## Author: Frederico Zornig

*Price elasticity models provide valuable insights into consumer behavior and market dynamics. By quantifying the responsiveness of demand to price changes, businesses can develop more effective pricing strategies, optimize promotional efforts, and drive revenue growth with dynamic pricing. Understanding and applying price elasticity concepts empower businesses to make informed pricing decisions in an ever-changing marketplace, as the author explains. Frederico Zornig (fzornig@quantiz.com.br) is the CEO of Quantiz Pricing Solutions.*

*The Pricing Advisor, June 2024*

Price elasticity models are essential tools for businesses seeking to optimize pricing strategies, understand consumer behavior, and forecast sales and revenues. By quantifying the responsiveness of demand to changes in price, these models enable businesses to make informed decisions regarding pricing adjustments, product positioning, and revenue maximization. In this article, I will delve into the concept of price elasticity, explore various elasticity models, provide formulas, and offer real-world examples to illustrate their application.

**What is Price Elasticity?**

Price elasticity measures the variation of demand for a product or service to changes in its price. It quantifies the percentage change in quantity demanded in response to a percentage change in price. When demand is highly responsive to price changes, we say it is very elastic, indicating that consumers are sensitive to price fluctuations. In this scenario, price increases should be managed with care. Conversely, when demand shows little response to price changes, it is inelastic. In this case you have a great opportunity to increase prices profitably.

**Types of Price Elasticity Models:**

**Price Elasticity of Demand (PED):** PED measures the responsiveness of quantity demanded to change in price while holding other factors constant. The formula for calculating PED is:

PED = (% Change in Quantity Demanded) / (% Change in Price)

Example: If a 10% decrease in the price of a product leads to a 20% increase in quantity demanded, the PED would be:

PED = (20%)/(−10%) = −2.

This indicates that demand is elastic, as the percentage change in quantity demanded is greater than the percentage change in price.

Despite the simplicity of this formula, some considerations need to be made. Price increases or reductions to measure elasticity need to be real and not nominal. For example, if you are measuring the elasticity of a product from one year to the next where the price increase was 5% but inflation or your competitor rose prices by 6%, in fact we cannot assume a price increase but rather a 1% price reduction.

Furthermore, the volumes observed may also vary depending on the market. Imagine a volume growing 20% per year because the market grows 20%. This growth needs to be considered. Another way to analyze volume changes, if we have the information, is to measure the variation in the market share of the product being analyzed. This way, price-independent demand fluctuations are better addressed, and we can have a more assertive result.

**Log-Log Price Elasticity:** Log-log elasticity models are a type of econometric model commonly used to estimate elasticities, including price elasticity of demand. They assume that the relationship between variables, such as price and quantity demanded, is linear. In other words, the percentage change in one variable is proportional to the percentage change in another variable when both are expressed as logarithms.

Basic Formulation: Let’s consider a simple log-log model for estimating price elasticity of demand:

ln(Q)=β0+β1ln(P)+ϵ

Where:

- Q is the quantity demanded of the product.
- P is the price of the product.
- β0 and β1 are coefficients to be estimated.
- ϵ is the error term representing factors not accounted for in the model.

Taking the natural logarithm (ln) of both sides of the equation linearizes the relationship between quantity demanded and price. The coefficient β1 represents the elasticity of quantity demanded with respect to price. If β1 is greater than 1, demand is elastic; if it is less than 1, demand is inelastic.

**Advantages of Log-Log Models:**

**A. Linearization:** By taking the natural logarithm of both sides of the equation, the log-log model transforms the non-linear relationship between price and quantity demanded into a linear relationship, making it easier to estimate using linear regression techniques.

**B. Elasticity Interpretation:** The coefficient β1\beta_1β1 directly represents the price elasticity of demand, providing a clear and interpretable measure of responsiveness.

**C. Flexibility:** Log-log models can capture non-linear relationships between price and quantity demanded more effectively than simple linear models, allowing for more accurate estimation of elasticities.

**Important Considerations to use this model:**

**Assumption of Linearity:** Log-log models assume a log-linear relationship between price and quantity demanded. While this assumption holds well for many products in practice, it may not be appropriate for all cases.

**Data Quality:** As with any statistical model, the quality of the estimates depends on the quality of the data and the appropriateness of the model specification. Care should be taken to ensure that the data used for estimation are accurate and representative of the market.

In conclusion, log-log elasticity models provide a powerful framework for estimating price elasticity of demand and other elasticities. By transforming the relationship between variables into a linear form, these models enable economists and analysts to estimate elasticities more accurately and interpret the results more intuitively, aiding in decision-making for businesses and policymakers alike.

**Double Machine Learning Elasticity: **(Double ML) is a somewhat new methodology used for estimating causal effects. While it’s primarily used in various fields, including economics, its application to estimate price elasticity is particularly interesting.

Understanding Double ML: In traditional econometrics, estimating causal effects, such as price elasticity, often involves controlling for confounding variables that might bias the estimates. Double ML provides a robust framework to address this issue by combining techniques from machine learning with established econometric methods.

**Double ML calculation involves two main steps:**

**One – Prediction:** In this stage, machine learning algorithms are used to predict the outcome variable (e.g., quantity demanded) and the treatment variable (e.g., price) separately. Each prediction model is estimated independently.

Using historical data, machine learning models are trained to predict the quantity demanded based on various factors such as price, income, demographics, and other relevant variables. This model estimates the expected quantity demanded for each observation.

Another machine learning model is trained to predict the price of the product or service based on factors such as cost, competition, seasonality, and other market conditions. This model estimates the expected price for each observation.

**Tw0 – Estimation:** In this stage, the estimated treatment effect is calculated using the residuals obtained from the first stage predictions. Econometric methods, such as instrumental variables or propensity score weighting, are then applied to estimate the causal effect of interest (e.g., price elasticity) using the predicted and residualized data. After obtaining the predicted quantity demanded and predicted price for each observation, residuals are calculated by subtracting the predicted values from the actual values.

Statistics techniques, such as variable regression or propensity score weighting, are then applied using these residuals to estimate the price elasticity of demand.

**Benefits of Double ML for Price Elasticity Estimation:**

- Reduced Bias: By controlling for confounding variables in the prediction stage, Double ML reduces bias in the estimation of price elasticity, providing more accurate results.
- Flexible Modeling
**:**Double ML allows for flexibility in modeling complex relationships between price, quantity demanded, and other relevant factors, capturing nonlinearities and interactions more effectively than traditional methods. - Robustness: Double ML provides robust estimates even in the presence of unobserved confounding variables or omitted variable bias, enhancing the reliability of the estimated price elasticity.

In conclusion, Double ML offers a powerful framework for estimating price elasticity by leveraging the strengths of both machine learning and statistical methods. Its ability to address confounding variables and produce robust estimates makes it a valuable tool for businesses seeking to understand consumer behavior and optimize pricing strategies.

**Real-World Applications:**

- Price Optimization: Companies can use price elasticity models to determine the optimal price for their products. By understanding the price sensitivity of consumers, businesses can set prices to maximize revenue or market share.
- Promotional Planning: Price elasticity models help businesses evaluate the effectiveness of promotional activities. For example, a company can assess the impact of a discount or promotion on sales volume and revenue.
- Product Development/Adjustments: Price elasticity analysis informs product development decisions by identifying consumer preferences and willingness to pay for new features or enhancements.

**Conclusion:**

Price elasticity models provide valuable insights into consumer behavior and market dynamics. By quantifying the responsiveness of demand to price changes, businesses can develop more effective pricing strategies, optimize promotional efforts, and drive revenue growth with dynamic pricing. Understanding and applying price elasticity concepts empower businesses to make informed pricing decisions in an ever-changing marketplace.

**References:**

Rafizadeh, Nima, “Debiasing the Price Elasticity of Gasoline Demand with Double Machine Learning (March, 2024)”.

Available at SSRN: https://ssrn.com/abstract=4771121 or http://dx.doi.org/10.2139/ssrn.4771121

Kenneth Benoit, “Linear Regression Models with Logarithmic Transformations (March, 2017)”. London School of Economics.

Available at SSRN: https://kenbenoit.net/assets/courses/ME104/logmodels2.pdf