Problem 12 - Applied Calculus

Question

Values of a linear cost function are in Table \(1.25 .\) What are the fixed costs and the marginal cost? Find a formula for the cost function. $$ \begin{array}{l} \text { Table } 1.25\\\ \begin{array}{c|c|c|c|c|c} \hline q & 0 & 5 & 10 & 15 & 20 \\ \hline C(q) & 5000 & 5020 & 5040 & 5060 & 5080 \\ \hline \end{array} \end{array} $$

Step-by-Step Solution

Answer

Fixed cost is 5000, marginal cost is 4, cost function is \( C(q) = 5000 + 4q \).

Step 1: Identify the Fixed Costs

The fixed cost represents the cost when production quantity \( q = 0 \). From the table, when \( q = 0 \), \( C(q) = 5000 \). Thus, the fixed cost is \( 5000 \).

Step 2: Calculate the Marginal Cost

The marginal cost can be found from the change in cost over the change in quantity produced. Here, observe that as \( q \) increases from \( 0 \) to \( 5 \), \( C(q) \) increases from \( 5000 \) to \( 5020 \). The change in cost \( \Delta C = 5020 - 5000 = 20 \), and the change in quantity \( \Delta q = 5 \). Thus, the marginal cost \( MC = \frac{\Delta C}{\Delta q} = \frac{20}{5} = 4 \).

Step 3: Formulate the Cost Function

The linear cost function is typically of the form \( C(q) = F + MC \times q \), where \( F \) is the fixed cost and \( MC \) is the marginal cost. From Step 1, we have \( F = 5000 \), and from Step 2, \( MC = 4 \). Substituting these values into the cost function gives \( C(q) = 5000 + 4q \).

Key Concepts

Fixed Costs

Fixed costs are expenses that do not change with the level of production or sales, meaning they remain constant regardless of the quantity produced. In the context of the exercise provided, the fixed costs can be directly identified when the quantity produced, denoted as \( q \), is zero. You can think of fixed costs as the basic overheads required just to keep operations running, such as rent, salaries, and utilities. From the table, it is clear that when \( q = 0 \), the cost \( C(q) \) is \( 5000 \). Thus, the fixed costs amount to \( 5000 \). Fixed costs are a crucial part of the cost structure as they set the minimum expenditure threshold for any production activity. Recognizing your fixed costs is essential for budgeting and financial management, as they represent the amount you need to cover, even when no units are produced. This consistent figure is beneficial when planning in the long term for lower production periods.

Marginal Cost

Marginal cost represents the additional cost that arises from producing one additional unit of a product. It's essential for decision-making regarding pricing and output levels, showing how much the total cost changes with incremental production adjustments. In linear cost systems like the one we're discussing, the marginal cost is constant.

To calculate the marginal cost from the exercise, you observe the changes in cost \( C(q) \) as production \( q \) increases. For instance, the cost increased from \( 5000 \) to \( 5020 \) as the quantity changed from \( 0 \) to \( 5 \). This means the marginal cost \( MC \) is calculated as the change in cost \( \Delta C \) divided by the change in quantity \( \Delta q \):

• \( \Delta C = 5020 - 5000 = 20 \)
• \( \Delta q = 5 \)

Thus, the marginal cost \( MC = \frac{20}{5} = 4 \).

The ability to calculate and understand marginal cost helps businesses to make efficient production decisions and optimize for profitability, ensuring every additional unit produced contributes positively to overall revenue.

Cost Function Formula

A cost function provides a mathematical model to understand the relationship between the cost incurred by a business and the level of production. It combines fixed and marginal costs to show how total costs accumulate as output changes. In this exercise, the cost function is linear, which means costs increase at a constant rate as production rises.

The general form for a linear cost function is \( C(q) = F + MC \times q \), where:

• \( C(q) \) is the total cost for producing \( q \) units,
• \( F \) represents fixed costs, and
• \( MC \) is the marginal cost.

Using the values deduced:

• \( F = 5000 \)
• \( MC = 4 \)

Substituting these into the formula gives us the specific cost function: \( C(q) = 5000 + 4q \).

Understanding this formula allows businesses to accurately predict and allocate costs based on desired production levels, making it a fundamental tool in cost management and financial planning. It simplifies complex financial decisions by providing clarity on how costs behave as production scales.