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Rev. Jun. 20, 2014
This note was prepared by Rajkumar Venkatesan, Bank of America Research Associate Professor of Business Administration, and Paul W. Farris, Landmark Communications Professor of Business Administration. Copyright ¤ 2011 by the University of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail to [email protected]. No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying, recording, or otherwise—without the permission of the Darden School Foundation.
DESIGN OF PRICE AND ADVERTISING ELASTICITY MODELS
Introduction
The marketing mix that a manager may deploy can affect the sales of a product and can be categorized under the traditional four Ps of marketing (product, price, promotion, and placement). But the perennial question managers face concerns the combination of these different marketing-mix variables that will give them maximized sales, highest share, lowest inventory, or maximized margins. Quite often, these questions are answered by historical data: for example, past sales or market share for different levels of expenditures on these marketing- mix variables. In this note, we consider the design of models that allow managers to obtain robust price and advertising elasticity estimates.
Consider the following scenario: Belvedere vodka was introduced in the United States in
- This vodka traced its roots back to the Warsaw suburb of Żyrardów, Poland, and its production process went back more than 600 years. Lately, it had begun to observe a decline in its overall share of the vodka market. The company suspected the cause to be new market entrants that were capturing market share with effective advertising. To sustain the growth rate and defend its share from the competition, Belvedere was considering two options: increasing its advertising expenditure and/or reducing the pricing. Such a scenario is very common for most brands during the various stages of their brand (or product) life cycles. The first step toward solving this issue is to estimate the elasticity of a brand to its price and advertising.
Price Elasticity of Demand
Pricing is one of the most critical variables that marketers have problems with. Based on common sense, consumers tend to buy more of a product as its price goes down, and using the same logic, they will buy less if the price goes up. Price elasticity of demand ( Figure 1 ) is a measure to show the responsiveness of the quantity demanded of a good (or service) to a change in its price; it gives the percentage change in quantity demanded in response to a 1% change in
price (holding constant all the other variables in the marketing mix).^1 A product with a price elasticity above 1 is said to be elastic, as changes in demand are relatively large compared with changes in price. Correspondingly, a product whose elasticity goes below 1 is deemed inelastic.
Figure 1. Price elasticity of demand.
Source: Created by case writer.
Price elasticity of demand (PED) can be calculated using Equation 1 :
PED = [Change in Sales ÷ Change in Price] × [Price ÷ Sales] = ( ∆Q ÷ ∆P ) × ( P ÷ Q ) (1)
Or if we have a sample of historical sales and price data, then we can regress the sales against price, and the coefficient of this regression will give price elasticity as shown in Equation 2 : 2
PED = Coefficient of price when Ln (sales) is regressed on Ln (price) (2)
Here, we are assuming that the Ln - Ln model (i.e., dependent Ln (sales) regressed on independent Ln (price)) gives us a better linear model, which historically has been the case with most models, such as that in Equation 3 :
Ln (sales) = α 1 + β 1 × Ln (price) + H 1 (3)
where β 1 represents the price elasticity in the above case and H 1 is the random error term drawn
from a normal distribution (the standard assumption in a linear regression model).
(^1) This is essentially never the case—as will be explored in more detail later in this note. (^2) The coefficient and price elasticity are by definition not the same, but they are very closely related to each other, and in most cases, the coefficient is a close proxy for the elasticity.
Price elasticity can be derived as the ratio of change in quantity demanded (%∆ Q ) and percentage change in price (%∆ P ).
sensitivities) can be used for short-term advertising effects^3 —values less than 0 imply negative returns to advertising and greater than 1 imply the firm is underadvertising. So the value should range from 0 to 1. In our simple regression model above, we took advertising expenditure as one simple independent variable by combining expenditures for all possible media. But different media (e.g., print, display, in-store, television) may have varied effects on the demand of a product based on its characteristics. More analysis is required to study the effect of different media, and if required, more than one variable should be incorporated in the regression model to get a better-fitting model and help the marketing manager decide on the advertising expenditure—both the total amount and its distribution across different media.
Building a Comprehensive Model
If both PED and AED are significant, the regression model should include both price and advertising as independent variables ( Equation 6 ):
Ln (sales) = α + β 1 × Ln (price) + β 2 × Ln (advertising) + H 1 (6)
“Bias” is a commonly used term to describe the effect of omitted variables. It is used where there are systematic differences in the estimated elasticity (due to errors in estimation, not environmental differences) and the true elasticity in the market. This bias may be caused by omission of variables, which may be correlated with those included in the equation. The decision to include or omit certain variables in the model other than price and advertising will therefore depend on the correlation of a variable with the dependent variable and its correlation with other independent variables. If advertising elasticity is higher than the true value, then it is said to be a positive bias, but if it is lower than the true value, then it is called a negative bias. Conversely, if price elasticity is more negative than the true value, then it is said to be a positive bias, and if it is less negative, then it is called a negative bias (see Figure 2 ).
(^3) In the case of multiplicative models, the coefficients were elasticities, whereas in the case of linear models, elasticities can be estimated by multiplying the regression coefficient by the ratio of means of the dependent variable and the advertising measure.
Figure 2. Building a model.
If the true model is as follows:
Ln ( Y ) = α 0 + α 1 × Ln (price) + α 2 × Z + H,
but we estimate the model to be
Ln ( Y ) = β 0 + β 1 × Ln (price) + H,
the true value of coefficient β (^) 1 will be the sum of the estimated coefficient β (^) 1 and the bias β 1 true^ = β 1 + bias.
If r is the covariance between independent variables, Ln (price) and Z ,
then the bias can be proven to be the product of the coefficient of the omitted variable ( D 2 ) and
some function of covariance of independent variables [ f ( r )]:
Bias = α 2 × f ( r ).
x If the dependent variable is not related to the omitted variable, then there is no bias (=0).
x If the included independent variable (here, Ln (price)) is not correlated to the omitted variable (covariance is zero), then there is no bias (=0).
x If α 2 (correlation between the omitted variable and the dependent variable) and r (covariance between independent variables) are of same sign, then the bias is positive.
x If α 2 and r are of different signs, then the bias is negative.
Source: Created by case writer.
3
1
2
1 = true model, with all variables included 2 = estimated model with omitted variables—positive bias 3 = estimated model with omitted variables—negative bias
Linear regression model
Carry-over effect of advertising
Advertising rarely has an immediate effect on sales. If we take into account the effect of advertising on sales for the current period, more often than not, those effects would be in the form of spikes and they would be relatively small (i.e., quite fragile) as compared with other marketing variables. Some research indicates that the current effect of price is 20 times larger than the current effect of advertising. The portion of advertising that retains its effect and affects consumers even beyond the period of its exposure is known as the carry-over effect. Depending on the product type, consumer segment, and firm’s strategy, there could be several reasons for this carry-over effect: delayed consumer response due to their backup inventory, delayed exposure to the ad, shortage of retail inventory, and so on. Therefore, to account for the total effect of advertising, include both the current effect and all the carry-over effect.
The Koyck model provides a way to capture the carry-over effect of advertising: It enhances the basic linear marketing-mix model, by including a lagged dependent variable as an additional independent variable. So, as per the enhanced model, sales of the current period depend on sales of the prior period and all the independent variables that caused prior sales, plus the current values of the same independent variables.
If the original model (before Koyck) was as shown in Equation 7 :
Ln ( Yt ) = α + β 1 × Ln ( At ) + β 1 × Bt + H t (7)
then Equation 8 is the enhanced model (by Koyck):
Ln ( Yt ) = α + λ × Ln ( Yt – 1 ) + β 1 × Ln ( At ) + β 1 × Bt + H t. (8)
In this model, β 1 captures the current effect of advertising, while β (^) 1 × λ ÷ (1 − λ ) can be calculated to be the carry-over effect of advertising. The higher the value of factor λ , the longer the effect of advertising will be. Similarly, the smaller the value of λ , the shorter the effects of advertising will be (i.e., sales depend more on current advertising). The total effect of advertising is the sum of current and carry-over effects; that is, β (^) 1 ÷ (1 − λ).
If the advertising effects are positively correlated from one period to the next (i.e., the last period’s advertising has a positive correlation with current period’s advertising), and if the past advertising has a positive correlation with the current period’s sales, then the omission of the carry-over effect will result in a positive bias.
Contextual factors
Another factor that may come in to play is the disposable income of consumers in a region where a product is being sold. Consumers in countries (or regions) with high disposable income may be less price sensitive. If so, then higher income would lead to lower price elasticity
(i.e., less negative). At the same time, better-informed customers in a region (as well as stronger regulations and antitrust laws) may lead to increased price sensitivity.
Overall, the exogenous variables (e.g., GNP and sociodemographics such as average family income, family size) generally have a positive correlation with sales, and their exclusion could have a positive bias on the model. The regional context may also have a correlation with advertising, for example, due to differences in preferences, production cost structures, and restrictions.
Table 1 summarizes the effect of bias due to the omission of different variables from the marketing mix:
Table 1. Effect of bias in price and advertising elasticity.
Factor
Bias in Price Elasticity
Bias in Advertising Elasticity Product Quality + Distribution – Brand Life Cycle—Early + Absolute Sales + Time Series – – Include Carry-over + Contextual Factors (income, family size, etc.) + Source: Created by case writer.
Other factors to consider while designing the model include:
Promotion
Promotional activities can take one of two forms: (1) increasing product awareness through displays, campaigns, demonstrations, and so forth or (2) incentivizing consumers to try a company’s products through coupons, rebates, and so on. Normally, firms tend to run the incentive programs along with their strategy of charging higher prices: higher prices for existing customers and rebates to acquire new customers. In such a case, prices and promotions would be positively correlated. On the other hand, the other form of promotions (increasing awareness) is generally used concurrently with lower prices. The goal is to maximize consumer awareness, and in this case, the prices and promotions would be negatively correlated. In either case, inclusion of promotion characteristics is necessary to obtain a better distinction between price effects and promotion effects on sales.
Exhibit 1 DESIGN OF PRICE AND ADVERTISING ELASTICITY MODELS Regression of Ln (Sales) versus Ln (Price)
Year
Sales (thousands of units)
Ln (Sales) (thousands of units)
Price (dollars)
Ln (Price) (dollars)
Advertising (thousands of dollars)
Ln (Advertising) (thousands of dollars) 2007 410 6.016 215.44 5.373 20486.1 9. 2006 381 5.943 211.45 5.354 2923.5 7. 2005 365 5.900 207.45 5.335 4826.3 8. 2004 369 5.911 240.87 5.484 13726.6 9. 2003 339 5.826 241.33 5.486 10330.2 9. 2002 306 5.724 247.55 5.512 13473.6 9. 2001 273 5.609 240.48 5.483 9264.6 9.
Regression Statistics Multiple R 0. R Squared 0. Adjusted R Squared 0. Standard Error 0. Observations 7 Coefficients Standard Error t-Stat P-value Intercept 12.686 3.340 3.798 0. Ln (Price) −1.259 0.615 −2.048 0.
Source: Created by case writer.
6
5.3 5.35 5.4 5.45 5.5 5.
Ln
(Sales)
Ln (Price)
Ln (Sales) Linear (Ln (Sales))
Exhibit 2 DESIGN OF PRICE AND ADVERTISING ELASTICITY MODELS Regression of Ln (Sales) versus Ln (Advertising)
Regression Statistics Multiple R 0. R Squared 0. Adjusted R Squared −0. Standard Error 0. Observations 7 Coefficients Standard Error t-Stat P-value Intercept 5.963 0.850 7.018 0. Ln (advertising) −0.013 0.093 −0.137 0.
Source: Created by case writer.
6
8 8.5 9 9.5 10 10.
Ln
(Sales)
Ln (Advertising)
Ln (Sales) Linear (Ln (Sales))
