Grades K-2, 3-5, 6-8, 9-12

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# Rates of Change

### Objective

Students will be able to:

- Define and compute marginal cost.
- Describe the relationship of marginal cost to the slope of the total cost and variable cost curves when those costs are plotted relative to the quantity of output produced.
- Describe the effect of increasing variable costs as fixed cost remains constant.
- Explain the difference between and the uses of marginal cost and average cost.
- Define and compute average cost relative to the quantity of output.
- Explain that Marginal Product (MP) and Marginal Cost (MC) are inversely related.
- Explain that the ratio of the total cost and the quantity of output produced is average cost of production.

### Concepts

In this personal finance lesson, students will analyze the relationship between differing costs using the concept of slopes.

### Resources

- Activity 1- Exploring Average Cost and Marginal Cost, one copy per student
- Activity 1 Answer Key- Exploring Average Cost and Marginal Cost
**,**one copy for the teacher - Rates of Change Powerpoint | PDF File
- One Ream of paper or scrap paper
- Stapler
- 3-hole punch
- Two to three desks arranged end-to-end at the front of the classroom

### Procedure

Profit maximizing firms use marginal analysis to determine whether employing an extra resource is feasible. At its most basic, Marginal Cost (MC) is simply a measure of the rate of change between the Total Cost (TC) and the Quantity of output produced (Q). Using the concept of cost, this lesson explores and connects the concepts of slope, costs, and revenue from an economics point of view.

The lesson uses MC as context for the underlying meaning of the slope as rate of change between two points on the TC curve. Students will calculate sums and differences of costs, the rate of change between two points, and contrast these with averages such as Average Variable Cost (AVC) and Average Cost (AC). By comparing and contrasting Average Cost and Marginal Cost the students will understand the difference between finding average versus finding the rate of change, which economists call marginal.

- Play the video clip “Lucy’s Famous Chocolate Scene” After the video clip discuss:
- What was the problem in the video? [
*The workers couldn’t keep up with the number of candies needing to be wrapped.*] - What could have helped them solve the problem? [
*Add more workers*.] - How many workers should be hired? [
*Answers will vary*.]

- What was the problem in the video? [
- Create the following chart on the board:

Round | Number of Workers | Quantity of Output |

- Explain that students will participate in a simulation. Describe the roles students will play.
- Quality Controller- This student (or small team of students) assesses the quality of the output produced. The definition of acceptable quality should be set after a practice round of production takes place so all of the students are familiar with the standards. This student must observe the number of workers and their collective acceptable output as each additional worker is added to the production process.
- Data Recorder- This student records what is reported by the Quality Controller on the chart on the board.
- Workers- These students will put together hole-punched and stapled packets from the papers supplied. After each 30-second round one more student will be added to the team.Note to teacher: continue with 30-second rounds (adding a worker in each round) until the Marginal Product declines significantly. It is not necessary to have the Marginal Product approach zero or a negative value. Eventually this will happen, assuming there are enough workers using a fixed amount of capital goods (e.g., tools). The number of workers (rounds) will vary. Typically marginal product begins to decline after five to eight rounds.
- Timekeeper- This student times each 30-second round. The workers are not allowed to start production until instructed by the timekeeper and workers must cease production when instructed by the timekeeper.

- Demonstrate how the papers should be stapled and hole-punched using these steps.
- Put two sheets of paper together.
- Hole-punch the two papers.
- Staple the two sheets together. Staple must be in the upper left corner above the punched holes.

- After the demonstration, explain that the goal is to make as many properly completed sets of stapled and hole-punched paper sets in each 30-second round as possible. Select students for each of the roles. Arrange the tables in the front of the room. Point out that the tables or desks represent a factory. Tell the timekeeper to begin round one.
- At the end of each round, have the Data Recorder record the information from the round on the chart on the board and ask students to think about the trend that they are noticing about total output as the number of workers increases.
- Have students look at the output data in the table and ask them what happened as more workers were added.

[*In early rounds, students should note that output was increasing at an increasing rate. Eventually, they should note that output is increasing at a decreasing rate*.] - Explain that Marginal Product (MP) is the additional output produced by the added worker. Add a MP column to the table on the board. Have students compute the MP for each worker. Ask the students what happened to MP.
*[First it increased and then it began to decrease.]*Discuss the following:- What caused this change in output?
*[At first, having more workers may increase efficiency because the workers help one another and specialize. Eventually, there won’t be enough space, equipment (stapler and hole-punch) or tasks to keep all the workers busy*.*]* - When do you think the company should stop hiring workers on the assembly line?
*[When the MP begins to decline; when there isn’t enough equipment for each worker.]*Note to teacher: student responses may include that hiring more workers does not always mean more profit. The students may erroneously suggest that diminishing marginal product diminishes the firm’s profit; although, this is possible over some ranges of workers, virtually all profit-maximizing firms produce in the range where marginal product is diminishing.

- What caused this change in output?
- Show Slide 1. Define the different types of productive costs that firms face when producing a good or a service as follows:
**Fixed Costs (FC)**are the costs associated with a business whether or not it produces a product. An example of a fixed cost would be leasing a building (e.g., a factory or a storefront). A business/firm must pay this cost whether or not they produce any goods or services.**Variable Costs (VC)**are the costs associated with inputs that can be increased (or decreased) by the owner or manager of the firm. An example of a variable cost is deciding the number of employees to hire.

- Ask students to give examples of FC and VC in context of the classroom production activity. [
*Students should respond that the fixed costs are the costs of the tables or desks which represented the factory and variable costs are the costs of hiring employees and buying production materials*.] - Display Slide 2. This slide illustrates how fixed and variable costs are used to calculate
**Total Cost (TC)**and**Average Cost (AC)**. Explain the following:- TC is the sum of fixed and variable costs (FC+VC=TC); it is a broad measure that gives an overall cost picture for the firm.
- AC gives a business an idea of its per unit cost of production; it is calculated by taking the firm’s TC and dividing by its corresponding level of output or Q (AC=TC/Q). Average cost is a ratio of the cost associated with the number of items produced.
- A
**ratio**is the quantitative relation between two numbers showing the number of times one value contains or is contained within the other. A ratio shows the relative sizes of two or more values.

- Display Slide 3. Review the concepts of marginal cost and revenue.
- Display Slide 4. Define the formula for calculating
**Marginal Cost (MC).**Marginal cost is the cost that is incurred by the production of an additional unit of output. The marginal cost of an additional unit of output is the cost of the additional inputs needed to produce that output. Marginal cost is calculated by dividing the change in TC by the change in output or Q (MC = ΔTC/ΔQ). Explain that measuring MC allows a firm to know the cost associated with producing an additional unit of output. - Display Slide 5. Remind students what rate of change means and define the slope of a line as the rate of change between two points on the line. Write the equations on the board if needed. Discuss the idea of slope. Remind students that the slope of a linear equation is the (horizontal change) / (vertical change), which is sometimes called rise / run or (y2-y1) / (x2-x1). If necessary, plot points (1,2) and (5,6) on the board and ask students to draw the line and find the slope of the line.
- Distribute a copy of Activity 1 to each student and discuss the following:
- What are fixed costs?
*[The costs associated with a business whether or not it produces a product.]* - What are the fixed costs for this firm?
*[$100]* - What are variable costs?
*[The costs associated with inputs that can be increased or decreased.]* - Why are the variable costs for this firm zero when the firm produced no output?
*[Because the firm is not buying any inputs, such as flour and sugar or hiring any workers*.*]* - What are total costs?
*[The sum of fixed and variable costs.]*

- What are fixed costs?
- Divide students into pairs. Allow time for students to complete Activity 1. Circulate the room to help pairs complete the task by citing equations and information from the slides.
- After students have completed their work
**,**discuss the following:- What happens to output as workers are hired?
*[As more workers are hired, output increases*.*]* - What happens to total costs as the firm moves from zero workers to ten workers?
*[Total cost increases.]* - What happens to average costs as the firm hires more workers?
*[Average cost decreases for workers one through seven and then begins to increase with worker eight.]* - What happens to marginal cost as the firm hires more workers?
*[Marginal cost decreases for workers one through four. With the addition of worker five, marginal cost begins to increase*.*]*

- What happens to output as workers are hired?
- Explain that firms maximize or earn the greatest profit when the marginal cost of producing the next unit is equal to the marginal revenue or the extra revenue the firm earns from selling an additional unit of a product. Tell the students that the price per cookie is $5. Point out that this means the marginal revenue (the revenue earned from the sale of an additional cookie) is $5 (the price of the cookie.). Tell students to add a column to the end of the table and label the column Profit. Tell them to calculate profit earned with each worker using a price of $5 and the total costs for that number of workers.
*[$60, $5, $95, $210, $300, $365, $405, $420, $420, $415]*In addition, it is worth noting that the greatest profit of $420 occurs twice. This is because the TC cost function is not continuous (e.g., it jumps from 20 to 22 cookies). If this is an issue, ask the students what would happen if the price of cookies was $5.01. They should be able to recognize that producing 22 cookies is profit maximizing. Emphasize this Marginal Revenue=Marginal Cost (i.e., MR = MC) rule for maximizing profit. - Point out that the firm will produce an extra unit of output as long as the extra revenue from producing and selling it is greater than or equal to the extra cost of it. Continue the discussion by reviewing the questions on Activity 1 using the answer key provided.
- (Optional) Ask the students if the same principle (i.e., using marginal analysis) would apply to other things such as hiring workers. Explain that the firm should hire an extra worker as long as the extra cost of hiring the worker is less than or equal to the extra revenue earned from the sale of the extra cookies the worker produces.

- Discuss the following to review the key points of the lesson:
- What can companies learn by looking at average cost and marginal cost?
*[Companies can use average cost to better understand the profit (or loss) they earn on a typical unit of output; they do this by comparing Average Cost with the Average Revenue they earn by selling a specific Quantity of output. Marginal Cost illustrates the impact of adding an additional employee, or input, upon the costs of production*.*Companies can use this information to make production and hiring decisions.]* - What will happen to output when more workers are added to a fixed amount of space?
*[Eventually individual workers will not have enough space to work as effectively as they did and Marginal Product will decrease*.*]* - What are the mathematical concepts behind Average Cost and Marginal Cost in terms of the company’s production data?
*[Average cost is a ratio of the cost associated with the number of items produced and marginal cost is a rate of change of cost associated with the rate of change of items produced.]* - Ask students if they have worked at or seen the results of a local business mismanaging staffing and capital levels. Be sure to call upon students who work after school and use their personal experiences as an example.
*[Some companies might not hire enough workers (or they may hire too many workers) and so they do not maximize their production and companies might hire too many workers to maximize profit.]*

- What can companies learn by looking at average cost and marginal cost?

### Assessment

- What level of output should a firm produce in order to maximize profit?
- At the output where the MC equals the AC.
*[At the output where MC equals MR.]*- At the output where MC equals AVC.
- At the output where TC equals TR.

- What is the difference between marginal cost and average cost?
- Marginal cost is the sum of costs while average is a per unit measure.
- Marginal cost is the difference in costs while average cost measures overall costs.
*[Average Cost is a per unit measure of cost while marginal cost is a comparison of the change in total costs divided by the change in output.]*- Average Cost is the difference between marginal costs and fixed costs while marginal costs are the product of total cost and quantity.

- Which of the following describes marginal cost?
- Sum the variable and fixed costs and divide by the total amount produced.
*[The extra cost associated with producing one more unit of output.]*- Sum the variable costs and divide by the total amount produced.
- Divide the fixed costs by the output produced.

- Short Answer. Why should a company consider using marginal cost instead of average cost when deciding how much to produce?

*[The firm maximizes profits when it produces the output at which MC=MR, or, when it hires workers up to the point that the extra cost of the output the worker produces (Change in TC/Change in Output) equals the additional revenue from the sale of the additional output.]*

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