Grades 35, 68
Concepts
This lesson creates a connection between derivatives and marginalism. Students will be engaging in a set of scaffolding activities that explore the Marginal Cost Function, Marginal Revenue Function, and the implications that these functions have on production. A short video clip will provide an example of why the intersection of Marginal Cost and Marginal Revenue yields maximum profit.
Time Required
90 minutes
Will Be Able To
 Define marginal, total, variable, and fixed costs.
 Describe the relationship between marginal cost and the slope of total cost (and/or variable cost) curve.
 Define marginal revenue.
 Describe the relationship between marginal revenue and total revenue.
 Define average fixed, variable, and total costs.
 Compare average total (or variable) cost with marginal cost.
 Use calculus to derive the production quantity at which marginal cost equals marginal revenue.
 Recognize that profit maximization (or loss minimization) occurs when marginal cost equals marginal revenue (unless the firm should close immediately).
 Use the definition of limit to connect marginal analysis with derivative functions.
 Determine the amount of profit (or loss) a firm would earn at the point at which marginal cost equals marginal revenue.
Materials

Activity 1, one copy per student

Answer Key for teacher

Answer Key for teacher

Activity 2, one copy per student

Answer Key for teacher

Answer Key for teacher

Activity 3, one copy per student

Answer Key for teacher

Answer Key for teacher

Activity 4, one copy per student

Answer Key for teacher

Answer Key for teacher

Activity 5, one copy per student

Answer Key for teacher

Answer Key for teacher

Deriving Marginalism presentation PowerPoint  pdf File

Graphing Calculators
 Access to YouTube
Assessment Activity

Why is the derivative of the cost function the most precise estimate of marginal cost? (Hint: because the limit approaches zero with the derivative.)

An analyst has estimated the total cost of producing x items for a certain company as C(x) = 1800 + 10x + 0.02x^{2} and the demand for producing x items for a certain company as p = 60 + 0.01x. Find the number of items the company should produce to maximize profit or minimize loss. How much will the company profit from producing such a quantity?

An analyst found that the company’s marginal cost is less than the marginal revenue associated with producing 5000 units of a good for the previous year. What implications does this data have for the company to maximize profit this year?

The company should produce less than 5,000 units.

The company should produce more than 5,000 units.

The company should produce 5,000 units.

There are no implications in regards to level of production.

The company should produce less than 5,000 units.

An analyst found that the company’s marginal cost is equal to the marginal revenue associated with producing 5000 units of a good. What implications does this information have for profit maximization?

The company should produce less than 5,000 units.

The company should produce more than 5,000 units.

The company should produce 5,000 units.

There are no implications in regards to level of production.

The company should produce less than 5,000 units.

An analyst found that the company’s marginal cost is more than the marginal revenue associated with producing 5000 units of a certain good. What implications does this information have for profit maximization?

The company should produce less than 5,000 units.

The company should produce more than 5,000 units.

The company should produce 5,000 units.
 There are no implications in regards to level of production.

The company should produce less than 5,000 units.
Conclusion

Review the important content in the lesson with the following questions:

What are fixed costs? [Costs that do not change with an increase or decrease in amount of goods or services produced.]

Give some examples of fixed costs. [The cost of leasing a building.]

Why do these costs remain fixed regardless of the quantity produced? [These are costs that it takes to run the company regardless of how many items the company produces.]

What are variable costs? [Expenses that vary with the level of output.]

Give some examples of variable costs. [Raw materials, shipping cost, labor, electric power.]

Why do these costs vary with the quantity produced? [The more goods that are produced, the more raw materials and electricity a company might use. The more a company produces the more it may ship.]

What are marginal costs? [The costs associated with producing one additional unit.]

What is the relationship between marginal cost and total cost? [Total cost takes into account the cost of producing all units. Marginal cost is the cost of producing the next additional unit. Marginal cost represents the additional total cost that results from producing one more unit of a good or service.]
 What is the relationship between total revenue and marginal revenue? [Total revenue takes into account all receipts from the sale of units produced. Marginal revenue is the revenue associated with the sale of an additional unit of production.]

What are fixed costs? [Costs that do not change with an increase or decrease in amount of goods or services produced.]
Overview
Profit maximizing firms use marginal analysis to determine whether to produce an additional unit of output or to employ an extra resource. In its most basic sense marginal cost is simply a measure of the rate of change between the total costs and the quantity of output (or in another context the amount of a variable input). Using the metric of cost this lesson explores the concept of slope from an economic point of view.
The lesson uses Marginal Costs and Marginal Revenue as context for the underlying meaning of the derivative function. Students make the connection between the difference quotient and marginalism leading to the application of derivatives to find marginal functions. Students also apply their findings to a firm’s marginal functions to decide how much the firm should produce to maximize profit.
Assessment

Display Slide 1. Ask students if they see any relationship between the columns, if there are any functions, and what they think each column might represent for a company. [Answers will vary.]

Display Slide 2 and distribute a copy of Activity 1 to each student. Refer students to Table A. Point out that the columns are labeled Fixed Cost, Variable Cost, and Total Cost. Give the definitions of each using the points below:

Fixed costs are costs that do not change with an increase or decrease in the amount of goods or services produced. An example of a fixed cost is a company’s lease on a building.

Variable costs are expenses that vary with the level of output. Typical examples of variable costs are raw materials, labor, and shipping cost.

Total costs are all costs associated with producing a good or service; the sum of total fixed costs plus total variable costs.

Fixed costs are costs that do not change with an increase or decrease in the amount of goods or services produced. An example of a fixed cost is a company’s lease on a building.

Discuss the following:

Why do variable costs change as the quantity of goods produced changes? [Producing more requires using more variable inputs such as raw materials.]

Why don’t fixed costs change as the quantity of goods produced changes? [The company is responsible for these expenses regardless of how much they produce]

Why do variable costs change as the quantity of goods produced changes? [Producing more requires using more variable inputs such as raw materials.]
 Display Slide 3. Direct students attention to Table B on Activity 1. Model for the class the process of finding the cost of producing one additional item for the first two rows in the table for the entire class.
Ex. for production of 1^{st} item.
Ex. for production of 2^{nd} item.

Assign students to groups of three or four. Tell group members to work together and determine the costs for each additional item produced in Table B.

Display Slide 4 and review student answers. Tell students that this column is called Marginal Cost. Display Slide 5 and review the definition of marginal cost. Marginal cost is the increase in a producer’s total cost when output is increased by one unit.

Discuss the following:

What are the implications of the new data? [The EXTRA cost of producing one more unit of a good initially decreases and then may increase.]

Why is it important for a business to analyze marginal cost? [To decide when the best time is to stop producing goods.]

Does the marginal cost resemble the first derivative of the total cost function? [It appears to be slope of the curve at that particular point, which essentially is the derivative of the function.]

What are the implications of the new data? [The EXTRA cost of producing one more unit of a good initially decreases and then may increase.]

Display Slide 6 Marginal Cost Function as a Derivative of C(x). Revisit the main points of the previous activity and emphasize that the Marginal Cost function is the derivative of the Total (or Variable) Cost function. Display Slide 7 and review the alternative marginal cost function expositions.

Revisit slope as a rate of change and connect that to the rate of change of price per single increase in unit of production.

Revisit slope as a rate of change and connect that to the rate of change of price per single increase in unit of production.

Distribute a copy of Activity 2 to each student. Have students complete Activity 2. Review student answers using Activity 2: Answer Key.

Display Slide 8, and define total revenue and marginal revenue.

Why is it that the total revenue as a function of the quantity sold is often not a linear function? [Almost always, the quantity demanded varies inversely with the price of a good or service; the result is a nonlinear total revenue function. This is easy to confirm with an example. Total revenue {TR} is linear in veryfew special cases; one special case is called pure competition, which is for firms like farmers who are forced to accept or reject the market price. For example, if the price of corn is $4/bushel, then if the farmer sells 1 bushel, TR is $4, and if the farmer sells 100 bushels, the TR is $400. This creates a linear total revenue line. In almost all other cases, the total revenue is nonlinear. This is because most firms must reduce the price of a product to sell a larger quantity, which is exemplified by a (typical) negatively sloped demand curve.]

Discuss demand function p = f(x) where p denotes the unit price as a function of the quantity demanded. (This is important for students to recognize that the variation in the quantity demanded is inversely related to the price of an item.)

Explain that the price of an item can be translated as a function of the quantity produced. If the supply of a given product increases with no change in the demand, a temporary surplus will emerge and the price of the item will decrease.

Discuss R(x) = px. The revenue generated by the company is the selling price of the item multiplied by the quantity sold.

Why is it that the total revenue as a function of the quantity sold is often not a linear function? [Almost always, the quantity demanded varies inversely with the price of a good or service; the result is a nonlinear total revenue function. This is easy to confirm with an example. Total revenue {TR} is linear in veryfew special cases; one special case is called pure competition, which is for firms like farmers who are forced to accept or reject the market price. For example, if the price of corn is $4/bushel, then if the farmer sells 1 bushel, TR is $4, and if the farmer sells 100 bushels, the TR is $400. This creates a linear total revenue line. In almost all other cases, the total revenue is nonlinear. This is because most firms must reduce the price of a product to sell a larger quantity, which is exemplified by a (typical) negatively sloped demand curve.]

Distribute a copy of Activity 3 to each student. Have students work through the activity. Review student answers using Activity 3: Answer Key.

Show video: “How Much to Produce”
www.youtube.com/watch?v=WGEZAthjCk
Stop the video at 1:31 — Ask students to think about what implications marginal revenue and marginal cost have in relation to how many apples a farmer should pick.

Review questions about the video:

Why are there varying costs of harvest for different amounts of apples? [The apples at the bottom are cheaper to pick because it requires less resources; the marginal revenue is greater than the marginal cost.]

Which apples should farmers pick? [The apples at the bottom and in the middle because they are cheaper to pick.]

Why wouldn't the farmer pick apples at the top of the tree? [His marginal cost is greater than his marginal revenue.]

How does a farmer decide how many apples to pick to maximize his profit or minimize his loss? [Farm workers should pick apples until the marginal cost of picking the next apple is equal to the marginal revenue for selling that apple.]

Why are there varying costs of harvest for different amounts of apples? [The apples at the bottom are cheaper to pick because it requires less resources; the marginal revenue is greater than the marginal cost.]

Display Slide 9: Profit Function, and explain that this is what it looks like when MC = MR. Discuss that this is when a company should not expand production because this is where its profit is maximized (or loss is minimized).

Distribute a copy of Activity 4, one copy per student, and have them work through the problems. Review student answers using Activity 4: Answer Key.
 Distribute a copy of Activity 5 to each student and have them work through the problems. Review answers with Activity 5: Answer Key.
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