Problem 39 - Calculus Single Variable

Question

If the cost of producing \(x\) units of a commodity is \(C(x)\) then the average cost \(\bar{C}(x)\) of producing those \(x\) units is \(\bar{C}(x)=\mathrm{C}(x) / x .\) Prove the Minimum Average Cost Principle: When minimized, the average cost equals the marginal cost.

Step-by-Step Solution

Answer

To minimize average cost, it equals marginal cost, as derived from setting the derivative of average cost to zero.

Step 1: Understand Given Functions

We are given two functions: the cost function \(C(x)\), which represents the total cost of producing \(x\) units, and the average cost function \(\bar{C}(x)\), which is given by \(\bar{C}(x) = \frac{C(x)}{x}\). Our goal is to show that when \(\bar{C}(x)\) is minimized, it equals the marginal cost, \(C'(x)\).

Step 2: Set Up the Derivative of Average Cost

To find where the average cost is minimized, we take the derivative of \(\bar{C}(x)\) with respect to \(x\). We use the quotient rule: if \(f(x) = \frac{u(x)}{v(x)}\), then \(f'(x)=\frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\). Here, \(u(x) = C(x)\) and \(v(x) = x\).

Step 3: Compute the Derivative

Apply the quotient rule to \(\bar{C}(x)\):\[\bar{C}'(x) = \frac{C'(x) \cdot x - C(x) \cdot 1}{x^2} = \frac{C'(x)x - C(x)}{x^2}.\]

Step 4: Set the Derivative to Zero

To find the critical points where average cost is minimized, set the derivative \(\bar{C}'(x)\) to zero:\[\frac{C'(x)x - C(x)}{x^2} = 0.\]This implies \(C'(x)x - C(x) = 0\), so \(C'(x)x = C(x)\).

Step 5: Solve for Marginal Cost Equals Average Cost

From \(C'(x)x = C(x)\), divide both sides by \(x\) to isolate \(C'(x)\):\[C'(x) = \frac{C(x)}{x} = \bar{C}(x).\]Thus, when \(\bar{C}(x)\) is minimized, \(C'(x) = \bar{C}(x)\), proving the Minimum Average Cost Principle.

Key Concepts

Cost Function

In economics and production, understanding the cost function is crucial for businesses to manage their expenses effectively. The cost function, denoted as \(C(x)\), describes the total cost of producing \(x\) units of a commodity. It's a mathematical relationship that accounts for both fixed and variable costs involved in the production process.

Fixed costs are expenses that don't change with the level of output, such as rent or salaries. Variable costs, on the other hand, fluctuate with production output, like raw materials and utilities necessary for manufacturing. As a firm increases its production, the cost function shows how these costs accumulate.

By examining \(C(x)\), businesses can make informed production decisions and identify efficient cost levels. It's foundational for further economic analysis, including calculating average and marginal costs. Understanding this function allows firms to strategize, maximizing profit while minimizing unnecessary spending.

Average Cost

Average cost, represented as \( \bar{C}(x) \), is an important metric for evaluating the efficiency of production. It indicates the cost associated with producing each unit of a commodity. Mathematically, it's the total cost \(C(x)\) divided by the number of units \(x\).

Here's the formula: \( \bar{C}(x) = \frac{C(x)}{x} \). This gives businesses insight into how cost-effective their production processes are. If the average cost is declining as output increases, it usually suggests that the firm is benefiting from economies of scale—producing more reduces the cost per unit.

However, it’s essential to find the optimal level of production where the average cost is minimized. At this point, the company can operate most efficiently, balancing production scales and costs effectively. Monitoring average costs helps in price setting and maximizing profit margins over time.

Marginal Cost

Marginal cost is a key concept in decision-making and resource allocation. It refers to the cost of producing one additional unit of a commodity. Mathematically, it's the derivative of the cost function, expressed as \(C'(x)\).

The marginal cost indicates the change in total cost when production output is increased by one unit. Understanding this helps businesses decide whether increasing production is profitable. For instance, if the revenue from an additional unit exceeds the marginal cost, it might be beneficial to increase production.

The Minimum Average Cost Principle connects marginal cost to average cost. When average cost is minimized, it equals marginal cost: \(C'(x) = \bar{C}(x)\). This equality is crucial for determining optimal production levels. Operating at this point ensures resources are used most efficiently, without overspending or under utilizing capacity. Recognizing and calculating marginal cost aids businesses in optimizing their production strategies.