Seoul Journal of Buine Volume 16, Number 1 (June 2010) Item Aggregate and Price Elaticity INSEONG SONG *1) Seoul National Univerity Seoul, Korea Abtract Thi tudy provide analytical reult on the ytematic relationhip between the elaticitie obtained from item-aggregated data and thoe from SKU level data. It i hown that the brand level (or any aggregate level) elaticitie are hare-weighted average of SKU elaticitie. A SKU in a brand are ubtitute each other in general, the own brand elaticitie would be maller in magnitude than the own SKU elaticitie and the crobrand elaticitie will be larger than cro-sku elaticitie. It i alo found that when the SKU level demand function i given by a homogeneou logit model with the latent utility being linear in price, the price enitivity parameter etimated from the brand level data hould be the ame a that from SKU level data if the brand level model fit data well and the brand price are given by the weighted average of SKU price with the weight being within-brand SKU hare. Keyword: Item Aggregate, Price Elaticity, Logit Model INTRODUCTION Mot conumer packaged good categorie have hundred of item called tockkeeping unit (SKU) or UPC in a category. Such a large number of SKU in a category poit challenge to marketing reearcher in modeling the demand for product in the category. One imple approach would be to limit the analyi to a few elected item with mot hare o that the data et ha a manageable number of alternative. Such a data pruning practice can invite * Aociate Profeor of Marketing, College of Buine Adminitration, Seoul National Univerity, 599 Gwanangno, Gwanak-gu, Seoul, Korea, 151-916 (iong@ nu.ac.kr), Phone: +82-2-880-2649
46 Seoul Journal of Buine ignificant bia in parameter etimate and implied elaticitie (Zanutto and Bradlow 2006). Alternatively, marketing reearcher have modeled demand for the product in a category motly at the brand level or at the brand-ize level rather than at the SKU level in order to reduce the number of choice alternative. In brand level or brand-ize level tudie, aggregate meaure uch a aggregate price and aggregate demand are contructed even when SKU level canner data are available. Typically, the item aggregation would be motivated by practical reaon uch a the computational requirement involved in the model etimation proce or by the nature of data uch a pare obervation at SKU level or volatile choice et due to the frequent entry and exit of SKU (Bucklin and Gupta 1999). More importantly, it i alo often the cae that the item aggregation i required by the nature of the reearch quetion being involved in the tudy. For example, in order to ae the impact of the merger between two brand in a category on the market demand and the equilibrium pricing, one need to figure out the ubtitution pattern at the brand level rather than at the SKU level. The outcome of brand level tudie would include the etimate of own and cro-brand price elaticitie. One of the potential iue would be how the etimate obtained from the data of item aggregate can be related to the demand characteritic for diaggregate unit, SKU. It ha been empirically oberved that item aggregation would have an impact on the etimate of demand repone for marketing activitie. For example, the meta analyi by Bijmolt, van Heerde, and Pieter (2005) report that brand level tudie produce ignificantly maller price elaticitie than SKU level tudie, -2.50 for brand v. -2.97 for SKU. In thi paper, I invetigate the ytematic relationhip between the brand level price elaticity and the SKU level price elaticity. I try to reconcile the difference between elaticitie from different aggregation level and to provide mathematical relationhip between them. I focu on price elaticity for everal reaon. Firt, price elaticity i one of the key pricing iue identified by practitioner (Bucklin and Gupta 1999). Manager recognize that undertanding price elaticity i the baic tarting point for better pricing. Second, price elaticity i alo a quantity of interet to policy maker a it play a key role in everal iue in indutrial organization tudie uch a market tructure, merger and acquiition, and o on. Third, a documented well in Telli (1988) and Bijmolt, van Heerde, and
Item Aggregate and Price Elaticity 47 Pieter (2005), price elaticity ha been one of the key reearch iue among academic. Moreover, although I limit my analyi to price elaticity in thi paper, the tructure of the analyi can alo be applicable directly to the elaticity meaure for other marketing activitie uch a promotion. Unlike Allenby and Roi (1991), the aggregation iue I invetigate in thi paper i the aggregation acro product item, not acro houehold or acro tore. While relatively little attention ha been given to the iue of item aggregation, there i relatively rich literature on the iue of data aggregation acro houehold or acro tore. For example, Chriten et al. (1997) how that the etimate of promotion effect calibrated from linearly aggregated market level data would be ubtantially different from thoe obtained from tore level data. Gupta et al. (1996) find that panelit houehold in houehold level canner data are not repreentative enough to reflect the demand characteritic of tore level data from the ame community, although the average price elaticity etimate are cloe for both data et. Unlike Chriten et al. (1997) and Gupta et al. (1996), my interet in thi paper i how the price elaticitie obtained from item aggregate (brand) can be related to the price elaticitie of demand for individual item (SKU). While any difference in etimation outcome between aggregate data (market level data) and diaggregate data (tore level or houehold level data) would be conidered bia in their tudie, I view the difference between the price elaticitie from item aggregate data and thoe from individual item data a natural and try to reconcile the difference. The iue of item aggregation ha been tudied in marketing area. The main theme of uch tudie i how to etimate the SKU level preference parameter. Fader and Hardie (1996) and Ho and Chong (2003) provide way to analyze houehold level SKU choice by building a parimoniou model o that the number of parameter would not explode with the number of SKU conidered. Thee tudie would impoe a particular et of retriction on the tructure of SKU level demand in order to keep the number of parameter manageable. Bell, Bonfrer, and Chintagunta (2005) utilize tore level data and provide a way to recover SKU level preference parameter from the market hare model etimated from item aggregate. They exploit the particular tructure of the functional form of the demand logit. Unlike thoe tudie, the focu of my tudy i not on how to
48 Seoul Journal of Buine etimate the SKU level elaticity. The key iue in thi paper i how and why the brand level (or any other item aggregate level) elaticity i different from the SKU level elaticity. Without impoing any particular tructure or functional form on the SKU level demand, I how how the price elaticity obtained from item aggregate can be related to the price elaticity obtained from individual item o that the finding can be applied to a general et of demand model. The remainder of the paper i organized a follow: In the next ection, an analytical model on the impact of item aggregation on the price elaticity i preented. Then, I explore how the functional form for the demand would be affected by item aggregation. Analytical reult along with ome imulation reult are provided. Finally, a brief concluion follow. GENERAL MODEL Conider a model for SKU level demand in a category. Suppoe there are M brand in the category and J m SKU in brand m=1,..,m. While I ue the ubcript m to denote brand, it can be any level of item aggregate uch a brand-ize or product line. I ue a general demand function for SKU a follow: Q = Q ( p,..., p, p,..., p,..., p,..., p ) mj mj 11 1J1 21 2J 2 M1 MJ M (1) where Q mj i the demand for SKU j in brand m and p mk i the unit price of SKU k in brand m. Note that the unit price i ued in the demand function o that the price of item aggregate are meaningful. Similarly, the SKU demand i alo meaured by the common unit uch a ounce, not by the number of item old. I ue the following notation to denote the SKU level own and cro price elaticitie: η Q p Q p mj mj mj nk mj, mj = and ηmj, nk = pmj Qmj pnk Qmj (2) where the upercript tand for SKU level elaticity. Now conider the demand for item aggregate (brand), where the brand level data are the linear aggregation of SKU level data,
Item Aggregate and Price Elaticity 49 (,,..., ) Q = Q p p p (3) m m 1 2 M where Q m i the demand for brand m and p m i the price of brand m. The linear aggregation indicate that the brand level data are given a follow: Q m J m = Q (4) j = 1 mj Jm (5) Jm p = w p, w 0, m, j, and w = 1. m mj mj mj mj j= 1 j= 1 The brand demand i the um of SKU demand and the brand price i a weighted average of SKU price. Note that a different weighting cheme, w mj, for the brand price contruction implie a different demand function for (3). Different weight would reult in ytematically different brand price while the demand quantitie in (4) in the aggregate data are not affected by the weighting cheme. So the parameter and/or even the functional form of the demand function in (3) would be dependent upon the weighting cheme ued in the item aggregation proce. In order to derive the brand level price elaticitie, I need to clarify the meaning of the change in brand price. Price elaticity i conidered a thought experiment where the percentage change in demand i meaured a a conequence of one percent change in price while other thing are held contant. So the experiment i meaningful only when reearcher compare two point within a demand function. Therefore, when reearcher change the brand price in the experiment, they hould not change the weight, w mj, given to SKU price. Change in weight reult in a comparion acro two different demand function, not within a demand function, becaue the nature of the brand price in (3) would not be the ame. That i, the weight hould be the ame between two regime before price change and after price change. The weightpreerving change in brand price i accomplihed by the identical percentage change in price of all SKU in the brand. That i, in order to be a legitimate thought experiment, the manipulation, one percentage change in brand price, mut be accomplihed by one percentage change in all SKU. Mathematically, thi implie
50 Seoul Journal of Buine dlog p = dlog p, j = 1,.., J. (6) m mj m The change in the demand for brand m due to the change in the price of brand n i given by Q dq dq dp Jm Jm Jn mj m = mj = nk j= 1 j= 1k= 1 pnk = Jm Jn Qmj pnk Q p Q p mj j= 1k= 1 nk mj nk dp nk Jm Jn mj, nkqmjd pnk j= 1k= 1 = η log. (7) So the percentage change in brand demand i given by Q dlogq dlog p. (8) Jm Jn m mj m = dq mj, nk nk Q = η m j= 1k= 1 Qm Denote the within-brand SKU hare λmj Qmj / Qm and ue the weight-preerving condition for brand price change in (6) to derive the own and cro brand level elaticitie a follow: J log m Jm b d Qm m, m mj mj, mk dlog pm j= 1 k= 1 = = (9) η λ η J log m Jn b d Qm m, n mj mj, nk dlog pn j= 1 k= 1 = =. (10) η λ η where the upercript b indicate the brand level elaticity. The expreion in (9) and (10) how that the brand level elaticity i a weighted um of SKU level elaticitie with the weight given by the within-brand SKU hare. In an extreme cae where SKU demand are independent ( η mj, mk = 0, j k), the own brand elaticity i nothing but the hare-weighted average of own SKU elaticitie. But in mot real world marketing application where SKU within a brand are expected to be ubtitute each other in general ( η mj, mk > 0, j k), the own brand level elaticitie will be maller than the hare-weighted average of own SKU level elaticitie in
Item Aggregate and Price Elaticity 51 magnitude. Similarly, the cro-brand elaticity will be larger than cro-sku elaticity. If the price of 128oz Tide detergent increae, conumer can witch to a different ize of Tide detergent or to a ame or a different ize in other brand. When the price of all Tide detergent product increae, conumer will witch only to other brand. So the price elaticity would be maller (in magnitude) for Tide brand than for 128oz Tide. The expreion in (9) and (10) provide an analytical relationhip of price elaticitie between different level of aggregation. They imply that the ubtitutability hould be the only factor related to any difference between the brand level elaticity and the SKU level elaticity if the analyi i done for the ame demand group (a group of houehold or a market). In general, a broader definition of a product would reult in a maller own price elaticity for the aggregated item. A longer time frame in a demand model would reult in maller own elaticitie. It ha been reported empirically that a tatic demand model ignoring intertemporal ubtitution would produce larger etimate for own elaticitie (Hendel and Nevo 2006) and maller cro elaticitie (Erdem, Imai, and Kean 2003). The demand tructure and the nature of item aggregation alo provide information on the relationhip among brand level elaticity, SKU level elaticity, and the elaticity of within-brand SKU hare. In order to derive the elaticity of the within-brand SKU hare, firt note that the percentage change in relative hare of SKU in a brand i given by Qmj dlog λ mj = dlog = dlogqmj dlogqm. (11) Q m Combining (11) with (6) yield an expreion for the elaticity of within-brand SKU hare a follow: η λ mj, n dlogq J log n mj d Q dlogq m mj dlogq = dlog p dlog p dlog p dlog p n n k = 1 nk n Jn b ηmj, nk ηm, n. k = 1 = m (12) where the upercript λ indicate the elaticity of within-brand SKU hare. Rewriting (12) produce an expreion for the elaticity of a SKU relative to a brand price a follow:
52 Seoul Journal of Buine J n b λ mj, n mj, nk = m, n + mj, n k = 1 η η η η. (13) The expreion in (13) i intuitive in the ene that the effect of brand-wide price change of all SKU in brand n on the demand for a SKU in brand m (in the ame or other brand) can be decompoed into two effect: brand witching effect ( η mn, ) and within-brand SKU λ witching effect ( η mj, n ). Note that although I ue the term witching, the brand witching effect can include primary demand effect uch a the change in overall conumption level while the SKU witching effect refer to pure hare-adjuting effect. It can be eaily verified that the hare-weighted um of the within-brand SKU witching effect i zero by multiplying λ mj both ide of (12) and umming over j. Equation (9) and (10) provide a way to compute brand level elaticitie from SKU level elaticitie. However, in general it i not feaible to recover SKU level elaticitie from brand level elaticitie uing (9) and (10) unle reearcher impoe a particular et of retriction on the tructure of SKU level demand a in Fader and Hardie (1996), Ho and Chong (2003), or in Bell, Bonfrer, and Chintagunta (2005). b ITEM AGGREGATES AND FUNCTIONAL FORMS FOR DEMANDS Another iue related to item aggregation i the functional form for the demand function in (1) and (3). In general, the functional form doe not remain the ame a item are aggregated. Conider, for example, linear demand function for SKU demand. Suppoe there i only one brand with 2 SKU in the market. The SKU level linear demand i given by 1 α1 β11 1 β12 2 ε1 2 α2 β21 1 β22 2 ε2 Q = + p + p +, and Q = + p + p +. (14) So the true demand function for the item aggregate (brand) i 1 2 α1 α2 β11 β21 1 β12 β22 2 ε1 ε2 Q = Q + Q = + + ( + ) p + ( + ) p + +. (15)
Item Aggregate and Price Elaticity 53 I the demand function in (15) i linear in brand price? The (poibly ill-pecified) linear demand function for the brand demand baed on aggregate price would have the following form: b b b b b b b 1 1 2 2. Q= α + β p+ ε = α + β wp + β w p + ε (16) Becaue the brand price i given by p = w 1 p 1 + w 2 p 2 from (5). The two linear demand function, (15) and (16), are equivalent only when β b 11 + 21 12 + 22 β β β β = =. (17) w w 1 2 That implie that if the weight ued to contruct aggregate price do not atify the condition w1 / w2 = ( β11 + β21)/( β12 + β22) then there doe not exit a linear function for brand demand when the true SKU level demand i characterized by a linear function. The ituation get wore when it come to nonlinear demand function uch a the multiplicative demand function or the log-log function, which i frequently ued to etimate price elaticity in many marketing application. Conider the following log-log demand function for the SKU, 1 = α1 + β11 1 + β12 2 + ε1 logq log p log p, 2 = α2 + β12 1 + β22 2 + ε2 logq log p log p. (18) If (18) i the true model, then the true aggregate demand i given by 1 2 α1 β11 1 β12 2 ε1 Q = Q + Q = exp( + log p + log p + ) 2 21 p1 22 p2 1 + exp( α + β log + β log + ε ). (19) The demand in (19) cannot be expreed a log Q = α b + β b log b ( wp 1 1+ w2p2) + ε regardle of the weight. It i expected that when the true demand for SKU are given by a nonlinear demand function the brand level demand function will not follow the ame functional form. Note that it i poible that there exit a correct functional form for brand level demand which i likely to be different from the functional form for SKU level demand. While
54 Seoul Journal of Buine thi iue i beyond the cope of thi tudy, intereted reader are directed to Deaton and Muellbauer (1980) Chapter 5. What would happen if reearcher impoe the ame functional form for demand at different level of aggregation? While uch iue ha been tudied little for mot nonlinear function, one notable exception i the logit model. A the extreme value ditribution i maintained under maximization (i.e., the maximum of independent extreme value ditributed random variable i alo extreme value ditributed), if the SKU level market hare follow a logit model, the brand level market hare i alo characterized by a logit model. However, one cannot ay it ha the ame functional form. In fact, the true utility tructure i no longer linear in the brand price index in the model for the item aggregate even if the true utilitie in the SKU level model are linear in SKU price. It i analogou to the neted logit tructure. (See Chapter 9 of Ben-Akiva and Lerman (1985) for a detailed dicuion.) Interetingly, I find that impoing the ame linear-in-price tructure on the utility in the model for item aggregate would impoe a retriction on the price enitivity parameter. Conider a SKU level logit model where the market hare of the SKU j of brand m ( mj ) i given by mj = exp( α + β p ) M Jn n= 1k= 1 mj nk exp( α + β p ) mj nk. (18) The true brand level hare i obtained by umming the hare of all SKU. The brand hare ( m ) and the within-brand SKU hare (λ mj ) are given a follow: Jm exp( ) J αmj + β pmj m j = 1 m = mj = M Jn j = 1 αnk + β pnk n= 1k= 1 mj exp( αmj + β pmj ) mj = = Jn m exp( αmk + β pmk ) k = 1 λ exp( ), and (19) Uing the reult in (9), it can be eaily hown that the true own brand elaticity for the demand in (19) i given by
Item Aggregate and Price Elaticity 55 J m b ηm, m = β λmj pmj ( 1 m ) (20) j = 1 Suppoe the following linear-in-price tructure i ued to model brand level market hare a a a exp( αm + β pm) m = M a a n n n = 1 exp( α + β p ) (21) where the upercript a indicate the aggregate level model which i poibly mipecified. The model in (21) would produce the following expreion for own brand price elaticity: ( ) J 1 m ( 1 ) a a a a a m, m = pm m = wmj pmj m j = 1 η β β. (22) If the model in (21) fit the data well ( m m) and the weight ued to contruct brand price indice are equal or cloe to withinbrand SKU hare ( wmj λmj ), then the price enitivity parameter a in (21) hould be cloe to that in (18), β β. I conduct a mall cale imulation to verify it. In the imulation, there are 2 brand with 2 SKU in each brand. The SKU level utility i given by Umjt = αmj + βpmjt + εmjt where the true parameter are α = {0.5, 0.6, 0.7, 0.8} and β = -2. SKU price are generated by adding independent uniform random number, u(0, 1), to the mean price of 4 SKU, {0.5, 0.7, 0.8, 1.0}. I generate 10000 obervation. I etimate the SKU level logit model in (18). And then I aggregate data into brand level and etimate the brand level logit model in (21) uing the brand level data. The aggregate price are weighted average of SKU price where weight are within-brand SKU hare computed acro all obervation. I repeat the imulation 50 time. A not all product intercept are eparately identified in the logit model, I normalize the intercept for the firt product to zero. Note that the SKU level model i the true model ued to generate the data. The imulation reult in table 1 how that the brand level model produce almot identical reult a the SKU level model doe. Although Ben-Akiva and Lerman (1985: 259) alo expect a
56 Seoul Journal of Buine Table 1. Simulation Reult with Same Number of SKU SKU Level Model (β ) Brand Level Model(β a ) Average Etimate acro replication -2.0070-1.9959 Std. Dev. acro imulation replication 0.0525 0.0893 Average Standard Error 0.0399 0.0715 Average Difference (β - β a ) -0.0111 Std. Dev. of Difference (β - β a ) 0.0738 Table 2. Simulation Reult with Different Number of SKU SKU Level Model (β ) Brand Level Model(β a ) Average Etimate acro replication -1.9948-2.0023 Std. Dev. acro imulation replication 0.0483 0.0833 Average Standard Error 0.0405 0.0859 Average Difference (β - β a ) 0.0075 Std. Dev. of Difference (β - β a ) 0.0730 uch reult when the ize and the variance meaure are adjuted properly, I do not control for uch factor in the brand level model in the imulation. I do not include the variance meaure in the brand level model o that the brand level model i mipecified. However, the ize and the variance meaure are quite imilar acro brand, o they may be cancelled out in the imulation model. I conduct another et of imulation where the number of SKU i different acro brand. In thi et of imulation, brand 1 ha 3 SKU and brand 2 ha only one SKU. Under thi cae, the ize and the variance are different acro brand o they would not be cancelled out. I keep the other imulation parameter the ame a before. The reult in table 2 indicate that the etimate of price enitivity parameter from the aggregate data i again almot identical to that obtained from SKU level data, which verifie well the relationhip in (20) and (22). The analytical reult in (20) and (22) together with the imulation reult indicate that, a long a the brand level logit model fit data well, the price enitivity parameter obtained from the brand level data will be the ame a (cloe to) the price enitivity parameter
Item Aggregate and Price Elaticity 57 from the SKU level data in a logit model. Although there i little literature that empirically invetigate the impact of item aggregation on the etimate of price enitivity parameter, Bell, Bonfrer, and Chintagunta (2005) provide ome reult on the iue. However, they report finding inconitent with my reult. They found that the price enitivity calibrated from SKU level toothpate data i -5.47 wherea the price enitivity etimate from brand level data i -6.18. While it i not a key reearch iue in their paper, they alo report the etimate of price enitivity obtained from aggregate data baed on variou aggregation cheme uch a flavor, form, function, and ize. Their price enitivity etimate dramatically differ along the dimenion ued to aggregate item. For example, when the data were aggregated into flavor level (i.e., SKU with ame flavor are aggregated into a choice alternative), the price enitivity etimate wa -8.74. Such finding i inconitent with the property of the homogeneou logit model. Specifically, if the average utilitie are well pecified,, the parameter of the choice model are not dependent on the definition of the aggregate alternative. (Ben- Akiva and Lerman 1985: 259) A they ue market-hare weighted average price for the aggregate price, their reult might be due to either (1) relatively poor data fit at the aggregate model or (2) the poibility that the homogeneou logit model i not the true model underlying SKU level data generating proce, unlike the aumption in my imulation. In order to check the impact of the model mipecification at the SKU level, I conduct one more et of imulation where the error term in the utility function follow normal ditribution while other imulation parameter are the ame a in table 2. To make the model very different from logit, I aume that error term are heterocedatic but till independent acro choice alternative. The tandard deviation of the error term are 1, 3, 4, and 6 for 4 SKU repectively. A expected, the reult in table 3 how that the etimate are far from the true value. In addition, the etimate of price enitivity obtained from the SKU level data i ignificantly different from that from brand level data. It implie that if the homogeneou logit i not the true data generating proce for SKU level data, the etimate obtained from item aggregate would be different from thoe from SKU. It alo implie that comparing price enitivity etimate obtained from different aggregation level might be a indirect way to check whether the homogeneou logit model i
58 Seoul Journal of Buine Table 3. Simulation Reult with Mipecified SKU Level Model SKU Level Model (β ) Brand Level Model(β a ) Average Etimate acro replication -0.7877-0.5042 Std. Dev. acro imulation replication 0.0366 0.0662 Average Standard Error 0.0355 0.0641 Average Difference (β - β a ) -0.2835 Std. Dev. of Difference (β - β a ) 0.0544 Table 4. Simulation Reult with Larger Price Variance SKU Level Model (β ) Brand Level Model(β a ) Average Etimate acro replication -1.9984-1.967 Std. Dev. acro imulation replication 0.0374 0.0559 Average Standard Error 0.0318 0.053 Average Difference (β - β a ) -0.0313 Std. Dev. of Difference (β - β a ) 0.0551 far from the true model. What would happen if price become more volatile? In order to invetigate if the qualitative implication would remain the ame even when price variance are large, I conduct another imulation where the price i randomly drawn from u(0,1.5) while keeping other imulation etting the ame a in table 1. A preented in table 2, the new imulation produce a imilar reult a the firt imulation hown in table 1. That i, the brand level model produce almot identical reult a the SKU level model doe even when price variance i larger. One intereting iue would be to check what difference will be oberved between SKU level etimate and brand level etimate from real data. I conduct a mall cale empirical analyi uing a real data et from a panel of conumer who purchaed ground coffee product at a tore in Chicago area. The data et conit of 203 purchae obervation from 69 conumer over 62 week tarting from June 1991. A preented in table 5, I elected even major SKU from top three brand Folger, Hill Brother, and Maxwell Houe. According to the data et, Hill Brother ha the
Item Aggregate and Price Elaticity 59 Table 5. Etimation Reult from Coffee Data Decriptive Statitic Etimate Brand SKU SKU Choice Share SKU Unit Price (per oz) Brand Choice Share Brand Unit Price (per oz) SKU Level Model Brand Level Model Mean Std. Dev. Mean Std. Dev. Et. S.E. Et. S.E. Folger 29oz 0.0246 13.44 0.81 0 Fixed Folger 0.2463 14.21 1.56 0 Fixed Folger 26oz 0.2217 14.29 1.71 2.7754 0.4640 Hill Brother Hill Brother 26oz 0.1773 13.48 2.64 1.9873 0.4677 Hill Brother 24oz 0.0542 15.69 2.71 0.5616 13.20 1.47 1.7962 0.5507 Hill Brother 39oz 0.3300 12.65 1.66 2.2218 0.4432 0.3189 0.2042 Maxwell Houe Maxwell Houe 26oz 0.1379 14.72 1.05 2.6067 0.4839 0.1921 15.05 1.07 0.3520 0.2397 Maxwell Houe 24oz 0.0542 15.91 1.13 2.3843 0.5587 Price Coefficient -0.6257 0.0607-0.7355 0.0910
60 Seoul Journal of Buine larget relative hare among the three. Within a brand, SKU are different only in their package ize. SKU level choice hare vary acro SKU even within a brand. The price variation among different package ize within a brand indicate that firm pricing behavior are conitent with volume dicounting. In term of the brand level price, Hill Brother i the cheapet among the three while Maxwell Houe i relatively expenive. I etimate a logit brand choice model where the utility pecification i the ame a in the erie of imulation. That i, the utility of a product conit of the product pecific dummy and the price effect. Given uch utility pecification, I need to etimate product pecific intrinic preference and a price coefficient. Given the conditional choice pecification, I need to normalize the intrinic preference parameter for a product to zero. All the product pecific intrinic preference parameter hould be interpreted a the relative preference over the normalized product. For the SKU level model, I normalize the intrinic preference for Folger 29oz to zero. For the brand level, the normalized brand i Folger. All of the intrinic brand preference parameter are ignificant in the SKU level etimation reult. That i, all the other SKU are preferred over Folger 29oz. It i intuitive a thi product ha a mallet hare even though it i relatively cheap. On the contrary, none of the intrinic preference parameter etimate i ignificant in the brand level model. So the data et doe not provide any evidence that Hill Brother or Maxwell Houe i preferred over Folger. Next, what happen to the price coefficient? Although the etimate of the price coefficient look different between the two model, the 95% confidence interval of the SKU level etimate include the brand level etimate. Alo the 95% confidence interval of the brand level etimate include the SKU level etimate. Thi reult, combined with the implication of the imulation reult preented in table 3, might imply that the homogeneou logit model i a good candidate to decribe the underlying data generating proce for the coffee data ued in the analyi. CONCLUSION I how how the elaticitie of item aggregate are related to the elaticite for individual item. I find that that the price elaticitie
Item Aggregate and Price Elaticity 61 for item aggregate are hare weighted average of the elaticitie for diaggregate unit, indicating the ubtitutability i the only factor related to the difference between elaticitie for different aggregation. A SKU are ubtitute in general, the own brand elaticity will be maller than the own SKU elaticity and the cro-brand elaticity will be larger than cro-sku elaticity. The difference between brand elaticity and SKU elaticity i by no mean a bia. It i a pure effect of within-brand SKU ubtitutability. While in general the functional form of demand function would not be preerved a item are aggregated, the homogeneou logit model turn out to be robut to uch item aggregation and price enitivity parameter etimate i not affected by item aggregation a long a the logit model i the true model underlying the SKU level data generating proce. Although thi paper i mainly focued on methodological iue, it ha managerial implication. Mot of all, thi paper provide a theoretical ground on why manager hould take into account the cro price elaticitie when etting bae price. Economic theorie ugget that it i optimal for a monopoly firm to et the price at a level equal to the invere of own price elaticity. If uch principle i blindly applied a a rule of thumb to all SKU produced by the monopoly firm without taking the cro price elaticitie among SKU into account, the firm will end up underpricing SKU. It i eential to meaure inter-sku ubtitutability within a brand in order to optimize pricing. In addition to the bae pricing iue, thi paper alo provide an analytical framework to ae the price implication of a merger between firm. A the merged firm would optimize it price level by taking into account the cro price elaticitie, one can compute the optimal pot-merger price even before the merger happen. While uch analye have been done in empirical etting in literature, thi paper provide an analytical framework behind uch analye. Although thi paper doe not try to provide a way to recover SKU level elaticity from brand level elaticity, it would be an intereting venue for future reearch. The ability to recover eaily SKU elaticity from item aggregate would not only make it eay to analyze the demand for thouand of SKU but alo enable reearcher to infer the pricing behavior in a micro etting. Although the iue ha been tudied by Bell, Bonfrer, and Chintagunta (2005), it would be intereting to tudy how to recover SKU elaticitie from item
62 Seoul Journal of Buine aggregate with a more general demand function. Another important iue that i not explicitly analyzed here i the poibility of miing information. If ome of SKU are not included in the analyi, uch omiion can create bia. Unlike in analye of imulated data, reearcher cannot have all poible SKU in their empirical analye of real data. Zanutto and Bradlow (2006) how that pruning the data to a manageable number of SKU can create bia epecially when model fit i poor, when random utility error are correlated with covariate, or when the model i mipecified. While imulation reult in thi paper are free from uch bia, empirical analye are ubject to uch bia. The interaction between data pruning and item aggregation would be an intereting venue for future reearch. REFERENCES Alleby, G. M. and P. E. Roi (1991), There I No Aggregation Bia: Why Macro Logit Model Work, Journal of Buine and Economic Statitic, 9(January), 1-11. Ben-Akiva, M. and S. R. Lerman (1985), Dicrete Choice Analyi: Theory and Application to Travel Demand, MIT Pre, Cambridge, MA. Bell, D. R., A. Bonfrer, and P. K. Chintagunta (2005), Recovering Stockkeeping Unit-Level Preference and Repone Senitivitie from Market Share Model Etimated on Item Aggregate, Journal of Marketing Reearch, 42(May), 169-182. Bijmolt, T. H. A., H. J. Van Heerde, and R. G. M. Pieter (2005), New Empirical Generalization on the Determinant of Price Elaticity, Journal of Marketing Reearch, 42(May), 141-156. Bucklin, R. E. and S. Gupta (1999), Commercial Ue of UPC Scanner Data: Indutry and Academic Perpective, Marketing Science, 18( 3), 247-273. Chriten, M., S. Gupta, J. C. Porter, R. S., and D. R. Wittink (1997), Uing Market-Level Data to Undertand Promotion Effect in a Nonlinear Model, Journal of Marketing Reearch, 34(Augut), 322-334. Deaton, A. and J. Muellbauer (1980), Economic and Conumer Behavior, Cambridge Univerity Pre. Erdem, T., S. Imai, and M. P. Keane (2003), Brand and Quanitity Choice Dynamic Under Price Uncertainty, Quantitative Marketing and Economic, 1(1), 5-64. Fader, P. S. and B. G. S. Hardie (1996), Modeling Conumer Choice among SKU, Journal of Marketing Reearch, 33(November), 442-452.
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