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Profit Maximum

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About This MicroSim

This MicroSim is the natural "Part 2" following the Revenue Maximum simulation. While revenue maximization finds the price that generates the most total income, profit maximization finds the price that generates the most money after costs are subtracted.

Key Insight

The profit-maximizing price is higher than the revenue-maximizing price whenever production has costs. This is because selling fewer units at a higher margin can be more profitable than selling many units at a low margin.

Two-Panel Design

Left Panel: Demand Curve with Profit Rectangle

  • The diagonal line shows the demand curve (higher prices → fewer sales)
  • The green rectangle shows profit when Price > Marginal Cost
  • The red rectangle shows loss when Price < Marginal Cost
  • The orange dashed line shows the Marginal Cost level

Right Panel: Revenue vs Profit Curves

  • Blue curve: Total Revenue (same parabola as Revenue Maximum)
  • Green curve: Total Profit (revenue minus costs)
  • Blue vertical line: Revenue-maximizing price (always $100)
  • Green vertical line: Profit-maximizing price (shifts with marginal cost)

How to Use

  1. Adjust the Price slider to explore different selling prices
  2. Adjust the Marginal Cost slider to see how production costs affect optimal pricing
  3. Toggle Show Revenue Curve to compare revenue and profit directly
  4. Watch how the green profit-max line moves right as marginal cost increases

The Key Formula

\[P^* = 100 + \frac{MC}{2}\]

Where \(P^*\) is the profit-maximizing price and \(MC\) is the marginal cost per unit.

Marginal Cost Revenue-Max Price Profit-Max Price Difference
$0 $100 $100 $0
$20 $100 $110 $10
$40 $100 $120 $20
$60 $100 $130 $30

Lesson Plan

Learning Objectives

By the end of this lesson, students will be able to:

  1. Distinguish between revenue and profit as business metrics
  2. Calculate profit using the formula: Profit = Revenue - Total Cost
  3. Explain why profit-maximizing price differs from revenue-maximizing price
  4. Predict how changes in marginal cost affect optimal pricing
  5. Identify break-even points where profit equals zero
  6. Apply profit maximization concepts to real-world business decisions

Target Audience

  • High school economics students (grades 10-12)
  • AP Microeconomics students
  • Business and entrepreneurship classes
  • Students who have completed the Revenue Maximum lesson

Prerequisites

  • Understanding of the Revenue Maximum MicroSim
  • Basic algebra (solving equations, working with variables)
  • Concept of costs in business (what it costs to make something)

Key Vocabulary

Term Definition
Revenue Total income from sales (Price × Quantity)
Marginal Cost (MC) Cost to produce one additional unit
Total Cost Marginal Cost × Quantity produced
Profit Revenue minus Total Cost
Break-even Point Price where Profit = 0
Profit Margin Price minus Marginal Cost (profit per unit)

Lesson Activities

Activity 1: Connecting to Revenue Maximum

Setup: Start with MC = $0

  1. Set the Marginal Cost slider to $0
  2. Find the price that maximizes profit
  3. Compare to the revenue-maximizing price ($100)
  4. Discussion: When costs are zero, why are they the same?

Activity 2: Discovering the Shift

Exploration: Increase marginal cost gradually

  1. Slowly increase Marginal Cost from $0 to $60
  2. Watch the green "Profit Max" line move
  3. Record the profit-maximizing price at MC = $0, $20, $40, $60
  4. Pattern: What's the relationship between MC and optimal price?

Activity 3: Understanding the Profit Rectangle

Focus on the left panel:

  1. Set Price = $100 and MC = $40
  2. Observe the green profit rectangle
  3. Calculate: Height = Price - MC = $60, Width = Quantity = 100
  4. Area = $60 × 100 = $6,000 (profit)
  5. Now set Price = $120 (the profit-max). What happens to the rectangle?

Activity 3b: Watching Profit Change in Real Time

Dynamic exploration of the profit rectangle:

Keep your eyes on the green rectangle in the left panel as you slowly drag the Price slider. The area of this rectangle is the profit—height times width, margin times quantity.

  1. Set MC = $30 and start with Price = $50
  2. Slowly drag the Price slider upward while watching the green rectangle
  3. Notice how the rectangle gets taller (higher margin) but narrower (fewer sales)
  4. Find the "sweet spot" where the rectangle has maximum area
  5. Key observation: The rectangle is tallest at high prices but has zero width. It's widest at low prices but has zero height. Maximum area occurs somewhere in between.
  6. Verify that maximum area occurs at the profit-maximizing price shown by the green vertical line on the right panel

Activity 4: Finding Break-Even Points

Investigation:

  1. Set MC = $30
  2. Find the TWO prices where profit = $0 (hint: one is very low, one is very high)
  3. What happens to profit between these prices? Outside them?
  4. Real-world connection: Why do businesses care about break-even?

Activity 5: The Loss Zone

Exploring negative profit:

  1. Set MC = $50
  2. Set Price = $30 (below marginal cost)
  3. Observe the red rectangle (loss)
  4. Discussion: Why would a business ever price below cost? (Loss leaders, market share)

Discussion Questions

  1. The Profit Paradox: A company can increase profit by raising prices even though they sell fewer units. Explain why this works.

  2. Cost Sensitivity: If your supplier announces a 20% increase in costs, should you raise prices by 20%? What does the formula suggest?

  3. Revenue vs Profit: Why might a startup focus on revenue maximization instead of profit maximization? When might this strategy change?

  4. Real Examples:

  5. Why do movie theaters charge high prices for popcorn (low MC)?
  6. Why do airlines have complex pricing systems?

  7. Why do luxury brands price higher than cost-based calculations suggest?

  8. Break-Even Analysis: A food truck has MC = $5 per item. Using our model, at what two prices would they break even? What price maximizes profit?

Common Misconceptions

Misconception Clarification
"Just add a markup to cost" Optimal markup depends on demand elasticity, not just costs
"Sell as many as possible" Volume isn't always better—margin matters
"Highest price = highest profit" Price too high means few buyers, low total profit
"Revenue and profit move together" They peak at DIFFERENT prices when MC > 0

Assessment Ideas

Quick Check

  1. If MC = $40, what is the profit-maximizing price? (Answer: $120)
  2. Why is profit-max price always higher than revenue-max price when MC > 0?

Application Problem

A coffee shop has these characteristics:

  • At $0 per cup, they could give away 200 cups per day
  • At $10 per cup, no one would buy
  • Demand decreases linearly
  • Each cup costs $2 to make (MC = $2)

Calculate:

  1. The demand equation
  2. The revenue-maximizing price
  3. The profit-maximizing price
  4. Maximum daily profit

Reflection Prompt

"Explain to a friend who missed class why a business shouldn't just set the price that brings in the most revenue."

Extensions

For Advanced Students

  1. Non-linear costs: What if marginal cost increases with quantity? (Hint: the profit curve is no longer symmetric)

  2. Two products: If you sell two related products, how do you price them together?

  3. Competition: How would the presence of competitors change optimal pricing?

Cross-Curricular Connections

  • Mathematics: Quadratic optimization, derivatives, finding maxima
  • Business: Pricing strategy, cost accounting, margin analysis
  • Psychology: How do customers perceive price-quality relationships?

Relationship to Revenue Maximum

This MicroSim builds directly on the Revenue Maximum simulation:

Aspect Revenue Maximum Profit Maximum
Objective Maximize P × Q Maximize P × Q - MC × Q
Optimal Price Always $100 $100 + MC/2
Considers Costs No Yes
Real-world Use Market share focus Profitability focus

Students should complete Revenue Maximum first to understand the baseline, then use Profit Maximum to see how costs change the optimal decision.

References

  1. Khan Academy - Profit Maximization - Video explanation of profit concepts
  2. Revenue Maximum MicroSim - Prerequisite simulation (Part 1)
  3. Principles of Economics - OpenStax - Chapter on firm behavior and profit
  4. Marginal Analysis - Investopedia - Business perspective on marginal thinking