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.

Understanding the Marginal Rate of Technical Substitution (MRTS) is essential for firms looking to efficiently allocate resources. The MRTS reflects the rate at which one production factor (like labor) can be substituted for another (like machinery) while keeping output constant.

In simple terms, if you picture a production process that can swap labor hours for machine hours without affecting the number of goods produced, the MRTS tells you how many machine hours you can cut for every additional hour of labor employed. It's calculated by dividing the quantity of labor by the quantity of capital (machine hours), which in the provided exercise resulted in an MRTS of 3.

An isocost line represents the different combinations of factors a firm can purchase with a specific total cost. In our exercise, the chair manufacturer has costs related to labor and machinery, each with their own hourly rates. The slope of the isocost line is the ratio of the factor costs, and this slope reflects the trade-off the firm faces when choosing between more labor or more machinery.

The steeper the line, the more expensive labor is relative to machinery. The existing isocost line has a slope of 2, given the cost of labor () and machine (). A firm reaches optimal production when the isocost line is tangent to the isoquant, indicating the most efficient use of resources for a given cost.

An isoquant represents all the combinations of inputs that yield the same level of output. Imagine a map where each line traces combinations of labor and machinery that produce the same number of chairs. In the exercise, the isoquant could be depicted as a straight line, since labor and machinery are perfect substitutes.

The firm's current situation uses 3 hours of labor for every hour of machine time. This point is on the isoquant, but it does not minimize costs because it's not where the isocost line is tangent to the isoquant. Thus, while the firm is producing efficiently in a technical sense, they are not doing so in an economically efficient way.

Optimization in production is all about producing goods in the most cost-effective way without sacrificing quality or quantity of output. In our chair manufacturer's scenario, achieving cost minimization requires adjusting the input combination of labor and machinery until the MRTS equals the ratio of input costs.

Since the firm's MRTS is higher than the cost ratio, they should use more labor (which is cheaper relative to machinery) until the MRTS falls to match the cost ratio of labor to machinery. This process will result in the optimal isocost line becoming tangent to the isoquant, which is where optimum production at minimum cost is achieved.