Price elasticity, forecasting and new product diffusion
Estimate how demand responds to price, forecast sales with exponential smoothing and ARIMA, and model the adoption of new products with the Bass model.
Three questions come up in every planning cycle: what happens to sales if we change the price, how much will we sell next year, and how fast will a new product be adopted? This lesson covers the standard analytics answer to each.
Price elasticity of demand
Price elasticity is the percentage change in quantity sold for a 1% change in price. An elasticity of −2 means a 1% price increase reduces quantity by 2%. Demand is elastic when the elasticity is below −1 (price cuts raise revenue) and inelastic between −1 and 0 (price increases raise revenue). The standard estimate comes from a log-log regression of units sold on price, with controls for promotions, competitor prices and seasonality: the coefficient of ln(price) is the elasticity.
Estimate price elasticity
- 156 weeks of unit sales, own price, competitor price and promotions.
- Read the own-price elasticity and its CI, and the cross-price elasticity.
- Look at the table of price changes and the profit-maximising price.
In the example (156 weeks), the own-price elasticity is −1.74 (95% CI [−1.89, −1.59]): demand is elastic. A 10% price cut raises quantity by 20.1% and revenue by 8.1%, but lowers profit by 3.1% because of the unit cost of 2.20. The cross-price elasticity with the competitor’s price is +0.67: when the competitor raises its price by 1%, our sales rise by 0.67%, so the products are substitutes. Promotion weeks sell about 23% more (a coefficient of 0.21 on the log scale, e0.21 − 1 ≈ 0.23). The profit-maximising price, by the Lerner rule P = c·ε / (1 + ε), is 5.19.
One caution matters more than all the others: elasticities from observational data are biased if prices respond to demand (for example, if you raise prices when demand is high). Price experiments, or at least variation driven by costs rather than demand, give more credible estimates.
Sales forecasting
Forecasts drive inventory, staffing and budgets. Most business series combine a level, a trend and seasonality. Exponential smoothing models (simple, Holt’s linear and damped trend, Holt-Winters additive and multiplicative) and ARIMA models capture these components in different ways.
Forecast monthly sales
- 72 months of sales with trend and seasonality.
- Compare the models on the 12-month holdout.
- Read the forecast and its prediction intervals.
The tool fits every model to the first 60 months and forecasts the last 12, a holdout test that shows how each model would have performed in reality. Holt-Winters additive wins clearly with a holdout MAPE (mean absolute percentage error) of 1.8%, compared with 14.0% for simple exponential smoothing and 8.9% for ARIMA, because the series has strong seasonality. The chosen model is refitted to all 72 months and forecasts 28,148 units for the next 12 months, 9.7% up on the last 12. The Ljung-Box test (p = .798) shows that the residuals are white noise: the model has captured the systematic patterns.
Always report prediction intervals, which widen the further ahead you forecast: the 95% interval for the final month runs from 2,677 to 3,190. A point forecast without an interval invites overconfidence.
The Bass diffusion model
For a new product there is no sales history to extrapolate. The Bass model (Bass, 1969) describes adoption as the result of two forces: innovation (p), adoption driven by external influence such as advertising, and imitation (q), adoption driven by previous adopters through word of mouth. Together with the market potential m, they produce the familiar S-shaped cumulative adoption curve.
Model the adoption of a new product
- 14 years of adoptions of a new durable product.
- Read p (innovation), q (imitation) and m (market potential).
- Look at the forecast of the remaining adoptions.
Fitted to 14 years of adoption data (R² = .978), the model gives p = 0.019, q = 0.413 and m = 48,336 adopters. Imitation is 21 times stronger than innovation, so adoption is driven mainly by word of mouth. Adoption peaked about 7.1 periods after launch at roughly 5,466 adopters per period, and 94.8% of the market potential has already adopted; the forecast for the coming years is a long tail of late adopters (827 next year, falling to 43 by 2032). Typical values across durable goods are p ≈ 0.03 and q ≈ 0.38; for a product not yet launched, analysts often use p and q from analogous products together with a survey-based estimate of m.
Reporting
A log-log demand model (n = 156 weeks, R² = .82) estimated an own-price elasticity of −1.74, 95% CI [−1.89, −1.59], and a cross-price elasticity of 0.67. Holt-Winters additive exponential smoothing, selected by holdout accuracy (MAPE = 1.8%), forecast 28,148 units over the next 12 months. A Bass diffusion model (R² = .98) gave p = 0.019, q = 0.413 and m = 48,336.
Key takeaways
- Elasticity below −1 is elastic: price cuts raise revenue, but check profit with costs.
- Cross-price elasticity above zero means substitutes; observational elasticities can be biased.
- Choose forecasting models on holdout accuracy and always report prediction intervals.
- The Bass model separates innovation from imitation and estimates the market potential.
Check your understanding
Answer each question to see the explanation. Finishing the quiz marks this lesson as complete.