Variations in price elasticities

E L S E V I E R European Journal of Operational Research 88 (1996) 13-22

EUROPEAN JOURNAL

OF OPERATIONAL RESEARCH

Case Study

Variations in price elasticities

Jennifer George a Alan Mercer b,* He len Wilson c

a Caminus Energy Limited, Cambridge, CB30RA, UK b Department of Management Science, Lancaster University, Lancaster, LA1 4YX, UK

c Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4177, UK

Received 18 July 1995

Abstract

Multilevel analyses have been performed on weekly scanner data obtained from 74 stores of a large retailing chain for 30 competing items of a household product group. Discrete price change parameters for the brand being modelled reflect the fact that prices do not change from week to week. Their estimates show that the price elasticity is not constant. Variance components models for four of the products suggest that a brand's inherent attractiveness is affected by the social and demographic characteristics of the retail outlet's catchment area. However, random slopes models indicate that there are no systematically different price effects between stores.

Keywords: Pricing; Marketing; Retailing

1. Introduct ion

The two principal objectives of an empirical study set by the management of a leading fast moving consumer goods manufacturer were to 1. obtain a more comprehensive View of the price

response curve than is represented by the constant elasticity model,

2. determine whether any particular subset of stores is consistently more or less price sensi- tive and identify any factors which appear to influence price elasticity by store. Weekly scanner data for a year were available

from each of 74 stores of one of the UK's largest retailing chains. The household-product group se-

* Corresponding author.

lected was dominated by the retailer's own label (A) and the brands of two major competitors (B and C); together they accounted for 90% of the sales through this particular chain. There were three forms of the basic product; the original ready-to-use, refill and concentrate. The retailer and competitor C also marketed a deluxe, pre- mium priced variety in the orfginal and concentrate forms. As a large and small size were usually sold, scanner data for a total of 30 products were analysed. Although the products are used in only about 60% of UK homes, with the interpurchase interval being approximately one month, there was no evidence that the overall market size changed dur ing the year when the data were recorded.

The price of a product was set by the retailer to be normally the same in each store and cer-

0377-2217/96/$09.50 © 1996 Elsevier Science B.V. All rights reserved SSDI 0377-2217(95)00203-0

i4 z George et al. /European Journal of Operational Research 88 (1996) 13-22

tainly for there never to be more than two different prices charged across the chain's outlets at any one time. Consequently there were only 54 price increases and 28 decreases during the year for all the 30 products. Bonus packs of the 10% extra type were also available for five of the products during some of the weeks, when the proportion of a product's sales in bonus packs was known for each store.

The two manufacturers of the branded products advertised extensively on television, with C having distinct campaigns for its basic and deluxe varieties. The advertising schedules and therefore the television viewing ratings (TVRs) differed between the television areas of the UK. Manage- ment had determined from earlier studies that the weekly advertising retention factor be taken as 0.8, so that this value was used to calculate weekly adstocks from the daily TVRs. The best selling products were always available on the shelves but the remainder suffered from occa- sional stockouts.

2. Background literature

Tellis (1988) carried out a meta-analysis of price elasticities estimated by econometric modelling and reported in the literature during the period 1960-1985. Using 367 elasticities from about 220 different brands/markets, he concluded that there were variations due to factors such as product category, period of the life-cycle and the country from which the data originated. Significantly, the econometric studies and their analysis were based on the assumption that the price elasticities are constant at any particular point in time. There is also growing evidence, summarised by Gatignon (1993), that other ele- ments of the marketing mix, such as advertising, affect price elasticities.

Eastlack and Rao (1986) detected extreme price sensitivity in the sales of V-8 cocktail veg- etable juice for a short period of time after a major price increase, thereby inferring that the price elasticity was not constant. From a study limited to 133 people, who were given the oppor- tunity each week to purchase on a door-to-door

basis one brand from each of three product fields, Motes (1987) concluded that their reactions differed between equivalent price increases and decreases. He also speculated that implementing a large price increase in a series of small incre- ments would be less damaging than a single step. The same conclusion was drawn by Brearley and Mercer (1995) in a study of head-to-head compe- tition between two bus operators, whose own price elasticities increased in magnitude as the price difference resulting from any change be- came larger.

The availability of scanner data has allowed previously unaddressed problems to be considered but it has also presented problems of how best to analyse extensive data sets. Hoch et al. (1995) argued that previous attempts to relate store level price elasticities to the catchment area demographic and competitive variables had failed because consumer panel data were used, only one or two product categories were considered and the effects were swamped by the various promotional activities. Instead, they used scanner data for 18 product categories collected over a period of 160 weeks by a major chain of 83 supermarkets located in the Chicago metropolitan area. These data were reduced to manageable proportions by grouping the 4636 individual lines into 265 so- called items, giving an average of 15 items per category. The logarithm of an item's unit sales was modelled as a linear additive function of the logarithms of the prices of the items in the same category, promotional variables and lagged sales volume; temporary price reductions and deals were represented by dummy variables. Even at the item level, if all the parameters were store specific, their number would be extremely large. Consequently only the intercept and own price elasticity were allowed to vary across both items and stores. Then these elasticities were combined to give the 18 product category price elasticities, which were related to the catchment area statistics.

When Blattberg and George (1991) applied ordinary least squares to estimate the price and promotion elasticities from two years weekly scanner data for 4 brands of toilet tissue sold in 3 retail chains, they found that the 9 parameters in

J. George et al. /European Journal of Operational Research 88 (1996) 13-22 15

each model varied too much between the 12 different chain-brands, frequently having the wrong sign. The problem was overcome by using shrinkage estimation, whereby a prior distribution was imposed on selected parameters to shrink each towards a hyperparameter, although none was constrained to equal another. However, the constraints were selected with the intention of partitioning the models into homogeneous groups having the most similar coefficients. The shrinkage estimates were more managerially credible and the predictions were improved, so that the parameters formed the basis of practical marketing recommendations. Notably, the shrinkage estimation was based on hierarchical models, which recognise that weekly sales are nested within specific stores.

Statisticians recoguised s o m e t i m e ago that carrying out any analysis which does not acknowl- edge the existence of a hierarchy creates poten- tially serious difficulties, because clustering will usually result in the standard errors of regression coefficients being underestimated. For example, Aitkin a n d Longford (1986) demonstrated that ignoring the nesting of students in schools produced misleading conclusions. Moreover, fitting separate regression equations to the marketing data for each store is both inefficient and inap- propriate for generalising. It is inefficient because the procedure involves estimating many more coefficients than multilevel modelling and not generalisable due to the fact that no information is produced about the underlying population of stores in the retail chain. The problems resulting from the correlations amongst observations emanating from a multilevel structure remained intractable until the development by Dempster et al. (1981) of the EM algorithmic approach to variance components models, which is so-called because each iteration of the algorithm consists of an expectation step followed by a maximisation step. Further developments for hierarchical mixed linear models, such as described by Goldstein (1987), allow the coefficient of any explanatory variable to be random at any level of the hierarchy and the random coefficients at each level to have any pattern of variances and covariances. By contrast with shrinkage estimation, which links

parameters across models, the method for fitting mixed linear models to data with a nested structure is unconstrained.

3. Pr ice r e s p o n s e curves

The study by Hoch et al. (1995) is an example of the modelling of the logarithm of a sales measure as a linear additive function of various marketing pressures, frequently in logarithmic form. A similar basic model was applied in the present empirical study but the dependent variables were taken to be the market shares of the 30 products in each of the 74 stores. However, there were two modifications which distinguish the model from those appearing in the marketing literature. In a multilevel representation with level-1 weeks being nested within level-2 stores, the only random parameters of a variance components model were the intercept variances at each level, reflecting the variation between weeks within stores and between stores. This had the effect of reducing the number of models to be fitted from 2220 to 30. The second modification was to recognise that a product's own price has few changes and to represent those by dummy variables. Lagged market share was not included as an independent variable because in addition to the arguments of Mercer (1991), Maddala (1971) has shown that parameter estimates can be bi- ased seriously when the lagged dependent variable is included in variance components models as an independent variable.

The prices of the 29 products competing with that being modelled were treated as continuous variables, in order to restrict the number of parameters. Perfect correlation between the prices of some products prevented all being included. Thus the model is given by the expression

In Si j k = a i -Jr E ~ i l X i j k l -[- E ~ir In Pryk + eij -k fiyk I r

+ terms for bonus packs, distribution

and advertising,

where Si j k is the market share of product i in store j" during week k, a i is the average intercept

16 J. George et aL / European Journal of Operational Research 88 (1996) 13-22

for product i across all stores, Xijkt is a dummy variable taking value 0 before price change l for product i in store j and value 1 after the change, [3il is the proportional change in share of product i for price change l, Prik is the equivalent unit volume price of competing product r in store j at time k, Yir is the price cross-elasticity of competing product r, eq is the level-2 residual, assumed to have zero mean and constant variance, repre- senting the departure of the j-th store's actual intercept from the overall value a i, fijlc is the level-1 residual for product i in store j during week k, assumed to have zero mean and constant variance and to be uncorrelated with eij.

Bonus packs were modelled in the usual man- ner by the inclusion of (0, 1) dummy variables but no carry-over effects due to stocking up were allowed. A distribution term for each product not present in all the stores every week was also

included as a dummy variable, which for those weeks when the product was present was equal to the share of the product averaged over the weeks when it was available in the particular store, and otherwise was zero. Advertising for the three brand groups seen by consumers in the television area where a store is located was represented by the logarithms of the adstocks.

The ML3 software described by Prosser et al. (1991) was used in the study. This is an iterative generalised least squares algorithm, which pro- vides consistent estimates of model parameters, and maximum likelihood estimates when normal- ity assumptions are valid. After fitting a saturated model, the level-I residual plots were examined for heteroscedasticity and consequently a further random term gi ikQjk was added to the model, where Qi~ is the total sales of all products in store j during week k. This had the effect of

A ( - 2 8 . 4 ~ 6 2 . 2 ) A

- 1 0 - f l - 6 -4 -2

-111

S h a r e

change (~1

P r i c e change (~ )

2 ~ 6 B 10 12 14

Fig. 1. T h e pr ice r e s p o n s e r e l a t ionsh ip for r e t a i l e r A ' s own labe l products . Key: Basic, r egu l a r - t r iangle ; De luxe - inver ted t r i ang le ; Ref i l l - squa re su r round ; C o n c e n t r a t e - c i rcu la r su r round ; S m a l l / l a r g e size - h o l l o w / s o l i d t r iangle .

Z George et al./European Journal of Operational Research 88 (1996) 13-22 17

introducing two additional random parameters; a variance of go e and a covariance between the level-1 random variables fijk and giie" The likelihood ratio statistic was then used to test the null hypothesis of zero parameter values. Complex variation at level-I was found for some but not for all of the products. Then stepwise elimination was applied to the appropriate model. First, negative cross-price, negative own distribution and positive cross-distribution terms were removed because the sign was wrong. However, no hypothesis was made about the correct sign of the ad- stock terms. Second, all non-significant terms were removed after comparing each parameter estimate to its standard error.

The only parameters of managerial interest in the present empirical study were the proportional changes in market share,/3it, for the various price changes of all the products. The significant estimates for the retailer's own: label (A) and the two major competitors' brands (B and C) are given in Figs. 1-3 respectively as a function of the price

change, of which there are 43 increases and 23 decreases. The proportional price change is defined to b e twice the difference between the prices before and after the change, divided by their sum. Thus a price change and its reversal are defined to have the same proportional price change, although of opposite signs. Consequently shares before and after these two price changes are equal if the elasticities, as defined, have the same magnitude but opposite signs. Because the product is available in various combinations of the original ready-to-use, refill and concentrate forms, basic and deluxe varieties, large and small sizes, the numbers sold by A, B and C are 7, 6 and l0 respectively, Consequently there are insuf- ficient price changes to examine their effects on the changes in share separately for each product of the three brands. However, a visual examina- tion of Figs. 1-3 in turn reveals such close simi- larities between different products tha t all may be considered to contribute to a single relation~ ship for each brand. The percentage change in

Zg (-1B.4, 64.1)

- 10 - 8 - 6 - 4 -2

- 10

-20~

5 h a r e

change (~)~

10

Price change (9)

4 6 13 ~f l 12 14 l f i Ii~

®

-50

Fig. 2. The price response relationship for competitor B. Key as for Fig. 1.

18 J. George et aL ~European Journal of Operational Research 88 (1996) 13-22

Table 1 The relationships between the percentage changes in price and share Product Price Best form Price coefficient

change of price Value Std. error

A Increase Square - 0.235 0.032 Decrease Square 0.084 0.020

B Increase Linear - 1.595 0.350 Decrease Square 0.189 0.007

C Increase Cube - 0.026 0.003 Decrease Square 0.249 0.030

share was fitted by a linear function through the origin to the percentage change in price, its square and its cube, separately for price increases and decreases, using weighted least squares to reflect the accuracies of the estima.ted share changes. Table 1 contains the results, all of which are extremely significant statistically. In four of the instances, the best statistical fit is obtained with

the square of the price change, for which the likelihood ratio compared with the linear price change is 15% higher for decreases of brand A and 100% or more for increases of brand A and for decreases of brands B and C. For increases of brand B, the linear form is preferred, with a 500% increase in the likelihood compared to the squared form. Similarly for increases of brand C, the likelihood ratio for the cubic price change is 74% higher than for the square. Consequently there is very real evidence to indicate that the price elasticity in this market varies with the magnitude of the proportional price change and the six most likely relationships are shown in Figs. 1-3.

Having assumed that all products contribute to each relationship, this premise has been checked by calculating whether the observed changes in share are greater or less than their expectations. The results are given in Table 2, where it can be seen that the assumption is justified, so that whilst

g a i

-10 -g -6 - 4 -2

S h a r e change (%)

3O

2 4 B 8 10

~ 12 P r i c e change (%)

Fig. 3. The price response relationship for competitor C. Key as for Fig. 1.

J. George et al. / European Journal of Operational Research 88 (1996) 13-22 19

Table 2 Number of products with share change greater and less than estimate

Product Greater Less

Variety Basic 27 27 Deluxe 7 4

Form Ready-to-use 19 17 Refill 6 6 Concentrate 9 8

Size Small 14 11 Large 20 20

the price elasticity relationships differ between brands, they are unaffected by the product variety, form or size within a brand.

From Table 1, the loss of share for a given price increase of brand A is always larger than the increase in share for the corresponding price decrease. However for brand B, the loss for a price increase only exceeds the gain in share for the corresponding price decrease if the change is less than 8.4%. Conversely, the loss for a price increase of brand C is larger than the gain in share from a decrease whenever the change is more than 9.8%. Interestingly, company C has restricted its price changes to less than about 10%, which suggests that it might have reached the correct conclusions from practical experience. By contrast, almost half of company B's changes are less than 8.4% but it can also be making entirely right decisions, depending on the pur- poses of the price changes.

For permanent price increases, frequent small changes are preferable to one large increase, irrespective of brand, and conversely for permanent price decreases. However, the situation is different for price promotions, which involve a price decrease t o be reversed later by the same price increase. Retailer A should not hold price promotions for its own label products, because it loses more share when it restores the price than it gained at the t i m e of the price reduction. Obviously, the reason is that the own-label brand is the cheapest. At the other extreme, brand B's promotions should never involve price changes of less than 8.4%. Its prices are higher than those of

the middle-priced brand, C, so that any smaller price cut will not have sufficient impact on pur- chasers generally, whilst the subsequent increase simply restores price levels normally associated with the brand. With the brand leader C attempt- ing to differentiate its products from the own-label brands and maintain a price differential compared with B, its price changes during promotions should be less than 9.8%. In summary, the results for the three brands indicate that price elasticities are not constant. How they vary is influenced by the prices of competing products, which agrees with the conclusions of Brearley and Mercer (1995).

4. Store effects

4.1. I n h e r e n t a t t r a c t i v e n e s s

The intercept for product i in store j, which represents the inherent attractiveness of product i to consumers shopping at the j-th store, is given by oL i -I- eij. Consequently values of eii calculated by the ML3 software have been examined for four products. Three were the retailer's own label (A) and the two competitors' (B and C) brands of the large size, ready-to-use form of the basic product. The fourth was the small size, refill of company B. Whilst caution must be exercised in interpreting any coefficients, those from the multilevel analysis are more reliable than estimates from OLS regression, because the model uses data for all the weeks and stores in the. sample.

Each store was classified according to the television area in which it was located and 6 were found to be scattered across four TV areas, Anal- yses of the level-2 residuals for the remaining 68 stores produced the same order of the five television areas containing 7 or more stores, when ranked separately according to the average value of ei j for the ready-to-use products of A and C. Moreover, the average values for A and C were not different for any TV area, even at the 25% significance level, so they were combined. The rank of the resulting level-2 residuals, with the percentage levels of statistical significance be-

20 J. George et al. / European Journal of Operational Research 88 (1996) 13-22

tween consecutive values, gives the following order for television areas.

Highest: North West (10) Midlands (5) Yorkshire (25) London (10) North East: Lowest

The order for the comparable product of company B is different, having the highest above average value in the London area and the lowest in Yorkshire. However the order of television areas for company B's refill is the exact opposite of' that for the ready-to-use products of A and C, apart from a non-significant reversal between London and Yorkshire, which is significantly be- low the North East at almost the 1% level.

The manufacturer had previously carried out studies to cluster retailer A's stores according to the social and demographic characteristics of the catchment areas. Thus the cluster number, rang- ing from 1 to 7, was known for 55 of the 74 stores but the details of the cluster analysis were un- known. Analyses of the level-2 residuals showed that as a store's catchment area becomes more down-market, the inherent attractiveness (with significance levels in 'parentheses) 1. increases for A's own label ready-to-use prod-

uct (5%), 2. decreases for the ready-to-use product of brand

B (8%), 3. is unaffected for brand C, 4. decreases for company B's refill (3%), if the

two most down-market clusters are ignored; one of these clusters contains only four stores and the other cluster is also an outlier in the analyses for the ready-to-use products, sug- gesting that perhaps the particular cluster is not well-defined. Company B advertises heavily to promote a

quality image and its prices are correspondingly

high. Company C is the brand leader and whilst it also advertises, its products are priced closer to retailer A's own label products than to those of B, in order to protect its market share. It is therefore not surprising that both products of B have a higher inherent attractiveness to the more up-market customers, whereas A has greater appeal to the less affluent shopper and C, being between the two, is equally attractive to all classes of customer. It is fallacious to associate TV area with social grade, because retail outlets in the same TV area can have very different catchment areas; no interaction was sought in the analysis, as the cluster number was not known for all stores. The manufacturer 's management described the orderings of the inherent attractive- nesses as reflecting historical marketing activities. Company B's general advertising policy has been to concentrate on areas where most customers can be reached. As the TV area with the largest number of households has the highest inherent attractiveness for its ready-to-use product, and so on for the top three TV areas, it appears that there may be a long-term advertising effect. The consistency of the ordering for the other three products indicates a very real effect, although the reason must be speculative. The inherent attrac- tivenesses of the other two ready-to-use products reflect the retailer's geographical expansion, sug- gesting that the appeal of both a brand leader and own label item increase with familiarity. The refill, which was launched by B as a value-for-mo- ney product to attack C, seems to be more attractive where its competitors are relatively less ap- pealing.

4.2. Price effects

The preceding analyses have assumed that the price effects are the same for all the stores. The variation in the proportional change in share of product i for price change l for the stores in the whole population was examined for the four particular products by adding the term Ehij l gijkl t o

the previous expression for In Sij k. In this random slopes model, hij t is a level-2 residual for store j, assumed to have zero mean and constant variance and to be correlated with eij, as both vary at

J. George et al. ~European Journal

the same level. The ML3 software was again used to calculate the multi-level predictions of the residuals and the values of the proportional change in share of product i in store j for price change l were estimated by (~i l + hiyt), denoted by/3iy t.

Analyses of variance were performed on the values of flijl for all four products, with the same results. As would be expected, given the form of the price response curves, the differences between the various price changes were extremely significant. However, neither the television area in which a store is located nor the socio-demographic categorisation of a store's catchment area had any effect. Identical conclusions were reached when the price elasticities derived from the values of flijl were analysed. Thus it must be concluded that in this UK market, price effects do not differ systematically between stores, so that pricing is a national marketing tool. Indeed, major U K retailing chains almost invariably imple- ment such a policy. This contrasts with the differences in promotional pricing strategies between the Chicago area supermarkets studied by Hoch et al. (1995), who found that two-thirds of the store-to-store variation in price elasticities was explained by eleven demographic and competitive variables, with the latter being of relatively little importance. Apart from the differences between the UK and US retailing environments, there are two other possible factors to explain the opposite conclusions. The catchment areas for the UK stores would most likely be larger and the retailer would probably have a wider choice of sites in relation to the total number of outlets, so that relatively more demographically homogeneous lo- cations could be selected than those of the 83 Chicago supermarkets. However, using the seven generally defined dusters provided by the UK manufacturer would be less likely to identify any real differences between the price elasticities of the stores than would specifically fitting price elasticities to a set of demographic measures.

5. Concluding remarks

Multi level analyses have facilitated the estimation of reliable parameters capable of market-

of Operational Research 88 (1996) 13-22 21

ing interpretation. By using a discontinuous representation of a product 's own price, it has been shown that price elasticities are not constant but vary with the percentage price change. The impli- cations have considerable practical importance for a company wishing to increase its profit mar- gins over time and for how price promotions are presented to the customer. Careful analysis and interpretation are needed, because the elasticities for the changes at the start and end of short term promotions are likely to be relatively inaccurate, especially if the interpurchase interval is compar- atively long.

The variations in price elasticity with the percentage price change differ between brands according to how each brand is positioned in the market, being affected generally by the competitors' pricing policies. This conclusion implies t h e results obtained in recent years by researchers using an experimental approach, either by having consumers make purchases in their own homes at r egu la r intervals over a period of time or by making a series of choices on a single occasion, are liable to be invalid unless the competitive purchasing environment was represented realisti- cally. Pricing research in face-to-face situations, using purchase probability scales like the Juster Scale, are even more likely to give incorrect re- suits.

Further analyses for four of the largest selling products in this group suggest that the level of a product 's share varies between stores due to both the catchment area and long term developments by the manufacturer and the retailer. By contrast, price changes appear to have short term effects, which are not significantly different either region- ally or for customers with different backgrounds, so that national pricing policies may be imple- mented.

The results from the modelling include a wealth of other information, including the price cross-elasticities. These were assumed to be constant with continuous price variables, because the software is incapable of estimating the very large number of parameters, which would result from all price changes having distinct values. However, the most interesting additional results of marketing significance concern the advertising effects.

22 J. George et al. ~European Journal of Operational Research 88 (1996) 13-22

F o r example , c o m p a n y C wou ld typical ly a l te r - na t e b e t w e e n adver t i s ing campa igns las t ing sev- e ra l mon ths for its bas ic and de luxe var ie t ies . I t was found tha t the adver t i s ing o f the de luxe b r a n d had a nega t ive effect on its own sales bu t in- c r ea s ed those for C 's bas ic var ie ty and the sales of r e t a i l e r A ' s own labe l p roduc t s . A p p a r e n t l y consumers we re a t t r ac t ed by C 's adver t i s ing of the de luxe var ie ty and many r e m e m b e r e d the b r a n d n a m e b u t p u r c h a s e d C 's c h e a p e r bas ic p roduc t s , when c o n f r o n t e d by the pr ices at t he po in t -of -sa le . O t h e r s s imply chose the much c h e a p e r own labe l p roduc t s , which a r e invar iab ly d i sp layed m o r e p r o m i n e n t l y on the re ta i l ou t l e t shelves.

Acknowledgements

T h e au tho r s a re i n d e b t e d to D a v i d Pegg for the d a t a p r e p a r a t i o n and p re l im ina ry analyses and to K a t i a Lazza r in i for he r help. T h a n k s a r e also due to t he m a n a g e m e n t o f t he c o m p a n y which m a d e the d a t a avai lable .

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