Product Offerings and Product Line Length Dynamics

Product Offerings and Product Line Length DynamicsXing Li

Department of Economics, Stanford University, 579 Serra Mall, Stanford, CA 94305-6072. [email protected]

This paper provides a model that uses preference heterogeneity to rationalize the cross-sectional and intertem-

poral variation in a firm’s horizontal product differentiation strategies. Product-line dynamics arise from

shocks to preference heterogeneity. For example, in the potato chip category I study, consumer concerns

over fat levels in foods created two desirable alternatives (low fat and zero fat) for each flavor. On the

supply side, firms learn about these changing tastes and adapt product lines accordingly. For tractability,

the heterogeneity in preference is captured by the nesting parameter in an aggregate nested logit demand

model. I find greater preference heterogeneity for chips in smaller packages and for markets with more demo-

graphic diversity. The dominant firm in the market bases its decisions primarily on its past experience in the

market, with the latest preference shocks representing only 30% of the influence in product-line decisions.

Gross margins are increased by 5% if firms have perfect information about preference heterogeneity. Costs

for product line maintenance constitute about 2% of total revenue. Sunk costs incurred when expanding the

product line are estimated to be four times the per-product fixed cost, thereby limiting the flexibility of

product-line adjustment. The probability of line length adjustment grows from 70% to 90% under a smooth

cost structure.

1. Introduction

One of the central decisions firms make is the level of their product differentiation. Product

differentiation can be exercised in two dimensions: vertically or horizontally. Vertical differ-

entiation means providing an upgraded or downgraded model and charging a different price,

for example, Apple iPhone5S and iPhone5C. Within the same “model”, firms can further

differentiate horizontally by providing different features of colors, flavors, or designs. Apple

1

Li: Product Offerings and Product Line Length Dynamics2

offers iPhone5S with three choices of colors; Dannon produces 6oz yogurt in different flavors.

Both companies are doing horizontal product differentiation within the same model.

There are at least two difference between these two types of product differentiation strate-

gies. First, vertical differentiation is mainly driven by leaps in R&D success (e.g., Goettler and

Gordon 2011), whereas horizontal differentiation is largely initiated by consumer tastes (e.g.,

Draganska and Jain 2005) that vary across different markets and over time. Furthermore,

vertical differentiation usually involves higher fixed costs of adapting production processes,

whereas horizontal differentiation typically utilizes the same process as existing products.

For both reasons, horizontal differentiation is more flexible and therefore creates more vari-

ation in a firm’s decisions. This variation in horizontal differentiation is what motivates my

investigation into firms’ extensions and contractions of their product lines.

Acknowledging the fact that consumers preference heterogeneity on the demand side is

the primary driver of horizontal product-line, I propose the following framework to ratio-

nalize both cross-sectional and intertemporal variation in product-line decisions. The extent

of preference heterogeneity varies across markets for reasons such as the concentration of

different demographic groups.1 Firms will provide a richer set of (horizontally differentiated)

products in markets with a more heterogeneous preference to serve a larger proportion of

consumers and make more profits. Within each market, firms can also adjust their product

lines over time in response to changes in preference heterogeneity.2 Firms are more likely to

expand (contract) their product lines when preference heterogeneity increases (decreases).

The main mechanism to support the above argument is as follows. Preference hetero-

geneity affects the tradeoff between cannibalization and new sales creation when expanding1 In this paper, “preference heterogeneity” is an aggregate statistic for both variety seeking within individuals and

preference heterogeneity among individuals.2 For example, manufacturers of potato chip will consider the immigration of Asian and Hispanic population. They

will also be aware of the consumers’ growing concern for their own health.


the product line. For multi-product firms, the newly launched product brings in new con-

sumers and cannibalizes market shares from existing products. When consumer’s preference

is quite homogenous, it is difficult to initiate extra sales to new consumers by new product

launching, and cannibalization effect dominates new sales creation effect. As a result, firms

may not maintain a long product line. On the other hand, when consumer’s preference is

quite heterogeneous, new sales creation effect dominates and expanding product line is more

profitable.

To formalize and quantify the above argument, I model the demand side using Nested

Logit. Different products (features) from the same brand are clustered in one nest (line) in

the choice structure. The nesting parameter has the same behavioral interpretation as the

heterogeneity of preference, which is an aggregate measure of both variety seeking within an

individual and preference heterogeneity across individuals.3 When products are more nested

within the line, they are closer substitutes; consumers agree on the preference ranking among

these products and the preference is more homogenous.4 On the other hand, when products

are less nested within the line, they are less substitutes; consumers have more varied views

on their favorite products, and preference is more heterogenous.

On the supply side, the multi-product firm chases time-varying preference heterogeneity

by adjusting product line length,5 which is described by an in-market learning model based

3 This modeling idea is rooted in the early motivation of nested logit model, that is specifying distributions of

unobserved heterogeneity to capture substitution pattern.4 Intuitively, consider two products that are equally favorable, and split the market. Suppose the price of one product

rises. When these two products are closer substitutes, the market share of the second product will increase more,

which means more people agree on the which product is their favorite.5 In reality, the firm decides on the number and the contents of all products in line simultaneously. However, I only

model the line length decision in this paper for the following three reasons. First, taste for potato chips depends on

numerous flavors, so that it is difficult to write down a characteristic-based demand model to predict the demand


on Hitsch (2006). Firms decide on product differentiation based on their belief of preference

heterogeneities, and realized market outcomes help them to update their belief. On top of

the standard in-market learning model, I allow the preference heterogeneity to evolve over

time, which explains the frequent adjusting behavior for an experienced firm operating in a

matured market for years.6 The elegant representation of nested logit model that is linear in

the nesting parameter makes modeling supply-side learning framework tractable.

I apply the model to the potato chip market, where there is a leading firm (I call it

Company A hereafter) with a market share of 60%. I find a more heterogeneous preference

for potato chips in smaller packages, which implies consumers are more willing to try new

flavors when buying small-packaged potato chips. I also find that a more diverse population

in the local market will tend to exhibit more preference heterogeneity, which is confirmed by

the estimation with a series of measures for population diversity in that market, including

the dispersion of income and age distribution, and the diversity of ethnic groups. On the

supply side, Company A applies in-market learning on preference heterogeneity to adjust his

product line length. I find that Company A bases its decisions primarily on past experience in

the market, with the latest preference shocks representing 30% of the influence. The marginal

cost of offering one additional product is estimated to be $3,560 per million households by

quarter; the total maintenance cost is estimated to be 2% of total revenue for an average

line with a length of 22. I also estimate the sunk cost incurred when expanding the product

differentiation to be three times the usual maintenance cost, which may limit the flexibility

of product-line adjustments.

for new products. Second, the action space will become extremely large if I model the content of products as well,

i.e., for an average product line in my sample with a length of 20, the action space is {Offer ,NotOffer}20. Third, the

dynamics of product line that I am focused in this paper requires more tractability.6 Technically speaking, by allowing time-evolving preference heterogeneity, the precision of belief does not explode to

infinity so that the belief will still change in response to market outcome, and so is product line strategies.


Based on the estimates, two counterfactual exercises evaluate the firm’s optimal line length

decisions under different product-line specific policy experiments. In the first exercise, I

simulate the optimal line length decisions without the existence of extra cost for line length

expansion. This removes the restrictions on Company A’s flexibility in adjusting line length,

and the probability of line length changes grows from 70% to 90%. In the second exercise, I

consider the situation where Company A knows the precise value of preference heterogeneity

at the time of product line length decisions. She can make a better decision based on the

true value instead of some guess, and the gross margin is increased by 5%. A byproduct

of the second counterfactual is to test the hypothesis of learning or knowing preference

heterogeneity when making line length decisions. I construct a test based on gross margin, and

the test result supports the assumption of learning rather than knowing about heterogeneity.

Both simulations shed lights on firm’s potential gain from product-line related improvement.

The first one relates to a more efficient cost for product line maintenance, say, a more flexible

contract on shelf-space and a better distribution system, and the second one relates to a

better knowledge about consumers from sources other than market realizations.

This paper is related to several strands of literature. First, there is a growing literature

on firm’s product differentiation and product-line design, both theoretically and empirically.

Theoretical works have discovered varies factors to determine product differentiation, includ-

ing communication cost to consumers (Villas-Boas 2004), vertical structure of distribution

(Liu and Cui 2010), consumer’s deliberation on their preference (Guo and Zhang 2012), vari-

ety preference and purchase cost (Bronnenberg 2014) and other rational interpretations as

well as behavioral explanations such as cognitive overload (Iyengar and Lepper 2000), artic-

ulated preference (Chernev 2003a,b), and contextual effects (Orhun 2009). However, there

are relatively few empirical papers in this field. Among those, most works take the prod-

uct line as given and evaluate its implications in consumers demand (Hui 2004, Draganska


and Jain 2006). The only work I know that explains the product line design is Draganska

et al. (2009) that offer a supply-side model of product provisions for horizontally differen-

tiated products. However, they restrict to a small subset of products and mainly attribute

the cross-sectional variation in product line to various supply-side competition environment.

This paper complements their work by offering a supply-side model for both cross-sectional

and inter-temporal variation in the length of the whole product line where the driving force

is preference heterogeneity on the demand side.

Second, this paper contributes to the in-market learning literature by proposing a new

learning object: preference heterogeneity. Numerous papers study demand-side learning

about the quality of new products (Roberts and Urban 1988, Erdem and Keane 1996, Ching

et al. 2013, Lin et al. 2014). On the supply side, Urban and Katz (1983) and Urban and Hauser

(1993) address firms’ market experimentation in designing new products. Hitsch (2006) stud-

ies firms’ learning the quality of new products when making exit decisions. Another series of

papers(Crawford and Shum 2005, Narayanan and Manchanda 2009) considers physicians and

patients learning about the effectiveness of drugs when making prescription decisions. This

paper differs from those empirical learning papers in two perspectives. First, the learning

object is preference heterogeneity rather than mean preference in most existing literature.

Second, the learning object is evolving over time whereas in standard learning framework,

the learning object is constant.7

Third, this paper is also related to researches on variety seeking. Models for variety seeking

find negative state dependence on past choice (Chintagunta 1998, 1999, Seetharaman et al.

2005, Dubé et al. 2009, 2010). This paper uses a different, but related metrics for the variety

seeking in the aggregate level. Finally, the supply side is modeled similarly as researches

7 Lovett et al. (2009) also model time-evolving learning parameters.


on empirical entry and product positioning.8 Early research on empirical entry infer firms’

profitability from their entry decisions (Reiss and Spiller 1989, Bresnahan and Reiss 1990,

1991, Berry 1992). Later research treats as endogenous variables the marketing mix other

than price (Berry and Waldfogel 2001, Mazzeo 2002, Berry et al. 2004, Seim 2006, Crawford

et al. 2011, Ryan and Tucker 2012). This paper contributes to this strand of literature by

proposing a tractable model of product line length dynamics for multi-product firms.

The rest of the paper is organized as follows. Section 2 introduces the data and some

reduced-form evidences on line length dynamics. Section 3 provides an empirical model to

quantify firm’s optimal line length decisions driven by preference heterogeneity. Section 4

describes the full specification and identification. Section 5 shows the results, and section 6

concludes.

2. Product Offerings in US Potato Chip Market

In this section, I will provide an overview of potato chip industry and description on the IRI

Academic Dataset (Bronnenberg et al. 2008) that I use.9 The last part of the section shows

some reduced-form evidence on product line length dynamics.

2.1. The Potato Chip Market

Potato chips can be found in most American households. An average US household will

spend $80 a year in salty snacks. Potato chips have a dollar share of 30% in the industry of

salty snacks, which means an average household will spend around $24 each year on potato

chips (First-Research 2011).

Chip manufacturers anticipate and respond to changes in consumer preferences. First of

all, in the potato chip industry, the ability to be innovative and differentiate a product is the

8 Dubé et al. (2005) provide an excellent summary of these papers.9 All estimates and analyses in this paper based on Information Resources Inc. data are by the author and not by

Information Resources Inc.


key to competition. As a result, manufacturers offer different choices of potato chips with

different flavors, fat contents, and cut types. Furthermore, consumer’s tastes vary by region

and over time. For example, Joon (2013) states that “consumers in the Midwestern region

prefer thick cuts and consumers in the southwestern states prefer bold and spicy flavors.” At

the same time, many exogenous factors drive the evolution of tastes over time. Population

migration is one such factor (Bronnenberg et al. 2012). Manufacturers are creating new

spicy flavors catering to a growing Hispanic and Asian population (First-Research 2011).

Consumers’ awareness of the health cost of eating potato chips high in trans fat and salt is

another factor. To capitalize on this shift, leading manufacturers have introduced a number

of new products with reduced fat and low salt content (Joon 2013). A third factor is the

change in taste for (new) flavors. Firms can elicit this change by inviting consumers to submit

their newly designed flavors.10 With the existence of diehard fans of classically flavored

potato chips, the regional and temporal variations of tastes imply changes in preference

heterogeneity and have corresponding implications on product differentiation decisions.

A second feature of potato chip industry is that it is highly concentrated, with a leading

player (Company A) having a market share of 60%. The second largest player has a market

share of only 5.2% (Joon 2013). Company A does not worry too much about potential

entrants. First, consumers have strong brand preference in picking potato chips. They are

willing to pay extra for branded chips. In addition, operating firms in this industry need to

have good relations with upstream suppliers and downstream retailers. They use long-term

contracts to hedge against the volatile prices for potatoes, sugars, oils, and fats from their

suppliers, and they are competing for the best shelf spaces in grocery stores.

10 For example, Frito Lay holds the contest called “Do us a flavor” in each year to invite consumers to submit their

newly designed flavors and launches the winners. The winning flavor will be awarded 1 million dollars.


2.2. Data

I use the IRI Academic Dataset from 2001 to 2007 to estimate the model.11 The IRI academic

dataset provides scanned sales data from a sample of grocery stores at the level of UPC-store-

week across 50 IRI markets. I restrict the analysis in this paper to the Salty Snack - Potato

Chip industry. I restrict to 8-13 serving size packages in my analysis. For potato chips offered

by firms other than the leading firm (Company A), I aggregate into the level of market-

quarter. For potato chips by Company A, I aggregate to the level of feature-market-quarter,

where one feature is defined as the triplet of flavor-cut-fat.12

Company A has wide variation in the length of its product line, defined as the count of

features sold in one market-quarter. Table 1 and Figure 1 present the distribution of line

length. The average line length is 22.09 with a standard deviation of 3.86. The shortest line

is in Raleigh/Durham - 2001q1, with a length of 8, whereas the longest line is in Chicago -

2002q2, that supplied 30 different features. Variation in line length derives from two sources:

cross-sectional and inter-temporal. Chip lines vary widely in line length in both sources.

Cross-sectionally, Pittsfield has the shortest line, with an average length of 16.89, whereas

Houston has the longest line, with an average length of 25.05. Line length also changes over

time, as is shown in Table 1 and right panel of Figure 1. Line length is quite sticky, with

about 30% cases there is zero changes, and in 85% the changes are within 2 features, but

there are still cases where Company A is quite aggressive in line length adjustment.

11 Although the IRI Academic Dataset is available from 2001 to 2011, I only make use of seven years for the following

reasons. First, the 2008 financial crisis heavily drove up prices of potato chips, which makes pricing decisions non-

trivial and complicate the model. Second, Company A did a national launch of zero-trans fat in 2008. The reasons

for the timing and scale of such a big event are beyond the scope of this project. Moreover, the concurrence of the

two events further complicates the analysis.12 Please refer online appendix for justification of data aggregation process.


I supplement the IRI dataset by merging with the IPUM CPS data to get the demograph-

ics. Among 302 MSAs (Metropolitan Statistical Area) in CPS, I have identified 98 that can

be merged with IRI markets. In terms of population, those matched MSAs constitute half

of the total population nation-wide. I proxy the total market size by number of households

in 2007, whereas other demographics that may correlate with preference heterogeneity are

calculated in a quarter by city level.

2.3. Reduced-form Evidence on Dynamic Product Offering

Before going into the structural estimation, I will show some reduced-form evidence on firms’

changing differentiation decisions based on market outcomes over time. When preference is

homogenous, consumers tend to agree on the preference ranking of all features within a line,

and in-line market shares for features are concentrated. In the extreme case, consumers fully

agree on the preference ranking, and the in-line market share is 1 for the most preferred

feature and 0 for others. In these cases, the model predicts that firms will contract the

product line. On the contrary, when preference heterogeneity becomes high, consumers have

various opinions about the most favorable feature, and in-line market shares become less

concentrated. In this case, the model predicts that firms will expand the product line.

To illustrate the argument above, I run the following regression:

LineLengthmt = ↵0 +↵1HHIm,t�1 +↵2LineLengthm,t�1 + cm + dt + "mt

where m indexes market, t denotes quarter, LineLengthmt is the length of the product

line, HHImt is the Herfindahl Index for in-line market share (i.e., HHImt =P

f s2f |l,mt,

where sf |l,mt is the in-line market share for feature f in market-time mt). cm and dt are

market and time fixed effects to control for geographic unobservables and seasoning effects.

Baseline regression confirms the model prediction (Table 2, Column 1). The higher the market


concentration, the shorter the product line in response. A one standard-deviation change in

HHI (0.03) will lead to a change in line length of 0.02.

One of the challenge for the interpretation of the estimate is that the measure of HHI is

mechanically decreasing in line length, and the estimated correlation is artifactual.13 The

worry is partly true as shown in Online Appendix, and I use another measure of market

concentration: the standard deviation of log in-line market share defined as

StdLnShareInLinemt = Std�lnsf |l,mt

�

which is not mechanically correlated with line length (Online Appendix). When in-line mar-

ket shares are more concentrated, the standard deviation is high. Regression results still

support our conjecture. A one standard-deviation increase in this concentration measure will

lead to a 0.36 increase in line length (Table 2, Column 2).

One alternative interpretation of the above findings is that firms will automatically with-

draw losing features that are unpopular. To deal with this challenge, I change the dependent

variable to be the indicator of line expansion. Regression results also confirm the initial the-

ory proposed above (Table 2, Column 3,4). The higher the market concentration, the less

likely the line gets expanded. A one standard-deviation decrease in HHI will lead to a higher

chance of line expansion by 4.74% (Table 2, Column 3) and 4.60% (Table 2, Column 4).

Compared to an average chance of line expansion of 34%, this increase is economically large.

A final caveat is that all these reduced-form evidences are correlational, not causal. The

complete model allows firms to adjust their line length based on all past market realizations

rather than just the last one. To quantify the above mechanism, we will estimate a structural

model with a richer set of specifications.

13 I regress line length on one-period lagged HHI. The mechanics in calculating HHI will contaminate the inference

only when firms have some inertia to adjust line length.


3. A Model of Product Line Length Dynamics

In this section, I propose a model that is structural in both demand and supply to capture

the effect of preference heterogeneity on the tradeoff between cannibalization and new sales

creation when firms are making product line length decisions. For simplicity, I assume that in

each market m, there is a separate monopolist. Within each market, the monopolist provides

a line of nt products indexed by j 2 {1,2, ..., nt} to compete with one single outside good

j = 0 in each period t.

3.1. Demand Side

For each market m and period t (suppressed temporarily), the utility for consumer i from

consuming Company A chip j 2 {1,2, ..., n} and outside goods j = 0 is

uij = (a+ "i)+ (c̄j + ✓"ij)�↵pj = �j +("i + ✓"ij)

ui0 = "i0

where (a+ "i) is consumer’s brand preference for company A, which includes average level

a and consumer heterogeneity "i; (c̄j + ✓"ij) is consumer i’s utility for product j, which also

includes the mean value c̄j and consumer’s heterogeneity ✓"ij; pj is the price for product j.

After some rearrangement, the utility for consumer i consuming product j equals the mean

utility level �j = a+ c̄j �↵pj and consumer’s heterogeneity ("i + "ij). Following Berry (1994)

and Cardell (1997), both "ij and ("i + "ij) follows i.i.d. type I extreme value distribution.

From the representation of cij = c̄j + ✓"ij, the value of ✓ measures consumer’s preference

heterogeneity over product j.14 When ✓ is high, cij varies a lot across different individual i,

and the preference is heterogenous. On the other hand, lower ✓ implies more homogenous

preference.

14 Here I am measuring the average level of preference heterogeneity across all products. Ideally we can assign ✓j for

each product j, but the data lack statistical power to identify all ✓js


The nesting parameter is an aggregate statistics of both individual level variety seeking and

cross-individual preference heterogeneity. If we think about the repeated purchases of one

individual as different purchase occasions, the variety seeking behavior can be rationalized

as the low correlation for individual-specific demand shocks among different products, which

is captured as high ✓ in current model. With market level data, I cannot identify between

variety seeking within individual or preference heterogeneity among individuals. But these

two channels should have similar implication for product assortment decisions, which is

presented later.

The Nested Logit model proposed here can be easily estimated in linear GMM. Within

each market m,

lns1t � lns0t = �jt + ✓t�� lnsj|l,t

�

= a+ cj �↵pjt + ✓t�� lnsj|l,t

�+ ⇠jt (1)

where s1t is the market share for all Company A products, s0t is the market share for all

non-Company-A products, sj|l,t is the in-line market share, which equals sjt1�s0t

. Following the

standard model, I allow the taste for product j to vary by time, with c̄jt = cj + ⇠jt, where cj

is the product fixed effects, and ⇠jt is the unobserved demand shock, which is distributed as

N�0,��1

⇠

�.

3.2. Static Profit when ✓ is Known

I assume that at the time of product line length decision, Company A does not know the

precise value of mean utility �jt so that she is taking expectation on over some distribution

F� (·). There are three reasons behind. First, product line length decisions are made prior to

the realization of demand, so Company A is ignorant about the demand shock ⇠jt. Second,

retailers can observe the demand shock and adjust the retail price pjt, so pjt is also unknown


before market realization. Third, when Company A launches some new product, the value of

cj is also unknown to her.15 By making this assumptions, I abstract away the identity of each

products in line and focus mainly on the length of product line. The total market share for

Company from offering a product line with length n follows the nested logit representation

with

s (n,✓) =E�

✓exp (I)

1+ exp (I)

◆

where

I = ✓ · lnnX

j=1

exp

✓�j✓

◆

The total market share s (n,✓) is increasing in n, increasing in ✓, and super-modular in n

and ✓ for most of the distributions F�.16 Super-modularity implies when expanding the line

length, the marginal gain in total share is larger when the preference is more heterogenous.17

Formally, let C (n, l) denote the cost of launching a line with length n while the line length

in the last period is l. A myopic firm will choose n to maximize

w ·M · s (n,✓)�C (n, l)

15 This also assumes out the product launching in the vertical sense or mass market strategy. (Johnson and Myatt

2006). When a company decides to launch a new product, she can either play mass market strategy so that the

new product is attractive to all consumers (i.e., with a high value of �) or niche market strategy that the feature is

attractive to a set of consumers (i.e., similar �). In the potato chip industry, it is quite difficult to launch a potato

chip that is favorable to all consumers and play mass market strategy.16 Super-modular means s (n+1,✓) � s (n,✓) is increasing in ✓. Proof of these properties are provided in Online

Appendix.17 This is consistent with the standard interpretation of price elasticity in the nested logit model. When products are

more nested within line, the price elasticity is higher within nests than between nests. Lowering price for one feature

will have larger “cannibalization effect” that will consume the market share of other products within the line than

“new sales creation effect” that will increase the total share of all products in line. The same logic applies to the

strategy of line expansion. The “cannibalization effect” of expanding the line dominates the “business stealing effect”

when features are more nested, and in this case firms are less likely to expand the line.


where w is the manufacture margin, M is the market size. Let n⇤(✓,m) be the optimal line

length choice made, and super-modularity means n⇤ is increasing in ✓t.

3.3. Dynamic Learning on Time-evolving ✓

As mentioned earlier, preference heterogeneity evolves over time due to many exogenous

factors including population migration, health concerns, as well as evolving tastes for new

flavors. I further assume that firms do not know the true value of preference heterogeneity

when making line length decisions. Instead, they have some beliefs on this value and update

their beliefs based on market realizations.18

3.3.1. Learning from Market Realizations Suppose at the beginning of period t, Com-

pany A has a prior belief on ✓t, which is modeled as a truncated normal with mean µt and

precision �t, truncated at unit interval (0,1), which is denoted as TN�µt,�

�1t

�. After market

gets realized, the market shares on all products are observed, and Company A can observe

one signal from each product j as derived from (1):

'jt = lns1t � lns0t � a� cj �↵pj = ✓t�� lnsj|l,t

�+ ⇠jt

Aggregate signals from all products about the same ✓t will get an aggregate signal19

t =

Pj

�� lnsj|l,t

�·'jtP

j ln2 sj|l,t

= ✓t +

Pj

�� lnsj|l,t

�· ⇠jtP

j ln2 sj|l,t

with precision

ht =

X

j

ln

2 sj|l,t

!·�⇠

A nice property for truncated normal belief is that it is also a conjugate prior for normal

data generation process, which is shown in the next theorem18 There is no direct test about the informational assumption that firms do not know the exact value of preference

heterogeneity because the stationary learning model (as described below in this paper) and complete information

model are not nested with each other. However, I will show some indirect test result based on simulation in later

section.19 For convenience, the notation ln2 sj|l,t means

�ln sj|l,t

�2.


Theorem 1. Suppose the prior is truncated normal

✓t ⇠ TN�µt,�

2t = ��1

t

�

and an unbounded signal is observed with value t and precision ht, then the posterior belief

is also truncated normal

✓t|t, ht ⇠ TN⇣µ0t, (�

0t)

2= (�0

t)�1⌘

with

µ0t =

�t

�t +ht·µt +

ht

�t +ht·t (2)

�0t = �t +ht (3)

Proof is shown in Appendix ??.

3.3.2. Evolution of ✓t The next step is to model the time-evolution of preference hetero-

geneity ✓t. The reason for allowing ✓t to evolve over time is two-folds. First, in the potato chip

industry, we do observe preference heterogeneity changes over time and chip manufactures

responds by adjusting their product line strategies. Second, for modeling perspective, if the

preference heterogeneity is constant over time, as an experienced firm operating in a mature

market, Company A is sophisticated enough to know the true value of preference preference

heterogeneity and no intertemporal variation in product line should be observed. The large

intertemporal variation in product line length motivates the assumption of time-evolving

preference heterogeneity.

If ✓t is not truncated, a natural candidate model is random walk, with

✓t+1 = ✓t +�✓⌫t


where ⌫t ⇠N (0,1) is the evolution error, or equivalently,

✓t+1|✓t ⇠N�✓t,�

2✓

�

In the truncated case, I propose the following “quasi” random walk

✓t+1|✓t ⇠ f (·|✓t;�✓)

which is similar to the random walk process as for unbounded case with acceptance-rejection

at unit interval. Convoluted with the truncated normality on ✓t, we can approximate the

prior belief of ✓t+1 as TN�µt+1,�

2t+1 = ��1

t+1

�with

µt+1 = µ0t (4)

�2t+1 = (�0

t)2+�2

✓ (5)

Details are included in Appendix ??.

3.4. Line Length Dynamics Chasing Time-evolving Preference Heterogeneity

When we combine the above two pieces of dynamic learning and ✓ evolution, we can have

the full description of firm’s dynamic problem. The action-specific flow profit

⇡n (µt,�t, lt) =w ·M ·E (s (n,✓t) |µt,�t)�C (n, lt)

and the value value function is

Vn (µt,�t, lt) = ⇡n (µt,�t, lt)+� ·E (V (µt+1,�t+1, lt+1) |µt,�t, lt, n)

V (µ,�, l) = Emax

n(Vn (µ,�, l)+�✏✏n)

where the state variables are the belief mean, belief precision, as well as last period line

length, and the transition probability is defined as (2) (3) (4) (5), with an additional one for

lt+1 = n.


4. Empirical Specification and Identification

In this section, I will present the full empirical specification and identification of the model.

Similar to Hitsch (2006), I apply two-step estimation, where the demand side is estimated in

linear GMM, and its parameters are plugged in to the supply side. I estimate the dynamic

supply model by maximizing likelihood. This section ends with a discussion on the identifi-

cation of the model.

4.1. Demand Side

The demand side is modeled as a nested logit of with two nests where all Company A chips

of different features are nested in one line, and all non-Company A chips are treated as

homogenous outside products. Based on (1), for each market m,

lns1mt � lns0mt = am + cj �↵pjmt + ✓mt

�� lnsj|l,mt

�+ ⇠jmt (6)

Both pjmt and lnsj|l,mt are endogenous, because they are correlated with the unobserved

demand shock ⇠jmt. I employ the following sets of instruments for the two endogenous vari-

ables:

• The summation of characteristics (flavors, fat content and cut type) of other Company

A chips sold in the same market-time

X

j0 6=j

xj0mt

• Average price of the same feature sold in other geographical markets in the same time

1

#

X

m0 6=m

pjm0t

• Other cost for raw materials, including potatoes, sugar, soy bean oil, edible butter, and

edible tallow


• Number of competitor brands and number of competitor UPCs other than Company A

chips within the same market-time

The first set of instruments are widely known as BLP instruments, which Berry et al.

(1995) started to use. The underlying assumption is that the characteristics are exogenous

to demand shocks. In the current model, the upstream wholesalers make product assortment

decisions whereas downstream retailers make pricing decisions. In reality, grocery stores and

manufacturers jointly decide what to display in advance. If some of the features do not sell

well, grocery stores will lower prices to sell out the storage. In this case, it is natural to

assume the assortment decision is made prior to the realization of local demand shock.

The second set of instruments are known as Hausman instruments which Nevo (2001)

started to use in demand estimation. The underlying assumption is that demand shocks are

independent over different markets, but there are factors that may affect the pricing for all

markets. These factors include, but are not restricted to, common cost shifters that affect

the pricing decisions across markets.

The last set of instruments consider the competition environment that was used in Bres-

nahan et al. (1997). The argument is that competition environments affect firms’ pricing

decisions, which is orthogonal to demand shocks. In this project, I can also exploit the huge

variation in the competition environment across different markets measured by the number

of competitor brands and UPCs.

4.2. Supply Side - Flow Profit

In each market m, the action-specific flow payoff of Company A is

⇡n,m (µ,�, l) = wm ·Mm ·E (sm (n,✓) |µ,�)�Hm · c (n, l)

sm (n,✓) = E�⇠Fm

✓exp (I)

1+ exp (I)

◆

I = ✓ · lnnX

j=1

exp

✓�j✓

◆


In other words, I allow a market-specific value profit function and calibrate the parameters

as follows:

• wm: manufacturer’s margin, calibrated from average price in that market, adjusted by

retailer’s markup (15%), distributor’s markup (25%) and manufacturer’s gross margin (30%),

i.e., wm = p̄m · 0.85 · 0.75 · 0.3

• Mm: market size, calibrated from total number of household Hm, with assumption that

an average household spend X dollars per quarter in buying potato chips, where X is cal-

culated from $24 spent by an average household in a year in potato chip consumption,

adjusted, by quarters and market shares of large package sized chips, i.e., Mm = Hm · 6 ·

ShareLargem/p̄m

• Cost of line length maintenance: assume a per-capita cost, i.e., Cm (n, l) =Hm · c (n, l).

In the estimation, I tried two specifications of the per-capita cost: linear and kink. In the

linear specification, c (n, l) = c ·n. In the kink specification, c (n, l) = (c1 + c21 (n> l)) ·n

• Fm: distribution of mean utility �, assume normality, with mean and variance calibrated

by the empirical distribution of {�jmt}j,t

The only parameters to estimate in the flow profit is the cost parameter {c1, c2}.

4.3. Supply Side - Dynamics

Firm’s dynamic problem is described as

Vn,m (µt,�t, lt) = ⇡n,m (µt,�t, lt)+� ·Em (Vm (µt+1,�t+1, lt+1) |µt,�t, lt, n)

Vm (µ,�, l) = Emax

n(Vn,m (µ,�, l)+�✏✏n)

The unspecified parameters are initial belief (µ1,m,�1,m), the evolution rate �✓,m as well

as the scale of random fixed cost �✏. All parameters are identified as shown from below, but

I still impose the following cross-market restrictions to simplify the calculation.


• �1m: initial prior precision is assumed to be proportional to the precision of signal.

This is justified by stationary assumptions in the learning process. For markets with a more

precise signal, the learning speed is expected to be fast. However, this is only valid if the

belief precision is the same. I equalize the learning speed across all markets by assuming that

the prior belief is proportional to signal precision, i.e., �1m = k� · hm, where hm =

1#

Pt hmt

be the average precision.

• �✓,m: evolution rate of preference heterogeneity. From stationary assumption, �✓,m =

k� · (k� +1) ·hm after combining stationarity and (5)

1

�1m=

1

�1m +hm+�✓,m

• µ1m: initial prior mean, integrated from calibrated normal distribution, with mean and

variance estimated from {mt}t20

So the dynamic parameters to identify is {k�,�✏}

4.4. Identification

This section briefly shows the identification of of supply side parameters without imposing

any cross-market restriction, i.e., market-specific parameters are separately identified. In

the current version, we assume that initial prior mean µ1 is known (and integrated out in

the estimation). However, the identification does not rely on this assumption. A stronger

identification result without knowing prior mean is described in Appendix ??.

In our data, we can observe actual line length decisions, signal values and precisions, as

well as prior mean

{nt,t, ht, µ1}

20 Note that initial prior mean is also identifiable as is shown in Appendix ??. However, I follow the convention of

learning literature to integrate out this value.


Based on these information, I will show the non-parametric identification of preference evo-

lution rate, initial belief precision, and line length maintenance cost, and scale of fixed cost

for launching21

{�✓,�1, c}

4.4.1. Preference evolution rate �✓ and prior precision �1 Signal evolution rate �✓

measures how fast ✓ evolves over time. Intuitively, ✓t can be estimated from demand, and

this rate is identified by the demand side estimation ˆ✓t. Equivalently, the signal value t is

calculated based on demand estimation, and �✓ is identified from Var (t+1|t), because t+1

deviates from t by three reasons: signal error in period t, signal error in period t+1, and

the deviation of ✓t+1 from ✓1. The precision of the first two errors are known, so the rate of

evolution is identified.

Initial precision is identified by stationary assumptions that the precision belief does not

explode. From the following equation

1

�1=

1

�1 +¯h+�✓

we can pin down �1. The intuition is that when making line length decisions, Company A

cannot rely too much on market signal, because signal is noisy, measured by ¯h. She can

neither rely too much on her prior belief, because ✓ evolves over time, as is measured by

�✓. The optimal balancing between these two sources pin down the belief precision in the

stationary level.

21 A final supply side parameter �✏ is a nuisance parameter which is not non-parametrically identified. But since

we have impose functional form assumption on the value function, including the estimation of this parameter will

improve the model fit a lot.


4.4.2. Cost of line length maintenance c From the last part, I have shown identification

of �1 and �✓. With the knowledge of µ1, I can calculate the whole process of belief process

{µt,�t}, and the state variable is known. The cost parameter is identified by the standard

argument of Conditional Choice Probability E (nt|µt,�t, lt) proposed by Magnac and Thes-

mar (2002). Intuitively, fixing the belief precision, when the cost is low, optimal line length

is more responsive to changes in belief mean, as is shown in Figure 2. The cost is identified

by “regressing” actual line length nt on the belief mean µt, controlling for �t.

5. Results

This section shows the model estimates and various simulation results based on estimates

obtained.

5.1. Demand Estimation

In the demand side, I estimate a Nested Logit model specified in (6). I report the average

estimates of preference heterogeneity by imposing ✓mt = ✓ in this part, but in the supply

side, I allow preference heterogeneity to vary by market and time.

Table 3 reports the estimation result from the demand side. Column (1) disregards the

existence of endogeneity problem and directly estimate the equation by OLS. Column (2)

overcome this problem by applying three sets of instruments as described before. By compar-

ing column (1) and column (2), I find that instrumental variables work well as expected. Both

preference heterogeneity and price elasticity will be under-estimated without controlling for

endogeneity, and the characteristic vectors only become significant in 2SLS specification.

Note that the first two columns in Table 3 use characteristic vectors (flavor fixed effects,

cut types, fat contents) to describe one product. In column (3), I replace with a more precise

control, that is product fixed effects. The estimates for price elasticity does not change too

much (-2.38 in Column 3 compared to -2.53 in Column 2), but the estimates for preference


heterogeneity almost doubled. As mentioned, the characteristics vectors cannot capture con-

sumer’s preference completely, so I take the product fixed effects estimates as benchmark

case, where the preference heterogeneity is estimated to be 0.41 (with a standard error of

0.02, Column 3, Table 3). In Column (4), I allow price elasticity to vary by demographics. I

find that price is less elastic in markets with a richer population measured by median income,

or older population measured by median age, which coincides with most previous findings.

The main parameter of interest is the preference heterogeneity in this paper, so in Table

4, I explore the source of preference heterogeneity by interacting with different observables.

Column (1) copies the Column (3) from Table 3 to serve as a benchmark case. In Column

(2), I estimate the same model but in the data for small-package-sized potato chips. I find

that preference is more heterogenous (0.67 in Column 2 compared to 0.41 in Column 1)

and price is more elastic (2.74 in Column 2 compared to 2.38 in Column 1). This extra

heterogeneity in preference may come from the fact that consumers are more willingness to

try new flavors when buying small sized potato chips. There are two sources of preference

heterogeneity estimated in this paper: one is the preference heterogeneity between consumers,

and the other is the preference heterogeneity within consumer but in different purchase

occasions. I cannot separately identify these two sources with only market level data, but I

believe that the second source is more significant in markets for small packaged potato chips.

The difference in heterogeneity estimation supports the existence of heterogeneity within

consumers in different purchase occasions, and this is related to variety seeking behavior.

Another source of preference heterogeneity comes from population diversity. In Column

(3)-(7) of Table 4, I explore to what extent population diversity can explain preference

heterogeneity. The results are robust to a series of diversity measures. In Column (3), I uses

interquartile of income distribution to measure the population diversity. I find that in markets


with a more disperse income distribution, the preference heterogeneity is significantly higher.

To quantify this estimates, I take out two markets with minimum (0.04) and maximum

(0.10) diversity measure, and the implied difference in heterogeneity is 0.09,22 or 20% of

the baseline heterogeneity of 0.41. In column (4), the diversity measure is the dispersion

of age distribution, and the implied difference in heterogeneity is 0.07, or 17% of baseline

value. Other than the above two dispersions, the preference heterogeneity is also explained

by diversity of ethnic groups. In Column (5), I use Asian population ratio in that market

and find that in markets with a 10% higher Asian population ratio, the preference is more

heterogenous by a measure of 0.047 out of baseline value of 0.41. In Column (6), I use

Hispanic population ratio, and the interaction term is not significant. This is because there

is a wide range of Hispanic population measure from 0 to 53%. If the true functional form is

non-linear, using linear function form to approximate may not get significant result. Instead,

I discretize the measure using a dummy for above median, and the estimates is reported in

Column (7). In markets with above-median Hispanic population ratio, the preference is more

heterogenous by a measure of 0.12 out of baseline value of 0.41.

5.2. Supply Estimation

I plug in the coefficients and estimate the supply side by maximum likelihood. Solving the

original problem with brute force is difficult, because calculating the line share sm (n,✓), the

flow payoff ⇡n,m (µ,�, l) and the state transition f (µt+1,�t+1|µt,�t, n) all requires simulation.

However, I can employ numerical methods to further simplify the calculations.

For sm (n,✓), I use power polynomials to approximate. Because it does not contain any

parameters to estimate, the approximation needs to be calculated only once. The reason

for using polynomials is the ease for preserving monotonicity and super-modularity in the

22 This is calculated by (0.1� 0.04)⇥ 1.48


approximated function, which is the key for identification.23 To calculate ⇡n,m (µ,�, l), I

use quadrature to calculate the expectation with respect to ✓ although ✓ is distributed in

truncated normal instead of normal. When the precision is quite high, and the mean is far

from the boundary, the truncated normal can be approximated by standard normal because

the probability of ✓ lying outside the boundary is low. In terms of state transition probability,

because the line length stays at a high level (for the large package size, the line length ranges

from 8 to 30, with an average of 22), and the precision does not explode because of the

time-varying ✓, I simply assume the state transition probability does not depend on action n,

which relieves the computation burden. Finally, I use Chebyshev polynomials to approximate

the value function and estimate the single-agent dynamic game with unobservable and time-

varying state variables.24

Table 5 reports the estimation results. I estimate the model in two specifications. In the first

specification, I assume the maintenance cost per capita (1M household) is linear in the line

length, whereas in the second specification, the marginal cost is higher when manufactures

are expanding their lines. In the first specification, the marginal cost of expanding a line

by length one is $3,560 per million of household. For an average line length of 22, the total

(variable) cost of maintaining a line length in an average-size city with 2.63 million household

is approximately $0.2 million.25 As a comparison, the industrial in an averaged-sized city

with average line length selling at average price is $8.96 million,26 the product line related

cost constitutes about 2% of total revenue.

23 I use CVX to get the approximation, which is a regularized optimization package (Grant et al. 2008). See Appendix

?? for details.24 The recent development of MPEC (Dubé et al. 2013) is also applicable to this model.25 $3,560⇥ 22⇥ 2.63 = $0.2M, all numbers are taken from Table 1.26 $0.25⇥ 0.03⇥ 22⇥ 54.31M= $8.96M, all numbers are taken from Table 1.


In the second specification, the cost is nonlinear, and I find an extra cost ($6.14K compared

to $2.08K) of expanding the product line. This extra cost comes from the inflexibility of

displaying, distributing, storing or advertising additional products. The extra cost limits the

flexibility of line length adjustment in two senses. First, it restricts the possibility of line

expansion because expanding the product line may incur this extra cost. Second, it also

restricts the possibility of product line contraction, because when Company A considers

withdrawing some products, she might worry about the future cost of pulling them back

again. Counterfactual analysis in the next subsection may quantify this inflexibility caused

by non-linear cost structure.

The precision ratio between belief and signal is estimated to be around 2.5 in both specifi-

cations. Note that this ratio determines the linear weight for prior and signal when updating

the belief. From the estimation, Company A places 30% of decision weight on in-market sig-

nal and 70% weights on past experience, summarized by prior belief. Even as an experienced

player in a matured market, Company A is still leveraging heavily on the in-market learning,

because of the evolutionary nature of preference heterogeneity. The market signal is a bit too

noisy, so Company A cannot rely completely on the market signal. Counterfactual analysis

in the next subsection will show the gross margin Company A may achieve if she knows the

true value of heterogeneity in advance.

5.3. Model Fit

In order to evaluate how the model fit the data, I simulate the line length decisions in all

50 markets. Within each market, the prior mean µ1 is drawn from known distribution, and

prior precision �1 is known from estimation, initial line length n1 = l2 is taken as given.

After specifying the initial condition, beliefs are updated from signals (1, h1) to get belief

in period 2 (µ2,�2), and the optimal line length n2 is simulated, and the process goes on to

the end of data period.


I run simulations to check how the model fit the data. In the first simulation, signals (t, ht)

are taken from data. In the second simulation, I simulate these signals. Figure 3 compares

actual and simulated line length in two markets, and Figure 4 compares the whole distribution

of line length and line length changes for actual and simulated data. Both simulations fit the

data quite well in most markets. The first simulation fits the data almost perfect, because

it makes use of most information from the data. The second simulation also fits well. In

the model, there are three factors that determines the optimal line length choices. They are

evolution of preference heterogeneity, signaling error caused by demand shocks, and random

fixed cost of product line adjustment. The first simulation only average out random fixed

cost, and simulation result confirms that this cost is not the driving force for actual line

length patterns. The second simulation averaged out both random fixed cost and signaling

error. The only remaining force that determines the line length pattern is the evolution of

preference heterogeneity, which is the main mechanism in this paper. In the remaining part

of this paper, I will always implement the second simulation.

5.4. Counterfactuals

I run two sets of counterfactual simulations to evaluate Company A’s optimal line length

responses to product-line related policy changes. In the first counterfactual exercise, I evalu-

ate firm’s optimal line length decisions under a smooth cost structure; in the second counter-

factual exercise, I estimate firm’s improvement in gross margin under complete information

about preference heterogeneity when making line length decisions. A byproduct of of the sec-

ond counterfactual exercise is to provide some indirect test on the information assumptions

of the firm: does he know or learn?

5.4.1. Smooth cost structure The non-linearity of cost structure restricts firm’s flexibil-

ity to adjust product line. This simulation quantify how much. In this simulation, I take the


cost structure as linear in the first specification from supply side and simulate market signals

as well as firm’s optimal responses. The results is illustrated in Figure 5. The distribution

of line length does not change too much, as is shown from the left panel, whereas the distri-

bution of line length changes becomes more dispersed in the right panel, which means that

Company A is more likely to adjust line length aggressively in the smooth cost structure.

To further quantify this change, the probability of line length adjustment grows from 70%

in the raw data to 90% in simulation.

The effect is quite symmetric in line length expansion and line length contraction, as is

shown in the right panel. Under a smooth cost structure, probability of line length expansion

and line length contraction both increases significantly. As is mentioned before, the increase

in line length expansion reflects the static concern that expanding the product line will incur

more cost, whereas increase in line length contraction reflects the dynamic concern that the

firm is more cautious in withdrawing some flavor because they might worry about the future

cost of pull them back again. Simulation result confirms the existence of both effects that

restricts the flexibility of line length adjustment.

5.4.2. Perfect information on preference heterogeneity Figure 6 shows the simula-

tion result for complete information on preference heterogeneity when making line length

decisions. The actual line length decisions under complete information deviate a lot from

the baseline case with learning heterogeneity. This is simply because Company A adapts

instantly to the time-evolving heterogeneity rather than chasing time-varying heterogeneity

under the learning model. The resulting gross margin is increased by 5% under complete

information. On the other hand, the change in line length adjustment does not change a lot.

Based on this simulation result, I can indirectly test the information hypothesis that

Company A learns rather than knows the true value of preference heterogeneity. First note


that the two hypothesis are not nested in the model of stationary learning,27 so there is no

direct test based on some parameters. Motivated by the fact that with complete information,

Company A will enjoy a higher gross margin, I propose the following test based on gross

margin.

In the data, we can calculate the gross margin across 50 cities over 28 quarters, which

gives us a vector gm with a length of 1,400. Let FK denote the distribution of gm generated

in model where firms knows heterogeneity, and FL denote the distribution of gm generated

from the model where firms learns heterogeneity. To test the assumption of learning, it is

equivalent to test

H0 : gm⇠ FK , H1 : gm⇠ FL

It is quite difficult to calculate a test statistics in testing high-dimensional vector, but at

least we can sacrifice some of the power and focus on some statistics. Figure 7 reports the

test result for the median level of gross margin. We can see that the two distributions are

quite separated, and the actual data is observed to come from FL. We can reject the null

and tend to believe in the information assumption, that Company A learns about preference

heterogeneity when making product line decisions.

6. Conclusion

This paper links product line length decisions with heterogeneity of preference and rational-

izes its cross-sectional and intertemporal variation. Preference heterogeneity in this paper

includes both preference heterogeneity across individuals and variety seeking within indi-

viduals, and it is measured by the nesting parameter in the standard nested logit model.

Cross-sectional variation in preference heterogeneity, which is partly driven by the diver-

sity of population demographics, explains differentials in line length among different cities.

27 In the standard learning framework, the two hypothesis is nested. In order to test whether agent knows the true

value, it is equivalent to test whether the initial belief precision is infinity. (Hitsch 2006)


Within one city, a firm’s in-market learning of preference heterogeneity drives line length

adjustment.

I apply the model to the potato chip industry, where Company A is the lead player. The

preference heterogeneity is estimated to be 0.41 in large package size chips and 0.67 in small

package size chips, which means preference for small packages is more heterogenous. This is

driven by more intensive variety seeking for small package chips. I also find that preference

is more heterogeneous in markets with a more diverse population, measured by dispersion

of income and age distribution and complication of ethnic groups.

On the supply side, Company A, as an experienced firm in a mature market, also applies

in-market learning about preference heterogeneity to adjust differentiation decisions. I find

Company A bases its decisions primarily on past experience in the market, with the most

recent market realization representing only one-third of the influence on product-line deci-

sions. The cost for maintaining an average line length constitutes about 2% of total revenue.

I estimate the sunk cost incurred when expanding product differentiation to be three times

the usual maintenance cost, which may limit the flexibility of product-line adjustment.

Counterfactual analysis based on the estimates evaluate firm’s optimal line length deci-

sions under smooth cost and in cases with complete information rather than learning about

preference heterogeneity. In the first case, Company A is found to be more aggressive in line

length adjustment under a smooth cost structure; in the second case, Company A’s gross

margin is increased by 5% when she knows the true value of preference heterogeneity. The

result for the second counterfactual also helps to test the information assumption that firms

learns rather than knows the preference heterogeneity at the time of line length decisions.

The test statistics supports the information assumption of learning.

The whole model is easily applicable to other industries in which product differentiation

is a key decision. One example is the two MP3 players produced by Apple: iPod Classic


and iPod Nano. iPod classic provides a limited choice of colors—always black or white—but

iPod Nano offers a longer line of colors. The length of the Nano line also varies over time,

from two in the first generation to nine in the fourth generation and back to six in the most

recent one. The mechanism in this paper explains the difference between two MP3 players,

as most consumers of the iPod Classic are professional music lovers who care more about

sound quality, control convenience, and storage and less about colors, whereas consumers

buying iPod Nano are younger on average and care more about colors and have more diverse

views on their favorite one. The time-varying changes in line length for the Nano can be

attributed to Apple gradual learning about preference heterogeneity.

The model simplifies the measure of preference heterogeneity. I use a nesting parameter in

the nested logit model for two primary reasons. First, nested logit is simple and interpretable.

A more sophisticated heterogeneity pattern can be carried by a mixed logit, but getting one

statistic to measure the diversity of preference is difficult. Second, the linear representation

of the nested logit model makes the supply-side learning tractable. Further research should

be directed toward finding a better way to model preference heterogeneity and link it to

product differentiation decisions.

Another shortcoming of the paper is the only focus on product line length and abstracting

away the identify of each products. This is a natural result from a less restrictive metric

of preference heterogeneity that is not associated with any of the characteristics or flavors.

The identity of the product will be captured by a more complicated model with preference

heterogeneity specified to each products.

A third limitation is that I make the monopoly assumption. This assumption is justifiable

in the potato chip market, but in other markets with competition, the supply-side learning

model needs to be modified.


Acknowledgments

I am grateful to my advisors, Tim Bresnahan, Wes Hartmann, and Petra Moser, for their invaluable guidance,

discussion, and encouragement. I would also like to thank Chris Colon, Chen Cheng, Oystein Daljord, Michael

Dickstein, Liran Einav, Pedro Gardete, Daniel Grodzicki, Han Hong, Mike Kruger, Brad Larsen, James

Lattin, Anqi Li, Harikesh Nair, Sridhar Narayanan, Joe Orsini, Qiusha Peng, Peter Reiss, Gregory Rosston,

Navdeep Sahni, Stephan Seiler, Stephen Teng Sun, Paul Wong, Yiqing Xing, Constantine Yannelis, Pai-Ling

Yin, and seminar participants at Stanford Department of Economics, Stanford Marketing WIP, Marketing

Science Conference in 2014 Atlanta for their helpful comments. The usual disclaimer applies.


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Table 1 Summary Statistics

Obs. Mean S.D. Min MaxSales and Prices

Market Share 30930 0.03 0.05 0.00 0.38Market Share In Line 30930 0.05 0.06 0.00 0.53Price ($/oz) 30930 0.25 0.07 0.12 0.43Fat Free 30930 0.09 0.28 0.00 1.00Reduced Fat 30930 0.15 0.35 0.00 1.00Ruffle Cut 30930 0.28 0.45 0.00 1.00Wavy Cut 30930 0.13 0.33 0.00 1.00Line Length (number of features) 1400 22.09 3.86 8.00 30.00Change in Line Length 1350 0.19 2.11 -8.00 9.00Line Expansion 1350 0.39 0.49 0.00 1.00HHI for In-line Market Share 1400 0.13 0.03 0.07 0.36Std. for Log In-line Market Share 1400 1.28 0.23 0.72 2.06Number of competitor firms 1400 7.41 2.77 3.00 20.00Number of competitor UPC 1400 51.77 24.71 12.00 166.00Market Size (Million Oz) 50 54.31 57.10 6.06 278.27

Demographics

Median Income (1K $) 1400 56.81 8.98 23.10 89.09Median Age 1400 35.08 2.89 26.00 48.33Interquartile Income (1M $) 1400 0.06 0.01 0.04 0.10Interquartile Age (10 years) 1400 3.40 0.20 2.75 4.47Asian (%) 1400 0.04 0.04 0.00 0.31Hispanic (%) 1400 0.10 0.11 0.00 0.53Number of Households (Million) 50 2.63 3.10 0.26 17.10

Cost Shifters

Potato Price ($/100lb) 28 12.37 4.29 7.42 21.90Refined Sugar Price (cent/lb) 28 45.20 3.58 41.93 51.93Soy Bean Oil Price (cent/lb) 7 28.28 11.58 16.46 52.03Edible Butter Price ($/lb) 7 1.41 0.27 1.11 1.82Edible Tallow Price (cent/lb) 7 19.60 5.53 13.71 30.76

Notes: Sales and prices data for 58 Company-A features (a unique combination of 36 flavors, 3 fat contents - regular,

reduced fat, fat free, and 3 cut types - flat, ruffle, wavy) across 50 markets, over 28 quarters in 7 years (2001-2007)

are aggregated from IRI Academic dataset. Features that have positive sales for less than 12 weeks are dropped from

the sample and their market shares are proportionally allocated to other features within serving sizes - city - quarter.

Demographic data over 50 cities and 28 quarters are merged from IPUM CPS dataset. Cost shifters for 28 quarters

or 7 years depending on the data availability are collected from various year books published by Bureau of Labor

Statistics and Department of Agriculture.


Table 2 Reduced Form Evidence for Line Length Dynamics

FE, dependent variable is

Line Length,t+1 1(Line Expansion,t+1)

(1) (2) (3) (4)HHI for In-line Market Share -6.78*** -1.58***

(1.41) (0.41)sd Ln ShareInLine -1.57*** -0.20***

(0.25) (0.07)Line Length 0.86*** 0.79*** -0.03*** -0.05***

(0.01) (0.02) (0.00) (0.01)Market fe Yes Yes Yes YesQuarter fe Yes Yes Yes YesObservations 1350 1350 1350 1350Adjusted R2 0.86 0.73 0.34 0.34Standard errors in parentheses* p < 0.10, ** p < 0.05, *** p < 0.01

Notes: This table illustrates the reduced-form evidence for line length adjustment in response to time-evolving pref-

erence heterogeneity. Preference heterogeneity is inversely correlated with concentration for in-line market shares,

i.e., concentrated in-line market-share means homogenous preference. All columns are panel data regressions with

market fixed effects. The dependent variables are next-quarter line-length in columns (1) and (2) and next-quarter

dummy for line length expansion in columns (3) and (4). Line length is the count of features (flavor-cut-fat) within

each market-quarter after dropping transient ones with less than 12 weeks of positive sales. All data come from IRI

Academic Dataset.


Table 3 Demand Estimation

Dependent Variable is Ln(Share1) - Ln(Share0)

OLS 2SLS

(1) (2) (3) (4)Preference Heterogeneity 0.02*** 0.23*** 0.41*** 0.49***

(0.00) (0.01) (0.02) (0.02)Price -0.13** -2.53*** -2.38*** -49.50***

(0.05) (0.19) (0.22) (2.92)⇥ Ln(Median Income) 3.08***

(0.30)⇥ Ln(Median Age) 3.33***

(0.60)Ruffle cut -0.01*** -0.11***

(0.00) (0.01)Wavy cut 0.01 0.03***

(0.01) (0.01)Fat free 0.01 -0.06***

(0.01) (0.01)Reduced fat -0.01 -0.15***

(0.01) (0.01)Flavor fe Yes Yes No NoProduct fe No No Yes YesQuarter fe Yes Yes Yes YesObservations 30930 30930 30930 30930Adjusted R2 0.90 0.86 0.81 0.76Standard errors in parentheses* p < 0.10, ** p < 0.05, *** p < 0.01

Notes: This table shows the demand estimation induced by nested logit model. The dependent variable for all columns

are the difference between logarithm of total Frito Lay shares and total shares from outside goods. Column (1) uses

OLS, column (2) - (4) uses 2SLS, with three sets of instrumental variables including BLP instruments (summation

of flavor, cut and fat dummies for other features in the same serving-city-quarter), Hausman instruments (average

price sold for the same feature in other city within serving-quarter, price of materials including potatoes, sugar, soy

bean oil, edible butter and edible tallow) and competition environment (number of competitor firms and number of

competitor UPCs other than Company-A chips within serving-city-quarter).


Table 4 Demand Estimation with Varieties of Preference Heterogeneity

2SLS, Dependent Variable is Ln(Share1) - Ln(Share0)

Baseline SmallPackage

IqrInc(1M)

IqrAge(10 yrs) Asian Hispan Hispan

>p50

(1) (2) (3) (4) (5) (6) (7)Preference Heterogeneity 0.41*** 0.67*** 0.36*** 0.30*** 0.41*** 0.42*** 0.36***

(0.02) (0.04) (0.02) (0.04) (0.02) (0.02) (0.02)⇥ Diversity Measure 1.48*** 0.04*** 0.47*** -0.09 0.12***

(0.19) (0.01) (0.15) (0.06) (0.01)Price -2.38*** -2.74*** -2.99*** -2.62*** -2.57*** -2.30*** -2.87***

(0.22) (0.29) (0.24) (0.24) (0.23) (0.22) (0.23)Product fe Yes No Yes Yes Yes Yes YesQuarter fe Yes Yes Yes Yes Yes Yes YesObservations 30930 9155 30930 30930 30930 30930 30930Adjusted R2 0.81 0.41 0.79 0.80 0.80 0.81 0.78Summary statistics of population diversity measure

Mean 0.06 3.40 0.04 0.10 0.49Min 0.04 2.75 0.00 0.00 0.00Max 0.10 4.47 0.31 0.53 1.00

Standard errors in parentheses* p < 0.10, ** p < 0.05, *** p < 0.01

Notes: This table shows the demand estimation of nested logit model allowing preference heterogeneity to vary by

observables. The dependent variable for all columns are the difference between logarithm of total Frito Lay shares

and total shares from outside goods. All columns are estimated using 2SLS with three sets of instruments: BLP

instruments, Hausman instruments, and competition environments. Column (1) is the baseline estimates for large

package size potato chips, which is identical to Column (3) in Table 3. Column (2) reports the estimates with identical

specification but in small-sized package chips (1-4 serving sizes). Column (3)-(7) allow preference heterogeneity to

vary by different measures of population diversity, where Column(3) uses interquartile of income, Column (4) uses

interquartile of age, Column(5) uses Asian population ratio, Column (6) uses Hispanic population ratio, and Column

(7) uses discretized Hispanic population ratio, which is the dummy for above-median Hispanic population ratio.

Table 5 Supply Estimation

Linear Cost Nonlinear Cost

b se b seCost c1 (1K$ / 1M HH) 3.56 (0.86) 2.08 (1.38)Cost c2 (1K$ / 1M HH) 6.04 (0.65)Precision Ratio �1/h 2.55 (0.02) 2.42 (0.02)Scale of fixed cost �" 0.14 (0.00) 0.02 (0.00)Prior mean µ Integrated IntegratedLog Likelihood -83.37 -63.61

Notes: The cost function for linear specification is c (n) = c1 ·n„ while the cost function for nonlinear specification is

c (nt, nt�1) = (c1 + c2 (nt >nt�1)) ·nt.


Figure 1 Distribution of line length and line length changes

0.0

5.1

.15

10 15 20 25 30Line Length

Line Length

0.1

.2.3

-10 -5 0 5 10Change in Line Length

Change in Line Length

Note: Left figure plots the distribution of line length among 50 markets over 28 quarters, and right

figure plots the distribution of change in line length, which is first difference for line length over

two consecutive quarters within one market. Line length is defined as the count of products (unique

combination of flavor-fat-cut) within the city-quarters. Products with positive sales for less than 12

weeks within city-quarters are not counted.


Figure 2 Identification line length maintenance cost

.3.4

.5.6

.7.8

Tota

l Sha

re

5 10 15 20Line length (n)

Low c

.3.4

.5.6

.7.8

Tota

l Sha

re5 10 15 20

Line length (n)

High c

µH µM µL

Note: This figure shows the identification of line length maintenance cost. In each plot, the thick

curves are the total market share as a function of line length. I plot three curves with identical

variance but different mean value of preference heterogeneity ✓. We can see that the total market

share is increasing in line length, preference heterogeneity and super-modular in the two parameters.

Straight lines are cost function, and the slope represents the marginal cost of expanding the line

length. The tangent point of cost line and market share curve represents the optimal line length

decisions. We can see that the implied optimal line length is higher when preference heterogeneity

is higher. The two plots differ in marginal cost, and we can see that when cost is lower, line length

decisions are more responsive to change in mean for heterogeneity, which completes the identification

for cost.


Figure 3 Model fit - two markets

1520

2530

2001q3 2003q1 2004q3 2006q1 2007q3

BOSTON

1520

2530

2001q3 2003q1 2004q3 2006q1 2007q3

DETROIT

Actual Signal from simulation Signal from data

Note: This figure shows how the model fits the data in two cities: Boston and Detroit. Solid lines are

actual line length decisions, and two dashed lines are line length decisions from simulation. In the

first simulation, “signal from data”, market signals are taken from the data; in the second simulation,

“signal from simulation”, market signals are also simulated from the model. Prior mean in the first

periods are drawn from the known distribution, prior precision in the first periods are estimated.


Figure 4 Model fit - distribution

0.0

5.1

.15


Line Length

0.1

.2.3



Actual Signal from simulation Signal from data

Note: This figure shows how the model fits the distribution of line length and line length changes.

Solid bars are distribution of actual line length decisions, two lines are kernel density of simulated

line length. In the first simulation, “signal from data”, market signals are taken from the data; in

the second simulation, “signal from simulation”, market signals are also simulated from the model.

Prior mean in the first periods are drawn from the known distribution, prior precision in the first

periods are estimated.


Figure 5 Counterfactual - smooth cost

0.0

5.1

.15


Line Length

0.1

.2.3



Actual Simulated, step cost Simulated, smooth cost

Note: This figure shows evaluates optimal line length decisions under a smooth cost structure. Solid

bars are distribution of actual line length decisions, and two lines are kernel density of simulated

line length: dashed line represents simulated line length in original model with nonlinear cost,

whereas solid line represents simulated line length under linear cost structure. In both simulations,

market signals are taken from simulation; prior mean in the first periods are drawn from the known

distribution, prior precision in the first periods are estimated.


Figure 6 Counterfactual - known heterogeneity ✓

0.0

5.1

.15


Line Length

0.1

.2.3



Actual Simulated, learning θ Simulated, knowing θ

Note: This figure shows evaluates optimal line length decisions when firms know the precise value of

time-varying preference heterogeneity. Solid bars are distribution of actual line length decisions, and

two lines are kernel density of simulated line length: dashed line represents simulated line length

in original model with learning heterogeneity, whereas solid line represents simulated line length

assuming known heterogeneity. In both simulations, market signals are taken from simulation; prior

mean in the first periods are drawn from the known distribution, prior precision in the first periods

are estimated.


Figure 7 Testing learning assumption based on gross margin

05

1015

2025

2.1 2.15 2.2 2.25 2.3Gross margin in median market (1M $)

Learning θ Knowing θ

Note: This figure plots the distribution of simulated median level gross margin in two simulations:

learning preference heterogeneity and knowing heterogeneity. In both simulations, market signals

are taken from simulation; prior mean in the first periods are drawn from the known distribution,

prior precision in the first periods are estimated. Vertical line is the observed median gross margin

from the data.

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Product Offerings and Product Line Length Dynamics

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