Problem 54 - University Calculus: Early Transcendentals

Question

Production level Prove that the production level (if any) at which average cost is smallest is a level at which the average cost equals marginal cost.

Step-by-Step Solution

Answer

Minimum average cost occurs where AC equals MC.

Step 1: Understand the Problem

The problem asks us to show that the minimum average cost occurs where the average cost (AC) is equal to the marginal cost (MC). This is a result that often emerges in economics, particularly in the analysis of production and cost functions.

Step 2: Define Average and Marginal Costs

The average cost (AC) is given by the function \( AC = \frac{C(q)}{q} \), where \( C(q) \) is the total cost function and \( q \) is the quantity of output. The marginal cost (MC) is the derivative of the total cost function with respect to quantity, \( MC = \frac{dC}{dq} \).

Step 3: Set Up the Condition for Minimum AC

To find the production level that minimizes the average cost, we set the derivative of the average cost with respect to quantity (AC'(q)) to zero. Thus, we calculate \( AC'(q) = \frac{d}{dq}\left(\frac{C(q)}{q}\right) \) and set it equal to zero.

Step 4: Differentiate the Average Cost

To differentiate \( AC = \frac{C(q)}{q} \), we use the quotient rule: \( AC'(q) = \frac{q \cdot C'(q) - C(q)}{q^2} \). Simplifying gives \( AC'(q) = \frac{C'(q)q - C(q)}{q^2} \).

Step 5: Set the Derivative to Zero

For AC to be minimized, \( AC'(q) = 0 \). This means \( C'(q)q - C(q) = 0 \). Solving for this, we have \( C'(q)q = C(q) \), which simplifies to \( C'(q) = \frac{C(q)}{q} \), showing that \( MC = AC \).

Step 6: Interpret the Result

When the average cost is minimized, the marginal cost is equal to the average cost. Therefore, the production level at which average cost is smallest is indeed the level where AC equals MC.

Key Concepts

Marginal Cost

The concept of marginal cost is crucial in understanding how production costs evolve as the production level changes.
Marginal cost (MC) is the additional cost incurred by producing one more unit of a good. It is derived mathematically as the derivative of the total cost function with respect to the quantity of output produced. This means that marginal cost tells us how much the total cost increases when output is slightly increased.
In formula terms, if the total cost function is represented as \( C(q) \), where \( q \) is the quantity of output, then the marginal cost \( MC \) is given by:

• \( MC = \frac{dC}{dq} \)

Marginal cost is significant for decision-making in production, as it helps to determine the most cost-effective production levels. If producing additional units means a lower marginal cost, it can be profitable for businesses to increase production. Conversely, if the marginal cost rises, it could serve as a signal to reconsider the quantity of output produced.

Quotient Rule

The quotient rule is a calculus method used to differentiate functions that are expressed as a ratio of two other functions. This rule is essential when dealing with average cost functions, which are typically in the form \( \frac{C(q)}{q} \).
The quotient rule can be stated as follows:
If you have a function \( y = \frac{u}{v} \), where both \( u \) and \( v \) are functions of \( q \), then the derivative \( \frac{dy}{dq} \) is:

• \( \frac{dy}{dq} = \frac{v \cdot \frac{du}{dq} - u \cdot \frac{dv}{dq}}{v^2} \)

In the context of the average cost function \( AC = \frac{C(q)}{q} \), the quotient rule helps to find \( AC'(q) \), the derivative of average cost with respect to \( q \). Using the quotient rule allows economists to analyze changes in average cost as production levels vary, giving insights into cost efficiency thresholds.

Production Level

Production level refers to the quantity of goods produced and plays a critical role in cost analysis. In the context of minimizing average costs, it is important to identify the production level at which costs are optimized.
The goal is to find the quantity, denoted by \( q \), that minimizes the average cost function \( AC = \frac{C(q)}{q} \). This is done by setting the derivative of the average cost function with respect to \( q \) to zero and solving for \( q \).
When this condition is satisfied, the average cost is at its lowest point, and interestingly, the average cost equates to the marginal cost. This occurs because when the derivative of the average cost is set to zero, it implies \( C'(q) = \frac{C(q)}{q} \), leading to the equality \( AC = MC \).

• Knowing the optimal production level is vital for businesses as it ensures efficient use of resources while minimizing costs.
• This balance is crucial for financial sustainability and achieving competitive pricing in the market.

Understanding production levels can help businesses align their output with market demand, optimize supply chains, and maintain profitability.