Estimation of Demand Systems Based on Elasticities of Substitution

Transcription

1 Estmaton of Demand Systems Based on Elastctes of Substtuton by Germán Coloma (CEMA Unversty, Buenos Ares, Argentna) * Abstract Ths paper develops a model for demand-system estmatons, whose coeffcents are own-prce Marshallan elastctes and elastctes of substtuton between goods. The model satsfes the homogenety, symmetry and, eventually, addng-up restrctons mpled by consumer theory, and s prmarly useful for the estmaton of the demands of several goods of the same ndustry or group of products. The characterstcs of the model are compared to other exstng alternatves (logarthmc, translog, AIDS and QUAIDS demand systems). The model s fnally appled to estmate the demands for several carbonated soft drnks n Argentna, and ts results are presented, together wth the ones obtaned wth the other estmaton methods. Resumen Este trabao desarrolla un modelo para estmacones de sstemas de demanda, cuyos coefcentes son elastcdades-preco marshalanas drectas y elastcdades de susttucón entre benes. El modelo satsface las restrccones de homogenedad, smetría y, eventualmente, adtvdad surgdas de la teoría del consumdor, y es útl prncpalmente para estmar demandas de varos benes de la msma ndustra o grupo de productos. Las característcas del modelo se comparan con las de otras alternatvas exstentes (sstemas logarítmcos, translogarítmcos, AIDS y QUAIDS). Fnalmente, el modelo se aplca a la estmacón de la demanda de bebdas gaseosas en la Argentna, y se presentan sus resultados, unto con los que se obtenen de aplcar los otros métodos de estmacón. JEL Classfcaton: C30, C51, D12, L66. Keywords: Demand Systems, Elastcty of Substtuton, Smultaneous Equatons, Carbonated Soft Drnks. 0. Introducton Ths paper develops a model for demand-system estmatons, based on a logarthmc form. The basc coeffcents to estmate, therefore, are demand elastctes. To avod certan estmaton problems, and to ncorporate several restrctons mpled by consumer theory (namely, homogenety, symmetry and, eventually, addng-up), the orgnal coeffcents of the model are transformed, and the equatons end up as lnear functons of the own-prce Marshallan elastctes of the dfferent goods and the elastctes of substtuton between those goods. The model s prmarly useful for the estmaton of the demands for several goods of the same ndustry or group of products, rather than for demand estmatons of large * The vews and opnons expressed n ths publcaton are those of the author and are not necessarly those of 1

2 consumpton categores. The paper s organzed as follows. In secton 1 we revew the theoretcal concept of elastcty of substtuton, and ts relatonshps wth the Marshallan and Hcksan demand elastctes. In secton 2 the model s presented, and n secton 3 ts man characterstcs are compared wth the ones of other alternatve demand systems. In secton 4 the model s appled to a database of supermarket sales of carbonated soft drnks n Argentna, and ts results are tested and compared to the ones generated by the alternatve demand systems. Fnally, n secton 5 we summarze the man conclusons of the whole paper. 1. The concept of elastcty of substtuton The concept of elastcty of substtuton, created by Allen (1938), measures the relatve change n the rato between the quanttes of two goods consumed by a certan ndvdual as a response to a relatve change n the rato of the prces of those goods. It s defned for a gven level of the ndvdual s utlty,.e., for a stuaton where that ndvdual s located at a certan ndfference curve 1. For two arbtrary goods and, consumed at quanttes Q and Q and bought at prces P and P, the elasttcty of substtuton between those goods (σ ) s defned as: d(q / Q ) /(Q / Q ) σ = (1). d(p / P ) /(P / P ) As one of the basc mplcatons of consumer theory, whch holds for dfferentable utlty functons, s that prce ratos are equated to margnal utlty ratos, t s possble to wrte (1) n the followng alternatve form: d(q / Q ) /(Q / Q ) σ = (2) ; d(u / U ) /(U / U ) where U and U are the margnal utltes of goods and evaluated at Q and Q. 2 If the correspondng utlty functon s homogeneous, moreover, ths equaton can be transformed to reach the followng expresson: CEMA Unversty. 1 The concept of elastcty of substtuton can also be appled n producton theory. In that case t refers to ratos of nput quanttes and nput prces, evaluated at a fxed output level. 2 In fact, the defntons of σ under (1) and (2) are dentcal for a case of two goods. If there are more than two commodtes, then the two defntons may dffer. For more detals about ths, see Blackorby and Russell (1989). 2

3 U U U U σ = = = σ (3) ; U U U U where U = U s the symmetrc second dervatve of the utlty functon wth respect to Q and Q. As we can see n (3), the elastcty of substtuton s a symmetrc concept, whch s the same whether we are measurng the substtuton of good for good or the substtuton of good for good. The elastcty of substtuton between goods and can also be related to the cross elastctes of demand for those goods. Let us consder, for example, the Hcksan demand elastcty of good wth respect to good (ε ), whch s defned for a gven level of utlty. It can be shown that: Q P ε = = σ s (4) ; P Q where s s the share of good n consumer s total expendture. But as the Hcksan demand elastcty and the ordnary, or Marshallan, demand elastcty (η ) are related n the followng way by the so-called Slutsky equaton : η = ε η s (5) ; Y where η Y s the ncome elastcty of good, then we can combne (4) and (5) to obtan the followng alternatve expresson: η = s ( σ η ) (6) ; Y whch s expressed n terms of an ncome elastcty and a symmetrc substtuton elastcty 3. As we wll see, ths formula wll be useful to estmate a partcular class of demand systems, where elastctes of substtuton wll be related among themselves. 2. The substtuton elastcty demand system Let us defne a system of N demands, each of whch has the followng form: ln( Q ) = α + η ln(p ) + η ln(p ) + η Y ln(y) (7) ; where Y s consumer s ncome. Due to the logarthmc nature of the model, ts coeffcents (η, η, η Y ) are Marshallan demand elastctes. 3

4 Let us now substtute (6) nto (7). What we obtan s: ln( Q ) = α + η ln(p ) + σ s ln(p ) + ηy ln(y) s ln(p) (8). Let us now recall that Marshallan demands are homogeneous of degree zero n prces and ncome, and wrte the correspondng restrcton n elastcty form: ηy = η η (9). Substtutng (6) nto (9), ths mples: η s σ η Y = (10) ; s whch, replaced nto (8), generates the followng demand system: ln( Q + ) = α + η ln(p ) σ s ln(p ) (11). s s ln(y) s ln(p ) ln(y) s ln(p ) The system of N equatons defned by (11), whch we wll call substtuton elastcty demand system (SEDS), s a lnear system whose coeffcents are the own-prce Marshallan demand elastctes and the elastctes of substtuton between goods. As those elastctes of substtuton are symmetrc (that s, σ = σ ), ths system dsplays the symmetry property, together wth the homogenety property mpled by (9). The ncluson of the homogenety and symmetry restrctons n ths demand system model reduces the number of elastcty coeffcents from N (N+1) to N+N (N-1)/2. Ths s the result of the N+N (N-1)/2 restrctons mposed to the system. SEDS s also capable to ncorporate the so-called addng-up restrcton of consumer theory. In order to do that, t s useful to wrte that restrcton n a way that relates Marshallan own-prce elastctes and cross-prce elastctes. Ths form s usually called Cournot aggregaton condton, and t mples that: η s η = 1 (12). s 3 For a more complete explanaton of these relatonshps, see Barten (1993). 4

5 η Combnng (9), (10) and (12), t s possble to obtan that: = 1 η σ s (13) ; k k and ths can be substtuted nto (11). Wth ths substtuton we can elmnate the η coeffcent n one of the N equatons of the model 4, and we therefore have a system wth N-1 own-prce elastcty coeffcents and N (N-1)/2 elastctes of substtuton. 3. Characterstcs of SEDS The man characterstcs of the proposed model are nherted from the fact that t s orgnated n a logarthmc demand system and from the restrctons mposed to t. Probably the most notceable one s that ts man coeffcents are drect estmates of dfferent elastcty concepts (namely, own-prce and substtuton elastctes). Ths allows for a straghtforward nterpretaton of ts results, whch s somethng that does not occur when we use other more ndrect models. Another characterstc of SEDS s that ts equatons do not come from the maxmzaton of an explct utlty functon subect to a budget constrant, but they rather are a local approxmaton of the results generated by an arbtrary functon. Ths approxmaton s nevertheless meanngful, snce the sgns and magntudes of the estmated coeffcents can be tested for consstency wth dfferent postulated utlty functons 5. Due to the restrctons mposed, we know that those estmates wll also be consstent wth some general propertes of consumer demand functons, namely homogenety of degree zero, symmetry of the Slutsky matrx and, f ncluded, the addng-up restrcton. Compared to a more general logarthmc demand system, the man advantage of SEDS s that t can ncorporate the symmetry restrcton n a very natural way. As cross-prce elastctes are generally not symmetrc, one of the man problems of logarthmc demand systems s that they typcally volate symmetry. They also generally volate the addng-up property, unless that constrant s mposed through a set of Cournot aggregaton condtons that apply to each of the equatons to be estmated. Homogenety restrctons, conversely, are 4 Note that (13) cannot be substtuted nto the N equatons separately, snce t s n fact a sngle constrant and not a set of N ndependent restrctons. 5 It s possble to check, for example, f estmates are consstent wth the demands generated by a Cobb-Douglas utlty functon, that dsplay own-prce elastctes equal to 1 and elastctes of substtuton equal to 1. Other utlty functons wth constant elastctes of substtuton (for example, the ones that consttute the CES famly) do not generate demand functons wth constant own-prce elastctes, but they can nevertheless be tested usng the average elastcty values mpled for the data set under analyss. 5

6 easly mposed on logarthmc demands, and they are also easly ncluded n the SEDS model. Compared to other more sophstcated demand systems based on the so-called flexble functonal forms, SEDS has the advantage that t s more effcent n the use of nformaton. Consder, for example, three common specfcatons such as the translog demand system, orgnated n the work of Chrstensen, Jorgenson and Lau (1975), the almost deal demand system (AIDS), proposed by Deaton and Muellbauer (1980), and the quadratc almost deal demand system (QUAIDS), created by Banks, Blundell and Lewbel (1997). They can all be thought of as part of the same famly of demand systems, bult upon a seres of equatons whose dependent varables are expendture shares. For the case of the translog demand system, those equatons have the followng form: s = α + β + β (14) ; ln(p ) ln(p) whle, for the case of AIDS, they have the followng form: Y s = α + β ln(p ) + β ln(p ) + βy ln (15) ; PI1 where PI 1 s an arthmetc prce ndex, and, for the case of QUAIDS, they have the followng form: s = α + β ln(p ) + β ln(p ) + β Y ln Y PI 1 λ + PI Y 2 ln Y PI 1 2 (16) ; where PI 2 s a geometrc prce ndex. To fulfll the homogenety, symmetry and addng-up propertes of demand functons derved from consumer theory, these systems have to be estmated mposng certan restrctons on the coeffcents, whch are bascally the followng: α = 1; β = 0 ; β = 0 ; β = β ; β = Y 0; λ = Y 0 (17). The mposton of those condtons, however, reduces the number of coeffcents n such a way that makes one of the N equatons redundant. Therefore, we end up wth systems of N-1 equatons, each of whch has the expendture shares of N-1 goods as ther dependent varables. In comparson wth these systems whose estmaton s performed usng expendture share equatons, SEDS has the advantage of beng more effcent. Ths s because t does not 6

7 lose one equaton, and because t keeps the nformaton about total quanttes, nstead of transformng those quanttes nto expendture shares. The parameters estmated are also easer to nterpret, snce they are drect estmates of own-prce and substtuton elastctes, nstead of coeffcents that have to be transformed n order to be nterpreted as elastctes. The man dsadvantage of SEDS wth respect to the other demand systems mentoned n ths secton, however, s that ther dependent varables are not exogenous. Ths s because those varables are not prces and ncome but transformatons of those varables, whch also nclude expendture shares n ther formulae. But as expendture shares are based on prces and quanttes, and quanttes are supposed to be the consumers decson varables, then all the varables bult usng expendture share nformaton are, at least partly, endogenous to the model. In order to obtan consstent estmatons of the coeffcents, therefore, t s necessary to use nstrumental varables. The choce of those nstrumental varables, however, s more or less obvous, snce we are basng our analyss n the behavor of consumers who take prces and ncome as gven. Usng prces and ncome as nstrumental varables, and estmatng the system of equatons through a method that ncorporates those nstrumental varables (such as two-stage least squares, or three-stage least squares), we are able to obtan a set of consstent and unbased estmators for the elastcty coeffcents embedded n the model 6. SEDS also provdes a natural way to smplfy the estmaton when we are workng wth a set of goods that we have some addtonal nformaton about. Let us magne, for example, that we can pool the goods nto dfferent groups and classes (based on obectve characterstcs of those goods). We can assume, for example, that two goods that belong to the same class may have the same elastcty of substtuton wth respect to another good that belongs to a dfferent class, and that hypothess can be easly ncorporated nto the estmaton of SEDS. In other models, those smplfcatons are much more dffcult to handle, snce they mply redefnng the ndependent varables of the regresson 7. Beng a model that does not come from the maxmzaton of an explct utlty functon, we thnk that SEDS s more sutable for demand systems that nclude several related goods (for example, goods from the same ndustry) but not large consumpton categores. Ths s the case, for example, of estmatons based on supermarket scanned data for products 6 In fact, the assumpton that prces and ncome are exogenous varables whle quanttes are endogenous s determned by the dea that we work usng ndvduals level data. If we are workng wth aggregate data, however, prces are exogenous f supples are perfectly elastc and demands adust to clear the market. For a deeper analyss of ths assumpton and alternatve ones, see Moschn and Vssa (1993). 7 There s a verson of the AIDS model, called PCAIDS, whch essentally makes an assumpton lke that. In order to ncorporate that assumpton to an econometrc model, however, t s necessary to multply and dvde the coeffcents by the expendture shares of the goods under analyss, and ths mples a change n the specfcaton 7

8 that belong to the same ndustry, n whch we can make the assumpton that ther demands are related among themselves but bascally ndependent from the demands of other goods 8. In a context lke that, the mposton of homogenety and symmetry restrctons s very mportant, but addng-up may be less mportant or even nconvenent. Ths s because demands are supposed to be functons of ncome and all the avalable prces, and substtuton patterns between goods are supposed to be symmetrc. However, there s no need to assume that, when ncome changes, the rato between expendture (n those goods) and ncome wll reman the same. Imposng an addng-up restrcton s equvalent to assume that the average ncome elastcty of the estmated goods s equal to one, and ths may not be reasonable f we are dealng wth a group of goods from an ndustry that represents a relatvely small fracton of total consumers ncome. 4. Applcaton to the Argentne carbonated soft-drnk ndustry In ths secton we wll apply SEDS to a data set of 93 weekly observatons from the Argentne carbonated soft-drnk ndustry, durng the years 2004 and That data set s propretary. It was bult by a frm that specalzes n market research, usng scanned data from the man supermarket chans that operate n Argentna. To avod possble confdentalty problems, we have pooled the data nto eght commodty categores. Each of them represents a partcular taste and varety of carbonated soft drnks, but t ncludes nformaton from several dfferent brands and companes. The categores defned n that way are normal cola (good 1), lght cola (good 2), normal lme soda (good 3), lght lme soda (good 4), normal orange soda (good 5), lght orange soda (good 6), normal grapefrut soda (good 7) and tonc water (good 8). For each of the goods n our database we have prce and quantty data. Quantty s measured n lters sold n each week, whle prce s measured n Argentne pesos per lter, and s obtaned from dvdng total sales of the correspondng good by total quantty of that good. We also have two addtonal varables to be ncluded as demand shfters n all the equatons. One of them s the consumers nomnal ncome, estmated by multplyng the Argentne Monthly Estmator of Economc Actvty (EMAE) and the Argentne Consumer Prce Index of the model tself. For more detals about PCAIDS, see Epsten and Rubnfeld (2002). 8 The use of SEDS to estmate the demands for several goods of the same ndustry also makes convenent to defne expendture shares as the rato between the expendture on each good and the total expendture n the group of goods whose demands are estmated. Usng a measure of expendture shares that relates expendture on each good to total consumers expendture n every good of the economy, conversely, may generate a problem of measurement of the elastctes of substtuton, frst ponted out by Frsch (1959). 8

9 (CPI) 9. As those ndces are publshed monthly, we had to nterpolate them to obtan weekly seres. The other demand shfter s the average daly maxmum temperature n the Buenos Ares area for each of the weeks of the data set, measured n Celsus degrees 10. Ths s supposed to be an mportant determnant of soft-drnks consumpton. Other varables used n our regressons come from transformng the orgnal varables. The expendture shares, for example, are the ratos between the product of prce tmes quantty dvded by total expendture n carbonated soft drnks. The average prce ndces requred for the estmaton of the AIDS and QUAIDS models, smlarly, are arthmetc and geometrc means of the eght products prces, whch use average expendture shares as weghtng factors. The man nformaton about the data set used s summarzed n table 1. In t we see that the average carbonated soft drnk prces vary consderably accordng to the dfferent tastes and varetes, and have followed an ncreasng path durng the perod n Argentna. On average, they have grown 20% between the second quarter of 2004 and the fourth quarter of 2005, whch s a perod where the CPI ncreased 14%. We can also see that some tastes and varetes have always been more expensve than others, but the evoluton of prces was not homogeneous. For example, the lght sodas are always more expensve on average than the normal sodas wth the same taste. However, the lght lme soda was cheaper than the tonc water n the frst two quarters of 2004, but t became more expensve n the last quarter of 2004 and durng the year Table 1 also contans nformaton about market shares, calculated as the expendture share of each good n the total sales of the database. That nformaton shows us that the normal cola s by far the most mportant carbonated soft drnk, wth a share that oscllates between 44% and 46%, followed by the lght cola and the normal lme soda, wth market shares around 15%. The next most mportant carbonated soft drnk s the normal orange soda, wth a share between 8% and 9%, followed by the lght lme soda (7%) and the grapefrut soda (5%). Fnally, the lght orange soda and the tonc water are the two categores wth the smallest consumpton (around 2% each). 9 The source for these two ndces s the Argentne Natonal Insttute of Statstcs and the Census (INDEC). 10 The source for ths nformaton s the Argentne Natonal Meteorologcal Offce. 9

10 1. DESCRIPTION OF THE DATA Concept Mar-Jun/04 Jul-Sep/04 Oct-Dec/04 Jan-Mar/05 Apr-Jun/05 Jul-Sep/05 Oct-Dec/05 Mar04-Dec05 Prces (Arg$/lt) Normal Cola (P1) 1,2226 1,2499 1,2762 1,3354 1,3701 1,4059 1,4591 1,3239 Lght Cola (P2) 1,4033 1,4568 1,4881 1,5620 1,6024 1,6336 1,6641 1,5357 Normal Lme (P3) 1,2039 1,2402 1,2760 1,3276 1,3530 1,3824 1,4277 1,3086 Lght Lme (P4) 1,3899 1,4261 1,4793 1,5619 1,5986 1,6263 1,6532 1,5249 Normal Orange (P5) 1,0026 1,0189 1,0604 1,1194 1,1446 1,1935 1,2801 1,1086 Lght Orange (P6) 1,5436 1,5811 1,6634 1,7201 1,7456 1,8112 1,8630 1,6937 Grapefrut (P7) 0,7692 0,7895 0,8341 0,9191 0,9574 1,0188 1,0568 0,8973 Tonc Water (P8) 1,4162 1,4460 1,4589 1,5331 1,5565 1,5522 1,6031 1,5034 Average Prce 1,2273 1,2596 1,2929 1,3565 1,3898 1,4256 1,4742 1,3387 Expendture Shares (%) Normal Cola (S1) 44,41 45,12 44,86 44,63 45,74 46,13 44,77 45,07 Lght Cola (S2) 15,60 16,67 15,43 14,75 15,91 15,85 15,73 15,70 Normal Lme (S3) 14,70 14,30 15,22 15,67 14,76 15,08 15,96 15,06 Lght Lme (S4) 7,15 6,70 7,03 6,94 6,62 6,22 6,55 6,76 Normal Orange (S5) 8,46 8,82 8,57 8,56 8,14 8,28 8,02 8,42 Lght Orange (S6) 1,75 1,58 1,79 1,88 2,03 2,05 2,00 1,86 Grapefrut (S7) 5,95 5,07 5,23 5,46 4,98 4,71 5,07 5,24 Tonc Water (S8) 2,00 1,75 1,88 2,12 1,83 1,68 1,90 1,88 Other varables Quantty (thous lt) 2442,8 2431,7 2837,2 3094,9 2379,4 2461,4 2826,9 2626,7 Expendture (thous $) 2920,4 2999,2 3594,9 4107,0 3258,1 3465,2 4122,4 3457,1 Argentne CPI 147,53 148,08 150,98 154,85 159,20 162,48 167,95 155,25 Real Income (EMAE) 119,36 120,25 123,35 116,51 132,52 131,26 134,30 124,91 Temperature (ºC) 21,03 17,01 24,87 28,41 19,51 16,96 24,12 21,62 10

11 Although relatvely stable, these market shares exhbt some changes n the perod under analyss. For example, the lght cola had, on average, a largest market share than the normal lme soda, but that stuaton was the opposte n the frst and fourth quarters of the year Smlarly, the tonc water had a larger market share than the lght orange soda untl the frst quarter of the year 2005, and a smaller one n the last three quarters 11. The last rows of table 1 show some addtonal nformaton that was used n the regresson of the demand equatons for the eght products under analyss. We can see, for example, that the total quantty sold by the supermarket chans ncreased almost 16% between the second quarter of 2004 and the last quarter of 2005, whle the economc actvty of Argentna, measured by the EMAE, grew 12.5%. We can also see that the combnaton of the ncreases n prce and quantty experenced by the carbonated soft drnks of our database generated an ncrease n total expendture of 41% durng the perod under analyss. On table 2 we report the man results of the estmaton of a demand system that follows the SEDS model developed n secton 2 and summarzed by equaton (11). To perform that estmaton we used the prces and market shares of the eght goods of our database, together wth our estmaton of the nomnal ncome varable (EMAE tmes CPI), and the natural logarthm of temperature as an addtonal demand shfter. We also ncluded an autocorrelaton correcton n the form of an AR(1) process, whch reduced the number of avalable observatons to 92. To estmate the equatons we used teratve three-stage least squares (3SLS), that acheved convergence after 63 teratons. The nstrumental varables used were the logarthms of the eght prces, together wth the logarthm of the nomnal ncome and the logarthm of temperature. 11 Ths last phenomenon may be due to the fact that the lght orange soda s a relatvely new product, whle the tonc water s much more tradtonal n Argentna. 11

12 2. SEDS ESTIMATION RESULTS Concept Coeffcent Std Error t-statstc Probablty Own-prce elastctes Normal Cola (η 11 ) -0, , ,5556 0,0000 Lght Cola (η 22 ) -0, , ,5347 0,0000 Normal Lme (η 33 ) -0, , ,6146 0,0000 Lght Lme (η 44 ) -0, , ,9092 0,0000 Normal Orange (η 55 ) -0, , ,1195 0,0000 Lght Orange (η 66 ) -1, , ,7289 0,0000 Grapefrut (η 77 ) -1, , ,9302 0,0000 Tonc Water (η 88 ) -0, , ,6451 0,0000 Substtuton elastctes NCola/LCola (σ 12 ) 0, , ,6169 0,0000 NCola/NLme (σ 13 ) 0, , ,1444 0,0000 NCola/LLme (σ 14 ) 0, , ,1163 0,0000 NCola/NOrange (σ 15 ) 0, , ,7387 0,0000 NCola/LOrange (σ 16 ) 1, , ,0289 0,0000 NCola/Grapefrut (σ 17 ) 1, , ,4861 0,0000 NCola/Tonc (σ 18 ) 0, , ,7905 0,0000 LCola/NLme (σ 23 ) 0, , ,0360 0,0000 LCola/LLme (σ 24 ) 0, , ,7434 0,0000 LCola/NOrange (σ 25 ) 0, , ,4877 0,0000 LCola/LOrange (σ 26 ) 1, , ,8012 0,0000 LCola/Grapefrut (σ 27 ) 1, , ,1124 0,0000 LCola/Tonc (σ 28 ) 0, , ,5548 0,0000 NLme/LLme (σ 34 ) 0, , ,8506 0,0000 NLme/NOrange (σ 35 ) 0, , ,5760 0,0000 NLme/LOrange (σ 36 ) 1, , ,0886 0,0000 NLme/Grapefrut (σ 37 ) 1, , ,2297 0,0000 NLme/Tonc (σ 38 ) 0, , ,6335 0,0000 LLme/NOrange (σ 45 ) 0, , ,1168 0,0000 LLme/LOrange (σ 46 ) 1, , ,8174 0,0000 LLme/Grapefrut (σ 47 ) 1, , ,0533 0,0000 LLme/Tonc (σ 48 ) 0, , ,8977 0,0000 NOrange/LOrange (σ 56 ) 1, , ,3491 0,0000 NOrange/Grapefrut (σ 57 ) 1, , ,3717 0,0000 NOrange/Tonc (σ 58 ) 0, , ,6458 0,0000 LOrange/Grapefrut (σ 67 ) 1, , ,5403 0,0000 LOrange/Tonc (σ 68 ) 1, , ,4141 0,0000 Grapefrut/Tonc (σ 78 ) 1, , ,0527 0,0000 AR(1) coeffcents Eqn 1 (Normal Cola) 0, , ,2021 0,0000 Eqn 2 (Lght Cola) 0, , ,1534 0,0000 Eqn 3 (Normal Lme) 0, , ,0596 0,0000 Eqn 4 (Lght Lme) 0, , ,1799 0,0000 Eqn 5 (Normal Orange) 0, , ,7653 0,0000 Eqn 6 (Lght Orange) 0, , ,7575 0,0000 Eqn 7 (Grapefrut) 0, , ,7246 0,0000 Eqn 8 (Tonc Water) 0, , ,5138 0,

13 As we see, the results obtaned are very reasonable and precse. All the own-prce elastctes have the rght sgns and are sgnfcantly dfferent from zero at any possble probablty level, wth values that range from 0.90 to The substtuton elastctes also have the expected sgns and they are also sgnfcantly dfferent from zero at any possble probablty level, wth values that range from 0.96 to We also see that the estmaton resduals seem to have an mportant autocorrelaton (somethng that s expected, due to the weekly frequency of the seres), whch averages Wth the results reported on table 2, we have calculated all the mplct ncome and cross-prce elastctes of the model, followng equatons (10) and (6). They are the ones that appear on table 3. All the ncome elastctes have the expected postve sgn, wth values that range from 0.76 to Sx of them are statstcally dfferent from zero at a 1% level of sgnfcance, but the remanng two are not statstcally dfferent from zero at a 10% level of sgnfcance 13. They correspond to the demand equatons for lght orange soda and tonc water, whch are the two products wth the smallest expendture shares. The Marshallan cross-prce elastctes mpled by our SEDS estmaton, correspondngly, have also the expected postve sgn, wth values that range from 0.10 to Twenty-one of them are statstcally dfferent from zero at a 1% level of sgnfcance, sx of them are statstcally dfferent from zero at a 10% level of sgnfcance, and the remanng twenty-nne are not statstcally dfferent from zero at a 10% level of sgnfcance. In partcular, we can see that the cross elastctes that correspond to the demands of the goods wth the smallest market shares (lght lme, normal orange, lght orange, grapefrut and tonc water) tend not to be sgnfcantly dfferent from zero. 12 These results mply relatvely small prce elastctes, n comparson wth other studes of the carbonated soft drnk ndustry. For example, usng an AIDS specfcaton, Dahr and others (2005) fnd own-prce elastctes for these products that vary from 2 to 4. These elastctes, however, correspond to the US market and were calculated for partcular brands and not for dfferent product categores. 13 To estmate f these elastctes were statstcally dfferent from zero, we frst calculated ther mplct standard devatons, usng the standard devatons of the parameters estmated by the model. We then calculated ther correspondng t-statstcs, and obtaned the p-values for a stuaton wth 676 degrees of freedom (that s, 92 observatons tmes 8 equatons mnus 60 coeffcents). 13

14 3. MARSHALLIAN ELASTICITIES IMPLIED BY THE SEDS ESTIMATION Equaton/Varable P1 P2 P3 P4 P5 P6 P7 P8 YN Normal Cola (Q1) -0, , , , , , , , , P-value (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) Lght Cola (Q2) 0, , , , , , , , , P-value (0,0026) (0,0000) (0,0023) (0,0115) (0,0021) (0,0004) (0,0003) (0,0037) (0,0000) Normal Lme (Q3) 0, , , , , , , , , P-value (0,0014) (0,0019) (0,0000) (0,0045) (0,0020) (0,0003) (0,0003) (0,0025) (0,0000) Lght Lme (Q4) 0, , , , , , , , , P-value (0,1614) (0,2148) (0,1802) (0,0000) (0,1948) (0,0910) (0,0766) (0,1998) (0,0000) Normal Orange (Q5) 0, , , , , , , , , P-value (0,0962) (0,0990) (0,1040) (0,1535) (0,0000) (0,0546) (0,0463) (0,1011) (0,0000) Lght Orange (Q6) 0, , , , , , , , , P-value (0,7087) (0,7028) (0,7021) (0,6972) (0,7010) (0,0000) (0,7318) (0,6927) (0,1593) Grapefrut (Q7) 0, , , , , , , , , P-value (0,2617) (0,2521) (0,2592) (0,2293) (0,2435) (0,3139) (0,0000) (0,2271) (0,0001) Tonc Water (Q8) 0, , , , , , , , , P-value (0,6983) (0,7122) (0,7066) (0,7350) (0,6987) (0,6469) (0,6339) (0,0000) (0,1719) 14

15 To check f the estmates generated by the SEDS model were good and reasonable, we compared them wth the ones produced by other alternatve specfcatons. One frst natural experment was to compare them wth the ones produced by other less restrctve logarthmc forms. These forms were an unconstraned logarthmc system, a logarthmc system to whch we mposed N homogenety restrctons gven by equaton (9), and a logarthmc system to whch we mposed the same homogenety restrctons plus N addng-up restrctons gven by equaton (12). The man results of these three alternatve specfcatons appear on the frst three columns of table 4. In t we see that the unrestrcted logarthmc and homogeneous logarthmc regressons generate very poor estmatons of the own-prce elastctes of the dfferent carbonated soft drnks, snce only one of the eght estmated coeffcents s sgnfcantly dfferent from zero at a 5% level of probablty (and that s the same coeffcent n both cases: the own-prce elastcty of the grapefrut soda). We also see that one coeffcent, whch corresponds to the own-prce elastcty of the lght lme soda, dsplays the wrong sgn n both estmatons 14. When we move to the estmates generated by the logarthmc specfcaton that ncludes both the homogenety and addng-up restrctons, whch appear on the thrd column of table 4, the results mprove, snce all the estmated own-prce elastctes have the rght sgn and are statstcally dfferent from zero 15. They also end up beng around the same range of values (from 0.99 to 1.24), somethng that dd not happen wth the unrestrcted logarthmc and homogeneous logarthmc regressons. The estmates for the cross-prce elastctes, not reported on table 4, are nevertheless dsappontng, snce only 16 out of 48 coeffcents dsplay the expected postve sgn, and only 7 of them are statstcally dfferent from zero at a 5% level of sgnfcance. 14 Many other coeffcents estmated by these models, not reported on table 4, are also not sgnfcant and/or dsplay the wrong sgn. These two systems were estmated usng teratve seemngly unrelated regressons (SUR), snce endogenety s not an ssue n those models and therefore t was not necessary to use 3SLS. 15 Ths model, unlke the two prevous ones, was estmated usng teratve 3SLS, snce some of ts varables are functons of the expendture shares, whch are endogenous to the demand systems. 15

16 4. COMPARISON WITH ALTERNATIVE SPECIFICATIONS Concept Logarthmc Log homog Log hom add SEDS SEDS add Translog AIDS QUAIDS Own-prce elastctes Normal Cola (η 11 ) -2, , , , , , , , P-value (0,0778) (0,0836) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) Lght Cola (η 22 ) -0, , , , , , , , P-value (0,5128) (0,4625) (0,0000) (0,0000) (0,0000) (0,4786) (0,3775) (0,7850) Normal Lme (η 33 ) -1, , , , , , , , P-value (0,0929) (0,0680) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) Lght Lme (η 44 ) 0, , , , , , , , P-value (0,5442) (0,8125) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) Normal Orange (η 55 ) -0, , , , , , , , P-value (0,8911) (0,7972) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) Lght Orange (η 66 ) -1, , , , , , , , P-value (0,2161) (0,4495) (0,0000) (0,0000) (0,0000) (0,5173) (0,5456) (0,0963) Grapefrut (η 77 ) -1, , , , , , , , P-value (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) Tonc Water (η 88 ) -0, , , , , , , , P-value (0,2620) (0,1576) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) (0,0000) R2 coeffcents Eqn 1 (Normal Cola) 0, , , , , , , , Eqn 2 (Lght Cola) 0, , , , , , , , Eqn 3 (Normal Lme) 0, , , , , , , , Eqn 4 (Lght Lme) 0, , , , , , , , Eqn 5 (Normal Orange) 0, , , , , , , , Eqn 6 (Lght Orange) 0, , , , , , , , Eqn 7 (Grapefrut) 0, , , , , , , , Eqn 8 (Tonc Water) 0, , , , , , , , System 0, , , , , , , ,

17 All logarthmc models produce relatvely hgh R 2 coeffcents n most equatons. As we expect, these coeffcents are generally hgher n the unrestrcted model, and a bt lower n the homogeneous one. In the homogeneous model wth addng-up restrctons they are even lower, and ths s partcularly notceable for the case of the equatons that represent the demand functons of lght cola and lght lme soda. Compared to the R 2 coeffcents generated by the SEDS regressons, moreover, they are lower n seven out of eght equatons. Ths s partcularly notceable snce the homogeneous model wth addng-up restrctons s a model wth 80 coeffcents, whle SEDS s a model wth only 60 coeffcents. To compare the goodness of ft of the dfferent models we have also estmated the systems R 2 coeffcents, based on the methodology proposed by McElroy (1977). Comparng those coeffcents among themselves we can conclude that the SEDS model performs only slghtly worse than the unrestrcted and homogeneous logarthmc models (that are models wth 96 and 88 coeffcents, respectvely), but better than the homogeneous logarthmc model wth addng-up restrctons. The next alternatve specfcaton whose results are reported on table 4 s a varety of the SEDS model that ncludes one addng-up restrcton gven by equaton (13). Its results, obtaned after performng teratve 3SLS regressons, are relatvely smlar to the ones generated by the SEDS model wthout the addng-up restrcton, wth the partcularty that the estmated own-prce elastctes are hgher. The estmated substtuton elastctes, not reported on table 4, are also hgher n general, and they all dsplay the rght postve sgn and are statstcally dfferent from zero at any reasonable level of sgnfcance. The only weakness of ths model seems to be ts goodness of ft, snce the estmated R 2 coeffcent of the system s consderably lower than the one produced by the SEDS model wthout the addng-up restrcton. Ths may be due to the fact that mposng that restrcton s equvalent to force the average ncome elastcty of the eght goods to be equal to one. Ths may generate a relatvely hgh dstorton, consderng that our prevous estmates for those ncome elastctes were on the range between 0.76 to The last three columns of table 4 show the results produced by the three flexble functonal forms that run expendture share regressons to estmate the demand parameters (.e., translog, AIDS and QUAIDS). They were all made usng teratve SUR equatons, and measurng ncome usng the varable of total expendture n carbonated soft drnks, nstead of the EMAE tmes CPI varable used n the prevous models 16. The average elastctes reported 16 Ths s a theoretcal partcularty of the AIDS and QUAIDS models (the translog model does not use ncome as an ndependent varable), related to the need that expendture n all the goods whose demands are estmated 17

18 were n all cases calculated usng the followng formula 17 : η = + (18) ; s 1 β and ther correspondng p-values were obtaned usng the same method reported n footnote 13. The R 2 coeffcents obtaned correspond to the seven equatons of the model (that regress the expendture shares of the frst seven goods), plus the R 2 coeffcent of an equaton for the expendture share of tonc water. Ths last coeffcent was obtaned from runnng the system agan, ncludng the tonc water share equaton and excludng the normal cola one. The results produced by the translog, AIDS and QUAIDS models are relatvely smlar among themselves, and clearly worse than the ones generated by SEDS. The estmated elastctes dsplay the rght sgns for seven out of the eght goods, but one of them (the one correspondng to the demand of lght cola) s not statstcally dfferent from zero. The estmated demand elastcty whose sgn s postve (lght orange soda) s not statstcally dfferent from zero, ether, and ths s a feature that appears n the three alternatve models. Many cross-prce coeffcents, moreover, dsplay wrong (negatve) sngs, and ths s also a pervasve feature of the three models under consderaton. The correspondng R 2 coeffcents, fnally, are not consstently hgher than the ones produced by SEDS but, as expected, are always hgher n the QUAIDS model, slghtly lower n the AIDS model, and even lower n the translog model 18. The system R 2 coeffcents generated by the three models, fnally, are smaller than the one that corresponds to the SEDS model, although, for the AIDS and QUAIDS models, they are hgher than the one produced by the SEDS wth the addng-up restrcton 19. A last experment that we performed s the one whose results appear on table 5. It conssts of runnng alternatve SEDS models wth dfferent aggregaton levels for our commodtes. Apart from our benchmark model wth eght commodtes, we also estmated a demand system for only four commodtes. In t, we pooled together the normal cola and the adds up to the total consumer s ncome. We nevertheless tred alternatve regressons usng EMAE*CPI as a measure of nomnal ncome and the results dd not vary a lot. 17 In fact, ths formula s exact only for the translog system, and t s one of the possble lnear approxmatons for the own-prce elastcty under the AIDS and QUAIDS models. For other alternatve specfcatons, see Alston, Foster and Green (1994). 18 Ths rankng has to do wth the fact that the translog model can be seen as a partcular case of AIDS (for whch all β Y = 0), and AIDS can be seen as a partcular case of QUAIDS (for whch all λ Y = 0). 19 In fact, the use of the systems R 2 coeffcents to contrast the dfferent models s only a relatvely quck method to compare the results. For a more sophstcated analyss of ths queston, appled to the comparson of logarthmc and AIDS models, see Alston, Chalfant and Pggott (2002). 18

19 normal orange soda to create a new composte commodty (good 1), and we dd the same wth the lght cola and the lght orange soda (good 2), the normal lme and grapefrut sodas (good 3), and the lght lme soda and the tonc water (good 4). We further reduced the number of commodtes to two, poolng together all the normal carbonated soft drnks (cola, lme, orange and grapefrut) to create a sngle composte commodty (good 1), and the lght carbonated soft drnks (cola, lme, orange and tonc water) to create another composte commodty (good 2). On one sde, we ended up wth a system of four equatons, wth four own-prce elastctes and sx substtuton elastctes. On the other sde, we have a system of two equatons wth two own-prce elastctes and one substtuton elastcty. 5. COMPARISON OF DIFFERENT AGGREGATION LEVEL RESULTS Concept Eght commodtes Four commodtes Two commodtes Coeffcent P-value Coeffcent P-value Coeffcent P-value Own-prce elastctes Normal Cola (η 11 ) -0, ,0000-0, ,0000-0, ,0000 Lght Cola (η 22 ) -0, ,0000-0, ,0000-0, ,0000 Normal Lme (η 33 ) -0, ,0000-0, ,0000-0, ,0000 Lght Lme (η 44 ) -0, ,0000-0, ,0000-0, ,0000 Normal Orange (η 55 ) -0, ,0000-0, ,0000-0, ,0000 Lght Orange (η 66 ) -1, ,0000-0, ,0000-0, ,0000 Grapefrut (η 77 ) -1, ,0000-0, ,0000-0, ,0000 Tonc Water (η 88 ) -0, ,0000-0, ,0000-0, ,0000 Substtuton elastctes NCola/LCola (σ 12 ) 0, ,0000 0, ,0000 0, ,0000 NCola/NLme (σ 13 ) 0, ,0000 0, ,0000 NCola/LLme (σ 14 ) 0, ,0000 0, ,0000 0, ,0000 LCola/NLme (σ 23 ) 0, ,0000 0, ,0000 0, ,0000 LCola/LLme (σ 24 ) 0, ,0000 0, ,0000 NLme/LLme (σ 34 ) 0, ,0000 0, ,0000 0, ,0000 By lookng at the results reported on table 5, we see that they respond to what economc theory predcts. When we nclude more products n the defnton of a commodty, own-prce elastctes become smaller n absolute value, and substtuton elastctes become lower. Ths s because the redefned commodtes are now poorer substtutes among themselves, and ther demands must therefore be more nelastc than the ones estmated under a more precse commodty dentfcaton 20. Table 5 also shows that all the estmated elastctes contnue to dsplay the expected sgns and to be statstcally dfferent from zero at any possble level of sgnfcance. 20 For a more detaled explanaton of the economc logc of ths, see Werden (1998). 19

20 5. Conclusons The man conclusons of our analyss can be summarzed as follows: a) The concept of elastcty of substtuton s a good nstrument to ntroduce symmetry n the estmaton of a system of demand functons. b) Relyng on t, t s possble to buld a lnear system of logarthmc demand equatons whose man coeffcents are own-prce elastctes and substtuton elastctes, whch s capable of ncorporatng the homogenety, symmetry and, eventually, addng-up restrctons of consumer demand theory. c) Ths system of equatons, whch we call SEDS, has to be estmated usng nstrumental varables (for example, through three-stage least square regressons), snce ts ndependent varables are functons of prces, ncome and expendture shares, and these shares are endogenous varables n the demand equatons. d) Despte ths endogenety problem, SEDS has some advantages over the most common demand systems based on flexble functonal forms (namely, translog, AIDS and QUAIDS models), snce t s more effcent and t generates estmates that are easer to handle when we want to mpose addtonal estmaton restrctons. e) It s also better than the less restrcted logarthmc models, snce t s capable of ncorporatng the symmetry property n a way that s more consstent wth consumer theory. It s also less lkely to generate coeffcents wth the wrong sngs or coeffcents that are not statstcally sgnfcant. f) All ths makes SEDS partcularly sutable for the estmaton of demand systems of products that belong to the same ndustry, n whch we can make the assumpton that ther demands are related among themselves but are bascally ndependent from the demands of other goods. g) Wth ths dea n mnd, we have appled the model to a database of weekly observatons of prces and quanttes of eght dfferent carbonated soft drnks, n order to estmate the correspondng demand equatons. We have obtaned reasonable and hghly sgnfcant estmates for all the own-prce and substtuton elastctes. h) Usng those estmated coeffcents, we have also obtaned reasonable estmates for the mpled ncome and cross elastctes between the products. ) Compared to other alternatve estmaton methodologes (unrestrcted logarthmc, translog, AIDS, QUAIDS) the results of the SEDS model perform notceably well, snce the alternatve methodologes always produce wrong sngs for some elastctes, less 20

21 sgnfcant coeffcents, or a lower goodness of ft. ) The model also performs well aganst dfferent versons of tself. For example, when we aggregate the commodtes to run a system wth four equatons and a system wth two equatons, own-prce elastctes become smaller n absolute value, and substtuton elastctes become lower. References Allen, R. G. D. (1938). Mathematcal Analyss for Economsts. London, Macmllan, Alston, Julan; James Chalfant and Ncholas Pggott (2002). Estmatng and Testng the Compensated Double-Log Demand Model ; Appled Economcs, vol 34, pp Alston, Julan; Kenneth Foster and Rchard Green (1994). Estmatng Elastctes wth the Lnear Approxmate Almost Ideal Demand System ; Revew of Economcs and Statstcs, vol 76, pp Banks, James; Rchard Blundell and Arthur Lewbel (1997). Quadratc Engel Curves and Consumer Demand ; Revew of Economcs and Statstcs, vol 79, pp Barten, Anton (1993). Consumer Allocaton Models: Choce of Functonal Form ; Emprcal Economcs, vol 18, pp Blackorby, Charles and Robert Russell (1989). Wll the Real Elastcty of Substtuton Please Stand Up? ; Amercan Economc Revew, vol 79, pp Chrstensen, Laurts; Dale Jorgenson and Lawrence Lau (1975). Trascen dental Logarthmc Utlty Functon ; Amercan Economc Revew, vol 65, pp Dahr, Trtha; Jean-Paul Chavas, Ronald Cotterll and Bran Gould (2005). An Econometrc Analyss of Brand-Level Strategc Prcng Between Coca-Cola Co. and PepsCo. ; Journal of Economcs and Management Strategy, vol 14, pp Deaton, Angus and John Muellbauer (1980). An Almost Ideal Demand System ; Amercan Economc Revew, vol 70, pp Epsten, Roy and Danel Rubnfeld (2002). Merger Smulaton: A Smplfe d Approach wth New Applcatons ; Anttrust Law Journal, vol 69, pp Frsch, Ragnar (1959). A Complete Scheme for Computng All Drect and Cross Demand Elastctes n a Model wth Many Sectors ; Econometrca, vol 27, pp McElroy, Marore (1977). Goodness of Ft for Seemngly Unrelated Regressons ; Journal of Econometrcs, vol 6, pp

22 Moschn, Gancarlo and Anuradha Vssa (1993). Flexble Specfcatons of Mxed Demand Systems ; Amercan Journal of Agrcultural Economcs, vol 75, pp 1-9. Werden, Gregory (1998). Demand Elastctes n Anttrust Analyss ; Anttrust Law Journal, vol 66, pp Acknowledgements I thank comments by Rcardo Bebczuk, Marana Conte Grand, Marana Marchonn, Jorge Streb, and partcpants at semnars held at CEMA Unversty and the Natonal Unversty of La Plata. 22