Grade by Grade
B. Derivatives and Their Applications · Applications of Derivatives: Motion, Rates and Optimization

Marginal Cost, Revenue and Profit

B2.2 B2.3About 65 minutes
Learning goals
  • I can build revenue and profit functions from a price (demand) function and a cost function.
  • I can find and interpret marginal cost, marginal revenue and marginal profit.
  • I can compare marginal cost with average cost.
  • I can explain why the derivative estimates the cost of one more item.

Words to know

cost function C(x)
The total cost of producing x items.
demand (price) function p(x)
The price at which x items will be sold.
revenue R(x)
Money taken in: R(x) = x × p(x).
profit P(x)
P(x) = R(x) − C(x).
marginal cost
C′(x): approximately the cost of producing one more item.
average cost
A(x) = C(x)/x: the cost per item.

What does one more cost?

A maple syrup producer already makes 200 litres a week. A restaurant asks for one more litre. What will it cost to make it? Economists call that the marginal cost, and it is (very nearly) a derivative.

In this lesson you will use derivatives to analyse cost, revenue and profit. Every business course in university, from first-year economics on, uses these ideas, and they set up the business optimization problems later in this unit.

The business functions

Let x be the number of items made and sold.

  • Cost C(x): the total cost of making x items. It usually has a fixed cost C(0) (rent, equipment) plus costs that grow with x.
  • Price (or demand) function p(x): the price at which exactly x items will sell. Usually it decreases: to sell more, you lower the price.
  • Revenue R(x) = x × p(x): the money taken in.
  • Profit P(x) = R(x) − C(x).
  • Average cost A(x) = C(x) ÷ x: the cost per item.

The marginal cost, revenue and profit are the derivatives C′(x), R′(x) and P′(x). They tell you about how much each quantity changes when you make and sell one more item, because

C(x + 1) − C(x) ≈ C′(x) × 1.

Worked example

Example 1: Marginal cost

A small sugar bush has weekly cost C(x) = 2000 + 15x + 0.01x² dollars for x litres of syrup. (a) Find the marginal cost at x = 200 and interpret it. (b) Compare with the actual cost of the 201st litre. (c) Find the average cost at x = 200.

  1. (a) C′(x) = 15 + 0.02x, so C′(200) = 15 + 4 = $19 per litre. When 200 L are being made, one more litre costs about $19.
  2. (b) C(201) − C(200) = [2000 + 3015 + 404.01] − [2000 + 3000 + 400] = 5419.01 − 5400 = $19.01. Very close.
  3. (c) A(200) = C(200) ÷ 200 = 5400 ÷ 200 = $27 per litre.

Marginal cost $19 per litre (the 201st litre actually costs $19.01). Average cost $27 per litre: the fixed $2000 is spread over every litre, so the average is higher than the marginal.

200400600500010 00015 000x (litres)C ($)(200, 5400)tangent slope C′(200) = 19Zoom near x = 200one more litre (Δx = 1)ΔCcurve: ΔC = 19.01tangent: 19.00
The cost curve with its tangent at x = 200. Zoomed in, one more litre raises the cost by ΔC = 19.01 along the curve, almost exactly the tangent's rise of 19. (The zoom is not to scale.)
Worked example

Example 2: Marginal revenue

A concert promoter finds that x tickets sell when the price is p(x) = 50 − 0.02x dollars. (a) Find R(x) and R′(x). (b) Find R′(1000) and R′(1500), and interpret each.

  1. (a) R(x) = x(50 − 0.02x) = 50x − 0.02x². R′(x) = 50 − 0.04x.
  2. (b) R′(1000) = 50 − 40 = $10 per ticket. At 1000 tickets, selling one more (at a slightly lower price) raises revenue by about $10.
  3. R′(1500) = 50 − 60 = −$10 per ticket. At 1500 tickets, selling one more lowers revenue by about $10: the price cut on every ticket costs more than the extra sale brings in.

R′(1000) = $10 per ticket (revenue rising); R′(1500) = −$10 per ticket (revenue falling). Revenue is largest where R′(x) = 0, at x = 1250.

500100015002000250010 00020 00030 000R ($)x (tickets)R′ = 0 at x = 1250R′(x) > 0: selling more raises revenueR′(x) < 0: revenue falls
Revenue R(x) = 50x − 0.02x². Marginal revenue is positive (green band) until x = 1250 and negative (yellow band) after, where selling more tickets at a lower price brings in less money.
Worked example

Example 3: Marginal profit

For the concert, the costs are C(x) = 8000 + 12x dollars. Find the profit function and the marginal profit at 800 tickets and at 1100 tickets.

  1. P(x) = R(x) − C(x) = 50x − 0.02x² − 8000 − 12x = −0.02x² + 38x − 8000.
  2. P′(x) = −0.04x + 38.
  3. P′(800) = −32 + 38 = $6 per ticket. P′(1100) = −44 + 38 = −$6 per ticket.
  4. Marginal profit is marginal revenue minus marginal cost: R′(x) − C′(x) = (50 − 0.04x) − 12, the same thing.

At 800 tickets, profit is still rising (about $6 per extra ticket); at 1100 tickets it is falling (about −$6 per ticket). Somewhere between, at x = 950, profit is largest: you will find maximums like this in the optimization lessons.

Worked example

Example 4: Is the average cost falling?

For the sugar bush, C(x) = 2000 + 15x + 0.01x². Is the average cost falling or rising at x = 300?

  1. A(x) = C(x)/x = 2000x⁻¹ + 15 + 0.01x.
  2. A′(x) = −2000x⁻² + 0.01.
  3. A′(300) = −2000/90 000 + 0.01 ≈ −0.0222 + 0.01 = −0.0122.
  4. Compare: C′(300) = 15 + 6 = $21 and A(300) = 7400/300 ≈ $24.67. The marginal cost is below the average, so each new litre pulls the average down.

A′(300) ≈ −0.012 dollars per litre per litre: the average cost is still falling at 300 L.

Marginal versus average

If the next item costs less than the current average, making it pulls the average down. If it costs more, the average goes up. So the average cost is lowest where marginal cost = average cost. (It is like a class average: a mark above the average raises it.)

Just for fun

Economists love the word marginal so much that a slightly better cup of coffee is called a "marginal improvement". Calculus students just call it a small Δ.

Hands-on activity

Bake-sale pricing survey

You need: paper, a pencil, a calculator, and 4 to 6 family members or friends.

  1. Pretend you will sell homemade cookies. Ask each person: "How many cookies would you buy this week at $0.50 each? At $1.00? At $1.50? At $2.00?"
  2. Add up the answers at each price to get total demand x.
  3. Make a table with columns price p, quantity x and revenue R = px.
  4. Plot p against x and draw a straight line that fits roughly. Find its equation p(x) = a − bx.
  5. Write R(x) = x(a − bx), find R′(x), and find the number of cookies (and the price) where R′(x) = 0.

What to look for: demand should go down as price goes up. The revenue usually rises and then falls, and your R′(x) = 0 point should be near the best revenue in your table. Real businesses gather demand data the same way, with surveys and price tests.

Your turn

1

C(x) = 500 + 8x + 0.05x². Find the marginal cost at x = 40 and interpret it.

2

For the same C(x), find the exact cost of the 41st item.

3

A demand function is p(x) = 30 − 0.5x. Find R(x) and R′(x).

4

For R(x) = 30x − 0.5x², find R′(20) and R′(40). What do the signs mean?

5

R(x) = 40x − 0.1x² and C(x) = 600 + 10x. Find the marginal profit at x = 100.

6

C(x) = 900 + 6x + 0.04x². Find the average cost at x = 150.

7

For the same cost, is the average cost rising or falling at x = 150? Use marginal cost.

More practice: warm-up

1

Write the revenue if x items sell at price p each.

2

Profit equals revenue minus…

3

C(x) = 1200 + 25x. What is the marginal cost?

4

C(x) = 1200 + 25x. What is the fixed cost?

5

R′(x) < 0 at some level of sales. What happens to revenue if you sell one more?

More practice: on your own

1

A company's cost is C(x) = 0.001x³ − 0.3x² + 40x + 1000. Find C′(100) and C′(200), and say where marginal cost is lowest.

2

Demand for a bike rental is p(x) = 60 − 0.4x dollars per day, for x rentals. At what number of rentals is marginal revenue zero, and what is the price then?

3

C(x) = 3000 + 20x + 0.03x². Find A′(x) and show the average cost is falling at x = 200.

4

A tour company has R(x) = 120x − 0.5x² and C(x) = 2000 + 40x. Find P′(x) and the sign of P′ at x = 70 and x = 90.

5

Explain why marginal cost is the slope of the tangent to the cost curve, while average cost is the slope of the line from the origin to the point on the cost curve.

Challenge

1

Prove that if A(x) = C(x)/x has a minimum where A′(x) = 0, then C′(x) = A(x) there.

2

For C(x) = 2000 + 15x + 0.01x² (the sugar bush), find the production level that minimizes average cost, and the minimum average cost.

3

A demand function is p(x) = 80e^(−0.01x). Find the marginal revenue and the number of items for which it is zero.

Check your understanding

Quiz

1.

C(x) = 400 + 9x + 0.02x². What is the marginal cost at x = 50?

2.

Marginal revenue is negative at a certain sales level. What does that mean?

3.

For a demand function p(x) = 20 − 0.1x, what is R(x)?

4.

If the marginal cost is less than the average cost, then making one more item…

5.

Which expression gives the marginal profit?

6.

C(x) = 1000 + 5x. What is the average cost of 200 items?

For parents and teachers

Big idea (B2.2, B2.3): in business, "marginal" means "derivative". Marginal cost, revenue and profit estimate the effect of making and selling one more item, and comparing marginal cost with average cost shows whether costs per item are rising or falling.

How to help: use a real example: if your family runs a garage sale or a small business, ask your teen what the "fixed costs" and "per-item costs" are, and whether lowering a price could raise or lower revenue.

Common difficulties:

  • Confusing the price function p(x) with revenue R(x) = x·p(x).
  • Thinking negative marginal revenue means the business loses money.
  • Mixing up average cost C(x)/x with marginal cost C′(x).
  • Forgetting the fixed cost when computing average cost.