Approximating the Cost-of-Living Index for a Storable Good Matthew Osborne∗ November, 2014

Abstract This paper estimates a cost of living index (COLI) using a dynamic structural model for a storable product category with periodic promotions, and examines what types of simple price indexes approximate it well. A price index derived from the COLI averages about 79% over the course of one year. Standard fixed weight indexes such as the Laspeyres and Geometric indexes provide poor approximations to the structural index, averaging as much as 15% higher. A simpler index proposed by Feenstra and Shapiro (2003) provides a close approximation, and is within 3% of the average structural index.

∗

University of Toronto, Institute for Management and Innovation and School of Management. Email:

[email protected]. I thank Christoph Bauner, Andrew Ching, John Greenlees, Masakazu Ishihara, Marshall Reinsdorf, Emily Wang and Lionel Wilner for providing valuable comments on earlier drafts of the paper. I also thank seminar participants at the 2013 International Industrial Organization Conference and the Bureau of Labor Statistics for comments. A previous version of this paper circulated under the title “Estimation of a Cost-of-Living Index for a Storable Good Using a Dynamic Structural Model.”

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1

Introduction

The Consumer Price Index (CPI) tracks the price movements of many different household food products and is widely used in economic measurement and policy. As an example, when constructing real measures of food purchased for consumption at home, the Bureau of Economic Analysis deflates revenues in a product category by the corresponding CPI.1 Using accurate price indexes is therefore a topic that is of great importance for accurate measurement of economic output. The increasing availability of household and storelevel scanner data, as well as the development of techniques that allow for the estimation of sophisticated dynamic models of household behavior, can facilitate the improvement and accuracy of price indexes. This paper addresses two questions related to price indexes. First, how large are biases in the standard fixed base price indexes used by statistical agencies when consumers store products for future consumption? Second, to the extent such biases exist, is there a way to correct them using an alternative index that is simple to calculate? To answer these questions one needs to compute the price index that correctly accounts for changes in the cost of living when consumers anticipate periodic promotions and stockpile. The approach taken in this paper is to estimate a dynamic structural model of consumer stockpiling behavior for a frequently-purchased product category which is storable, and to use the model to construct a cost-of-living index (COLI) for the category. A COLI measures how much one would need to compensate consumers for price changes over time, relative to some base period. Although standard price indexes are not constructed as COLIs, according to the Bureau of Labor Statistics (Bureau of Labor Statistics 2007) the measurement objective underlying the CPI is that of a COLI, and the CPI is often used as such in practice.2 In order to construct a COLI, one needs to be able to compute consumer utility. Structural econometric methods are ideal for 1

See Chapter 5 of the NIPA handbook located at http://www.bea.gov/national/pdf/ch5%20PCEforposting.pdf,

page 5-13. Measured in 2005 dollars, food purchased for off premises consumption totaled about 106 billion dollars for the first quarter of 2012 (source: NIPA tables at http://www.bea.gov/national/txt/dgpa.txt, Table 1.5.3.). 2

Bureau of Labor Statistics (2007), pg 2 states “The concept of COLI provides the CPI’s measurement

objective.”

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this, because they produce estimates of the parameters of a consumer’s utility function. With the estimated parameters in hand, constructing a COLI is straightforward, and a simple transformation of the COLI allows me to compare it to a standard price index. I find that the differences between the standard indexes and the structural model-derived price index are as large as 15%. Dynamic consumer behavior can cause a price index derived from a COLI to diverge from standard price indexes.3 In this paper, the dynamics I consider are driven by consumer stockpiling of a storable product. For these types of products, observed prices stay at a high level most of the time, and occasionally drop to a low, promotional price for a short period of time. When the product is on promotion, consumers stockpile it for future consumption. This type of behavior is problematic when one tries to compute price indexes because it creates a bias akin to substitution bias: when consumers are strategic they anticipate periodic promotions, and can consume out of inventory when prices are high. To develop intuition for why standard indexes may not perform well in the presence of intertemporal substitution, consider a market for a storable product category that is observed for three periods. Suppose that prices are high in period 1, they drop in period 2, and in period 3 the prices of all products return to their period 1 values. Standard fixed base prices indexes such as the Laspeyres index accurately reflect changes in the cost of living when households solve a static problem.4 From the perspective of cost of living changes, a price index is measured as a consumer’s expenditure function in period t evaluated at the prices in period t and the consumer’s utility in period 1, divided by the expenditure function evaluated at period 1 prices and at period 1 utility. In our simple market, the static price index will be below 1 in period 2, since consumers observe lower prices and need less income to achieve period 1 utility, and it will be exactly 1 in period 3 since prices are unchanged from the period 1 level. Now contrast the static index with a price index that reflects changes in the cost 3

See Gowrisankaran and Rysman (2012) or Reis (2009) for examples in different applications.

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The Laspeyres index will accurately measure cost of living changes under the assumption that household

optimization is static and preferences are Leontief. The BLS uses a fixed base Geometric index rather than a Laspeyres since the Laspeyres tends to overstate the cost of living. As I demonstrate below both indexes are biased upwards in the presence of intertemporal substitution.

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of living experienced by consumers who stockpile, which I will call the dynamic index. Suppose in period 1 all consumers start with no inventory, and no consumers purchase more than is needed for current consumption in that period. In period 2, the dynamic index will be less than 1, since prices are lower than period 1. Additionally, in period 2 strategic consumers will build up inventory. The presence of inventories will imply that the period 3 dynamic index will be lower than 1, the value of the static index in period 3. The reason for this is that in period 3, consumers will receive some utility from consumption out of inventory; As a result, the level of expenditures needed in period 3 to achieve period 1 utility will be lower than the static case. To quantify the magnitude of the divergence between the COLI and standard price indexes, I specify and estimate a dynamic structural model of consumer stockpiling behavior using one year of household level scanner data of canned tuna purchases, recovering the parameters underlying consumer utility. I compute a COLI from my structural model by calculating a sequence of aggregate taxes or subsidies that keep average flow utility equal to flow utility in a base period, which I choose to be the first week of the data. I then turn the COLI into a price index by adding the average base period price to the COLI, and dividing by the result by average base period price.5 During the course of the year, canned tuna prices fall. The index derived from my structural model averages 79% over the course of the one-year sample, consistent with falling prices. Consistent with the intuition in the previous paragraph, a fixed-base Laspeyres index is much higher, averaging about 96% over the course of the year.6 A fixed base Geometric index is biased upwards by a similar magnitude. The second question this paper addresses is whether it is possible to construct a simple index that approximates the structural index reasonably well. This question is motivated by the fact that the structural approach has two drawbacks from the perspective of practitioners in statistical agencies: First, estimation of the structural model is computationally intensive, and is only feasible to implement for a few product cate5

This procedure for transforming a COLI into a price index follows Gowrisankaran and Rysman (2012).

Since my COLI is by definition 0 in the base period, the price index derived from the COLI will be 1 in the base period. 6

I construct indexes at the weekly level.

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gories. Statistical agencies must construct indexes for many products. Second, statistical agencies highly value the use of techniques that are transparent to non-economists. The structural approach to computing a price index is potentially more accurate and consistent with more realistic modeling assumptions than simpler approaches; however, since it relies on a more complicated consumer model and more sophisticated estimation techniques it is less transparent to users who are not economists. A simple index that potentially can address this issue is a COLI proposed by Feenstra and Shapiro (2003) (hereby abbreviated FS). The FS COLI was originally proposed as to address biases in standard indexes that might arise from intertemporal substitution, and it will correctly measure the cost of living for forward-looking consumers under three assumptions: First, that consumers are homogeneous, second, that consumers know the path of future prices (as well as other exogenous variables) exactly, and third, that consumer purchases are made under a planning horizon that is known to the researcher. On the surface, these assumptions seem quite restrictive, and the dynamic model I estimate relaxes them: I model unobserved heterogeneity in tastes and price sensitivities, I assume consumers know the distribution of future prices, but not the exact values, and the model is couched in a standard infinite horizon dynamic programming framework. However, I find that on average, the the FS COLI is within about 3% of my dynamic index, significantly closer than the fixed weight indexes. This is a useful finding because it suggests that although the assumptions behind the FS COLI seem restrictive, in practice they produce a COLI that provides a reasonable approximation to the dynamic COLI. Note that one could not verify this without estimating the structural model and comparing it to the FS COLI . By construction, the FS COLI is a measure of the average change in the cost of living over a consumer’s planning horizon, relative to some base period. The FS COLI is closer to the average of the dynamic index than the average Laspeyres index because of the way that it assigns weights to promotional and nonpromotional periods when averaging price relatives over time. To see this, first consider the average of a weekly Laspeyres index over the course of a long period of time, such as a year. Such an average will equally weight the index value in each week; if in the base period prices are high (which they likely will be) then the index will be close to 1 in most weeks, and hence the average Laspeyres will also be close to 1. The dynamic COLI will have a yearly average

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that is lower than the Laspeyres due to the fact that in post-promotional periods, the index will be below the Laspeyres due to build-up of prior inventory.7 The FS COLI is a measure of the average change in the cost of living over the predefined planning horizon, and is defined as a weighted geometric average of the ratio of a product’s price in period t divided by its base period price. Importantly, the average is taken over all time periods in the planning horizon, as well as all products in the sample. The weight is the expenditure for a product in period t divided by total expenditure for all products over the entire planning horizon. Since in many storable goods most expenditures occurs during promotions, in the FS COLI promotional periods will receive a disproportionately high weight as compared to periods where prices are high. Note that a drawback to the FS COLI is that it cannot be computed for a short period of time (such as a week), like the dynamic index can: it can only be computed for the planning horizon. As a final note, although I only investigate the comparison between the FS COLI and the dynamic index for a single product category, I would expect that for most storable products the FS COLI should come closer to the structural index than the fixed-base indexes, for the reasons outlined above. In addition to evaluating the FS COLI, I also investigate how close the dynamic index is to other indexes that allow expenditure weights to vary over time. A fixed base Tornqvist index is closer to the dynamic index than the Laspeyres or Geometric, averaging about 89% over the sample period. Chained indexes also allow for weights to vary over time, however I find that these indexes do not do a good job of approximating the true COLI: a chained Laspeyres averages 219%, whereas a chained Tornqvist averages 61%. The chained Laspeyres index is upwards-biased due to it failing the “time reversal” test.8 The chained Tornqvist index has weights that average slightly less then 1, but 7

Note that this argument is dependent on what the distribution of inventory is in the base period. If it

is high then the index will be above 1 in most periods. However, since promotions occur infrequently and inventory will run down after promotions, it is reasonable to expect that aggregate inventories would be low from any randomly selected period. 8

If prices in period t and period t + 2 are p, and prices are equal to p˜ in period t + 1, a price index satisfies

the time reversal test if a two-period chained index between periods t, t + 1 and t + 2, P (p, p˜)P (˜ p, p) is 1. It can be shown that this index is bigger than or equal to 1 for the chained Laspeyres, which results in an upward bias.

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when this is cumulated the overall index drops sharply over time. My results also appear to be robust to the choice of base period. To check this, I replicate the exercise above for two different base periods: the first month of the sample and the year prior to the estimation sample.9 For both these cases I find that fixed base indexes are still higher than my dynamic index, but the Tornqvist index is on average closer to my dynamic index - the average difference between the indexes is less than 2%. The Tornqvist index is likely close to the dynamic index due to its similarity to the FS index: both indexes have a similar functional form. However, from a theoretical perspective the FS index may be preferred since it can be motivated by dynamic behavior, while the Tornqvist index correctly measures the cost of living under static optimization. The rest of the paper proceeds as follows. Section 2 describes related literature. Section 3 introduces and describes the data. Section 4 provides some reduced form evidence that stockpiling behavior is important in my data. Sections 5 and 6 describe the stockpiling model, and how I model consumer price expectations. Section 7 describes the estimation technique, Section 8 provides an overview of identification, and Section 9 describes the estimation results. Section 10 describes the dynamic price index and compares the estimated index to alternatives.

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Overview of Prior Literature

The idea that intertemporal substitution can impact price indexes has been a subject of recent research. One example is van der Grient and de Haan (2010), who propose using scanner data to construct the Dutch CPI. Interestingly, they also document periodic promotions and purchasing patterns consistent with stockpiling for many products, and argue that standard indexes will be biased due to incorrect weighting. To correct the problem, they propose to construct the price index of an individual product using an unweighted geometric mean. Concurrent work by Sillard and Wilner (2014) constructs a price index for consumer goods in France that adjusts for intertemporal substitution. Consistent with my own findings, their paper finds that Laspeyres price 9

I observe price and quantity data for the year prior to the estimation sample, but data on household store

visits which is necessary for estimation is missing. Thus that year of data is not used in estimation.

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indexes overstate average price growth relative to an adjusted index. Their adjusted index is based on a structural model of consumer behavior that is fit to aggregate purchase data. To implement their approach, simplified modeling assumptions are required: representative consumers, and perfect foresight about future prices. Additionally, the driver of intertemporal substitution in their paper is different than mine. In Sillard and Wilner (2014), consumers substitute purchases across time in the sense that if a product category is expensive in some period, they will substitute to a different category. In work, intertemporal substitution is driven by inventory behavior. There is no storage of products in Sillard and Wilner (2014) to keep the aggregate model tractable. Additionally, recently there has been increasing interest in using structural models to impute COLIs. In concurrent work, Wang (2013) uses a dynamic stockpiling model to quantify the welfare impact of adopting a dynamic COLI instead of the Consumer Price Index to adjust Supplemental Nutrition Assistance Program payments. The paper finds that such an adjustment would likely result in harm to low-income households, as lowincome households tend to benefit less from stockpiling than a representative household would. Others include Handbury (2013), who uses a discrete choice model to quantify the impact of accounting for non-homotheticity on regional price indexes, and Nevo (2003) who uses the estimates of a differentiated products demand to assess the impact of quality changes on price indexes. In the context of durable goods, Gowrisankaran and Rysman (2012) construct a cost-of-living index (COLI) using aggregate scanner data and the estimated parameters of a dynamic structural model of consumer behavior. There are some key differences between my work and that of Gowrisankaran and Rysman (2012). First, since I examine the canned tuna category my implications are likely to hold in other product categories where repeated purchases occur, and the product is storable. Gowrisankaran and Rysman (2012) examine durable goods where a product is likely only purchased once, or very infrequently. Although Gowrisankaran and Rysman (2012) find that their price index differs significantly from a BLS-style index, the dynamics that drive the difference are somewhat different than in my case. In their case, the difference arises due to a “new buyer” problem identified by Aizcorbe (2005): low value consumers enter their market near the end of their sample when prices are low, creating concavity in the price index. In contrast, in a market where stockpiling is important consumers are timing their purchases to coincide with promotions. As I described above, this behavior

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creates a wedge betweeen standard fixed-base price indexes, which weight by base period shares, and my COLI, which will give higher weight to promotional periods. Methodologically my paper incorporates some improvements over prior work involving the structural estimation of stockpiling models. One difference between my work and most prior work is that because I only need to model consumption decisions over two brands, in solving the consumer dynamic programming problem I include inventories of each brand as a state variable (an exception is Sun (2005), who models separate inventories and endogenous consumption in a similar model to the one I propose here). This approach is feasible in my case but in situations where consumers could choose among many different brands some simplifications to the consumer state space would have to be made. If one was willing to assume that consumption was exogenous, rather than endogenous, one could track a quality-weighted index of inventory, following Erdem, Imai, and Keane (2003). If one wanted to allow for endogenous consumption (as I do in this paper), one could apply the methods developed in Hendel and Nevo (2006a). If one were willing to assume that consumers look ahead a finite number of periods, rather than solving an infinite horizon problem, then new techniques introduced in Hendel and Nevo (2013) could be used. Because solving my dynamic model involves integrating out unobservables such as consumer heteroegeneity in tastes, price sensititivities and initial inventories, I apply the Bayesian approach of Imai, Jain, and Ching (2009) (IJC). One additional advantage of this approach over the aforementioned approaches, which involve estimation of model parameters by optimization of an objective function such as a likelihood, is that I can feasibly estimate parameters that govern consumer price expectations along with the other model parameters10 Typically, one estimates price expectations before estimating the dynamic model and then uses the estimates as inputs into the model. Technically, the standard errors of the dynamic estimates should be adjusted since they are functions of previously estimated parameters, but this is rarely done in practice (an exception is Hendel and Nevo (2006a)). The two step approach is taken because the processes that govern price expectations are typically functions of many parameters, and maximizing 10

As is standard in the literature, I assume consumer expectations are rational, so that consumer price

expectations coincide with observed prices. This assumption seems reasonable for products like canned tuna that have been in existence for a long time, and where consumers are likely familiar with store pricing.

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the model’s likelihood over a high dimensional parameter space using standard methods such as the nested fixed-point method is computationally infeasible.11 In the nested fixed point approach, one must solve for consumer value functions every time a parameter is changed. Because the IJC approach only involves solving for the value function once, rather than many many times, it can accommodate more parameters.

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The Data Set

The data set is household level Nielsen scanner data on canned tuna purchases from Sioux Falls, SD. I focus the analysis on the two most popular brands of canned tuna, Starkist and Chicken of the Sea, which comprise over 90% of all purchases. Although canned tuna is available in different package sizes, the most popular size by far is the standard 6 ounce can. Thus, for computational simplicity the analysis focuses on households who only purchase the 6 ounce can size. As I show in Table 1, the two brands have roughly equal market shares by volume. Additionally, prices are very similar, although Starkist is slightly more expensive than Chicken of the Sea. Starkist has a lower standard deviation of prices than Chicken of the Sea, indicating it goes on sale less often, and its sales are less deep than those of its competitor. In this paper, promotions are defined as dips in the observed shelf price. Coupons may also be a part of a firm’s promotional strategy; however, in this market there is very little observed coupon use. Coupons are used in less than 10% of purchases. Due to the computational issues that arise due to including coupons, I do not include them in the analysis.12 My data set spans 3 years, 1985 to 1988. The data tracks purchases of canned tuna over this period for a sample of about 1500 households at 19 different stores. When a household makes a purchase at a store, the store’s identity is recorded; however, household store visits where no purchase is made are only recorded for the final 51 weeks of the data. I restrict most of the analysis to that period because to estimate the model of household stockpiling that will be presented below it is necessary to know the price of a product when a store is visited, but no purchase is made. I also remove some households 11

See Rust (1994) and Rust (1987) for an overview of the nested fixed point algorithm.

12

See Osborne (2011) for discussion on including coupons in dynamic structural models.

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from the data where the quality of the data may be suspect. First, households for whom I observe less than 25 store visits during the 51 weeks are removed. Households who were surveyed in the data were given swipe cards that they were required to bring with them on shopping trips, and it is likely that households for whom the frequency of store visits is low were forgetting to use their card, or frequently shopping at a smaller store that was not included in the Nielsen data. Additionally, to keep the model’s state space tractable I limit the sample to households who purchase at most ten cans of tuna in a week. This is not a strong limitation: among household-week observations where a purchase is observed, purchases of 11 cans or more comprise about 0.2%. I also restrict the sample to households who purchase 10 or more cans of tuna over the entire year. The behavior of households who never purchase tuna or who only purchase it very infrequently is unlikely to be well explained by a stockpiling model. After these cuts, I am left with a final sample of about 600 households.13 In the left panel of Figure 1 I plot quantity weighted average prices for my sample on a weekly basis. There is a significant amount of volatility in prices over time - the standard deviation in weekly average prices is roughly 10% of its mean. A downward trend in prices can be seen in the data - a regression of log price on a time trend produces a coefficient that implies prices drop by an average of 0.3% per week. Overall quantities, displayed in the right panel of Figure 1, display a similar amount of volatility. Periodically total quantity sold spikes upwards - these spikes are at least partially driven by consumers purchasing the product on promotion. To make this obvious, I plot the Starkist brand’s weekly prices and quantities for all households over the longer 1985-1988 sample period in the left panel of Figure 2 for a typical store in the data. The price, shown in the top panel, stays relatively flat at around 60 cents for most of the time period, but occasionally it drops significantly for a short period of time. The bottom panel shows the quantity sold, measured in the number of 6 ounce cans. The quantity sold is on average about 100 cans per week, but when promotions occur it jumps significantly to over 300 cans. This behavior is consistent with consumer stockpiling behavior: stores keep the price high most of the time, but recognize that price sensitive consumers will 13

The numbers in Table 1 were computed for all households, rather than the reduced sample. Prices and

shares for the reduced sample are similar to the overall sample.

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run down their inventories. To draw these consumers into the market, a sale eventually occurs. As discussed in the previous section, accounting for this type of behavior when measuring price indexes can be important: when the BLS constructs price indexes, it randomly samples prices at stores across the United States. However, as can be seen from the figure, and as has been pointed out by Erdem, Imai, and Keane (2003), the average offer price will differ from the average accepted price since most purchases occur on promotion. Some evidence that the two brands may be using temporary promotions as a method of competition is shown in the right panel of Figure 2. This figure shows the price of Starkist in black for the same store as the previous figure, and the price of Chicken of the Sea as the red dotted line. Notice that it is rarely the case that both brands go on promotion at the same time. Often when Starkist has a sale, Chicken of the Sea has a promotion soon afterwards, and vice versa. This observation will help to motivate my specification for consumer price expectations, which will include competitor reactions.

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Evidence of Stockpiling Behavior

The time-series patterns of price and quantity in Figure 2, while suggestive of inventory behavior, are not totally conclusive of its presence. Alternative explanations have been put forward to explain periodic sales (such as variation in store inventories). The large spikes in demand may just reflect the overall price sensitivity of consumers in the market; stockpiling is not necessary to explain them. To offer more evidence of stockpiling, I run a regression of the total quantity a household purchases during a given week on a measure of the household’s inventory, the price, a dummy variable for whether a sale occurs, the interaction of that dummy variable with price, and feature and display variables.14 In reality, a household’s inventory is unobserved, so I estimate household inventories by assuming that household consumption rates are constant. One can construct an estimate of the household’s inventory during a given week as the sum of quantity purchased prior to that time, minus total consumption, plus initial inventories. I estimate a household’s consumption rate by dividing the total quantity consumed by the number of weeks the 14

This type of regression analysis follows similar work in Hendel and Nevo (2006b).

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household is observed. Initial inventories are also unobserved, so a household fixed effect is included to control for them. Quantity is measured in ounces, while inventory is measured in ounces divided by 100. Price is measured in dollars per ounce. Table 2 shows the estimates of this regression. Inventory is negative and significant, which suggests that when a household’s inventory increases, the quantity they purchase decreases. Further, when a product goes on sale, households purchase more of the product, and have a larger price coefficient, which is consistent with stockpiling. A sale is a dummy variable that is one when the price is observed to be 5% or more lower than the modal price of the product in the store. The regression also includes store and product dummy variables.

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Model of Consumer Stockpiling

Consumers in the model make purchase and consumption decisions over two classes of goods: canned tuna and a composite good. These decisions are made to maximize the consumer’s expected discounted utility, subject to a lifetime budget constraint that the present discounted value of all purchases are less than or equal to the present discounted value of income. A period in the stockpiling model presented in this paper corresponds to a week. I assume that each week can be split up into 3 sub-periods, which are shown in Figure 3. In the first subperiod, the consumer observes prices and realizations of a vector of errors, where there is an error corresponding to each possible purchase decision consumer i can make in period t, qit . Purchase decisions are indexed by q. In the second subperiod, the consumer makes a purchase decision. J different brands of the product are available to the consumer, and consumers can purchase up to Nu units of canned tuna every period. Formally, consumer i’s purchase decision in period t is a J-vector of quantities of canned P tuna, xit = (x1it , ..., xJit ), such that 0 ≤ Jj=1 xjit ≤ Nu , as well as an amount of the composite good zit . Note that the dimensionality of the purchase error corresponds to the dimensionality of the number of purchase options. In the final subperiod, a consumer makes an optimal consumption decision conditional on her purchase decision and her inventory at the beginning of week t. Consumption is expressed as a vector cit = (c1it , ..., cJit ). To facilitate estimation of the model three simplifying assumptions are made about

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consumption and inventory. First, I assume that consumption is integral, that is, consumers cannot eat fractions of a unit in a period and store the rest for use in a future period. This is a reasonable assumption for canned tuna, since contents of a tuna will spoil before the week is over if only part of a can is used. Note that because consumption is integral, and because purchases are also integral, inventory can also be expressed as a J-dimensional integer vector ιit = (ι1it , ..., ιT it ). In order to solve the consumer problem outlined below it will be necessary to put an upper bound on the amount of inventory a consumer can hold. The second assumption is that consumers have a maximum storage P space of 2Nu , which means that Jj=1 ιjit ≤ 2Nu . It will be convenient to denote the P consumer’s total inventory as Iit = Jj=1 ιjit . Iit evolves as follows:

Iit = Iit−1 +

J X

xjit −

j=1

J X

cjit

(1)

j=1

The third assumption is that consumers do not buy different brands of canned tuna in a single purchase occasion. This assumption implies that size of the choice set in subperiod 2 (the number of values q can take) is 2Nu + 1; a choice set that included all bundles would be of size (Nu + 1)(Nu + 2)/2. This is a reasonable simplification since in my data there are very few consumers who ever purchase more than a single brand in a single purchase occasion. I now turn to the consumer’s utility and the formulation of the consumer dynamic programming problem. I denote pjit as the price per can of canned tuna for brand j in period t that is observed by consumer i. Unlike some storable goods, in my data set I have not been able to find significant evidence of quantity discounts - pjit does not depend on the number of cans purchased. The vector of prices for all brands is denoted as pit . In each period, a consumer’s flow utility is quasilinear in cit and the composite good zit , and takes the following form

U (cit , xit , ιit , zit ) =

J X

γij u(cjit ; β) + αi zit − sc0 Iit − sc1 Iit2

j=1

− CC 1{

J X

(2) xjit > 0} + qit .

j=1

In this function, the utility from consuming a given product is γij u(cjit ; β), where β ≤ 0

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is a parameter that impacts the shape of this subutility. I assume that flow utility for each product is quadratic, so that

u(c; β) = c + βc2 .

(3)

The parameter αi is consumer i’s price sensitivity. The consumer’s inventory holding costs are assumed to be quadratic, with sc0 on the linear term and sc1 on the quadratic. The term CC is a carrying cost, and it represents the disutility of purchasing and carrying the product. One difference between my model and those of Sun (2005) and Hendel and Nevo (2006a) is that I do not model unobserved shocks to the marginal utility of consumption. I experimented with including such shocks and in my application I found it difficult to identify the distribution of the shocks. Although unobserved consumption shocks can intensify stockpiling behavior by generating random stockouts, they are not necessary to produce it; periodic promotions are enough. If consumers observe a low price today, they know it is likely that the price of the product will be high tomorrow. This price variation generates an incentive to stockpile as a means of avoiding paying high prices. In unreported work I have verified this through simulation and solution of the stockpiling model without consumption shocks. Consumers are assumed to be forward-looking with rational expectations, and they discount the future at a discount rate of δ = 0.95.15 To understand the solution to the consumer’s problem within a period it is useful to think about solving it backwards. In the third subperiod, the consumer chooses optimal consumption conditional on her purchase quantity xit . In the second subperiod, the consumer chooses the optimal xit given the error draw it , knowing how much she will consume. There are three state variables which each consumer keeps track of every period. One is the current price vector, pit . Related to this is a state variable that tracks whether a promotion occurs in period t, sit ; this is a vector of length J containing 1 in position j if product j is on promotion, and 0 otherwise. I assume that a product is on promotion if its price is observed to be below some level pj . Promotions evolve over time according to a discrete Markov process, S(sit |pit−1 , sit−1 ). The probability of a promotion occurring today is 15

The discount factor is usually difficult to identify in forward-looking structural models, so it is common

practice to assign it a value (Magnac and Thesmar 2002).

15

a function of whether or not the product was on sale in the previous week, whether its competitors were on sale in the previous week, and what last week’s prices were. Given that sales occur sporadically and are usually short, one would expect that the probability of a sale occurring given no sale last week would be low, and the probability of no sale occurring given a sale last week would be higher. Conditional on a sale occurring today, prices evolve over time according to a Markov process P (pit |pit−1 , sit−1 , sit ). Although it is technically redundant to include s as an argument in P , I feel it eases the exposition to specify the price process conditional on the current promotion state. This is because if a sale occurs, the price distribution is truncated at pj . The last state variable is the consumer’s inventory, ιit . Inventories for individual brands evolve analogously to Equation (1):

ιjit = ιjit−1 + xjit − cjit

(4)

Denote the set of state variables as Σit = (pit , sit , iit ), and denote the vector of utility parameters as θ = (γi1 , ..., γiJ , β, αi , sc0 , sc1 , CC). I follow Chintagunta, Kyriazidou, and Perktold (2001) and assume that markets are complete so that a consumer’s budget constraint in period 0 can be expressed as a single equation ∞ X

  J ∞ X X δt  xijt pijt  + δ t pzit zit = P V Ii ,

t=0

(5)

t=0

j=1

where pzit is the price of the composite good in period t. Following Chintagunta, Kyriazidou, and Perktold (2001), I assume that pzit is known to consumers and is equal to 1 in every period. This implies that the marginal utility of income in every time period is equal to αi , which is standard in discrete choice models. The consumer’s problem is to maximize the present discounted value of utility subject to the budget constraint in equation (5):

max

J ∞ {{xijt }J j=1 ,{cijt }j=1 ,zit }t=0

E0

∞ X

h i ˜ (cit , xit , ιit ) + αi zit , δt U

(6)

t=0

˜ (cit , xit , ιit ) = U (cit , xit , ιit , zit ) − αi zit . E0 is the expectation of future where I define U variables (prices and the error term) in time 0. Because I assume utility is quasilinear

16

in the composite good and consumption of tuna, and because the cost of canned tuna is small relative to a consumer’s total income, the consumer will choose the xit vector optimally and will spend the rest of her income on zit . This implies that the consumer’s problem can be rewritten as

max

J ∞ {{xijt }J j=1 ,{cijt }j=1 ,zit }t=0

E0

∞ X

 ˜ (cit , xit , ιit ) − αi δ t U

t=0

J X

 pjit xjit  + αi E0 P V Ii ,

(7)

j=1

where αi E0 P V Ii is a constant that does not affect the consumer’s purchase decision. This term can be ignored during estimation - because it is constant over choices and over time it drops out of consumer choice probabilities. I also ignore it when I compute the COLI for the same reason. As I will explain further in Section 10, my COLI is a sequence of taxes that will keep an average consumer’s flow utility fixed over time. When computing the COLI I am only concerned with the difference in the consumer’s utility between some base period and period t, so an additive constant will have no impact on the calculation. In the exposition that follows it will be easier to work with the consumer problem in Bellman equation form. In this form the structure of the consumer problem in period t can be written as follows. Denote the value function as V (Σit ; θ). In the third subperiod the consumer chooses an optimal consumption, c∗ , conditional on xit : ˜ (cit , xit , ιit ) + δEP (p |p ) V (Σit+1 (cit ); θ)}]. c∗ (Σit , xit ) = max{U it+1 it cit

(8)

Here we have written the next period state Σit+1 (cit ) explicitly as a function of consumption because next period’s inventory is a function of consumption. The value function will be defined by

˜ (c∗ (Σit , xit ), xit , ιit ) − αi V (Σit ; θ) = E [maxxit { U

J X

pjit xjit +

(9)

j=1

δEP (pit+1 |pit ) V (Σit+1 (c∗ (Σit , xit )); θ) }]. As a final note, I index the utility parameters γij and αi by i, which indicates that they are heterogeneous across the population. I assume that these parameters are lognormally distributed across the population. In the estimation I experimented with letting the other

17

model parameters (β, SC0 , SC1 , and CC) be heterogeneous, but found it was difficult to identify the variances of those parameters.

6

Price Expectations

Consumer price expectations are an important component of a stockpiling model. I do not actually observe price expectations in the data, so I have to make an assumption about how they look. I assume that consumers are rational: their expectations about future prices correspond to the actual evolution of prices. My approach is to estimate a price process from the data, and to assume that process corresponds to how consumers expect future prices to evolve. There are several important features of the observed price process which the estimated price process should capture. First, as can be seen in Figure 2, the price series is relatively flat for most of the time, but periodically promotions occur for a short period of time. It is easy to see from the figure when a promotion occurs, but it is more difficult to define a rule which splits prices into promotional and regular prices. I find that defining a promotion as any price below the median price works well - if one overlays the median price (which is 59 cents) on the price series graphs, the flat areas all lie above it and the promotions lie below it. The price process that I propose models the probability a promotion occurs using a discrete Markov process. A second important feature is that, when a product is not on promotion, its price is flat for long periods of time. I model the price process during these periods using a discrete- continuous Markov process that is similar to that of Erdem, Imai, and Keane (2003). Conditional on no promotion occurring in weeks t−1 and t, there is a probability that the price changes. If the price does change, a truncated regression is used to predict that change. The third feature that my price process captures is the existence of competitor reactions: when one firm’s price drops, the competitor drops shortly afterwards. This can be seen in the right panel of Figure 2, which shows the price paths for Starkist and Chicken of the Sea in one store. My price process allows the probability of promotions to depend on the competitor’s prices and promotional behavior. The state variable governing promotions, sjt is one when product j is on promotion

18

in week t, and 0 otherwise. I estimate the Markov transition process on the probability of a sale occurring using a probit model:

P (sjt = 1|sjt−1 = k)  h i pr pr pr pr pr = 1 − Φ − ψ0jk + ψ1jk pjt−1 + ψ2jk p−jt−1 + ψ3jk s−jt−1 + ψ4jk sjt−1 p−jt−1 .

(10)

The superscript pr stands for promotion, and the subscript k ∈ {0, 1}. The probability of transitioning from the nonsale state into the sale state, or vice-versa, is governed by the competitor’s previous price, the product’s own previous price, and whether the competitor’s product was on promotion. When a product is not on sale for two consecutive weeks, its price will often stay constant for a few weeks. To account for this, I model the probability the pjt = pjt−1 using a probit process as

P (pjt = pjt−1 |sjt = 1, sjt−1 = 1) =  s 
s s s s 1 − Φ − ψ0j + ψ1j pjt−1 + ψ2j p−jt−1 + ψ3j s−jt−1 + ψ4j sjt−1 p−jt−1 .

(11)

When the price of a product changes, I assume that the change is distributed according to a truncated normal distribution. If the product is transitioning into the non-sale state, then its distribution is censored at the median price, mj ; If it transitions into the sale state, I truncate from above at mj − 1. I model this using a Tobit model, where I allow the parameters to depend on whether the product was previously on promotion (sjt−1 = k): c c c c c yjt = ψ0jk + ψ1jk ln(pjt−1 ) + ψ2jk ln(p−jt−1 ) + ψ3jk s−jt−1 + ψ4jk sjt−1 ln(p−jt−1 )   yjt (12) if yjt > ln(mj ) ln(pjt ) =  ln(m ) if y ≤ ln(m ). j

jt

j

A similar specification is run when the product transitions to a sale state. The inclusion of the competing brand’s prices and promotions allow for competitor reactions. As a final note, I estimate all the parameters of consumer price expectations along with utility parameters in the model. Earlier work typically estimates the process for price expectations prior to estimating the model. Although doing this will produce consistent estimates, there could be inconsistency in the standard errors of the utility coefficients,

19

due to them being functions of the estimated parameters of the price process. My approach avoids this problem.

7

Estimation Technique

This section outlines the two major parts of the estimation. I use the Bayesian technique of Imai, Jain, and Ching (2009) to estimate my model. This approach has some computational and statistical advantages over classical approaches such as simulated maximum likelihood, or method of simulated moments. The primary computational difficulty that arises when estimating dynamic discrete choice models is that one must solve the Bellman equation that gives rise to the consumer value function. Typically this is done via a nested fixed point algorithm: every time one changes the parameter values when optimizing the objective function, one solves the contraction mapping that defines the value function using an iterative procedure.16 Solving for the value function over and over again significantly increases the computational complexity of estimating dynamic discrete choice models. In contrast, in the Bayesian technique of Imai, Jain, and Ching (2009) one only solves for the value function once over the course of the estimation. In every step of the Gibbs sampler, one performs a single value function update. As the parameters of the Gibbs sampler converge to draws from the posterior distribution, the updated value functions at each Gibbs draw converge to the solution of the Bellman equation. Overall, the Gibbs chain has three main steps in it: step one draws population-varying coefficients, step two draws population-fixed coefficients, and step three draws price expectations coefficients. The following section discusses the first three steps, and the next discusses the fourth step.

7.1

Gibbs Steps for Utility Parameters

First I consider the Gibbs chain for the demand parameters. The vector of utility coefficients being estimated is (γi1 , ..., γiJ , β, αi , sc0 , sc1 , CC). Denoting the vector of ˜ ∼ N (b, W ), population-varying parameters as θ˜i = (γi1 , γi2 , αi ), I assume that ln(θ) where the matrix W is diagonal. I denote the vector of population-fixed parameters 16

See Rust (1994) and Rust (1987) for an overview of the nested fixed point algorithm.

20

as θ = (β, sc0 , sc1 , CC) and the vector of price expectations coefficients, which enter consumer expectations, as ψ. In many of the Gibbs steps, it is necessary to compute the probability of a consumer’s sequence of observed choices conditional on a set of draws on initial inventories, ιi0 , observed prices, pi0 , ..., piT , and consumer utility coefficients θ. An observation is a household-week, and the household’s purchase decision, xit , is observed. In what follows I describe how to construct the probability of each household’s sequence of purchase decisions. Household consumption decisions are unobserved, but conditional on a value of xit , and all the previous parameters, these can be calculated as

cit ∗ = arg max cit

 J X 

2 γij u(cjit ; β) − sc0 Ijit − sc1 Ijit + EV (Σit ; θ˜i , θ, ψ)

 

.

(13)



j=1

An approximation to EV (Σit ; θ˜i , θ, ψ) is computed using a nearest neighbor algorithm which is described in Section 7.3. Inventories are unobserved in period t, but they can be computed conditional on the initial inventories in period 0 (I describe how I compute period 0 inventories at the end of this section). In other words, conditional on all possible choices in period 1, one can compute period 1’s consumption. Then the inventory at the beginning of period 2 is the period 0 inventory, plus period 1’s observed purchase, xi1 , minus the optimal consumption, c∗i1 . Period 2’s inventory can be constructed similarly, and so on. Denote the utility from the optimal consumption in period t as

ν(xit ) = max cit

 J X 

j=1

− CC 1{

  2 ˜ γij u(cjit ; β) − sc0 Ijit − sc1 Ijit + EV (Σit ; θ i , θ, ψ) 

J X

xjit > 0} − αi

j=1

J X

xjit pjit

j=1

I assume that the choice specific error term, qit , is logit. Denote the observed xit in q period t as xobs it , and denote each possible value of xit , indexed by q, as x . Then a

consumer’s sequence of choices can be computed as

P ri (θ˜i , θ, ψ) =

T Y

exp(ν(xobs )) . P2NU +1 it exp(ν(xq )) q=1 t=1

21

(14)

The sum in the denominator in equation (14) goes from q = 1 up to 2Nu + 1 because I assume that consumers only purchase a maximum of Nu units in a single purchase occasion, there are 2 brands, and consumers do not purchase different brands in a single occasion. To draw the utility parameters, I first draw θ˜i for each consumer i, conditional on θ, ˜ {ιi0 }N i=1 and and ψ. The posterior distribution of θ i does not have a closed-form solution, so I use the Metropolis-Hastings (MH) algorithm to draw a new θ˜i . Denoting the 0 ˜ 1 ∼ N (ln(θ) ˜ i , ρ2 W ), previous draw on θ˜i as θ˜i , my procedure is to draw a candidate ln(θ) i

and to accept the new draw with probability 1 P ri (θ˜i , θ, ψ) . 0 P r(θ˜i , θ, ψ) 1 The parameter ρ is adjusted every period so that about 30% of the candidate θ˜i draws

are accepted every period (see Train (2003)). Once I have drawn {θ˜i }N i=1 , I draw out b and W conditional on the logarithm of the draws. I assume that the prior on b is normal with a prior variance of 100I, so the prior are proper but relatively uninformative. This prior generates a normal posterior distribution for b, which I can draw from using standard methods. Similarly, I put an inverse gamma prior on each of the diagonal elements of W with scale parameter 3 and shape parameter 0.5, which results in an inverse gamma posterior for the elements of W .17 N Next I draw θ given the drawn {θ˜i }N i=1 , the previous draw’s {ιi0 }i=1 and ψ. I

also use a random walk MH for this step, with a parameter ρ2 on the variance that periodically updates to keep the acceptance rate at 30%. Given a previous draw θ 0 , I draw a candidate θ 1 where

θ 1 ∼ N (θ 0 , ρ22 I). The candidate draw is accepted with probability 17

Most of the mass of this prior distribution is between 0 and 0.5, which would be too tight if I was not

exponentiating the utility parameters. However, my utility parameters are lognormal, which means a relatively small variance in the underlying normal can lead to a large variance in the resulting lognormal. For example, the exponential of a normal distribution with mean 1 and variance 0.5 has variance of about 13.7.

22

I Y P ri (θ˜i , θ 1 , ψ) i=1

P ri (θ˜i , θ 0 , ψ)

!

k(θ 1 ) , k(θ 0 )

where k is the prior distribution on θ. I assume that k is normal, with a variance matrix of 100I so that priors are close to uninformative. The prior means are set to the average parameter estimates from a similar model presented in Osborne (2010). My choice probabilities above depend on inventories prior to period 1, ιi0 , which are unobserved. I integrate them out using simulation. My approach is to simulate ιi0 conditional on θ˜i , θ and ψ for each consumer. The initial inventories are drawn using the incomplete data prior to the final year of the data, where consumer purchases are observed but store visits are unobserved when purchases are not made. I proceed by first drawing a sequence of store visits for each household where a visit is unobserved from the empirical density of observed store visits for that particular household.18 I then assume that at the beginning of the data (73 weeks prior to the beginning of the sample) initial inventories are zero. I then compute optimal consumption in each period, conditional on the parameter draws, prices and quantities chosen in each period, which gives us end of period inventories. ιi0 is then taken to be the level of inventories computed at the end of the 73rd week. A new draw on the initial inventories is taken in each Gibbs step.

7.2

Gibbs Step for Price Expectations

Consumer price expectations are assumed to coincide with the distribution of prices that are observed in the data. To see how I draw the parameters governing these, consider first the posterior distribution for the ψ pr , the parameters associated with the probability of a price change. I use standard data-augmentation techniques, with an adjustment 18

A slightly different procedure was presented in Osborne (2010) where the ιi0 ’s were assigned a distribution

and draws were taken on the ιi0 in a separate Gibbs step to rationalize the final year of choice data. The procedure used in Osborne (2010) produces a distribution of initial inventories that is zero for more than 99% of households. Although the approach of Osborne (2010) is more nonparametric, my approach produces a distribution of initial inventories which more closely mimics the distribution of inventories I compute at the very end of the sample, which I feel is more realistic. Since the consumers in the sample are fairly regular purchasers of canned tuna, it seems unlikely that almost all of them would have no tuna in inventory at the beginning of the sample. The distribution of initial inventories may be difficult to identify nonparametrically.

23

that accounts for consumer choice probabilities. For a given consumer i in week t and a vector of observables for brand j, wj,i,t = (1, pj,i,t−1 , p−j,i,t−1 , s−j,i,t−1 , sj,i,t−1 p−j,i,t−1 ), consider a latent variable formation of the probit model

zj,i,t = w0i,t ψ pr j + ej,i,t , ej,i,t ∼ N (0, 1)   0 if zj,i,t ≤ 0 . yj,i,t =  1 otherwise

Given the prior iteration’s draw on ψ pr , which I call ψ pr,0 , I draw out the z’s. I then draw a new ψ pr,1 using the MH algorithm. As a proposal density, I use the density of ψ pr,1 conditional on the zj,i,t ’s and the data on prices observed by all consumers, but not consumer choices.19 Note that this density is just standard normal. Then I accept Q the new draw with probability Ii=1 P ri (ψ pr,1 ; θ, ιi0 )/P ri (ψ pr,0 ; θ, ιi0 ). I also need to draw ψ c and ψ s . Note that if it were not for the consumer choices, I could draw both of these parameters using standard Bayesian procedures for the Tobit and probit models. I therefore use the same type of proposal distributions with both these parameters - I draw any latent unobservables, and then draw candidate parameters conditional on the latent unobservables and the store level data. I accept or reject the vector of ψ pr ’s, ψ c ’s, and ψ s ’s jointly to save computational time. The acceptance rate in this step tends to Q be high, as the ratio Ii=1 P ri (ψ pr,1 ; θ, ιi0 )/P ri (ψ pr,0 ; θ, ιi0 ) is typically close to 1. 19

I assume that the price expectations I estimate accurately capture the price process that consumer i

observes. The prices observed by the consumer will depend on both the pricing policies of the canned tuna firms, and the choice of store that the consumer makes. The equations describing price expectations outlined in Section 6 will therefore capture the composition of both these random variables, and the correct data to use will be the data on consumer prices at the level of the consumer. As discussed by Erdem, Imai, and Keane (2003), an alternative approach is to estimate the price process at the store level and to adjust for store visit probabilities afterwards. Their paper finds that both of these ways of modeling price expectations generate very similar parameter estimates. Note that this specification assumes that the choice of store is exogenous to the decision to purchase canned tuna.

24

7.3

Approximating the Expected Value Function

One of the most computationally intensive parts of the estimation procedure is com1 puting an estimate of the value function, which I need to draw the parameters θ˜i , θ, ψ,

to compute simulated quantities qˆ, and to update the value function (I describe the updating procedure in the next section). Here I follow the Imai, Jain, and Ching (2009) procedure of starting with a guess of the value function, and computing one update to the value function at each Gibbs iteration. In each update, it is necessary to compute an estimate of the value function at the current Gibbs draw. This is done by averaging over value functions saved in previous Gibbs draws where the parameters are “close” to the current draw. I use a nearest neighbor approach to selecting close parameter draws. Another alternative is to use a kernel-weighted average of past value functions, where the kernel weights measure the closeness of the current draw to past draws. Norets (2009) discusses advantages and disadvantages of each approach. At step g of the Gibbs sampler, I will have N (g) saved draws on θ˜i for every household and N (g) saved draws on θ, ψ. There will also be N (g) saved value function estimates for each state space point at each of those draws.20 Denote the draw on θ (θ˜i ) from iteration k as θ k (θ ki ), k = g − N (g), ..., g − 1 and the observed price expectations draw k as ψ k . Denote the vector composed of θ˜i , θ k and ψ k as Γki = (θ˜i , θ k , ψ k ).

For each i and k there is an associated saved value function, Vi,k (ι, p). I will have a saved value function for each value of ι, since ι is discrete, but not for p, since p is continuous. Instead what I do is draw a set of Np prices in each iteration from an N ,g−1

p importance distribution h(·). This means I have a set of {pg,s }s=1,g=g−N (g) saved prices.

Suppose now that I need an estimate of the expected value function at some parameter vector Γ, a price p, and an inventory ι. I do this in two steps. First, for household i I ˜ of them as follows, indexing them as find the closest Γki ’s to Γ. I will find m = 1, ..., N ˜lm 20

When starting the algorithm, I begin with an estimate of the value function that is zero in all states.

25

˜l1 =

{kΓ − Γli k}

min

l∈{1,...,N (g)}

˜l2 =

min

l∈{1,...,N (g)}/˜ l1

{kΓ − Γli k}

.. . ˜lN˜

=

{kΓ − Γki k},

min

l∈{1,...,N (g)}/{˜ l1 ,...,˜ lN˜ −1 }

where the kk function indicates the Euclidean norm and / indicates set subtraction. The estimated value function will average over the ˜lm ’s, as well as the saved pg,s ’s at each Gibbs iteration associated with ˜lm , which I will call g(˜lm ). Note that I need to average over the saved pg,s ’s, because when I compute the expected value function at a current price p, I must integrate over the distribution of possible prices tomorrow. To average over the saved pg,s ’s, I use importance sampling. I compute the probability that pg,s occurs tomorrow given today’s price p using the transition density at the current ψ, P (pg,s |p, ψ), and use P (pg,s |p, ψ)/h(pg,s ) as the importance weight. The estimated expected value function is then ˜ N 1 X ˆ EV i (ι, p; Γ) = ˜ N m=1

7.4

PNp

s=1 Vi,˜ lm



ι, pg(˜lm ),s

g(˜ lm ),s |p,ψ) h(pg(˜lm ),s )

 P (p

P (pg(˜lm ),s |p,ψ) s=1 h(pg(˜lm ),s )

PNp

.

Updating the Value Function

The last step of the algorithm is to update the value function Vi,k (ι, pg,s ) at every ι and pg,s combination. Note that every iteration, I only perform a single value function update, as outlined in Imai, Jain, and Ching (2009). Over the course of running the Gibbs sampler the value function will converge. Updating the value function is relatively straightforward. At the end of iteration g, I will have a new parameter vector θ g , as ˆ (ι, pg,s ; Γ) for all well as a draw on ψ g . First, given some Γ = (θ gi , θ g , ψ g ) I compute EV possible ι values. Then, I compute optimal consumption at all possible x choices, which I will denote c∗ (xq ). Net of the error term, utility at this point will be

q

νˆ(x ) =

J X

J J X X q 0 ˆ γij u(c ; β)−sc0 I−sc1 I +δ EV (ι , pg,s ; Γ)−CC1{ xj > 0}−αi xqj pj,g,s , ∗

2

j=1

j=1

26

j=1

where ι0 denotes period t + 1 inventory. The updated value function will then be  Vi,l (ι, pg,s ) = log 

Nu (Nu −1)/2

X

 exp(ˆ ν (xq )) .

q=1

8

Parameter Identification

In this section I present an informal argument for the identification of the model parameters. Most of the argument is similar to that laid out in Erdem, Imai, and Keane (2003), which presents a qualitatively similar model. First, consider the identification of the parameters on inventory costs. An decrease in the linear parameter on inventory costs, sc0 , will cause consumers to purchase smaller quantities more often: the overall incentive to stockpile will decrease. An increase in the quadratic term will cause consumers to avoid making large increases in inventory. Thus, if a consumer just purchased a large amount of a product, and the product is observed to go on sale again, I should observe the consumer to be unlikely to stockpile in response. Next consider the identification of the taste parameters. Conditional on a given amount in inventory, increasing the quadratic term will decrease the amount of tuna consumed.21 This parameter is analogous to a consumption rate. Changing this parameter will impact the duration dependence of the purchase hazard. The impact of changing this parameter is nonlinear: if β is high, then duration dependence increases - the time between purchases drops. If it is low, the time between purchases rises. Erdem, Imai, and Keane (2003) show that the multiplicative taste parameter (γj ) increases overall demand, but does not affect duration dependence. Thus, market shares will drive this parameter. Identification of the carrying cost parameter follows from the overall purchase frequency. The higher is the carrying cost, the less frequently will consumers make purchases. It is separately identified from the stockpiling costs because it does not impact the spacing between purchases - a consumer’s disutility due to carrying costs from making five purchases is the same whether the five purchases occur in five consecutive weeks, or over five months, each occurring once per month. However, making five consecutive 21

To see this, consider the case where consumers are not forward-looking, and there are no inventory costs.

A consumer’s optimal consumption of cj will be −1/(2β).

27

purchases will drive up stockpiling costs more than five purchases which spaced further apart. The heterogeneity in the price coefficient will be driven by heterogeneity in consumer response to promotions. In the data, we observe consumers making purchases over a year. Since different consumers shop at the same stores and observe the same prices, I will see if some consumers tend to be more likely to enter the market during promotions than other consumers. Heterogeneity in tastes will be driven by heterogeneity in consumer response to promotions for different products: I can infer that consumers who are unresponsive to promotions for Starkist, but are responsive to those for Chicken of the Sea have a preference for Chicken of the Sea, and vice versa.

9

Model Estimates

The Gibbs sampler is run for 13,000 iterations, where the first 3,000 draws are removed to reduce dependence on the starting values. I estimate the model on a 50% subsample of the data to reduce computational burden - the estimation data includes 317 households. I plot the time series of parameter draws in Figure 4, where it can be seen that a little ˜ = 30 after around 2,000 draws the parameter draws become relatively stable. I save N previous value function draws, choosing the 3 closest previous value function draws, and draw 20 new prices every iteration. Estimates of the utility parameters are shown in Table 3. The first column of the table shows the average of the draws for the population mean utility parameters. For population-varying coefficients, I present the average (over draws) of the draw mean of their lognormal distribution - if the draw on the underlying normal parameters are a mean of b and variance of w, then the draw on the mean of the lognormal is exp(b+w/2). For population-fixed parameters, the population mean is simply the draw. The second column shows the 5th and 95th percentiles of the parameter draws. The third shows the average of the draws for the population standard deviation parameters. Again, for the population-varying parameters I present the average of draws on the standard deviation of the lognormal.22 For population-fixed parameters these columns are dashed out. Turning to the estimates, the average tastes for Starkist and Chicken of the Sea 22

For an underlying normal draw with mean parameter b and variance w, the standard deviation of the

28

are similar, which is consistent with their marketshares being similar. The curvature parameter of about -0.8 implies that, absent forward-looking behavior, a consumer would use between one and two cans of tuna per week. The small estimates of the inventory cost parameters imply negligible holding costs. The price coefficient is about 0.028, which is consistent with significant price sensitivity. Since price is measured in cents, a increase in price of ten cents decreases consumer utility by about 0.28. Flow utilities average at around -0.6, so price is a significant determinant of the purchase decision. The carrying cost is large and significant, which is consistent with purchases of canned tuna being relatively infrequent. In dollar terms, the carrying cost is measured to be a little less than one dollar for an average consumer. The estimates of the standard deviations of the population-varying parameters are consistent with a significant amount of consumer heterogeneity. Note that although the standard deviation of the price coefficient is small in absolute terms, relative to the average price coefficient it is large, indicating significant unobserved heterogeneity in price sensitivity. Parameter estimates from the process governing consumer price expectations are available from the author upon request. The estimates are sensible in the sense that produce transition probabilities that mimic the observed patterns in the data well. This can be seen in the predicted sale probabilities in Table 4, which shows summary statistics of the model’s predicted probability of a sale for sale or nonsale observations in the store level data.23 Conditional on being on sale, a product has about a 50 percent chance of being on sale the next period; if it is not on sale, though, the product is unlikely to go on sale. If Starkist is on sale, the likelihood it stays on sale the next period is 0.5; if it is not, then the likelihood it will go on sale is only 0.16, and Chicken of the Sea looks similar to this. The raw transition probabilities of sale and nonsale states in the data closely match my model predictions. This type of time-series variation in prices will help to drive stockpiling behavior: when consumers observe a promotional price, they will know it is likely to be short-lived and will stockpile in response to it. lognormal is 23

p (exp(w) − 1) exp(2b + w).

The predicted probabilities are computed for each observation at the mean of the posterior estimates.

29

10

Comparison of the Dynamic Index to Simpler

Indexes 10.1

Construction of the Dynamic Index

This section presents the price index for canned tuna I derive from my estimated stockpiling model. The cost-of-living index was defined by Kon¨ us (1939) in terms of compensating variations. Consider a consumer who observes a vector of prices p0 in the base period, and pt in period t. The Kon¨ us COLI is the amount of money that one needs to compensate a consumer in period t to keep her utility at the same level as in period 0, assuming all other time-varying factors are unchanged. My approach follows the spirit of the Kon¨ us thought experiment, with one difference. Due to consumers’ dynamic optimization behavior, consumer inventories will evolve over time in addition to prices. The Kon¨ us thought experiment assumes that consumers behave in a static manner, and incorporating dynamic behavior into COLIs is a subject of current research, so there currently is not a “correct” way to account for the fact that state variables will evolve over time. The approach taken in my paper follows that of Gowrisankaran and Rysman (2012), who face a similar issue in their setting.24 The Gowrisankaran and Rysman (2012) approach to computing a dynamic COLI is to construct a sequence of taxes or subsidies that hold average flow utilities constant over time. From equations (6) and (7) a consumer’s flow utility in period t will be equal to a constant plus

˜ (c∗it , x∗it , ιit ) − αi U it = U

J X

xjit pjit ,

j=1

where c∗it , and x∗it , are the optimal values of consumption and purchases, which are functions of period t prices and state variables. We can keep a consumer’s flow utility in period t at the same level as the period 0 utility by adding U i0 − U it αi

(15)

to the consumer’s period t income. 24

Gowrisankaran and Rysman (2012) construct a COLI for a durable product, camcorders, where consumer

holdings will also evolve over time.

30

Operationally, the way I construct my dynamic COLI is as follows: in a given draw of the Gibbs sampler I compute the flow utilities needed to construct the compensating variation in equation (15) for each consumer, and then I compute the average compensating variation across consumers for each period, which I denote τt .25 In the context of durable goods, Gowrisankaran and Rysman (2012) use a similar formulation of the COLI to that proposed here. If a social planner were to impose a sequence of taxes or subsidies equal to τt on each consumer, average flow utilities would be fixed over time. Moreover, because the sequence of taxes/subsidies is not dependent on individual behavior, it will not affect individual consumption choices.26 Price indexes such as those produced by the BLS are not typically presented as COLIs, although as discussed in the Introduction, they are motivated by COLIs. In order to compare the sequence of taxes, which is a COLI, to a price index I need a way to map one to the other. As discussed in Gowrisankaran and Rysman (2012), to convert a price index to a COLI one multiplies the period t index by period 0 expenditures, divides the result by 100, and subtracts the result from the initial average price.27 This index provides the change in income relative to the initial period needed to purchase the 25

Note that I compute the estimated COLI at the same time that I draw the parameters, following the

procedure outlined in Ching, Imai, Ishihara, and Jain (2012) for computing counterfactuals with dynamic models. The standard approach taken in classical models is to first obtain parameter estimates, and then to compute the counterfactual at the estimates. The Bayesian approach offers an advantage in this regard in that the counterfactual COLI is computed along with the estimation with little added computational time, and at the end of the estimation a small sample distribution of COLIs is produced, which makes computing the estimated precision of the counterfactual straightforward. 26

An alternative way to construct a dynamic price index is presented in Reis (2009). That paper sets up

an overlapping-generations consumption savings problem and defines a price index to be a factor πt that multiplies asset values in period t such that period t expected discounted utility is equal to period t − 1 expected discounted utility. A difference between the index proposed by Reis (2009) and the one used in my paper is that changes in the Reis index will affect consumer expectations about future prices and hence current consumption choices. 27

Gowrisankaran and Rysman (2012) make a modification to the standard practice to account for the fact

that in the market they analyze prices drop rapidly while total quantity sold rises rapidly. They include the outside good as a product in their indexes which has a normalized price. Because in the market I analyze I do not observe these sorts of dynamics in prices and quantities, I follow the standard approach.

31

same basket of goods. Rather than taking the price indexes I compute and turning them into COLIs, I invert the transformation above and turn my COLI into a price index. I compute a base period average price, p0 , then add this average price to the COLI each week, and divide the result by the initial average price:

Ptdynamic =

τt + p0 . p0

This index will be 1 in the initial period, and will be low when prices are low (τt is negative) and above 1 when prices rise above the base level (τt is positive). The presentation of the index that I will focus on in the paper uses week 1 as a base period. For the week 1 base period I choose U i0 to be week 1 utility, and p0 to be the average week 1 price observed in stores (prices are weighted by quantities sold). To make sure that my results are robust to changes in the way the base period and prices are calculated, I will also present two other different versions of the dynamic cost of living index. The second version of the index varies the base period, and uses the first four weeks of the data rather than the first week. For this version of the index I choose U i0 to be average utility over the first four weeks of the data, and the base price is chosen to be the average of the modal prices of each brand-store combination in the first 4 weeks, where the weights used are the average quantities sold of each brand-store combination over the 4 week period. The 4 week base version of the index will be less sensitive to weekly price and quantity movement in the initial period of the data.28 The third version of the index keeps utility fixed at the week 1 value but varies the initial average price p0 . In this third version of the index I compute base prices in the same way that FS do. I choose the modal price of each SKU in each store in the year prior to the one in which I estimate my model, and I use as quantity weights the average quantity sold of the SKU in the store at the modal price. In all 3 cases, to compute the average price index in each week, I average the price index over saved Gibbs draws. Figure 5 shows the how the estimated price index evolves over time for the week 1 base period. The dotted red lines around the index show 95% confidence intervals for each 28

There is a significant amount of price volatility over time, and quantities sold vary a lot as well; additionally,

in some weeks the quantity of a brand of canned tuna sold in a specific store is 0 (for week 1, 10 out of 24 brand-store combinations are 0). The quantity weights in the initial four weeks are never 0.

32

period, which I can compute from the Gibbs draws. The index is quite volatile, which is consistent with the significant amount of volatility I observe in prices. Additionally, the index is usually below 1, consistent with prices falling over time which was documented earlier. The second and third schemes for constructing the index produce indexes that look qualitatively similar to the week 1 index. Using the first month as a base period produces an index that looks similar to the first week as a base, only the index is shifted upwards because overall prices in the first month of the data are lower than those in the first week. Using the FS scheme for constructing initial prices produces an index that looks very similar to the first specification, but is slightly less volatile.

10.2

Formulas for Simpler Indexes

This section presents formulas for several alternative price indexes to my dynamic index: a fixed-base Laspeyres, a geometric index, a Tornqvist index, chained Laspeyres and Tornqvist indexes, and an alternative COLI index proposed by Feenstra and Shapiro (2003) (FS).29 The index formulas I present below correspond to those used for the week 1 base period. During the time that my data was collected (in 1988), the price index for food products used by the BLS was a Laspeyres index:

PtLaspeyres =

X i

wi0

pit . pi0

In the above equation, i indexes a brand-store combination. wi0 is the base period expenditure share for product i, and pi0 is the modal price in the previous year. In 1999 the BLS started using a geometric price index to compute the CPI. I also compute a (fixed-base) geometric index, which is defined as

PtGeo

= exp

( X i



pit wi0 ln pi0

) .

The third fixed-base index I compute is a Tornqvist index, which allows weights to vary over time: 29

The alternative indexes are computed using data from the full sample of 600 households, rather than the

50% subsample the estimation was done on.

33

PtT ornqvist

= exp

( X i



pit 0.5(wi0 + wit ) ln pi0

) .

The weights in the base period 0 are the quantities sold for the first week of the data. For the monthly index, the weights correspond to the expenditure shares of the modal prices; if we define pmonthly to be the modal price of product i over the first month of i0   P P monthly monthly monthly 4 4 1{p = p } , then q 1{p = p } / the data, and qi0 = it it it t=1 t=1 i0 i0 the expenditure weight for product i is pmonthly q monthly monthly wi0 = P i0 monthlyi0 monthly .30 qi0 i pi0 The expenditure weights for the index that uses the prior year as a base period are constructed in a similar manner to those that use the prior month. I compute two chained indexes as well. The chained Laspeyres index31 is

PtC,Laspeyres =

X i

wi,t−1

pit , pi,t−1

and the chained Tornqvist index is

PtC,T ornqvist = exp

( X

 0.5(wit + wi,t−1 ) ln

i

pit pi,t−1

) .

The final index I use as a comparison is a COLI index proposed by FS for frequently purchased, storable products. FS develop a representative consumer model where it is assumed that consumption and purchases differ, and consumers have perfect foresight over future prices. In the FS model, a key assumption is that consumers make decisions over a finite planning horizon of T F S periods. In period 1, the consumer chooses consumption and purchases for periods 1 through T F S to maximize her present discounted utility for those periods; then, in period T F S + 1 the consumer solves the same problem again for periods T F S + 1 through 2T F S . The paper makes the simplification that the 30

An alternative to using modal prices would be to use average prices; I choose to use modal prices because

modal prices are used when constructing base weights for the FS COLI, which I describe below. 31

FS present a version of the chained Laspeyres index that holds fixed the expenditures weights at the base

period. This version of the index, which is also called a Young index, does not rise as quickly as the chained Laspeyres. I diverge from FS in this respect and present a standard chained Laspeyres.

34

problem solved at period T F S + 1 (or 2T F S + 1, etc) does not depend on information from periods 1 through T F S , corresponding to the prior planning horizon. Using results due to to Caves, Christensen, and Diewert (1982a) and Caves, Christensen, and Diewert (1982b), FS show that if one approximates the expenditure function arising from their model with a translog functional form the price index for planning horizon τ has the following formula:  COLFτ S = exp 

F S ∗τ TX

N X 1

t=1+T F S ∗(τ −1) i=1

2

 (si0 + sit ) ln





pit  , pi0

where si0 is the expenditure share of SKU-store combination i in the base period, where the share is taken over all weeks during the base period.32 I compute the FS COLI for two different planning horizons: a four week planning horizon and a one year planning horizon.33 When I compute the base period shares, I follow FS by finding the modal price of product i in the base period, multiplying by the average quantity sold at that price, and dividing by total expenditures on product-store combination i over the course of the base period. Similarly, sit is the expenditure share of product-store combination i in week t of planning period τ . For the monthly index, the base period is the first four weeks of the sample. For the yearly index, it is chosen to be the year of data prior to the estimation sample. Note that the expenditure weights for the Laspeyres, Tornqvist and Geometric indexes, as well as the average price used to rebase my dynamic COLI also depend on modal prices, which ensures they are comparable to the FS COLI.

10.3

Comparison of the Indexes

As discuss in the Introduction, the dynamic index is complicated to estimate and may not be feasible for statistical agencies to implement. This section compares the dynamic index to more standard indexes to ascertain which simpler indexes can approximate the dynamic index well. Table 5 summarizes the comparison of my dynamic index to the 32

Technically the FS COL is a price index rather than a COLI. I adopt their notation here.

33

The monthly version of the index is computed over the 51 week period of the data where complete data

on store visits is available. Because 51 weeks cannot be divided evenly into 4 week periods, I define the last “month” of the data to be the final 3 weeks.

35

alternatives for the three different base periods. An entry in the table shows either the mean or standard deviation of each index for my sample period, excluding the base period. The dynamic index, shown in the first row, is on average 79%. All of the fixed base indexes are much higher - the Laspeyres averages about 96%, the geometric averages 95%, while the Tornqvist averages 90%. This finding is consistent with the intuition I laid out earlier in the paper that fixed base indexes should tend to overstate inflation, because they will overstate the increase in the cost of living that arises when the price of a product goes up after a promotion. The dynamic index correctly measures the cost of living because it allows consumers to carry inventories: inventory built up in a promotional period can provide consumption utility in periods when prices are higher, lowering the cost of living and the implied price index. The chained indexes display much wider variation. The chained Laspeyres rises significantly over time, ending at a value of almost 4.34 As discussed in the introduction, the chained Laspeyres is known to have a significant upward bias. The chained Tornqvist, on the other hand, drops significantly over time. As I will discuss below, this is likely due to the Tornqvist overweighting price falls relative to price rises. As a final note, the FS COLI for the weekly model is identical to the Tornqvist index, so that row is dashed out. I observe similar patterns for the indexes that use the first month as a base period, and the prior year as a base period. The results for the monthly index are shown in the third and fourth columns. Because modal prices in the first month tend to be low relative to the rest of the year, the indexes are all above 1. The Laspeyres and Geometric indexes are furthest from my dynamic index, while the average Tornqvist and FS index are closest.35 Similarly, for the indexes that use the prior year expenditure 34

As I described in footnote 31, I present a standard chained Laspeyres rather than a Young index. In my

sample, the Young index rises over time, and averages at about 1.15 over the course of the year. 35

The FS index is computed at a monthly level, while all the other indexes are computed at a weekly level.

To compare the FS index to the other indexes I expand the FS index by replicating each month’s value 4 times (3 times for the final month). An alternative way to compare the indexes would be to compute a monthly average for each index, and to compare those monthly average indexes to the monthly FS index. The averages of the monthly indexes look almost identical to the overall averages of the weekly indexes (they differ slightly because the sample year includes 51 weeks rather than 52 weeks, so the final “month” of the monthly index is only 3 weeks long). The standard deviations of the monthly indexes all look similar more similar to each

36

weights as a base, the Laspeyres index is the highest, and the closest average indexes are the Tornqvist and the FS index. As discussed in the Introduction, the fact that the FS index is closer to the average value of the dynamic index than the Laspeyres or Geometric arises due to the way it averages price changes over time. A yearly average of the fixed base index puts equal weight on each week’s index value. Turning to the FS index, the expenditure weights for periods that are promotional are 21% higher than nonpromotional periods, meaning that when the average index is constructed, more weight will be put on periods where prices are low than in the average fixed base index. The average Tornqvist likely matches the dynamic index due to its similar functional form to the FS index; if the weights for a product in a particular period are higher in promotional than nonpromotional periods, the index will be lower than the Laspeyres or Geometric. Which of the alternative indexes are statistically different from my estimated index? To answer this, I compute the average dynamic index index for each of the Gibbs draws, and then plot a histogram of the index estimates in Figure 6. The distribution of the predicted index does not appear to be normal, which suggests the sample is too small to rely on asymptotics. The estimated distribution of the index has two modes, one at around 1, and one at around 0.8, which is where most of the mass is. The hump around 1 (the estimated support of this hump lies between 0.98 and 1.04) contains a little bit more than 10% of the mass of the estimated dynamic index - the other 90% of the mass is between about 0.7 and 0.85. All of the average fixed-base indexes lie in the range 90 to 95, where estimated distribution of the dynamic index has no support, which suggests that the indexes are statistically different.36 For the dynamic index with other than the standard deviations of the weekly indexes. 36

To be more precise, one could think of constructing a p-value for the hypothesis test that the average

dynamic index is equal to one of the average indexes using an approximation to the empirical distribution of the average index. If we denote fˆ as the estimated distribution of the index and µ0 as the value we are testing, a p-value could be constructed as Z

fˆ(x)d x.

x:fˆ(x)
I compute fˆ for the average index by computing a kernel density of the empirical distribution produced by the Gibbs sampler. For the weekly index, the p-values of the average Laspeyres, the average geometric,

37

the 1 month base period, the estimated distribution looks similar, only the two modes are shifted to the right, with the first mode between 0.97 and 1.06, and the second between 1.11 and 1.14. The average Laspeyres, geometric and FS COL indexes fall just in the region of no support, which suggests that again the dynamic index is statistically different from the fixed base indexes.37 The Tornqvist index falls closer to the mode of the average index, suggesting that on average, that index is statistically similar to the dynamic index.38 Similar results are obtained for the indexes that use the prior year as the base period, where only the Tornqvist index is statistically similar (on average) to the dynamic index.39 Although the Tornqvist, and to some degree the FS COLI, are both close to my index on average, they both understate volatility in the dynamic index derived from the true COLI. This can be seen in the standard deviations of the indexes for the 1 month base and the prior year base, and in Figure 7, where I plot the dynamic index with the month 1 base period against the FS index and the Tornqvist index. There is much more week to week variation in the dynamic index as compared to the other two. The increased volatility of the dynamic index arises due, to time series variation in inventories which are not captured by the Tornqvist index. By construction, the FS index cannot capture this type of short term variation, since it is only constructed for longer time horizon. Before concluding, it is worth making some comparisons between my findings and data set and the findings of and data used by FS. My results are generally in line with FS, who find that fixed base indexes approximate their COLI better than chained indexes. However, I suspect that the reasons for this differ from those found in FS. FS find that in their data, most purchases of canned tuna occur in the later weeks of a promotion. Those purchases also coincide with feature and display advertising. I checked my data to see whether the same phenomenon occurred. First, I categorized promotions by the and average Tornqvist indexes are 0.0001, 1e−8 , and 1e−19 respectively, indicating that they are statistically unlikely under the estimated sampling density for the average index. 37

Approximate p-values for these indexes are 0.02, 0 and 0 respectively. See Footnote 36.

38

An estimated p-value for the Tornqvist index is about 0.39. See Footnote 36.

39

The p-value of the average Tornqvist is about 0.13, while the FS is about 0.008. The other two fixed base

indexes occur with essentially zero probability under the estimated sampling distribution. See Footnote 36.

38

number of weeks the promotion lasted.40 I found that about 80 percent of promotions are two weeks long, and about 65 percent of the quantity purchased during a two week sale happened in the first week.41 Additionally, for the two week promotions, the product being promoted was on advertised 99 percent of the time in the first week, but only 13 percent of the time in the second week. FS also note that the incidence of purchases near the end of a promotion tends to drive a significant amount of upward bias in chained indexes - their chained Laspeyres and chained Tornqvist indexes rise 37 to 45 percent over the course of the year.42 In contrast, although my chained Laspeyres also rises, my chained Tornqvist drops significantly over time. This may be due to the fact that the chained Laspeyres uses period t − 1 weights, while the weights in the chained Tornqvist index are an average of weights in periods t and t − 1. Because I observe purchases occurring primarily at the beginning of promotions, the chained Tornqvist will overweight initial price falls relative to price rises.

11

Discussion

I derive a price index for storable goods using the estimates of a dynamic structural model. My results suggest that the average of the structural index can be approximated well by the FS index, or the Tornqvist. Standard fixed base indexes are, on average, significantly higher than my dynamic index because they underweight the importance of promotional periods in the cost of living. All fixed base indexes underweight time series volatility in the cost of living relative to my dynamic index. My findings suggest that if the measurement objective of a price index is to capture changes in the cost of living over time, for storable goods the FS index may do a better job of capturing this than 40

I denote a price drop as a promotion if the price of one of the products drops below the previous period’s

price and subsequently returns to a higher price. I limit the analysis to promotion spells between 1 and 3 weeks. Temporary price cuts seldom lasted longer than 3 weeks. I inspected graphs of prices for some stores and my procedure seemed to do a good job of identifying temporary price promotions. 41

3 week promotions occur about 10 percent of the time, and about 80 percent of purchases occur in the

first or second week. 42

As noted in footnote 31, FS present a chained Laspeyres index that holds expenditure weights fixed over

time, which does not rise as quickly as the canonical chained Laspeyres.

39

a Laspeyres or Geometric index. This is a useful result because the FS index is derived under fairly restrictive assumptions on consumer behavior. There is a significant amount of future work to be done to quantify the importance of stockpiling behavior in economic measurement. Although I have shown that stockpiling is important in purchases of canned tuna, it is likely to matter for many other products. Purchases of food at home, which is a significant part of Personal Consumption Expenditures as produced by the Bureau of Economic Analysis, are an aggregate measure of the value of all food at home products, weighted by their respective price indexes. A useful future exercise would be to investigate the impact of using alternative indexes, such as Tornqvist or FS index, on measurement of output. Additionally, I found that my dynamic price index displayed more volatility from week to week than BLS type indexes - in Table 5 the standard deviation of my index is about two or three times that of the other three fixed base indexes. This suggests that month to month volatility in food at home purchases may be greater than what is implied by the estimates produced by the Bureau of Economic Analysis. Looking further, it is likely that other types of dynamics will have a significant impact on measures of prices. Gowrisankaran and Rysman (2012) provide an example with durable goods; dynamics such as learning or variety-seeking also will play an important role for many consumer products.

References Aizcorbe, A. (2005). Price deflators for high technology goods and the new buyer problem. Unpublished manuscript, Board of Governors of the Federal Reserve Bank. Bureau of Economic Analysis (2011, November). NIPA Handbook. Technical report. Bureau of Labor Statistics (2007). BLS Handbook of Methods. Technical report. Caves, D., L. Christensen, and W. E. Diewert (1982a). The economic theory of index numbers and the measurement of input, output, and productivity. Econometrica 50(11), 1393–1414. Caves, D., L. Christensen, and W. E. Diewert (1982b, March). Multilateral comparisons of output, input and productivity using superlative index numbers. Economic Journal 92, 73–86.

40

Ching, A., S. Imai, M. Ishihara, and N. Jain (2012). A practitioner’s guide to bayesian estimation of discrete choice dynamic programming models. Quantitative Marketing and Economics 10(2), 151–196. Chintagunta, P., E. Kyriazidou, and J. Perktold (2001). Panel data analysis of household brand choices. Journal of Econometrics 103(1-2), 111–153. Erdem, T., S. Imai, and M. Keane (2003). A model of consumer brand and quantity choice dynamics under price uncertainty. Quantitative Marketing and Economics 1(1), 5–64. Feenstra, R. and M. Shapiro (2003, August). High-Frequency Substitution and the Measurement of Price Indexes. Univ. of Chicago and NBER. in Robert Feenstra and Matthew Shapiro (eds.), Scanner Data and Price Indexes. Gowrisankaran, G. and M. Rysman (2012). Dynamics of consumer demand for new durable goods. Journal of Political Economy 120, 1173–1219. Handbury, J. (2013). Are poor cities cheap for everyone? non-homotheticity and cost of living across u.s. cities. Working Paper. Hendel, I. and A. Nevo (2006a). Measuring the implications of sales and consumer inventory behavior. Econometrica 74, 1637–1673. Hendel, I. and A. Nevo (2006b). Sales and consumer inventory. The RAND Journal of Economics 37, 543–561. Hendel, I. and A. Nevo (2013). Intertemporal price discrimination in storable goods markets. American Economic Review 103(7), 2722–2751. Imai, S., N. Jain, and A. Ching (2009). Bayesian estimation of dynamic discrete choice models. Econometrica 77(6), 1865–1899. Kon¨ us, A. A. (1939). The problem of the true index of the cost of living. Econometrica 7(1), 10–29. Magnac, T. and D. Thesmar (2002). Identifying dynamic discrete decision processes. Econometrica 20(2), 801–816. Nevo, A. (2003). New products, quality changes, and welfare measures computed from estimated demand systems. The Review of Economics and Statistics 85(2), 266–275.

41

Norets, A. (2009). Inference in dynamic discrete choice models with serially correlated unobserved state variables. Econometrica 77, 1665–1682. Osborne, M. (2010). Frequency versus depth: How changing the temporal process for promotions impacts demand for a storable good. Working Paper. Osborne, M. (2011, March). Consumer learning, switching costs and heterogeneity: A structural examination. Quantitative Marketing and Economics 9(1), 25–70. Reis, R. (2009). A dynamic measure of inflation. Working Paper. Rust, J. (1987). Optimal replacement of gmc bus engines: An empirical model of harold zurchner. Econometrica 55, 993–1033. Rust, J. (1994). Structural Estimation of Markov Decision Processes. Elsevier. Handbook of Econometrics Vol. 4 Engle R. and McFadden D (eds.). Sillard, P. and L. Wilner (2014). An estimation of the elasticity of intertemporal substitution on micro data: Implications for the consumption price index. Working Paper. Sun, B. (2005). Promotion effect on endogenous consumption. Marketing Science 24, 430–443. Train, K. (2003). Discrete Choice Methods with Simulation. Cambridge University Press, New York. van der Grient, H. and J. de Haan (2010). The use of supermarket scanner data in the dutch cpi. Technical report, Statistics Netherlands. Wang, E. (2013). The effect of more accurate cost-of-living indexes on welfare recipients. Working Paper.

42

Table 1: Summary of Data Starkist

Chicken of the Sea

Market Shares

48.2 %

51.8 %

Average Prices

$ 0.63

$ 0.61

Std Dev of Prices

$ 0.9

$ 0.11

Table 2: Test for Inventory Behavior: Household Level Regression of Quantity on Inventory Regressor

Estimate

Std Err

Inventory

-2.60

0.119

Price

-0.985

0.046

Sale

6.48

0.525

Price*Sale

-0.583

0.053

Display

0.113

0.134

Feature

0.141

0.115

Regression includes household, store, and brand fixed effects.

43

Table 3: Estimates of Utility Coefficients

Coefficient

Mean

95% Conf. Int.

Variance

95% Conf. Int.

SK Taste (γ)

5.862

(5.44, 6.316)

1.895

(1.618, 2.209)

COS Taste (γ)

5.763

(5.36, 6.193)

1.749

(1.526, 1.994)

Curvature (β)

-0.797

(-0.823, -0.768)

-

-

Inv Cost Linear

-5e-04

(-0.00072, -0.00032)

-

-

-1.6e-05

(-2.3e-05, -9e-06)

-

-

Price (αi )

0.028

(0.026, 0.03)

0.005

(0.0046, 0.006)

Carrying Cost

-2.346

(-2.456, -2.245)

-

-

Inv Cost Quadratic

This table shows the average estimates of population mean coefficients (column 1) and of variance coefficients (column 3), as well as 95% confidence intervals around those coefficients (columns 2 and 4). Dashes in the variance columns indicate a coefficient is fixed across the population. The means and confidence intervals are taken over 10,000 Gibbs draws. 13,000 Gibbs draws are taken in total, and the first 3,000 are excluded to remove the influence of starting points.

44

Table 4: Predicted Transition Probabilities between Sale and Non-Sale Weeks Starkist: Prob(sale|sale) Min.

1st Qu.

Median

Mean

3rd Qu.

Max.

0.3965

0.4814

0.4849

0.4896

0.4987

0.6121

Starkist: Prob(sale|nonsale) Min.

1st Qu.

Median

Mean

3rd Qu.

Max.

0.07765

0.13746

0.16493

0.15863

0.18402

0.24716

COS: Prob(sale|sale) Min.

1st Qu.

Median

Mean

3rd Qu.

Max.

0.3479

0.4889

0.5645

0.5355

0.5944

0.7125

COS: Prob(sale|nonsale) Min.

1st Qu.

Median

Mean

3rd Qu.

Max.

0.1224

0.1900

0.3399

0.2846

0.3499

0.3600

45

Table 5: Summary of Price Indexes

Base

First Week

First Month

Prior Year

Average

S.D.

Average

S.D.

Average

S.D.

Dynamic Model Index

79.36

18.68

104.28

21.3

81.97

16.32

Laspeyres

95.87

6.19

110.91

10.2

86.68

6.92

Fixed Base Geo.

94.96

5.83

109.31

9.7

85.98

6.94

Fixed Base Tornqvist

89.73

6.97

103.68

9.66

83.24

7.43

Chained Laspeyres

218.59

82.14

-

-

-

-

Chained Tornqvist

60.35

19.19

-

-

-

-

-

-

106.59

6.06

84.31

-

Feenstra and Shapiro (2003) Index

Entries describe the average index (columns 1, 3 and 5) or the standard deviation (columns 2, 4 and 6) over the course of the sample year, excluding the base period. In columns 1 and 2 the base period is the first week of the data, in columns 3 and 4 it is the first month of the data, and in the last two columns purchases made in the prior year and modal prices in the prior year are used to construct expenditure weights. For the Prior Year columns, all indexes are rebased so that in the first week of the data they are 1, and the first week is excluded in the average and standard deviation columns. The dynamic index differs slightly from the dynamic index in the first two columns because the average price in the prior year is used to rebase the index. More details on the index calculations are provided in the main text of the paper.

46

300 250 200 100

150

Quantity (Cans)

65 60 50

50

55

Price (Cents)

70

Observed Trend Line

0

10

20

30

40

50

0

10

Week

20

30

40

50

Week

80 75

70 60

70

50

60

80

100

55

300 100

50

Starkist COS

0

Quantity

Week

65

40

Price (cents)

20

60

Price (cents)

80

Figure 1: Average Prices (Left) and Quantities (Right) Over Sample Period

20

40

60

80

100

20

Week

40

60

80

100

Week

Figure 2: Left Panel: Prices and Quantities of Starkist for a Single Store. Right Panel: Prices of Starkist and Chicken of the Sea for a Single Store

47

!

Beginning! of!period!t!

Choose! quantity,!xijt!of! each!brand!j!

Observe!εit!

Choose! consumption,!cijt!

End!of! period!t!

4

6

0.000 0.005 0.010 0.015 0.020 0.025 0.030

Figure 3: Timing of Purchase and Consumption within a Period

−2

0

2

γ1 γ2 β CC

0

2000

4000

6000

8000

10000

12000

sc0 sc1 α

0

2000

4000

Draw

6000 Draw

Figure 4: Plots Of MCMC Draws

48

8000

10000

12000

1.4 1.2 1.0 Index

0.8 0.6 0.4 0

10

20

30

40

50

Week

Figure 5: Dynamic Price Index for Canned Tuna. Base period used is the first week of sample.

2000 1500 0

500

1000

Frequency

2500

3000

Dotted lines indicate upper and lower 95% confidence intervals.

0.7

0.8

0.9

1.0

Average Estimated Index

Figure 6: Distribution Of Average Dynamic Index Across Gibbs Draws (Base period: first week of data)

49

1.4 1.2 Index

1.0 0.8 0.6

Dynamic Index FS Index Tornqvist 0

10

20

30

40

Week

Figure 7: Comparison of Dynamic Index to FS and Tornqvist Indexes (Base period: first month of data)

50