Question

Suppose a firm is maximizing profits in the short run with variable factor \(x_{1}\) and fixed factor \(x_{2} .\) If the price of \(x_{2}\) goes down, what happens to the firm's use of \(x_{1} ?\) What happens to the firm's level of profits?

Short answer

Use of \(x_1\) stays the same; profits increase.

Step-by-step solution

Analyze the Initial Condition

Initially, the firm is maximizing profits, which means it is producing at an optimal level of input utilization where marginal cost (MC) equals marginal revenue (MR). The firm uses a variable factor \(x_1\) and a fixed factor \(x_2\). The price of \(x_2\) initially affects profits but does not affect how much \(x_1\) is used, as \(x_2\) is fixed in the short run.

Impact of Price of Fixed Factor Decrease

When the price of the fixed factor \(x_2\) decreases, the costs for this factor reduce. The fixed cost component of the firm’s cost structure decreases, making the production process more cost-effective.

Determining Use of \(x_1\)

Since \(x_2\) is fixed and there is a decrease in its cost, there is no direct impact on the optimal use of \(x_1\) in the short run, given profit-maximizing conditions. \(x_1\) remains the same in the short run as it is not directly tied to the cost of \(x_2\).

Effect on Profit Levels

The decrease in the price of \(x_2\) reduces fixed costs, thereby increasing overall profits. Since the firm's revenue remains the same and its cost decreases, given \(x_2\) is fixed, profits will rise due to the reduction in expense.

Key concepts

Short Run Analysis

The short run in economic terms refers to a period where certain factors of production are fixed, meaning they do not change. In this context, a firm is examining its production capabilities and profitability when some factors can be adjusted—like labor or raw materials—but others cannot, such as machinery or land.
In the short run, the firm focuses on optimizing the use of variable factors to align with the fixed factors. This means they aim to produce goods or services at a point where the marginal cost of production exactly matches the marginal revenue they earn. This balance is critical for maximizing profits. If the price of a fixed factor, such as machinery (denoted as \(x_2\)), changes, it may lead to shifts in cost but does not initially affect the production-related decisions involving variable factors like labor (\(x_1\)).
Thus, while profit maximization is still the goal, the nature of fixed and variable inputs dictates how easily a firm can adapt to changes within the short run.

Fixed and Variable Factors

In economic production, inputs can be categorized into fixed and variable factors. Fixed factors (\(x_2\)) are resources or inputs that remain constant in the short run. Examples include capital equipment, factories, or long-term leases. Their constancy means they cannot be increased or decreased rapidly or without incurring new costs.
Variable factors (\(x_1\)) are those that a firm can adjust on short notice. These usually involve labor, raw materials, or utilities, which can be altered as per the production needs.
Understanding how each input affects production is crucial. Because fixed factors like \(x_2\) do not change in the short run, their cost reductions do not alter how much of the variable input \(x_1\) is used. Instead, they impact the firm’s overall cost, potentially increasing profit margins if costs decrease. This shows the delicate balance a firm must maintain in adjusting its variable resources around the fixed ones.

Cost Structure Analysis

Cost structure is the combination of fixed and variable costs that a firm incurs during production. Fixed costs are those that remain constant regardless of how much is produced, like rent or salaries of permanent staff. Variable costs fluctuate with production volume, such as raw materials or hourly wages.
When a firm experiences a decrease in the price of a fixed factor (\(x_2\)), its fixed costs go down. This reduction does not affect how much variable factor (\(x_1\)) is utilized but directly impacts profitability.
A lower cost structure means that for the same level of output, the firm's expenses are reduced. As revenue stays constant, and costs decrease, the firm observes an increase in profits. This is a key strategy in profit maximization: managing costs to increase the bottom line.
Understanding cost structures helps firms make informed decisions about scaling production and adapting to market changes. By analyzing their cost setup, businesses can better anticipate and respond to changes in fixed input prices, optimizing their profit strategy.