Problem 60 - Calculus Single Variable

Question

Let \(C(x)\) denote the cost of producing \(x\) units of a commodity. Although \(x\) usually refers to a nonnegative integer, it is common in economics to treat \(x\) as a continuous variable. The marginal cost \(M(x)\) refers to the cost of producing an \(x+1^{\text {st }}\) unit after \(x\) units are produced. a. Describe \(M\) in terms of \(C\). b. Write down the limit definition of \(C^{\prime}(x) .\) What is the relationship between this quantity and \(M(x) ?\) c. Under what circumstances can we plausibly use \(C^{\prime}(x)\) as an expression for the marginal cost?

Step-by-Step Solution

Answer

a. \(M(x) = C(x+1) - C(x)\); b. \(C'(x) = \lim_{h \to 0} \frac{C(x+h) - C(x)}{h}\), \(C'(x) \approx M(x)\); c. When \(C(x)\) is continuous and differentiable.

Step 1: Define Marginal Cost (M) in Terms of Cost (C)

Marginal cost, \(M(x)\), is the cost of producing one additional unit after \(x\) units have already been produced. Mathematically, we describe \(M(x)\) as the derivative of the cost function, essentially being the change in cost when production changes from \(x\) to \(x+1\), that is: \[ M(x) = C(x+1) - C(x) \]

Step 2: Define the Derivative of C(x) Using Limit Definition

The derivative of \(C(x)\), denoted \(C'(x)\), describes the instantaneous rate of change of the cost function at a particular \(x\). The limit definition of this derivative is: \[ C'(x) = \lim_{h \to 0} \frac{C(x+h) - C(x)}{h} \]

Step 3: Relationship Between C'(x) and M(x)

The derivative \(C'(x)\) provides the marginal cost when we allow \(h\) to become very small (approaching zero), effectively becoming \(M(x)\) as \(h \to 1\). Thus, under continuous production, \(C'(x)\) is approximately equal to \(M(x)\) and represents the marginal cost.

Step 4: Using C'(x) as an Expression for Marginal Cost

We use \(C'(x)\) as an expression for the marginal cost when cost function \(C(x)\) is differentiable. This is common in cases where \(x\) is treated as a continuous variable, and production costs change smoothly. Therefore, when \(C(x)\) is smooth and continuous on the interval considered, \(C'(x)\) is a plausible representation of the marginal cost.

Key Concepts

Derivative

A derivative is a fundamental concept in calculus that helps us understand how functions change. In simple terms, the derivative of a function at a point measures how fast the function's value changes near that point. It is like capturing a snapshot of the function’s slope there.
For cost functions like our example with production costs, the derivative helps us understand how costs change with the production of each additional unit. The formula for the derivative under the limit definition is important:

• The derivative not only tells us the rate of change, but how marginal costs behave as production scales.

By determining the derivative of the cost function, represented as \(C'(x)\), we identify the instantaneous rate of change in cost as we scale production from one unit to another. This is crucial for efficiently managing and predicting production expenses.

Cost Function

The cost function is a mathematical expression that describes the total cost incurred by producing a given number of units of a product. The function is typically represented as \(C(x)\), where \(x\) represents the number of units produced.
Understanding this function is essential because:

• It provides insight into how much it costs to produce a particular number of units.
• It helps businesses decide on pricing, production levels, and compare expenses with revenue.

In our context, the cost function plays a significant role in determining the marginal cost, or the additional cost of producing one more unit. Mathematically examining changes in the cost function via derivatives is essential for calculating marginal costs.

Limit Definition

The limit definition in calculus provides a precise way of defining the concept of a derivative. This definition is used extensively in economics to examine how functions behave as they approach certain points. In terms of a cost function \(C(x)\), the derivative is defined as:

• \[ C'(x) = \lim_{h \to 0} \frac{C(x+h) - C(x)}{h} \]

This formula calculates the derivative by observing the function's behavior as the incremental change \(h\) approaches zero.
Utilizing the limit definition ensures accuracy in deriving marginal costs because it gives us the smooth, continuous variation in cost. The derivative under this definition is crucial in identifying how marginal costs proceed in circumstances where production functions behave continuously.

Continuous Variable

A continuous variable allows us to treat certain economic functions, like cost functions, as smooth and unbroken. In most practical applications, production is measured in whole numbers, but in economic theory, using continuous variables helps simulate real-world scenarios where changes are fluid and ongoing.
Economics often employs continuous variables to:

• Model realistic production levels without the restriction of discrete units.
• Simplify mathematical modeling, making differential calculus applicable.
• Enable the computation of derivatives which describe the behavior of costs as production scales progressively.

Treating \(x\) as a continuous variable permits the seamless application of calculus tools to analyze cost functions, making it a cornerstone concept in determining efficient production strategies.