Percentage Change in Total Cost

Oct 13, 2025 by talynt.org 13 views

Introduction

In economics, the cost function is a crucial concept used to determine the total cost of producing a certain quantity of goods or services. The cost function is typically represented by the equation C(x) = 5,000 + 20x - (1/4)x^2, where x is the quantity produced and C(x) is the total cost. In this article, we will perform three tasks related to the cost function: calculating the marginal average cost when x = 20, using the result to estimate the total cost when x = 21, and calculating the percentage change in total cost.

Task 1: Calculate the Marginal Average Cost when x = 20

The marginal average cost is the rate of change of the total cost with respect to the quantity produced. It can be calculated by taking the derivative of the cost function with respect to x.

Cost Function Derivative

To calculate the marginal average cost, we need to find the derivative of the cost function C(x) = 5,000 + 20x - (1/4)x^2.

Using the power rule of differentiation, we get:

dC/dx = d(5,000)/dx + d(20x)/dx - d((1/4)x^2)/dx = 0 + 20 - (1/2)x

Marginal Average Cost

The marginal average cost is given by the derivative of the cost function:

MC(x) = dC/dx = 20 - (1/2)x

Now, we need to calculate the marginal average cost when x = 20.

Marginal Average Cost when x = 20

Substituting x = 20 into the marginal average cost equation, we get:

MC(20) = 20 - (1/2)(20) = 20 - 10 = 10

Therefore, the marginal average cost when x = 20 is 10.

Task 2: Estimate the Total Cost when x = 21

Using the result from Task 1, we can estimate the total cost when x = 21.

Estimating the Total Cost

Since the marginal average cost is 10, we can estimate the total cost when x = 21 by adding the marginal average cost to the total cost when x = 20.

Total Cost when x = 20

To estimate the total cost when x = 21, we need to calculate the total cost when x = 20.

Substituting x = 20 into the cost function equation, we get:

C(20) = 5,000 + 20(20) - (1/4)(20)^2 = 5,000 + 400 - 100 = 5,300

Estimating the Total Cost when x = 21

Now, we can estimate the total cost when x = 21 by adding the marginal average cost to the total cost when x = 20:

C(21) = C(20) + MC(20) = 5,300 + 10 = 5,310

Therefore, the estimated total cost when x = 21 is 5,310.

Task 3: Calculate the Percentage Change in Total Cost

To calculate the percentage change in total cost, we need to find the difference between the estimated total cost when x = 21 and the total cost when x = 20.

Percentage Change in Total Cost

The percentage change in total cost is given by:

Percentage Change = ((C(21) - C(20)) / C(20)) x 100 = ((5,310 - 5,300) / 5,300) x 100 = (10 / 5,300) x 100 = 0.187%

Therefore, the percentage change in total cost is 0.187%.

Conclusion

Introduction

In our previous article, we explored the concept of marginal average cost and total cost estimation using the cost function C(x) = 5,000 + 20x - (1/4)x^2. We calculated the marginal average cost when x = 20, estimated the total cost when x = 21, and calculated the percentage change in total cost. In this article, we will answer some frequently asked questions related to the cost function and its derivatives.

Q&A

Q: What is the marginal average cost, and how is it calculated?

A: The marginal average cost is the rate of change of the total cost with respect to the quantity produced. It can be calculated by taking the derivative of the cost function with respect to x.

Q: How do I calculate the marginal average cost when x = 20?

A: To calculate the marginal average cost when x = 20, you need to substitute x = 20 into the marginal average cost equation: MC(x) = 20 - (1/2)x.

Q: What is the estimated total cost when x = 21?

A: The estimated total cost when x = 21 is 5,310. This is calculated by adding the marginal average cost to the total cost when x = 20.

Q: How do I calculate the percentage change in total cost?

A: To calculate the percentage change in total cost, you need to find the difference between the estimated total cost when x = 21 and the total cost when x = 20, and then divide by the total cost when x = 20 and multiply by 100.

Q: What is the significance of the marginal average cost in economics?

A: The marginal average cost is a crucial concept in economics, as it helps businesses and policymakers understand the relationship between the quantity produced and the total cost. It can be used to make informed decisions about production levels and pricing strategies.

Q: Can I use the marginal average cost to estimate the total cost for any value of x?

A: Yes, you can use the marginal average cost to estimate the total cost for any value of x. However, the accuracy of the estimate will depend on the value of x and the shape of the cost function.

Q: How do I determine the shape of the cost function?

A: The shape of the cost function can be determined by examining the second derivative of the cost function. If the second derivative is positive, the cost function is concave up, and if it is negative, the cost function is concave down.

Q: What are some common applications of the marginal average cost in economics?

A: The marginal average cost has many applications in economics, including:

Production planning and control
Pricing strategies and revenue management
Cost-benefit analysis and decision-making
Economic forecasting and modeling

Conclusion

In this article, we answered some frequently asked questions related to the cost function and its derivatives. We hope that this Q&A article has provided you with a better understanding of the marginal average cost and its applications in economics.