Question

Explain why, at an interior optimal solution to the firm's cost-minimization problem, the additional output that the firm gets from a dollar spent on labor equals the additional output from a dollar spent on capital. Why would this condition not necessarily hold at a corner point optimal solution?

Answer 1

To understand why at an interior optimal the additional output the firm gets from a dollar spent on labour must equal the additional output the firm gets from a dollar spent on capital, assume these were not equal. For example, suppose the firm could get more output from a dollar spent on labour than on a dollar spent on capital. Then the firm could take one dollar away from capital and reallocate it to labor. Since the firm gets more output from a dollar of labor than from a dollar of capital, it will require the firm to spend less than one dollar on labor to offset the decline in output from taking one dollar away from capital. This implies the firm can keep output at the same level but do so at a lower cost. Therefore, if these amounts are not equal the firm is not minimising cost.

This requirement does not necessarily hold at a corner solution. While the firm could potentially reduce cost by reallocating spending to the more productive input, at a corner solution, by definition, the firm is not using one of the outputs. There is no further opportunity to reallocate spending if the firm is spending nothing on one of the inputs?i.e., the firm cannot move to a point where one of the inputs is negative.