Problem 46 - Thomas Calculus

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

The average cost is minimized when it equals the marginal cost.

Step 1: Define Average Cost and Marginal Cost

The average cost (AC) is defined as the total cost (TC) divided by the quantity (q) produced: \( AC = \frac{TC}{q} \). The marginal cost (MC) is defined as the derivative of the total cost with respect to quantity: \( MC = \frac{d(TC)}{dq} \).

Step 2: Determine the Critical Points of Average Cost

To find the production level where average cost is smallest, take the derivative of the average cost function with respect to quantity and set it to zero: \( \frac{d(AC)}{dq} = 0 \). This will give us the critical points of the average cost function.

Step 3: Use the Relationship between Total Cost, Average Cost, and Marginal Cost

We know that \( AC = \frac{TC}{q} \). Taking the derivative of both sides with respect to \( q \) gives \( \frac{d(AC)}{dq} = \frac{d}{dq}\left( \frac{TC}{q} \right)\). By applying the product rule, we get: \( \frac{d(AC)}{dq} = \frac{1}{q} MC - \frac{TC}{q^2} \).

Step 4: Set the Derivative of Average Cost to Zero

Substituting \( \frac{d(AC)}{dq} = 0 \), we have \( \frac{1}{q} MC - \frac{TC}{q^2} = 0 \). Simplifying gives \( MC = \frac{TC}{q} \). Hence, \( MC = AC \).

Step 5: Conclusion

From the above algebra, we conclude that at the production level where average cost is minimized, the average cost is equal to the marginal cost.

Key Concepts

Marginal Cost

Marginal cost (MC) is a vital concept in economics, helping us understand the cost of producing one more unit of a good. In simpler terms, it shows the additional cost incurred when producing one extra item. The formula for marginal cost is represented as the derivative of the total cost (TC) with respect to quantity (q):\[MC = \frac{d(TC)}{dq}\]Why is this important? Knowing the marginal cost helps businesses make critical production decisions. It tells manufacturers the exact cost to increase production by one unit, enabling them to balance production costs against potential profits. When marginal cost aligns with average cost, it indicates optimal production efficiency, as will be discussed in later sections. Businesses often aim to produce where their costs are minimized, maximizing overall efficiency.

Critical Points

Critical points in calculus are values in the domain of a function where the derivative is zero or undefined. These points can represent either a maximum, a minimum, or a point of inflection in a function. For identifying the production level where average cost is smallest, we take the derivative of the average cost (AC) function with respect to quantity, set this derivative equal to zero, and solve for the critical points:\[\frac{d(AC)}{dq} = 0\]These solutions reveal the critical points, helping us spot where important changes in the shape of the cost curve occur. In practice, economists and business leaders use these to find minimum costs, securing an economical cost of production. This step is crucial to optimize production and cost-effectiveness.

Total Cost

Total cost (TC) represents the entire sum of costs associated with the production of goods or services. It encapsulates both fixed and variable costs that a company incurs during production. The formula for calculating average cost (AC) is:\[AC = \frac{TC}{q}\]where \( q \) denotes the quantity of goods produced. Understanding total cost is vital because it allows businesses to assess whether their production levels are sustainable and profitable. By analyzing total costs alongside average and marginal costs, companies can find the balance that optimizes production without overspending. Observing how these costs interact and influence each other supports effective financial planning and strategic decision-making.

Production Level

Production level refers to the scale at which goods are manufactured or production activities are undertaken. It directly influences several cost measures, including average cost (AC) and marginal cost (MC).The point at which AC is minimized is crucial. It represents production efficiency, balancing the total cost distribution over quantity, ensuring cost-effectiveness. By setting \( \frac{d(AC)}{dq} = 0 \), we find the production level where the average cost is minimized. At this critical production level, the average cost equals the marginal cost:\[MC = AC\]This equality marks an optimal production condition. Operating at this level ensures that a firm's resources are used efficiently, maximizing output without unnecessary expenses. Production level decisions are, therefore, pivotal to optimizing business operations and enhancing competitiveness.