An Empirical model of Demand with (Super)Market. Selection. 1 Introduction. Alessandro Iaria

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1 An Empirical model of Demand with (Super)Market Selection Alessandro Iaria 27/9/2012: Preliminary and incomplete, please do not cite nor circulate. 1 Introduction 1.1 Motivation A casual visit to any supermarket shows us that, in each product category (e.g., cheese, ready-to-eat cereals, coee, etc.) many similar (but not identical) products are oered for sale. An additional trip to a dierent supermarket shows us that, in each product category across supermarkets: 1. Identical products are oered on sale at dierent prices (i.e., price competition). 2. The range of products oered on sale are dierent (i.e., variety competition). Most of the recent Empirical IO literature focused on (1): the issue of price competition. Point (2) is not at all new in economics, for instance Spence in 1976 wrote: A signicant fraction of the [social] cost of imperfect competition may be due to the currently unmeasured [social] cost of having too many, too few, or Special thanks for the constant support and suggestions to my advisors: Valentina Corradi, Andrés Carvajal, and Gregory Crawford. I am indebted for the insightful discussions and useful comments to Dan Ackerberg, Rachel Grith, Michelle Sovinski and the participants of the Pizza Workshop (Warwick) and NIE Colloquium (Nottingham). All errors are mine. Department of Economics, University of Warwick. a.iaria@warwick.ac.uk 1

2 1.1 Motivation 1 INTRODUCTION the wrong products. Notwithstanding important exceptions (see section (1.2)), the challenge raised by Spence went somehow unnoticed in the Empirical IO literature. Because of its self-evident diusion in everyday life and its potential economic relevance (from Spence (1976)), in the current paper I develop a discrete choice model of demand that explicitly deals with point (2) above. Such a demand model is necessary to address the empirical question of measuring the share of social cost of imperfect competition due to variety issues (i.e., choice set heterogeneity). In other words, I wish to empirically test Spence's Hypothesis. The majority of the existing empirical papers which focus on the variety or choice set heterogeneity aspect adopts a reduced form approach for the demand-side of the market. Dierent consumers are allowed to face dierent ranges of products during dierent choice situations, but usually dierent choice sets are matched to dierent people for reasons exogenous with respect to the decision process itself. I propose a structural demand model in which the selection of the market (and thus the range of products and their prices) where the choice situation will actually take place is endogenously determined by the agent. To be more concrete, the typical choice situation I have in mind is the following. An individual wishes to purchase, say, a Nivea spray deodorant at the lowest possible price. In the person's neighborhood there are three dierent supermarkets. Before actually being in any of the supermarkets, the individual cannot know for sure which of the three supermarkets sells right now her favorite deodorant (stock-outs happen), and even if all of them have it, where she can nd the lowest price (there could always be unexpected promotions in some stores). Basing her decision on her own beliefs about the availability of the Nivea spray deodorant and its price in each of the three supermarkets, the individual will choose a store. Once there, the uncertainty about the availability of the product and its price will resolve and the person will make her deodorant choice. From each visit to any supermarket the individual learns something about the set of products on sale there 2

3 1.2 Dierences from Existing Literature 1 INTRODUCTION and their prices; this additional information contributes to the evolution of the beliefs of the person and will aect her future store choices. The supermarket endogeneity problem can be readily framed as follows. The probability with which individual i chooses alternative k in store j is: Pr [product k, store j θ i ] = j Pr [product k store j, θ 1i ] Pr [store j θ 2i ], where θ i = [θ 1i, θ 2i ] is a group of parameters. θ 1i is the vector of preference parameters and θ 2i is the vector of parameters governing the supermarket matching process. In demand estimation, the researcher's objective is usually θ 1i. The joint probability of choosing product k from store j can be broken down into two parts. The rst is the familiar conditional model of discrete choice, conditional on a given supermarket j. The second part represents the supermarket matching process. If the structural model above is accepted, then an analysis conducted only on Pr [product k shop j, θ 1i ], in the hope of making inference about θ 1i, will be valid only if store j is matched to individual i exogenously with respect to θ 1i. Only if θ 1i and θ 2i are functionally independent, then store j will be exogenously matched with respect to θ 1i. In other words, only if θ 1i and θ 2i have nothing to do one with the other, then knowledge of Pr [shop j θ 2i ] is not required for conducting valid inference on θ 1i. Notice that even if only one of the elements of θ is common to both θ 1i and θ 2i, then endogeneity arises. Thus, whenever this further exogeneity condition does not seem to hold, we must say something about Pr [shop j θ 2i ] even if θ 2i is not our direct interest, just to be able to successfully recover θ 1i. 1.2 Dierences from Existing Literature The rst economic paper which addressed the topic of variety or choice set heterogeneity in demand estimation was Sovinski (2008). The market under investigation is that of Personal Computers (PC). Choice set heterogeneity aects the demand-side of the model exogenously with respect to people's preferences for PCs. On the other hand, the range of products a person faces depends on people's preferences for media (TV, newspapers, 3

4 1.2 Dierences from Existing Literature 1 INTRODUCTION magazines, etc.). Also, choice set heterogeneity is endogenously determined by the supplyside of the market through rms' decisions of where (i.e., which media) to advertise their PCs. Other two articles which are similar in spirit (for my purposes) to Sovinski (2008) are Draganska, Mazzeo & Seim (2009) and Conlon & Mortimer (2010). Indeed, in the rst of the two papers, the demand-side of the ice-cream market is aected by exogenous choice set heterogeneity, but the supply-side is fully structural in determining simultaneously prices and ranges of products. In Conlon & Mortimer (2010), the demand-side of the vending-machine market is aected by exogenous stock-outs which, when they happen, restrict the sets of products sold by the vending-machine. Koulayev (2010) studies the online demand for hotels. This was the rst paper in the economic literature to allow for choice set heterogeneity to be endogenous with respect to preferences. Endogeneity is obtained through a search model. Endogeneity adds a lot of complexity to the demand model which, furthermore, requires very specic internet data (what the customers see on the screen during each step of their hotel search) to be estimated. As a consequence, Koulayev's method although very elegant, does not seem to be widely applicable. A similar contribution (again, for my purposes) came from the marketing literature: Mehta, Rajiv & Srinivasan (2003). They develop a model of demand for ketchup with in-store endogenous choice set heterogeneity. Analogously to Koulayev (2010), they obtain endogeneity through a search model, but the choice set heterogeneity they allow for is only valid within each store. Basically, aligned to the marketing tradition of consideration set formation, the authors try to model the costly search for products people undertake once they are in a specic store. Jacobi & Sovinski (2012) investigate the demand for cannabis. The authors allow for choice set heterogeneity to be endogenous to preferences but in a reduced form way. Endogeneity is obtained through two channels: rst, through a group of regressors contained simultaneously in both the choice and the selection equation; second, through correlation amongst the error terms of the model. A second strand of empirical literature on demand, relevant to my purposes, is that 4

5 2 THE MODEL on bayesian learning started in marketing by Erdem & Keane (1996). In the Empirical IO literature, the classic references on learning are Ackerberg (2003) and Crawford & Shum (2005). See Ching, Erdem & Keane (2011) for a comprehensive survey of the empirical literature on demand estimation with learning. The main idea is that consumers are not perfectly informed about some features of the products they consider for purchase (e.g., price, quality, etc.). Through advertisement, experience and word-of-mouth (for example) they learn, period after period, something about the imperfectly observed features. The more people learn (i.e., the less their information is imperfect), the better informed their future choices will be. The current article contributes to the existing empirical literature in two ways. First, I propose a demand model which allows for choice set heterogeneity to be endogenous with regard to preferences. The endogeneity is fully structural (i.e., no reduced form) and obtained through imperfect information and bayesian learning. Importantly, the resulting model has two practical advantages with respect to the available alternatives. It is signicantly simpler when it comes to estimation and it does not require any specic dataset beyond the widely used scanner panel data. Second, to the best of my knowledge, my model is the rst to allow for imperfect information and learning about the set of available products in dierent stores. 2 The Model 2.1 Primitives The indirect utility individual i = 1,..., I obtains from purchasing product k = 1,..., K in store j = 1,..., J is: 5

6 2.1 Primitives 2 THE MODEL U ijk (p jk ) = β ik α i ln (p jk ) + g (γ i, s j ) + ν ijk = Ṽijk (β ik, α i, γ i ) + ν ijk, (1) U ij0 = g (γ i, s j ) + ν ij0 where β ik = f (ρ i, x ik ) is some function of individual-specic parameters (ρ i ), demographics (x i ), and product-specic characteristics (x k ). g (γ i, s j ) is a function of individualspecic parameters (γ i ) and store-specic characteristics (s j ). In what follows, the observables X i = (x i1,..., x ik ) and S = (s 1,..., s J ) are assumed to be exogenously given and perfectly known by each decision maker i. p jk is the price of product k in store j. α i is the price sensitivity of individual i. ν ijk is an independent stochastic term distributed Gumbel [0, 1] which is known to individual i when making her choice but unobservable to the econometrician. U ij0 Gumbel [g (γ i, s j ), 1] is the indirect utility associated with the alternative of going to store j and purchasing products dierent from k = 1,..., K. r jk {0, 1} is a dummy variable which equals 1 if product k is available in store j. Alternative 0 is available in every store j (r j0 = 1, j). Dierently, product k = 1,..., K can be missing from store j. In that case r jk = 0 and individual i cannot purchase it. De- ne the normalized level of systematic utility as V ijk (β ik, α i ) Ṽijk (β ik, α i, γ i ) g (γ i, s j ). 6

7 2.1 Primitives 2 THE MODEL The key points of the proposed model can be summarized as follows. Any store j is allowed to supply any set of products identied by r j = (r j1,..., r jk ) and, for any product k which is supplied (r jk = 1), j can set any strictly positive price p jk. 1 Dene p j (r j ) as the vector of prices charged by store j for the set of products it sells, the ones in r j. Individuals, in order to be able to purchase any product k, are required to visit a store j which has product k on sale. The decision of which store j to visit for shopping is based on what individuals think the oer ( r j, p j (r j ) ) will be in each j = 1,..., J. Given these beliefs about product availabilities and prices, each individual will then choose to visit the store whose expected indirect utility is highest. Once the individual is in store j, she will see what the actual oer ( r j, p j (r j ) ) is and she will use this newly acquired information in two ways. First, the individual will make her utility maximizing product choice over the set of available products (remember that, if the individual is extremely disappointed by the actual oer, she can always decide not to purchase any product k = 1,..., K). 1 The current model does not encompass the supply-side of the market, which is assumed to be exogenous. Completing the proposed demand model with a supply-side where supermarkets choose their assortments and prices necessary to address Spence's Hypothesis is what I am working on at the moment. 7

8 2.2 Product Probability given Store Visit 2 THE MODEL Second, she will update her beliefs with respect to ( r j, p j (r j ) ). In section 2.2, I will briey describe the standard Pr [product k shop j, θ 1i ]. In section 2.3, I will derive from (1) the expected indirect utility of shop j given individual i's beliefs about availabilities and prices, and consequently Pr [shop j θ 2i ]. In section 2.4, I will outline a system of beliefs and bayesian learning consistent with the derivations of section Product Probability given Store Visit Given (1) and conditional on being in a specic store j, the demand model for any product k is a standard Mixed Multinomial Logit: 2 Pr [product k shop j, θ 1i ] [ = Pr k = arg ] max {U ijs (p js )} { s r js =1} = exp (V ijk (β ik, α i )) r jk 1 + K s=1 exp (V ijs (β is, α i )) r js, (2) = exp (f (ρ i, x ik ) α i ln (p jk )) r jk 1 + K s=1 exp (f (ρ i, x is ) α i ln (p js )) r js where the vector of preference parameters θ 1i = (ρ i, α i ) is the main object of our inference. Notice that because of additive separability, conditional on being inside store j, the part of the systematic utility dependent on the store-specic characteristics, g (γ i, s j ), does not inuence the product choice (i.e., cannot be identied). 2.3 Probability of Store Visit From (1) it follows that, given ( r j, p j (r j ) ), individual i's indirect utility associated to visiting store j is: 2 Rember that, once inside store j, individual i perfectly observes the oer ( r j, p j (r j ) ). 8

9 2.3 Probability of Store Visit 2 THE MODEL U ij ( rj, p j (r j ) ) = max { k r jk =1} {U ijk (p jk )} [ K ) ] = ln k=0 (Ṽijk exp (β ik, α i, γ i ) r jk + ε ij, (3) where ε ij is an independent stochastic term distributed Gumbel [0, 1]. 3 The indirect utility (3) is conditional on (β ik, α i, γ i, s j ): the known product/store characteristics and individual preferences; and on ( r j, p j (r j ) ) : the unknown availability of products and their prices in store j. Individual i evaluates (3) guessing a value for ( r j, p j (r j ) ), her belief. In order to compute Pr [shop j θ 2i ], I need to derive the expectation of (3) with respect to ( rj, p j (r j ) ). Let me start from p j (r j ). Fenton (1960) showed that if we have k = 1,..., K independent normal random variables, y k Normal [µ k, σ k ], then z = ln [ K ] k=1 exp (y k) is approximately distributed Normal [µ z, σ z ], where: σ 2 z K k=1 = ln exp (2 µ k + σk 2) [exp (σ2 k ) 1] ) K k=1 (µ exp k + σ2 k 2. (4) [ ( )] K µ z = ln k=1 exp µ k + σ2 k σ2 z 2 2 Fenton's approximation relies on the assumption of independence across the y k 's. Further research by Safak (1993), Ho (1995), Pirinen (2003) and Wu et al. (2005) extended the result to the more complex case of correlated y k 's. Even though, for simplicity, I will not deal with the correlated case in the current exposition, it is possible to allow for it. Assume the following: (A1) Conditional on (β ik, α i, γ i, s j ), Ṽijk (β ik, α i, γ i ) Normal [ µ v ijk, σv ijk] and in- 3 The second equality follows from the properties of the Gumbel distribution (see, for instance, pages of Ben-Akiva & Lerman (1985)). 9

10 2.3 Probability of Store Visit 2 THE MODEL dependent j, k. Assumption (A1) says that person i's belief with respect to the price of any product in any shop, p jk, is a draw from an individual-specic lognormal distribution. This is so because prices enter the indirect utilities in logarithmic form, ln (p jk ), j, k. As I said earlier, the independence assumption can be relaxed, even though I will not for the time being. In section 2.4 I will outline a more primitive set of assumptions which induces (A1). From (3), (A1) and Felton's approximation, we obtain: E pj (r j,ε ij ) [ ( Uij rj, p j (r j ) )] [ ( ) = E pj r j ln r j exp (Ṽ ij (β i, α i, γ i ) + + exp (g (γ i, s j )))] + ε ij, (5) = µ z ij (r j ) + ε ij where β i = (β i1,..., β ik ), Ṽ ij (β i, α i, γ i ) = (Ṽij1 (β i1, α i, γ i ),..., ṼijK (β ik, α i, γ i )) and µ z ij (r j ) is derived from (4). Notice that the set of available products r j determines the number of summands in (4). For any given system of beliefs on prices, dierent r j 's will imply dierent µ z ij (r j )'s. Thus, the value of the expectation in (5) is conditional on the specic set of products individual i thinks will be available in store j. A possible alternative to (A1) is to assume that people's beliefs with respect to prices correspond to the empirical distribution of those prices. The empirical distribution would then be a discrete distribution and the expectation in (5) would be computed as a weighted sum, where the weights would be the relative frequencies associated to each realized vector p j. In that case we would not need any approximation à la Fenton, but only to specify the bayesian learning mechanism through which individuals update the weights they associate to each realized vector p j any time they obtain new price information. { Dene CS r j {0, 1} K } K k=1 r jk > 0 and notice that #CS = 2 K 1. The only requirement on the elements of CS is that at least one of the K products must 10

11 2.3 Probability of Store Visit 2 THE MODEL be thought of as available (in addition to alternative 0, the choice not to buy any of the K products, which is always available). Call the probability individual i associates with the event product k is available in store j, Pr i [r jk = 1] = π ijk ; so that Pr i [r jk ] = π r jk ijk (1 π ijk) (1 r jk). Assume the following: (A2) Pr i [r j ] = K k=1 Pr i [r jk ] = K k=1 πr jk ijk (1 π ijk) (1 r jk), rj CS. The assumption implies that the availabilities of any two products are considered as independent events by each individual i. This independence assumption is required because of computational feasibility, having a fully exible categorical distribution over 2 K 1 possible events is not aordable (yet). As for (A1), a set of more primitive assumptions consistent with (A2) will be presented in section 2.4. From (5) and (A2) we can integrate out the beliefs with respect to product availability: E [ ( U ij rj, p j (r j ) ) ] [ ( εij = E(rj,p Uij rj j ) ε ij, p j (r j ) )] = E rj ε ij [E pj (r j,ε ij ) [ Uij ( rj, p j (r j ) )]] = E rj ε ij [ µ z ij (r j ) + ε ij ], (6) = r j CS [ µ z ij (r j ) K k=1 πr jk ijk (1 π ijk) (1 r jk) ] + ε ij = Ṽij ( µ z ij, π ij ) + εij where µ z ij = ( µ z ij (r j ) ) and π r j CS ij = (π ij1,..., π ijk ). Furthermore, allowing individuals for the possibility of not visiting any of the J stores, dene E [U i0 ε i0 ] = Ṽi0 + ε i0. ( ) The normalized level of expected systematic utility for store j is V ij µ z ij, π ij ( ) Ṽ ij µ z ij, π ij Ṽ i0. From (6) it follows that: 11

12 2.3 Probability of Store Visit 2 THE MODEL Pr [shop j θ 2i ] [ { [ ( = Pr j = arg max E Uig rg, p g (r g ) ) ]} ] εig g = exp ( V ij ( µ z ij, π ij )) 1 + J g=1 exp ( V ig ( µ z ig, π ig )), (7) which is, like (2), a standard Mixed Multinomial Logit with parameters θ 2i = ( µ z ij, π ij ) J j=1. It appears clear from (4) and (A1) how, in the proposed model, θ 2i is indeed a function of the vector of preference parameters, θ 1i. Notice that this would be the case even if there were no learning in the model, i.e., individuals knew perfectly ( r j, p j (r j ) ) J j=1 when choosing their favorite store. As it will be clearer from section 2.4, the fact of having learning enhances the supermarket endogeneity problem with an additional endogenous path-dependence issue: not only, at any point in time, individual preferences over product characteristics inuence store choice, but also previous store choices through the updating of personal believes about ( r j, p j (r j ) ) J j=1 aect the current store choice. Finally, from expressions (2) and (7), we obtain the joint probability of purchasing product k in store j, k, j: Pr [product k, shop j θ i ] = Pr [product k shop j, θ 1i ] Pr [shop j θ 2i ] = exp (V ijk (β ik, α i )) r jk 1 + K s=1 exp (V ijs (β is, α i )) r js, (8) exp ( V ij ( µ z ij, π ij )) 1 + J g=1 exp ( V ig ( µ z ig, π ig )) From (8), it might seem that the proposed model has the avor of a Nested Logit model. In reality, this is not the case. The Nested Logit model depicts a decision process over a set of alternatives, {1,..., K}, which can be partitioned into sub-groups according to an unobservable feature that makes some alternatives more similar amongst themselves than to others. The sub-groups 12

13 2.4 Beliefs and Bayesian Learning 2 THE MODEL of similar alternatives are called nests. Notice that the universal set of alternatives, {1,..., K}, is partitioned into dierent nests (i.e., the same alternative k cannot belong to two dierent nests j 1 and j 2 ). In The Nested Logit, the similarity amongst alternatives is driven by unobservable factors, so that the indirect utilities of the alternatives belonging to a same nest end up having correlated error terms. In other words, the main point of the Nested Logit model is to relax, in a parsimonious fashion, the IIA property inherent in the Multinomial Logit model. My model represents a decision process over the pair (alternative k, nest j) where each dierent nest (i.e., store) j could, in principle, contain all the available alternatives {1,..., K} (i.e., nests do not partition the universal set of alternatives). Furthermore, in the proposed model there is not any correlation amongst the error terms of the alternatives belonging to a same nest. The main point of my model, dierently from the Nested Logit model, is to address the endogenous nest selection. 2.4 Beliefs and Bayesian Learning For an account of the statistical claims contained in the current section, see chapter 9 of DeGroot (2004) Beliefs on Prices In this section I will describe a system of beliefs on prices which is consistent with assumption (A1). To reiterate, if (A1) appears to be too strong or not adequate for other reasons, I could always go down the road of the empirical distribution of prices. Sampling Distribution. Assume the sampling distribution: p jk (µ jk, σ jk ) LogNormal [µ jk, σ jk ], j = 1,..., J, k = 1,..., K, independent both over j and k. 4 4 Here, µ jk and σ jk are the mean and the standard deviation of the normally distributed ln (p jk ). 13

14 2.4 Beliefs and Bayesian Learning 2 THE MODEL Notice that the previous assumption coincides to saying that the ln (p jk )'s are normally distributed. In what follows, I will adhere to this last interpretation. As said earlier, the independence assumption is not essential, I maintain it here because of expositional simplicity. Person i is assumed to know the true σ jk, j, k; but not to know any of the true µ jk 's. She will have to guess, or form beliefs on, what the average log-prices of products in the dierent stores are. Also in this respect, I do have some degrees of freedom. I could instead assume that individuals know the means of the log-prices but not their variances, so that the objects of people's beliefs would be the log-price volatilities rather than means. Or, more realistically, I could assume that people do not know neither the means nor the variances of the log-prices, so that they would have to form beliefs on both. In the current document I stick to the easiest option in the hope that the underlying intuition will go through more easily. On the other hand, it seems possible to have, as sampling distribution, a multivariate normal (e.g., each store j sets its K prices jointly, and not each one independently) whose true mean vector and variance-covariance matrix are both unknown to the decision makers. Prior Distribution. Given that individuals do not know the true µ j = (µ j1,..., µ jk ), j = 1,..., J, assume their prior distribution on each µ jk is: µ jk (m jk, τ jk ) Normal [m jk, τ jk ], independent both over j and k. Posterior Distribution. Each time period t = 1,..., T during which individual i visits store j, she will observe and learn (by assumption) all the true product prices p jt = (p j1t,..., p jkt ). 5 As a consequence, after T visits to shop j, individual i will update 5 Also here there is some modeling decision to be made. To one extreme, I assumed each person, during any visit to store j, learns about the prices of all the products on sale there. Another option would be 14

15 2.4 Beliefs and Bayesian Learning 2 THE MODEL her prior beliefs (the beliefs the individual had prior to gathering any new price information) into the following posterior, for each µ jk : µ jk p jk Normal σ2 jk m jk + T τjk 2 ln ( p T jk σjk 2 + T τ jk 2 (p jk1,..., p jkt ) and ln ( ) p T 1 jk = T t=1 T ln (p jkt). ), ( τ 2 jk σ 2 jk σ 2 jk + T τ 2 jk ) 1 2, where p jk = Notice how the posterior mean is a convex combination of the sample mean of log-prices, ln ( p T jk), and the prior mean, mjk. In the convex combination, the weights are given by, respectively, T τjk 2 and σ2 jk. The vaguer, in relative terms, is the prior information on µ jk available to the individual (i.e., a high T τ jk 2 ), the more the person will rely on the σjk 2 sample information personally gathered during her T visits to store j, ln ( p T jk), relatively to the prior mean, m jk. After each information update, the posterior distribution of µ jk p jk becomes the prior distribution of µ jk (m ijk, τ ijk ) that individual i will use for her next choice. In other words, the parameters of the posterior distribution become the new m ijk and τ ijk. Notice that the new prior distribution is individual-specic. In fact, the sequence of stores visited by individual i will determine, through the bayesian updating described above, the evolution of the person's believes. Even if two dierent individuals have the same sequence of store visits at any point in time, they still could hold dierent price believes unless they also visited each (same) store simultaneously. Indeed, each store j is expected to vary the prices of its products from time to time (at least due to promotions and/or discounts). In addition, each individual i's store visit is the outcome of a choice, therefore the evolution of people's beliefs is endogenous in the proposed model. Updated Sampling Distribution. Given the assumptions made with respect to the sampling and the prior distribution, it can be shown that, for i = 1,..., I: to restrict, somehow, the learning ability of individuals. For instance, the opposite extreme would be to assume that people learn only about the prices of the products they purchase. In this sense we could use simultaneously both extrema in order to get an upper and a lower bound, given that the true learning is likely to be somewhere in between. 15

16 2.4 Beliefs and Bayesian Learning 2 THE MODEL [ p jk (m ijk, τ ijk, σ jk ) LogNormal m ijk, ( τijk 2 + ) ] 1 σ2 2 jk, j = 1,..., J, k = 1,..., K, independent both over j and k. From this, (A1) follows. In fact, given (β ik, α i, γ i, s j ), Ṽijk (β ik, α i, γ i ) = β ik α i ln (p jk )+ g (γ i, s j ) Normal [ µ v ijk, ijk] σv and independent j, k; where µ v ijk = β ik α i m ijk + g (γ i, s j ) and ( σijk) v 2 = α 2 i (τ ijk 2 + jk) σ2. As mentioned before, a very dierent stand could be made with regard to assumption (A1). For each product k in each store j, it could be assumed that p jk has a categorical distribution dened over the set of realizations of p jk that we observe in our data (i.e., all the dierent prices for which product k was sold by j as we can tell from our data). Assume that, in the data, p jk takes M jk dierent values (e.g., 2.5, 2.99, 1.75, etc.). Then p jk would have a categorical distribution with M jk possible outcomes, only one of which could be drawn during each trial of the price random variable. It follows that the sampling distribution of p jk would have, for each j and k, M jk parameters (e.g., the true probability of each of the M jk price values to realize). Individual i would be assumed not to know the parameters of the categorical distribution. Her prior belief of these parameters would then have to be a Dirichlet distribution with M jk parameters, j, k. As a consequence, the updated sampling distribution would be a categorical over the set of realized prices with higher weights (i.e., probabilities) on those values more frequently experienced by the individual. This alternative assumption seems less restrictive than (A1), but, on the other hand, the number of parameters involved in the belief system would grow substantially. Indeed, with (A1) two parameters are enough to characterize the sampling distribution of each p jk. Under the current alternative, instead, M jk parameters are required in order to pin down the sampling distribution of each p jk. Therefore, parsimony appears to suggest, at least for the time being, to adhere to assumption (A1). 16

17 2.4 Beliefs and Bayesian Learning 2 THE MODEL Beliefs on Product Availabilities In this section I will outline a system of beliefs on the availability of products which is consistent with assumption (A2). The contents and remarks in this section parallels closely those in the previous one, thus I will be brief. Sampling Distribution. Assume the sampling distribution: r jk π jk Bernoulli [π jk ], j = 1,..., J, k = 1,..., K, independent both over j and k. Furthermore, assume that people do not to know the true π jk, j, k. They will need to have beliefs about the probabilities that any product is sold by each store, π jk. Prior Distribution. Given that individuals do not know the true π j = (π j1,..., π jk ), j = 1,..., J, assume their prior distribution on each π jk is: π jk (λ jk, ω jk ) Beta [λ jk, ω jk ], λ jk, ω jk > 0, independent both over j and k. For the beta distribution, which is dened over the interval (0, 1), we have: E [π jk (λ jk, ω jk )] = λ jk λ jk + ω jk. Var [π jk (λ jk, ω jk )] = λ jk ω jk (λ jk + ω jk ) 2 (λ jk + ω jk + 1) Posterior Distribution. Each time period t = 1,..., T during which individual i visits store j, she will observe and learn (by assumption) all the true product availabilities r jt = (r j1t,..., r jkt ). 6 As a consequence, after T visits to shop j, individual i will update 6 Considerations similar to those regarding the learning assumption about prices hold here. 17

18 3 CONCLUSIONS her prior beliefs (the beliefs the individual had prior to gathering any new availability information) into the following posterior, for each π jk : [ π jk r jk Beta λ jk + T t=1 r jkt, ω jk + T ] T t=1 r jkt, where r jk = (r jk1,..., r jkt ). Similarly to the posterior distribution on prices, it can be seen how the parameters of the posterior distribution on product availability evolve endogenously with the store choices each individual made. For this reason, the updated parameters of the beta distribution, at any point in time, are individual-specic: λ ijk and ω ijk. Updated Sampling Distribution. Given the assumptions made with respect to the sampling and the prior distribution, it can be shown that, for i = 1,..., I: [ r jk (λ ijk, ω ijk ) Bernoulli both over j and k. λ ijk λ ijk + ω ijk ], j = 1,..., J, k = 1,..., K, independent From this, (A2) follows with π ijk = λ ijk λ ijk +ω ijk. Notice that individual i thinks that the probability of not nding product k on the shelves of store j is to store j, the same belief will evolve to ω ijk+t T t=1 r jkt λ ijk +ω ijk +T ω ijk λ ijk +ω ijk. After T visits. If in her personal experience individual i has almost never found product k available in store j (i.e., many of the r jkt 's are equal to zero), then she will revise her initial belief upward, ω ijk λ ijk +ω ijk < ω ijk+t T t=1 r jkt λ ijk +ω ijk, +T and this will clearly aect her future choices of which store to visit for her shopping. 3 Conclusions In the current paper, I develop an empirical demand model for (super)market products which explicitly accounts for endogenous (super)market selection. Individuals choose rst a (super)market and then a product amongst the many oered by the chosen (super)market. The selection of a (super)market is based on the imperfect information about 18

19 REFERENCES REFERENCES the actual oers available held by each individual. Once in a store, the individual learns (in a bayesian sense) about the actual oer available in that specic store and makes her product choice. The updated information will be used by the individual during her next (super)market decision and so on. The resulting demand model contributes to the existing empirical literature in two respects. First, notwithstanding addressing the endogenous choice set heterogeneity issue, it is relatively simple (see (8) above) and it does not require any exclusive or peculiar dataset for its estimation. Second, to the best of my knowledge, it is the rst to allow for imperfect information and learning about the set of available products in dierent stores. The objective outlined in the introduction is more ambitious than what has been done so far in the current article. In order to test for Spence's Hypothesis, it is necessary to complement the current demand model with an appropriate supply-side. Knowing about the decision process people go through any time they go shopping to supermarkets, it seems natural to assume that supermarkets in turn choose strategically both their assortment of products and their prices so to maximize prots. Once a model of consumer surplus and of supermarket prots as a function of products variety is available; then the question of 'Which is the optimal set of products that can be sold?' could be nally addressed. As soon as the supply-side model is completed, in order to show the practical relevance of the newly developed tools, a real data application will be explored. For this purpose, I plan to use individual level scanner-data. References [1] Ackerberg, D. Advertising, learning and consumer choice in experience goods markets: A structural empirical examination. International Economic Review 44 (2003), [2] Ching, A. T., Erdem, T., and Keane, M. P. Learning models: An assessment of progress, challenges and new developments. Working Paper (2011). 19

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