DECOMPOSING PURCHASE ELASTICITY WITH A DYNAMIC STRUCTURAL MODEL OF FLEXIBLE CONSUMPTION. Tat Chan. Chakravarthi Narasimhan.

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1 DECOMPOSING PURCHASE ELASTICITY WITH A DYNAMIC STRUCTURAL MODEL OF FLEXIBLE CONSUMPTION Tat Can Cakravarti Narasiman Qin Zang 1 August 26, Te autors are Assistant Professor of Marketing, Pilip L. Siteman Professor of Marketing at Wasington University in St. Louis, and Assistant Professor of Marketing at University of Texas at Dallas. Autors are listed in alpabetical order. Corresponding autor: Cakravarti Narasiman, narasiman@olin.wustl.edu, Tel:

2 ABSTRACT It is well known tat in package goods categories a temporary price cut of a brand leads to increase in te sales of tat brand in te current period. Under te assumption of stable consumption rate, previous literature as identified tat te sources for te increase in sales are brand switcing, purcase acceleration, and increase in quantity. However, tere are very few studies tat ave formally modeled te impact of price promotions on consumption wen consumption rate in a category is not constant. In tis paper we offer a metodology to decompose te effects of price promotions into brand switcing, stockpiling and cange in consumption and explicitly allow for consumer eterogeneity in brand preferences and consumption needs. A dynamic structural model of a ouseold tat decides wen, wat, ow muc to buy as well as ow muc to consume to maximize its expected utility over an infinite orizon is developed. By making certain simplifying assumptions we are able to reduce te dimensionality of te problem. We estimate te proposed model using a scanner panel data of 1000 ouseolds on canned tuna purcases for 12 product alternatives over a two-year period. Te results from te model sed insigts on te decomposition of te price elasticity into its components. Tis could elp managers make inferences about wic brands sales are most responsive to ouseold stockpiling and consumption expansion as well as understand ow temporary price cuts affect future sales. Contrary to previous literature, we find tat brand switcing is not te dominant force for te increase in sales. We also find tat a ouseold s brand preference as significant impact on its stockpiling and flexible consumption. More specifically, we sow tat brand loyals respond to a price promotion mainly wit stockpiling for future consumption wile brand switcers do not stockpile at all. We also find tat eavy users 2

3 stockpile more but teir consumption rate does not increase as muc as for te ligt users wen tere is a price promotion. Key words: flexible consumption, decomposition of price elasticity, consumer eterogeneity, dynamic structural model. 3

4 1. INTRODUCTION Modeling and understanding te demand for a product is at te core of applied marketing researc. In te applied literature, following Guadagni and Little's (1983) seminal work, researcers ave empasized te need for and advantages of using individual level data to model demand. Subsequently researc as evolved in te direction of breaking down te demand into its sub components of purcase incidence, brand coice and purcase quantity (Gupta (1988), Ciang (1991), Cintagunta (1993)), modeling observed and unobserved eterogeneity in te demand parameters (Cintagunta et. al. (1991), Gonul and Srinivasan (1993)), incorporating consideration sets in coice (Andrews and Srinivasan (1995), Ciang et. al. (1999)), and modeling state dependence on coice (Seetaraman et. al. (1998), Roy et. al. (1999)). In tese streams of researc te quantity modeled is te quantity purcased. In te case of packaged goods most of te witin ouseold variance in quantity bougt is due to price promotions. If ouseolds anticipate suc promotions to be temporary, te quantity bougt would be a complex function of its consumption, inventory on and, inventory carrying cost, current marketing mix, and its expectations about future. Terefore it seems natural to parcel out te effect of current and future marketing mix on consumption and quantity purcased due to te reasons we explore below. Moreover, weter and ow a ouseold responds to price promotions would also interact wit te ouseold brand preferences and consumption needs. A ouseold tat is particular about its brand or consumes more in te category is more likely to stockpile tan a similar ouseold tat is indifferent across many brands or consumes little. Our focus in tis paper is to develop a structural model of utility maximizing ouseolds, allowing observed and unobserved eterogeneity, tat simultaneously decides on consumption and purcase decisions taking into account not only te current but also future utility derived from consumption. 4

5 Underlying te economic model is te notion tat ouseolds maximize utility subject to a budget constraint. Te standard static model does not distinguis between consumption and purcase quantity since watever is purcased is assumed to be consumed by tat ouseold witin te period. But, for many frequently purcased packaged goods, te ouseold enters te market periodically and past literature as documented tat current prices or promotions affect not only wic brand and ow muc te ouseold purcases today but also te amount consumed (Ailawadi and Neslin (1998), Bell et. al. (2002), Sun (2003)). It is tis effect tat needs to be properly understood from a manufacturer or a retailer perspective and is te major focus of tis paper. For example, does a promotion lead to an increase in consumption, wic we define as an increase in consumption due to eiter te income or te substitution (away from oter categories) effect? Or does a promotion simply lead to ouseolds stockpiling te product for future use witout any increase in teir consumption? Te manufacturer of a product wants to increase te expected (current and discounted future) profits of is product wile te retailer wants to increase te expected profits from te category in is store. How promotions affect sales clearly ave different implications for te manufacturer and te retailer in terms of managing te pricing and promotion policies. An increase in consumption benefits bot manufacturers and retailers. A pure increase in stockpiling urts te long term profits of bot manufacturers and retailers. Manufacturers benefit from brand switcing from oter brands, wile weter retailers can benefit from brand switcing or not will depend on te profit margins of different brands tey stock. Researcers and managers interested in evaluating te competitive position of a brand relative to oters in a category ave relied on te concepts of clout and vulnerability (Kamakura and Russell (1989)). To compute tese measures we need estimates of own and cross price 5

6 elasticities. Since elasticity estimates for most packaged goods are based on temporary price promotions, our discussion above points to anoter reason wy understanding elasticities and teir decomposition due to flexible consumption are important. For example, a brand's clout and vulnerability indexes need to be reevaluated if te increase in elasticity is primarily troug stockpiling vs. troug increase in consumption. Anoter reason wy we believe tat tis problem is important is te implication on competitive strategies wen ouseolds are eterogeneous in teir response to price promotions. Marketing as a ric tradition of analytical models tat explain te existence of consumer and trade promotions tat occur in frequently purcased package goods (Jeuland and Narasiman (1985), Narasiman (1988), Lal (1990), Rao (1991), Lal et. al. (1996)). In tese models ouseolds are segmented into brand loyals and brand switcers (non-loyals) but teir purcasing beavior in terms of quantity purcased is assumed to be te same. However, if tis is not true, te difference in quantity purcased ten needs to be considered in te profit implications of competitive strategies. Te analyses from tis paper sould terefore be of interest to analytical modelers wo would want to enric teir models to provide deeper understanding of te strategic games among manufacturers and retailers. Te findings on ouseold eterogeneity respect to price promotion responses can furter elp managers to design better pricing and promotion strategies. Housolds brand preferences and consumption needs, inventory cost, and expectations about future determine weter tey will respond to price promotions wit switcing brands, stockpiling or increasing consumption as well as determine te relative magnitude of tese responses. A general concern tat managers ave wen tey design a pricing/promotion strategy will be making sure tat te strategy will discourage te group(s) tat will respond in a way tat does not lead to improvement in profits 6

7 from responding wile encourage te oter group(s) to do so. Managers can also determine weter a pricing/promotion strategy sould be used given te composition of ouseolds on te market. Our goal is to address tese issues and provide some specific guidelines. In summary, tere are tree issues tat we attempt to address in tis paper: 1. Wat is te composition of sales increase induced by price promotions? Furter, is te increase mainly due to brand switcing, stockpiling or due to an increase in consumption? Is te composition different across brands? 2. Is tere eterogeneity across ouseolds in teir response to price promotions? Specifically, do brand loyals stockpile more since tey want to save for future consumption wen te price of teir favorite brand could be ig? Do brand switcers increase consumption and don t stockpile since tey can always buy te least expensive brand. How do consumption needs affect ouseolds consumption flexibility and stockpiling beavior? 3. Given te results from te above two issues, wat are te managerial implications? Specifically, ow can manufacturers or retailers use te information to formulate teir pricing strategies? How can manufacturers or retailers price discriminate among different ouseold types to maximize teir long term profits? To address tese issues we develop a dynamic structural model of a ouseold tat maximizes discounted utility from its consumption in a category. Consumption in eac period is endogenous and is allowed to vary based on current price, inventory levels, and consumption preferences. In deciding to buy today, te ouseold trades off te inventory cost wit te opportunity cost of buying te product in te future at a iger price. We allow a ouseold s inventory cost and price responsiveness to be functions of its demograpics, and also allow 7

8 brand and consumption preferences to vary across ouseolds. We apply our model to scanner panel data on canned tuna category wit 12 product alternatives. By imposing simplifying beavioral assumptions te infinite planning orizon dynamic programming problem is reduced to a finite orizon problem. We propose a solution to overcome te dimensionality problem in model estimation. We run simulations (policy experiments) using te estimates from te proposed model as input. We are able to parcel out te effect of increase in sales due to a price promotion into its components of consumption increase, brand switcing occurring in bot current and future periods, and stockpiling. Our analysis reveals several interesting insigts: (1) contrary to wat is sown in most of relevant previous literature (Gupta (1988), Ciang (1991), Sun (2003) etc.) 2, brand switcing is not te dominant effect; (2) majority of promotion induced sales increase from larger sare brands is attributed to stockpiling; (3) te consumption effect is greater for smaller sare brands. We furter sow tat a ouseold s brand preference as significant impact on its stockpiling and flexible consumption beavior. More specifically, brand loyals mainly respond to a price promotion wit stockpiling wile brand switcers do not stockpile at all. We also find tat systematic differences exist between eavy and ligt users in te category on teir responses to a promotion. Heavy users stockpile more for future consumption and increase less consumption tan ligt users do. Tese findings provide managerial implications for pricing and promotion strategies of bot manufacturers and retailers. Te rest of te paper is organized as follows. In te next section we describe related researc and color our contribution wit respect to te literature. Following tis we describe our econometric model. Te section following tat describes te data we use to estimate te proposed model, discusses te estimation as well as te simulation results. Also in tat section 2 Our finding is consistent wit tat in Erdem et al. (2003), Van Heerde et al. (2003). 8

9 we provide specific suggestions to managers based on our findings. Finally, we conclude te paper wit some extensions and suggestions for future researc. 2. BACKGROUND In tis section we describe prior literature to position our paper and its contributions. We argued earlier tat it is important to model flexible consumption in order to understand te spike in sales more fully. Moreover, we argued tat tere may be systematic differences across consumers 3 based on brand preferences and teir consumption needs (brand loyals vs. switcers and ligt vs. eavy users) in teir responses to price promotions. Tere are tree but not necessarily separate streams of literature tat are relevant to wic we potentially contribute. Tese are (1) researc tat focuses on breaking down te purcase elasticity into its components, (2) researc tat focuses on modeling consumption beavior wit consumer future expectations, and (3) dynamic structural models accounting for flexible consumption. At te end we will also briefly discuss our contributions from a metodology perspective. 2.1 Breaking down total elasticity into its components Te sales increase of a brand under temporary price promotions is due to several factors: (1) consumers wo would normally buy some oter brand migt switc to buy te promoted brand (brand switcing); (2) consumers migt purcase more in current period for future consumption (stockpiling); consumers migt also consume more in current or future periods (consumption increase). Gupta (1988) was te earliest to decompose te promotional response (using promotional elasticity as te metric) into brand switcing, purcase (incidence) 3 In our paper, consumer and ouseold are used intercangeably. 9

10 acceleration and stockpiling effects using coffee data. He found tat te dominant force was brand switcing accounting for 84% of te cange in response wile purcase (incidence) acceleration accounted for 14% and stockpiling accounted for 2%. Similar results were reported by Ciang (1991). Bell et. al (1999) pursue tis furter using a muc broader dataset involving 13 categories and more tan 170 brands. Wile teir results in general confirm Gupta s findings tey also report substantial variances in te brand switcing effect (69% to 81%) across categories. Te above papers assume tat consumption rate is fixed. Sun (2003) relaxes tis assumption and allows consumers to decide on consumption rate eac period based on teir inventories and new purcases. Se finds tat te brand switcing effect of a promotion is overestimated wen consumption is fixed but it is still te dominant effect. Van Heerde et. al (2003) proposes a different decomposition measure based on sales units. Using te same dataset as Gupta (1988) did, tey found tat only 33% of unit sales increase is due to losses of oter brands. Taking off from tis stream, we breakdown te promotional effect into te brand switcing, stockpiling and consumption effects. We measure te impact of a price promotion not just in te promotion period but over a long term orizon. We find tat most sales increase due to a price promotion can be attributed to consumption increase and stockpiling rater tan brand switcing. 2.2 Modeling Consumers Consumption Beavior under Price Uncertainty Wen consumers perceive costs of purcasing in future different from purcasing today, tey are likely to cange teir beavior today. For example, based on teir expectation about future prices, consumers migt cange te amount tey buy today and te amount to stockpile (Golabi (1985), Meyer and Assuncao (1990), Krisna (1992, 1994)). Following te framework 10

11 of Golabi (1985), Assuncao and Meyer (1993) develop a normative model of a consumer wo endogenously cooses te optimal level of consumption by assuming a particular price process for te future. Tey model a consumer as maximizing an inter-temporal utility function and optimally coosing a consumption and purcase stream. Teir model predicts tat consumption is an increasing function of inventory and tis as subsequently found empirical support in te works of Wansink (1996), Ailawadi and Neslin (1998), Erdem et al (2003), and Sun (2003) etc. Bell et. al (2002) develop an analytical model tat tries to capture te effect of flexible consumption on te part of te consumers and explores te pricing strategies of identical retailers. Like tese papers we also model forward looking consumers and allow tem to rationally coose not only te quantity to be bougt but also teir consumption stream. We take tis stream furter by empirically exploring differences across customer groups (brand loyals vs. brand switcers and ligt vs. eavy users) in te impact of promotions on teir consumption and stockpiling beavior. 2.3 Dynamic Structural Models wit Flexible Consumption It as been demonstrated in te marketing literature tat a dynamic structural model as its advantages wen modeling a consumer s purcase, stockpiling and consumption beavior under price uncertainty in package goods categories (e.g. Gunul and Srinivasan (1996), Erdem et al (2003), Hendel and Nevo (2003), Sun et al (2003) and Sun (2003)). Tere is a stream of literature tat accounts for flexible consumption under price uncertainty using dynamic structural models. Erdem et al (2003) assume tat consumers ave an exogenous usage requirement eac period wic is revealed to consumers after teir purcases. Consumers are allowed to stock out but tere is a stock-out cost. In teir model tere are a fixed number of segments in terms of 11

12 usage rate and consumers can move from one segment to anoter following a Markov process. Hendel and Nevo (2003) and Sun (2003) relax te assumption of exogenous consumption rates and model consumption as a decision variable of consumers. In Hendel and Nevo (2003), te assumption tat utility of a brand is derived at te point of purcase enables tem to model consumers optimization of purcase quantity in a dynamic programming setting and ten separately model consumers brand coice decisions in a static framework. Tis results in significant computational simplification but also comes at a cost (see te discussions in Erdem et al (2003)). Moreover, consumers are assumed to coose package sizes rater tan units, limiting te applications of te model to markets were package sizes rater tan multiple purcases are considerations of consumers 4. Sun (2003) as sown tat a dynamic structural model wit endogenous consumption rate improves te goodness of fit to data as well as model prediction. Te focus of er paper is on te impact of promotion on consumption in general. Se confirms tat consumers consumption rate on canned tuna is not constant and tat sort term consumption increases wen tere is a price promotion. We follow tis stream of researc on endogenous consumption, but we propose a edonic approac to model consumer utility. Te focus of our study is te decomposition of price elasticity to its components of brand switcing, stockpiling and consumption increase over te planning orizon 5. Anoter distinct feature in our model is tat we also explicitly allow consumer eterogeneity in terms of brand preferences, consumption needs, and consumption flexibility. We are interested in ow tese affect teir responses to price promotions. For example, wic types of consumers (brand loyals vs. brand switcers, ligt users vs. eavy users) will mainly increase consumption or mainly stockpile wen facing temporary price promotions? 4 For example, it is not suitable to model te purcases in categories like canned tuna and yogurt. 5 As will become clear later on, we contrast our results to te sort term perspective were te decomposition is done for te period of promotion only. Our long term perspective is defined over te planning orizon of consumers. 12

13 From metodology perspective, we demonstrate tat te infinite orizon dynamic problem can be reduced into a finite one, elping reducing te computational burden. We impose furter beavioral assumptions 6 under wic te problem is furter simplified. Simulated Metod of Moments (SMM) is used to estimate our model. Altoug our model as its limitations due to te simplifying assumptions, it offers an alternative solution to overcome te problem of curse of dimensionality in dynamic optimization problem. Hence, it is useful to researcers wen tey are facing ig-dimension problems or dealing wit large datasets. 3. MODEL 3.1 THE HOUSEHOLD S PROBLEM Assume tat tere are J products or brands, and H ouseolds in te market. Let y t be a J 1 vector of ouseold s continuous consumption quantity, x t a J 1 vector of continuous quantity purcased, in week t, and u( ) be te utility function of consumption 7. In week t, ouseold as to decide wic product to purcase, ow muc to purcase, as well as ow muc to consume. Since quantity bougt at time t can be eld over and consumed in future periods, te infinite orizon planning problem can be stated as follows: sup [ ( ) s. t. I = x + I y J s t { xs, ys } Et γ u ys λ p s xs c Isj σ t s= t j= 1 s s s 1 s x, y, I 0 s s s (1) were p s is a J 1 vector of prices, and I s a J 1 vector of te inventory level of products in week s. We use λ to denote te marginal utility of income, c te inventory cost of one 6 We will discuss tese assumptions in detail in section 3. 7 Its functional form will be specified in detail in equation (2) in section

14 and Finally, standardized unit for ouseold, and γ te discount rate (assumed to be equal across ouseolds in a standard way). E t { σ t } is te expectation operator conditional on te information set at t, period, It 1 σ t. Te information set includes te inventory inerited from previous, past marketing variables suc as prices, features and displays, and ouseold demograpic variables. Te endogenous variables in (1) include { x, x,...; y, y,...} t t+ 1 t t DETAILS OF THE INDIRECT UTILITY FUNCTION We assume, for s t, a utility function of consumption as follows: α E u ( y ) = ( Ψ ' A' y + ϕ ) (2) t s s were, A is a J ( C + J ) caracteristic matrix. Te first C columns represent observable product attributes including brand names, flavor etc. Te last J columns of te matrix A is an identity matrix of dimension J. Ψ is a vector of preferences and tastes coefficients consisting of a C 1 vector of time-invariant ouseold s preferences for te observed attributes, ψ, and a J 1 vector of time-invariant ouseold specific taste preferences for products, ξ. 8 Hence, Ψ = ( ψ ξ) in (2). We assume tat ξ in (1) are i.i.d. continuously distributed over ouseolds. Specifically, we assume approac by assuming tat ξ normal σξ 2 (0, ). we use a random coefficient ψ = ψ + η (3) were η is a vector of random variables tat are continuously distributed. In te model 2 estimation, we assume η normal(0, ση ), tey are i.i.d. over ouseolds. 8 We note tat state-dependence like variety seeking beavior is not modeled ere. 14

15 Te parameters α and ϕ control te rate of diminising marginal returns and translation in te utility function 9, were α is restricted to be in te unit interval. To allow for eterogeneity across ouseolds in consumption, we assume tat α is a function of ouseold demograpics. In te estimation, we assume tat tere exist K types of ouseolds; for type k, α exp( α + α FMY ) k,0 k,1 = (4) exp( α α FMY ) k,0 k,1 were FMY represents te family size of ouseold, and α k,0 and α k,1 are parameters to be estimated. Houseolds wit larger α are more likely to purcase te product category and consume larger quantity compared to tose ouseolds wit smaller α. To make sure tat marginal utility does not go to infinity wen te consumption level is zero, ϕ is restricted to be positive. In tis case corner solutions are allowed. Following Kim et. al (2002), we fix ϕ = 1 in te equation because it is difficult to identify α and ϕ separately. To allow for eterogeneity in price sensitivity, we furter assume tat λ = 1 + λ ( INCOME AvgINCOME) + λ EMPLOY (5) 1 2 were INCOME is te income of ouseold, AvgINCOME is te average of ouseold income, and EMPLOY is te employment status of te female ouseold ead. To allow for eterogeneity in inventory costs, we assume tat c = c + c ( INCOME AvgINCOME) + c RESIDENCE (6) were oters. RESIDENCE is te residence type of te ouseold suc as single family ouse or 9 For more detailed explanation see Kim, Allenby, and Rossi (2002). 15

16 (7) We do not observe te initial inventory tat ouseold as at te time wen we observe its very first purcase. Intuitively, if ouseold makes a purcase wen te price in tat period is iger tan its regular price, it could indicate tat tis ouseold s inventory is low at te beginning of tat first period. In te estimation model we assume tat 0,0 = exp( υ ), and υ ( ρ (,1, j j ),1) I normal p p were I,0 is te initial inventory of, p,1, j is te observed price of product j in te period wen ouseold made te first purcase in data, and 0 p j is te perceived cost of purcasing of product j. 10 ρ is a parameter to be estimated. If ρ is negative, a iger first purcase price will indicate a lower initial inventory. 3.3 SOME SIMPLIFYING ASSUMPTIONS AND PROPOSED SOLUTION Using Bellman s equation, equation (1) can be re-written as 11 : J V ( σ t ) = sup { x, } [ ( ) ( ) ] ( ) (,, ) t y E t t u yt λp t xt c Itj σ t + γ V σ dp σ σ t xt yt j= 1 σ were P(, x, y) (8) is te Markov transition kernel for { σ } conditional on actions {, } t x y. Optimal controls { x, y } are solved from (8) under te constraints of inventory and nonnegativity. In practice, equation (8) is difficult, if not infeasible, to solve because of te curse of dimensionality. Tere are J 2 continuous actions (i.e., x, y ), ence, te dimension of controls are infinite. Moreover, we do not observe inventory or consumption rate. t t 10 We will discuss ow to derive te perceived cost of purcasing later in tis section. 11 For simplicity we omit te subscript ereafter. 16

17 We propose a solution to overcome te problems mentioned above. Suppose c > 0 for all, p tj > 0 for all t and j. Let p j be te igest future price tat product j will possibly carge. We assume p p < b < for all t and j. In Appendix A we sow tat tere exists a j tj finite time period, T, suc tat, ouseolds will not purcase at t and stockpile for te consumption of periods beyond t can rewrite te problem in (1) as a finite orizon problem, + T no matter ow muc inventory tey are olding. Tus, we sup [ ( ) s. t. I = x + I y t+ T J s t { xs, ys } Et γ u ys λ p s xs c Isj σ t s= t j= 1 s s s 1 s x, y, I 0 s s s (9) * * Te optimal solutions { x, y } for te week t in (9), wic are te focal decision variables of te t t problem, are equivalent to te optimal solutions we would obtain from solving te infinite orizon problem in (1). Tis implies tat empirical researcers may start from a reasonably large number of T and solve tis finite orizon dynamic programming problem. Tis substantially reduces te computational burden compared to te traditional metod of successive approximations in solving te Bellman equation in (8). 12 Altoug te problem is reduced to a finite orizon dynamic programming problem, solving for te optimal decisions in (9) is still not an easy task, particularly, since { x, y ; s = t, t + 1,...} are J-dimensional vectors. We impose two more assumptions to simplify s s te problem. We first adopt an assumption used by Erdem et. al (2003). l Assumption 1: 12 For example, see a detailed explanation of different solution metods in Rust (1994). 17

18 Houseolds use eac product in teir inventory proportionately. Tat is, given teir inventory after te purcase, It 1 + xt, ouseolds will consume a proportion t+ T δ s for eac product j, were δ s 0 and = 1, for eac future period s. δ s= t s Under te above assumption, ouseolds consumption from teir inventory at s is y s = δ s (I t-1 + x t ). Here δ s is a scalar. We ten only need to solve for a T 1 vector of δ s instead of a T J matrix of y s in (9). Second, we assume tat ouseolds use a simple yet, intuitive updating rule regarding wat tey perceive as te future cost of purcasing. l Assumption 2: In any given week t<s, ouseolds form a perceived cost of purcasing for buying brand j in week s, 0 pt s, j, using current and past prices. Tese prices are expected to remain constant for all weeks l>t, i.e., 0 0 pt l, j = pt, j. 13 Te above assumption implies tat, facing a trade-off between buying now and buying later for future consumption, ouseolds would compare te perceived cost of purcasing wit te current observed prices. Suc a decision rule is a simplification from te traditional optimization algoritm, were ouseolds consider all possible future price pats and maximize te expected discounted utility over tem. Since te focuses of our paper are on flexible consumption and stockpiling, tis simplifying assumption will elp us capture te main effects wile te implied decision rules are still consistent wit intuitive beavior. A flexible specification for te perceived cost of purcasing sould provide an approximation to te 13 Tis assumption still allows te flexibility of incorporating different updating rules for future prices. For example, in our empirical analysis we assume ouseolds update teir perceived future cost of purcasing using: p = p + ω ( p p ), were 0 p j is te regular price of product j, t, j p te observed price at time t, and ω a parameter to be estimated. t, j j j t, j 18

19 solution of a dynamic planning problem under price uncertainty. Note tat te perceived cost of purcasing sould not be confused wit te expected future prices. In particular, if ouseolds are risk-averse and prefer not to commit to consuming any product in te future by olding inventory, 0 p t can be lower tan te expected future prices in model estimation. Under te assumptions of stationary perceived cost of purcasing and positive inventory cost, it is not optimal for ouseolds to make purcases in a future period s and old inventory for te consumption in a furter future period u, were u wit Assumptions 1 and 2, equation (9) can be re-written as > s. Based on tis reasoning and along sup { E u( δ ( I + x )) λ p ' x c (1 δ ) ( I + x ) t t t 1 t t t t t 1, j t, j { xt,..., xt + T ; δt,..., δt+ T } j= 1 t+ T J 0 Etu δ s It 1 + xt + xs λ pt xs c δt δ s It 1, j + xt, j s= t+ 1 j= 1 + ( ( ) ) ' (1... ) ( ) s.t. x,..., x, δ,..., δ 0, δ = 1 t t+ T t t+ T t+ T s= t s J (10) Note tat x s in (10) only enters te expected utility function in period s and does not affect te expected inventory in any future period. Te dynamic programming problem in a finite orizon of T is ten vastly simplified. Next, let us examine ow a ouseold decides weter to buy, wic product to buy, ow muc to consume and ow muc to buy. To simplify te analysis, first suppose tat te ouseold does not old inventory in week t. Te utility function in (2) implies tat, after observing p t, a ouseold will coose at most one product j* suc tat, for all k = 1,, J, MU j* (0) α ψ ' Aj* α ψ ' Ak MUk (0) = = λ p λ p λ p λ p ψ ' A p t, j* t, j* t, k t, k j* ψ ' A p t, j* t, k k (11) 19

20 were MU (0) j* is te marginal utility level wit respect to j* at yk = 0, k = 1,..., J. Since we assume tat a ouseold s preference is stationary over time, j* will also be te product cosen if te ouseold purcases now and olds it for future consumption. Corner solution exists wen α ψ ' Ak max{ } < 1, for all k. In tis case we ave x t = 0. Tis occurs wen te ouseold finds { k} λ p t, k current prices too ig to purcase te product category at week t Wit te perceived cost of purcasing equal to p 0 t and te utility function as specified in (2), te ouseold expects to coose at most one product j 0 suc tat, for all k, MU (0) α ψ ' A α ψ ' A MU (0) λ p λ p λ p λ p ψ ' A p 0 0 j j k k = = t, j t, j t, k t, k ψ ' A 0 j k 0 0 t, j pt, k (12) α ψ ' Ak Again, wen max{ } < 1, for all k, te expected purcase at te perceived cost of { k} 0 λ p t, k purcasing will be zero. Tis occurs wen te ouseold does not normally purcase or consume te category, and will only purcase wen tere is a big price promotion. As implied by (10), te ouseold will purcase in week t and old it for consumption in week s, were s>t, only if te following two conditions are satisfied: s t (i) j* t, j* s t u γ MU (0) λ p c γ 0 u= 0 and s t u s t 0 (ii) γ MU (0) λ p c γ γ [ MU 0 (0) λ p 0 ] s t j* t, j* j t, j u= 0 20

21 Condition (i) ensures tat discounted consumption utility in week s net of purcasing costs in week t and discounted inventory cost is non-negative. Condition (ii) ensures tat it is wortwile to buy now and old inventory until week s. 14 Suppose te above two conditions are satisfied, i.e., te ouseold purcases in week t and olds it for consumption in week s. Te optimal purcase quantity ten satisfies te tird condition tat is derived from te first-order condition: s t (iii) t, s, j* t, j* s t u γ MU ( y ) λ p c γ = 0 u= 0 were y t, s, j* is te optimal quantity purcased in week t for te consumption in week s. Te optimal level of x t in (10) is equal to te sum of y t, s, j*, s=t,,t, in te j*-t row and zero elsewere. Tus, te optimal level of * δ s is y t, s, j* x t, j*. Note tat our model sould not be treated as a substitute for te dynamic models wit rational expectations because of te simplifying beavioral assumptions we make, but it offers an alternative solution to capture te main effects wile overcoming te problem of te curse of dimensionality in dynamic optimization problem. Wen te ouseold olds positive inventory in week t, solution y t, s, j* for all s cannot be solved separately as in conditions (i) to (iii), since te ouseold as to additionally consider te 14 Tis decision rule is similar to tose in reference price literature (Winer (1986), Mayew and Winer (1992), Mazumdar and Papatla (2000), etc.). But in our setting ouseolds are comparing current prices wit te perceived cost of purcasing in te s t s t 1 s t 1 0 future. Indeed we may treat min{ { MU *(0) c u s t }, { [ MU u s t j j* (0) MU 0 (0)] c γ γ γ γ + γ λ p 0 }} j t, j λ λ u = 0 u = 0 as an intertemporal reference price of buying j* at week t for consumption at future period s. In tis case te inter-temporal reference price is a function of brand and consumption preferences, inventory cost, and perceived cost of purcasing. 21

22 benefit and cost of consuming te inventory vs. buying in current week. However, basic principles of te solution concept discussed above still apply. In our estimation algoritm, we directly solve te non-linear constrained optimization problem in (10), given parameters θ and inventory level It 1. Our estimator is a non-linear least square estimator using te Metod of Moments. Details of te estimation algoritm will be provided in Appendix B IDENTIFICATION OF DIFFERENT TYPES OF CONSUMPTION BEHAVIOR Depending on (1) preferences for te attributes of different products, (2) inventory cost, and (3) flexibility in consumption as determined by α, a price promotion will ave different effect on different ouseolds. One major identification issue for model estimation is tat we, as researcers, do not observe consumption and inventory of ouseolds in te data. We only observe weter a ouseold makes a purcase and if so wic product it buys and te quantity it purcases in eac period. Still, we can identify te parameters of a ouseold s utility function from te variations in its purcasing pattern over time: Brand switcing patterns of ouseolds over time elp identify te differences in product preferences of ouseolds, and variations in purcase quantity and time-intervals between purcases elp identify te inventory cost and consumption rate canges. For example, suppose tat tere are two ouseolds, A and B, wo buy one unit of te product in eac period at te same price. Suppose te price is cut by 10 percent in te current period, and bot A and B increase teir purcase from 1 to 2 units. If ouseold A comes back to purcase 1 unit again in te next period, but ouseold B does not make a purcase until period 3, we can infer tat A increases its consumption and does not stockpile in te current period, wile ouseold B does te opposite. In tis case ouseold A as 15 In tis application we fix te number of finite periods to T=12, and te number of simulation draws to just one. Since we ave a large number of observations, estimator efficiency is not our major concern. 22

23 a flexible consumption rate but a iger inventory cost tan ouseold B. Suppose tere is anoter ouseold, C, wic buys 4 units of te product during promotion, and only comes back to market in period 3. Ten we infer tat ouseold C may ave a more flexible consumption rate tan ouseold B but a lower inventory cost tan ouseold A. Hence, te parameters are identified if tere are enoug variations in responses to prices, even toug we just observe te quantity purcased. Given tese parameter values, effects of stockpiling, brand-switcing and consumption increase due to temporary price promotions can also be identified. For example, wen tere is a price promotion ouseold A will sow a larger consumption effect but a smaller stockpiling effect tan ouseold B does, wile ouseold C will sow a larger consumption effect as well as a stockpiling effect. 4. EMPIRICAL ANALYSIS 4.1 DESCRIPTION OF DATA We estimate te proposed model using te A. C. Nielsen scanner panel data on canned tuna. Te reason we coose tis category is tat canned tuna is easily storable and potentially a good candidate for stockpiling and flexible consumption. Te dataset on canned tuna contains weekly data on prices, feature advertising, and displays from 19 stores in Sioux Falls, Sout Dakota for a period of 123 weeks from January 1985 to May It also provides purcase istory information of 4308 ouseolds. Tere are totally 10 package sizes in te data, among wic te one size, 6.5 oz., as te largest sare, accounting for 94.2% of te total quantity sold, and 93.7% of te total purcase occasions. Also, among te 4308 ouseolds, 3250 or 75.4% of total ouseolds, purcases canned tuna of tis size only. Given te obvious dominance of tis package size, in order to avoid dealing wit te issue of 23

24 quantity discount wit te existence of multiple package sizes, we focus our analysis on te 6.5 oz size and purcases from tese 3250 ouseolds. Tere are total 33 SKUs of 6.5 oz size in te data. We use product attributes to group te SKUs. We identify tree main product attributes: brand, water/oil based, and ligt/regular. Tere are total 10 brands among wic four brands, Star-Kist, Cicken of te Sea (CKN), 3 Diamond, and CTL (store controlled brand) account for 99.5% of all purcases. Te grouping of SKUs by product attributes generates 12 product alternatives (brands*attributes), of wic 11 product alternatives represent te SKUs tat belongs to one of te four brands, water or oil, and ligt or regular. We combine te rest of te SKUs tat belong to a brand oter tan te four brands into te 12t product alternative 16. Table 1 provides te summary statistics for tese 12 products. To keep te size of te dataset manageable, we randomly cose 1000 ouseolds from te 3250 ouseolds. Tese ouseolds made purcases during te sample periods. Te average number of units per purcase occasion was 2.15 units and te average inter-purcase time was 9.84 weeks. Figure 1 gives te istogram of inter-purcase times of te 1000 ouseolds over te 123 weeks. Te price, feature, and display of eac product assumed to be observed by a ouseold in eac week is constructed as follows: l For a product tat a ouseold purcases in a week, price, feature and display are constructed as te weigted average over te SKUs tat belong to te product alternative. Te weigt used is te quantity of purcased; 16 In eac purcase occasion, a ouseold purcases one of te 12 product alternatives, so tere is no underestimate of ouseold inventory. Hencefort we will use te term product to refer to one of tese 12 alternatives. 24

25 l For a product tat a ouseold does not purcase in a week, te price, feature and display are constructed as te numerical average over all te SKUs tat belong to te product alternative in te ouseold's preferred store. Te dataset also contains te demograpic caracteristics of te ouseolds suc as family size, income, te employment status of te female ead of te ouseold, and type of residency, etc. We incorporate tese variables in te estimation of our proposed model. [Insert Table 1 and Figure 1 ere] 4.2 EMPIRICAL RESULTS We estimate two model specifications: In model A, te perceived costs of purcasing are assumed to be te regular prices 17 in te data. In model B, a ouseold updates its perceived cost of purcasing after observing current price. Specifically, we assume tat: p = p + ω ( p p ) t, j j j t, j were 0 p j is te regular price of product j, t, j p te observed price at time t, and ω a parameter to be estimated. If ω is negative, it suggests tat a ouseold lowers its perceived cost of purcasing product j because of te price promotion at time t. Since model B yields a lower value of te objective function tan model A, we report te results for model B in Table 2. From te table, we can see tat te estimate of mean preference for all ouseolds for te category of canned tuna is However, te preference is eterogeneous among ouseolds since te variance of te normally distributed preference parameter is 0.50, wic is statically significant. Te most preferred brand is Star-Kist wile te least preferred brand is 3 Diamond. Furter, ouseolds preferences for Star-Kist and CKN are 17 For eac ouseold, te regular price of eac product is te average price over te 123 weeks. 25

26 eterogeneous (i.e., σ, σ are statistically significant) wile tose for 3 Diamond and CTL are not. Te estimates for ouseolds mean preferences for water based and ligt tuna are positive (1.83 and 0.33 respectively), implying tat ouseolds like water-based more tan oil-based tuna, and like ligt more tan regular tuna. Consistent wit our expectations, te ouseolds also ave positive responses to feature advertising and store display (1.53 and 2.47 respectively). Te ouseold specific taste preferences for products are assumed to be i.i.d normal distribution wit zero mean, over ouseolds and product alternatives. Te estimation sows tat tis preference is eterogeneous among ouseolds (i.e., 2 σ is 1.89 and statistically significant). Te coefficient of price is normalized to one for te model parameters to be identified. Te coefficients of income and employment status of te female ead of te ouseold are not statistically significant. Te estimate for inventory cost of canned tuna is $0.03 per can per week. It is witin a reasonable range and consistent wit our expectation. Te income and te residence type a ouseold as in do not seem to ave significant impact on its inventory cost. One explanation for tis could be tat because canned tuna is relatively easy to store ouseolds do not run into space constraints caused by income or residency types. exp( α0 + τ FMYSize) Te consumption level in our model is determined by α =. Our 1 + exp( α + τ FMYSize) estimation indicates tat tere are two segments wit respect to α 0 andτ. Te first segment 0 accounts for 76.43% of all te ouseolds. Interestingly, te negative sign for τ implies tat smaller families ave a larger consumption rate for tis category. Te estimate of ρ is negative (-2.22), implying tat if a ouseold buys at a iger price during its first observed purcase, it indicates tat te ouseold as a lower inventory level. Tis is consistent wit our intuition. 26

27 4.3 SIMULATION AND DISCUSSIONS In order to answer te questions tat we raised earlier regarding te effects of price promotions on brand switcing, stockpiling and consumption, we conduct simulations using te parameter estimates from our dynamic structural model. One advantage of structural models is tat te underlying beavioral process of a rational ouseold is explicitly specified, tus te estimated parameters are invariant to marketing activities and are suitable for policy experiments. In te simulations, we used te 1000 ouseolds observed in data, and for eac ouseold we take 10 random draws out of te distribution. Terefore, we ave 10,000 observations. We first simulate te purcases and consumption of tese ouseolds wen tey face te regular prices 18 for all products for 12 continuous weeks. Ten, we simulate te purcases and consumption for te same ouseolds for te 12 weeks wen all te product alternatives tat belong to a brand (te focal brand) offer an equal amount of price discount in te first week. Te comparisons of te sales before and after te price cut allow us to decompose te effects of price promotions into tose of consumption increase, brand switcing and stockpiling. We define te total effects of te price cut as te sales increases in week one. Tat is, b = b1 b b1 = 1 TE x ( p ) x ( p ) were 0 p is te price vector wen all brands carge teir regular prices, and 1 p b is te price vector wen brand b as a price cut wile oter brands remain teir regular prices, and x ( ) 1 te quantity of brand b tat ouseold purcases in week one under te corresponding price vector. Te consumption effect is defined as te difference in te total sales in all 12 weeks before and after te price cut. Tat is: b 18 We use average prices of eac product alternative from te data. 27

28 b = bt ( b) bt ( ) t= 1 = 1 CE x p x p Note tat increase in consumption comes from tose ouseolds tat purcase only on promotions and ouseolds tat buy during non-promoted weeks as well. Since we assume tat te planning orizon of ouseolds are 12 weeks, i.e., ouseolds will not purcase some quantity in week one and old tem as inventory for future consumption beyond te 12 t week, given tat prices in all weeks after week one are te same regular prices 19, it is valid to use te difference in total purcases for all 12 weeks as te measure for canges in total consumption 20. Te brand switcing effect is measured by te decrease in total sales of oter brands (wose prices remain te same) in all 12 weeks as te focal brand cuts its price in week one. We furter decompose te brand switcing effect into two components: brand-switcing tat occurs in te first week ( BS b1 ) and in later weeks ( BS = x ( p ) x ( p ) 0 1 b1 1 1 b H b1 H b1 BS bl = t t b t= 2 H bt t= 2 H bt BS x ( p ) x ( p ) bl ). Tat is: were H are tose ouseolds wo purcase brands oter tan b in week t. Notice tat since bt tese ouseolds are paying te same prices for te oter brands before and after te price cut of brand b, tere is no consumption or stockpiling effect in x p ; tus te differences must be 1 t ( b) pure brand switcing effects. Te residual is ten te stockpiling effect. Tat is: SP = TE CE BS BS b b b b1 bl 19 So from week 2 to 11, ouseolds will purcase tuna only for consumption of tat week witout stockpiling for future weeks. 20 We can also measure te consumption effect by comparing te total quantity consumed during te 12 weeks before and after te price cut. However, it is simpler to use te metod we mentioned ere, and te results are equivalent. 28

29 tis is derived from te fact tat total sales increase of a brand due to a reduction in price in week 1 as to come from increase in consumption of tuna, ouseolds brand switcing from oter brands, and ouseolds stockpiling for future consumption Decomposition of Price Elasticities We decompose demand elasticities 21 into te above components. We first examine te case wit a price cut of 10 cents. Te decomposition for all 1,000 ouseolds is given in Table 3. [Insert Table 3 ere] First, we notice tat te brand switcing effect for all brands is muc smaller tan wat is identified in most previous relevant literature (e.g., Gupta (1988), Ciang (1991)), and is not te dominant force. Te discrepancy is due to two differences between our model and tose of Gupta (1988) and Ciang (1991): 1. In our model, a ouseold is assumed to maximize its consumption utility over a planning orizon (e.g., in our estimation of canned tuna category, we let it be 12 weeks) wile in te models of Gupta (1988) and Ciang (1991), a ouseold is assumed to maximize its purcase utility only in current period. 2. In our model, consumption is a decision variable for te ouseolds wile te consumption rate is fixed in Gupta (1988) and Ciang (1991). Moreover, te brand switcing effect in our model is computed as te lost sales of oter brands in te current period as well as in later periods wile in Ciang (1991) and Gupta (1988) it is calculated as te increased coice probability in current period for coosing focal brand 21 In eac table, we report te decomposition of elasticities bot in terms of numerical values as well as in percentages relative to te total elasticities. Te latter is te number in parenteses. 29

30 (conditional on purcase incidence 22 ). Since te consumption is restricted to be fixed, te brand switcing effect in teir models in fact includes te increased quantity purcased by te ouseold for consuming more or and stockpiling for future consumption, wile tose two effects are separately identified in our model. Terefore, te brand switcing effect in teir models is inflated. 23 Second, we notice te different responses exibited between larger sare brands (i.e., Star- Kist and CKN) and smaller sare brands (i.e., 3 Diamond and CTL): 1. Smaller sare brands ave iger price elasticity tan larger sare brands. Tis is consistent wit te findings in Cintagunta (1993). 2. Majority of sales increase from te two large brands is due to stockpiling effects (62% and 67% respectively) wile te stockpiling effects for smaller sare brands are smaller (37% for 3 Diamond and 39% for CTL). 3. Brand switcing effect is relatively small for larger sare brands (8% for bot brands) but substantially iger for smaller sare brands (26% for 3 Diamond and 25% for CKN) Consumption effect is significant for all brands. But tis effect is iger for smaller sare brands tan for larger sare brands. Te comparisons imply tat, from a manufacturer s perspective, te strategy of temporarily cutting prices to steal sales from oter brands migt not be very effective for large sare brands. 22 Tis is te condition in te model of Ciang (1991). Gupta (1988) assumes tree decisions to be independent. 23 Notice tat in Sun (2003) a ouseold is also assumed to decide on consumption eac period to maximize total consumption utility over a long period of time, but se still finds te brand switcing effect is te dominant force. We suspect te discrepancy is due to te fact tat in er model te price elasticity is calculated base on te sales increase (caused by te price cut) in te current period (i.e., te period wen a price cut occurs) only, wile in our model te price elasticity is calculated based on te sales increase occurring during te planning orizon (i.e., 12 weeks) 24 Since te calculation of price elasticity is based on te sales at regular prices (for smaller sare brands, tey ave muc fewer sales tan tose of larger sare brands), one sould be cautious to draw te conclusion tat smaller sare brands can draw more sales by price promotions tan larger sare brands by simply comparing te brand switcing elasticity (or te corresponding percentage measure). 30