Single-Point Price Optimizer

Based on Estimated Elasticity and a Single Price/Demand Point

Theoretical Background

1. Origin: The Log-Log Model

When historical data is available, analysts run a regression on the natural logs (ln) of Price and Sales.

ln(Sales) = a + b · ln(Price)

Why use logs? While the raw relationship between Price and Sales is a curve, the relationship between ln(Price) and ln(Sales) is a straight line. This transformation allows us to use standard linear regression to find the elasticity (slope b).

To use this for prediction (in real units like units/dollars), we perform an algebraic transformation:

Step 1 Sales = e^{(a + b·ln(P))} We take the exponential (exp) of both sides. Step 2 Sales = e^a · (e^ln(P))^b Using algebra rules to separate terms. (Note: e^ln(P) = P). Step 3 Q = A · P^E We simplify by relabeling:
• Sales → Q
• e^a → A (Scale Factor)
• b → E (Elasticity)

2. Result: The Power Demand Curve

The result of that derivation is the Power Demand Curve. Unlike a linear model (which subtracts units), this model multiplies by percentages. This ensures demand never drops below zero, even at very high prices.

Figure 1: Illustrative Power Demand Curve

3. Prediction Example

Because elasticity is constant, we predict sales using the ratio of the price change:

Q_new = Q_old × (P_new / P_old)^E

Numerical Example:
• Start: Price = $20, Sales = 1,000 units.
• Change: Raise Price to $22. Elasticity = -2.0.

1. Ratio: 22 / 20 = 1.10 (10% increase)
2. Power: 1.10^-2 = 0.8264
3. Result: 1,000 × 0.8264 = 826 units

Strategic Context: When to use this tool?

This tool bridges the gap when you face a Data Limitation:

Scenario A: You have historical data. (Sales at $10, $12, $15...).
You don't need this calculator. Run the Ln-Ln regression above to find your exact parameters.
Scenario B: You have NO data. (You only know you sell 1,000 units at $20).
You cannot run a regression. Instead, you estimate an elasticity (using Tool 2) and use the calculator below. It uses the Power Function logic to extrapolate the entire demand curve from your single data point.

Calculator Inputs

Enter your single known data point and estimated elasticity.

Current Price ($)

Current Quantity

Estimated Elasticity (E)

Must be negative (e.g., -2).

Variable Cost ($)

Current Profit

Optimal Price