Maximum Revenue Calculator
This tool helps you find the price point that is likely to generate the highest revenue, assuming a linear relationship between the price you charge and the quantity customers will buy (a linear demand curve).
Enter your current price and the quantity sold at that price. Then, indicate how the quantity sold changes for a specific change in price (e.g., "Sales decrease by 10 units for every $1 increase").
Enter Current Sales Data and Price Sensitivity
Understanding Maximum Revenue and Linear Demand
What is Maximum Revenue?
Revenue is simply the total income generated from sales: Revenue = Price × Quantity Sold. Maximum revenue is the highest possible revenue you can achieve by choosing the optimal price for your product or service.
The Linear Demand Curve Assumption
This calculator assumes a linear relationship between price (P) and the quantity demanded (Q). This relationship can be expressed as: Q = a + m*P, where 'm' is the slope of the demand curve (how much quantity changes for a one-unit change in price) and 'a' is the quantity demanded if the price were zero.
From your inputs (Current Price P₀, Current Quantity Q₀, Price Change ΔP, Quantity Change ΔQ), we can determine this linear equation:
- The slope
m = ΔQ / ΔP. (Note: For typical products, as price increases, quantity decreases, so ΔQ is negative when ΔP is positive, resulting in a negative slope 'm'). - Using your current data point (P₀, Q₀) and the slope, we find 'a':
Q₀ = a + m*P₀, soa = Q₀ - m*P₀.
The demand curve is then fully defined as Q = (Q₀ - m*P₀) + m*P.
Finding the Maximum Revenue
Revenue (R) is R = P * Q. Substituting the linear demand curve Q = a + m*P:
R(P) = P * (a + m*P) = a*P + m*P²
This is a quadratic equation for Revenue as a function of Price. Since 'm' (the slope ΔQ/ΔP) is typically negative for a demand curve, this parabola opens downwards, and its maximum point (the vertex) represents the maximum revenue.
The price (P) at which this maximum occurs is given by the vertex formula P = -B / (2A) for a quadratic AP² + BP + C. In our case, A=m and B=a.
Optimal Price Formula: P_optimal = -a / (2m)
Once the optimal price (P_optimal) is found, the optimal quantity (Q_optimal) is found by plugging P_optimal back into the demand equation: Q_optimal = a + m * P_optimal.
Finally, the Maximum Revenue is simply: Revenue_max = P_optimal * Q_optimal.
Maximum Revenue Examples
Click on an example to see the step-by-step calculation:
Example 1: Basic Price Increase Impact
Scenario: A store currently sells 100 items per week at $20 each. They observe that for every $1 increase in price, they sell 5 fewer items.
1. Known Values: P₀ = $20, Q₀ = 100 units, ΔP = $1, ΔQ = -5 units.
2. Calculate Slope (m): m = ΔQ / ΔP = -5 / 1 = -5.
3. Calculate 'a': a = Q₀ - m*P₀ = 100 - (-5 * 20) = 100 - (-100) = 100 + 100 = 200.
Demand Curve: Q = 200 - 5P.
4. Calculate Optimal Price (P_opt): P_opt = -a / (2m) = -200 / (2 * -5) = -200 / -10 = $20.
5. Calculate Optimal Quantity (Q_opt): Q_opt = a + m*P_opt = 200 + (-5 * 20) = 200 - 100 = 100 units.
6. Calculate Maximum Revenue (R_max): R_max = P_opt * Q_opt = $20 * 100 units = $2000.
Conclusion: Based on this linear model, the current price of $20 already achieves the maximum revenue of $2000.
Example 2: Higher Price Sensitivity
Scenario: Same store, but now for every $1 price increase, they sell 10 fewer items.
1. Known Values: P₀ = $20, Q₀ = 100 units, ΔP = $1, ΔQ = -10 units.
2. Calculate Slope (m): m = ΔQ / ΔP = -10 / 1 = -10.
3. Calculate 'a': a = Q₀ - m*P₀ = 100 - (-10 * 20) = 100 - (-200) = 100 + 200 = 300.
Demand Curve: Q = 300 - 10P.
4. Calculate Optimal Price (P_opt): P_opt = -a / (2m) = -300 / (2 * -10) = -300 / -20 = $15.
5. Calculate Optimal Quantity (Q_opt): Q_opt = a + m*P_opt = 300 + (-10 * 15) = 300 - 150 = 150 units.
6. Calculate Maximum Revenue (R_max): R_max = P_opt * Q_opt = $15 * 150 units = $2250.
Conclusion: Lowering the price to $15 increases quantity sold, leading to a maximum revenue of $2250 (higher than the $2000 at $20).
Example 3: Price Decrease Observation
Scenario: An online service has 500 subscribers at $50/month. When they lowered the price by $5, they gained 100 subscribers.
1. Known Values: P₀ = $50, Q₀ = 500, ΔP = -$5, ΔQ = +100.
2. Calculate Slope (m): m = ΔQ / ΔP = 100 / -5 = -20.
3. Calculate 'a': a = Q₀ - m*P₀ = 500 - (-20 * 50) = 500 - (-1000) = 500 + 1000 = 1500.
Demand Curve: Q = 1500 - 20P.
4. Calculate Optimal Price (P_opt): P_opt = -a / (2m) = -1500 / (2 * -20) = -1500 / -40 = $37.50.
5. Calculate Optimal Quantity (Q_opt): Q_opt = a + m*P_opt = 1500 + (-20 * 37.5) = 1500 - 750 = 750 units.
6. Calculate Maximum Revenue (R_max): R_max = P_opt * Q_opt = $37.50 * 750 units = $28125.
Conclusion: The optimal monthly price is $37.50, potentially bringing in $28125 in maximum revenue.
Example 4: Low Price Point
Scenario: Selling keychains at a fair. You sell 250 at $2 each. You think increasing the price by $0.50 might cause you to sell 75 fewer.
1. Known Values: P₀ = $2.00, Q₀ = 250, ΔP = $0.50, ΔQ = -75.
2. Calculate Slope (m): m = ΔQ / ΔP = -75 / 0.50 = -150.
3. Calculate 'a': a = Q₀ - m*P₀ = 250 - (-150 * 2) = 250 - (-300) = 250 + 300 = 550.
Demand Curve: Q = 550 - 150P.
4. Calculate Optimal Price (P_opt): P_opt = -a / (2m) = -550 / (2 * -150) = -550 / -300 ≈ $1.83.
5. Calculate Optimal Quantity (Q_opt): Q_opt = a + m*P_opt = 550 + (-150 * 1.8333) ≈ 550 - 275 = 275 units.
6. Calculate Maximum Revenue (R_max): R_max = P_opt * Q_opt ≈ $1.8333 * 275 units ≈ $504.16.
Conclusion: A slight price decrease to $1.83 might increase quantity sold and result in slightly higher maximum revenue.
Example 5: Service Pricing
Scenario: A consultant charges $150/hour and gets 40 billable hours a month. If they raise the rate by $20, they expect to lose 5 hours of work.
1. Known Values: P₀ = $150, Q₀ = 40, ΔP = $20, ΔQ = -5.
2. Calculate Slope (m): m = ΔQ / ΔP = -5 / 20 = -0.25.
3. Calculate 'a': a = Q₀ - m*P₀ = 40 - (-0.25 * 150) = 40 - (-37.5) = 40 + 37.5 = 77.5.
Demand Curve: Q = 77.5 - 0.25P.
4. Calculate Optimal Price (P_opt): P_opt = -a / (2m) = -77.5 / (2 * -0.25) = -77.5 / -0.5 = $155.
5. Calculate Optimal Quantity (Q_opt): Q_opt = a + m*P_opt = 77.5 + (-0.25 * 155) = 77.5 - 38.75 = 38.75 hours.
6. Calculate Maximum Revenue (R_max): R_max = P_opt * Q_opt = $155 * 38.75 hours = $6006.25.
Conclusion: Increasing the rate slightly to $155/hour could yield maximum revenue, even with a small drop in hours.
Example 6: Subscription Box Pricing
Scenario: A new subscription box has 1000 subscribers at $30/month. Market research suggests if they drop the price by $2, they could gain 200 subscribers.
1. Known Values: P₀ = $30, Q₀ = 1000, ΔP = -$2, ΔQ = +200.
2. Calculate Slope (m): m = ΔQ / ΔP = 200 / -2 = -100.
3. Calculate 'a': a = Q₀ - m*P₀ = 1000 - (-100 * 30) = 1000 - (-3000) = 1000 + 3000 = 4000.
Demand Curve: Q = 4000 - 100P.
4. Calculate Optimal Price (P_opt): P_opt = -a / (2m) = -4000 / (2 * -100) = -4000 / -200 = $20.
5. Calculate Optimal Quantity (Q_opt): Q_opt = a + m*P_opt = 4000 + (-100 * 20) = 4000 - 2000 = 2000 units.
6. Calculate Maximum Revenue (R_max): R_max = P_opt * Q_opt = $20 * 2000 units = $40000.
Conclusion: A price point of $20/month appears to maximize revenue for this subscription box model.
Example 7: High-End Product
Scenario: Selling custom art pieces. You sold 5 last month at $500 each. A small increase of $50 might lose you 1 sale.
1. Known Values: P₀ = $500, Q₀ = 5, ΔP = $50, ΔQ = -1.
2. Calculate Slope (m): m = ΔQ / ΔP = -1 / 50 = -0.02.
3. Calculate 'a': a = Q₀ - m*P₀ = 5 - (-0.02 * 500) = 5 - (-10) = 5 + 10 = 15.
Demand Curve: Q = 15 - 0.02P.
4. Calculate Optimal Price (P_opt): P_opt = -a / (2m) = -15 / (2 * -0.02) = -15 / -0.04 = $375.
5. Calculate Optimal Quantity (Q_opt): Q_opt = a + m*P_opt = 15 + (-0.02 * 375) = 15 - 7.5 = 7.5 units.
6. Calculate Maximum Revenue (R_max): R_max = P_opt * Q_opt = $375 * 7.5 units = $2812.50.
Conclusion: Lowering the price to $375 could increase sales and maximize revenue, but selling exactly 7.5 units isn't practical; this suggests the linear model is an approximation, and pricing near $375 might be optimal.
Example 8: Event Ticket Pricing
Scenario: An event sold 100 tickets at $75 each. For every $10 decrease in price, they expect to sell 30 more tickets.
1. Known Values: P₀ = $75, Q₀ = 100, ΔP = -$10, ΔQ = +30.
2. Calculate Slope (m): m = ΔQ / ΔP = 30 / -10 = -3.
3. Calculate 'a': a = Q₀ - m*P₀ = 100 - (-3 * 75) = 100 - (-225) = 100 + 225 = 325.
Demand Curve: Q = 325 - 3P.
4. Calculate Optimal Price (P_opt): P_opt = -a / (2m) = -325 / (2 * -3) = -325 / -6 ≈ $54.17.
5. Calculate Optimal Quantity (Q_opt): Q_opt = a + m*P_opt = 325 + (-3 * 54.1667) ≈ 325 - 162.5 = 162.5 units.
6. Calculate Maximum Revenue (R_max): R_max = P_opt * Q_opt ≈ $54.17 * 162.5 units ≈ $8804.12.
Conclusion: A price point around $54.17 could maximize ticket revenue, selling around 162 or 163 tickets.
Example 9: Digital Product (Low Marginal Cost)
Scenario: Selling an e-book for $10, you sell 50 copies/month. You speculate that dropping the price to $8 might double sales to 100 copies.
1. Known Values: P₀ = $10, Q₀ = 50, ΔP = -$2, ΔQ = +50.
2. Calculate Slope (m): m = ΔQ / ΔP = 50 / -2 = -25.
3. Calculate 'a': a = Q₀ - m*P₀ = 50 - (-25 * 10) = 50 - (-250) = 50 + 250 = 300.
Demand Curve: Q = 300 - 25P.
4. Calculate Optimal Price (P_opt): P_opt = -a / (2m) = -300 / (2 * -25) = -300 / -50 = $6.
5. Calculate Optimal Quantity (Q_opt): Q_opt = a + m*P_opt = 300 + (-25 * 6) = 300 - 150 = 150 units.
6. Calculate Maximum Revenue (R_max): R_max = P_opt * Q_opt = $6 * 150 units = $900.
Conclusion: With a potential for high volume increase at lower prices (low marginal cost typical for digital goods), $6 could be the optimal price maximizing revenue.
Example 10: Price Point Resulting in Negative Quantity (Illustrating Model Limits)
Scenario: You sell 10 items at $50. If you increase the price by $1, you lose 5 sales (highly sensitive!).
1. Known Values: P₀ = $50, Q₀ = 10, ΔP = $1, ΔQ = -5.
2. Calculate Slope (m): m = ΔQ / ΔP = -5 / 1 = -5.
3. Calculate 'a': a = Q₀ - m*P₀ = 10 - (-5 * 50) = 10 - (-250) = 10 + 250 = 260.
Demand Curve: Q = 260 - 5P.
4. Calculate Optimal Price (P_opt): P_opt = -a / (2m) = -260 / (2 * -5) = -260 / -10 = $26.
5. Calculate Optimal Quantity (Q_opt): Q_opt = a + m*P_opt = 260 + (-5 * 26) = 260 - 130 = 130 units.
6. Calculate Maximum Revenue (R_max): R_max = P_opt * Q_opt = $26 * 130 units = $3380.
Conclusion: In this (perhaps unrealistic) scenario where demand is extremely sensitive, the optimal price is significantly lower at $26, aiming for a much higher quantity (130) to maximize revenue. This example highlights how the results depend heavily on the accuracy of the input data and the assumption of linearity over a wide range.
Important Considerations
The linear demand model is a simplification. Real-world demand curves are often non-linear and can be influenced by many factors not included here (competition, marketing, seasonality, perceived value, etc.). This calculator provides a theoretical maximum based *only* on the linear relationship derived from your two data points.
Frequently Asked Questions about Maximum Revenue Calculation
1. What is the core formula used by this calculator?
It uses the principle that for a linear demand curve (Q = a + mP), the revenue R = P * Q = P * (a + mP) = mP² + aP. The maximum of this quadratic function occurs at the price P = -a / (2m).
2. What inputs do I need?
You need your current sales data (Price and Quantity Sold) and information on how a specific change in price affects the quantity sold (Change in Price and the resulting Change in Quantity).
3. What outputs does the calculator provide?
It calculates the Optimal Price for maximum revenue, the Optimal Quantity expected to be sold at that price, and the Maximum Possible Revenue.
4. Why is the 'Change in Price' input important?
This input, along with the 'Change in Quantity', allows the calculator to determine the slope of the demand curve (m = ΔQ / ΔP), which is crucial for defining the linear relationship between price and quantity.
5. What does a negative 'Change in Quantity' mean?
A negative change in quantity resulting from a positive change in price (e.g., sales decrease when price increases) indicates a typical downward-sloping demand curve, which is necessary for a finite, positive maximum revenue in this model.
6. What if I enter a 'Change in Price' or 'Change in Quantity' of zero?
If the 'Change in Price' is zero, the slope cannot be calculated (division by zero). If both 'Change in Price' and 'Change in Quantity' are zero, the calculator cannot determine how price affects quantity. The tool will show an error in these cases as it cannot define the linear demand curve.
7. What if the calculated Optimal Price or Quantity is negative?
A negative result suggests that the mathematically derived vertex of the revenue function occurs at a price or quantity below zero. In reality, price and quantity must be non-negative. This usually indicates that based on the linear model derived, the maximum realistic revenue might occur at a boundary (e.g., a price of zero, assuming positive quantity at P=0). However, this calculator focuses on the theoretical vertex; if results are negative, treat the linear model's applicability with caution in that range.
8. Does this calculator find the price for maximum profit?
No, this calculator only finds the price for maximum *revenue*. To find maximum *profit*, you would need to include cost information (fixed and variable costs) in your calculation, as Profit = Revenue - Total Costs.
9. What are the limitations of assuming a linear demand curve?
Real-world demand is rarely perfectly linear across all possible prices. Factors like price thresholds, competitor pricing, and market saturation can cause demand to behave differently. The results are most reliable when your input data points and the calculated optimal price are within a relevant range of observable behavior.
10. What units should I use for inputs?
Use consistent units. If 'Current Price' is in dollars, 'Change in Price' should also be in dollars. 'Quantity' units (e.g., units, items, subscribers) just need to be consistent for 'Current Quantity' and 'Change in Quantity'. The output 'Maximum Revenue' will be in (Price Unit) * (Quantity Unit).
11. How can I test the model's accuracy?
The best way is through A/B testing different price points in the market. The results from this calculator can give you a starting point or a hypothesis to test.
