Question

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

(a)

Cost is minimized when MPK/MPF = PK/PF

MPK = Q/K = 10 x 0.5 x (F/K)0.5

MPF = Q/F = 10 x 0.5 x (K/F)0.5

MPK/MPF = [10 x 0.5 x (F/K)0.5] / [10 x 0.5 x (K/F)0.5] = F/K = PK/PF = 8/4 = 2

F = 2K

In following graph, cost is minimized at point E where isoquant Q0 is tangent to isocost line AB, with optimal bundle being (K0, F0).

(b)

Isocost: 1200 = 8K + 4F

300 = 2K + F

300 = 2K + 2K = 4K

K = 75

F = 2 x 75 = 150

Q = 10 x (75 x 150)0.5 = 10 x 106.07 = 1,060.7

Therefore, the given budget will produce less than current demand of 5,000.

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