Revenue, cost and price
| Term | Formula |
|---|---|
| Revenue | price × quantity |
| Cost | fixed + variable |
| Profit | revenue − cost |
Setting the right price is a balancing act: too high and customers buy less; too low and each sale earns little.
Pricing strategies
| Pricing strategy | How it works |
|---|---|
| Cost-plus (markup) | add a percent to the cost |
| Competitive | match or beat competitors |
| Value-based | price by what customers think it’s worth |
| Penetration | start low to win customers, raise later |
| Premium | price high to signal quality |
| Psychological | $9.99 instead of $10 |
Markup and margin
Markup on cost = (price − cost)/cost
Margin on price = (price − cost)/price
Example: Cost $40, price $60.
- Markup = 20/40 = 50%
- Margin = 20/60 ≈ 33.3%
Same dollars, different percents: know which one is being quoted.
Demand and price
Usually, as price rises, demand (quantity sold) falls.
Example: A theatre sells 400 tickets at $20. For every $1 increase, it sells 10 fewer.
- Price: 20 + x
- Tickets: 400 − 10x
- Revenue: R(x) = (20 + x)(400 − 10x) = −10x² + 200x + 8,000
Finding maximum revenue
R(x) is quadratic and opens down. The vertex gives the maximum revenue.
- x = −200/(2 × −10) = 10
- Price: $30; tickets: 300
- Maximum revenue: 30 × 300 = $9,000
Raising the price beyond $30 loses so many customers that revenue falls.
Maximizing profit
Profit = revenue − cost. If each ticket costs the theatre $4 to serve plus $2,000 fixed:
P(x) = (20 + x)(400 − 10x) − 4(400 − 10x) − 2,000
The best price for profit may differ from the best price for revenue.
Break-even with pricing
Changing the price shifts the break-even point. Higher prices lower it, if customers still buy.
Worked examples
Example 1: Cost $25, 60% markup. Price? 25 × 1.6 = $40.
Example 2: 500 units at $12. Revenue? $6,000.
Example 3: R(x) = (10 + x)(200 − 5x). Price for max revenue? Expand: −5x² + 150x + 2,000 → x = 15 → $25.
Common mistakes
- Confusing markup and margin.
- Assuming higher prices always increase revenue.
- Ignoring costs when choosing a price.
Try it yourself
- Cost $80, price $100. Markup and margin?
- 1,000 items at $8, then 800 items at $11. Which earns more revenue?
- R(x) = (50 + x)(300 − 3x). Find the price for maximum revenue.
- Name two pricing strategies.
Answers: 1) 25% and 20% 2) 800 at $11 ($8,800 vs $8,000) 3) x = 25 → $75 4) e.g. cost-plus, competitive
Quick recap
- Revenue = price × quantity; profit = revenue − cost.
- Choose a pricing strategy; know markup vs margin.
- Demand falls as price rises; quadratic models find maximum revenue.