Hong Life CalculatorsUnderstanding the optimal price calculator — concept and principle
This calculator estimates the profit-maximizing price from two price tests. It fits a straight-line demand curve through the two results, then finds the price where marginal revenue equals marginal cost. It shows the best price, units and profit at that price, the gain over the first price, the revenue-maximizing price, the choke price where sales stop and the price elasticity at the optimum. Small sellers and pricing analysts use it after A/B price tests.
How it is calculated
- Units lost per 1 of price: b = (Q1 − Q2) ÷ (P2 − P1)
- Demand line: Q = a − b × P, with a = Q1 + b × P1
- Price where sales fall to zero = a ÷ b
- Profit-maximizing price P* = (a ÷ b + unit cost) ÷ 2
- Units at P* = a − b × P*; profit = (P* − unit cost) × units − fixed costs
- Revenue-maximizing price = (a ÷ b) ÷ 2
- Elasticity at P* = −b × P* ÷ units
The principle
Profit is highest where the revenue from one more sale equals its cost, that is, where marginal revenue equals marginal cost. On a straight-line demand curve, marginal revenue falls twice as fast as price, which leads to a neat rule: the best price lies halfway between the choke price and the unit cost. Fixed costs lower profit but do not change the best price, because they are paid whatever price you choose. The revenue-maximizing price ignores costs and is always lower than the profit-maximizing price when unit cost is positive.
Worked example
Defaults: 1,000 units at 20; 800 units at 24; unit cost 10; fixed costs 3,000.
- b = 200 ÷ 4 = 50 units lost per 1 of price; a = 1,000 + 50 × 20 = 2,000.
- Choke price = 2,000 ÷ 50 = $40.00.
- P* = (40 + 10) ÷ 2 = $25.00; units = 2,000 − 1,250 = 750.
- Revenue = $18,750; profit = (25 − 10) × 750 − 3,000 = $8,250.
- Profit at the first price = (20 − 10) × 1,000 − 3,000 = $7,000; gain = $1,250.
- Revenue-maximizing price = $20.00; elasticity at P* = −50 × 25 ÷ 750 = -1.67.
| Variable cost per unit (marginal cost) | Profit-maximizing price |
|---|---|
| 2.5 | $21.25 |
| 5 | $22.50 |
| 7.5 | $23.75 |
| 10 | $25.00 |
| 12.5 | $26.25 |
| 15 | $27.50 |
| 20 | $30.00 |
How to read the result
The suggested price raises profit even though it sells fewer units. Because the line is fitted through only two points, test a price close to P* before committing. An elasticity at the optimum always below −1 is expected: a profit-maximizing seller never prices where demand is inelastic.
Common mistakes
- Using tests run at different times or seasons, so other factors changed.
- Including fixed costs in the unit cost.
- Trusting a price far outside the tested range.
- Ignoring competitor reactions to a price change.
Frequently asked questions
Why do fixed costs not change the best price?
They are the same at every price, so they shift profit up or down without moving the peak.
Why is the profit-maximizing price above the revenue-maximizing price?
Each unit costs money to make, so selling fewer units at a higher margin can earn more profit than maximizing revenue.
How many price tests do I need?
Two give a line; three or more show whether demand is really straight.
Sources: OpenStax, Principles of Economics, 3rd ed. (2022), section 9.2 "How a Profit-Maximizing Monopoly Chooses Output and Price" · Saylor Academy, Principles of Economics v2.0, section 10.2 "The Monopoly Model"