A firm has a production function of Q=0.25 K L^0.5, the...

A firm has a production function of $Q=0.25 K L^{0.5}$, the rental rate of capital is $\$ 100$, and the wage rate is $\$ 25 .$ In the short run, $\bar{K}$ is fixed at 100 units. a. What is the short-run production function? b. What is the short-run demand for labor? c. What are the firm's short-run total cost and shortrun marginal cost?

A firm has a production function of $Q=0.25 K L^{0.5}$, the rental rate of capital is $\$ 100$, and the wage rate is $\$ 25 .$ In the short run, $\bar{K}$ is fixed at 100 units.
a. What is the short-run production function?
b. What is the short-run demand for labor?
c. What are the firm's short-run total cost and shortrun marginal cost?

Key Concepts

Production Function

A production function is a mathematical representation that describes the relationship between the inputs used by a firm and the output produced. It explains how different quantities of inputs such as labor, capital, and materials combine to produce a certain amount of goods or services, serving as a fundamental tool for analyzing production efficiency and technological effectiveness.

Short-run Production Function

In the short run, at least one factor of production, such as capital, is fixed while other inputs can be varied. The short-run production function reflects this constraint by incorporating the fixed input along with the variable inputs to show how output can change when only certain inputs are adjustable. This concept is critical for understanding how firms operate under limited flexibility in input choices over a given period.

Fixed Input (Capital) in the Short Run

A fixed input, such as capital in the short-run context, is an input whose quantity cannot be changed due to contractual, physical, or time-based constraints. The presence of a fixed factor affects decision-making and cost structure by forcing the firm to adjust production levels through the use of variable inputs like labor, and it plays a key role in shaping the short-run cost curves.

Short-run Demand for Labor

The short-run demand for labor is the optimal amount of labor that a firm chooses to hire when some inputs are fixed. This decision is made by equating the marginal product of labor, which indicates the additional output obtained from an extra unit of labor, with the marginal cost of hiring, typically represented by the wage rate. This concept helps firms determine the cost-effective level of labor needed to meet production goals given the fixed level of capital.

Cost Functions

Cost functions are mathematical models that capture the total cost of production as a function of output level, considering both fixed and variable costs. In the short run, these functions are influenced by the presence of fixed inputs, leading to a combination of costs that remain constant (fixed costs) and costs that vary with production (variable costs). Understanding cost functions is essential for analyzing production decisions and profit maximization.

Marginal Cost

Marginal cost is the additional cost incurred when producing one more unit of output. It is derived from the short-run cost function by calculating the change in total cost as output increases incrementally. This concept is crucial for making production decisions, as firms use marginal cost analysis to determine the point at which producing an extra unit is beneficial or detrimental to overall profitability.

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