What is ’Firm Heterogeneity’ in Trade Models? The Role of
Quality, Scope, Markups, and Cost∗

Colin Hottman
Columbia University†
Stephen J. Redding
Princeton University and NBER‡David E. Weinstein
Columbia University and NBER§
September 23, 2014

Abstract
We develop and structurally estimate a model of heterogeneous multiproduct firms that canbe used to decompose the firm-size distribution into the contributions of costs, quality, markups,
and product scope. Using Nielsen barcode data on prices and sales, we find that variation in firm
quality and product scope explains at least four fifths of the variation in firm sales. We show thatthe imperfect substitutability of products within firms, and the fact that larger firms supply moreproducts than smaller firms, implies that standard productivity measures are not independent ofdemand system assumptions and probably dramatically understate the relative productivity of
the largest firms. Although most firms are well approximated by the monopolistic competition
benchmark of constant markups, we find that the largest firms that account for most of aggregate
sales depart substantially from this benchmark, and exhibit both variable markups and substantial
cannibalization effects.

JEL CLASSIFICATION: L11, L21, L25, L60
KEYWORDS: firm heterogeneity, multiproduct firms, cannibalization effects∗ We are grateful to Andrew Bernard, Swati Dhingra, Cecilia Fieler, Oleg Itskhoki, Bernard Salanié and conference andseminar participants at ERWIT, Dartmouth and Princeton for helpful comments. Thanks to Ildiko Magyari for providing
excellent research assistance. David Weinstein would like to thank the NSF (Award 1127493) for generous financial support. We also would like to thank GS1 for providing us with a concordance between the barcode data and firm identifiers.
The usual disclaimer applies. All results are calculated based on data from The Nielsen Company (US), LLC and provided
by the Marketing Data Center at The University of Chicago Booth School of Business.
† 420 W. 118th Street, MC 3308, New York, NY 10027. Email: cjh2164@columbia.edu.
‡ Fisher Hall, Princeton, NJ 08544. Email: reddings@princeton.edu.
§ 420 W. 118th Street, MC 3308, New York, NY 10027. Email: dew35@columbia.edu.

11Introduction

Why are some firms larger than others? Some companies, such as the Coca Cola Corporation, generate billions of dollars of sales and dominate the markets in which they operate. Other companies
account for only a small fraction of the sales of their larger competitors. What explains these vastdifferences in firm performance? Answering this question is important for quantifying equilibrium
models of firm heterogeneity developed in the recent trade and macro literatures and for understanding the relationship between microeconomic firm performance and macroeconomic outcomes.
Recent research on firm heterogeneity in trade and macroeconomics (e.g. Melitz 2003, Feenstra 2014, Manova and Zhang 2012) points to four components of firm heterogeneity: costs, quality,
markups and product scope (i.e., the number of products produced by firms). We develop a structural model of heterogeneous multiproduct firms that can be used to decompose the firm-size distribution into the relative contributions of each component. We first use this framework to structurally
estimate elasticities of substitution across varieties between and within multiproduct firms. We next
implement our model-based decomposition for the just over 50,000 firms that supply goods with barcodes in the Nielsen HomeScan Database in a typical quarter. This decomposition uses the structureof the model to isolate different margins in the data without making assumptions about how thosemargins are related to one another (as in the business cycle decomposition of Chari, Kehoe and McGratten 2007 in the macroeconomics literature). Our framework requires only price and expenditure
data and hence is widely applicable. We separate out the contributions of price and quality using theexclusion restriction that marginal cost only affects firm sales through price. In contrast, quality is ademand shifter that shifts sales conditional on price.1
Our results point to quality differences, which we define as average consumer utility per physicalunit of output, as being the principal reason why some firms are successful in the marketplace andothers are not. Depending on the specification considered, we find that 50-70 percent of the variancein firm size can be attributed to differences in quality, about 23-30 percent to differences in productscope, and less than 24 percent to cost differences. When we turn to examine time-series evidence,
the results become even more stark. Virtually all firm growth can be attributed quality improvements with most of the remainder due to increases in scope. These results suggest that most of what
economists call differences in revenue productivity reflects differences in quality rather than cost.
Our framework uses a nested constant elasticity of substitution (CES) utility system that allows
the elasticity of substitution between varieties within a firm to differ from the elasticity of substitution
between varieties supplied by different firms. Our choice of this CES demand structure is guided byits prominence, tractability and empirical feasibility. Across international trade, economic geography,
and macroeconomics, there is little doubt that this framework is the preferred approach to modelingproduct variety. Since our approach nests the standard CES-monopolistic competition model as a
1 Our definition of quality is standard in the literature. For example Sutton (1986) defines “’vertical’ product differentiation...has the defining property that if two distinct products are offered at the same price, then all consumers prefer thesame one (the higher-quality product)”2

special case, we can compare our results with those that would obtain in a standard trade, macro or
economic geography model. However, we generalize this standard model to allow firms to supply
multiple products (a pervasive feature of our data) and to have market power (since the largest firms
in our data are far from being measure zero).
Incorporating these features into our structural model yields a number of additional insights.
Our model makes clear a conceptual problem in the estimation of firm productivity that is likely
to bias existing estimates. Most productivity estimates rely on the concept of real output, which is
calculated by dividing nominal output by a price index. However, the formula for any economicallymotivated price index, which is the same as a unit expenditure function, is dependent on implicit
assumptions about how the output of firms enters utility. Thus, one cannot move from nominal output
to real output without imposing assumptions about the structure of the demand system.2 Our results show
theoretically and empirically that the CES measure of a multiproduct firm’s price is highly sensitive
to how differentiated its output is and how many products it supplies. The sensitivity of the CES
price index to demand parameters, such as the elasticity of substitution and whether multiproductfirms exist, implies that estimates of real output are equally sensitive to these demand parameters.
We show that if demand has a nested CES structure, conventional measures of real output will havea downward bias that rises with firm size with an elasticity of around one third. In other words, real
output variation is substantially greater than nominal output variation. This bias also implies thattrue productivity differences are much larger than conventionally measured productivity differences.
The bias is driven by two features of reality that are typically ignored in most analyses. First,
most analyses treat producers as single-product firms so they can avoid complications arising fromthe challenges of measuring the real output of multiproduct firms. However, our results indicatethat multiproduct firms are the norm. For example, we document that 70 percent of firms supply
more than one barcode and these firms account for more than 99 percent of output in their sectors.3
Therefore truly single product firms account for a negligible share of sales in our data. Second, weshow that if the output of multiproduct firms is differentiated, the common assumption that total
firm output is simply the sum of the output of each good understates real output for multiproductfirms, and the degree of this downward bias rises in the number of products supplied.
Our framework also provides a new metric for quantifying the extent to which a firm’s products
are differentiated from those of its rivals. If a firm’s products are perfectly substitutable with eachother but not with those of other firms, then 100 percent of a new product’s sales will come fromthe firm’s existing sales, which implies a cannibalization rate of one. However, if a firm’s market
share is negligible (as it is for most firms) and its products are as differentiated from each otheras they are from products supplied by other firms, none of a new product’s sales will come at the2 This point was stressed by Aristotle in his Nicomachean Ethics (Book V, Section 5): ”Demand holds things together asa single unit.... In truth it is impossible that things differing so much should become commensurate, but with reference todemand they may become so.” (see http://classics.mit.edu/Aristotle/nicomachaen.5.v.html)
3 By contrast, U.S. Census of Manufactures data indicates that producers of multiple five-digit Standard Industrial Classification (SIC) products account for 37 percent of firms and 87 percent of shipments (Bernard, Redding and Schott 2010).
The difference comes from defining single product firms as one-industry firms as opposed to one-barcode firms.

3expense of the firm’s other products, which implies a cannibalization rate of zero. We estimate thatthe cannibalization rate for the typical firm is about 50 percent, indicating that although products
supplied by the same firm are more substitutable with each other than with those of other firms, it is
not correct to think of them as perfect substitutes.
We find that the monopolistic competition benchmark of atomistic firms with constant markups
provides a good approximation for the vast majority of firms. The reason is simple. Most firms
have trivial market shares and hence are unable to exploit their market power. However, there issubstantial variation in markups for the very largest firms that account for disproportionate sharesof aggregate sales. This variation is greater under quantity competition than under price competition.
In most sectors, the largest firm has a market share above 20 percent, which enables it to charge amarkup that is thirty percent higher than that of the median firm under price competition and doublethat of the median firm under quantity competition. We use the model to undertake counterfactuals,
in which we show that these variable markups for the largest firms have quantitatively relevant
effects on aggregate welfare and the firm sales distribution.
The remainder of the paper is structured as follows. Section 2 reviews the related literature.
Section 3 discusses the data. Section 4 introduces the model. Section 5 uses the structure of the model
to derive moment conditions to estimate elasticities of substitution and undertake our decompositionof firm sales. Section 6 presents our estimation results. Section 7 undertakes counterfactuals. Section
8 concludes.

2

Related LiteratureOver the last decade, the fields of international trade and macroeconomics have undergone a transformation as the dissemination of micro datasets and the development of new theories has led toa shift in attention towards firm heterogeneity. Existing research has suggested a number of candidate explanations for differences in firm performance, including differences in production efficiency(e.g. Melitz 2003), product quality (e.g. Eslava, Fieler and Xu 2014, Johnson 2012, Khandelwal 2010,
Schott 2004), markups (e.g. De Loecker and Warzynski 2012, De Loecker, Goldberg, Khandelwal
and Pavcnik 2014), fixed costs (e.g. Das, Roberts and Tybout 2009), and the ability to supply multiple products (e.g. Arkolakis and Muendler 2010, Bernard, Redding and Schott 2010, 2011, Eckel,
Iacavone, Javorcik and Neary 2013, Eckel and Neary 2010, Mayer, Melitz and Ottaviano 2014). While
existing research typically focuses on one or more of these candidate explanations, they are all likely
to operate to some degree in the data. We develop a general theoretical model that incorporates eachof these candidate explanations and estimate that model structurally using disaggregated data on
prices and sales by firm and product. We use the estimated model to provide evidence on the quantitative importance of each source of firm heterogeneity and the ways in which they interact with one
another.
In much of the literature on firm heterogeneity following Melitz (2003) productivity and product4

quality are isomorphic. Under the assumption of CES preferences and monopolistic competition,
productivity and product quality enter equilibrium firm revenue in exactly the same way. However,these different sources of firm heterogeneity have different implications for firm revenue conditionalon prices (e.g. Berry 1994, Khandelwal 2010). While productivity only affects firm revenue throughprices, product quality is a demand-shifter that shifts firm revenue conditional on prices. An advantage of our approach is that we observe prices and sales in our data, and hence we are able toseparate these two sources of dispersion of firm sales.4
Most of the existing research on firm heterogeneity in trade and macroeconomics has assumed
that firms are atomistic and compete under conditions of monopolistic competition (e.g. Melitz 2003
and Melitz and Ottaviano 2008). In contrast, a small number of papers have allowed firms to be large
relative to the markets in which they operate (e.g. Atkeson and Burstein 2008, Amiti, Itskhoki andKonings 2012, and Edmond, Midrigan and Xu 2012). When firms internalize the effects of their decisions on market aggregates, they behave systematically differently from atomistic firms. Even under
CES demand, firms charge variable mark-ups, because each firm internalizes the effects of its pricing
decisions on market price indices and these effects are greater for larger firms.5 Furthermore, sincefirms are of positive measure, idiosyncratic shocks to these “granular” firms can affect aggregate
outcomes, as in Gabaix (2011) and di Giovanni and Levchenko (2012). In contrast to these papers,
we structurally estimate a model of heterogeneous firms, and show how it can be used to recover thedeterminants of the dispersion of firm sales and undertake counterfactuals.
To the extent that such large firms supply multiple products, they take into account the effect of
introducing new varieties on the sales of existing varieties. Most of the existing theoretical researchon multiproduct firms in trade and macroeconomics has abstracted from these cannibalization effects
by again assuming atomistic firms (e.g. Agur 2010, Allanson and Montagna 2005, Arkolakis andMuendler 2012, Bernard, Redding and Schott 2010, 2011, Mayer, Melitz and Ottaviano 2014 andNocke and Yeaple 2006). Important exceptions that explore cannibalization effects theoretically are
Feenstra and Ma (2008), Eckel and Neary (2010), and Dhingra (2013). In contrast to these theoreticalstudies, we develop a structural model that can be used to provide quantitative evidence on how
important it is to introduce such cannibalization effects into models of firm behavior.
Most existing empirical research on multiproduct firms has measured products using production
classification codes (e.g. around 1,500 five-digit Standard Industrial Classification (SIC) categories
in Bernard, Redding and Schott 2010) or trade classification codes (e.g. around 10,000 Harmonized
System codes in Bernard, Jensen and Schott 2009). In contrast, we measure products at a much finerlevel of resolution using what we term “barcodes”—either 12-digit Universal Product Codes (UPCs)
4 Our CES formulation of market demand can be derived from a discrete choice model of the demands of individual
consumers, as shown in Anderson, de Palma and Thisse (1992).
5 Our model generates variable mark-ups without a “choke price” above which demand is zero. Therefore this model
lies outside the classes considered by Arkolakis, Costinot and Rodriguez-Clare (2012) and Arkolakis, Costinot, Donaldson
and Rodriguez-Clare (2012), in which aggregate statistics such as the trade share and trade elasticity are sufficient statistics
for the welfare gains from trade.

5

or 13-digit European Article Numbers (EANs)—in scanner data.6 This measure corresponds closely
to the level at which product choice decisions are made by firms, because it is rare for an observablechange in product attributes to occur without the introduction of a new barcode.
Our econometric approach builds on the literature estimating elasticities of substitution andquantifying the contribution of new varieties to welfare following Feenstra (1994) and Broda andWeinstein (2006). We extend this estimation approach to allow firms to be of positive measure relative to the market and to supply multiple products. We show how this extended approach can beused to recover demand heterogeneity, marginal cost heterogeneity, variable markups, and cannibalization effects. While a number of other studies have used scanner data, including Broda andWeinstein (2010) and Chevalier et al. (2003), so far these data have not been used to estimate astructural model of heterogeneous multiproduct firms and quantify the sources of dispersion in thefirm-size distribution.

3

Data

Our data source is the Nielsen HomeScan database which enables us to observe price and sales
information for millions of products with a barcode.7 Barcode data have a number of advantages forthe purpose of our analysis. First, since barcodes are inexpensive but provide sellers access to storeswith scanners as well as internet sales, producers have a strong incentive to purchase barcodes for
all products that have more than a trivial amount of sales.8 This feature of the data means that it islikely we observe all products supplied by firms. Second, since assigning more than one product toa single barcode can interfere with a store’s inventory system and pricing policy, firms have a strong
incentive not to reuse barcodes. This second feature of the data ensures that in general identicalgoods do not have different barcodes. Thus, a barcode is the closest thing we have empirically to thetheoretical concept of a good. Finally, since the cutoff size for a firm is to make a sale rather than anarbitrary number of workers, we actually observe something close to the full distribution of firms.
Nielsen collects its barcode data by providing handheld scanners to on average 55,000 households
per year to scan each good purchased that has a barcode.9 Prices are either downloaded from thestore in which the good was purchased, or they are hand entered, and the household records any
deals used that may affect the price. These households represent a demographically balanced sampleof households in 42 cities in the United States. Overall, the database covers around 30 percent of all
6 Recently, the 12-digit UPCs have been upgraded to 13-digit EAN-13s (European Article Numbers). The extra digit of

the EAN-13 allows for more products and firms.
7 Our results are calculated based on data from The Nielsen Company (US), LLC and provided by the Marketing Data
Center at The University of Chicago Booth School of Business. Information on availability and access to the data is available
at http://research.chicagobooth.edu/nielsen
8 GS1 provides a company with up to 10 barcodes for a $250 initial membership fee and a $50 annual fee. There are deep
discounts in the per barcode cost for firms purchasing larger numbers of them (see http://www.gs1us.org/get-started/imnew-to-gs1-us)
9 The data for 2004 through 2006 come from a sample of 40,000 households, while the data for 2007 through 2011 comefrom a sample of 60,000 households.

6expenditure on goods in the CPI.10 We collapse the household dimension in the data and collapse
the weekly purchase frequency in order to construct a national quarterly database by barcode on the
total value sold, total quantity sold, and average unit value.11
Defining products as goods with barcodes has a number of advantages over defining goods byindustry classifications. While industry classifications aggregate products produced by a firm within
an industry, our data reveals that large firms typically sell hundreds of different products within evena narrowly-defined sector. In other words, while prior work equates multiproduct firms with multiindustry firms, our data hews extremely closely to what an economist would call a multiproductfirm. In principle, it could be appropriate to aggregate all output of a firm within an industry into asingle good (Is whole milk the same as skim milk? Is a six-pack of soda the same as a two-liter bottle?)
However, as we will show theoretically, the validity of this procedure depends on the elasticity of
substitution between the products supplied by a firm, which is an empirical question.
Instead of relying on product data for a single industry, we observe virtually the entire universe ofgoods purchased by households in the sectors that we examine. Our database covers approximately
1.4 million goods purchased at some point by households in our sample. The data were weighted byNielsen to correct for sampling error. For example, if the response rate for a particular demographic
category is low relative to the census, Nielsen re-weights the averages so that the price paid and thequantity purchased is representative of the United States as a whole.
Nielsen organizes the barcodes into product groups according to where they would likely bestocked in a store. The five largest of our 100 product groups are carbonated beverages, pet food,paper products, bread and baked goods, and tobacco. Output units are common within a productgroup: typically volume, weight, area, length, or counts. Importantly, we deflate by the number ofunits in the barcode, so prices are expressed in price per unit (e.g., price per ounce). When the units
are in counts, we also deflate by the number of goods in a multipack, so, for instance, we would
measure price per battery for batteries sold in multipacks. While about two thirds of these barcoded
items correspond to food items, the data also contains significant amounts on information about
non-food items like medications, housewares, detergents and electronics. The first few digits of thebarcode identify the manufacturer.12
Table 1 presents descriptive statistics for our sample of firms and barcodes in the 100 productgroups. We weight the data by the sales of the product group in each year and average across years
10 For further discussion of the Nielsen data, see Broda and Weinstein (2010).
11 One question that naturally arises is whether it is appropriate to think of US firms competing in a national market orin different regional markets. Handbury and Weinstein (forthcoming) argue there is not that much heterogeneity in theprice levels of barcoded goods across cities. Similarly, we find that national and regional expenditure shares and prices arestrongly correlated in the data. National brands and firms (brands and firms that sell in 10 or more cities) account for 97
and 98 percent of expenditure respectively in cities with more than 1,000 households in the AC Nielsen data. Regressing
city prices in these cities on national prices, quarter-year dummies, and barcode fixed effects, we find a partial R2 (i.e., an
R2 not inclusive of the fixed effects) of 0.88 and 0.73 for price levels and price growth respectively. These results suggestthat most of the variation in U.S. consumer prices reflects national shocks not local ones.
12 Firm prefixes are usually seven digits but they can be as long as eleven digits, so we use GS1 data to map the barcodes
into firm identifiers. We could not obtain a firm identifier for just under 5 percent of the barcodes, and so we dropped
these products.

7

because sectors like carbonated beverages are much larger economically than sectors like “feminine
hygiene.” There are on average 518 firms in each product group with 90 percent of the productgroups having more than 200 firms. We see enormous range in firm product scope. The median
number of products supplied by a firm is 3 and the average is 13. On average, 670,000 different
UPCs were sold each quarter.
One of the most striking facts displayed in this table is the degree of firm heterogeneity. Thisis manifest in the skewness of the size and barcode distributions. The largest firm in an industry
typically sells 2600 times more than the median firm. We see similar patterns in terms of productscope and sales per product. The firm with the most products typically has 134 times more products
than the firm with the median number of barcodes, and the barcode with the most sales on average
generates almost almost a thousand times more revenue than the revenue of the median barcode.
Table 1: Sample Statistics

No. Firms per Product Group
Firm Sales
Log Firm Sales
No. of UPCs per Firm
UPC Sales

MeanMedianStd. Dev.

518
4141
12
13
314

416
152
12
3
40

309
25836
3
32
1224

10th
Percentile
200
5
8
1
2

90th
Percentile
942
4599
15
32
650

Max Value1434
402350
19
405
36590

Note: Weighted by product group. Firm and UPC sales in thousands. Calculations on data from The Nielsen Company (US), LLC andprovided by the Marketing Data Center at The University of Chicago Booth School of Business.

We see in Table 2 that almost 90 percent of sales in a product group was produced by firms withsales in the top decile of sales. Table 3 provides a more detailed description of this firm heterogeneity
by focusing on the ten largest firms in each product group (where we weight the averages by thesales of the product group). Table 3 reveals an almost fractal nature of firm sales. Around two-thirdsof all of the sales of firms in the top decile is produced by the ten largest firms (which on average only
account for 1.9 percent of firms in this decile). While on average half of all output in a product groupis produced by just five firms, 98 percent of firms have market shares of less than 2 percent. Thus, thetypical sector is characterized by a few large firms and a vast competitive fringe comprised of firms
with trivial market shares. A second striking feature of the data is that even the largest firms are notclose to being monopolists. The largest firm in a product group on average only has a market shareof 23 percent. Finally, the data reveal that firms in the top decile of sales are all multiproduct firms,
firms on average supply 67 different goods, and the largest firms supplying hundreds of goods.
The extent of multiproduct firms can be seen more clearly in Table 4, which shows the results ofsplitting the data by the number of UPCs supplied by a firm. While single product firms constituteabout one third of all firms on average, these firms account for less than one percent of all output.
In other words virtually all output is supplied by multiproduct firms. Moreover, the fact that overninety percent of output is sold by firms selling eleven or more varieties and two-thirds of all output
is supplied by firms selling more than fifty varieties, suggests that single product firms are more the8Table 2: Size Distribution by Decile
Ranked Decile
1
2
3
4
5
6
7
8
9
10

Decile Market
Share

Mean Firm
Market Share

Mean Log
Firm Sales

Avg. Std. Dev.
log UPC Sales

87.82
7.03
2.67
1.25
0.63
0.32
0.17
0.08
0.03
0.01

2.65
0.22
0.08
0.04
0.02
0.01
0.00
0.00
0.00
0.00

16.4
14.5
13.5
12.7
12.0
11.4
10.7
9.9
8.9
7.3

1.8
1.6
1.6
1.5
1.4
1.3
1.2
1.1
0.9
0.6

Mean No.
UPCs per
Firm
67.1
25.5
14.4
8.9
5.8
3.9
2.8
2.0
1.5
1.2

Median No.
UPCs per
Firm
44.1
20.5
11.6
7.1
4.5
3.1
2.2
1.5
1.1
1.0

Note: Largest decile is ranked first. Weighted by product group. UPC counts and mean firm sales are across firm within each decile.
Calculations on data from The Nielsen Company (US), LLC and provided by the Marketing Data Center at The University of Chicago
Booth School of Business.

Table 3: Size Distribution by Firm Rank
Firm Rank
1
2
3
4
5
6
7
8
9
10

Firm Market Share (%)

Log Firm Sales

No. UPCs per Firm

22.7
12.5
7.8
5.4
4.1
3.3
2.8
2.4
2.1
1.9

19.4
18.8
18.4
18.1
17.8
17.6
17.4
17.2
17.1
17.0

296.7
193.8
155.2
119.5
115.5
111.2
105.3
92.8
85.8
73.3

Note: Largest firm is ranked first. Weighted by product group. Calculations on data from The Nielsen Company (US), LLC and provided
by the Marketing Data Center at The University of Chicago Booth School of Business.

9

exception than the rule.
Large firms not only sell more products but they also sell a lot more of each product. The penultimate column of Table 4 documents that while the typical barcode sold by a single-product firm
only only brings in $65,4000 in revenue, the typical barcode sold by a firm selling over one hundredbarcodes brings almost twice as much ($122,500). In other words, large firms not only supply moreproducts but they sell more of each product. If firms differed only in the fixed cost of adding new
varieties, one would not expect to see large firms sell more of each variety. The fact that they dostrongly suggests that large firms must also differ in the marginal cost or quality of their output.
Table 4: Size Distribution by Number of UPCs
No. of UPCs
1
2–5
6–10
11–20
21–50
51–100> 100

No. of Firms

Share of Value (%)

Mean Sales(Thousands)

Median Sales(Thousands)

Avg. St. Dev. UPC
Sales

165
171
63
48
44
18
10

0.9
3.7
4.9
8.9
19.1
18.7
43.9

73
327
1207
2924
7823
23385
123156

65447
69802
56096
66186
66474
119190122528

1.31
1.50
1.57
1.65
1.73
1.80

Note: Weighted by product group. To get last column, calculate standard deviation over log UPC sales by product group, then takeweighted average across product groups. Calculations on data from The Nielsen Company (US), LLC and provided by the Marketing
Data Center at The University of Chicago Booth School of Business.

In sum, our overview of the data reveals some key features that we model in our empirical exercise. First, the vast majority of firms have trivial market shares, which means that if we believethat all firms may have some market power, we need to work with demand systems that do not imply that firms with trivial market shares have trivial markups. Second, the fact that most economic
output is produced by multiproduct firms impels us to build this feature directly into the estimation
system and allow for both differences in the fixed cost of developing new products as well as costand quality differences among products. Finally, the fact that there are firms with non-trivial market
shares implies that at least some firms are likely to have market power. To the extent that these large
firms internalize the effects of their price choices on market aggregates, this concentration of market
shares will induce departures from the monopolistic competition benchmark.

4

Theoretical Framework

Our choice of functional forms is motivated not only by the data issues we identified above but
also by some theoretical concerns. Since much of the theoretical literature in international tradeand economic geography has worked with CES models, we want our results to nest this case (atleast within product groups), so that our results easily can be compared with existing work in trade,
macro and regional economics. However, we also need a framework that allows the elasticity of
substitution for products supplied by the same firm to be different than that for products suppliedby different firms, thereby to allow for the possibility of cannibalization effects. Finally, we need a10

setup that can be applied to firm-level data without imposing implausible assumptions or empiricalpredictions. This last requirement rules out two common demand systems: linear demand and thesymmetric translog. A linear demand system would be problematic in our setting because one needs
to impose the assumption that income elasticities are equal to zero and the estimation of marginal
costs derived from a linear demand system can often result in negative values. While the symmetrictranslog demand system improves on the linear demand system in this regard, it is a difficult system
to implement at the firm level because it has the undesirable result that firms with negligible market
shares have negligible markups.
Our estimation strategy therefore is based on an upper level Cobb-Douglas demand system
across product groups with CES nests below it. The upper-level Cobb-Douglas assumption dramatically simplifies the estimation process because its implication of constant product group expenditure
shares allows us to assume that firms supplying different classes of products, e.g. carbonated beverages and pet food, do not interact strategically. However, the nested CES structure within broadly
defined “product groups” allows for strategic interactions among firms supplying similar products.
Within this structure the real consumption of each product group is composed of the real consumption of each firm’s output, which itself is made up of the consumption of each of the varieties suppliedby the firm.

4.1

DemandIn order to implement this approach, we assume that utility, Ut , at time t is a Cobb-Douglas aggregateof real consumption of each product group, Cgt :Ut = ∏ Cgtgt ,∑ ϕgt = 1,

ϕ

g∈ G

g∈ G

where g denotes each product group, ϕ gt is the share of expenditure on product group g at time t.13
Within product groups, we assume two CES nests for firms ( f ) and UPCs (u). The first nest for firms
enables us to connect with the existing literature on measuring firm productivity. The second nest for
UPCs enables us to incorporate multi-product firms. Therefore the consumption indices for productgroups and firms can be written as follows for tier of utility j ∈ { g, f }:"Cjt =∑ ( ϕkt Ckt )σKg −1σKg# σ σKg−1
Kg,

σKg > 1, ϕkt > 0,(1)

k ∈ Ktwhere k is the tier of utility below j (i.e., if j corresponds to firm f , k corresponds to UPCs u, and if jcorresponds to a product group g, k corresponds to firms f ); Kt is the set of varieties k; Ckt denotes
consumption of variety k; ϕkt is the perceived quality of variety k; σKg is the constant elasticity of
substitution across varieties k for product group g. In other words, the real consumption in any
13 While we allow the Cobb-Douglas parameters ϕ to change over time, we find that product-group expenditure sharesgt
are relatively constant over time, which suggests that a Cobb-Douglas functional form with time-invariant parameters
would provide a reasonable approximation to the data.

11

product group, g, is a function of the consumption of each firm’s output, C f t , weighted by the quality
consumers assign to that firm’s physical output at time t, ϕ f t , and adjusted for the substitutability ofthe output of each firm, σFg .14 Similarly, we can write the sub-utility derived from the consumptionof a firm’s output, C f t , as a function of the consumption of each UPC (i.e. barcode) supplied by that
firm, Cut , multiplied by the quality assigned to the physical output of that barcode, ϕut , and adjusted
by the substitutability between the various varieties supplied by the firm, σUg .15
There are a few features of this specification that are worth noting. First, if the elasticity of substitution across varieties supplied by a firm, σUg , is finite, then the real output of a multiproduct firm
is not equal to the sum of the outputs of each product. For example, if the only reason firms differ insize is that larger firms supply more varieties than smaller firms, then assuming firm real output isthe sum of the output of each variety will tend to understate the relative size of larger firms. This size
bias is a topic that we will explore in much more detail later. Second, for much of what follows, weare focused on the sales decomposition within a given product group (g), so for notational simplicity
we can suppress the g subscript on σFg and σUg until we need it again to pool results across sectors
in Section 4.6. Third, we would expect (but do not impose) that the elasticity of substitution acrossvarieties is larger within firms than across firms, i.e., σU ≥ σF . When the two elasticities are equal,
our system will collapse to a standard CES at the product-group level, and if the inequality is strict,
we will show that our setup features cannibalization effects. While our nested CES specification provides a parsimonious and natural approach to modeling multi-product firms, we return to consider
the robustness of our results to alternative nesting structures in subsection 6.6 below.
Third, we allow firms to be large relative to product groups (and hence internalize their effects on
the consumption and price index for the product group). But we assume that the number of productgroups is sufficiently large that each firm remains small relative to the economy as a whole (andhence takes aggregate expenditure Et and factor prices as given). Finally, since the utility function ishomogeneous of degree one in quality it is impossible to have a firm-quality, ϕ f t , that is independentof the quality of the varieties produced by that firm, ϕut . We therefore need to choose a normalization.
It will prove convenient to normalize the geometric means of the ϕut for each firm and the ϕ f t foreach product group to equal one:
14 A small number of firms operate across multiple product groups. We assume that consumers view each of these firm-

product-groups as a separate firm and that pricing and UPC introduction decisions are made for each firm-product-groupseparately. The Cobb-Douglas assumption for the upper tier of utility implies that no firm would have an incentive to
price strategically across firm-product-groups, because product group expenditure shares are fixed by parameters. From
now onwards, we refer to firm-product-groups as firms. These assumptions are also consistent with product groups being
quite distinct from one another (e.g. Carbonated Beverages versus Office Supplies) and the vast majority of firms being
active in only one product group.
15 Our definition of quality is the utility per common physical unit (e.g. utility per ounce of a firm’s output). However,variation in quality could either be interpreted as a difference in utility per physical unit or as variation in the number ofidentical-quality unobservable sub-units within a physical unit. For example, it is isomorphic to say that Firm A produces
products with twice the utility per ounce as Firm B and to say that 1/2 an ounce of Firm A’s product generates as much
utility as an ounce of Firm B’s product. In the latter case, an ounce of Firm A’s product would contain two “1/2 ounce
sub-units” each of which has identical quality to Firm B’s product. We think our interpretation of utility per physical unit
is the most natural for our data.

12

 N1



 N1



ft

gt

=  ∏ ϕ f t

 ∏ ϕut u ∈U f t

= 1,

(2)

f ∈ Fgt

where U f t is the set of varieties supplied by firm f at time t, and N f t is the number of elements of thisset; Fgt is the set of firms in product group g at time t, and Ngt is the number of elements of this set.
Thus, firm average quality (ϕ f t ) corresponds to a demand-shifter that affects all products suppliedby the firm proportionately, while product quality (ϕut ) is a demand-shifter that determines the relative
sales of individual products within the firm.
We can gain some intuition for this framework by by using the product group “carbonated beverages” as an example. Aggregate utility depends on the expenditure share (given by ϕ gt ) and amountof consumption of goods in the carbonated-beverages product group, (given by Cgt ). The utility derived from the consumption of carbonated beverages depends on the quality of Coke versus Perrier(ϕ f t ), the amounts of each firm’s real output consumed (C f t ), and the degree of substitutability between Coke and Perrier (σF ). Finally, the real amount of Coke or Perrier consumed (C f t ) depends on
the number of different types of soda produced by each company (U f t ), the quality of each of thesetypes of soda (ϕut ), the consumption of each variety of soda (Cut ), and how similar varieties of Coke(or Perrier) products are with other varieties offered by the same company (σU ).
It turns out that it also will be useful to also work with the exact price index for consumption:
"
Pjt =

∑

k∈K



Pkt
ϕkt

1−σk # 1−1σk.

(3)Without loss of generality, we index varieties k in a given year t from the largest in terms of sales tothe smallest, and we denote the variety with the largest sales in a given year by k.
Using the properties of CES demand, the expenditure share of variety k within tier j (Skt ) is equal
to the elasticity of the price index for tier j with respect to the price of variety k and is given by thefollowing expression:
Skt =dPjt Pkt( Pkt /ϕkt )1−σK=.
1−σK
dPkt Pjt∑k∈Kt ( Pkt /ϕkt )(4)Equation (4) makes clear exactly how we conceive of quality in this setup. Holding fixed prices, a
UPC with higher quality will have higher market shares. Similarly, holding fixed prices, firms that
supply higher quality goods will have greater market shares.
The role of both firm and product quality also can be seen by writing down the demand for the
output of each variety:σF −1 σU −σF −σU
Cut = ϕσf Ft −1 ϕσutU −1 Egt Pgt
Pf t
Put ,

(5)

where Egt denotes total expenditure on product group g at time t. Equation (5) is critical in determining how we can use this framework for understanding the different roles played by costs andquality for understanding the sales of a firm. While quality has a direct effect on consumer demand13independent of price, cost only affects consumer demand through price. Thus, the specification of
how cost and quality affect firm pricing decisions is crucial for our identification strategy.

4.2

Technology

We allow the costs of supplying products to the market to vary across UPCs, firm-product-groups
and firms. This specification encompasses both heterogeneity in productivity across firms (as in
Melitz 2003) and heterogeneity in productivity within firms (as in Bernard, Redding and Schott 2011).
All costs are incurred in terms of a composite factor input that is chosen as our numeraire. We assume
that the variable cost function is separable across UPCs and that supplying Yut units of output of
1 + δg

UPC u incurs a total variable cost of Aut (Yut ) = aut Yut, where aut is a cost shifter and δg > 0

parameterizes the convexity of marginal costs with respect to output. Once again, we will suppress
the g subscript on δg until Section 4.6. In addition, each firm faces a fixed market entry cost of H f t > 0(e.g. the fixed costs of headquarters operations) and a fixed market entry cost for each UPC suppliedof h f t > 0 (e.g. the fixed costs of product development and distribution).

4.3

Profit Maximization

In our baseline specification, we assume that firms choose prices under Bertrand competition, though
we also report results in which firms instead choose quantities under Cournot competition. Under our assumption of CES preferences, the decisions of any one firm only affect the decisionsof other firms through the product group price indices (Pgt ). Each firm chooses the set of UPCsnou ∈ u f t , . . . , u f t to supply and their prices { Pu } to maximize its profits:uftmax Π f t =N f t ,{ Put }∑ [ Put Yut − Au (Yut )] − N f t h f t − H f t ,(6)

u=u f twhere we index the UPCs supplied by the firm from the largest to the smallest in sales, and the total
number of goods supplied by the firm is denoted by N f t , where u f t = u f t + N f t .
Multiproduct firms that are large relative to the market internalize the effect of their decisions
for any one variety on the sales of their other varieties. From the first-order conditions for profit
maximization, we can derive the firm markup for each UPC, as shown in Appendix A.1:
ε ft
µft =
,
ε ft − 1(7)

where we define the firm’s perceived elasticity of demand as
ε f t = σF − (σF − 1) S f t = σF 1 − S f t + S f t ,(8)

and the firm’s pricing rule as
Put = µ f t γut ,δ
γut = (1 + δ) aut Yut,

where γut denotes marginal cost.

14

(9)

One of the surprising features of this setup is that markups only vary at the firm level within productgroups.16 The intuition is that the firm internalizes that it is the monopoly supplier of its real output,
which in our model equals real consumption of the firm’s bundle of goods, C f t . Hence its profit
maximization problem can be thought of in two stages. First, the firm chooses the price index (Pf t )to maximize the profits from supplying real consumption (C f t ), which implies a markup at the firmlevel over the cost of supplying real output. Second, the firm chooses the price of each UPC tominimize the cost of supplying real output (C f t ), which requires setting the relative prices of UPCs
equal to their relative marginal costs. Together these two results ensure the same markup across all
UPCs supplied by the firm within a product group. However, for firms that operate across multiple
product groups, prices are chosen separately for each product group. Therefore markups vary withinfirms across product groups.
The firm’s perceived elasticity of demand (8) is less than the consumer’s elasticity of substitution
between firms, σF . The reason is that each firm is large relative to the market and hence internalizes
the effect of its pricing choices on market price indices. When the firm raises the price of a UPC(Put ) and hence the firm price index (Pf t ), it reduces demand for that UPC and raises demand forthe firm’s other UPCs in (5). Furthermore, the rise in the firm price index also increases the productgroup price index (Pgt ), which raises demand for each UPC and for the firm as a whole in (5). Finally,
our assumption of a Cobb-Douglas functional form for the upper tier of utility together with theassumption that each firm is small relative to the aggregate economy ensures that total expenditureon each product group (Egt ) in demand (5) is constant and unaffected by firm decisions.
Although consumers have constant elasticity of substitution preferences (σF ), each firm perceives
a variable elasticity of demand (ε f t ) that is decreasing in its expenditure share (S f t ), as in Atkeson andBurstein (2008) and Edmond, Midrigan and Xu (2012). As a result, the firm’s equilibrium pricing rule(9) involves a variable markup (µ f t ) that is increasing in its expenditure share (S f t ). For a positive
equilibrium price (9), we require that the perceived elasticity of demand (ε f t ) is greater than one(firms produce substitutes), which requires that the elasticity of substitution between firms (σF ) issufficiently large. As a firm’s sales becomes small relative to the product group (S f t → 0), the markup (7) collapses to the standard constant mark-up of price over marginal cost under monopolisticcompetition with atomistic multiproduct firms and nested CES preferences (see for example Allanson
and Montagna 2005 and Arkolakis and Muendler 2010).
Using the above equilibrium pricing rule, overall UPC profits (Πut ) are equal to UPC variable
profits (πut N f t ) minus fixed costs. UPC variable profits in turn can be written in terms of UPC
revenues (Put Yut ), the markup (µ f t ), and the elasticity of costs with respect to output (ζ u ):
Πut = πut N f t − h f t ,
(10)



ζuµ f t − 1
dAu (Yut ) Yut
πut N f t = Put Yut − Au (Yut ) =Put Yut , where ζ u == 1 + δ. (11)ζuµ f tdYut Au (Yut )That is, πut N f t denotes the variable profits from UPC u when the firm supplies N f t UPCs.

.

16 This result would also hold if we substituted nested logit demand for nested CES

15

4.4

Cannibalization Effects

The number of UPCs supplied by each firm, N f t , is determined by the requirement that the increasein profits from introducing an additional UPC, u f t + 1, minus the reduction in profits from reducednosales of existing UPCs u ∈ u f t , . . . , u f t is less than the fixed cost of introducing the new UPC, h f t .
If a firm makes N f t products in equilibrium, then it must be the case that if it were to introduce a new
good, its profits would fall, i.e.,u f t +1

∑ πut N f t + 1 − N f t + 1 h f t

u=u f tuft<

∑ πut N f t − N f t h f t

u=u f t

uft





⇔ πu f t +1,t N f t + 1 − ∑ πut N f t − πut N f t + 1< h f t.

(12)

u=u f tIn the case where the number of UPCs is large and can be approximated by a continuous variable,
we obtain after some manipulation (see Appendix A.2) an expression for the “cannibalization rate”:

 
 
∂Yut N f t
σU − σF
σF − 1−=+S f t S N f t N f t > 0, f or σu ≥ σ f > 1.
(13)
∂N f t YutσU − 1σU − 1
where S N f t is the share of firm revenues from the new product when firm supplies N f t products, andS f t is the share of consumer expenditure on firm f . This cannibalization rate is defined as the partial
elasticity of the sales of existing products with respect to the number of products. This partial elasticity captures the direct effect of the introduction of a new product on the sales of existing products,
through the firm and product group price indices, holding constant the prices and marginal costs ofthese existing products.
The first term on the right hand-side captures the cannibalization rate within firms: The introduction of a UPC reduces the firm price index (Pf t ), which reduces the revenue of existing UPCs if
varieties are more substitutable within firms than across firms (σU > σF ). The second term captures
the cannibalization rate across firms: The introduction of the new UPC reduces the product-groupprice index (Pgt ), which reduces the revenue of existing UPCs if varieties are more substitutablewithin product-groups than across product-groups (σF > 1).
There are two useful benchmarks for understanding the magnitude of the cannibalization rate inequation (13). In both cases, it useful to think of the introduction of a “standardized” product thathas a market share equal to the average market share of the firms other goods (S N f t N f t = 1). If firms
are monopolistic competitors, S f t ≈ 0. Moreover, if we also assume that all products supplied by afirm are as differentiated among themselves as they are with the output of other firms, i.e., σU = σF ,then the cannibalization rate will be zero because all sales revenue arising from introducing a new
product will come from the sales of goods supplied by other firms. Thus, a world with monopolisticcompetition and equal product differentiation is a world with no cannibalization. Clearly, the cannibalization rate will rise if firms cease being small S f t > 0 or if goods supplied by the same firm
are more substitutable with each other than with goods supplied by different firms: σU > σF . At the16other extreme, we can assume that goods are “perfect substitutes” within firms, i.e., σU = ∞ > σ f , sothat varieties are differentiated across firms but there is no difference between varieties supplied by
the same firm. In this case the cannibalization rate will be 1 because any sales of a new product will
be exactly offset by a reduction in the sales of existing products. Thus the cannibalization rate provides a measure of where in the spectrum ranging from perfect substitutes to equal differentiation
the output of a firm lies.

4.5

The Sources of Firm Heterogeneity

In this section, we use the model to quantify the contribution of the different sources of firm heterogeneity to the dispersion in sales across firms. Nominal firm sales, E f t , is the sum of sales across
UPCs supplied by the firm:
E f t ≡ ∑ Put Cut .
ueU f tUsing CES demand (5), firm sales can be re-written as:
σF −1 σU −σF
Pf t
E f t = ϕσf Ft −1 Egt Pgt∑(
ueU f tPut 1−σU
.
)
ϕut

(14)

Using the firm price index (3) to substitute for PfσU −σF and taking logarithms, we obtain:




1−σU

Put
1 − σF,ln  ∑
ln E f t = (σF − 1) ln ϕ f t + ln Egt + (σF − 1) ln Pgt +1 − σUϕutueU
ft

where the final term summarizes the impact of productivity, markups, UPC quality, and the numberof products on firm sales. This final term can be further decomposed into the contributions of thenumber of UPCs supplied by a firm (N f t ), the geometric mean of marginal costs (eγ f t ), relative qualityeut ) /ϕut ), and the firm markup (µ f t ):adjusted marginal costs across UPCs ((γut /γln E f t = ln Egt + (σF − 1) ln Pgt
(15)




σF − 1ln N f t+ (σF − 1) ln ϕ f t +σU − 11−σU γ/γ̃σF − 11utft
 + (1 − σF ) ln µ f t ,+− (σF − 1) ln γ̃ f t +
ln 
∑


σU − 1N f t ueU
ϕut
ft

where
 N1



ft

γ̃ f t =  ∏ γut u ∈U f t

δ denotes marginal cost.
and recall that γut = (1 + δ) aut Yut

Equation (15) decomposes firm sales into seven terms that capture the various margins throughwhich firms can differ in sales. Clearly these margins are related to one another, since both thenumber of products and the markup are endogenous to firm quality, product quality and marginal17

cost. Nonetheless, the decomposition (15) isolates the direct effects of firm quality, product quality
and marginal cost from their indirect effects through the number of products and the markup.
Although our decomposition is exact for any co-movement in variables, it is easiest to obtain
intuition for this equation if we consider one-at-a-time, small movements in each variable so that
we can safely ignore interactions between different variables. We therefore will explain the intuition
for this equation in terms of small movements in each variable, and note that in general and in ourempirical implementation we will allow all variables to move simultaneously.
The seven terms in equation (15) can be grouped into four main elements. The first element,
contained in the first set of braces, captures market size and relative pricing. Our demand system
is homogeneous of degree one in product group expenditures, so firm sales rise one to one withaggregate expenditures. The second term captures the impact of the product group price index thatsummarizes the prices of competing varieties. Holding fixed a firm’s characteristics, an increase in
the product group price level of one percent will cause the firm’s sales to rise by (σF − 1) percent.
Here, the elasticity of substitution between firm’s output and the output of other firms, σF , plays the
crucial role of explaining how much a relative price movement affects firm sales.
“Total firm quality” is captured by the second two terms in braces, which capture the impacts of
the average quality of the firm’s output, “firm quality,”and the number of products the firm offers
holding fixed average quality, “product scope”. Consider two firms that supply the same numberof products, but one firm supplies higher quality varieties (measured in utility per unit), meaningthat ϕ f t > ϕ f 0 t and N f t = N f 0 t . Firm f will then have a higher market share; how much depends
on the elasticity of substitution between firm output, σF . For a larger value of this elasticity, a given
difference in firm average quality will translate into a larger difference in firm market share.
Now consider two firms that supply products of identical quality but one firm supplies more
UPCs than another (N f t > N f 0 t ). Here, it is easiest to think about this term in a symmetric worldin which all goods and firms have identical quality (ϕut = ϕ f t = 1) and identical marginal cost
(γut = γt ), so we can just focus on the role played by product scope. Although firms have identicalqualities, they do not have identical market shares because they differ in the number of products they
offer, i.e., ln N f t > ln N f 0 t . For example, if consumers treated all UPCs identically regardless of which
firm supplied them, i.e., σU = σF , firm f would sell ln N f t /N f 0 t percent more output than firm f 0 .
More generally, if the products supplied by a firm are more substitutable with each other than withthose of other firms, σU > σF , the percentage gain in sales accruing to a firm that adds a productwill be less than one reflecting the fact that a new product will cannibalize the sales of its existingproducts. Indeed the degree of cannibalization will depend on the magnitude of σU ; as this elasticity
approaches infinity, the cannibalization rate will approach one, and all sales of new products will
come from the sales of the firm’s existing products. Hence, in this limiting case of σu → ∞, adding
product scope will have no impact on sales.
The terms in the third set of braces capture the role played by marginal costs. These costs canbe divided into average marginal cost (γ̃ f t ) and “cost dispersion”. Average marginal costs (γ̃ f t )18

is the more conventional measure which captures the fact that high cost firms have lower sales inequilibrium. The second term captures the fact that a firm sells more in equilibrium as the dispersion
in the cost-to-quality ratio across its products increases. The intuition is straightforward. The termin logs is a form of Theil index of dispersion with the numerator being the cost of production ofeach variety relative to the average and the denominator being the quality of the variety relative
to the average (which is always normalized to one). Consider a firm that supplies N f t varieties of
e f t = 1 . Now consider a comparative static
identical quality at identical cost so that ϕut = γut = γ
in which we multiply γut by τ > 1 and divide γkt for all k 6= u by τ N f t −1 . This change has thee f t , equal to one while increasing the dispersion in theproperty of keeping average marginal costs, γcost of providing each element of the production bundle. Even though average costs are unaffectedby construction, firm sales will rise because the increase in dispersion allows the firm to supply its
production bundle more cheaply by shifting its output towards the sales of cheaper varieties. This
cost-dispersion term indicates that a firm sells more if its costs relative to quality are less evenly
distributed across its varieties. Here, we also see the first instance of the insidiousness of demand forunderstanding multiproduct firms—in the nested CES case, one cannot express firm-level marginal
costs without reference to the demand parameters σF and σU .
Finally, the last term captures the role played by firm markups (µ f t ), which are themselves afunction of the elasticity of substitution between firms (σF ) and firm market share (S f t ).

4.6

Firm Sales Decompositions

While the decomposition given in equation (15) is exact for each sector, it is difficult to use thatspecification to understand the general determinants of firm size because aggregate product groupexpenditures are a major determinant of firm sales. Therefore we decompose firm sales in each product group relative to the geometric mean for that product group. Notationally, the algebra is clearer
if we reintroduce the g subscript for all the relevant variables. We can write the sales decomposition
for the average firm in product group g asln E gt = (σFg − 1)ln ϕ gt − (σFg − 1)ln γ̃ gt + ln Egt + (σFg − 1) ln Pgt 1−σUg
γut /γ̃ f t
1 − σFg 1 +ln N gt + 1 − σUg ln µ gt + ln ,∑
1 − σUg
Nu f t ueU
ϕut
ft

(16)

gt

where we define X̄gt = F1t ∑ f ∈ g X f t and Ft is the number of firms in the product group. If we subtract
equation (16) from equation (15) we obtain

19

n

o



1 − σFg
1 − σUg

hiln E f t − ln E gt = (σFg − 1) ln ϕ f t − ln ϕ gt +ln N f t − ln N gt
(17)
o

n

−(σFg − 1) ln γ̃ f t − ln γ̃ gt + 1 − σFg ln µ f t − ln µ gt
 


 




1−σUg1−σUgγut /γ̃ f t
γut /γ̃ f t
1 − σFg 1
 − ln  1 ∑
 
+
ln 
,
∑
1 − σUg
Nu f t ueU
ϕut
Nu f t ueU
ϕut
ft

ft

gt

which can be written more compactly as
1 − σFgg∆ ln E f t = (σFg − 1)∆ ln ϕ f t +∆ ln N f t
1 − σUg






1−σU 

 g

γ
/
γ̃
1 − σFg1ut
ft
∆ g ln − σFg − 1 ∆ ln γ̃ f t ++∑
1 − σUg
N f t ueU
ϕutft
+ 1 − σFg ∆ g ln µ f t .
g



g



(18)

The left-hand side of equation (18) is the sales of firm f relative to average firm sales in the sector,
and the five terms on the right-hand side tell us the importance of relative firm average quality, scope,
average marginal costs, cost dispersion, and markups in understanding cross-sectional differencesin firm size. Moreover, we can also undertake this decomposition for a firm’s sales growth rateby taking the first difference of equation (18), which in our notation simply involves replacing ∆ gwith ∆ g,t in the equation. We now can decompose the cross-sectional variation in firm sales using
a procedure analogous to Eaton, Kortum, and Kramarz’s (2004) variance decomposition commonlyused in the international trade literature. In particular, we regress each of the components of log firmsales on log firm sales as follows:(σFg − 1)∆ g ln ϕ f t− σFg − 1 ∆ g ln γ̃ f t


!

1−σU
γ
/
γ̃
1 − σFg1ut
ft
∆ g ln ∑
1 − σUg
N f t ueU
ϕut
ft!
1 − σFg∆ g ln N f t
1 − σUg
1 − σFg ∆ g ln µ f t=

α ϕ ∆ g ln E f t + ε ϕ ,
g

(19)

=

αγ ∆ ln E f t + ε γ ,

(20)

=

α D ∆ g ln E f t + ε a ,

(21)

=

α N ∆ g ln E f t + ε N ,

(22)

=

αµ ∆ g ln E f t + ε µ ,(23)where each firm’s log sales is differenced relative to the largest firm in its product group. By theproperties of OLS, α ϕ + αγ + α D + α N + αµ = 1. The values for each of the α’s provide us with ameasure of how much of the variation in the distribution of firm sales can be attributed to each factor.
If we replace ∆ g with ∆ g,t in the above equations, we can determine how much of the variance in sales
growth rates can be attributed to each factor. Thus, estimation of these five equations provides us
with a simple way to decompose firm sales in the cross section and in the time series.
20

4.7

Decomposing Changes in firm average quality

When decomposing firm growth, we can go one step further and understand how much of a firm’schange in quality is due to the introduction of new products and how much is due to demand factors
for existing products. In order to see how to do this decomposition, note that our normalization for
UPC quality implies that the geometric mean of the ϕut equals one for each firm in each period, i.e.,
 N1



ft

ϕ f t = ϕ f t  ∏ ϕut (24)u ∈U f t

We can then can take log differences of this equation over time to yield:∆ g,t ln ϕ f t ≡ ∆ ln ϕ f t − ∆ln ϕ gt = ∆ ln ϕ f t +11
ln ϕut −∑N f t u ∈U
N f ,t−1 u∈∑
Uln ϕu,t−1 − ∆ln ϕ gt , (25)f ,t−1ft

where our normalization (2) implies that the second and third terms are both equal to zero.
To isolate the sources of changes in firm average quality, we distinguish between UPCs that aresupplied in both periods versus those that are supplied in only one of the two periods. Let I f t =U f t ∩ U f ,t−1 denote the set of UPC’s that are supplied by firm f in both periods t and t − 1. Similarly,
define U +f t to be the set of newly introduced UPCs, i.e., the set of UPCs in U f t but not in U f ,t−1 , andU−
f t to be the set of disappearing UPCs, i.e., the set of UPCs in U f ,t−1 but not in U f t .
−Noting that U f t = I f t + U +f t and U f t−1 = I f t + U f t , we can decompose the change infirm average quality into quality changes for a constant set of products and quality changes from theadding and dropping of UPCs:






 I



I
1
1
ft
ft
g,t




∆ ln ϕ f t =∑ ln ϕut − U∑ ln ϕu,t−1 + ∆ ln ϕ f t − ∆ln ϕgt  Uft
I f t u∈ I f t
I f t u∈ I f t
f ,t−1
(26)


 U+
ft
 1
+
 Uft
U+



∑ ln ϕut  −+f t u ∈U f t

U−
ft





∑− ln ϕu,t−1  .

 1
U f ,t−1U−
f t u ∈U f t

The first term in braces captures quality changes for a constant set of products, and includes thechange in firm average quality (∆ ln ϕ f t ), as well as the difference between average product quality
for the common set of products (in square brackets) weighted by their relative importance in the setof products in each year. This difference adjusts changes in our measure of firm quality for changes
in our normalization due to the entry and exit of new goods. We refer to the first term in braces inequation (26) as the “demand effect”. It represents the change in consumers perceived quality of the
UPCs present in both periods. If the firm does not add or eliminate any products, the summation
terms will be zero since our normalization (2) means that average log product quality must be zeroin all periods.
21

However, if the set of products changes, the summation terms will not necessarily be zero. Product turnover can influence sales either through product upgrading, which is measured in the second
term in braces in equation (26), as well as through changes in the number of products, which entersseparately into our decomposition of log firms sales (15) above. The “product upgrading effect” (thesecond term in braces in (26) captures quality changes arising from the adding and dropping of products. Product upgrading depends on average product quality for the entering and exiting products(in square brackets) weighted by their relative importance in the set of products in each year. For
example, suppose a firm introduces a new high-quality product and retires a low-quality productleaving the number of products unchanged. In this case, the second term in braces will be positivereflecting the fact that the firm upgraded the average quality of its product mix.

5

Structural Estimation

Our structural estimation of the model has two components. First, given data on expenditure sharesand prices Sut , S f t , Put and known values of the elasticities of substitution {σU , σF }, we show
how the model can be used to determine unique values of firm average quality (ϕ f t ), product quality (ϕut ), and marginal cost shocks (aut ) up to our normalization of quality. These correspond to
structural residuals of the model that are functions of the observed data and parameters and ensurethat the model exactly replicates the observed data. We use these structural residuals to implement
our decomposition of firm sales from Section 4.5 with our observed data on expenditure shares andprices.
Second, we estimate the elasticities of substitution {σU , σF } using a generalization of Feenstra(1994) and Broda and Weinstein (2006) to allow allow firms to be large relative to the markets inwhich they operate (which introduces variable markups) and to incorporate multiproduct firms (sothat firm pricing decisions are made jointly for all varieties). This estimation uses moment conditionsin the double-differenced values of the structural residuals {ϕ f t , ϕut , aut } and also has a recursive
structure. In a first step, we estimate the elasticity of substitution across UPCs within firms for eachproduct group {σU }. In a second step, we use these estimates for UPCs to estimate the elasticity of
substitution across firms for each product group {σF }.

5.1

Structural Residuals

We begin by showing that there is a one-to-one mapping from the observed data on expenditureshares and prices Sut , S f t , Put and the model’s parameters {σU , σF , δ} to the unobserved struc
tural residuals ϕut , ϕ f t , aut .
Given known values for the model’s parameters {σU , σF , δ} and the observed UPC expenditure
shares and prices, we can use the expression for the expenditure share given in equation (4) to determine UPC qualities { ϕut } up to our normalization that the geometric mean of UPC qualities isequal to one. These solutions for UPC qualities and observed UPC prices can be substituted into the22
CES price index (3) to compute firm price indices Pf t . These solutions for firm price indices andobserved firm expenditure shares can be combined with the CES expenditure share (4) to determinefirm qualities ϕ f t up to our normalization that the geometric mean of firm qualities is equal toone. Furthermore observed firm expenditure shares and the CES markup (7) are sufficient to recover
firm markups µ f t . These solutions for markups and observed UPC prices and expenditures canbe substituted into the CES pricing rule (9) to determine the marginal cost shock (aut ).
Finally, our solutions for markups and observed UPC expenditures can be combined with CES
variable profits (11) to obtain upper upper bounds to the fixed costs of supplying UPCs (h f t ) and thefixed costs of operating a firm (H f t ). The upper bound for UPC fixed costs for each product group(h f t ) is defined by the requirement that variable profits for the least profitable UPC within a productgroup must be greater than this fixed cost. Similarly, the upper bound for firm fixed costs for eachproduct group (H f t ) is defined by the requirement that variable profits for the least profitable firmmust be greater than this fixed cost.
This mapping from the observed data on expenditure shares and prices Sut , S f t , Put and themodel’s parameters {σU , σF , δ} to the unobserved structural residuals ϕut , ϕ f t , aut does not impose assumptions about the functional forms of the distributions for the structural residuals or about
their correlation with one another. When we estimate the model’s parameters {σU , σF , δ} below, we
impose some identifying assumptions on the double-differenced values of these structural results
but not upon their levels. Therefore, having recovered these structural residuals, we can examine thefunctional form of their distributions and their correlation with one another.

5.2

UPC Moment Conditions

We now discuss our methodology for estimating the elasticities of substitution {σU , σF } and theelasticity of marginal costs with respect to output (δ), which uses moment conditions in double
differenced values of the structural residuals ϕ f t , ϕut , aut . This estimation again has a recursive
structure. In a first step, we estimate the elasticity of substitution across UPCs within firms for eachproduct group {σU } and the marginal cost elasticity (δ). In a second step, we use these estimates for
UPCs to estimate the elasticity of substitution across firms for each product group {σF }.
In the first step, we double-difference log UPC expenditure shares (4) over time and relative tothe largest UPC within each firm to obtain the following equation for relative UPC demand:

4k,t ln Sut = (1 − σU )4k,t ln put + ωut ,(27)

where4k,t is the double-difference operator such that 4k,t ln Sut = 4 ln Sut − 4 ln Skt ; u is a UPC
supplied by the firm; k corresponds to the largest UPC supplied by the same firm (as measured
by the sum of expenditure across the two years); Sut and put are directly observed in our data;ωut = (1 − σU ) [4 ln ϕkt − 4 ln ϕut ] is a stochastic error. Since we double difference the market
shares of two UPCs supplied by the same firm, we eliminate all demand shocks that are common
across a firm’s UPCs, which leaves only demand shocks that affect the sales of one a firm’s UPCs
23

relative to another. For example, random demand shocks and the timing of holidays, weekends, andmeteorological events might affect the success of certain products in a firm’s lineup relative to others.
Double-differencing the UPC pricing rule (9) enables us to obtain an equation for relative UPC
1+ δ), and the relationship between output and revsupply. Using the cost function (Au (Yu ) = aut Yutenue (Yut = Sut /Put ), the UPC pricing rule given in equation (9) can be re-written as:
1

1

1

δ

1+ δ
1+ δ
Put = µ f t γut = µ f1+t δ (1 + δ) 1+δ aut
Sut.

Taking logs and double-differencing, we obtain the following equation for relative UPC supply:

4k,t ln put =δ4k,t ln Sut + κut ,
1+δ

(28)

where the markup µ f has differenced out because it is the same across UPCs within the firm; Sut andput are again directly observed in our data; κut = 1+1 δ [4 ln aut − 4 ln aut ] is a stochastic error. Since
the firm markup differences out and we observe prices, our estimation approach is the same undereither price or quantity competition, and hence is robust across these different forms of competition.
Once again the double differencing within a firm eliminates shocks that are common across a firm’s
products, and the fact that a barcode uniquely identifies a product means that changes in observableproduct attributes will be manifest in a change of barcode but not in a change in aut . Therefore thesupply-side shocks κut correspond to factors that affect one variety supplied by firm but not another.
These might be problems in individual plants, exchange rate movements in the countries supplying
those products, etc. We assume that these idiosyncratic shocks to marginal cost, κut , are orthogonal
to the idiosyncratic demand shocks, ωut .
Following Broda and Weinstein (2006), the orthogonality of idiosyncratic demand and supply
shocks defines a set of moment conditions (one for each UPC):
G ( β g ) = ET vut ( β g ) = 0,


(29)



σU
and vut = ωut κut . For each product group, we stack all the moment conditions
δto form the GMM objective function and obtain:

where β g =β̂ g = arg min G ∗ ( β g )0 WG ∗ ( β g )

∀ g,

(30)

βg

where G ∗ ( β g ) is the sample analog of G ( β g ) stacked over all UPCs in a product group and W is apositive definite weighting matrix. As in Broda and Weinstein (2010), we weight the data for each
UPC by the number of raw buyers for that UPC to ensure that our objective function is more sensitive
to UPCs purchased by larger numbers of consumers.
The moment condition (29) for each UPC involves the expectation of the product of the doubledifferenced demand and supply shocks: υ = ωut κut . From relative demand (27) and relative supply(28), this expectation depends on the variance of prices, the variance of expenditure shares, the covariance of prices and expenditure shares, and parameters. Our identifying assumption that this24

expectation is equal to zero defines a rectangular hyperbola in (σU , δ) space for each UPC, along
which a higher value of σU has to be offset by a lower value of δ in order for the expectation to beequal to zero (Leontief 1929). Therefore, this rectangular hyperbola places bounds on the demandand supply elasticities for each UPC, even in the absence of instruments for demand and supply. Furthermore, if the variances for the double-differenced demand and supply shocks are heteroskedastic
across UPCs, the rectangular hyperbolas are different for each pair of UPCs, and their intersection
can be used to separately identify the demand and supply elasticities (Feenstra 1994). Consistent
with these identifying assumptions in Broda and Weinstein (2006, 2010) and Feenstra (1994), we findthat the double-differenced demand and supply shocks are in general heteroskedastic.17

5.3

Firm Moment Conditions

We use our estimates of the UPC elasticities of substitution {σU } from the first step to solve for UPC
quality { ϕut } and compute the firm price indices Pf t using equations (3) and (4). In our second
step, we double difference log firm expenditure shares (4) over time and relative to the largest firm
within each product-group, f , to obtain the following equation for relative firm market share:∆ f ,t ln S f t = (1 − σF ) ∆ f ,t ln Pf t + ω f t ,(31)

where the stochastic error is ω f t ≡ − (σF − 1) ∆ f ,t ln ϕ f t .
Estimating equation (31) using ordinary least squares would be problematic because changes infirm price indices could be correlated with changes in firm average quality: Cov ∆ f ,t ln Pf t , ∆ f ,t ln ϕ f t 6=
0. To find a suitable instrument for changes in firm price indices, we use the structure of the model towrite changes in firm price indices in terms of the underlying UPC characteristics of the firm. Using
the CES expenditure shares (4), we can write relative UPC expenditures in terms of relative UPC
prices and relative UPC qualities:
Sut=
S̃ f tPut /ϕut
P̃f t / ϕ̃ f t

!1−σU
,

u ∈ Uf ,(32)

where here we compare each UPC to the geometric mean of UPCs within the firm, which is denotednonoby a tilde so that S̃ f t = exp N1u f t ∑u∈U f t ln Sut and P̃f t = exp N1u f t ∑u∈U f t ln Put . Using this expression for relative expenditure shares to substitute for product quality (ϕut ) in the CES price index (3),we can write the firm price index solely in terms of observed relative expenditures and the geometricmean of UPC prices:


1
S
ut 
ln Pf t = ln P̃f t +ln  ∑,
1 − σUS̃u ∈U f t f t(33)

where we have used our normalization that ϕ̃ f t = 1.
17 In a White test for heteroskedasticity, we are able to reject the null hypothesis of homoskedasticity at conventionalsignificance levels for 94 percent of product groups.

25Equation (33) decomposes the firm price index into two terms. The first term is entirely conventional: the geometric mean of the prices of all goods supplied by the firm. When researchers
approximate this firm price index using firm-level unit values or the prices of representative goods
supplied by firms they are essentially capturing this component of the firm price index. The second
term is novel and arises because the conventional price indexes are not appropriate for multiproductfirms. The log component of the second term is a variant of the Theil index of dispersion.18 If theshares of all products are equal, the Theil index will equal ln N f t , which is increasing the number of
products supplied by the firm, N f t . Since the Theil index is multiplied by (1 − σU )−1 ≤ 0, the firm’s
price index falls as the number of goods supplied by the firm rises. This Theil index also increases as
the dispersion of the market shares across UPCs within the firm rises.
One can obtain some intuition for this formula by comparing it to the conventional price indexes
commonly used in economics. Firm price indexes are typically constructed by making at least oneof two critical assumptions—firms only supply one product (N f t = 1), or the goods supplied byfirms are perfect substitutes (i.e., σU = ∞). Either of these assumptions is sufficient to guaranteethat the firm’s price level equals its average price level. However, both of these assumptions arelikely violated in reality. The average price of a firm’s output prices overstates the price level for a
multiproduct firms because consumers derive more utility per dollar spent on a firm’s production ifthat production bundle contains more products. Thus, the (geometric) average firm price is a theoreticallyrigorous way to measure the price level of firms that produce perfect substitutes, it overstates the prices of firmsthat produce differentiated goods, and this bias will tend to rise as the number of products supplied by the firm
increases.
The structure of the model implies that the dispersion of the shares of UPCs in firm expenditure(Sut ) only affects the shares of firms in product group expenditure (S f t ) through the firm price indices(Pf t ). Double-differencing equation (33) for the log firm price index over time and relative to thelargest firm within each product-group, we obtain:∆ f ,t ln Pf t = ∆ f ,t ln Pef t +



1
Sut 
,
∆ f ,t ln  ∑e
1 − σUu ∈U f t S f t(34)

where the model implies that the second term on the right-hand side containing the shares of UPCs
in firm expenditure is a suitable instrument for the double-differenced firm price index in (31). We
find that this instrument is powerful in the first-stage regression (34), with a first-stage F-statistic forthe statistical significance of the excluded exogenous variable that is substantially above the recommended threshold of 10 from Stock, Wright and Yogo (2002).
18 The standard Theil index uses shares relative to simple average shares, while ours expresses shares relative to thegeometric mean.

26

6

Estimation Results

We present our results in several stages. First, we present our elasticity and quality estimates andshow that they are reasonable. Second we use these estimates to examine cannibalization, and finally
we present our results on the sources of firm heterogeneity.

6.1

Estimated Elasticities of Substitution

Because we estimate 100 σU ’s and σF ’s, it would needlessly clutter the paper to present all of them
individually. Table 5 shows goods supplied by the same firm are imperfect substitutes. For UPCs, theestimated elasticity of substitution ranges from 4.7 at the 5th percentile to 17.6 at the 95th percentile
with a median elasticity of 6.9. These numbers are large compared with trade elasticities reflecting
the fact that products supplied by the same firm are closer substitutes than products supplied bydifferent firms. The median elasticity implies that a one percent price cut causes the sales of that
UPC to rise by 6.9 percent.19
Table 5: Distribution of 100 GMM Estimates
Ranked Percentile

σU

σF

σU − σF

δ

1
5
10
25
50
75
90
99

24.4
17.6
14.0
8.6
6.9
5.4
5.0
4.4

12.8
8.5
7.3
5.1
3.9
3.1
2.6
2.1

16.8
9.4
6.6
4.0
2.6
2.2
1.8
1.2

0.62
0.42
0.30
0.20
0.16
0.09
0.06
0.02

Note: Percentiles are decreasing: largest estimate is ranked first. Calculations on data from The Nielsen Company (US), LLC andprovided by the Marketing Data Center at The University of Chicago Booth School of Business.

The fact that we estimate the different products supplied by a given firm to be imperfect substitutes for each other has profound implications for how we should understand firm pricing andproductivity. All productivity estimates are based on a concept of real output, which, in index number theory, equals nominal output divided by the minimum expenditure necessary to generate a unitof utility. According to equation (1) the quality-adjusted flow of consumption from a firm’s output
is ϕ f t C f t , so the corresponding expenditure function for a unit of quality-adjusted consumption is
Pf t /ϕ f t . However, for multiproduct firms, the formula for Pf t depends on the demand system, andtherefore the formula for a firm’s real output also depends on the demand system. In other words,
for multiproduct firms, the concept of real output is not independent of the demand system, so all attempts tomeasure productivity based on a real output concept contain an implicit assumption about the structure of the
19 The UPC own price elasticity derived in equation (39) is given by∂Cut Put= (σF − 1) S f t Sut + (σU − σF ) Sut − σU .
Put Cut27

demand system.20
This problem is more pernicious than simply saying that firm prices are measured with error; theerrors are likely to be systematic. Equation (33) indicates that, in the CES system, the use of average
goods prices to measure firm-level prices will overstate the price level (and understate real output)
more for large multiproduct firms than for small single-product ones. Therefore, if the true modelis CES but a researcher uses a quantity-weighted average of the prices of the goods supplied by the—the results are
firm to measure firm prices—what we term a “conventional” price index or PfConvtlikely to underestimate productivity differences.
But how big is this problem in practice? If we denote the consumer’s expenditure on a firm’s
Conv , and theoutput by E f t , we can write the conventional measure of real output, QConv
f t , as E f t /Pf t

CES measure of real output, QCES
f t , as E f t / Pf t /ϕ f t . These measures will vary across sectors dueto the units, so in order to compare size variation across sectors we will work with unitless shares
CES
Conv
Conv
of each firm in total output (i.e., QCES
f t / ∑ f ∈ g Q f t and Q f t / ∑ f ∈ g Q f t ). Figure 1 plots these two

measures of real output. As one can see in Figure 1, the choice of price index matters enormously forthe computation of real output. Moreover, our estimate of σU suggests that there are large downward
biases in the measurement of real output of large firms. While, on average, the conventional measureof real output is quite close to the CES measure for small firms it systematically understates the valueof real output for firms with large market shares. We can econometrically measure the magnitude
of this bias by regressing ln PfConv
on ln E f t (with product group fixed effects) and ex/
P
/ϕ
ft
ft
t
amining the coefficient on log sales. If there were no bias, we would expect a coefficient of zero, but
we actually obtain a coefficient of 0.3447 (s.e. 0.0004, R2 = 0.3) indicating that failing to take into
account the multiproduct nature of firms causes us to significantly underestimate the size of large
firms. This estimate is economically significant as well. It implies that holding fixed a firm’s conventional price index, every one percent increase in firm sales is associated with a 1.3 percent increase in
its real output in a CES model. The difference between the two estimates reflects the importance ofassumptions about demand when constructing a firm price index.
A second striking feature of our estimated results is that the elasticity of substitution among varieties supplied by a firm is always larger than that that between firms (i.e., σU > σF ). It is importantto remember that this is not a result that we imposed on the data. Moreover, most of the elasticities are precisely estimated—in 82 percent of the cases, we can statistically reject the hypothesis that
σU = σF at the 5% level. The higher elasticities of substitution across UPCs than across firms imply
that varieties are more substitutable within firms than across firms, which implies cannibalization
effects from the introduction of new varieties by firms. Moreover, the fact that the estimated elasticities of substitution are always greater than one implies that firms’ varieties are substitutes, which is
required for positive markups of price over marginal costs (c.f., equation (9)).
In order to assess whether our elasticities are plausible, it is useful to compare our estimates with
20 Since firms supply differentiated products, the relevant measure of productivity is TFPR (Foster, Haltiwanger andSyverson 2008), which necessarily depends on prices and hence the specification of demand.

28

Figure 1: Bias in Conventional Real Output Share

Note: Calculated based on data from The Nielsen Company (US), LLC and provided by the Marketing Data
Center at The University of Chicago Booth School of Business.

those of other papers. In order to do this, we restricted ourselves to comparing our results withstudies that used US scanner data and estimated elasticities for the same product groups as ours. We
do this because, as Broda and Weinstein (2006) show, elasticity estimates for aggregate data can lookquite different than those for disaggregate data. Unfortunately, we did not find studies estimating
the elasticity of substitution within firms, but we did find studies that examined elasticities akin to
our cross-firm elasticity, σF . Figure 2 plots our estimates of σF against other estimates from Gordon
et al. (2013) and the literature review contained in the web appendix to their paper. Although theseother studies use different methods, we find that our estimates are very similar with a statisticallysignificant correlation of 0.59 and the points arrayed fairly close to the 45-degree line. Therefore,
while our empirical approach has a number of novel features in modeling multiproduct firms thatare of positive measure relative to the markets in which they operate, the empirical estimates generated by our procedure are reasonable compared to the benchmark of findings from other empiricalstudies.

6.2

Quality

The quality parameters are a second key component of our decomposition. Our estimation procedureallows us to estimate a different ϕut for every quarter in our dataset. We have strong priors that thequality of each UPC should be fairly stable across time. One way to gauge this stability is to regressthese quality parameters on UPC fixed effects in order to determine how much of a UPC’s quality is29

Figure 2: Price Elasticities Compared to Literature

Note: Calculated based on data from The Nielsen Company (US), LLC and provided by the Marketing Data
Center at The University of Chicago Booth School of Business.

common across time periods. When we do this and include time fixed effects to control for inflationand other common demand shocks, we find that the R2 is 0.84, which implies that very little of thevariation in our quality measures arise from changes in UPC quality. We should also expect firm
average quality, ϕ f t , to exhibit a strong firm component but to be less stable over time. This is exactly
what the data reveals. Running the same regression for firm average quality, we find that 83 percentof the variance can be explained by firm fixed effects. Consistent with these results, we also estimatethat firm average quality exhibits more variation overall (not just across time) than UPC quality or
marginal cost. This can be seen in Figure 3, which plots kernel density estimates of UPC quality, firm
average quality, and marginal cost.

6.3

MarkupsUsing our estimated elasticities of substitution, we can compute implied firm markups. As discussed
above, multiproduct firms internalize the fact that they supply a firm consumption index and hencechoose a price for each UPC that depends on the perceived elasticity of demand for the firm consumption index as a whole (ε f t ). Although demand exhibits a constant elasticity of substitution, thisperceived elasticity of demand differs across firms because they internalize the effect of their pricing decisions on market price indices. For any firm with a strictly positive expenditure share, theperceived elasticity of demand (ε f t ) is strictly less than consumers’ elasticity of substitution acrossvarieties (ε f t < σF ). Larger firms that account for greater shares of expenditure have lower per-30

Figure 3: Kernel Densities of Quality and Cost Estimates
UPC Quality (ϕut )Firm Quality (ϕ f t )δ)Marginal Cost (γut = (1 + δ) aut Yut
Note: The UPC quality is measured by ln ϕut . Firm quality is measured byln ϕ f t , and marginal cost is computed as ln γut /γ̃ f t . Calculated based on data from The Nielsen Company (US), LLC and provided by the Marketing Data Center at The University of Chicago
Booth School of Business.

31ceived elasticities of demand and hence charge higher markups of price over marginal costs (µ f t ), assummarized in equations (7) and (8) that are reproduced below:
µft =

ε ft
,
ε ft − 1

ε f t = σF − (σF − 1) S f t .
There are two important dimensions across which markups vary. The first is cross-sector variation with captures the fact that σF varies systematically across product groups, and the second iswithin-sector variation which captures the fact that larger firms have higher markups within a sector
than smaller firms. As discussed above, our procedure for estimating the elasticities {σU , σF } is robust to the assumption of either price or quantity competition, because the firm markup differences
out in (27) and we observe prices. However, the markup formula, and hence the decomposition ofprices into markups and marginal cost, depends in an important way on the form of competition.
In our baseline specification, we assume Bertrand competition in prices, but we report the derivation of the markup under Cournot competition in quantities in the appendix. A key takeaway fromthis comparison is that Cournot competition generates substantially higher markups for larger firms
than Bertrand competition if goods are substitutes. Additionally, as shown in the appendix, under
Bertrand competition, the relative markups of firms depend solely on the distribution of market
shares (and hence are robust to different estimates of the elasticity of substitution between firms). In
contrast, under Cournot competition, the relative markups of firms depend on both the distributionof market shares and the elasticity of substitution between firms.
Table 6 presents data on the distribution of markups across all firms in our 100 product groups.
The median markup is 33 percent, which is slightly lower than Domowitz et al. (1988) estimate of 36
percent for U.S. consumer goods, and slightly above the median markups estimated by De Loecker
and Warzynski (2012) for Slovenian data which range from 17 to 28 percent. This suggests that ourmarkup estimates are reasonable in the sense that they do not differ greatly from those found in prior
work.
Table 6: Distribution of Markups
Ranked Percentile
10
25
50
75
90

Cournot Using ε FBertrand Using ε F

Using σF

Using σU

0.57
0.45
0.33
0.24
0.18

0.57
0.45
0.33
0.24
0.17

0.57
0.45
0.33
0.24
0.16

0.24
0.23
0.17
0.14
0.09

Note: Markup=(Price-Marginal Cost)/Marginal Cost. Calculations on data from The Nielsen Company (US), LLC and provided by theMarketing Data Center at The University of Chicago Booth School of Business.

While the markups of most firms are essentially the same regardless of whether we assume firms
compete through prices or quantities, the markups of the largest firms are substantially different.
Table 7 shows the distribution of the markups for the largest and second largest firms by product32

group. In each case, we report the markup relative to the average markup in the product group,
so that one can see how different the markup of the largest firm is. The median largest firm within
product groups has a markup that is 24 percent larger than average if we assume competition is
Bertrand and almost double the average if we assume competition is Cournot. However, as one canalso see from this table, these markups drop off quite rapidly for the second largest firm. Therefore,both modes of competition suggest that very few firms can exploit their market power.
Table 7: Distribution of Markups Relative to Product Group Average
Cournot
Ranked Percentile
10
25
50
75
90

BertrandLargest Firm

Second Largest

Largest Firm

Second Largest

4.68
2.98
1.94
1.54
1.37

2.59
1.72
1.49
1.32
1.23

1.73
1.41
1.24
1.16
1.10

1.28
1.18
1.13
1.09
1.06

Note: Percentiles are decreasing: largest is ranked first. Markup=(Price-Marginal Cost)/Marginal Cost. Calculations on data from The
Nielsen Company (US), LLC and provided by the Marketing Data Center at The University of Chicago Booth School of Business.

6.4

Cannibalization Rate

Our estimated elasticities of substitution also determine the magnitude of cannibalization effects inthe model. As discussed above, when a multiproduct firm chooses whether to introduce a new variety, it takes into account the impact of this product introduction on the sales of its existing varieties.
The partial elasticity of the revenue of existing UPCs with respect to the introduction of a UPC isgiven by (13), which is reproduced below:
Cannibalization rate ≡σU − σF
σU − 1+σF − 1σU − 1


S f t SN f t N f t .

Since this cannibalization rate is defined as the partial elasticity of the sales of existing products withrespect to the number of products, and can be expressed in terms of observed expenditure shares, ittakes the same value under both price and quantity competition, as shown in the appendix.
As we mentioned earlier, a cannibalization level of 0 corresponds to equal differentiation of afirm’s products from its rivals and a level of 1 corresponds to a world in which the products of firms
are perfect substitutes. In Table 8, we present the average estimated markups and cannibalizationlevels for firms by decile of size. As one can see from the table, these are remarkably stable, whichreflects the fact that even in the upper decile of firms most firms have trivial market shares (S f ≈ 0).
Interestingly, we see that cannibalization levels lie almost exactly between the benchmarks of perfect
substitutes and equal differentiation. This cannibalization level implies that about half of the salesof a new product introduced by a firm comes from the sales of existing products and half fromthe sales of other firms. The fact that these cannibalization rates are much less than one, however,underscores the fact that it is not appropriate to treat the goods supplied by a multiproduct firm as
33

perfect substitutes nor is it appropriate to treat multiproduct firms as if the introduction of a new
good had no impact on the sales of existing goods.
Table 8: Markups and Cannibalization by Decile
Ranked Decile
1
2
3
4
5
6
7
8
9
10

Average Cournot Markup

Average Bertrand Markup

0.40
0.38
0.37
0.37
0.37
0.37
0.37
0.37
0.37
0.37

0.38
0.37
0.37
0.37
0.37
0.37
0.37
0.37
0.37
0.37

Average Cannibalization
Rate
0.51
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50

Note: Largest decile is ranked first. Weighted average across product groups. Calculations on data from The Nielsen Company (US),
LLC and provided by the Marketing Data Center at The University of Chicago Booth School of Business.

Moreover, there is a slight tendency for the markups and cannibalization levels to rise with firm
size reflecting the fact that large firms have higher markups and take into account the fact that moreof their sales of new products comes at the expense of their existing products. We can see this moreclearly in Table 9, where we focus on the ten largest firms in the sector. We estimate that the markupof the largest firm is on average 32 and 120 percent larger than the markups of firms in the firstnine deciles under price and quantity competition respectively (comparing Tables 9 and 8), but themarkup of the fifth-largest firm is not very different. Thus, the ability to exploit market power is afeature of very few firms on average. Similarly, cannibalization levels tend to rise for the the verylargest firms. We estimate that when the largest firm in a sector, which has an average market shareof 23 percent, introduces a new product, 62 percent of the sales of that product comes from from thesales of its existing products. Since we saw from Table 3 that the largest firms typically have the mostproducts per firm, this implies that cannibalization is likely to be a first-order issue for them.
Table 9: Markups and Cannibalization by Firm Rank
Firm Rank
1
2
3
4
5
6
7
8
9
10

Cournot Markup

Bertrand Markup

Cannibalization Rate

0.826
0.576
0.498
0.455
0.437
0.424
0.415
0.413
0.410
0.403

0.496
0.428
0.407
0.397
0.392
0.388
0.385
0.384
0.383
0.382

0.617
0.565
0.544
0.531
0.525
0.521
0.518
0.517
0.516
0.514

Note: Largest firm is ranked first. Weighted averages across product groups. Calculations on data from The Nielsen Company (US), LLC
and provided by the Marketing Data Center at The University of Chicago Booth School of Business.

34

6.5

Decomposing Firm Sales

The first panel of Table 10 presents the decompositions described in equations (19) to (23) for the fullsample of firms and the second presents the results for the largest firms (those with a market sharein excess of 0.5 percent). We run the results over both sets of firms to see if there are any differences
between them. Overall, the results suggest a powerful role for quality in accounting for the size offirms. The variance decomposition indicates that 76 percent of the overall size distribution can be
attributed to average quality and 21 percent is attributed to scope. Thus, total firm quality accounts
for 97 percent of variation in firm sales. If we restrict ourselves to looking at the data for the fifty
largest firms we still find that firm average quality explains on average 54 percent of the variation
with scope accounting for 26 percent. In other words, even if we restrict our sample to the largest
firms, quality variation accounts for 80 percent of the variation across firms.
What is perhaps most surprising is the small role played by cost in the determination of firm size.
There are two important features of this decomposition to bear in mind. First, a positive coefficientmeans that larger firms have lower costs: i.e., cost differentials help explain size differentials. Second,
our decomposition allows us to split firm-cost differentials into two components: average marginal
cost differentials and the dispersion of costs across products (“cost dispersion”). In all cross-sectional
decompositions, the cost dispersion term is positively correlated with firm size. The intuition is simple. Larger firms tend to sell more products and therefore are better positioned to benefit from costdifferentials in supplying their consumption index. The results for average marginal costs are more
mixed. In the full sample, we find almost no association between firm size and average costs indicating that larger firms have essentially the same average marginal costs as smaller firms. Their
total costs are lower because they can take advantage of the cost dispersion in their output. Interestingly, we find a much stronger association with average costs when we restrict our sample to only
the largest fifty firms. Here, we find that 18 percent of the firm size distribution can be attributed
to lower average costs of large firms and a quarter of of the overall distribution due to these costdifferences. Moreover, the results lend support to the managerial competence approach to thinking
about cost differences—large firms tend to have higher quality and lower costs. Finally markup variation, which depends on both quality and cost through expenditure shares, plays a very small rolein understanding the firm size distribution.21
Figure 4 presents the results of the cross-sectional decomposition for the largest 50 firms in eachproduct group. Each point represents the average contribution of a particular factor towards understanding the relative level of sales in a sector. For example, the column of points at the right of the
figure indicate that the largest firm in a product group is on average almost nine log units larger
than the average firm. Of this nine-log-unit difference the diamond-shaped point indicates that on
average about 4.9 log units (54 percent) can be attributed to average quality differences. The circular point reveals that about 2.3 log units (25 percent) is on average attributable to scope differences,
21 Consistent with these findings, Roberts et al. (2011) find substantial heterogeneity in demand across Chinese footwear

producers, while Foster, Haltiwanger and Syverson (2013) emphasize the role of demand in the growth of entering firms.

35

Table 10: Variance Decompositions
PANEL A: A LL F IRMS
Decomposition
Cross-SectionalFirm Growth

Coefficient
Std.Error
Coefficient
Std.Error

Quality

Scope

Average MC

Cost
Dispersion

MarkupUpgrading

0.758
0.002
0.905
0.009

0.20900.00020.15710.0012−0.0360.002−0.1310.008

0.07070.0001
0.06870.0010−0.00195
0.00001−0.00059
0.00004

0.00030.0014PANEL B: T OP 50 F IRMS
Decomposition
Cross-SectionalFirm Growth

Coefficient
Std.Error
Coefficient
Std.Error

Quality

Scope

Average MC

Cost
Dispersion

MarkupUpgrading

0.544
0.013
0.902
0.053

0.26370.0016
0.25970.00790.175
0.013−0.1630.051

0.06130.00080.02780.0082−0.04436
0.00016−0.02619
0.001310.02330.0138

Note: MC is marginal cost Calculations on data from The Nielsen Company (US), LLC and provided by the Marketing Data Center at
The University of Chicago Booth School of Business.

while the triangular and square points indicate that lower average costs and greater cost dispersion
account for about 1.3 (14 percent) and 0.6 log units (7 percent) of the sales. These percentages are
extremely similar to the percentages for larger firms that we saw in Table 10. However, the fact thateach set of points is relatively tightly arrayed along a line indicates that the results presented in Table
10 are not the result of any outlier but are a robust feature of the data.
There are two other features of the data that this plot also makes clear. First, we can also see thatwhile there is a tendency for average marginal costs to fall with size for the four largest firms, which
accounts for the upward trend in the sequence of triangular points, this upward trend is not a robustfeature of the data. This pattern helps explain why we found in Table 10 that average marginal costs
did not fall with firm size when we looked at the full sample of firms, but did fall when we restricted
our attention to the very largest firms. Moreover, we can see that the gains from increasing costdispersion are a very robust feature of the data.
A second feature of the data that suggests welfare implications of the ability of the largest firms
to exploit market power is also clearly seen in Figure 4. We saw in Table 10 that while markup
variation played almost no role in understanding the firm-size distribution when we focused on thefull sample of firms, it was negatively associated with firm sales when we examined the largest firms.
Figure 4 shows that this negative association comes from the fact that if it were not for the fact thatthe very largest two or three firms had higher markups (a result we also saw in Table 9), their output
levels would be higher. The implied inefficiently low levels of output of the largest firms suggeststhat the monopolistic competition model breaks down for these firms and motivates our attempt toquantify the welfare impact of this pattern in the data in Section 7.
As we showed in Section 4.6, we can also examine the time-series determinants of firm growthby differencing with respect to time (i.e., using ∆ g,t instead of ∆ g in the decompositions). As we cansee in Table 10, the importance of quality rises when understanding firm growth. Virtually all of a
36

Figure 4: Sales Decomposition by Firm Rank

Note: The graph contains the weighted average across product groups of the decomposition of firm sales
into firm average quality, marginal cost, markup and product scope for the fifty firms with the largest average
market share in 2004 and 2011. The vertical axis measures the contribution of each factor given by the left-handsides of equations (19) to (23). The horizontal axis is the log difference between the firm’s sales and that of theaverage firm defined by the right-hand side of these equations. Calculated based on data from The Nielsen
Company (US), LLC and provided by the Marketing Data Center at The University of Chicago Booth Schoolof Business.

firm’s growth can be understood in terms of increases in quality and scope with cost changes playing
almost no role. Interestingly, quality upgrading—i.e., the addition of new higher quality products—
plays almost no role in the typical firm’s expansion. Most of the expansion comes from shifts indemand towards the firm’s existing products, which may reflect marketing and other taste shifts.
Figure 5 presents the results from this decomposition for the fifty firms whose average market
shares in 2004 and 2011 were the highest. Thus, the column of points on the right of the figure
indicates that on average the fastest growing firm in a product group on average had sales growththat was 1.6 log units faster than the average firm (or about 26 percent per year). Of this faster
growth, the diamond-shaped point indicates that close to one log unit (about 50 percent) was due toquality improvements, with most of the remainder due to scope changes. The points in these scatters
are a bit less tightly distributed, indicating that there is more volatility in the estimated determinantsof firm growth than in firm shares, but the results consistently show that the two main determinantsof firm growth are firm average quality and product scope. These results are also quite similar toto those we saw in Table 10, with now the new piece of information being that we can see that theearlier results are not being driven by any particular firm or set of firms.

37

Figure 5: Growth Decomposition the Fifty Largest Firms

Note: The graph contains the weighted average across product groups of the growth decomposition of thegrowth rate in firm sales into estimated firm average quality, marginal cost, and product scope for the top
fifty firms by average market share in 2004 and 2011. The vertical axis measures the contribution of eachfactor given by the left-hand sides of equations (19) to (23). The horizontal axis is the log difference between
the firm’s sales and that of the average firm growth rate defined by the right-hand side of these equations.
Calculated based on data from The Nielsen Company (US), LLC and provided by the Marketing Data Centerat The University of Chicago Booth School of Business.

Taken together, the decomposition results are quite striking in their consistency about the sourcesof firm heterogeneity. Regardless of whether we examine firms in the cross-section or in the timeseries, improvements in firm average quality followed by firm scope are the most important driversof firm sales for both small and large firms alike. The role played by average cost differences tends
to be small and depend on the sample and decomposition method used. But we consistently finda positive contribution towards differences in firm size from the cost dispersion term. Finally, we
see that although markup differences are unimportant determinants of firm sales for most firms,
they do appear to be important for the very largest firms something that suggests aggregate welfare
implications from the exploitation of market power by these firms.

6.6

Robustness

One limitation of our approach is that to structurally estimate the model we need to assume a particular nesting structure for demand consisting of firms and products within firms. We choose thisnesting structure to connect with the existing literature on measuring firm productivity (which implicitly or explicitly treats firms as a nest) and as a natural approach to modeling multi-product38

firms. But it is reasonable to wonder whether our results are robust to other nesting structures. In
this section, we undertake a robustness test, in which we demonstrate that our main finding of theimportance of quality in explaining sales variation is not sensitive to our assumed nesting structure.
To do so, we use the CES functional form to derive a reduced-form equation that can be estimatedfor alternative nesting structures. Although this reduced-form equation cannot be used to recover
structural parameters, it can be used to provide a lower bound on the importance of quality variation
for alternative nesting structures.
To provide evidence on the role of quality variation under alternative nesting structures, we combine the following two results for CES demand. First, from (31), double-differenced expenditure
shares depend on double-differenced price indices for any tier of utility j. Second, from (33), theprice index for one tier of utility j can be expressed in terms of expenditure shares and the geometricmean of prices for the lower tier of utility k. Combining these results, and reintroducing the productgroup subscript, we obtain the following reduced-form equation that can be estimated using OLS:"

4 j,t ln S jgt = α jg + β 1jg 4 j,t ln P̃jt + β 2jg 4 j,t ln
where 4 j,t is again the double-difference operator; Pejt = exp

n

∑

Skgt

u∈b S̃ jgt#+ e jgt ,Njt
1
Njt ∑k =1 ln Pkt

o

(35)

and the error term e jgt

captures double-differenced quality.
In our structural model, the coefficient β 2jg can be related to structural parameters of the model(β 2jg = 1/ σkg − 1 ). In the reduced-form regression (35), this coefficient does not have a structuralinterpretation, because quality for tier j (e jgt ) can be correlated with the geometric mean of prices
in the lower tier k ( Pejgt ). Therefore, if the reduced-form regression (35) is estimated using OLS, thecoefficient β 2jg is subject to omitted variable bias and is not consistently estimated. For this reason, we
use our structural approach developed above to estimate the elasticities of substitution and elasticity
of marginal cost {σUg , σFg , δg } for our assumed nesting structure.
Nonetheless, the reduced-form specification (35) can be used to provide a lower bound on theimportance of quality for alternative nesting structures. The reason that this specification provides
a lower bound is that quality is captured in the estimated residual (be jgt ). Therefore, the componentof quality that is correlated with the explanatory variables is attributed to the coefficients {β 1jg , β 2jg }on these explanatory variables, leaving only the orthogonal component of quality in the estimatedresidual (be jgt ).
In Table 11, we report the average R2 across all product groups (g) from estimating this reducedform regression for alternative nesting structures, where one minus this R2 corresponds to the variance in double-differenced expenditure shares that is unexplained by the explanatory variables (andis instead explained by quality). We estimate the regression separately for each product group andyear, and summarize the mean and standard deviation of the R2 , weighting each product group andyear by their sales.
In the first row of the table, we report results for our assumed nesting structure. In the remaining

39

rows of the table, we report results for a wide range of alternative nesting structures. Some of thesealternative nesting structures disaggregate firms relative to our baseline specification (e.g. Specification II uses Product Group and Brand-Module instead of Product Group and Firm, where Brand
Module distinguishes between different brands supplied by a firm, such as Coke and Sprite soda for
the Coca-Cola corporation). Other alternative nesting structures omit firms as a nest altogether (e.g.
Specification IV uses Module and Brand Module, where Module corresponds to different types of
products within a product group, such as diet and regular soda). Across this wide range of specifications, we find that quality accounts for at least 43 percent of the variation in double-differenced
sales. Thus, regardless of one’s choice of nesting structure, it is hard to escape the conclusion thataround half of sales variation is attributable to quality variation. Since this lower bound also leaves
out the important role played by scope on demand, this further reinforces the importance of overall
quality (including scope) in explaining sales variation.
Table 11: Within Nest R2

Note:Specification

Nest 1

Nest 2

Nest 3

Nest 4

Mean

Std. Dev.

I
II
III
V
V
VI

Product Group
Product Group
Product Group
ModuleModuleModuleFirm
Brand-Module
Firm-Module
Firm-Module
Firm-Module
Firm-Module

UPC
UPC
UPC
UPC
UPC
Brand

N/A
N/A
N/A
N/A
N/A
UPC

0.53
0.46
0.47
0.48
0.49
0.57

0.05
0.07
0.07
0.06
0.05
0.07

Calculations on data from The Nielsen Company (US), LLC and provided by the Marketing Data Center at The University of

Chicago Booth School of Business.

7

Counterfactuals

We found in Table 9 that the monopolistic competition benchmark breaks down substantially for thelargest firms. Moreover, we learned from Figure 4 that the estimated exploitation of market powerby the largest firms implies reductions in these firms’ sales. We now use the structure of the modelto undertake counterfactuals to assess the quantitative relevance of our estimated departures frommonopolistic competition for aggregate welfare. We consider two types of counterfactuals. First,
we consider the implications for consumers from a regulator preventing large firms from exercising
their market power and thereby requiring each firm to compete under monopolistic competition.
Second, we consider the implications of allowing firms to supply a number of differentiated products. Thus, the second type of counterfactuals examines the gains to consumers from the different
varieties supplied by each firm.
In each of these counterfactual exercises, we examine the impact on consumers under both Bertrand
and Cournot competition. In all cases, we hold constant aggregate expenditure (Et ), the sets of
active UPCs and firms U f t , Fgt , and the sets of firm qualities, UPC qualities and cost shifters

ϕ f t , ϕut , aut . Therefore we abstract from general equilibrium effects through aggregate expen-

40

diture, factor prices, and entry and exit. Our goal is not to capture the full general equilibrium effectsof each counterfactual, but rather to show that our firm-level estimates are of quantitative relevancefor aggregate outcomes such as welfare.
In the first set of counterfactuals (Counterfactuals I-B and I-C), we compare the actual equilibrium in which firms exploit their market power with a counterfactual equilibrium in which a priceregulator requires all firms to charge the same constant markup over marginal cost equal to the monopolistically competitive firm markup:
µft =

σF
.
σF − 1(36)

Moving from the actual to the counterfactual markup reduces the relative prices of successful
firms with high market shares and hence redistributes market shares towards these firms. Therefore,
we expect these counterfactuals to increase welfare and the dispersion of firm sales, but the magnitude of these effects depends on the estimated parameters {σU , σF , δ} and the sets of firm qualities,
product qualities and cost shocks ϕ f t , ϕut , aut .
Given the assumed constant markup in equation (36), we solve for the counterfactual equilibriumusing an iterative procedure to solve for a fixed point in a system of five equations for expenditureshares and price indices Sut , Put , S f t , Pf t , Pgt . These five equations are the UPC expenditure share(given by equation (4) for k = u), the UPC pricing rule (equation (9)), the firm expenditure share(given by equation (4) for k = f ), the firm price index (given by (3) for k = f ) and the product groupprice index (given by equation (3) for k = g). We use the resulting solutions for product group priceindices together with aggregate expenditure to compute counterfactual aggregate welfare:Wt =Et
ϕ gt .
G∏ g=1 Pgt

(37)

In Counterfactuals II-B and II-C, we examine the quantitative relevance of multiproduct firms.
We compare the actual equilibrium in which firms supply multiple products to a counterfactual
equilibrium in which firms are restricted to supply a single UPC (their largest). Counterfactual II-B
undertakes this comparison for the actual equilibrium under Bertrand competition, while Counterfactual II-C undertakes the same comparison for the actual equilibrium under Cournot competition.
We again solve for the counterfactual equilibrium by solving for a fixed point in the system of fiveequations for expenditure shares and price indices discussed above.
Table 12 reports the results from both sets of counterfactuals. Each row of the table corresponds
to a different counterfactual. The second column reports the coefficient of variation of firm sales in
the counterfactual relative to that in the actual data. The third column reports aggregate welfare in
the counterfactual relative to that in the actual data.
In Counterfactual I-B, we find that moving from Bertrand to monopolistic competition increases
aggregate welfare by 4 percent, which is comparable to standard estimates of the welfare gains fromtrade for an economy such as the United States. In Counterfactual I-C, we find much larger welfare
effects of moving from Cournot to monopolistic competition, which increases aggregate welfare by41

Table 12: Counterfactual Exercises (Relative to Observed)
Coef. Variation (Firm Sales)

Aggregate Welfare

1.10
1.24
0.82
0.86

1.04
1.15
0.70
0.70

I-B: Bertrand to Monopolistic Comp.
I-C: Cournot to Monopolistic Comp.
II-B: Multiproduct to Single-Product Bertrand
II-C: Multiproduct to Single-Product Cournot

Note: Each statistic is expressed as the value in the counterfactual relative to the value in the observed data. Therefore relative coefficientvariation sales is the coefficient of variation of firm sales in the counterfactual divided by the coefficient of variation in the observed data.
Calculations on data from The Nielsen Company (US), LLC and provided by the Marketing Data Center at The University of Chicago
Booth School of Business.

around 15 percent. These much larger welfare effects reflect the fact that large firms charge muchhigher markups under Cournot than under Bertrand as we saw in Table 7. This result implies larger
price reductions from moving to monopolistically competitive markups, and hence larger increases
in welfare. In both Counterfactuals I-B and I-C, the relative prices of the largest firms fall and demandis elastic. Therefore, the coefficient of variation of firms sales increases, by around 10 percent for
Bertrand competition and around 24 percent for Cournot competition. Again the larger effects for
Cournot reflect the greater variation in markups under this mode of competition.
In both Counterfactuals II-B and II-C, we find quantitatively large effects of multiproduct firms.
Restricting firms to supply a single product under Bertrand competition reduces aggregate welfare
by around 30 percent and reduces the coefficient of variation of firm sales by around 18 percent. The
corresponding reductions in aggregate welfare and the coefficient of variation of firm sales under
Cournot competition are similar: 30 percent and 14 percent respectively. These results suggest that
consumers benefit enormously from the range of products supplied within a single firm.
All of these counterfactuals are subject to a number of caveats. In particular, they do not allow
for firm entry or exit, because they hold the set of firms constant, and they abstract from generalequilibrium effects by holding aggregate expenditure and factor prices constant. Nonetheless, thesecounterfactuals establish the quantitative relevance of our departures from the case of atomistic monopolistic competition with single product firms.

8

Conclusions

We develop and structurally estimate a model of heterogeneous multiproduct firms that can be used
to decompose the firm-size distribution into the contributions of costs, quality, scope and markups.
Our framework requires only price and expenditure data and hence is widely applicable. We use
these price and expenditure data to estimate the key parameters of the model, namely the elasticities
of substitution between and within firms, and the elasticities of marginal cost with respect to output.
We use the resulting parameter estimates and the structure of the model to recover overall firm
average quality, the relative quality of firm products, marginal costs, the number of products a firm
supplies and the firm’s markup.

42

Our results point to quality differences as being the principal reason why some firms are large andothers are not. Depending on the specification considered, we find that 50-75 percent of the variancein firm size can be attributed to quality differences, about 23-30 percent to differences in productscope, and less than 20 percent to average marginal cost differences. If we use a broad measure of
total firm quality, which encompasses both average quality and scope, we find that average quality
accounts for almost all of firm size differences.
We estimate substantially higher elasticities of substitution between varieties within firms thanbetween firms (median elasticities of 6.9 and 4.3 respectively), implying that a firm’s introduction of
new product varieties cannibalizes the sales of existing varieties. We estimate that the cannibalization rate for the typical firm is 0.45, roughly half-way in-between the extreme of no cannibalization(equal elasticities of substitution within and between atomistic firms), and the extreme of complete
cannibalization (varieties perfectly substitutable within firms).
These findings that firms supply multiple imperfectly substitutable varieties have important implications for the measurement of firm productivity, highlighting the role of assumptions about demand in the measurement of productivity for multiproduct firms. Conventional price indices basedon a weighted average of firm prices do not take into account that the true ideal price index for thefirm depends on the number of products it supplies. This introduces a systematic bias into the measurement of firm productivity, because larger firms supply more products than smaller firms. We
find this bias to be quantitatively large. An increase in firm sales is associated with around a one
third larger increase in true real output (using the ideal price index) than in measured real output(using the conventional price index).
We find that most firms charge markups close to the monopolistic competition benchmark of constant markups, because most firms have trivial market shares, and hence are unable to exploit theirmarket power. However, the largest firms that account for up to around 25 percent of sales within
sectors charge markups between 32 and 120 percent higher than the median firm. Using the estimated model to undertake counterfactuals, we find that these departures from the monopolisticallycompetitive benchmark have quantitatively relevant effects on aggregate welfare.

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47

A

Appendix A: Derivations

A.1

Derivation of Equations (7)-(9)The first-order condition with respect to the price of an individual UPC implies:u f t +Nf t 

Yut +

∑

k=u f tdAk (Ykt ) dYkt
dY
Pkt kt −dPut
dYkt dPut



= 0.

(38)

Using equation (5) and setting UPC supply equal to demand we have
 Y ∂Pgt Y ∂Pf t
Y ∂P∂Ykt= σ f − 1 kt
+ σu − σ f kt
− σu kt kt .
∂Put
Pgt ∂Put
Pf t ∂Put
Put ∂Put∂Ykas
We now can use equation (4) to solve for the elasticities and rewrite ∂P
u

∂Ykt
∂Put

=
=





 ∂Pf t Put Ykt∂Pf t Put Ykt∂Pgt Pf t
Y+ σu − σ f− σu kt 1{u=k}∂Pf t Pgt
∂Put Pf t Put∂Put Pf t Put
Put


Y
Y
Y
σ f − 1 S f t Sut kt + σu − σ f Sut kt − σu kt 1{u=k} .
(39)
Put
Put
Put

σf − 1

If we now substitute equation (39) into equation (38) and divide both sides by Yu , we get


P Y
P Y
1 + ∑ σ f − 1 S f t Sut kt kt + ∑ σu − σ f Sut kt kt − σu
Put Yut
Put Yutkk− ∑ σ f − 1 S f t Sut∂Ak (Ykt )∂Ykt Yktk

Put Yut− ∑ σu − σ f Sut
∂Ak (Ykt )∂Ykt Yktk

Put Yut

We define the markup at the firm or UPC level as µk ≡ Pk 

+ σu

∂Ak (Yk )∂Yk .

therefore ∑k Su PPuk YYuk = 1, we can rewrite equation (40) as







1 + σ f − 1 S f t + σu − σ f − σu − σ f − 1 S f t∑k∂Ak (Ykt )∂Ykt Ykt∑k Pkt Ykt− σu − σ f∂Ak (Ykt )
∂Ykt

Put

= 0.

(40)

Since Su Pu1Yu = ∑ P1k Yk andk ∑k∂Ak (Ykt )∂Ykt Ykt∑k Pkt Ykt

+ σu

1
= 0.
µut

Because we assume that σu is the same for all u produced by a firm, µut is the only u-specific term inthis expression. Hence, µut must be constant for all u produced by firm f in time t; in other words,
markups only vary at the firm level. Together these two results ensure the same markup across all UPCs
supplied by the firm.
We can now solve for µ f t by



 1
1
1
1 + σ f − 1 S f t + σu − σ f − σu − σ f − 1 S f t− σu − σ f
+ σu
= 0
µft
µft
µft

σf − σf − 1 S f t
⇒ µf =.
σf − σf − 1 S f t − 1

48

A.2

Derivation of Equation (13)σ −1 σu −1
σ −1 σu −σ −σu
ϕut Egt Pgtf Pf t f Put.

Equilibrium prices Pgt and Pf t , but not Put , are functions of this N f t . Recall Yut = ϕ f tf

Now suppose that the number of UPCs supplied by the firm (N f t ) can be approximated by a continuous variable and consider the partial elasticity of Yut with respect to this measure of UPCs:∂Yut∂N f t=== Y ∂Pf t
 Yut ∂Pgt
+ σu − σ f kt,
Pgt ∂N f t
Pf t ∂N f t
 Y N f t ∂Pf t
 Yut N f t Pf t ∂Pgt ∂Pf t+ σu − σ f kt,
σf − 1
Pgt Pf t N f t ∂Pf t ∂N f t
Pf t N f t ∂N f t





 ∂Pgt Pf t ∂Pf t N f t Ykt∂Pf t N f t Yut
σf − 1+ σu − σ f,∂Pf t Pgt
∂N f t Pf t N f t∂N f t Pf t N f t
σf − 1

where this partial elasticity holds constant the prices of existing UPCs. Using equation (3) for thefirm price index and (4) for the UPC expenditure share, we have:∂Pf t N f t=∂N f t Pf t

PN f t1−σu

ϕN f t1 − σu


Pfσtu

Nf t
Nf t
=
Pf t
1 − σuPN f t1−σu

ϕN f tNf t∑ n =1Putϕut

1−σu =Nf t
SN .
1 − σu f t

Recalling that expenditure shares are also the elasticities of the price index with respect to prices, we
obtain:Cannibalization ≡ −∂Yut N f t∂N f t Yut=−= Nf tNf t
σf − 1 S f t
S N f t − σu − σ f
SN
1 − σu1 − σu f t
 
 

σf − 1σu − σ ffor σu ≥ σ f > 1.
+
S f t S N f t N f t > 0,σu − 1σu − 1

Note that this derivation is based on solely on the derivatives of the CES price index with respect tothe measure of UPCs and observed prices. Since the assumption of Bertrand or Cournot competition
merely affects the decomposition of observed prices into markups and marginal costs, these derivatives are the same under both forms of competition. Therefore the derivation of the cannibalization
rate is the same under both forms of competition.

A.3

Cournot Quantity Competition

In our baseline specification in the main text above, we assume that firms choose prices under
Bertrand competition. In this appendix, we discuss a robustness test in which we assume insteadthat firms choose quantities under Cournot competition. Each firm chooses the number of UPCs
(N f t ) and their quantities (Yut ) to maximize its profits:u f +Nf t

max Π f t =N f t ,{Yut }∑ Put Yut − Au (Yut ) − N f t hut − h f t ,

(41)

u=u f

where we again index the UPCs supplied by the firm from the largest to the smallest in sales. From
the first-order conditions for profit maximization, we obtain the equilibrium markup:
49

ε ft
,
ε ft − 1

µft =

(42)

where the firm’s perceived elasticity of demand is now:ε ft =

1
1
σf −



1
σf − 1



,

(43)

Sft

and the firm’s pricing rule is:Put = µ f t

hidAu (Yut )δ= µ f t (1 + δ) aut Yut
.
dYut

(44)

Therefore the analysis with Cournot competition in quantities is similar to that with Bertrand
competition in prices, except that firms’ perceived elasticities of demand (ε f t ) and hence their markups
differ between these two cases. Since firms internalize their effects on product group aggregates,
markups are again variable. Furthermore, markups only vary at the firm level for the reasons discussed in the main text above.
Our estimation procedure for {σU , σF , δ} remains entirely unchanged because the firm markup
differences out when we take double differences between a pair of UPCs within the firm over time.
Therefore our estimates of the parameters {σU , σF , δ} are robust to the assumption of either Bertrand
or Cournot competition. Our solutions for firm and product quality { ϕU , ϕ F , δ} are also completely
unchanged, because they depend solely on observed expenditure shares and prices Sut , S f t , Put
and our estimates of the parameters { ϕU , ϕ F , δ}. Therefore whether we assume Cournot or Bertrand
competition is only consequential for the decomposition of observed UPC prices (Put ) into markups(µut ) and marginal costs (aut ). Markups are somewhat more variable in the case of Cournot competition, which strengthens our finding of substantial departures from the atomistic monopolisticallycompetitive markup for the largest firms within each product group.

A.4

Relative Firm Markups Under Bertrand Competition

For firms with non-negligible shares of expenditure within a product group (0 < S f t < 1):
µft − 1 =σF − (σF − 1)S f t
σF − (σF − 1)S f t − 1

−1

For a firm with a negligible share of expenditure within a product group (S f t ≈ 0):
µft − 1 =σFσF − 1

− 1.

The ratio of the two markups (for σF finite and S f t < 1) is:1.
1 − Sft
Therefore the dispersion of relative markups under Bertrand Competition depends solely on thedispersion of expenditure shares.

50

A.5

Relative Firm Markups Under Cournot Competition

For firms with non-negligible shares of expenditure within a product group (0 < S f t < 1):

1
σF −


µft − 1 = 

1
σF −

 1

 1



1
σF −1

1
σF −1





Sft

Sft

−1


 − 1.

For a firm with a negligible share of expenditure within a product group (S f t ≈ 0):
µft − 1 =σFσF − 1

− 1.

The ratio of the two markups (for σF finite and S f t < 1) is:
1 + S f t (σF − 1).
1 − Sft
Therefore, the dispersion of relative markups under Cournot Competition is increasing in the elasticity of substitution (σF ), and depends on both this elasticity and the dispersion of expenditure shares.

51