A Simple Model of Demand Accumulation

Transcription

1 A Simple Model of Demand Accumulation Igal Hendel Aviv Nevo March 17, 2009 Abstract In the presence of intertemporal substitution, static demand estimation yields biased estimates and fails to recover long run price responses. Previous work has documented the existence of intertemporal substitution, its implications and o ered ways to recover long run responses. These corrections are computationally intensive and data demanding. We present a simple model that maps, high frequency, market level data into preference. We estimate the model using data on soft drinks and nd a disparity between static and long run estimates of price responses. Alternatives solutions o ered in the literature peform poorly. 1 Introduction Demand estimation plays a key role in empirical Industrial Organization and in anti-trust. A typical exercise is to estimate a demand system and use it to infer conduct, simulate the e ects of a merger, evaluate a trade policy or compute cost pass through. 1 While for the most part the demand models used are static, there is ample evidence of demand dynamics. Demand dynamics can arise for di erent reasons including durability and storability of the product. Here we focus on storable products. Department of Economics at Northwestern University and NBER. Contact information: igal@northwestern.edu and nevo@northwestern.edu. 1 See, for example, BLP (1995, 1996), Goldberg (1995), Nevo (2001), HLZ, Hellerstien and Goldberg (2008). 1

2 A number of papers (Sun, Neslin and Srinivsan (2003), Erdem, Imai and Keane (2003), Hendel and Nevo (2006b)) estimate structural models of consumer inventory in order to simulate long run price responses for storable products. The estimation of these inventory models is computationally intensive and data demanding. These requirements have limited the use of dynamic demand models in policy analysis and academic work. This paper proposes a simple model to incorporate demand dynamics. We use a (simple) inventory model to separate purchases for current consumption from purchases for future consumption. That way we can relate consumption and prices, to recover preferences (clean of storage decisions) and translate short run responses to prices, observed in the data, into long run reactions, which are the object of interest in most applications. The way we impute purchases for storage is quite simple but intuitive. Its advantage is that it does not require solving the value function of the consumer and the estimation is straightforward. The intuition of the model can best be demonstrated by a simple example. Suppose (1) prices take on only two values: a sale and non-sale price; and (2) some consumers can store the product for one period (while others cannot store). Consider a non-sale period following a non-sale, all consumers purchase for consumption. Since it is not a sale there is no incentive to store, and since the previous period was also not a sale none of the consumers have any inventory. Similarly, during a sale that follows a non-sale only consumers that cannot store will purchase. The above example shows that the parameters of the model can be estimated in one of two ways. We could restrict attention to periods where dynamics do not play a role. In the above example these periods of two consecutive non-sale, as well as two consecutive sales. Alternatively, we could look at all periods and use the model to predict purchases, accounting for the fact that a fraction of the consumer may be stockpiling. The model we propose builds on the intuition of the example. We formalize the required assumptions and show that they simplify the state space. The problem remains dynamic, but easy to characterize. Solving the value function is not necessary. The approach can be seen as the exact solution to the speci c model or as a shortcut (approximation) to a complex dynamic inventory decision. 2

3 Using the model we describe the biases generated by neglecting dynamics. Estimated own price responses are upward biased. The reason is that estimates re ect a weighted average of long run price responsiveness (dictated by the underlying preferences) and short run inventory (intertemporal) considerations. In addition, the model suggests that standard static estimation controls for the wrong price of the competing goods. The "e ective" price (the actual opportunity cost of consumption) in this dynamic setup di ers from current price. We show that the implication of using the wrong price is to bias the estimated cross price e ect downward. For most antitrust applications the interest lies in long run elasticities. Both biases, own price upward cross price e ect downward, attenuate the assessed unilateral e ects in antitrust enforcement. We apply the model to weekly store-level data on 2 liter bottles of colas. The estimates using the simple model indeed deliver lower own and larger cross price responses than those from static estimates. Although the basic model requires fairly strong assumptions, it is relatively easy to check whether the assumptions are born in the data. Actually, the pricing patterns in a subset of the stores in the sample is inconsistent with the model s assumptions. In those stores sales come in long sequences. The state that determines purchasing behavior can no longer be characterized by four events, a richer model is needed. Interestingly, the estimates based on the basic model are unreasonable in the stores with the complex sale patterns. A slightly enriched model delivers demand estimates similar to the simple model on the conforming stores (stores with pricing patterns conforming to the assumptions of the basic model). In many applications dynamics are not the essence, but its presence may cast doubt on the estimates. The estimation we propose suggests a simple way, through a regression or just tabulation of purchases, to test for the presence of advanced purchases. We discuss alternative approaches in Section 6. Alternatives to dealing with dynamics include aggregating the data from weekly to monthly and quarterly frequency, or approximating the missing inventory by including lagged prices/quantities (and compute long run e ects using impulse response). We argue that the alternative methods also requires a model 3

4 to translates the estimated coe cients into preferences. We advocate the approach on this paper based on simplicity and interpretability of the ndings. Section 2 reviews the literature. The model is presented in Section 3, and the proposed solution in Section 4. Section 5 presents the application to soft drink scanner data. 2 Evidence of Demand Accumulation 2.1 Motivating Facts Several papers (discussed in the next sub-section) have documented demand dynamics. We rst look at typical scanner data for direct evidence on the relevance of intertemporal demand e ects. Figure 1 shows the price of a two liter bottle of Coke in a store over a year. The pattern is typical of pricing observed in scanner data: regular prices and occasional sales, with return to the regular price. Since soft-drinks are storable, pricing like this creates an incentive for consumers to anticipate purchases. Buy during a sale when the price is low and store for consumption in the future, when the price is high. 4

5 1 price week Figure 1: A typical pricing pattern Indeed, when we examine the quantity purchased we nd evidence of demand accumulation. Table 1 displays the quantity of 2 liter bottles of Coke sold during sale and non-sale periods (we present the data in more detail below). During sales the quantity sold is significantly higher (623 versus 227, or 2.75 times more). Interestingly, the quantity sold is lower if a sale was held in the previous week (399 versus 465, or 15 percent lower). The impact of previous sales is even larger if we condition on whether or not there is a sale in the current period (532 versus 763, or 30 percent lower, if there is a sale and 199 versus 248, or 20 percent lower in non sale periods). Below we survey the literature that applied various econometric tests to document the existence of consumer stockpiling. But to us the simple pattern present in Table 1 shows that dynamics are important and that consumers ability to store detaches consumption from purchases. Table 1 shows that purchases are linked to previous purchases, or at least, previous prices. 5

6 Table 1: Quantity of 2-Liter Bottles of Coke Sold S t 1 = 0 S t 1 = 1 S t = S t = Related Literature Numerous papers in Economics and Marketing document demand dynamics, speci cally, demand accumulation. Pesendorfer (2002) shows that aggregate demand increases in the duration from previous sales. Hendel and Nevo (2006a) document demand accumulation and demand anticipation e ects, namely, duration from previous purchase is shorter during sales, while duration to following purchase is longer for sale periods, suggesting intertermporal demand substitution. They also show that households frequency of purchases on sales correlates with proxies for storage costs. Sun, Neslin and Srinivsan (2001), Erdem, Imai and Keane (2003), Hendel and Nevo (2006b) estimate structural models of consumer inventory behavior. There is also an extensive Marketing Literature on the Post-Promotion Dip Puzzle (Blattberg and Neslin, 1990) that explores demand dynamics. Numerous explanations for the existence of sales have been suggested in the literature. Sales might be driven by changes in costs, demand or retailer inventory cost. Varian (1980) and Salop and Stiglitz (1982) proposed search based explanations which deliver mixed strategies strategy equilibria, interpreted as random prices. Sobel (1984), Conlisk Gerstner and Sobel (1984), Pesendorfer (2002), Narasimhan and Jeuland (1985) and Hong, McAfee and Nayyar (2002) present di erent models of intertemporal price discrimination. The main insight comes from Sobel (1984). Sobel (1984) shows that when consumers can time their purchases, say of a durable, the accumulation of low types (who have not yet purchased) increase sellers incentive to decrease price. In equilibrium the seller runs sporadic sales targeting the increasing mass of accumulating low types. 6

7 Hendel, Lizzeri and Nevo (2008) show that sales enhance the ability to non-linearly price a storable product. At a constant price consumers can, by smoothing consumption (out of stored units), undo sellers attempts to use non-linear prices. Sellers can increase pro ts by having infrequent low priced sales, which make more di cult for the consumer to skip purchases and smooth consumption. 3 The Model We start with the simplest model of product di erentiation with storage. While very simple the model conveys the main ideas. We later show the model can be generalized and discuss how to apply the proposed estimation to other demand systems, e.g., Berry, Levinsohn and Pakes (2005) (BLP). Assume quadratic preferences: U(q; m) = Aq q 0 Bq + m where q = [q 1 ; q 2 ; :::; q N ] is the vector of quantities consumed of the di erent varieties of the product (sodas in our application) and m is the outside good. Absent storage quadratic preferences lead to a linear demand system: q t i(p) = i p t i + X j ij p t j (1) In a multi-period set up with storage consumers can anticipate purchases for future consumption. We need to distinguish purchases from consumption: denote purchases by x and consumption by q: We make the following assumptions: A1: two prices: sale and non-sale (p S < p N ) 7

8 A key for our method to work is to be able to de ne a sale. When prices take on two values the de nition of a sale is easy. 2 In a more general setup one does not need to de ne a sale. Instead for any price, or history of prices, consumers may purchase for future consumption. Assumption A1 simpli es the determinants of demand anticipation. If current price is high then future prices will be (weakly) lower and there is no incentive to store. However, if the current price is low then future prices will be (weakly) higher and consumers will buy for inventory. The same logic applies even with a more general price process as long as we can properly de ne a sale. Next we make assumptions about the storage technology. A2: storage is free A3: inventory lasts for S = 1 periods (depreciates afterwards) In principle, we can relax both assumptions. However, depending on the alternative this might signi cantly complicate the analysis. Allowing inventory to last for a single period reduces the state space: we only have to consider whether there was a sale in the previous period. This assumption is testable. If longer periods of accumulation are apparent in the data a second period of storage can be introduced at little cost (just more accounting). A4: a proportion! of consumers do not store. A proportion of customers do not have access to the storing technology or they are just transient. 3.1 Purchasing Patterns and Preferences Consumers purchase for storage at p S, and never stores at p N : Thus, demand is quite simple to predict. We only need to de ne 4 events (or types of periods) to fully characterize consumer behavior: a sale followed by a sale (SS), a non-sale followed by a sale (NS), a sale followed by a non-sale (SN ), and two sale periods (SS). 3 2 Assumption A1 is stronger than we need. In practice, there can be several sale prices (as well as non-sale ones), and demand will depend on prices within each regime. As long as we can correctly de ne a sale we can easily control for prices in each regime. 3 When a sale follows a sale under our assumptions the consumer is indi erent between storing for consumption next period or buying then. We break the tie by assuming she buys now (a justi cation would be uncertainty about coming to the store in the next sale). 8

9 The four events just described may not be su cient to fully characterize consumer behavior if sales and storing events do not coincide. For example, if after a sale a deeper sale was expected, purchases in that rst sale period would di er from a period with the same price but not followed by a deeper sale. In such case, to characterize demand we would need to de ne more than four events. In the next section we show that some of the stores in our sample such pricing patterns, too complex for the basic model to work. We use those stores to show how the model can be enriched. Given Assumptions A1-A3, purchases, x i (p t ); by a storing consumer, are: 8 >< x i (p t ) = >: q i (p t i; p t i) 0 q i (p t i; p t i) + q i (p t i; p t+1 i ) q i (p t i; p t+1 i ) if p t 1 i p S i = p t 1 i p N i = p t 1 i p t 1 i = p t i = p N i < p t i = p N i > p t i = p S i = p t i = p S i (2) where q 1 (p 1 ; p 2 ) = p 1 + p 2 is the long run demand (we are after). The four rows represent events NN, SN, NS, and SS; respectively. At high prices there are no incentives to store, in which case purchases equal either: consumption (de ned by the static demand, i.e., long run preferences), or zero, if the previous period was a sale (i.e., event SN; in which case the buyer consumes out of storage). During sales preceded by a non-sale period purchases include current consumption as well as inventory. During periods of sale preceded by a sale, current consumption is out of storage, so purchases are for future consumption only, q i (p t i; p t+1 i ): Notice the di erence in the second argument relative to purchases during N N: Purchases for future consumption take into account the expected consumption of products i; which will be dictated by p t+1 i.4 A word on the storage technology. We assume that consumers can store for a pre-speci ed period of time. The assumption is probably reasonable for some products like milk or yogurt. For other products the relevant constraint on advance purchases is probably storage space at home or transportation from the store. Our assumption, although not ideal for sodas, is quite convenient. First, it simpli es the dynamics. Under a capacity constraint we would need 4 We can assume perfect foresight, else the consumer would use some expected price. However, the key here is that if a product is currently on sale we know its e ective next period price is p S (since the product will be stored today for consumption tomorrow). 9

10 to keep track of how much is left in storage at di erent prices. Second, it helps detach the storage decision of di erent products. If the binding constraint is storage capacity, storing one product restrains the ability to store the other. Under our assumption the optimal storage of one product depends on the e ective price of the other product (as can be seen in equation 2), but not on the quantity stored. Such link between products is quite easy incorporate, we just control for e ective prices as oppose to observed prices. Once we aggregate storer and non-storer demand the model predicts aggregate purchases x 1 (p 1 ; p 2 ), in each event: 8 >< x 1 (p 1 ; p 2 ) = >: q 1 (p t 1; p t 2) during NN!q 1 (p t 1; p t 2) during NS!q 1 (p t 1; p t 2) + (1!)(q 1 (p t 1; p t 2) + q 1 (p t 1; p t+1 2 )) during SN!q 1 (p t 1; p t 2) + (1!)(q 1 (p t 1; p t+1 2 )) during SS (3) where q 1 (p 1 ; p 2 ) is consumption (or long run demand). In the next section we impose the restrictions prescribed by the model across all di erent events. We jointly estimate! and preferences q 1 (p 1 ; p 2 ) that t purchases across the di erent regimes. 3.2 Responses to Price Changes Long Run Responses: Permanent Price Reductions Suppose we observe two price regimes for product one, with constant prices within each regime. Assume for now the prices of the other products are constant. Denote the di erences in prices across the regimes by p 1 : Since prices are constant (over time in each regime), there is no reason to store and therefore the di erence in purchases (and consumption) across regimes is q 1 = p 1 (4) q 2 = p 1 (5) 10

11 We can solve for and using data on q i and p 1 to recover preferences Short Run Responses: Temporary Price Reductions Instead of observing long lasting price di erences we may observe high frequency price changes, like in the case of sales. Consider for simplicity just three periods, and suppose product 1 0 s price decreases during the second period: p 2 1 < p 1 1 = p 3 1; while product 2 s price remains constant. Consumers have an incentive to store during period 2 for consumption in period 3. As a consequence x 6= q: Since storing is free consumers will purchase all of period 3 consumption, q 1 (p 2 1; p 3 2); in period 2: Notice the e ective price of product 1 in period 3 is actually the lowest of periods 2 and 3 prices, minfp 2 1; p 3 1g: The consumer can time her purchases to minimize expenses. Consumption in period three is determined by p 2 1: Quantities purchased by a storing consumer over the three periods (according to equation 2) are: x 1 = fq 1 1; 2(q 1 1 p 1 ); 0g (6) x 2 = fq 1 2; q p 1 ; q p 1 g (7) Should we estimate demand statically, using purchase data generated by 6, we would estimate the following price e ects: Own price k 2 + 3q1 1 2p 1 k > k k Cross price reaction 2 < There is an over estimation of long run own price reaction and underestimation of long run cross price e ects. 11

12 The over estimation of own price e ects is caused by attributing the response to a temporary price reduction as an increase in consumption, while the consumer is purchasing for storage. In addition, the price increase in period 3 coincides with a decline in purchases, which is also misconstrued as a decline in consumption. The cross price e ects are instead understated. The reason for the downward bias is that in period 3 the observed and e ective prices di er. The e ective price, which dictates consumption of good 1, is the period 2 purchase price. In the estimation we would instead interpret the price increase (observed in period 3), which is not accompanied by a decline in purchases of product 2, as lack of cross price reactions. We can view the problem as using the wrong p 3 1 or as lack of information on inventory. 4 Estimation 4.1 Intuition: How Do We Recover Preferences? Before presenting the formal estimation of the model we show intuitively how the model helps recover preference parameters. We o er two approaches, both are part the full estimation, but discussing them separately helps clarify what variation in the data identi es the parameters. The rst approach is based on events without storage, while the second approach imputes and purges storage from purchases. As a benchmark lets compute the price reaction recovered in static estimation, using information from Table 1: Timing Restrictions x N ST AT IC = x 1 = xs = p 1 p :4 = 988 According to the model during sale periods that follows a sale (event SS) purchases equal consumption: x(p) = q(p): Basically, after purchasing for storage, the pantry is lled, consumers (whether they are a storer or not) purchase for consumption only. Since neither NN and SS events involve storage we can rely on them to compute price responses. 12

13 We can use information from Table 1 to get as: x 1 = xss x NN = p 1 p SS pnn 0:4 = 710 Notice that we do not need to gure out!: The x is valid for any value of!: The disadvantage is that only part of the data is used. If sales are infrequent it is possible that we have only few observations of x SS Quantity "Accounting" Another way to map purchases into preferences is to use the model to impute storage. based on equation 3. We can decompose the extra purchases, x 1 ; that occur during during NS relative to N N; into a consumption e ect (i.e., a price response) and the quantity purchased for storage. The response to p 1 according to equation 6 is: x 1 = x NS x NN = (1!)(q j p 1 j) +! j p 1 j (8) Equation 8 says that the extra purchases can be decomposed into the consumption e ect of all buyers, and the inventory purchases of a proportion (1!) of the consumers. 5 We can proxy! by x SN =x NN ; the ratio of the quantity sold during SN over purchases during NN. The missing purchases (in SN relative to NN) re ects the consumers who stored during the previous period. According to Table 1 about 20% of consumers store (! = 199=248). From 8 we get = 970. These simple calculations, that either purge inventories or use non-storing periods, imputes between 2% to 30% of the price reaction to intertemporal e ects. 5 Note, if all consumers stored we should observe zero quantity sold during NS. As we see in Table 1, sales are positive in such event. 13

14 4.2 The Estimating Equations We estimate preferences in two ways. The rst uses data only from the NN and SS periods, where there is no storage. Linear estimation allows us to recover all the parameters of the model, except the fraction of consumers who store. The second approach uses data from all periods, and is therefore more e cient, but if we allow for store level xed e ects, to account for di erences across stores in the level of purchases, requires non-linear estimation. [to get w requires NLLS, even without FE]. To obtain the exact estimating equations we combine equations 1 and 3 and allow for a panel structure (that exists in the data we use below). The quantity of Coke sold in period t at store j is 8 >< x 1jt (p 1jt ; p 2jt ) = >: j + p 1jt + p e 2jt during NN and SS!( j + p 1jt + p e 2jt) during NS (2!)( j + p 1jt + p e 2jt) during SN (9) where p e 2jt is the "e ective" price of Pepsi, which is either the current price or the past price if Pepsi is consumed out of storage. To account for the store level xed e ects we de-mean the data. For prices this is straightforward. For quantities we have to account for the re-scaling in di erent regimes. We show in the Appendix how to modify the estimating equation to account for this rescaling. The adjustment is simple but makes the estimation non-linear in the parameters. We estimate all the parameters by least squares, linear or non-linear depending on the equation. In principle we could use instrumental variables to allow for correlation between prices and the econometric error term. However, we do not think correlation between prices and the error term is a major concern in the example below. 14

15 5 An Empirical Application: Demand for Colas The average numbers (from Table 1) used in the previous section hide within regime price variation which we can exploit. They also neglected properly controlling for the prices of substitute products. 6 We now estimate the model (equation 9) using all the events. 5.1 Data We use standard store level scanner data. We have weekly observations of prices and quantity sold at 729 stores that belong to 8 di erent chains throughout the Northeast. We focus on two-liter bottles of Coke, Pepsi and store brands. Our benchmark model is a static linear demand system, with store xed e ects. There is substantial homogeneity of pricing across stores of the same chain, but chains price quite di erently. As we show below, the model of Section 3 ts some chains better than others. We need a de nition of a sale, or more precisely, we need to identify periods of advance purchases. We need them to impute storage. After browsing the data we adopted a threshold of one dollar. Any price below a dollar is considered a sale, namely, a price at which storers purchase for future consumption. This is an arbitrary de nition. A more exible de nition may allow for chain speci c thresholds, or perhaps moving thresholds over time. For the moment we might err on the side of simplicity. 5.2 Pricing Patterns Our working assumptions conform with the pricing patterns of ve of the eight chains in the sample (the pricing shown in Figure 1). In those chains, it is easy to identify sales with periods of advance purchase. Namely, sales as we de ned them create incentives to purchase for future consumption. However, in the other three chains pricing is more complex. Figure 6 This is a serious concern since promotions of Coke and Pepsi are probably correlated, thus, a low Coke price may be also re ecting a high price of the closest substitute, thus contaminating the price reactions we infer. 15

16 1 price week Figure 1: Figure 2: Pricing patterns in non-conforming stores 2 shows the typical pricing patterns in those stores. We will refer to those stores as nonconforming (to our assumptions). Sales typically come in pairs, or longer sequences. 7 This is problematic for our model since not all sales are periods of advance purchases. Thus, purging storage from sales becomes tricky. The model predicts consumers store during N S periods. However, it would fail to predict large purchases during the second sale (event SS). To see the challenge of this pricing patter we recompute Table 1. While Table 1 shows that purchases are lower in periods that follow a sale (relative to non-sale histories), Table 2 shows the opposite. Consumers purchase more in event SS than in N S in non-conforming stores 7 One possible explanation to the pricing pattern is that the price changes in the middle of the rst week, Thus, the observed price is aweighted average of a sale and non-sale price. 16

17 Table 2: Quantity of 2-Liter Bottles of Coke Sold Non-conforming stores S t 1 = 0 S t 1 = 1 S t = S t = We need a richer model. Notice that absent long sales the basic model works (namely, properly associates purchases and events). Pricing like that in Figure 1 (i.e., in conforming stores) is handled well by the model. Thus, we expect the basic model to work for conforming stores but fail for nonconforming ones. It fails to predict storage in event SS, while it overstates purchases during N S: 5.3 Results Tables 3 and 4 show estimates for the conforming chains. The rst column shows estimates from a static model, with store xed e ects. Column 2 presents the x based on timing restriction only (namely, using the sub-sample with the events in which the model predicts no storage). Columns 3 and 4 present estimates of the full model. The di erence between the last two columns is that in column 3 we control for the current price of the competing products. Column 4 instead controls for the e ective price. According to the model the e ective price faced by a storer is the minimum of current and last period price. 8 Both xes lower own price e ects and increase cross ones, for both Coke and Pepsi. The estimated proportion of consumers who do not stockpile is around half the population, and is slightly higher for Coke. Consistent with this estimate, the di erence between the static and dynamic estimates are larger for Pepsi than for Coke. The estimates in column 3 are of no interest on their own. According to the model, the cross prices controls are incorrect. The model prescribes the use past prices during periods 8 For the non-storer the current price is the e ective one. Thus, the aggregate e ective price is the weighted average of the prices faced by storers and non-storer; weighted by the proportion of each type of buyer in the population. The price is recomputed as the estimation algorithm seaches for the optimal!: 17

18 preceded by a sale (i.e., the e ective price that dictates the consumption of storers is the lagged price). However, if the model is irrelevant, or demand dynamics absent, as we move from column 3 to column 4 we would be introducing noise in the price of the competing product. As such we would expect the coe cient of Pepsi in the Coke equation (and Coke s in the Pepsi equation) to be lower, due to measurement error (assuming the introduced noise will generate classical measurement error). Interestingly, both cross price e ects increase substantially as we replace current price by e ective price. Suggesting the latter is the correct control, and that indeed dynamics are present. Table 3: Demand for Coke - conforming chains FE Timing Only All Restrictions P Coke (11.1) (11.8) (11.2) (11.1) (11.3) P P epsi (11.9) (10.9) (11.0) (11.4) (10.4)! (fraction storers) (0.01) (0.01) (0.01) 1.05 (0.02) Cross price Corrections No Yes Yes All estimates are from least squares regressions. The dependent variable is the quantity of Coke sold at a store in a week. The regression in column (1) includes store xed e ects. The regression in column (2) is the same as in column (1) but using only the NN and SS periods. The regressions of columns (3)-(4) impose all the restrictions of the model using the actual and e ective price. The regression in column (5) uses all the restrctions of the enhanced model, described in section Standard errors are reported in pharenthesis. 18

19 Table 4: Demand for Pepsi - conforming chains FE Timing Only All Restrictions P Coke (12.2) (11.0) (10.5) (10.5) (10.1) P P epsi (13.1) (15.8) (11.6) (11.6) (11.1)! (fraction storers) (0.01) (0.01) (0.01) 0.66 (0.01) Cross Price Corrections No Yes Yes All estimates are from least squares regressions. The dependent variable is the quantity of Pepsi sold at a store in a week. The regression in column (1) includes store xed e ects. The regression in column (2) is the same as in column (1) but using only the NN and SS periods. The regressions of columns (3)-(4) impose all the restrictions of the model using the actual and e ective price. The regression in column (5) uses all the restrctions of the enhanced model, described in section Standard errors are reported in pharenthesis. The ndings are consistent with the biases predicted in Section 3. Both proposed xes lower own price e ects by purging purchases for storage. At the same time the xes raise cross price e ects, by accounting for e ective price variation (i.e., eliminating measurement problems in prices of substitute products). Tables 5 and 6 show estimates for the rest of the chains. Column 1 presents the static xed e ect estimates, columns 2 presents the timing corrections while column 3 present estimates of the full model. Interestingly neither x works in this case. The reason why the simple xes do not work is quite intuitive. The timing x is based on comparing purchases during events SS to NN, as neither event involve storage (thus, they re ect legitimate price reactions). However, since in the non-conforming chains event SS involves storage, the estimates are not free of storage 19

20 e ects. Actually, the timing x goes the wrong way because there is plenty of storing in event SS, which the model rationalizes as a price reaction. The full model estimates face similar problems. The numbers in columns 1 and 3 hardly di er. The reason being that the model imputes most purchases to non-storers, thus, interpreting all purchases as consumption which implies no x is done. The estimates predict almost nobody stores (! close to 1), in an attempt to rationalize large NS purchases. Moreover, to rationalize the large SS purchases it imputes large price e ects. In sum, the model we presented so far is very simple. It is able to capture purchasing behavior under very simple pricing processes, but it is expected to fail to describe buyers behavior under a slightly more complex price process like the one we observe in non-conforming stores. Indeed the xes based on the basic model fail in the richer data. We now modify the model to capture the non-conforming pricing patterns. Table 5: Demand for Coke - nonconforming chains FE Timing Only All Restrictions P Coke (12.8) (22.5) (13.9) (12.5) P P epsi (15.6) (28.7) (15.6) (12.2)! (fraction storers) (0.01) (0.01) 0.00 (0.02) All estimates are from least squares regressions using data from the stores non-conforming to our assumptions. The dependent variable is the quantity of Coke sold at a store in a week. The regression in column (1) includes store xed e ects. The regression in column (2) is the same as in column (1) but using only the NN and SS periods. The regressions in column (3) imposes all the restrictions of the model. The regression in column (4) uses all the restrctions of the enhanced model, described in section Standard errors are reported in pharenthesis. 20

21 Table 6: Demand for Pepsi - nonconforming chains FE Timing Only All Restrictions P Coke (8.8) (14.4) (13.5) (8.8) P P epsi (10.7) (15.0) (16.3) (11.1)!(fraction storers) (0.01) (0.01) 0.00 (0.03) All estimates are from least squares regressions using data from the stores non-conforming to our assumptions. The dependent variable is the quantity of Pepsi sold at a store in a week. The regression in column (1) includes store xed e ects. The regression in column (2) is the same as in column (1) but using only the NN and SS periods. The regressions in column (3) imposes all the restrictions of the model. The regression in column (4) uses all the restrctions of the enhanced model, described in section Standard errors are reported in pharenthesis Enhanced Model The convenience of the assumptions behind the basic model are that they enable to characterize purchasing (and storing) behavior within the context of 4 events, dictated by current and previous period state (sale or non-sale). The model associates sales with storing events. For more complex pricing processes a sale price may generate storing in some periods but not in others. For example, consumers may shop in a week classi ed as a sale but before the sale starts. The x works by purging storage out of purchases. In the so called non-conforming store consumers who shop in the rst part of second week with a sale are as likely to store as those that shopped in the previous week, unlike our basic model s prediciton. The model becomes unable to properly purge storage. Indeed, our basic model did not perform well in these stores. 21

22 In order to handle the non-conforming stores we account for the possibility that some consumers show up in the rst part of the same while others in the latter. Once we account for this possibility and derive predicted demand, we recover preference parameters and proportions in the population as we did in the previous section. The last column of Tables 5 and 6 show the ndings. The slight modi cation of the model enables to rationalize storing, thus, as predicted in section 3 lowering own price e ects. The estimates for Coke re ect a similar proportion of storers to the conforming stores, and a bit higher for Pepsi. We run the enriched model using the conforming stores (last column of Tables 3 and 4). Price responses and the proportions of storers is virtually unchanged, suggesting the richer model is not needed to t the data of the conforming stores. 6 Alternative Approaches There are several alternative approaches to recover preferences. One could estimate a structural inventory model (as in Sun, Neslin and Srinivsan (2003), Erdem, Imai and Keane (2004), Hendel and Nevo (2006b)). The estimation requires consumer-level data and it is quite computationally and time intense. We consider other alternatives next. Although several approaches are feasible, all of them require taking a position on what is behind the demand dynamics. Our approach is a simple one, other may work as well, but any approach requires a model to recover preferences. 6.1 Including Lagged Variables A common method often used to compute long run price responses is to include lagged variables, either lagged quantities or, more commonly, lagged prices. 9 Lagged quantities can be motivated through either a partial adjustment model or as approximating the missing inventory variable. Lagged prices can be seen as a way to control for dynamic e ects. In spirit, this is similar to our model, where we advocate de ning events based on lagged sales. However, it is model free and asking much more from the data. The long run e ect, in 9 See, for example,.. Wittik el at (Mktsci, 2003?), refernces for lagged prices... 22

23 either of these cases, is computed by tracing the e ect of a price change if lagged quantity is included, or by summing the coe cients of the lagged prices. It is worth notice that in some cases the lagged controls may help approximate a long run e ect, this is not generally the case. It is evident from equation 4 the lagged controls are not substitute for a model, namely, there is lagged structure that would recover the actual long run cross price e ects. We explore the performance of this method below. 6.2 Aggregating purchases over time A common approach to deal with demand anticipation is to aggregate observations over time. For example, from weekly to monthly data. In our example aggregation can solve the problem under some special conditions. Suppose that we had data from an additional period, t = 0, in which the prices and quantities were like period 1: If we aggregated the data from the rst two periods (t = 0; 1) and the last two periods (t = 2; 3) and de ned prices as revenue divided by quantity, then estimation based on the aggregate data would recover long run e ects. The success of this approach in recovering long run responses relies crucially on several assumptions. We focus on lack of heterogeneity in storage. We provide, in the Appendix, an analytic example that shows this. 6.3 Demand for Coke: Alternative Corrections We now apply these alternative corrections to the data from the conforming stores. The results for Coke are presented in Table 7. The rst two columns repeat the results from the store xed-e ects regression, and from our model. The next two columns present the long run e ect from models that include 1 and 4 lags, respectively. The results are not very promising. Both lagged prices models impact the own price elasticity in the "right" direction but the magnitude is smaller than our correction. The results for the cross price e ect are worse. The rst model does not change the cross-price e ect by much. The second, with more lags, does but estimates a negative cross-price elasticity. 23

24 The last two columns present the results from aggregating over time: into bi-weekly periods and then by month. In both cases the own price elasticities move in the right direction, in the case of monthly aggregation yielding numbers quite close to our estimates. However, both model estimate a negative cross price elasticity (although in one case the estimates is not signi cantly di erent from zero). Overall, these alternative methods do not seem to yield very sensible results. Table 7: Demand for Coke Alternative Corrections FE Our lag 1 p lag 4 p agg bi-w agg month P Coke (11.1) (11.2) (15.1) (22.9) (9.8) (14.5) P P epsi (11.9) (11.4) (17.1) (25.6) (11.1) (16.0)! 0.53 (0.01) All estimates are from least squares regressions using data from the stores conforming to our assumptions. The dependent variable is the quantity of Coke sold at a store in a week. Standard errors are reported in pharenthesis. 7 Concluding Comments We o er a simple model to account for demand dynamics due to consumer inventory behavior. The model can be estimated easily using store level data. An application to demand for Coke and Pepsi yields quite reasonable estimates. At the same time, corrections based on alternative methods, like aggregation or control for lagged variables, yield unreasonable estimates. The simple model seems to work well in the case where the assumptions on the pricing process are met. In stores where these assumptions were not met the so called nonconforming stores the model did not perform well. However, we showed how the model can be augmented to handle these stores as well. More generally our simple model can be augmented as needed. In some cases the model might need more consumer types, just as in 24

25 the augmented model we allowed for two types of consumers consumers. In other cases we might need to change the storage technology to allow for longer storage possibilities. This implies that the state space might become more complex: we cannot simply control for what happened last period. The application focused on linear demand curves. However, linearity is not essential to any of our methods 10. The main ideas can be applied to more exible demand systems. The main di culty, which in independent of linearity, has to do with the cross product inventory constraints. Suppose we want to study an industry will 10 di erent main brands say, laundry detergents. Do all brands share (at least partially) a common storage area? If they do then the quantity purchased today on one product, say Tide, will depend on past sales not just of Tide but of all brands. This is a problem for fully structural papers as well. For example, both Erdem, Imai and Keane (2003), and Hendel and Nevo (2006) have to make strong assumptions on the inventory process and the evolution of future prices in order to deal with this problem. Similar assumptions will be required here. 8 References Berry, Steven, James Levinsohn, and A. Pakes (1995), Automobile Prices in Market Equilibrium, Econometrica, 63, Berry, Steven, James Levinsohn and Ariel Pakes (1999) Voluntary Export Restraints on Automobiles: Evaluating a Strategic Trade Policy, American Economic Review, 89(3), Blattberg, R. and S. Neslin (1990), Sales Promotions, Prentice Hall. Conlisk J., E. Gerstner, and J. Sobel Cyclic Pricing by a Durable Goods Monopolist, Quarterly Journal of Economics, It does slightly simplify our estimating equations, but only slightly. 25

26 Erdem, T., M. Keane and S. Imai (2003), Consumer Price and Promotion Expectations: Capturing Consumer Brand and Quantity Choice Dynamics under Price Uncertainty, Quantitative Marketing and Economics, 1, Goldberg, Penny. "Product Di erentiation and Oligopoly in International Markets: The Case of the U.S. Automobile Industry," Econometrica, Jul. 1995, pp Godberg, Panny and Rebecca Hellerstein. "A Framework for Identifying the Sources of Local-Currency Price Stability with an Empirical Application," mimeo. Hendel, Igal, Alessandro Lizzeri and Aviv Nevo. "Non-Linear Pricing of a Storable Good" mimeo Northwestern University. Hendel, Igal and Aviv Nevo Sales and Consumer Inventory, Manuscript, Rand, 2006a. Hendel, Igal and Aviv Nevo Measuring the Implications of Sales and Consumer Inventory Behavior, Econometrica, 2006b. Hong P., P. McAfee, and A. Nayyar Equilibrium Price Dispersion with Consumer Inventories, Manuscript, University of Texas, Austin. Jeuland, Abel P. and Chakravarthi Narasimhan. Dealing-Temporary Price Cuts-By Seller as a Buyer Discrimination Mechanism Journal of Business, Vol. 58, No. 3. (Jul., 1985), pp Nevo, Aviv. Measuring Market Power in the Ready-to-Eat Cereal Industry, Econometrica, 69(2), , Pesendorfer, M. (2002), Retail Sales. A Study of Pricing Behavior in Supermarkets, Journal of Business, 75(1), Salop S. and J. E. Stiglitz. The Theory of Sales: A Simple Model of Equilibrium Price Dispersion with Identical Agents The American Economic Review, Vol. 72, No. 5. (Dec., 1982), pp Sobel, Joel. The Timing of Sales, Review of Economic Studies,

27 Sun, Baohong, S. Neslin and K. Srinivasan. "Measuring the Imapct of Promotions on Brand Switching Under Rational Consumer Behavior" Journal of Marketing Research, 40, (2003), p Varian Hal A Model of Sales The American Economic Review, Vol. 70, No. 4. (Sep., 1980), pp Appendix 9.1 Example where aggregation fails Consider the following example where aggregation fails. Suppose there are two types of consumers. Type A consumers can store for one period, type B cannot store. Assume four time periods with p 2 1 < p 0 1 = p 1 1 = p 3 1 (while p 0 2 = p 1 2 = p 2 2 = p 3 2): Following, equation x A 1 = 6 4 q1 1 q1 1 2(q1 1 p 1 ) 0 ; x A 2 = q2 1 q2 1 q2 1 + p 1 q2 1 + p 1 ; x B 1 = q1 1 q1 1 q 1 1 p 1 q 1 1 ; x A 2 = q2 1 q2 1 q p 1 q will be Assuming one type of each consumer, and aggregating over periods, aggregate purchases x 1 = 4 4q ; x 2 = 4 4q q 1 1 3p 1 4q p 1 If we used "consumer-week" weighted prices, i.e. p 1 1 and 1 4 p p2 1 = p p 4 1, we would recover the long run e ects. However, if we use unit value, i.e. revenue divided by quantity, prices will be p 1 1 and (1 w)p wp 2 1 = p (1 w)p 1, where w = 3(q1 1 p 1 )=(4q1 1 3p 1 ) > 3=4: 27

28 Because of aggregation the unit-value price will generate a price that is too low in the second period. Using this price (even if we know the true slopes of demand) will yield estimates that are biased towards zero. For own price e ects this is the right direction, giving the impression that the problem of demand dynamics has been attenuated. For cross price e ects this is the wrong direction. 28