Question

5.3 An electricity-generating firm estimates its production function to be:

Q = 10K^(0.5) F^(0.5) Where: Q = monthly electricity production measured in kilowatt-hours

              K = machine hours (capital) per month & F = fuel in gallons per month

     The rental cost of capital is $8 per machine hour and the cost of fuel is $4 per gallon.

a) State and illustrate the conditions that determine the firm’s cost-minimizing use of fuel and capital. Determine the firm’s optimal ratio of fuel to capital.

b) Suppose that this electricity firm is required by law to both operate within a fixed budget of $1200 per month as well as to produce enough that, at current prices, all demand is satisfied. It is estimated that the current demand is 5,000-kilowatt hours. Will the firm be able to satisfy both its budgetary and its demand requirements if it uses its inputs optimally?

Answer 1

Solution

Total Cost: C=8k+4F

(a)

The slope of isoquant= -MPk/MPF

=

=

So, slope of isoquant= -MPK/MPF= - F/K

the slope of isocost line= -Pk/PF= -8/4= -2

Cost is minimized when the slope of isoquant = slope of isocost line

-F/K=-2

F=2K

This is the optimal fuel-capital ratio.

(b)

we have, F=2k

when C=1200

1200=8K+4F

1200=8K+4(2K)

1200=16K

K=75

F=2*75=150

Therefore output=

= 10*8.66*12.25

= 1060

This falls short of 5000 demand

On the other hand, When Y=5000

K=353.55

F=2*353.55=707.1

then, C= 8*353.55+4*707.1=5656.8 which exceeds the budget.

So firm can not satisfy both budget and demand requirement at the same time.