Profit
Maximization

Learn how firms find their optimal output level where $ MR = MC $, and understand the logic behind the most fundamental decision in microeconomics.

AP Microeconomics

What Is Economic Profit?

In economics, profit is not just the money left over after paying bills. It accounts for all opportunity costs: the implicit costs of the resources an owner contributes.

Total Revenue

$ TR = P \times Q $: the total income from selling output.

Total Cost

$ TC = TFC + TVC $: all explicit and implicit costs of production.

Economic Profit

$ \pi = TR - TC $: revenue minus all costs, including opportunity costs.

Marginal Revenue and Marginal Cost

The key to profit maximization lies in understanding what happens when a firm produces one more unit. Two marginal concepts drive the decision:

Marginal Revenue (MR)

$ MR = \frac{\Delta TR}{\Delta Q} $: the additional revenue from selling one more unit.

Marginal Cost (MC)

$ MC = \frac{\Delta TC}{\Delta Q} $: the additional cost of producing one more unit.

Visualizing $ TR $ and $ TC $

The $ MR = MC $ Rule

The Formal Rule

A firm maximizes profit when it produces the quantity $ Q^* $ satisfying two conditions:

1

First-order condition: $ MR(Q^*) = MC(Q^*) $. The additional revenue from the last unit equals its additional cost.

2

Second-order condition: $ MC $ is rising at $ Q^* $ (the $ MC $ curve cuts the $ MR $ curve from below). This ensures a maximum, not a minimum.

Perfect Competition vs. Monopoly

	Perfect Competition	Monopoly
Price taker?	Yes: $ P $ is given by the market	No: firm sets the price
Demand curve	Perfectly elastic (horizontal)	Downward-sloping
$ MR $ relationship	$ MR = P $ (constant)	$ MR < P $ (falls faster than $ P $)
Profit-max rule	$ P = MC $	$ MR = MC $, then set $ P $ from demand
Long-run profit	Zero economic profit	Can sustain positive profit

The Shut-Down Decision

Even if $ MR = MC $ yields a loss, the firm may still produce in the short run: it depends on whether revenue covers variable costs.

Keep Producing

If $ P \geq AVC $: revenue covers variable costs and contributes to fixed costs. Loss is smaller than shutting down.

Shut Down

If $ P < AVC $: revenue doesn't even cover variable costs. The firm loses less by producing zero.

How to Find $ Q^* $: Step by Step

1

Identify $ MR $: In perfect competition, $ MR = P $. For a monopolist, derive $ MR $ from the demand function.

2

Identify $ MC $: Differentiate $ TC(Q) $ or read marginal cost from a table.

3

Set $ MR = MC $: Solve for $ Q^* $. If using a table, find where $ MR $ is closest to $ MC $ without $ MC $ exceeding $ MR $.

4

Check shut-down rule: Verify $ P \geq AVC $ at $ Q^* $. If not, optimal output is zero.

5

Calculate profit: $ \pi = (P - ATC) \times Q^* $. If $ P > ATC $, the firm earns positive economic profit.

Example 1: Perfect Competition

A firm in a competitive market faces $ P = \$12 $. Its cost function is $ TC = Q^3 - 6Q^2 + 15Q + 10 $. Find the profit-maximizing output.

1

$ MR = P = 12 $ (price taker).

2

$ MC = \frac{dTC}{dQ} = 3Q^2 - 12Q + 15 $.

3

Set $ MR = MC $: $ 12 = 3Q^2 - 12Q + 15 $ $\rightarrow$ $ 3Q^2 - 12Q + 3 = 0 $ $\rightarrow$ $ Q^2 - 4Q + 1 = 0 $.

4

Using the quadratic formula: $ Q = \frac{4 \pm \sqrt{16 - 4}}{2} = 2 \pm \sqrt{3} $. So $ Q \approx 3.73 $ or $ Q \approx 0.27 $.

Example 1: Profit Graph

Example 2: Using a Data Table

A competitive firm faces $ P = \$8 $. Use the table below to find profit-maximizing output.

Q	TR	TC	MR	MC	Profit
0	$0	$10	—	—	−$10
1	$8	$14	$8	$4	−$6
2	$16	$17	$8	$3	−$1
3	$24	$21	$8	$4	$3
4	$32	$27	$8	$6	$5
5	$40	$35	$8	$8	$5
6	$48	$45	$8	$10	$3

Example 3: Monopolist Pricing

A monopolist faces demand $ P = 50 - 2Q $ and has $ TC = 2Q^2 + 10Q + 20 $. Find the profit-maximizing price and quantity.

1

$ TR = P \cdot Q = (50 - 2Q)Q = 50Q - 2Q^2 $.

2

$ MR = \frac{dTR}{dQ} = 50 - 4Q $.

3

$ MC = \frac{dTC}{dQ} = 4Q + 10 $.

4

Set $ MR = MC $: $ 50 - 4Q = 4Q + 10 $ $\rightarrow$ $ 40 = 8Q $ $\rightarrow$ $ Q^* = 5 $.

5

Price from demand: $ P = 50 - 2(5) = \$40 $. Profit: $ \pi = TR - TC = 200 - 120 = $ $ \$80 $.

Example 3: Monopolist Diagram

Common Mistakes to Avoid

\u2715 Maximizing Revenue Instead

Producing where $ MR = 0 $ maximizes total revenue, not profit. Profit requires $ MR = MC $.

\u2715 Ignoring the Second-Order Condition

Choosing a quantity where $ MC $ is falling at the intersection gives a profit minimum, not maximum.

\u2715 Reading Price from MR (Monopoly)

For a monopolist, the price comes from the demand curve, not the $ MR $ curve. $ P > MR $ at $ Q^* $.

Key Concepts Summary

Concept	Rule / Formula	Key Insight
Profit-max output	$ MR = MC $	Applies to all market structures
Competitive firm price	$ P = MR = MC $	Price is given by the market
Monopolist price	Find $ Q^* $, read $ P $ from demand	$ P > MR $ at $ Q^* $
Shut-down rule (SR)	Produce if $ P \geq AVC $	Compare to variable costs only
Economic profit	$ \pi = (P - ATC) \times Q $	Includes all opportunity costs
Long-run exit	Exit if $ P < ATC $	No fixed costs in the long run

Question 1 of 5

A profit-maximizing firm should produce the quantity where:

The profit-maximizing rule requires $ MR = MC $ with $ MC $ rising at the intersection (second-order condition). Maximizing $ TR $ (where $ MR = 0 $) or minimizing $ ATC $ are different objectives entirely.

Question 2 of 5

A competitive firm has $ P = \$10 $, $ ATC = \$8 $, and $ AVC = \$6 $ at its profit-maximizing output of 100 units. What is the firm's economic profit?

Economic profit $ \pi = (P - ATC) \times Q = (\$10 - \$8) \times 100 = \$200 $. The $ AVC $ figure confirms the firm should stay open ($ P > AVC $), but profit is calculated using $ ATC $.

Question 3 of 5

A monopolist faces demand $ P = 100 - 5Q $. What is the marginal revenue function?

For a linear demand $ P = a - bQ $, total revenue is $ TR = aQ - bQ^2 $, so $ MR = a - 2bQ $. Here: $ MR = 100 - 2(5)Q = 100 - 10Q $. The $ MR $ curve has twice the slope of the demand curve.

Question 4 of 5

A firm finds that at its current output, $ MR = \$15 $ and $ MC = \$20 $. To maximize profit, the firm should:

When $ MC > MR $, each additional unit costs more than it earns. The firm is producing too much and should reduce output until $ MR = MC $. Shutting down requires checking if $ P < AVC $, which we don't know from this information alone.

Question 5 of 5

A competitive firm has $ P = \$5 $, $ ATC = \$7 $, and $ AVC = \$4 $ at its profit-maximizing output. In the short run, this firm should:

Since $ P = \$5 > AVC = \$4 $, the firm covers all variable costs and contributes $ \$1 $ per unit toward fixed costs. Shutting down would mean losing all fixed costs. The firm should continue operating in the short run despite the loss. A competitive firm cannot raise its price (it's a price taker).

Profit
Maximization

What Is Economic Profit?

Marginal Revenue and Marginal Cost

Visualizing \( TR \) and \( TC \)

The \( MR = MC \) Rule

The Formal Rule

Perfect Competition vs. Monopoly

The Shut-Down Decision

How to Find \( Q^* \): Step by Step

Example 1: Perfect Competition

Example 1: Profit Graph

Example 2: Using a Data Table

Example 3: Monopolist Pricing

Example 3: Monopolist Diagram

Common Mistakes to Avoid

Key Concepts Summary

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

	Perfect Competition	Monopoly
Price taker?	Yes: \( P \) is given by the market	No: firm sets the price
Demand curve	Perfectly elastic (horizontal)	Downward-sloping
\( MR \) relationship	\( MR = P \) (constant)	\( MR < P \) (falls faster than \( P \))
Profit-max rule	\( P = MC \)	\( MR = MC \), then set \( P \) from demand
Long-run profit	Zero economic profit	Can sustain positive profit

Concept	Rule / Formula	Key Insight
Profit-max output	\( MR = MC \)	Applies to all market structures
Competitive firm price	\( P = MR = MC \)	Price is given by the market
Monopolist price	Find \( Q^* \), read \( P \) from demand	\( P > MR \) at \( Q^* \)
Shut-down rule (SR)	Produce if \( P \geq AVC \)	Compare to variable costs only
Economic profit	\( \pi = (P - ATC) \times Q \)	Includes all opportunity costs
Long-run exit	Exit if \( P < ATC \)	No fixed costs in the long run

ProfitMaximization

What Is Economic Profit?

Marginal Revenue and Marginal Cost

Visualizing \( TR \) and \( TC \)

The \( MR = MC \) Rule

The Formal Rule

Perfect Competition vs. Monopoly

The Shut-Down Decision

How to Find \( Q^* \): Step by Step

Example 1: Perfect Competition

Example 1: Profit Graph

Example 2: Using a Data Table

Example 3: Monopolist Pricing

Example 3: Monopolist Diagram

Common Mistakes to Avoid

Key Concepts Summary

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Profit
Maximization