Question

Find the marginal cost for producing \(x\) units. (The cost is measured in dollars.) $$ C=55,000+470 x-0.25 x^{2} $$

Short answer

The marginal cost function for producing \(x\) units is \(C' = 470 - 0.5x\).

Step-by-step solution

Understanding the cost function

The given cost function is \(C=55,000+470x-0.25x^{2}\), where \(C\) is the total cost and \(x\) is the number of units produced. The constant term (55,000) is the fixed cost, the term 470x is the variable cost per unit, and the term -0.25x^2 represents economies of scale.

Differentiate the cost function

We now differentiate the cost function to obtain the marginal cost. The derivative of \(C=55,000+470x-0.25x^{2}\) with respect to \(x\) yields: \(C' = 470 - 0.5x\).

Interpret the marginal cost function

We found that the marginal cost function is \(C' = 470 - 0.5x\). This means that for each additional unit produced, the increase in cost decreases, which is due to economies of scale.

Key concepts

Cost Function

The cost function is a mathematical expression that helps us understand how the total cost of production is influenced by the number of units produced. In this particular exercise, the cost function is presented as:
\[ C = 55,000 + 470x - 0.25x^2 \]
Here, \(C\) represents the total cost, and \(x\) symbolizes the number of units produced. Let's decode this equation:

• The constant \(55,000\) is the fixed cost. This is the cost that remains constant regardless of production levels.
• The term \(470x\) stands for the variable cost. With more units produced, this cost changes linearly.
• Finally, \(-0.25x^2\) exemplifies the concept of economies of scale, which we'll explore next.

Understanding your cost function is crucial because it lays the foundation for identifying how various production levels impact costs. It also prepares the ground for calculating more complex concepts like marginal costs.

Economies of Scale

Economies of scale are a vital concept in production economics. They occur when increasing production leads to a lower cost per unit, due to the spreading of fixed costs over more units and improved operational efficiency. In the given cost function,
\[ -0.25x^2 \]
represents economies of scale. Here is how it benefits production:

• As you produce more units, the negative coefficient \(-0.25\) means that each additional unit costs less to produce.
• This reduction in cost happens because the negative squared term \(-0.25x^2\) grows faster than the positive variable cost \(470x\) after a certain point.

By understanding economies of scale, businesses can strategically increase production to maximize efficiency and minimize costs, taking advantage of reduced marginal costs as production scales up.

Differentiation

Differentiation is a powerful mathematical tool used to determine changes in a function's output given changes in its input. In this exercise, we differentiated the cost function to find the marginal cost, which shows the cost of producing one additional unit.
The original cost function is
\[ C = 55,000 + 470x - 0.25x^2 \]
By differentiating it with respect to \(x\), we get the derivative:
\[ C' = 470 - 0.5x \]
This derivative, \(C'\), is known as the marginal cost function. It tells us how the total cost changes with each additional unit produced. As seen here:

• The constant \(470\) suggests that if no additional economies of scale were in place, each unit would add $470 to the cost.
• The term \(-0.5x\) implies the decrease in cost per additional unit produced due to economies of scale.

By using differentiation, businesses can understand how their costs evolve as they increase production, which is fundamental for both pricing strategies and production planning.