91Ó°ÊÓ

The production function of a firm outlines how inputs are transformed into outputs. For the U.S. glass manufacturer, the production function is represented as \( q = 10 L^{0.5} K^{0.5} \). This function is of Cobb-Douglas form, which is common in economics due to its assumptions of constant returns to scale. Here, \( q \) is the quantity of output, \( L \) is the labor input, and \( K \) is the capital input.

The exponents \( 0.5 \) on both \( L \) and \( K \) indicate constant returns to scale because they sum to one. This implies that doubling both inputs will double the output, affirming efficient scaling in input application. Understanding this production function helps us grasp how variations in labor and capital affect production levels.

Marginal product refers to the additional output produced by using one more unit of an input while keeping other inputs constant. In our example, the marginal product of labor (MPL) is \( MP_L = \frac{5K^{0.5}}{L^{0.5}} = \frac{0.5q}{L} \), whereas the marginal product of capital (MPK) is \( MP_K = \frac{5L^{0.5}}{K^{0.5}} = \frac{0.5q}{K} \).

These equations highlight how each additional unit of labor or capital impacts the overall production. The analysis of marginal products is essential for cost minimization, enabling firms to make efficient input allocation choices. A key insight here is ensuring that the ratio of MPL to MPK matches the ratio of input prices. This accordance leads directly to optimal production costs and resource utilization strategies.

The long-run total cost curve represents the minimized cost of production for any level of output when all factors of production are variable. For the glass firm, the long-run total cost as a function of output \( q \) is derived as \( TC = \frac{2q}{5} \).

This equation is obtained by substituting the firm's optimal input ratio \( K = \frac{1}{4}L \) into its production function and simplifying. The derived cost curve shows that as production levels increase, the firm can adjust its input quantities proportionally, ensuring cost-effectiveness over time. The simplistic form of the cost curve also assists in forecasting expenses and making strategic long-term decisions. With an understanding of this curve, firms can better plan their resource allocation and predict economic outcomes.

The expansion path is pivotal when exploring how a firm adjusts its input combinations as output expands, adhering to cost efficiency. It is illustrated by the line \( K = \frac{1}{4}L \) in the \( K-L \) input space, starting from the origin.

This linear path indicates a consistent ratio of capital to labor usage as the firm scales its operations. By maintaining this fixed ratio, the firm ensures minimal production costs while increasing output. Visualizing this path helps in understanding how inputs must grow in tandem to maintain efficiency over various production scales. The expansion path is a strategic tool that guides the firm in aligning its input choices with its output goals, promoting sustainability in growth.