputs can be changed. So, total cost of production will change in long run as a result of decrease in r.