Question

The cost \(C\) of producing \(x\) units is modeled by \(C=v(x)+k\), where \(v\) represents the variable cost and \(k\) represents the fixed cost. Show that the marginal cost is independent of the fixed cost.

Short answer

The marginal cost is the derivative of the variable cost, and is independent of the fixed cost. It is shown through taking derivative of the total cost function, which only includes the variable cost, proving that marginal cost does not depend on the fixed cost.

Step-by-step solution

Understand the concept of marginal cost

Marginal cost is the derivative of the total cost function with respect to the quantity of units produced. It represents how the cost changes with every additional unit produced.

Differentiate the cost function

The cost function provided is \(C = v(x) + k\). The derivative of this function with respect to \(x\) gives the marginal cost, namely \(MC(x) = C'(x)\). Differentiate \(C(x)\) to find \(MC(x)\). The derivative of \(v(x)\) becomes \(v'(x)\) and the derivative of \(k\) (a constant) is 0.

Conclude that marginal cost is independent of fixed cost

The result of the differentiation is \(MC(x) = v'(x) + 0 = v'(x)\). Hence, the marginal cost, which is the derivative of the cost function, contains only the part corresponding to the variable cost. The fixed cost \(k\) doesn't appear in the derivative, proving that the marginal cost is independent of the fixed cost.

Key concepts

Cost Function

In economics and business, the cost function is a mathematical expression that describes the total cost incurred by a company to produce a certain level of output or number of products. It's essential for understanding how costs evolve in relation to production levels.
The general form of a cost function can often be expressed as:

• The sum of fixed costs: These are expenses that remain consistent regardless of the number of units produced.
• Variable costs: These fluctuate depending on production output levels.

The cost function formula can be written as:\[ C(x) = v(x) + k \]where:

• \( v(x) \) is the variable cost, dependent on the number of units \( x \) produced.
• \( k \) is the fixed cost, remaining constant across different output levels.

This formula helps companies to strategize and predict their expenses for producing goods or services, which is critical for budgeting and pricing strategies.

Derivative

The derivative is a fundamental concept in calculus, indicating how a function changes as its input changes. It essentially provides the rate at which a variable quantity (like cost) changes with respect to another variable (like production quantity).
When we take the derivative of a cost function like \( C(x) = v(x) + k \), we are looking for how the total cost changes as the number of produced units (\( x \)) changes. In this context, we're interested in the derivative because it gives us the marginal cost:

• Marginal cost, represented as \( MC(x) \), is calculated by differentiating \( C(x) \) with respect to \( x \).
• The derivative of the fixed cost \( k \), which does not depend on \( x \), is zero. Hence, it does not affect the marginal cost rate.
• The derivative of the variable cost, \( v'(x) \), shows the change in cost for each additional unit produced.

Thus, understanding derivatives allows businesses to measure and interpret the cost efficiency and pricing effectively.

Variable Cost

Variable costs are those expenses that vary directly with the level of output. These costs increase as more units are produced and decrease if fewer units are made. Common examples of variable costs include raw materials, direct labor, and utilities that fluctuate with production volume.
Variable costs are crucial to the cost function because:

• They determine the direct cost per unit, influencing pricing and marginal revenue.
• Their derivative, \( v'(x) \), plays a vital role in calculating the marginal cost, reflecting the cost of producing one more unit without incorporating fixed costs.

Understanding variable costs allow companies to make strategic decisions on scaling production, optimizing resource allocation, and identifying cost-saving opportunities. This knowledge can significantly impact profitability assessments.

Fixed Cost

Fixed costs are expenses that remain unchanged regardless of how much is produced. These costs include items like rent, salaries, and specific utilities that do not vary with production output.
In a cost function, fixed costs are denoted by \( k \), and they are important for several reasons:

• They are the baseline costs a company must cover to stay operational, regardless of sales volume.
• Unlike variable costs, they do not influence the marginal cost because their derivative is zero when calculating the total cost change per additional unit.
• Understanding fixed costs is essential for break-even analysis, ensuring businesses know the minimum production levels needed to cover all expenses.

While they don't impact short-term production decisions, managing fixed costs effectively can offer long-term financial predictability and stability for companies.