Estimating Automotive Elasticities from SegmentElasticities and First Choice/Second Choice Data

Robert F. BordleyJune 15, 2006

AbstractOf the share lost to one product because of a price change, diversionfractions are the fractions of that lost share going to each of the otherproducts. This paper expresses product cross-elasticities in terms of di-version fractions and a scaling factor.Since the automotive market includes more than 200 products, time-series data is insu�cient for estimating all elasticities. Instead this pa-per estimates automotive elasticities by specifying the diversion fractionsusing cross-sectional �rst choice/second choice data and estimating theremaining scaling factor and own-elasticities using more aggregate elas-ticities estimated from time-series.

1 INTRODUCTION1.1 Cross-sectional Data Constraining Time-Series Mod-

elsKnowing product demand-price elasticities(Hagerty, Carman and Russell,1988;Hauser,1988; Tellis,1988) is central to many corporate product planning andpricing decisions. The standard time-series approach for estimating elasticitiesspeci�es product demand as a parameterized function of price and estimatesthese parameters using time-series data. Because of the complicated patternsof cross-price interactions among di�erent products, a realistic demand modeloften involves many more parameters than can be estimated with existing timeseries data. This is especially true when non-stationarity and multicollinearityare present.

This problem of insu�cient data can be solved by placing restrictions on thecross-price interactions in a demand model1.

1Another approach, the hedonic, transforms the problem of estimating many product de-mand functions to the problem of estimating a smaller number of attribute demand function.

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Thus products might be sorted into segments using cross-sectional data withthe product demand models constrained to satisfy logit assumptions withinand across each segment(Kamakura and Russell,1989; Russell and Bolton,1988).But de�ning segments for which the logit assumptions are realistic is nearlyimpossible for some markets(e.g., the automotive.)

To develop an alternative approach to imposing restrictions on demand mod-els, note that all product elasticities are deducible from:

1. how much share a product loses when its own price increases (the own-elasticity),

2. the fraction of that lost share diverted to various other products (thediversion fraction.)

This diversion fraction is frequently independent of the amount of the pricechange and only a function of the cross-sectional similarity between di�er-ent products(Bordley,1985). Furthermore, as this paper's �rst section shows,standard economic conditions imply that all income-compensated demand-pricecross elasticities within a given market are deducible from diversion fractionsand a single scaling factor.

Hence a demand model could also be restricted by specifying its diversionfractions from cross-sectional data prior to estimating the model from time-series. For example, consider the Rotterdam demand model(Deaton and Muell-bauer,1986) on an economy with n products. Specifying the diversion fractionsprior to estimating the model on time-series would reduce the number of param-eters requiring estimation from n(n+2) (with n(n+1) parameters correspondingto demand-price elasticities) to 2n+1, a substantial reduction in the time-seriestask.

1.2 An Automotive ApplicationThis paper will focus on pricing in the automotive industry. Since vehiclesvary on many dimensions: size, paint, engine, interior upholstery etc., there aremany ways of de�ning what constitutes a distinct automotive `product.' Forthe purposes of competitive pricing, the de�nition of what constitutes a distinctproduct must distinguish between

� vehicles made by di�erent vehicle manufacturers,� vehicles with signi�cantly di�erent marginal costs,� vehicles targetted at di�erent kinds of customers.

We therefore de�ne a distinct product in terms of two factors:For discussion of the hedonic approach and some criticisms, see Berndt(1990), Rosen(1974),Mendelsohn(1984).

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� the vehicle's platform (which speci�es certain essential manufacturing at-tributes of the vehicle)

� the marketing division responsible for selling the vehicle (which speci�esthe vehicle's target buyer group and its manufacturer.)

Each speci�c combination of platform and division will constitute what we calla distinct `product' (or nameplate.) (Some examples include the ChevroletCorsica, the Pontiac LeMans, the Honda Accord etc.) Since many manufac-turers own several divisions (e.g., GM's North American operations include theChevrolet, Pontiac, Oldsmobile, Buick, Cadillac, GMC Truck and Saturn divi-sions) and produce several platforms, there are more than 200 distinct products.

Typically these products are grouped into seven segments(e.g., Economy,Small, Compact, Midsize, Large, Luxury and Sporty) not satisfying logit-likeassumptions. Thus the large car segment consists of the Chevrolet Caprice,the Pontiac Parisienne, the Olds Delta 88, the Buick LeSabre, the Ford CrownVictoria, the Ford Grand Marquis and the Chrysler New Yorker. Automotiveeconomists typically estimate a time-series model over the segments (which spec-i�es seven segment own elasticities and one market own elasticity.) Marketinganalysts then `explode' these aggregate forecasts into speci�c forecasts abouthow the more than 200 products within each segment will fare, generally usingcross-sectional data on product diversions.

This division of responsibility motivates the following variation on the method-ology previously discussed:

1. Estimate diversion fractions from cross-sectional data, speci�cally data oncustomer �rst and second choice product preferences,

2. Estimate the scaling factor and product own-elasticities using the segmentelasticity and market own-elasticity estimates.

This paper uses this technique to estimate more than 40,000 product elasticities.

2 ELASTICITIES FROM DIVERSION FRAC-TIONS & A SCALING FACTOR

2.1 The Critical TheoremLet qi; pj and Y denote product i's demand, product j's price and income re-spectively. Let wi = piqi. De�ne sij and the income-compensated elasticity,eij , by

sij = @qi@pj + qj @qi@Y ; eij = pj

qi sij (1)

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Of the sales revenue switching out of product j given a marginal increase inj's price, de�ne the diversion fraction, i.e., the income-compensated fractionswitching to product i versus any other product in the market M = f1:::ng, by:

fij = wieijPk 6=j;k2M wkekj (2)

De�ne wM =Pj2M wj and de�ne the intramarket cross-elasticity, EMM , byDEFINITION 1: EMM =

Pj;i2M;i 6=j wieijwMSlutsky Symmetry, (i.e., sij = sji, for all i; j), need not apply for aggregate

demand models. This paper therefore assumes that a limited version of SlutskySymmetry holds between products in di�erent markets, i.e.,

ASSUMPTION 1 : For all i 2M ,Pk 62M wieik =Pk 62M pipksik =Pk 62M pipkski =Pk 62M wkekiGiven Assumption 1, the critical Theorem is:THEOREM 1: Let (�1:::�n) solve:

Xj2M;j 6=i

fij�j = �iXj2M

�j = 1

Thenwjejj +

Xk 62M

wkekj = ��jwmEMMwieij = fij�jwMEMM

PROOF: The budget constraint(see Deaton and Muellbauer,1986,pg.46)implies X

k 6=j;k2Mwkekj + wjejj +

Xk 62M

wkekj =Xkpkpjskj = 0 (3)

and the economic homogeneity condition implieswieii +

Xj 62M

wieij +X

j 6=i;j2Mwieij =

Xjpipjsij = 0 (4)

Substituting (3) in (2) giveswieij = �fij(wjejj +

Xk 62M

wkekj); i 6= j (5)

Substituting (5) in (4) giveswieii +

Xj 62M

wieij �X

j 6=i;j2Mfij(wjejj +

Xk 62M

wkekj) = 0 (6)

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Using Assumption 1 in (6) giveswieii +

Xk 62M

wkeki �X

j 6=i;j2Mfij(wjejj +

Xk 62M

wkekj) = 0 (7)

If all automotive products sustained the same marginal percentage price in-crease, then the total sales revenue switching from one vehicle to another wouldbe proportional to Pj wjejj +Pk 62M wkekj . To focus on the fraction of thesediversions associated with sales revenue switching out of product j, de�ne

DEFINITION 2 : �j = wjejj+Pk 62M wkekjPj2M [wjejj+Pk 62M wkekj ]

Thus �j measures how elastic product j's share demand is relative to that ofother automotive products. De�nition 2 implies:X

j2M�j = 1 (8)

Substituting De�nition 2 into (7) gives�i =

Xj 6=i;j2M

fij�j i 2M (9)

Thus the homogeneity and aggregation conditions, when written in terms of di-version fractions, are equivalent to a Markov Process with the diversion fractionsas transition probabilities and �j as the Markov stationary probability.

Using (4) in De�nition 1 giveswMEMM = �X

j2M(wjejj +

Xk 62M

wkekj)

Substituting into De�nition 2 then giveswjejj +

Xk 62M

wkekj = ��jwMEMM (10)

Equations (5) and (10) then implywieij = �fij(wjejj +

Xk 62M

wkekj) = fij�jwMEMM (11)

which proves the Theorem.

2.2 Own-Elasticities from Diversion Fractions and a Scal-ing Factor

If diversion fractions for all products were known, then the market would includeall products and equations (10) and (11) become:

wjejj = ��jwMEMMwieij = fij�jwMEMM

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so that product own-elasticities are also deducible from diversion fractions anda single scaling factor.

If all products were strongly separable and if eiY was product i's income-elasticity, then

fij = wieijPk 6=j wkekj = wieiY wjejYPk 6=j wkekY wjejY = wieiY1� wjejY

�i = wieiY (1� wieiY )Pk wkekY (1� wkekY )Hence the elasticities become

wjejj = �wjejY (1� wjejY )Kwieij = wieiY wjejYK

with K a constant. This reproduces Frisch's results(1959).

3 ESTIMATING DIVERSION FRACTIONSThe author had access to a quarterly sample of 40,000 car buyers2 who wereasked to list the car they bought (their �rst choice) and the car they would havebought if their �rst choice car was made unavailable(their second choice.) Thisdata can be used to estimate fij by making the assumption:

ASSUMPTION 3: The fraction of j's potential switchers who woulddivert to

i, qieijPm;j2M;m 6=j qmemj

, equals the fraction of j buyers with i as a second choice.As Appendix III shows, this assumption holds if the income-compensated de-mand model describing the economy is the ratio of multilinear functions of price,i.e., if

qi = ai +Pnk=1 aikgk(pk) +Pnj;k=1 aijkgj(pj)gk(pk)b0 +Pnk=1 bkgk(pk) +Pnj;k=1 bjkgj(pj)gk(pk) (12)

where gj is an arbitrary monotonically increasing function of product j's price,ai; aik; aijk; b0; bk; bjk are constants and aijk = bjk = 0 for j = k.

Some special cases of this multilinear demand model are(I) Logit qi = aii euiPn

k=1 bkeuk where ui decreases with pi, satis�es (12) withgi = eui , aii = aii, bk =Pi aik and all other parameters equal to zero.

2In a four page survey sent out to 40,000 buy car buyers a few months after they purchasedtheir new car, buyers are asked to list the car they bought. They are also asked:`Suppose thecar you purchased was not o�ered for sale. What other vehicle (car or truck) make and modelwould you have purchased instead?' The sample was strati�ed to ensure that a statisticallysigni�cant number of buyers of each of more than 200 car and truck nameplates were sampled.Both domestic, Asian and European makes were sampled. This survey has been collected formore than ten years with no alteration in the �rst choice/second choice preference question.

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(II) Dogit qi = �0 eui+�i(Pnk=1 euk )Pn

k=1 euk (1+Pnj=1 �j) with �i > 0 satis�es (12) with gj = euj ,

aii = �0(1 + �i), aij = �0�i for i 6= j, bj = Pni=1 aij and zero for all otherparameters.(III) Additively Separable Sales Models:qi =Pj aijgj(pj) satis�es (12) with bk = bjk = 0.(IV) The Quadratic Cost Function: If p0 is a numeraire, then the costfunction, c(p0; p1:::pn) = p0Pij aij [ pip0 � ai][ pjp0 � aj ] + �u, implies a Hicksiandemand model linear in price for all products other than product 0(which formsa special case of III).(V) A Multilinear Generalized Extreme Value Model:qi = euiGi(eu1 ;eu2 ;:::eun )G(eu1 ;:::eun ) with G(eu1 ; :::eun) = Pnk=1 bkeuk +Pnj;k=1 bjkeujeukwith bjj = 0satis�es (12) with aii = bi; aiik = bij and all other parameters equal to zero.

For those problems in which �rst choice/second choice data is lacking, a moregeneral way of estimating diversion fractions involves constructing multiat-tribute product positioning maps(Hotelling,1929;Economides,1986; Hauser andShugan,1983) from cross-sectional information on product attributes3. In thiscase, Assumption 3 is unnecessary.

Given fij , (�1:::�n) can be deduced from solving (8) and (9) by standardtechniques4 in Markov Chains(Ross,1972). Then (11) speci�es all cross elastic-ities within the market up to a scaling factor, wMEMM .

3To construct multiattribute product positioning maps, a space of vectors, with each vectorhaving r arguments representing utility weights for each of r product attributes, is de�ned.The sales of a product are distributed among all the vectors whose utility weights would assignthat product higher utility than any other product. (A variation of this technique distributes�rst choice/second choice frequencies.) Once the number of people with various utility weightsis speci�ed, diversion fractions are easily deduced. (Such maps could, in principle, be used toinfer all elasticities but the author prefers to use them only to infer diversion fractions.)4Since the n + 1 equations are linearly dependent, drop any single equation other thanequation (8), and solve the remaining n equations for the �j 's by matrix inversion. Since thedeleted equation is just a linear combination of the remaining n equations, this solution tothe n remaining equations necessarily satis�es the one deleted equation.

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4 ESTIMATING SCALING FACTORS & OWNELASTICITIES

4.1 Estimating the Scaling FactorThe automotive market is divided into segments for which segment own elastic-ities, EI , and an overall market own elasticity, EM , were estimated5. Thesegments were constructed so that market and segment own-elasticities areweighted averages of product elasticities(Wold,1953), i.e.,

EM =Pi;j2M piqieij

wM ; wM =Xi2M

piqi (13)

EI =Pi;j2I piqieijPi;j2I wi ; wI =

Xi2I

piqi (14)

This condition is plausible since competitors in each segment have historicallytended to match each other's price changes. Drastic violations of this condi-tion(e.g., EI being a simple arithmetic average rule of the elasticities) can causeerrors of as much as 25%. But empirical results (reported in section 5) seemedrobust to small violations of this condition.

These de�nitions of segment and market own-elasticity implyXIwIEI � wMEM = �X

IX

j2I;i62Iwieij = �wMEMM

Xj;i 62I(j)

fij�j

where I(j) denotes the segment containing product j. Then the intramarketcross-elasticity, EMM , is computable from

EMM = �PI wIEI � wMEMwM (Pj;i 62I(j) fij�j) (15)

5For the automotive example, the best estimates of EI and EM came from the estimationof the segment-level demand model:QI =X

IJ

aIJpJ + bIY

where aIJ and bI and parameters and Y is income. All products were aggregated into sevensegments: economy, small, compact, mid-sized, large, luxury and sporty. The price index foreach segment was the average retail price of each car in that segment, corrected for rebates ina given year. The sales index was the total reported sales of cars in that segment in a givenyear. The regression was run over ten years. To make this model estimable, it was noted thataIJaJJ equalled the fraction of buyers in segment J who would switch to products in segment Iif all products in segment J were made unavailable. This quantity can be estimated from the�rst choice/second choice database. Hence the actual regression only involved the parametersaJJ and bI . Income-compensated segment elasticities can be deduced from this regression.The estimated regression satis�ed the standard diagnositicity checks.

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Thus the total revenue switching between segments (�(PI wIEI � wMEM ))equals the fraction of sales that would switch between segments given a pricechange (Pj;i 62I(j) fij�j) multiplied by the number of sales that would switchfrom one product to another given a price change (wMEMM ).

Given (15), equation (11) speci�es all cross-elasticities.

4.2 Estimating Own-ElasticitiesTHEOREM 2:

wjejj = ��jwMEMM +Pk 62M wkekjPk 62M;j2I wkekj (wMEMM

Xi62I;j2I

fij�j + wIEI)

PROOF: See Appendix I.De�ne the fraction of segment I switchers with second choices inside segmentI, i.e., segment I's exclusivity, by

FI =Pi;j2I;i6=j fij�jPj2I �j

De�ne segment I's share of all sales revenue divided by its share of potentialswitcher sales revenue, i.e., its loyalty, by

LI =Pj2I wjwPj2I �j

Then the average product own-elasticity in a segment is justCOROLLARY 1: eI = Pj2I wjwI ejj = EI � wMEMM

Pij2I fij�jwI = EI �FILIEMMThus the di�erence between the product's own-elasticity in a segment and the

segment own-elasticity increases with the segment's exclusivity and decreaseswith its loyalty.

To estimate product own-elasticities, an additional assumption is needed.This paper considers two alternative assumptions:

ASSUMPTION 2(a): For j 2M , Pk 62M wkekj = wjejY �Iwhere ejY is product j's income-elasticity and �I is a segment-speci�c con-stant. Thus each segment I is weakly separable(Deaton and Muellbauer,1986)with respect to products outside the market. Substituting in Theorem 2 gives

wjejj = ��jwMEMM + wjejYPk2I wkekY (wMEMMX

i 62I;j2Ifij�j + wIEI)

An alternative assumption is

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Table 1: ELASTICITIES BY SEGMENTAssump- Car Segment Car Line Elasticities(ejj)tion Segment Elasticity(EI) MIN AVE MAX LI FI2(a) Economy -1.9 -3.3 -4.7 -8.2 0.6 .632(b) -3.4 -8.12(a) Small -1.7 -1.9 -2.4 -3.1 0.8 .202(b) -1.7 -3.42(a) Compact -2.0 -2.1 -3.0 -4.9 0.9 .372(b) -2.2 -4.72(a) Midsize -2.3 -2.3 -3.3 -4.6 1.1 .482(b) -2.6 -4.22(a) Large -3.0 -3.1 -3.8 -4.3 1.3 .422(b) -3.5 -4.02(a) Luxury -2.4 -3.2 -3.7 -5.3 1.4 .702(b) -3.4 -4.52(a) Sporty -3.4 -2.6 -4.2 -6.5 1.3 .392(b) -3.4 -5.3Market Own(EM ): -1.0 Market Cross(EMM ): 2.6 Average Car Own: -3.6

ASSUMPTION 2(b): For j 2 M ,P

k 62M wkekjwjejj = 1��I�I within each seg-ment I

In other words, the fraction of buyers who switch out of the market is thesame for all products within each segment. As Appendix II notes, this implies

wjejj = �jPj2I �jwIeI j 2 I

This formula indicates that if product j has twice the number of switchersas product k for j; k 2 I, then the number of sales product j loses given apercentage price increase is twice the number of sales product k loses given apercentage price increase.

5 APPLICATION TO THE AUTOMOBILEMAR-KET

The segment own-elasticities varied by about 30% over the range of their 95%con�dence interval. If the �rst choice/second choice data describe the popu-lation of car buyers exactly, then these con�dence intervals imply con�denceintervals over the product elasticity estimates. Table 1 lists the estimates un-der both assumptions 2(a) and under 2(b). The loyalty index is higher for

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higher priced products (as would be expected since higher income buyers areless price-sensitive). The economy and luxury car segment are characterized bygreat exclusivity, i.e., economy car buyers do not view non-economy cars as closesubstitutes, luxury car buyers do not view non-luxury cars as close substitutes.This, also, is what would be expected. The combined e�ect of low loyalty andhigh exclusivity causes the demand for cars in the economy segment to be muchmore elastic than the economy segment own-elasticity. But note that luxury carbuyers are not, contrary to popular belief, substantially less price-sensitive thanother buyers. This re ects how competitive the luxury car market has become.

Income-uncompensated elasticities can be estimated from these elasticities ifproduct income elasticities are known. Bordley and McDonald(1993) estimateincome-elasticities from the marketshare of each vehicle within various incomegroups and the population income distribution by:

1. computing how much the number of people in each income bracket changeswhen all incomes increase by the same marginal fraction,

2. multiplying a product's sales per customer in each income bracket by thischange6 to get a product's sales change in each income bracket,

3. summing over all income brackets and dividing by the product's originalsales to get the income-elasticity.

Bordley and McDonald(1993) showed that this technique leads to very com-monsensical automotive income-elasticity estimates.

6 CONCLUSIONSThis paper developed and illustrated a method for estimating income-compensateddemand-price elasticities by:

� decomposing elasticities into diversion fractions, a scaling factor and own-elasticities,

� estimating the diversion fractions from cross-sectional data (speci�cally�rst choice/second choice data),

� estimating the scaling factor and own-elasticities from time-series data(speci�cally segment and market own-elasticities.).

This method led to very intuitive estimates of more than 40,000 demand-priceelasticities for the automotive industry.

6Thus a product's sales per customer in each income bracket is assumed not to change.

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References[1] Berndt, Ernst. The Practice of Econometrics. Addison-Wesley, Mas-

sachusetts(1990).[2] Bordley, Robert. \Relating Elasticities to Changes in Demand." Journal of

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Income Elasticities from the Population Income-Share Elasticity." Journal

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Cambridge University Press, London(1986).[5] Economides, Nicholas. \Nash Equilibrium in Duopoly with Products De�ned

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Demand Elasticities in a Model with Many Sectors." Econometrica. 27, 177-196,1959.

[7] Hagerty, Michael, James Carman and Gary Russell. \Estimating Elastici-ties with PIMS Data: Methodological Issues and Substantive Implications."Journal of Marketing Research. 25, 1(1988).

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ation in Perfect Competition." Journal of Political Economy. 82,34-55,1974.[14] Ross, Sheldon. Introduction to Probability Models. Academic Press, New

York(1972).

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[15] Russell, Gary J. and Ruth N. Bolton. \Implications of Market Structure forElasticity Structure." Journal of Marketing Research. 25, 199, 229-241,1988.

[16] Tellis, Gerard. \The Price Elasticity of Selective Demand: A Meta-Analysisof Econometric Models of Sales." Journal of Marketing Research 25(1988).

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APPENDIX I:PROOF OF THEOREM 2Using EMM in the results of Theorem 1 gives

wjejj = ��jwMEMM �Xk 62M

wkekj (16)

Given the de�nition of EI ,Xj2I

wjejj = wIEI �X

ij2I;i6=jwieij = wIEI �

Xij2I;i6=j

fijwMEMM (17)

Using (16) and (17) givesX

j2I;k 62Mwkekj = �wMEMM

Xi 62I;j2I

fij�j � wIEI (18)

so thatwjejj = ��jwMEMM +

Pk 62M wkekjPk 62M;j2I wkekj (wMEMMX

i 62I;j2Ifij�j +wIEI) (19)

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APPENDIX IITheorem 1 shows that

wjejj = ��jwMEMM �Xk 62M

wkekj (20)

Given Assumption 2(b),wjejj = ��I�jwMEMM )wjejj = � �jPj2I �j

Xj2I

wjejj )

wjejj = �jPj2I �j (wIEI � wMEMMX

i2I;j2Ifij�j) = �jPj2I �jwIeI

APPENDIX III:JUSTIFYING ASSUMPTION 3Consider the class of models:

qi = ai +Pnk=1 aikgk(pk) +Pnj;k=1 aijkgj(pj)gk(pk)b0 +Pnk=1 bkgk(pk) +Pnj;k=1 bjkgj(pj)gk(pk) (21)

where gj is an arbitrary monotonically increasing function of product j's price,ai; aik; aijk; b0; bk; bjk are constants and aijk = bjk = 0 for j = k.

First note that@qi@pj = g0(pj)aij � qibj +Pk gk(aijk � qibjk)

b0 +Pk bkgk +Pjk bjkgjgkThus @qi=@pj

@qj=@pj = aij � qibj +Pk gk(aijk � qibjk)ajj � qjbj +Pk gk(ajjk � qibjk) (22)

Now let Ni = ai+Pk aikgk+Pjk aijkgjgk and D = b0+Pk bkgk+Pjk bjkgjgkso that qi = NiD . Then the demand for product i if j became unavailable is

Ni � aijgj �Pk aijkgkgjD � bjgj �Pk bjkgkgj

Subtracting o� qi = NiD gives the change in demand for i when j becomesunavailable. This gives

gj aij � qibj +Pk gk(aijk � qibjk)D � gj(bj +Pk bjkgk)

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Dividing this by the change in j demand when j becomes unavailable givesaij � qibj +Pk gk(aijk � qibjk)ajj � qjbj +Pk gk(ajjk � qjbjk) (23)

This is just the fraction of j buyers who would switch to i if j became unavail-able. By inspection (22) and (23) are equal, proving the result.

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