9781118041581

economies of scale were found for most types of mines. The average elasticity

of output with respect to inputs was 1.24. (A 20 percent increase in all inputs

raised output by about 25 percent.) Typically, economies of scale were not

exhausted until an annual output level of 4.8 million tons of coal was reached—

a level higher than the actual operating scale of most mines. (Thus, further

increases in the scale of mineral extraction would seem to be warranted.) In

addition, there was evidence that increased use of large-scale, primary earth-

moving equipment greatly enhanced the degree of returns to scale. In short,

higher capital intensity implies greater returns to scale in mining.

Least-Cost Production

In the long run, the firm can vary all of its inputs. Because inputs are costly, this

flexibility raises the question: How can the firm determine the mix of inputs

that will minimize the cost of producing a given level of output? To answer this

question, let’s return to the case of two inputs, labor and capital. Here the

firm’s production function is of the form

where L is the number of labor hours per month and K is the amount of cap-

ital used per month. There are possibly many different ways to produce a given

level of output (call this Q 0 ), utilizing more capital and less labor or vice versa.

The optimal mix of labor and capital in producing output Q 0 depends on

the costs and marginal products of the inputs. Let’s denote the firm’s labor

cost per hour by PLand its cost per unit of capital by PK. Then the firm’s total

cost of using L and K units of inputs is

The firm seeks to minimize this cost, subject to the requirement that it use

enough L and K to produce Q 0. We now state the following important result

concerning optimal long-run production:

In the long run, the firm produces at least cost when the ratios of marginal prod-

ucts to input costs are equal across all inputs.

For the case of two inputs, we have

[5.4]

Equation 5.4 shows that when total cost is minimized, the extra output per

dollar of input must be the same for all inputs. To see why this must be true,

MPL/PLMPK/PK.

TCPLLPKK.

QF(L, K),

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