Problem 14 - Applied Calculus

Question

Show analytically that if marginal cost is less than average cost, then the derivative of average cost with respect to quantity satisfies \(a^{\prime}(q)<0\).

Step-by-Step Solution

Answer

If MC < AC, then \( a'(q) < 0 \), meaning average cost decreases with increased quantity.

Step 1: Understand the Relationship between Marginal and Average Costs

The marginal cost (MC) is defined as the additional cost of producing one more unit of a good. The average cost (AC) is the total cost divided by the number of units produced. The relationship between these two costs provides important insights into economics.

Step 2: Define Relevant Functions

Let MC be defined as \( MC = C'(q) \), where \( C(q) \) is the total cost function and \( q \) is the quantity. The AC is defined as \( AC = \frac{C(q)}{q} \).

Step 3: Differentiate Average Cost with Respect to Quantity

Find the derivative of average cost, \( a'(q) \), with respect to the quantity \( q \). Use the quotient rule: \[ a'(q) = \left( \frac{C(q)}{q} \right)' = \frac{qC'(q) - C(q)}{q^2} \]. This can be simplified to \( a'(q) = \frac{qMC - C(q)}{q^2} \).

Step 4: Analyze Conditions of the Problem

Since we know MC < AC, substitute AC in terms of \( C(q) \):\[ MC < \frac{C(q)}{q} \] which implies \( qMC < C(q) \).

Step 5: Determine Sign of the Derivative

Using the expression from Step 3: \( a'(q) = \frac{qMC - C(q)}{q^2} \). Given \( qMC < C(q) \), the numerator \( qMC - C(q) \) is negative. Since \( q^2 \) is positive, \( a'(q) < 0 \).

Step 6: Conclusion

The derivative of average cost with respect to quantity is negative when marginal cost is less than average cost, satisfying \( a'(q) < 0 \). This indicates that the average cost decreases as the quantity increases under these conditions.

Key Concepts

average cost

In business and economics, understanding the cost concepts is crucial. Let's delve into the concept of **Average Cost (AC)** first. Average cost represents the total cost of production divided by the number of goods produced. Mathematically, it is expressed as:\[ AC = \frac{C(q)}{q} \]where:

• \( C(q) \): Total cost as a function of quantity \( q \).
• \( q \): Quantity of goods produced.

The average cost gives us an overview of how costs behave as production scales up. Knowing your average cost is important because it helps businesses in setting prices and projecting profits.
Analyzing the behavior of average cost is key to making informed decisions. For instance, if average cost decreases as the quantity increases, a company might benefit from producing more units.

derivative with respect to quantity

Differentiation is a powerful tool in calculus, helping us to understand how a function changes as one of its variables changes. Here, we apply it to the **average cost (AC)** relative to the quantity \( q \). The derivative of average cost with respect to quantity \( a'(q) \) indicates how the average cost changes as more goods are produced. This is done using the quotient rule of derivatives.For the average cost, the formula for differentiation is:\[ a'(q) = \left( \frac{C(q)}{q} \right)' = \frac{qC'(q) - C(q)}{q^2} \]where:

• The numerator \( qC'(q) - C(q) \) is crucial in determining the sign of the derivative.
• Since \( q^2 \) is always positive, the sign of \( a'(q) \) relies on \( qC'(q) - C(q) \).

The derivative tells us if the average cost is increasing or decreasing.
This analysis is helpful in understanding the cost dynamics and optimizing production levels.

economics insights

Understanding the relationship between marginal cost (MC) and average cost (AC) brings about important **economics insights**. In the short run, the behavior of firms is often analyzed by looking at these costs. It's known that if marginal cost is less than average cost, the average cost is decreasing as more units are produced.This relationship can be visualized as:

• **Marginal Cost (MC)**: The cost of producing an additional unit, mathematically defined as \( MC = C'(q) \).
• Condition: If \( MC < AC \), it implies \( qMC < C(q) \).
• Outcome: \( a'(q) < 0 \) indicates that AC is trending downwards.

These insights are vital for decision-making. Knowing when increasing production leads to reduced average costs can guide firms in scaling up production efficiently.
Economics often shows us that higher efficiency is achieved when firms learn to fully exploit these cost behaviors.