Revenue Maximum

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About This MicroSim

This MicroSim asks one question: what price brings in the most money?

A seller faces a straight-line demand curve, quantity = 200 − price. At a price of $0 the seller gives away 200 units and takes in nothing. At a price of $200 nobody buys, so the seller again takes in nothing. The price with the most revenue is somewhere in between, and the learner's job is to find it.

Left panel: Demand Curve

• Price is on the vertical axis and quantity sold is on the horizontal axis.
• The blue rectangle under the current point is as tall as the price and as wide as the quantity, so its area is the revenue.
• A high price gives a tall, narrow rectangle. A low price gives a short, wide one.

Right panel: Revenue Curve

• Revenue is on the vertical axis and price is on the horizontal axis.
• The curve is not drawn when the MicroSim loads. It appears as the price changes, so the learner builds the hill and finds its top.
• An orange ring marks the highest revenue found so far.
• The height of the blue point always equals the area of the rectangle in the left panel.

The line under the charts shows the calculation with the current numbers, such as Revenue = Price × Quantity = $50 × 150 = $7,500. The line below it reports the highest revenue found so far.

Learning objective: The learner will identify the price that gives the most total revenue on a straight-line demand curve and explain why revenue falls at prices above and below it.

Bloom's taxonomy level: Analyze (verbs: identify, explain)

How to Use

1. Change the price. Drag the blue point along the demand curve, drag across the revenue chart, or move the Price slider.
2. Watch the rectangle. Its area is the revenue. The same number is the height of the point on the revenue chart.
3. Find the top of the hill. Keep changing the price until the revenue stops going up.
4. Press Sweep Price to clear the traced curve and watch the price move from $0 to $200. Press the button again to pause, and again to resume. The sweep only runs while the mouse is over the MicroSim, so it will not distract a reader who has moved on to the text.
5. Check Show Maximum to reveal the whole curve, the peak, and the dashed orange outline of the largest rectangle that fits under the demand curve.
6. Press Reset to return the price to $50, clear the traced curve and uncheck the box.

Keyboard Shortcuts

Click the MicroSim once so it has the keyboard focus.

Key	Action
→ and ←	Raise or lower the price by $1
Shift + → or ←	Raise or lower the price by $10
S	Start, pause or resume the sweep
M	Show or hide the maximum
R	Reset

What the Colors Mean

What you see	What it means
Crimson line	The demand curve: the quantity sold at each price
Blue rectangle	Revenue as an area: price (height) × quantity (width)
Blue curve	The part of the revenue curve traced so far
Black labels on the axes	The current price and the current quantity
Blue label on the revenue axis	The current revenue
Orange ring	The highest revenue found so far
Dashed orange lines	With Show Maximum checked: the peak of the revenue curve and the largest rectangle

What to Try

• Set the price to $40 and then to $160. Both give a revenue of $6,400. Use the two rectangles to explain why.
• Start at $50 and raise the price $10 at a time. Write down how much revenue you gain on each step. The gains are $900, $700, $500, $300 and $100. What happens on the next step, and why?
• Find the price where the rectangle is a square. What is special about that price?

Lesson Plan

Learning Objectives

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

1. Define revenue as the product of price and quantity sold
2. Explain the inverse relationship between price and quantity demanded
3. Identify the price point that maximizes total revenue
4. Analyze why neither the highest nor lowest price generates maximum revenue
5. Apply the concept of revenue optimization to real-world pricing decisions
6. Connect the geometric representation (rectangle area) to the algebraic formula

Target Audience

• High school economics students (grades 10-12)
• AP Microeconomics students
• College introductory economics courses
• Business and entrepreneurship classes

Prerequisites

• Basic algebra (multiplication, variables)
• Understanding of graphs and coordinate systems
• Concept of supply and demand (helpful but not required)

Key Concepts

Concept	Definition
Revenue	Total income from sales: Revenue = Price × Quantity
Demand Curve	Graph showing inverse relationship between price and quantity demanded
Revenue Curve	Parabolic graph showing revenue at each price point
Maximum Revenue	The highest possible revenue, occurring at the optimal price
Price Elasticity	How sensitive quantity demanded is to price changes

Lesson Activities

Activity 1: Discovery Exploration (10 minutes)

Instructions for students:

1. The MicroSim starts at a price of $50. Before you touch anything, predict: will revenue rise or fall if the price goes up?
2. Move the slider to Price = 0. What is the revenue? Why?
3. Move to Price = 200. What is the revenue now? Why?
4. Slowly move the slider from 0 to 200 and watch the revenue curve appear. At what price is revenue highest?
5. Record your observations in a table:

Price	Quantity	Revenue
0	?	?
50	?	?
100	?	?
150	?	?
200	?	?

Activity 2: The Revenue Rectangle (10 minutes)

Focus on the left panel:

1. Notice the shaded blue rectangle under the demand curve
2. The width represents quantity, the height represents price
3. The area of the rectangle equals revenue (length × width = P × Q)
4. Find the price where the rectangle has the maximum area
5. Check Show Maximum and compare your rectangle with the dashed orange outline
6. Discussion: Why does the rectangle get smaller at extreme prices?

Activity 3: Mathematical Connection (15 minutes)

Deriving the revenue formula:

Given the demand function: Q = 200 - P

Revenue = P × Q = P × (200 - P) = 200P - P²

This is a downward-opening parabola! The maximum occurs at:

\[P = \frac{200}{2} = 100\]

Maximum Revenue = 100 × (200 - 100) = 100 × 100 = 10,000

Verify with calculus (for advanced students):

\[\frac{dR}{dP} = 200 - 2P = 0\]

\[P = 100\]

Activity 4: Real-World Application (15 minutes)

Scenario: Movie Theater Pricing

A movie theater has a demand curve where:

• At $0, 1000 people would attend
• At $20, nobody would attend
• Demand decreases linearly with price

Questions:

1. Write the demand equation: Q = 1000 - 50P
2. Write the revenue equation: R = P × (1000 - 50P)
3. What price maximizes revenue?
4. What is the maximum revenue?
5. Should the theater charge this price? What other factors matter?

Discussion Questions

1. The Revenue Paradox: Why doesn't the highest possible price generate the most revenue?

2. Zero Revenue Points: The revenue curve touches zero at two points. What do these represent economically?

3. Business Strategy: If you were a business owner, would you always charge the revenue-maximizing price? What other factors might influence your decision?

4. Elasticity Connection: How does this simulation relate to the concept of price elasticity of demand?

5. Real Examples: Can you think of products where companies seem to price at the revenue-maximizing point? Products where they don't?

6. Shape of the Curve: Why is the revenue curve a parabola (symmetric curve) in this simulation? Would it always be symmetric in real life?

Assessment Ideas

Formative Assessment

• Exit ticket: "At what price is revenue maximized in our simulation, and why?"
• Partner discussion: Explain to your partner why revenue is zero at both P=0 and P=200

Summative Assessment

Problem: A concert venue has the following demand relationship:

• Maximum capacity: 5000 seats
• At $0, all 5000 seats would be filled
• At $100, no one would attend
• Demand is linear

Calculate:

1. The demand equation (5 points)
2. The revenue equation (5 points)
3. The revenue-maximizing price (5 points)
4. The maximum revenue (5 points)
5. Explain why the answer makes intuitive sense (5 points)

Answer key

Activity 1 table

Price	Quantity	Revenue
0	200	0
50	150	7,500
100	100	10,000
150	50	7,500
200	0	0

Activity 4 (movie theater): R = 1000P - 50P². Revenue is largest at P = $10, where 500 people attend and the revenue is $5,000.

Summative assessment (concert venue): Q = 5000 - 50P and R = 5000P - 50P². Revenue is largest at P = $50, where 2,500 seats are sold and the revenue is $125,000. The price is halfway between $0 (a full house that pays nothing) and $100 (an empty house).

Extensions

For Advanced Students

1. Non-linear demand: What if demand doesn't decrease linearly? How would the revenue curve change?

2. Cost consideration: Revenue isn't profit! If each unit costs $30 to produce, what price maximizes profit?

3. Multiple products: How would a company balance prices across multiple related products?

4. Dynamic pricing: Research how airlines and hotels use demand-based pricing

Cross-Curricular Connections

• Mathematics: Quadratic functions, optimization, calculus derivatives
• Business: Pricing strategy, market research, profit maximization
• Psychology: Consumer behavior, price perception
• Data Science: Demand forecasting, regression analysis

Common Misconceptions

Misconception	Clarification
"Higher prices always mean more revenue"	Revenue depends on BOTH price AND quantity sold
"The demand curve IS the revenue curve"	They're related but different. Revenue is the area of the rectangle under one point of the demand curve, not the area under the whole curve
"Maximum revenue = maximum profit"	Profit also considers costs, not just revenue
"This is how all real markets work"	This is a simplified model; real demand curves are rarely perfectly linear

Teacher Notes

• Time required: 45-60 minutes for full lesson
• Technology needs: Projector/smartboard for demonstration, student devices for exploration
• Differentiation: Pair struggling students with peers; provide extension problems for advanced students
• Prior knowledge check: Review multiplication and graph reading before starting
• Predict first: Leave Show Maximum unchecked until students have committed to a prediction. The revenue curve is hidden at the start for the same reason.

Revenue Maximization vs. Profit Maximization

Revenue Is Not Profit

A common trap here is to treat the top of the revenue hill as the best price for a business. Revenue ignores what each unit costs to make, so the price with the most profit is usually higher. Find the peak here first, then subtract the costs.

This MicroSim teaches revenue concepts, but businesses ultimately care about profit, which is what remains after costs are subtracted.

Next Steps

Ready to explore profit maximization? Continue to the Profit Maximum MicroSim to see how production costs change the optimal pricing decision.

Why Revenue ≠ Profit

Metric	Formula	What It Measures
Revenue	Price × Quantity	Total income from sales
Cost	Fixed Costs + (Variable Cost × Quantity)	Total expenses to produce goods
Profit	Revenue - Cost	Actual money earned

In our Revenue Maximum MicroSim, the optimal price is $100, generating revenue of $10,000. But what if each unit costs $30 to produce? At Price = $100, we sell 100 units:

• Revenue = $10,000
• Cost = 100 × $30 = $3,000
• Profit = $7,000

But at Price = $120, we sell 80 units:

• Revenue = $9,600 (lower!)
• Cost = 80 × $30 = $2,400
• Profit = $7,200 (higher!)

The profit-maximizing price is different from the revenue-maximizing price when costs are considered.

The Profit Maximum MicroSim

The design below was first written on this page as a proposal. It has since been built as the Profit Maximum MicroSim, which keeps the two-panel structure of this MicroSim and adds a marginal cost slider.

The original design proposal for the Profit Maximum MicroSim

A natural extension of this MicroSim would add production costs to demonstrate true profit optimization. The key design principle: keep the familiar two-panel structure to reduce cognitive load and enable direct visual comparison.

Two-Panel Design (Recommended)

┌───────────────────────┬───────────────────────┐
│    DEMAND CURVE       │   REVENUE vs PROFIT   │
│                       │                       │
│   Price               │   $                   │
│     │╲                │         ∩ Revenue     │
│     │ ╲               │        ╱ ╲  (blue)    │
│     │  ╲  Profit      │       ╱   ╲           │
│     │   ╲ Rectangle   │   ∩  ╱     ╲          │
│     │    ╲ (green)    │  ╱ ╲╱ Profit (green)  │
│     └─────────────    │  └────────────────    │
│        Quantity       │        Price          │
└───────────────────────┴───────────────────────┘
              Controls
   [Price]  [Marginal Cost]  [Toggle Revenue Curve]

Why Two Panels Instead of Three?

From an instructional design perspective:

1. Reduced cognitive load: Students track two visualizations, not three
2. Direct comparison: Overlaying revenue and profit curves on the same axes makes the key insight immediately visible—the peaks occur at different prices
3. Familiar structure: Matches the Revenue Maximum MicroSim, so students spend less time orienting and more time learning
4. Better responsiveness: Works well on mobile devices and classroom projectors

Design Principle

The core learning objective is simple: profit-maximizing price ≠ revenue-maximizing price. The UI should be equally simple. Save cost curve analysis (MC, ATC, AVC) for a separate, more advanced MicroSim.

Panel Descriptions

Left Panel: Demand Curve with Profit Rectangle

• Same demand curve as Revenue Maximum MicroSim
• Shaded area now shows PROFIT, not revenue
• Profit rectangle = (Price - Marginal Cost) × Quantity
• Green when profit > 0, Red when operating at a loss
• Horizontal dashed line shows the marginal cost level

Right Panel: Revenue and Profit Curves (Overlaid)

• Blue curve: Revenue (same parabola as Revenue Maximum)
• Green curve: Profit (shifted down and right)
• Blue vertical line: Revenue-maximizing price (always at P = 100)
• Green vertical line: Profit-maximizing price (shifts based on marginal cost)
• Students can visually see the gap between the two optimal prices

Interactive Controls

Control	Range	Purpose
Price Slider	$0 - $200	Set selling price
Marginal Cost Slider	$0 - $80	Adjust per-unit production cost
Show Revenue Curve	Toggle	Compare profit curve to revenue curve

Key Learning Moments

1. The Gap: When MC = $30, revenue peaks at P = 100, but profit peaks at P = 115
2. Why the Shift: Higher costs mean you need higher prices to maintain margins
3. Break-even Points: Two prices where profit = 0 (too low or too high)
4. Loss Region: Red shading when price is below marginal cost
5. Special Case: When MC = 0, both curves peak at the same price

Mathematical Foundation

Profit Function:

\[\pi(P) = P \cdot Q(P) - MC \cdot Q(P) = (P - MC) \cdot Q(P)\]

With our demand function \(Q(P) = 200 - P\):

\[\pi(P) = (P - MC)(200 - P)\]

\[\pi(P) = 200P - P^2 - 200 \cdot MC + MC \cdot P\]

\[\pi(P) = -P^2 + (200 + MC)P - 200 \cdot MC\]

Optimal Price (using calculus):

\[\frac{d\pi}{dP} = -2P + 200 + MC = 0\]

\[P^* = \frac{200 + MC}{2} = 100 + \frac{MC}{2}\]

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

Key insight: The profit-maximizing price is always \(100 + \frac{MC}{2}\), which is higher than the revenue-maximizing price whenever production has a cost.

Classroom Applications

1. What-if scenarios: "What happens to optimal price if our supplier raises costs by $10?"
2. Visual proof: "Why doesn't the highest price give the most profit?"
3. Break-even planning: "At what prices do we just cover our costs?"
4. Comparison exercise: Toggle the revenue curve on/off to see the relationship

This Profit Maximization MicroSim would serve as the natural "Part 2" to the Revenue Maximum MicroSim, completing the economic picture of business decision-making while maintaining a simple, learnable interface.

Version History

The first version of this MicroSim (December 2025) drew the whole revenue curve and an orange line at the maximum as soon as it loaded, and the slider started at $100, which is the answer. The current version starts at $50 and hides the curve so that the learner finds the peak. It also adds axis scales, value labels on the axes, dragging on both charts, keyboard shortcuts, the Sweep Price animation, the Show Maximum checkbox and a narrow-screen layout, and it runs on p5.js 2.3.2. The original version is saved in this repository's git history.

References

1. Total revenue - accessed 2026-09-30 - Wikipedia - Defines total revenue as price times the quantity sold and relates changes in revenue to price elasticity.
2. Demand curve - accessed 2026-09-30 - Wikipedia - Background on the curve in the left panel and on the law of demand behind its downward slope.
3. Price elasticity of demand - accessed 2026-09-30 - Wikipedia - Explains why a price increase raises revenue where demand is inelastic and lowers it where demand is elastic.
4. 5.3 Elasticity and Pricing - 2022 - Principles of Economics 3e, OpenStax - A free textbook section that uses a band's ticket price to ask whether a higher price brings in more revenue.
5. A Quick Guide to Value-Based Pricing - August 2016 - Harvard Business Review - How businesses set prices in practice, for the discussion of factors beyond the revenue-maximizing price.
6. Economic profit vs accounting profit - Khan Academy - A video on revenue, costs and profit that supports the revenue versus profit section.
7. Quadratic function - accessed 2026-09-30 - Wikipedia - The mathematics of the parabola R = 200P - P² and its vertex.
8. Desmos Graphing Calculator - Desmos - A tool for graphing the revenue function of the movie theater and concert venue problems.
9. p5.js Reference - accessed 2026-09-30 - p5.js - Documentation for the JavaScript library used to build this MicroSim.