A chair manufacturer hires its assembly-line labor for \(30 an hour and calculates that the rental cost of its machinery is \)15 per hour. Suppose that a chair can be produced using 4 hours of labor or machinery in any combination. If the firm is currently using 3 hours of labor for each hour of machine time, is it minimizing its costs of production? If so, why? If not, how can it improve the situation? Graphically illustrate the isoquant and the two isocost lines for the current combination of labor and capital and for the optimal combination of labor and capital.

The short-run isocost equation is of the general form TC = rK + wL, where rK is the cost of the capital, and wL is the cost of labor.

The slope of the isocost is:

$Slope=-\frac{w}{r}$

The price for labor per hour is $30 and that for capital per hour is $15.

The isocost equation is:

TC = 30L + 15K

$Slope=-\frac{30}{15}=-2$

Thus, the slope of the isocost is -2.

Also, the slope of a straight line is the ratio of its rise to run. Therefore, the slope of isocost is also given by:

$\mathit{S}\mathit{l}\mathit{o}\mathit{p}\mathit{e}\mathbf{=}\frac{\mathbf{△}\mathbf{K}}{\mathbf{△}\mathbf{L}}$

For cost-minimizing,

$\frac{\mathbf{△}\mathbf{K}}{\mathbf{△}\mathbf{L}}\mathbf{=}\mathbf{-}\frac{\mathbf{w}}{\mathbf{r}}$

A combination of 3 units of labor and 1 unit of capital gives the slope of 0.3334, which is not equal to 2 (= -w/r). Thus, the firm does not have minimum costs. It will cost $115 (= 30×3 + 15×1). Thus, the isoquant curve away from the origin will lead to higher costs.

The slope of an isoquant is equal to the marginal rate of technical substitution. The MRTS is the ratio of marginal productivity of labor to capital.

$\mathit{M}\mathit{R}\mathit{T}\mathit{S}\mathbf{=}\mathbf{-}\frac{\mathbf{△}\mathbf{K}}{\mathbf{△}\mathbf{L}}$

The firm can hire either a maximum of 4 hours of labor or 4 hours of capital at a time. Therefore, the X and Y axes intercepts are 4. Thus, the slope of the isoquant is:

$Slope=-\frac{4}{4}=-1$

Hence, the isoquant is a negatively sloped straight line.

As the isoquant curve and the idle isocost line, with slope -2, meet at a combination of 4 units of capital and zero labor units, the firm’s equilibrium lies at (0,4). Thus, the optimal solution for cost-minimizing is (0,4), which will give the minimum cost of $60 (= 30×0 + 15×4).

Thus, the firm will shift to the optimal combination on the isocost line closer to the origin.