Question

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.

Short answer

The chair firm is not minimizing its production costs. For optimal production, it needs to use twice as much machinery per hour as labor. This change would reduce costs per chair from \$105 to \$90. A graphical illustration of the isoquant and the two isocost lines would depict this optimization.

Step-by-step solution

Define the costs

The company is currently spending \(3 \times \$30 + 1 \times \$15 = \$105)\ per chair, where \$30 is the cost per hour of labor and \$15 is the cost per hour of machinery.

Establish optimal production levels

To optimally produce a chair with 4 hours labor or machinery, the firm should equate the marginal rate of technical substitution (MRTS) to the ratio of the input costs. The MRTS is the amount of one input that can be replaced with another input, while holding output constant. In this case, since it takes the same amount of time to produce a chair whether using labor or machinery, the MRTS equals 1.

Balance Cost and Production

Given the MRTS, we find the cost ratio of inputs to be \(\$30/\$15 = 2\). Therefore, for every hour of labor used, the company should use two hours of machinery to equate the marginal productivities per dollar of all inputs.

Graphical Representation

The isoquant is a curve showing all possible combinations of inputs that result in the production of a given level of output. Given the information, a 45-degree line is obtained from the origin: all points along this line indicate combinations where the total time is always 4 hours. For the isocost lines, plot one line depicting the current scenario at \(\$105) and another indicating the optimal scenario where for every hour of labor, two hours of machinery are used leading to a total cost of \(\$90). The point where the isocost line is tangential to the isoquant line gives the optimal combination of labor and machinery.