In the short run, a firm cannot vary its capital, K=2, but...

In the short run, a firm cannot vary its capital, $K=2,$ but can vary its labor, $L$. It produces output $q .$ Explain why the firm will or will not experience diminishing marginal returns to labor in the short run if its production function is a. $q=10 L+K$ b. $q=L^{0.5} K^{0.5} . \mathrm{C}$

Key Concepts

Short?Run Production

In the short run, at least one input is fixed (in this case, capital is fixed at K=2), while other inputs, such as labor, are variable. This setting emphasizes how the firm's output responds to changes in the variable input when one factor is held constant.

Marginal Product of Labor

The marginal product of labor measures the additional output produced by employing an extra unit of labor while keeping all other inputs constant. It is crucial for understanding how hiring more workers affects production in the fixed capital scenario.

Diminishing Marginal Returns

Diminishing marginal returns occur when the incremental gain in output from adding an extra unit of the variable input decreases as more of that input is employed, given that other inputs are fixed. This concept is fundamental in analyzing production efficiency and optimization.

Linear versus Nonlinear Production Functions

A linear production function implies constant marginal returns to the variable input because the additional output is the same regardless of the amount of labor already employed. In contrast, nonlinear functions (such as the Cobb-Douglas form) can exhibit diminishing marginal returns if the output increases at a decreasing rate as more labor is added.

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