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A Simple Model of Demand Anticipation
September 12, 2009
Abstract In the presence of intertemporal substitution, static demand estimation yields bi- ased estimates and fails to recover long run price responses. Our goal is to present a computationally simple way to estimate dynamic demand using aggregate data. Pre- vious work on demand dynamics is computationally intensive and relies on (hard to obtain) household level data. We estimate the model using store level data on soft drinks and Önd: (i) a disparity between static and long run estimates of price re- sponses, and (ii) heterogeneity consistent with sales being driven by discrimination motives. The modelís simplicity allows us to compute mark-ups implied by dynamic pricing.
1 Introduction
Demand estimation plays a key role in many applied Öelds. 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 evidence that product durability or storability may generate dynamics, which could contaminate estimates. Focusing on storable products, a number of papers (Erdem, Imai and Keane, 2003, and Hendel and Nevo, 2006b) use household level data to structurally estimate consumer inventory models and simulate long run price responses. The computational burden and (household level) data requirement have limited the use of these dynamic demand models. We propose an alternative model to incorporate demand dynamics. Our goal is to present a computationally simple way to estimate dynamic demand for storable products, or test for its presence, using aggregate, rather than household level, data. In many studies dynamics are not the essence. A test for the presence of dynamics may help rule them out. If dynamics are present their impact can be quantiÖed by comparing static estimates to estimates from our model. The model allows us 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. The latter 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. A key to the simplicity of the model is in the storage technology: consumers are assumed to be able to store for a pre-speciÖed number of periods. This assumption simpliÖes the solution to the consumerís problem. The intuition of the model can best be demonstrated by a simple example. Suppose there is a single variety of a product with (1) prices that take on two values: a sale and a non-sale price; and (2) some consumers can store the product for one period (while others cannot store). Given these assumptions the model deÖnes four states depending on the current and previous period price. The states determine whether there are purchases for storage or not, and whether consumption comes out of storage. Thus, (^1) See, for example, Berry, Levinsohn, and Pakes (1995, 1999), Goldberg (1995), Hausman, Leonard and Zona (1994).
than the static estimates. The order of magnitude of the bias is comparable to what Hendel and Nevo (2006b) Önd when they estimate a dynamic inventory model for laundry detergents. We discuss alternative approaches in Section 7. Alternatives to dealing with dynamics include aggregating the data from weekly to monthly and quarterly frequency, or approxi- mating the missing inventory by including lagged prices/quantities (and computing long run e§ects using impulse response). We show these alternatives perform poorly, yielding negative cross price e§ects. We argue that the alternative methods also require a model to translate the estimated coe¢ cients into preferences. Another advantage of the simplicity of the model is to make the supply side tractable. In principle, the presence of demand dynamics makes the pricing problem quite di¢ cult to solve. Especially so when there are multiple products sold by di§erent sellers. In contrast, the demand framework we propose leads to a simple solution to the sellersípricing problem. Studying the supply side is interesting in its own right, but it is particularly important in many applications. Demand elasticities are typically used in conjunction with static Örst order conditions to infer market power. Demand dynamics render static Örst order conditions irrelevant. A supply framework consistent with demand dynamics is needed. We show that sellersíoptimal behavior can still be characterized by Örst order conditions. Interestingly, the demand estimates show that consumers who store are signiÖcantly more price sensitive than non-storers, which is consistent with price discrimination being the motive behind sales. We use the estimated demand elasticities and the dynamic Örst order conditions to infer markups. Section 2 presents motivating facts and reviews the literature. The model is presented in Section 3 and the estimation in Section 4. Section 5 presents an application to soft drinks. Extensions of the model are presented in Section 6. Section 7 discusses alternative approaches.
2 Evidence of Demand Accumulation
2.1 Motivating Facts
Several papers (discussed in the next sub-section) have documented demand dynamics. We Örst look at typical scanner data for direct evidence on the relevance of intertemporal demand e§ects. Figure 1 shows the price of a 2-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 for future consumption.
.^
1
1 price
(^0 10 20) week 30 40 50
Figure 1: A typical pricing pattern
Quantity purchased shows evidence of demand accumulation. Table 1 displays the quan- tity 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 signiÖcantly higher (623 versus 227, or 2.75 times more). More importantly, the quantity sold is lower if a sale was held in the previous week (399 versus 465, or 15 percent lower).
3 The Model
In order to convey the main ideas we start with the simplest model of product di§erentiation with storage. We later show the model can be generalized in several dimensions. For exam- ple, the proposed estimation can be applied to more áexible demand systems, e.g., Berry, Levinsohn and Pakes (1995).
3.1 The Main Assumptions
Assume quadratic preferences:
U (q; m) = Aq q^0 Bq + m (1)
where q = [q 1 ; q 2 ; :::; qN ] is the vector of quantities consumed of the di§erent varieties of the product (colas in our application) and m is the outside good. Absent storage, quadratic preferences lead to a linear demand system:
qit (p) = (^) ipti +
X
j
ij ptj (2)
In a multi-period set up with storage, consumers can anticipate purchases for future consumption. We make the following assumptions: A1: two prices: sale and non-sale (pS^ < pN^ ) The transition between sale and non-sale periods could be random or deterministic. In the next section we discuss exactly how consumers form expectations regarding future prices. Assumption A1 is stronger than needed. A key for our method to work is to be able to deÖne a sale price, i.e., a price at which (some) consumers store for future consumption. When prices take on two values the deÖnition of a sale is immediate. There can be several sale prices (as well as non-sale ones), all we need is to correctly deÖne periods at which consumers store. Namely, periods of demand anticipation. Next we make assumptions on the storage technology. A2: storage is free A3: inventory lasts for T periods (depreciates afterwards)
Initially we will focus on the T = 1 case, which makes the analysis more transparent. 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. In Section 3.4 we show how to modify the estimation for T > 1. Notice the data can guide what the relevant T is, for example, by examining the e§ect of lagged sales on the quantity purchased. A4: a proportion! of consumers do not store. We assume that a proportion of customers do not have access to the storing technology. This assumption helps us explain why we see purchases in a non-sale period after a sale. If everyone stored in the previous period our model would predict no purchases. With two prices this assumption is not very restrictive, but as we add more prices it will have bite since it assumes that the fraction of non-storers does not change with price. In Section 3.4 we discuss the assumptions, their limitations, and possible generalizations.
3.2 Purchasing Patterns
We now characterize consumer behavior. To ease exposition we ignore discounting. The application involves weekly data, and therefore discounting does not play a big role. Consumers who store, purchase for storage at pS^ , and never store at pN^ : When they store, they do so for one period. Thus, to predict consumer behavior we only need to deÖne 4 events (or types of periods): a sale preceded by a sale (SS ), a sale preceded by a non-sale (NS ), a non-sale preceded by a sale (SN ), and two non-sale periods (NN ). We assume for now perfect price foresight, and later discuss (in section 6) behavior under rational price expectations. Assume, for a moment, that only product i is stored. Given Assumptions A1-A4 and perfect foresight, product i aggregate purchases, xi(p); are:
xi(pt) =
qi(pti; pt i) !qi(pti; pt i) !qi(pti; pt i) + (1 !)(qi(pti; pt i) + qi(pti; pt +1i )) !qi(pti; pt i) + (1 !)qi(pti; pt +1i )
if
pt i ^1 = pti = pNi pSi = pt i 1 < pti = pNi pNi = pt i 1 > pti = pSi pt i ^1 = pti = pSi (3)
who store is qi(pti; pet i) instead of qi(pti; pt i), where pet i is the e§ective price, in this case pt i 1. A similar adjustment is needed in all other states. An important implication is that current prices of other products are the wrong prices to control for in the estimation. Controlling for current price generates a bias in the estimated cross price e§ect.
3.3 Predicted Biases
We now explore the modelís implications for biases that may arise from neglecting dynamics. Suppose we observe several price regimes, with constant prices within each regime. Since prices are constant within each regime, there is no reason to store and therefore the di§erence in purchases (and consumption) across regimes helps recover preference parameters and : 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 1ís price decreases during the second period: p^11 = p^31 = pN 1 and pS 1 = p^21 < p^11 ; while product 2ís price remains constant at p 2. Denote by p 1 = p^21 p^11 = pS 1 pN 1 < 0 : Since storing is free, consumers (who store) will purchase all of period 3 consumption, q 1 (pS 1 ; p 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^21 ; p^31 g = pS 1 : The consumer can time her purchases to minimize expenses. In this case, period 3 consumption is determined by p^21 : Quantities purchased by a storing consumer over the three periods (according to equation
- are:
t = 1 t = 2 t = 3 pt 1 = pN^ pS^ pN x 1 = q^11 2(q 11 p 1 ) 0 x 2 = q^12 q^12 + p 1 q 21 + p 1
where q^11 = q 1 (pN 1 ; p 2 ) and q 21 = q 2 (pS 1 ; p 2 ). Should we estimate demand statically we would estimate the following price e§ects: Own price k 2 + 3 q 11 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. The over estimation of own price e§ects is caused by attributing the response to a tem- porary 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. While it is natural to expect an overestimation of own price responses, the impact of dynamics on cross prices responses is more delicate. Previous work has documented the e§ect on cross price responses, but did not show the expected bias theoretically. The model predicts cross price e§ects are understated. 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 an increase in purchases of product 2 , as lack of cross price reactions.
3.4 Discussion of the Main Assumptions
The model greatly simpliÖes the consumer problem. We now discuss the key assumptions that deliver the simplicity. A good point of comparison is the dynamic inventory model where the consumer maximizes the discounted expected áow of utility from consumption minus the price of the product and the cost of holding inventory. If we assume that prices follow a Örst order Markov process then the state variables are current prices and a vector of inventories.^3 We will refer to this model as the (standard) inventory model. Assumption A1 simpliÖes the determinants of demand anticipation. If current price is high then future prices will be (weakly) lower and there is no incentive to store. If instead current price is low then future prices will be (weakly) higher and consumers buy for inventory. The two-price support assumption is restrictive, and stronger than needed. What is really needed is that the periods of accumulation are well deÖned. The same logic (^3) The state space can be simpliÖed to include a smaller number of inventories. See Erdem, Imai and Keane (2003) or Hendel and Nevo (2006b).
The shortcut is based on equation 3 and predicted purchases for T = 2 (described in the Appendix). Some states (like SN and SS) are una§ected by longer histories, while others are. We are back to 4 states, knowing that predictions in states N N and N S are not exact. The shortcut seems to perform reasonably well. Assumption A4 holds the fraction of storers constant. If Assumption A1 holds in the sample this is not restrictive for the purpose of estimation, but if the price takes on additional values this assumption has some bite. Even if in the data there are only two prices, this assumption might be problematic for counterfactuals. The fraction of consumers who store might be a function of price. The current model does not allow it, but in principle we could extend the model to allow! to vary with prices. Section 6 discusses extensions like discrete choice models instead of linear demand and rational expectations about future prices.
3.5 Simulation Results
The model can be taken literally or as an approximation to a standard inventory model. In this section we apply the proposed estimation to data generated from an inventory model. The goal is to assess how well the proposed estimation does in recovering preferences when applied to data generated from an inventory model. The data was generated by computing the optimal (dynamic) behavior of a consumer with quadratic preference, linear inventory costs, and facing stochastic prices. We assume there are two prices, sales occur with probability 0.2, half (0.5) of the consumers store according to the standard inventory model while the other half have a static demand, and the discount factor is 0.995. The preference parameters used to generate the data imply = 4. We compute the consumerís value function and optimal policy, and then simulate purchases for 100, 200, and 500 periods for di§erent storage cost levels. We perform 1,000 repetitions of each estimation. Figure 2 shows consumption and storage predicted by the inventory model as a function of the storage parameter. The di§erent storage costs trace situations where storage is quite high (over twice the áow consumption) to no storage.
Simulated Consumption and Storage
0
2
4
6
8
10
12
14
0.08 0.11 0.14 0.17 0.2 0.23 0.26 0.29 0.32 0.35 0.38 0.41 0.44 0.47 0. Storage Cost
Storage Consumption
Figure 2: Optimal Dynamic Behavior as a Function of Storage Costs
Figure 3 displays the percent bias in the price coe¢ cient for OLS estimates and our Öx assuming T = 1 and T = 2. For moderate levels of anticipated purchases the proposed Öx does well. On the other hand, OLS shows substantial bias, about 60%, even for modest levels of storage. For very low storage costs all estimates overstate price responses. However, while the T = 1 Öx is o§ the mark by 40% the OLS estimate is over 160% o§. As expected the T = 2 Öx does better than the T = 1 Öx for very low storage costs.
Table 2: Monte Carlo Simulations Mean MSE Simulated Data OLS T = 1 T = 2 OLS T = 1 T = 2 c Consumption Storage N= 0.08 4.80 16.20 10.57 5.82 4.73 43.98 3.59 0. 0.17 4.19 9.31 8.36 4.89 4.07 19.26 0.92 0. 0.29 3.64 5.80 6.50 4.08 4.30 6.39 0.14 0. 0.38 3.54 5.05 6.16 3.91 4.37 4.75 0.15 0. 0.50 2.97 0 3.99 4.00 3.99 0.05 0.20 0. N= 0.08 4.80 16.20 10.50 5.71 4.68 42.59 3.02 0. 0.17 4.19 9.31 8.35 4.87 4.07 19.06 0.83 0. 0.29 3.64 5.80 6.50 4.07 4.29 6.32 0.07 0. 0.38 3.54 5.05 6.14 3.90 4.35 4.63 0.08 0. 0.50 2.97 0 4.00 4.01 3.99 0.03 0.09 0. N= 0.08 4.80 16.20 10.48 5.66 4.66 42.08 2.79 0. 0.17 4.19 9.31 8.32 4.84 4.05 18.69 0.73 0. 0.29 3.64 5.80 6.47 4.05 4.28 6.13 0.03 0. 0.38 3.54 5.05 6.13 3.89 4.33 4.52 0.04 0. 0.50 2.97 0 4.00 4.00 4.00 0.01 0.04 0. Note: Means and mean squared error of estimates of the slope coe¢ cient, beta, computed based on 1,000 repetitions of each estimation. The data was generated using with a slope parameter of 4. The storage level is the average storage conditional on being positive, as oppose to Figure 2 that shows the unconditional average storage.
4 IdentiÖcation and Estimation
4.1 How Do We Recover Preferences?
Before presenting the estimation we discuss intuitively how the model helps recover prefer- ences. We o§er two approaches, both are part of the full estimation, but discussing them
separately helps clarify what variation in the data identiÖes the parameters. The Örst ap- proach is based on events without storage, while the second approach imputes storage and purges it from purchases. For simplicity, assume a single product (and T = 1) in which equation 3 suggests that during N N and SS demand is given by q(pt), while during SN demand is scaled down by! and during N S it is scaled up by 2 !. This suggests two di§erent ways to recover the modelís parameters from the data. We will refer to the Örst as "timing" restrictions. According to the model during sale periods that follow a sale (event SS) purchases equal consumption: x(p) = q(p): Basically, after purchasing for storage, the pantry is Ölled, consumers (whether they are a storer or not) purchase for a single consumption event. Since both N N and SS events involve purchases dictated by q(p) we can rely on them to estimate preferences. Price variation across these states, across di§erent stores during these states or within these states (if there are more than 2 prices), can identify long-run responses. A di§erent way to map purchases into preferences is to take advantage of the data from all periods but use the model to adjust the predicted purchases to account for storage. We will refer to the additional restrictions as "accounting" restrictions. For example, during SN we need to scale down demand because only non-storers purchase, while during N S we have to scale up purchases due to storing. This approach is more e¢ cient, since it uses all the data, but it also adds additional parameters and imposes Assumption A4. We note that the model is over-identiÖed. For example, we can recover! by looking at the ratio of purchases during SN to purchases during N N , or by looking at 2 minus the ratio of purchases during N S to purchases during SS: In principle we can use the additional degrees of over-identiÖcation to enrich the model somewhat. To demonstrate how the di§erent restrictions work we can use the numbers in Table 1 to recover the demand parameters. As a benchmark, we note that the static estimate of the slope coe¢ cient is bStatic = xS ^ px N = 6230 : 4227 = 988, where p = 0: 4 is the price di§erence between sale and non-sale periods. Estimating the same slope using only the timing restrictions yields bT ime = x pSSSS^ xpN NN N = 5320 : 4248 = 710. Since the model is over-identiÖed, there are several ways to impose the accounting re- strictions. One way is to use only the information from N N , SS, and SN and recover ! = xN N^ =xSN^ = 0: 8 and b^ = !xSS !^ px SN= 708. Another way is to use the information from N N , SS, and N S, which implies! = 2 xN S^ =xSS^ = 0: 57 and b^ = xN S (2^ (2! )!)px N N = 713: More
5.1 Data
The data we use was collected by Nielsen and it includes store-level weekly observations of prices and quantity sold. The data set includes information at 729 stores that belong to 8 di§erent chains throughout the Northeast, for the 52 weeks of 2004. We focus on 2-liter bottles of Coke, Pepsi and store brands, which have a combined market share of over 95 percent of the market. There is substantial variation in prices over time and across chains. A full set of week dummy variables explains approximately 20 percent of the variation in the price in either Coke or Pepsi, while a full set of chain dummy variables explains less than 12 percent of the variation.^5 On the other hand, a set of chain-week dummy variables explains roughly 80 percent of the variation in price. Suggesting similarity in pricing across stores of the same chain (in a given week), but prices across chains look quite di§erent. As a Örst approximation it seems that all chains charge a single price each week. However, three of the chains appear to deÖne the week di§erently than Nielsen. This results in a change in price mid week, which implies that in many weeks we do not observe the actual price charged just a quantity weighted average. In principle we could try to impute the missing prices. Since this is orthogonal to our main point we drop these chains. We need a deÖnition of a sale, or more precisely, we need to identify periods of advance purchases. Figure 4 displays the distribution of the price of Coke in the Öve chains we examine below. The distribution seems to have a break at a price of one dollar, which we use as the threshold to deÖne a sale. Any price below a dollar is considered a sale, namely, a price at which storers purchase for future consumption. This is an arbitrary deÖnition. A more áexible deÖnition may allow for chain speciÖc thresholds, or perhaps moving thresholds over time. For the moment we prefer to err on the side of simplicity. Using this deÖnition we Önd that approximately 30 (36) percent of the observations are deÖned as a sale for Coke (Pepsi). Interestingly, sales are somewhat asynchronized with only 7 percent of the observations exhibiting both Pepsi and Coke on sale (compared to a 10.5 percent predicted if the sales were independent). (^5) These statistics are based on the whole sample, while the numbers in Table 2 below are based on only Öve chains as we explain next.
0
10
20
30
Percent
.5 (^1) Price of Coke 1.5 2
Distribution of the Price of Coke
Figure 4: The Distribution of the Price of Coke
For the analysis below we use 24,674 observations from Öve chains. The descriptive statistics for the key variables are presented in Table 3.
Table 3: Descriptive Statistics % of variance explained by: Variable Mean Std chain week chain-week QCoke 446.2 553.2 5.6 20.4 52. QP epsi 446.0 597.8 2.8 24.4 46. PCoke 1.25 0.25 7.1 29.7 79. PP epsi 1.19 0.23 7.5 30.7 79. Coke Sale 0.30 0.46 6.4 30.0 86. Pepsi Sale 0.36 0.48 9.3 29.2 89.
Note: Based on 24,674 observations for Öve chains, as explianed in the text. As sale is deÖned as any price below one dollar.
