Product Awareness and the Industry Life Cycle∗
Jesse Perla
University of British Columbia
March 12, 2013
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Does expanding consumer awareness of new products explain firm growth and
the industry life cycle? Output in many industries grows slowly and eventually stabilizes, while the market valuation peaks and then drops – all without
significant changes in productivity. To explain this pattern, I introduce a mechanism where consumers with heterogeneous preferences slowly become aware of
differentiated products in the industry. As the industry evolves and consumers
meet more firms, growing awareness generates two countervailing forces on firm
profits: 1) market power decreases due to more intense price competition, and
2) demand increases due to customers sorting into their preferred firms. Consequently, if increased awareness results in a strong enough reduction in market
power, profits and valuations peak and then decrease, even as output grows monotonically. When aggregated across firms in an industry, the slow expansion of
product awareness looks like a capacity constraint or an adjustment cost, while
the sorting of customers into preferred products gives the illusion of productivity
or quality growth.
Keywords: Macroeconomics, Industrial Organization, Firm Growth, Firm Dynamics, Firm Heterogeneity, Industry Equilibrium, Industry Life Cycle, Product
Differentiation
I would like to thank my committee Boyan Jovanovic, Tom Sargent, and Jess Benhabib. I would also
like to thank Heski Bar-Isaac, Randy Becker, Xavier Gabaix, Wayne Gray, Bart Hobijn, Ricardo
Lagos, Bob Lucas, Virgiliu Midrigan, Ben Moll, Edouard Schaal, and Daniel Xu.. Correspondence:
jesse.perla@ubc.ca
∗
1
1
Introduction
1
Empirics of the industry life cycle show that output in new industries tends to grow, peak,
and then remain fairly stable. Simultaneously, productivity usually starts high, grows modestly, and then flattens. In contrast, an industry’s stock market valuation often peaks and
then decreases significantly and slowly. These patterns lead to the following questions: If
productivity starts and remains high, then why would output take so long to grow? And, as
market capitalization is an expectation of future profits, why would valuations drop while
productivity and output remain high?1
Relative to Maximum Value
TFP
Output
0
Market Valuation
.1
1
Age Relative to Peak Real Market Cap
2
Figure 1: Average Life Cycle of Valuation, Output, and TFP for 439 Industries
Simple supply-side explanations in models with heterogeneous firm productivity have
difficulty answering these questions. In particular, slow output growth at the beginning of the
life cycle is difficult to reconcile with high initial levels of productivity. Similarly, in stylized
models where firm size and profitability are directly tied to productivity, profit margins and
industry market capitalization should not drop while productivity remains relatively high
and stable.
As an alternative, this paper asks whether a particular demand-side explanation can
better explain these facts: Does expanding consumer awareness of new products drive firm
growth and the industry life cycle? This paper models product awareness as each consumer
having incomplete information about firms. Over time, consumers are likely to expand their
1
See Figure 1 for empirical evidence of these puzzles.
2
awareness of firm existence and characteristics, but the process could be slow. In addition
to incomplete information, consumers have heterogeneous preferences for each good and
consider the quality of each match when making their consumption decisions.
With these model features, the slow expansion of product awareness explains the industry
life cycle through changes in both the number of customers and the demand per customer.
As new firms enter, information takes time to propagate to consumers, giving a simple
explanation for the slow growth of firms and industries as a time varying limitation on the
number of potential customers. Additionally, as consumers become aware of these new firms,
they increase their demand when they find a new good that they prefer.
For a firm, the expansion of product awareness causes two countervailing effects on profits. First, an increase in the degree of awareness among consumers increases the level of
competition, and hence decreases market power and prices. Intuitively, if all of a firm’s
customers only know of that firm, it would have monopoly pricing power over them. But
if some of those customers are aware of multiple firms, then a firm without the ability to
price discriminate needs to lower its prices to compete. Second, an increase in the degree of
awareness among consumers gives them more choice, and better matches on average between
firm and consumer. Over time, consumers sort into their preferred products, which increases
the demand for all firms. Whether profits and valuations are monotone over time depends
on whether changes in firm profits are dominated by 1) the decrease in market power due to
intensifying competition, or 2) the increase in demand due to customer sorting. If products
within an industry are not very differentiated in the consumer preferences, then consumers
do not make significant changes to their demand as they find their preferred products. Consequently, in industries with low differentiation, the increase in intensity of competition over
time explains why valuations in an industry can peak and then drop.
This paper is organized as follows: Section 2 summarizes new evidence on the industry life
cycle and interprets existing evidence from micro-studies of firm dynamics and growth rates.
A stylized model to explain this evidence and implement the product awareness mechanism
is discussed in Section 3, with aggregation to a macro-economy in Section 4. Simple examples
to highlight the mechanism are discussed in Section 5. Finally, the literature and summary
of contributions is detailed in Sections 6 and 7. Proofs and empirical results are expanded
further in the Appendix.
2
2.1
Evidence from the Industry Life Cycle
Aggregated Industry Data
The rise and fall of industries has been investigated in the context of product innovation
and technological progress. Generally, output monotonically increases, prices monotonically
decrease, and the number of producers peaks and then drops. In order to see if these results
are typical of most industries, data was collected across 439 industries and averaged over the
industry life cycle.2 To average across industries, it is necessary to “line up” the life cycle of
industries which peak at different years and evolve at different frequencies. In contrast to
2
The data comes from the NBER-CES Manufacturing Industry Database, Census Concentration Indices,
CRSP/Compustat, and BEA. Most of the data is sampled yearly, but the concentration index and number
of firms are sampled every 5 years, which accounts for the jagged data series when averaged across industries.
See Appendix F.1 for more details.
3
0
Relative to Maximum Value
1
Gort and Klepper (1982), which uses the highest number of firms as the peak of an industry,
I choose the maximum real market capitalization as the peak of the industry. The age of
each industry is rescaled such that at birth its relative age is 0 and at peak its relative age
is 1. With this normalization, and by rescaling variables relative to their maximum value, I
average the life cycle of a variety of industries.3 In addition to the number of firms, I show
the market share proportion of the top 20 of firms, which is a more direct measure of market
power.
.1
1
Age Relative to Peak Real Market Cap
Real Market Cap
Real Output
Concentration
2
TFP
Real Price
Num Firms
Figure 2: Average Industry Life Cycle with Market Concentration
The results in Figure 2 are stark: output grows slowly, and aggregate market capitalization often drops significantly and persistently after a peak. Moreover, Total Factor Productivity (TFP) starts high and levels off quickly, and prices drop but eventually stabilize.
While the exaggerated height of the peak market capitalization is partially driven by pricing
uncertainty, which this model does not attempt to address, the patterns on both sides of
the peak of market capitalization are fairly robust.4 The number of firms in the industry
3
This method has a direct analogy to comparing business cycles by measuring peak to trough and then
rescaling time to line up and compare each cycle. I also conduct robustness checks for other measures of
industry peak such as the market capitalization relative to aggregate stock market value. As an additional
robustness check on the age of the industry, I test a proportion of the maximum output as a threshold for
the normalized birth year.
4
A word of caution for interpreting the crude results here: the shape of the valuation data is complicated
by the underlying process for uncertainty. In particular, as the valuation data is normalizing the age to the
maximum, the average normalized valuation decrease on both sides of age = 1 with certainty if valuation data
4
does rise and then fall, though in general not as dramatically as the set of industries in other
studies, such as Gort and Klepper (1982).5
Since firm valuations are a measure of expectations of future profits, the drop in valuation
suggests a drop in future average prices or productivity. As productivity tends to be stable,
another explanation is a decrease in profit margins due to less market power. However, the
concentration index is weakly increasing over the life cycle, and the number of firms drops
after this peak. Thus, if changes in profit margins are driven entirely by changes in the
degree of industry concentration, markups should rise. This paper, however, will show that
markups can decrease entirely due to information frictions as customers become aware of
more products and competition becomes more intense.
2.2
Firm Micro-data
Studies where physical productivity can be measured directly, such as in Foster, Haltiwanger,
and Syverson (2008, 2012) and Hsieh and Klenow (2009), find a number of empirical regularities.
• Firms grow slowly: It often takes over 15 years for a new entrant to reach 73% of
an incumbent’s size
• Physical productivity is persistent: Average auto-correlation of idiosyncratic TFP
is approximately 0.80
• Small TFP advantage of entrants: Entrants start with higher TFP, but have
similar TFP to incumbents after 5 years. Entrant’s prices are lower, so entrants are
small “in spite of” having low prices and high productivity
• Large, sustained differences in productivity: For example, Hsieh and Klenow
(2009) finds that the ratio of the 75th and 25th percentile physical TFP is 5.0 in India
and 3.2 in the US
The striking result of these studies is that productivity is extremely important for determining profitability and selection, but does not explain firm size or growth rates especially well.
Small firms are frequently productive, have low prices, and yet grow slowly, while large firms
are often unproductive and only slowly removed through selection. In summarizing these
studies, Syverson (2011) cites understanding demand as an important approach for future
research in order to explain how these productivity disparities can be sustained over time.6
This paper contributes a model of demand to explain the industry dynamics of Section
2.1 that is consistent with the micro-evidence on firm dynamics.
is stochastic. This rough analysis also ignores the important dimension of firm debt and an age dependent
equity-debt ratio.
5
The number of firms and output start fairly high near the birth of the industry. This is partially due to
the left censoring of data from the creation of new SIC codes, but also selection since market capitalization
data is only available from public firms. If firms only slowly became publicly listed, it would help explain
why aggregate market capitalization stays fairly flat early in the life cycle. However, these issues have no
effect on the mechanism, as this paper adds no analysis of the low and flat pattern of early life cycle industry
valuation.
6
Demand factors in this model can be partially distinguishable from physical productivity without quality
unit pricing or physical production data. Section 5.2 discusses how this model breaks the typical “isomorphism” between representative consumer models with productivity versus demand shocks, as described in
Melitz (2003) and others.
5
3
The Model Economy
3.1
Summary
Time is discrete and infinite. There are two types of agents in the economy: consumers
and firms. Each firm produces a single good which is associated with a particular industry.
There are a continuum of industries, and a finite number of firms within each industry.
Consumers have permanent heterogeneous preferences for each product, and are only aware of
an evolving subset of the firms in the economy. Firms are identical except for the permanent
quality of their product, common to all consumers, and the set of consumers who are aware
of them.
Consumers have constant elasticity of substitution preferences across industries, but
quality-adjusted goods are perfectly substitutable within an industry. Firms compete by
simultaneously choosing a price each period (i.e., repeated Bertrand pricing). Given prices,
consumers choose the quantity demanded for each product that they are aware of. At the
end of the period, consumers become aware of new firms through an exogenous stochastic
process. The distribution of product awareness among consumers is the only time-varying
state in the economy.
Industries are indexed by m ∈ [0, 1]. Each m can be interpreted as a product category
that a consumer derives utility from. For simplicity, assume that there are a constant number
of firms, Nm , in each industry from birth.7 Within each industry, the set of firms is arbitrarily
indexed by i = 1, . . . Nm . Therefore, (i, m) uniquely denotes a firm in the economy. The set
of firms in industry m is defined as Im ≡ {1, . . . Nm }. A mass of measure 1 of infinitely lived
consumers are labeled by j ∈ [0, 1].
Assume that at time t = 0, a new industry is born with N > 0 firms and no future entry
or exit.8 The model will focus on the lifecycle of this new, infinitesimal industry.
Notation Simplification
For clarity, since the goal is to isolate a particular industry, drop the m subscript where
possible so that an i uniquely identifies a firm within the new industry. This abuse of
notation will simplify the exposition, as the continuum of industries is only necessary to
solve for the consumer preferences of Section 3.3.
Heterogeneity
Each firm has a quality for their good, qi > 0, common to all consumers. Additionally, there
is a quality of the idiosyncratic match between each product and each consumer, ξij > 0.
Both of these preference parameters stay constant after firm entry.9
Let ξj ∈ RN be the vector of preferences for consumer j. This time-invariant state is
distributed in the economy as ξj ∼ G with pdf g.
7
The full specification of the model with entry and exit is specified in the Technical Appendix D.
In equilibrium, with the baseline setup and deterministic firm quality, all firms would choose to enter at
t = 0.
9
As previously discussed, firm level quality and physical productivity parameters are often not identified.
In this model as well, qi would be indistinguishable from an idiosyncratic productivity per firm unless
quality unit pricing or physical output data was available. A more general equivalence between productivity
and demand parameters does not occur in this model due to the other sources of heterogeneity. These
considerations are discussed in Section 5.2.
8
6
Assumption 1 (Properties of the Idiosyncratic Preferences). Assume the following properties for ξj ∼ G
1. G is continuous and has no atoms10
2. The distribution of preferences is independent across industries: ξmj ⊥ ξm′ j for all
m 6= m′
Consumer j at time t is aware of a subset of the operating firms in each industry, Ajt ⊆ I.
Consumers start at time 0 with no awareness of the industry of interest, Aj0 = ∅ for all j.
The intuition for awareness is given in the Venn Diagram in Figure 3. In this duopoly
example, the intersection of the two sets is the mass of customers who are aware of both
firms.
Firm 1 and 2 Awareness
Firm 1 Awareness
Firm 2 Awareness
Consumers with No Awareness
Figure 3: Example Awareness in a Duopoly
Consumer Distribution
The joint distribution of consumers for a particular industry is Ψt (Ajt , ξj ) with pdf ψt (·).
The canonical case of the idiosyncratic distribution of preferences is the independent
product: ξ ∼ G(ξ1 ) × . . . G(ξN ) where G is the univariate Gumbel distribution.11 In that
case, the idiosyncratic preferences are independent across industries, firms, and consumers.
The marginal distribution of ξj for all products other than product i is G−ij (ξ−ij ).
The marginal pdf of A across consumers in the population is denoted ft (A) : 2I →
R, which is a discrete probability distribution due to the finite cardinality of the possible
10
This technical assumption is for convenience to ensure there is only a measure 0 of consumers with a
particular value for their idiosyncratic preferences.
11
This version of the Gumbel distribution is also known as the Extreme Value Type 1 Distribution. The
−ξ
−ξ
pdf is g(ξ) = e−e e−ξ and cdf is G(ξ) = e−e . A key property which enables analytic solutions for the
demand system is that the difference between two Gumbel random variables is distributed as a Logistic.
7
awareness states. As will be shown in Section 3.2, the distribution of A evolves independently
from ξ. Because of this independence, the joint pdf of these states in the economy is
ψt (Ajt , ξj ) = ft (Ajt )g(ξj )
(1)
Consumers have incomplete information of firms, as captured in Ajt . Firms have complete information of the consumer distribution Ψt (·), but cannot price discriminate based on
awareness or preferences. Firms also have complete information of the pricing and production
decisions of other firms in the industry.
Summary of Timing
The timing within a period is broken into the following steps:
1. Firms simultaneously post prices, pit
2. Consumers choose quantity demanded, yijt
3. Firms produce and markets clear
4. Consumers stochastically become aware, Ajt+1 evolves
5. The next period begins
3.2
Awareness Evolution
In order to isolate the effects of expanding awareness from the properties of a particular
awareness process, this model uses a simple, extensible law of motion for numerical examples.12 An alternative specification, which includes word-of-mouth contagion, is discussed in
Appendix E.1.
Baseline Awareness Process
Assume a probability 0 < θ < 1 that a consumer will meet a firm per industry per period.
All meeting arrivals are independent across j and m. If there are N firms in the industry,
the consumer has the same chance, N1 , to meet any of those firms.
Since agents do not have independent chances to meet every firm, this process captures
the limited capacity of consumers to evaluate products in any given period. If there are a
large number of firms, θ determines the capacity for evaluating the quality of the product
and the quality of the match value.13 This process also captures an increasing difficulty in
reaching new customers for maturing firms. For example, a large firm’s advertisements are
increasingly likely to be seen by customers already aware of the firm.14
12
The model’s novel contribution are the implications from evolving awareness, rather than the specification
of a particular process. Advertising, consumer search, and other motivations for the awareness process are
discussed in Section 6.1 with extensions in Technical Appendix G.
13
If search efforts were costly and endogenous to the consumer, then θ would decline over time, further
strengthening the core results of this paper. Since Stigler (1961), it has been known that returns to sampling
diminish with the size of the chosen sample, and therefore, when sampling is costly, the optimal sample size
will be finite.
14
This process also has the property that limits to large numbers of firms or to continuous time give a
non-degenerate version of the model. If, instead, each agent has independent draws from each firm each
8
Markov Process
This law can be formalized as a Markov process for the state Ajt for each consumer.
If consumer j is unaware of a particular firm i, then the probability to add i to her
information set next period is
Pr {Ajt+1 = Ajt ∪ {i} |i ∈
/ Ajt } = |{z}
θ
Meeting
1
N
(2)
|{z}
Meet firm i
The probability that a customer does not become aware of any new firms is the probability
that she receives no meeting or that she meets a firm already in her awareness set
Pr {Ajt+1 = Ajt } = 1 − θ +
| {z }
No Meeting
|A |
jt
θ
| N{z }
(3)
Meet Known Firm
where |Ajt | is the size of the discrete set Ajt .
The number of possible states for Ajt in the economy is finite and of cardinality N ≡ 2I .
If these states are ordered according to each product’s index i in the set, then the Markov
process can be written as a standard Markov chain.15 For example, with the previous duopoly
example, order the probability of being in each state as
fjt = Pr {Ajt = ∅} Pr {Ajt = {1}} Pr {Ajt = {2}} Pr {Ajt = {1, 2}}
With this ordering, the Markov transition matrix for the duopoly example is


θ
θ
0
1−θ
2
2
θ
 0
0
1 − 2θ
2
fjt+1 = fjt 
θ
 0
0
1 − 2 2θ 
0
0
0
1
(4)
Unconditional Distribution of Awareness
Given an initial distribution of awareness, derive the unconditional distribution in the economy for the continuum of mass 1 of consumers.
Proposition 1. If there is no initial awareness, i.e., Aj0 = ∅, ∀j then
I
1. The unconditional distribution of A, ft : R|2 | → R, is
ft = 1 0 . . . 0 · C t
(5)
2. In the limit there is total awareness:
lim ft = 0 . . . 1
t→∞
That is,
lim ft (I) = 1
t→∞
period, then in the limit as N → ∞, each consumer would be aware of an arbitrarily large number of firms
after only a single period. Independent draws for each firm would also contradict the results of consumer and
firm search models such as Crawford and Shum (2005), Dinlersoz and Yorukoglu (2008), Arkolakis (2010),
Eaton, Eslava, Krizan, Kugler, and Tybout (2011), and Gourio and Rudanko (2011).
15
See Appendix A for a more formal description of the setup for an arbitrary number of firms.
9
Proof. See Appendix A.2.
With the transition matrix in the duopoly example from equations 4 and 5, Figure 4
gives an example of the pdf for θ = 0.2. Note that the pdf of agents only aware of a single
firm peaks and then drops as awareness spreads. This behavior is crucial to the changes
in market power over time in the model. With different laws of motion than the Markov
process specified above, the evolution of this process could be different, but under fairly
general conditions, the basic property of a hump-shaped awareness of a single product over
time would be retained.
1
0.9
ft (∅)
0.8
ft ({1}) = ft ({2})
ft ({1, 2})
0.7
ft
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
t
20
25
30
Figure 4: Example Awareness Evolution
3.3
Consumers
Preferences
Consumers have constant elasticity of substitution (CES) preferences between industries
1
with an elasticity of κ ≡ 1−ρ
> 1. Define ȳmjt as the quality-adjusted quantity of products
consumed in industry m. The consumer’s period utility at time t is the standard Dixit-Stiglitz
aggregation.16
Z 1
1/ρ
ρ
ȳmjt dm
(6)
0
Goods within an industry are perfectly substitutable when adjusted for quality and quantity. Given a set of intensive demands yimjt > 0 for each consumer and product, the quality16
For this section, briefly reintroduce the m index when talking about multiple industries.
10
adjusted quantity consumed in industry m is
X
ȳmjt =
qim eσξimj yimjt
(7)
i∈Amjt
Differences from Standard Preferences
This utility specification has several differences from nested CES and discrete choice preferences.17
The preferences have an infinite elasticity of substitution between products at the industry level, which is similar to specifications of additive utility in discrete choice theory.18 In
equilibrium, this specification ensures that consumers have idiosyncratic bundles of products they consume depending on the equilibrium price vector pt , common quality qi , and
idiosyncratic preference ξij .19
Unlike standard discrete choice preferences, the idiosyncratic ξij term is multiplicative
rather than additive. Therefore, the optimal choice of quantity demanded for a consumer
conditional on purchasing the product is a function of both price and the idiosyncratic preference, yijt (ξij , pt ). Because of this state dependence, the preferences do not easily aggregate
to a representative consumer as is sometimes the case in discrete choice theory. Economically, this represents a consumer’s demand increasing as she meets firms that better match
her needs (i.e., an implication of customer sorting). The parameter σ > 0 determines the
degree of differentiation within an industry and has a direct analogy to the variance of the
random utility shocks in discrete choice theory.
Finally, the largest difference from standard preferences is the information friction, where
consumer j can only purchase from firms in the set Ajt . This constraint breaks aggregation to
a representative consumer with nested CES preferences, since the set of goods each consumer
can substitute between is idiosyncratic and evolving.
Consumer’s Problem
Assume that all consumers have a constant endowment Ω. As consumers’ choices do not
affect their future awareness (i.e., they have no built in habits), utility can be written as a
static optimization problem.
17
Classic references for discrete choice and location models are Salop (1979), Perloff and Salop (1985), and
McFadden (2001). In the context of this model, if ξ is a location specific preference, then σ is a measure of the
“distance” between firms. In a more elaborate setup of this model, σ could be a function of N , representing
less differentiation with a larger number of firms since they are spaced closer together. The multiplicative
form of these preferences is similar to Sattinger (1984), but with a CES implementation of the composite
good.
18
Perfect substitutability between products within an industry has a strong empirical connection to the discrete choice literature when scanner-level data is available. Discrete choice preferences with additive quality
and idiosyncratic shocks often aggregate to a representative consumer with CES preferences. See Anderson,
De Palma, and Thisse (1992) for theory describing aggregation from discrete choice to a representative agent
with CES preferences.
19
The preference specification here could also be similar to a disaggregated version of the nested CES
preferences in Atkeson and Burstein (2007). The intuition in this model remains the same, as is discussed in
Technical Appendix F.1. More directly, preferences in this model have similarities to Fajgelbaum, Grossman,
and Helpman (2011). Key differences in this paper include: 1) an assumption of multiplicative rather than
additive idiosyncratic preferences, which leads to homotheticity and dynamic effects on demand from customer sorting, and 2) incomplete information about product awareness introduces substitutability dynamics
and does not allow aggregation to market shares as shown in Proposition 4.
11
Given an awareness set Ajt , a set of idiosyncratic preferences ξj , and a set of prices pt
for all firms in the economy, the consumer chooses demand per good yimjt to maximize the
utility specified in equations 6 and 7

Z

max
yjt ≥0
Z
s.t.
1
0
1
0


X
i∈Amjt


X
i∈Amjt
ρ
1/ρ
qim eσξimj yimjt  dm

pimt yimjt  dm ≤ Ω
(8)
Proposition 2 (Intensive Demand). Given prices pt , for each industry m
1. Almost every consumer purchases from 0 or 1 firm per industry
2. A consumer purchases product i and no others if and only if
p
log qi′′t − log pqiti > σ (ξi′ j − ξij ) , ∀ i′ ∈ Ajt \ {i}
(9)
i
3. The demand for a consumer with state (ξj , Aj ) is
κ−1 σ(κ−1)ξij κ
P Ω̄,
yijt (ξij ) = p−κ
e
it qi
yi′ jt = 0, ∀ i′ ∈ Ajt \ {i}
for some aggregate price index common to all consumers, P and real income Ω̄ ≡
(10)
Ω
P
Proof. See Appendix B.1.
As is standard with CES utility functions, the primary goal of assuming a continuum
of infinitesimal industries is to ensure that this price index for a particular consumer does
not change based on consumption in a single industry. The exact formula for P depends
on the qim′ and Ajm′ t distribution in all industries m′ 6= m. For tractability, it is necessary
to assume conditions to ensure that the aggregate price index P is equal for all consumers.
The easiest additional assumption is that in all m′ not equal to the industry of interest m,
there is only a single product with price pm and normalized quality 1. In that case, since the
industry of interest is measure 0, integrate all of the other industries to form a price index.20
P ≡
Z
1
0
1−κ
′
pm
′ dm
20
1
1−κ
(11)
Alternatively, since consumers are infinitely lived and Amjt evolves independently for all consumers, the
prices faced by a particular consumer in one industry are orthogonal to those in any other industry. From
this assumption, laws of large numbers can be applied given a distribution of qim′ and Amjt to show that
Pj is identical for all consumers. While these considerations are necessary to solve the model in general
equilibrium with industry entry and exit, they provide no additional intuition for this mechanism.
12
Customers
Define the set of possible awareness states that contain product i as
A | i ≡ A |A ∈ 2I s.t. i ∈ A
Definition 1 (Number of Customers). Given the price vector p, the number of customers
for product i is defined as
Z
xit (p) ≡ 1{Choose i from Ajt given p and ξj }dΨt (Aj , ξj )
Using the choice inequalities in equation 9
Z
X
p
=
ft (A) 1{log qi′′ − log( pqii ) > σ(ξi′ j − ξij )|∀ i′ ∈ A \ i}dG(ξj )
i
A|i
Proposition 3 (Number of Customers for Gumbel Preferences). With ξj ∼ Gumbel:


p −1/σ
i
X
qi
ft (A)

xit (p) =
p ′ −1/σ
P
i′ ∈A
A|i
(12)
i
qi ′
Proof. The proof follows standard derivation of market shares in discrete choice theory. See
Appendix C.1.
To compare to standard discrete choice results, this is a weighted sum of the market shares
for every awareness set. Note that σ has no effect on market share if pi and qi are identical
for all firms. As the choices are deterministic and conditional on a particular information
set, the aggregated market shares cannot be easily interpreted as choice probabilities, as is
typical in discrete choice.
Total Demand
The quantity demanded conditional on purchasing product i is a function of ξi in equation
10. Figure 5 overlays the intensity of preferences on the duopoly example.
Prefer Firm 2
Prefer Firm 1
Figure 5: Intensity of Preferences
13
Firm 2 Customers
Average Intensity for Firm 1
Average Intensity for Firm 2
Firm 1 Customers
Figure 6: Intensity Conditional on Purchasing
The choice of a particular product depends on both the awareness set and the consumer’s
ξ vector in equation 9. Accordingly, the distribution of ξi conditional on purchasing product
i will not be the unconditional distribution of ξi in the economy, except for a subset of agents
only aware of firm i. This effect is shown in Figure 6, which overlays the optimal choice of
the product with the average intensity of ξ conditional on choosing product i.
To solve for the demand curve faced by firm i, integrate equation 10 to find the total
demand.
Definition 2 (Total Demand).
Z
yit (p) ≡ yij (p, ξij )1{Choose i from Aj given p and ξj }dΨt (Aj , ξj )
(13)
Using the choice inequalities in equation 9
Z
X
p
=
ft (A) yi (p, ξij )1{log qi′′ − log( pqii ) > σ(ξi′ j − ξij )|∀ i′ ∈ A \ i}dG(ξj ) (14)
i
A|i
Since demand is a function of the ξij conditional on purchasing product i, it is necessary
1
to ensure that the degree of differentiation is not too high, σ < κ−1
, or this expression will
21
not be defined.
Assumption 2 (Limits on the Degree of Differentiation). Assume that 0 < σ <
1
κ
The stronger requirement that σ < κ1 is to ensure that q̄it is defined for Proposition 5, but is not required
for Proposition 4. As should be clear with the calibration discussed in Appendix F.2, this property is similar
to the comparison of within and between industry substitutability in Atkeson and Burstein (2008).
21
14
Proposition 4 (Total Demand for Gumbel Preferences). With ξj ∼ Gumbel and Assumption
2,


−1/σ
pi

qi
π̄ X 


yit (p) =
(15)
ft (A) 1−σ(κ−1) 
pi
−1/σ


P
p′
A|i
i∈A
i
qi ′
Where π̄ ≡ Γ(1 − σ(κ − 1))P κ Ω̄, with Γ(·) the Gamma function.
If the firm is monopolist then
yi (pi ) = (1 − ft (∅))π̄qiκ−1 p−κ
i
(16)
Proof. See Appendix C.2.
Even with symmetric firms and prices, σ changes the total demand through consumer
sorting. The intuition is that the more intense the degree of product differentiation, the larger
the mean idiosyncratic match quality conditional on purchasing from a firm. Alternatively,
equation 16 can be compared to a typical model of monopolistic competition with an evolving
number of consumers, Popt ,
yi (pi ) = Popt P κ Ω̄qiκ−1 p−κ
i
Comparing the two, 1 − ft (∅) serves the role of an exogenous constraint
σξ on the number of
consumers. The distortion of Γ(1 − σ(κ − 1)) only occurs since E e
6= 1 for the Gumbel
distribution, and can be used to normalize qi without loss of generality. Also, the dispersion
of the preferences σ only appears through the Γ(·) distortion, which confirms the intuition
that with monopolists, no sorting effects occur and the quality is simply the unconditional
average.22
From the consumer’s preferences, conditional on consumer j choosing product i, the
quality per unit consumed is qi eσξij .
Definition 3 (Demand-Weighted Average Quality).
Z
1
q̄it ≡
q eσξij yijt (ξij )dΨt (Aj , ξj )
yit i
(17)
Proposition 5 (Average Quality for Gumbel Preferences). With ξj ∼ Gumbel and Assumption 2,
#
"
P
P pi′ −1/σ σκ−1
A | i ft (A)
i∈A qi′
pi Γ(1 − σκ)
#
"
q̄it =
(18)
Γ(1 − σ(κ − 1) P
P pi′ −1/σ σ(κ−1)−1
A | i ft (A)
i∈A q ′
i
Proof. See Appendix C.3.
22
When σ = 0, then Γ(1) = 1, perfectly nesting the monopolistic competition example.
15
As product awareness expands, the ratio of these weighted averages increases, which shows
that average quality is increasing in the average size of the awareness set. Therefore, since
TFP and product quality are usually not identified, this value would be some monotone
transformation of TFP, and the increase in average quality would appear as an increase
in average revenue TFP. If an econometrician misspecified this model as monopolistically
competitive with a representative agent, then estimates would give the illusion of quality or
TFP growth over time.
Prices distort average quality since they change both the quantity demanded and the
threshold above which agents consume. Because of this, lower prices could lead to lower
average quality.
3.4
Firms
Production Technology
All firms have identical constant returns to scale production technology in labor, with productivity normalized to unity. Output from firm i for labor choice l is
yi (l) = l
Given an exogenous wage w, all firms have a constant marginal cost of w.
Firm Value and Profits
Define the prices of all of the other firms in the industry at time t as
p−it ≡ {pi′ t |i′ ∈ I 6= i}
Using the total demand derived in equation 15, firm i’s profits in period t are
πit ≡ (pit − w)yi (pit , p−it )
(19)
Definition 4 (Firm Value). Given a discount factor 0 < β < 1 and the sequence of prices
{pt }∞
t=0 , the firm’s valuation is the present discounted value of profits
Vi (pi , p−i ) ≡
∞
X
t=0
β t (pit − w)yi (pit , p−it )
(20)
Firm’s Problem
Firms in an industry compete through repeated Bertrand competition, choosing a sequence of
prices {pit }∞
t=0 to maximize equation 20 given equilibrium pricing decisions of the other firms
in the industry.23 As with most dynamic games, a variety of Nash Equilibrium exist, but this
paper focuses on repetition of a pure-strategy equilibrium for the static Nash Equilibrium
23
Atkeson and Burstein (2007, 2008) and Edmond, Midrigan, and Xu (2012) investigate Cournot, Bertrand,
and perfectly competitive pricing within international trade and examine the endogeneity of markups. These
comparisons work well in setups with representative consumer preferences. The formulation of a Cournot
equilibrium is less obvious in a model with consumer preference heterogeneity. In Arkolakis (2010) and
Eaton, Kortum, and Kramarz (2011), consumers are heterogeneous over which firms they meet in every
period, but not over their preferences for a particular good.
16
(NE) in every period.24 By standard results of game theory, the strategy of repetition of the
NE of a stage game is a NE of the repeated game.
Given a set of prices for other firms, each firm chooses the price to maximize its period
profits. In the baseline law of motion, Ajt+1 is independent of consumption choices. Therefore, there is no profitable deviation for a firm to choose lower prices to attract customers
early in its life cycle since consumers can costlessly switch to any product in their information set.25 Instead, the empirical fact that new entrants have lower prices is explained by
differences in awareness and is explored in in Section 5.4.
pit = arg max {(p̃ − w)yit (p̃, p−it )}
p̃≥0
(21)
Definition 5 (Bertrand Nash Equilibrium(BNE) for the Dynamic Game).
A BNE is a pure-strategy NE for each stage game at each time period t. That is, a pt ∈
RN , ∀ t such that
pit = arg max {(p̃ − w)yit (p̃, p−it )} , ∀i ∈ I, t ≥ 0
(22)
p̃≥0
Unlike Bertrand Competition with undifferentiated products and symmetric firms, prices
are not driven to marginal costs with only a duopoly. Part of the reason is due to market
power inherent in product differentiation, as seen in the downward sloping demand curve of
Proposition 2. This ensures that an infinitesimal change in price does not result in a discrete
jump in profits. The more interesting reason in the context of this model is the information
friction. Even with very low product differentiation, a firm can extract monopolistic profits
from consumers only aware of its product, which ensures that deviations from marginal cost
are always profitable in the absence of full information. This force, and the existence of
pure-strategy equilibrium, is discussed in Technical Appendix E.1.
3.5
Equilibrium
As the evolution of awareness is deterministic from the perspective of a firm, the entry
decision could be modeled through standard approaches. For example, in equilibrium, given
the law of motion in Section 3.2, all entry would happen at t = 0. To isolate the more
interesting post-entry industry dynamics, focus on the equilibrium after this entry decision
has been made for an exogenous set of firms.
Definition 6 (Industry Equilibrium).
A (post-entry) industry equilibrium with history independent prices is a set of
1. Entrants at time zero: i = 1, . . . N , with quality: {qi }N
i=1
2. Demand functions: yijt (pt , At , ξj ) → R, ∀ i, t
24
Intuitively, this equilibrium is the least collusive and yields the competitive limit for a large number
of firms. Moreover, with a large number of firms and low product differentiation, this limit will approach
marginal cost. For a more general specification of the dynamic game, see Technical Appendix E.2.
25
This is unlike models with habits, such as Nakamura and Steinsson (2011), where firms have an incentive
to lower initial prices to build customer habits for their good. With more complicated laws of motion, such
as those discussed in Technical Appendix G, dynamic pricing could become optimal. For example, these
dynamics would occur with word-of-mouth contagion from consumers conditional on product usage, or with
advertising under cash-flow constraints.
17
3. Firm prices: pit , ∀ i, t
4. Consumer distributions: Ψt (Ajt , ξj )
5. A law of motion for awareness Ajt
Such that:
1. Given pt , yijt (·) is optimal for the consumer according to equation 10
2. Given yijt (·) and the aggregation in equation 15, pt are a BNE equilibrium of the
dynamic game as in equation 22
3. Ψt (Ajt , ξj ) evolves according to the law of motion discussed in Section 3.2
3.6
Symmetric Firms
Awareness
The number of awareness states, N, can get large as the number of firms increases. Fortunately, if firms have degrees of symmetry, a smaller state-space is sufficient for firms to
calculate profits and make optimal pricing decisions. In particular, if all firms enter at t = 0
and have the same qi , then only the distribution of the count of firms in the consumer’s
awareness set, with cardinality N + 1, is required to calculate firm profits and prices.26
Define the count of operating firms a consumer is aware of as n ≡ |Ajt | and the probability of being in each state as fˆt ∈ RN +1 . With this definition, the transition matrix
Ĉ ∈ R(N +1)×(N +1) from state n to n′ is

N −n
′

1 − N θ n = n
Ĉnn′ = NN−n θ
(23)
n′ = n + 1


0
otherwise
For example, if N = 2
fˆjt+1

1−θ
θ
ˆ

0
1−
= fjt
0
0
0
θ
2

θ
2
1
(24)
As in Proposition 1, the distribution of awareness of n firms in the economy is
fˆt+1 = fˆt Ĉ
The time evolution is
fˆt = 1 0 . . . 0 Ĉ t
(25)
and the ergodic distribution is full awareness of N firms, i.e., fˆ∞ = 0 . . . 1 .
26
As the awareness distribution evolves after entry, firms with the same qi are only truly symmetric if they
enter at the same date. If there are two firm qualities or two cohorts of entry, then a distribution of counts
over these two degrees of firm heterogeneity would be necessary, with cardinality N 2 + 1.
18
Zero-Removed Count Distribution Many calculations require sums and products of
the distribution of the firm count, n, except for n = 0. To remove the n = 0 point, take the
ˆ
pdf fˆt (n) and left-truncate it at n = 1. This gives a new pdf of 1−ftf(n)
ˆt (0) for n ≥ 1. Define this
as the random variable n̂, defined on n̂ ≥ 1. For any function, g(n) : N + → R, define the
expectation of the (0 truncated) awareness state among consumers
Et [g(n̂)] ≡
N
X
n=1
fˆt (n)
g(n)
1−fˆt (0)
Using this expression,
N
X
n=1
fˆt (n)g(n) = (1 − fˆt (0))Et [g(n̂)]
Solutions for Arbitrary Laws of Motion Given any time dependent distribution for
the random variables n, and a time dependent mass of agents with no empty awareness sets,
fˆt (0), Proposition 4 simplifies to the following
Proposition 6 (Total Demand for N Symmetric Firms). Maintain Assumption 2, ξj ∼
Gumbel, qi = 1 for all firms (without loss of generality), and that every firm i 6= i′ chooses
the symmetric price p̄. The demand for firm i choosing price pi is
yit (pi , p̄) =
π̄ −1−1/σ
p
N i
N
X
n=1
σ(κ−1)−1
−1/σ
+ (n − 1)p̄−1/σ
fˆt (n)n pi
In a symmetric Nash Equilibrium, where p ≡ pi = p̄,
yit (p) = (1 − fˆt (0)) Nπ̄ p−κ Et n̂σ(κ−1)
(26)
(27)
where π̄ ≡ Γ(1 − σ(κ − 1))P κ Ω̄
Proof. See Appendix C.4
Proposition 7 (Symmetric BNE Prices for N Symmetric Firms). If a symmetric purestrategy equilibrium exists according to Definition 5, then the equilibrium prices, profits, and
average quality are
σ(κ−1) (28)
yt = (1 − fˆt (0)) Nπ̄ p−κ
t Et n̂
Et [n̂σ(κ−1)−1 ]
1 + σ − (1 − σ(κ − 1)) E n̂σ(κ−1)
t[
]
pt =
w
(29)
σ(κ−1)−1
Et [n̂
]
1 − (1 − σ(κ − 1)) E n̂σ(κ−1)
t[
]
πt = (1 − fˆt (0)) π̄ (pt − w)p−κ Et n̂σ(κ−1)
(30)
t
N
Γ(1 − σκ)
Et [n̂σκ ]
q̄t =
Γ(1 − σ(κ − 1)) Et [n̂σ(κ−1) ]
Moreover,
1. Prices are increasing in σ
19
(31)
2. Prices are decreasing in N
3. Average quality is increasing in σ
4. Average quality is increasing in N
Proof. See Appendix C.4
The solutions in Proposition 7 hold for any law of motion where the fˆt (0) and n̂ random
variable can be defined at every t. For the Simple Law of Motion in Section 3.2, fˆt (0) = θt
and the expectation can be taken with the direct fˆt distribution evolving in equation 25.
Proposition 8 (Asymptotic Properties of Prices). For symmetric firms, if a pure strategy
equilibrium exists, then
1. p∞ (N ) ≡ limt→∞ pt =
2. pt (1) =
(N −1+σ(N κ−1))
w
N −1+σ(κ−1)
< p1 (N )
w
, ∀t
ρ
3. p1 (N ) =
w
ρ
4. p∞ (∞) ≡ limN →∞ p∞ = w(1 + σ)
Proof. See Appendix D.1.
3.7
Large Numbers of Firms
For large numbers of firms, the probability of meeting a firm already in At approaches 0.
For symmetric firms,
the Markov transition matrix in equation 23 has the following limit

′

1 − θ n = n
limN →∞ Ĉnn′ = θ
n′ = n + 1


0
otherwise
This law of motion is a standard Poisson counting process with arrival rate θ. For large t
or in the limit to continuous time, a Poisson Distribution with parameter θt is the standard
solution to the conditional expectation of a Poisson counting process.27
(θt)n e−θt
lim fˆt (n) =
N →∞
n!
(32)
Truncating, the pdf of n̂ is the Positive Poisson distribution.28
(θt)n 1
,
n! eθt −1
for n ≥ 1
(33)
The fractional moments in equations 29, 30, and 31 can be calculated with this distribution. For example,
∞
σ(κ−1) X
nσ(κ−1) (θt)n e−θt
lim Et n̂
=
N →∞
(eθt − 1)n!
n=1
27
Alternatively, for smaller t, this process is better approximated by the Binomial distribution for large
N . Figure 20 for numerical examples of the quality of approximation. Additionally, for sufficiently large t,
the distribution of n can approximated with a central limit theorem.
28
See Technical Appendix TA.G.1 for more details on the Positive Poisson Distribution.
20
Crucially, due to the Poisson approximation and the assumption of a large number of
firms, this expression is independent of any particular N . Equations 29 and 31 are expressed
as functions of these expectations, independent of either the value of N or the mass of Zeroawareness agents. Both equations 27 and 30 are functions of N , but only as the inverse
proportion. Hence, N can be removed from the model when looking at aggregate output
and profits for all N firms. Additionally, these two values are directly proportional to the
mass of agents with awareness of at least one firm in the industry, 1 − fˆt (0).
Proposition 9 (Mean Approximation for Large N ). For large N , define the average awareness set size as, n̄t ≡ Et [n̂],
σ(κ−1)
yt ≈ (1 − fˆt (0)) Nπ̄ p−κ
t n̄t
1 + σ − (1 − σ(κ − 1))/n̄t
pt ≈
1 − (1 − σ(κ − 1))/n̄t
σ(κ−1)
πt ≈ (1 − fˆt (0)) π̄ (pt − w)p−κ n̄t
(34)
q̄t ≈
(37)
N
t
Γ(1 − σκ)
n̄σ
Γ(1 − σ(κ − 1)) t
(35)
(36)
Proof. See Appendix D.1.′′ Higher-order Taylor approximations, such as using the expression
V [X] are also discussed. While the solutions in Proposition
E [f (X)] ≈ f (E [X]) + f (E[X])
2
9 are analytically convenient, they are systematically biased due to Jensen’s Inequality. See
Figure 20 for numerical examples showing the quality of the approximation.
3.8
Industry Age and Time-Varying Price Index
In Proposition 9, all industry aggregates can be calculated with information on the aggregate price index, P , and the age of the industry, which determines the average awareness
set size. Hence, industries with symmetric quality and rates of awareness growth are only
differentiated through their age.
Assume that industries can be born at different times, t, and have an age of a. Define
φ(t, a) to be the distribution of industry ages at time t where the mass of industries is
Z ∞
M (t) ≡
φ(t, a)da
(38)
0
If the number of industries, or the age distribution of industries, changes over time, then
the price index will be time dependent. Modifying equation 11 to weight for awareness and
quality growth
1/(1−κ)
Z ∞
σ(κ−1)
1−κ
φ(t, a) 1 − fˆ(a, 0) Ea n̂
p(a) da
P (t) ≡ Γ(1 − σ(κ − 1))
(39)
0
The 1 − fˆ(a, 0) adjustment is due to limited awareness in the economy of industries
depending on their age. Due to the continuum of industries, this can be calculated by the
probability that a consumer is aware of one or more firms in an industry. Furthermore, the
continuum of consumers and the independence of awareness in each industry ensures the
price index will be identical for all consumers. This adjustment to the normal price index
calculations shows the importance of calculating properly weighted new products within a
21
bundle by the degree of awareness in the economy (e.g. calculating the CPI by the BLS).
For example, if the proportion of new industries was over-weighted, then price inflation
may be overstated compared to a typical consumer bundle. The weighting by Ea n̂σ(κ−1)
encapsulates the quality growth due to the sorting effect.
4
Transition Dynamics in a Macro Equilibrium
4.1
Consumer’s Problem with Endogenous Labor Supply
CRRA preferences with a separable disutility of labor enables endogenous labor supply based
on the distribution of industry ages, equilibrium wage w, and profits from a perfectly diversified portfolio of firms Π(t).
Since industries are only differentiated by age, equation 8 can be modified using similar
organization to equation 39. Define the output for a composite good of all industry ages as
Y (t).29
To solve for P (t) and U (t) as in Section B.1, first note that in equation 8, all industries
of a particular age have the same price p(a). There are a continuum of industries for any
particular age, of mass φ(t, a). The consumer is aware of a proportion 1 − fa (0) of these
industries due to the independent evolution of awareness between industries, each with identical θ. Moreover, as each consumer is aware of a continuum of industries, each with the
same distribution of idiosyncratic preferences, the consumer’s utility will be identical for all
j by a law of large numbers. Using Proposition 2’s result that a single good will be chosen
per industry, the agent chooses an age dependent intensive demand per consumer, y(a, ξ).
The ξ of the agent is determined by the maximum draw within their awareness set. The
distribution of the maximum of n draws from the g(ξ) distribution is the order statistic,
g(n) (ξ). Define a composite good
Z
Y (t; y(·, ·)) ≡
∞
0
φ(t, a) 1 − fˆ(a, 0)
∞
X
n=1
fˆa (n)
Z
eσξ y(a, ξ)
ρ
g(n) (ξ)dξda
!1/ρ
(40)
If awareness evolves independently across all industries, and consumers start with no
awareness of each industry, then consumers will have identical welfare for some γ > 1 and
ν > 0. Given wages w(t) and aggregate profits Π(t), the consumer’s problem with endogenous
labor supply is to maximize their welfare
Z ∞
η
−rs
1−γ
1+1/ν
1
e
W (t) = max
Y (t + s; y(·))
− 1+1/ν H(t + s)
1−γ
y(·), H(·)
s.t.
Z
∞
0
0
φ(t, a) 1 − fˆ(a, 0)
∞
X
n=1
fˆa (n)
Z
p(a)y(a, ξ)g(n) (ξ)dξda ≤ w(t)H(t) + Π(t) ≡ Ω(t)
(41)
Firms will discount profits at the discount factor of consumers. The discount factor at
time t, s periods in the future is denoted Λ(t, s). The instantaneous interest rate implied by
this discount factor is r̃(t).
29
The use of a composite good here is for algebraic convenience and does not imply that a consumer
chooses demand from a final goods producer.
22
Proposition 10 (Transition Dynamics in Equilibrium). Assume the following
1. All firms in all industries have identical quality
2. A large number of firms enter at the birth of each industry, and no additional firm
entry takes place
3. The distribution of industry ages over time evolves exogenous with a pdf φ(t, a)
4. Firms die at a constant rate δ
5. Awareness evolves with the same independent stochastic process for all industries
Then consumer welfare is
W (t) = max
Y (·),H(·)
Z
∞
e
−rs
0
1
Y
1−γ
(t + s)
1−γ
−
η
H(t
1+1/ν
s.t. Y (s)P (s) ≤ w(s)H(s) + Π(s) ≡ Ω(s)
+ s)
1+1/ν
ds
for all s ≥ t
(42)
Where the following equations summarize the equilibrium in transition
p(a) =
1 + σ − (1 − σ(κ − 1))
Ea [n̂σ(κ−1)−1 ]
Ea [n̂σ(κ−1) ]
(43)
Ea [n̂σ(κ−1)−1 ]
1 − (1 − σ(κ − 1)) E n̂σ(κ−1)
a[
]
1/(1−κ)
Z ∞
1−κ
σ(κ−1)
p(a) φ(t, a)da
1 − fˆ(a, 0) Ea n̂
P (t) = Γ(1 − σ(κ − 1))
0
Z ∞
Γ(1 − σ(κ − 1))(1 − fˆ(a, 0))(p(a) − 1)p(a)−κ Ea n̂σ(κ−1) φ(t, a)da
Π̄(t) =
(44)
(45)
0
−1/(γν+1)
Ω(t) = η ν P (t)ν(1−γ) (1 − P (t)κ−1 Π̄(t))
Ω(t)
Y (t) =
P (t)
H̄(t)
Π(t)
H(t) =
Π̄(t)
Π(t) = Π̄(t)P (t)κ−1 Ω(t)
U (t) =
V (t) =
η
(t + s)
− 1+1/ν
H(t + s)1+1/ν
−γ
e−(δ+r)s YY(t+s)
Π(t + s)ds
(t)
1
Y
1−γ
Z ∞
0
1−γ
d log(Y (t))
dt
Proof. See Appendix D.1.
(46)
(47)
(48)
(49)
(50)
(51)
(52)
r̃(t) = r + γ
Equation 42 shows the distortions in the consumer’s utility due to the age distribution
of firms captured in the cost of living index. Consumer utility in this model benefits from
the sorts of variety effects discussed in Broda and Weinstein (2006).
Definition 7 (Equilibrium with Symmetric Industries).
Given an exogenous evolution of the distribution of firm ages, φ(t, a), an Equilibrium with
Symmetric Industries is a set of: p(a), P (t), H(t), Ω(t), and U (t), fulfilling Proposition 10.
23
4.2
Misspecified TFP Estimation
Assume that an econometrician misspecifies the model, and believes that the economy is
monopolistically competitive with firms and industries having symmetric quality and productivity. Importantly, the econometrician does not take into account partial awareness of
new industries and age varying markups, so all industries are fully symmetric. In a dynamic
equilibrium such as Definition 7, if the total mass of firms/industries, M (t), does not change,
then the econometrician would predict that a any time variation in aggregates occurs due to
changes in Z(t), as the age distribution of firms and industries is irrelevant.
The econometrician can estimate Z(t) from the following30
Proposition 11 (Misspecified TFP Estimation). If the econometrician observes M (t) and
a price index P (t), then the estimate of Z(t) is
Z(t) =
1
ρP (t)M (t)1/(κ−1)
If the econometrician instead observes M (t) and H(t), then the estimate of Z(t) is
ρ
!− γ−ρ
γ
1
1
H(t) ν + ρ M (t) 1−κ η
Z(t) =
ρ
(53)
(54)
Proof. See Appendix D.1.
5
Examples and Analysis
5.1
Sources of Market Power
In this model there are several separate sources of market power
1. Finite Number of Firms: This is the least novel source of market power in the
model. As N approaches infinity, this contribution disappears.31 However, unlike
many models, market power will still exist in the limit as the other sources are still
active.
2. Product Differentiation: This element of market power is time varying as it relies
on the distribution of awareness in the economy. Even if σ is arbitrarily low, prices are
only driven down to marginal costs in the asymptotic limit as the economy approaches
perfect awareness. Otherwise, there is a profitable deviation to charge the monopoly
price and gain positive profits from consumers only aware of the single firm.32
30
This example can be done assuming heterogeneous firms with a productivity z. Assuming a distribution, such as Pareto with tail parameter α, for productivity yields an price index of P (t) =
1/(1−κ)
w
α
M (t)1/(1−κ) ρZ(t)
. The qualitative mismeasurement of Z(t) are the same after aggregatκ+α−1
ing output. This is a standard result in monopolistic competition when decomposing the price index with
constant markup over marginal cost pricing.
31
See Figure 10 for an example of dynamics for large N . The dynamics do not change substantially after
around ten firms with the parametrization in this example.
32
Martins, Scarpetta, and Pilat (1996) compares differentiated versus non-differentiated industries in
OECD data and shows increasing market power with increasing differentiation, as is consistent with this
model.
24
3. Awareness: The novel contribution of this paper is that market power exists due
to the incomplete information of consumers. This source disappears as t approaches
infinity given the baseline Markov process described in this paper, but is active even for
low differentiation and a large number of firms. It interacts with the degree of product
differentiation, as discussed in Section 5.3.
5.2
Distinguishing Productivity from Demand
In models with a representative consumer and monopolistically competitive firms, heterogeneous firm quality and heterogeneous physical productivity are isomorphic in the sense that
parameters in a firm’s revenue, profits, and demand are not identified without quality unit
pricing data. This is discussed in detail in Melitz (2003) and Luttmer (2007), and derived
in the Technical Appendix F.2. Klette and Griliches (1996) discusses this in the context of
identification and shows how industry data can be used to disentangle the sources.
For this reason, many papers use a revenue-based productivity measurement and posit a
single exogenous process that captures the combination of these forces.
Use Revenue-based Productivity
z
}|
{
Physical Productivity vs. Product Quality vs. # Customers
|
{z
}
Demand
While appropriate for many aggregate economic questions, conflating the two is an unsatisfactory answer to understanding firm growth and the life cycle. For example, if firms
make investments to control their growth rates and profitability, then should econometricians
calibrate the returns to firm R&D expenditures or to advertising expenditures? Or is the
growth process largely out of firms’ control, driven through exogenous changes to demand
due to word-of-mouth contagion among consumers?
Many recent papers show how the distinction between demand and productivity is important in understanding firm growth and profitability. For example, Arkolakis (2010), Gourio
and Rudanko (2011), and Chaney (2011) highlight the importance of separating out the new
customer margin and the matching process between customers and firms. The role of customer markets and market power is discussed in Bils (1989) and Rotemberg and Woodford
(1991), and the connection between endogenous markups and changes to industry structure
is discussed in Jaimovich and Floetotto (2008).33
Here, quality and physical firm productivity still cannot be distinguished without qualityunit pricing data, as is consistent with the other literature. However, idiosyncratic total
demand factors, as defined in equation 15, and idiosyncratic productivity are no longer
econometrically equivalent. These factors can be disentangled through changes in the number
of customers and the degree of awareness, and the effects on market power.
5.3
New Industry with Symmetric Firms
Most of the intuition of the model can be evaluated using a duopoly.34
33
In contrast to Jaimovich and Floetotto (2008), in this model changes due to awareness evolution, rather
than entry and exit, determines the degree of competition between firms.
34
However, the techniques in this model work well for a large number of firms. See Appendix A for
equations using simple linear algebra for N firms, and Section 3.7 for asymptotics as N → ∞. Discussion
25
Normalized to Maximum
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Vt
yt
pt
q̄t
0.2
0.1
0
0
5
10
15
20
25
t
30
35
40
45
50
Figure 7: Life Cycle with Low Sigma
Figure 7 gives the results for an industry with low product differentiation which roughly
matches the average lifecycle of Figure 1.35 In particular, valuation peaks and then drops
as competition becomes more intense. The average quality q̄t increases due to the sorting
effect, but by a modest amount due to the low level of product differentiation. As discussed
in Section 5.2, if measured at the aggregate level, then q̄t is a monotone transformation
of TFP, which grows even though idiosyncratic firm quality and physical productivity are
constant.
A comparison between high versus low differentiation highlights the competing forces on
profits, as shown in Figure 8. In the case of high product differentiation, the valuation is
monotonically increasing. An explanation for this is shown in Figure 9. Since the products
are heavily differentiated, average quality grows significantly due to sorting. The average
quality increase dominates the effects of more intense competition due to increasing awareness, and hence, unlike in the low differentiation example, the prices do not decrease significantly. The high differentiation example suggests that increasing profitability per customer
over time is not necessarily evidence of either increasing productivity or of price collusion.
These asymptotic properties can be seen in the examples in Figure 10. In particular, the
forces of the model still exist for large numbers of firms. As N increases, only the market
power arising from a finite number of firms disappears, as discussed in Section 5.1.
of the κ and σ calibrations to aggregate data is given in F.2. The θ parameter only changes the timescale
and hence does not effect most of the qualitative results, but would need to be jointly determined with the
discount rate in the firm’s valuation.
35
This example solves for a duopoly with qi = 1, θ = 0.2, and compares σ = 0.05 versus σ = 0.3.
26
Normalized to Maximum
1
0.9
0.8
0.7
0.6
0.5
0.4
Vt for σl
yt for σl
Vt for σh
yt for σh
0.3
0.2
0.1
0
0
5
10
15
20
25
t
30
35
40
45
50
Figure 8: Life Cycle (low versus high σ)
Normalized to Maximum
1
0.95
0.9
0.85
0.8
0.75
0.7
pt
q̄t
pt
q̄t
0.65
0.6
0.55
0.5
0
5
10
15
20
25
t
30
35
40
Figure 9: Intensity of Competition (low vs. high σ)
27
for σl
for σl
for σh
for σh
45
50
Prices
1.4
1.35
1.3
0.07
N πt
1.2
pt
1.08
0.05
1.15
0.04
0.02
1.05
0.01
t
10
20
0
30
1.06
0.03
1.1
0
N=2
N=4
N=10
N=20
1.1
0.06
1.25
1
Average Quality
1.12
q̄t
ts
Industry Profits
0.08
N=2
N=4
N=10
N=20
1.04
N=2
N=4
N=10
N=20
0
10
t
20
1.02
1
30
0
10
t
20
Figure 10: Symmetric Equilibrium for Various N
5.4
Entry into a Monopoly
Section 2.2 presented evidence that, on average, entrants often have lower prices than incumbents and yet still grow slowly.
To show this, the following experiment solves the model with half of the population aware
of a monopolist at time t = 0, at which point a new entrant with identical quality enters.
The evolution of awareness is shown in Figure 11. Importantly, the ratio of awareness of
Firm 1 (the incumbent) to awareness of both firms is different than the ratio of the awareness
of Firm 2 (the entrant) to awareness of both firms.
1
ft (∅)
ft ({1})
ft ({2})
ft ({1, 2})
0.9
0.8
0.7
ft
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
t
25
30
35
40
45
50
Figure 11: Evolution of Awareness for Entry
This asymmetry in awareness makes the pricing decision dependent on firm age. For an
incumbent, decreasing the price decreases profits on existing customers who are unaware of
the entrant. In contrast, the entrant has monopoly power over very few of its customers and
in equilibrium can lower prices to temporarily attract the incumbent’s customers. This effect
is seen in Figure 12, where the entrant has weakly lower prices throughout the industry life
cycle.
28
30
The average quality of the incumbent and entrant shows that the entrant has higher
average quality during the expansion of awareness.36 This effect is due to sorting. The
incumbent starts with close to the unconditional average of the ξj for their consumers, while
the entrant’s customers are more sorted since it needs to compete against the incumbent.
This example shows that caution must be taken in interpreting increases in quality or TFP
after an increase in the number of competitors as evidence of competition spurring quality
innovation. In this case, the increase in quality and aggregate output of the industry after
entry is purely a passive process of consumer sorting.
These results are consistent with the evidence on firm entry discussed in Section 2.2.
Driven only by the information asymmetry in awareness, entrants have higher quality (or,
equivalently, revenue productivity) and lower prices than incumbents, but these differences
disappear over time. These facts highlight that firms are only symmetric if they have both
identical production technologies and customer awareness.
0.7
p1t
p2t
1.3
0.6
Customers
1.35
1.03
0.4
1.2
0.3
1.15
1.01
0.1
1.05
1
1.02
0.2
1.1
q̄1t
q̄2t
1.04
0.5
1.25
Prices
1.05
x1t
x2t
Average Quality
1.4
0
10
20
t
30
40
50
0
0
10
20
t
30
40
50
1
0
10
20
t
30
Figure 12: Customers, Prices, and Quality After Entry into a Monopoly
5.5
Asymmetric Firms
In order to understand why large differences in productivity and profitability can be sustained
within an industry, Figure 14 and Figure 13 show a duopoly with asymmetric firm quality.
To summarize the asymptotic results, the prices of the higher quality firm are only marginally
higher than those of the lower quality firm. However, the higher quality firm has over twice
the output and customers, and nearly five times the profits and valuation. The higher profits
are driven through both more customers and higher profits per customer. Increases in the
relative profit per customer, denoted by πxt , are due to the asymmetric effects of increasing
awareness of both firms. The asymmetry between the profit ratio and customer ratio explains
how a few firms in an industry can capture the vast majority of the industry’s profits, but
not necessarily capture the same proportion of customers or output.
Figure 14 compares asymmetric firms with low and high levels of differentiation. The
notable feature of these graphs is that a ratio of high versus low quality contemporaneous
firm fundamentals, such as profitability, revenue, prices, and customers, only diverges slowly
over time, while the valuation of the firms can be very different upon entry. This might
explain, for example, the dispersion of firm price-to-earnings ratios within an industry as
capturing differences in product quality that customers take time to become aware of. When
36
Other results of this experiment is shown in the Technical Appendix A.
29
40
50
Profit/Customer Ratio
1.07
2.5
1.06
Customers Ratio
p1t /p2t
Price Ratio
1.05
1.04
1.03
1.02
x1t /x2t
2
1.5
1.01
1
2.3
πx1t /πx2t
2.2
2.1
2
1.9
1.8
1.7
0
10
20
t
30
40
1
50
0
10
20
30
t
40
1.6
50
0
10
20
t
30
Figure 13: Prices, Customers, and Profit per Customer for Asymmetric Firms
5
V1t /V2t
V1t /V2t
4.5
4
3.5
2.3
π1t /π2t
π1t /π2t
4.5
Output Ratio
5
Profits Ratio
Valuation Ratio
degree of product differentiation is high, divergence is less evident since the idiosyncratic
preferences dominate the asymmetry in the common quality.
4
3.5
3
3
2.5
2.5
2
2
y1t /y2t
y1t /y2t
2.2
2.1
2
1.9
1.8
1.7
1.6
1.5
1.4
1.5
0
10
20
t
30
40
50
1.5
0
10
20
t
30
40
50
1.3
0
10
20
t
30
40
50
Figure 14: Valuations and Output for Asymmetric Firms (low versus high σ)
5.6
Macroeconomic Experiments
The following experiment uses the results of Section 4 and the equilibrium in Definition
7. See Appendix F.2 for details on calibration, and Technical Appendix C for a similar
experiment with monopolistic competition.
The state of the aggregate economy is determined by an exogenous process of industry
ages. Assume that through an unmodeled process, industries die at a constant death rate
δ > 0, and a flow of industries of size δ is born at a = 0 each period. The pde for the age
= − ∂φ(t,a)
− δφ(t, a), with an initial condition φ(0, a), and the
distribution is then ∂φ(t,a)
∂t
∂a
total mass of industries is defined in equation 38.37
37
This approach emphasizes the creation and destruction of new products and varieties rather than firms.
Changes in the varieties available to consumers is discussed in Broda and Weinstein (2010). In this paper,
while firms produce only one product, at the aggregate level this distinction does not matter, and the
30
40
50
The solution to the evolution of the age distribution from any initial condition is
(
φ(0, a − t)e−δt for t ≤ a
φ(t, a) =
δe−δt
for 0 ≤ a < t
(55)
Hence, limt→∞ M (t) = 1 and the stationary distribution of industry ages is
φ(∞, a) ≡ δe−δa
(56)
The experiment is a variation of an impulse response to the age distribution, using the law
of motion derived in equation 56. Assume that from the stationary equilibrium, an exogenous
shock occurs as a large cohort of new firms enters due to changes in unmodeled incentives
to new industry formation (e.g. government subsidies to small business, or opening to trade
where production takes place within the country). led incentives to new industry formation
(e.g. government subsidies to small business, or opening to trade where production takes
place within the country).
Set t = 0 to be after this exogenous shock has spurred growth for one year, at which
point birth and death ratios stabilize immediately to δ. One approximation to this shock
is to add an additional quadratic set of young industries to the stationary distribution of
φ(∞, a) = δe−δa .38
(
(.25 − (a − .5)2 ) 0 ≤ a ≤ 1
φ(0, a) = δe−δa +
0
a>1
With these parameters, there are approximately 16.7% more industries at time 0 than in
the stationary distribution. The magnitude of this shock is not intended have of economic
significance, other than to show the responses for a very large shock. Figure 15 gives an
example of how the age distribution, φ(t, a), evolves at t = 0, 5, and 10.
0.35
φ(0, a)
φ(5, a)
φ(10, a)
0.3
φ(t, a)
0.25
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
a
12
14
16
18
20
Figure 15: Propagation of Shock in the Industry Age Distribution
Figure 16 shows the transition dynamics for this example. All values are plotted relative
to their stationary value. Define the real profits and real income as Πr (t) ≡ Π(t)/P (t)) and
organization of products among firms is indeterminate as long as multi-product firms has no span of control
or production externalities.
38
Alternatively, the shock could be a Dirac delta impulse, with potentially more difficult calculations.
31
Ωr (t) ≡ Ω(t)/P (t), respectively. RDefine the effective number of varieties in a consumer’s
∞
consumption bundle as M e (t) = 0 (1 − fˆ(a, 0)φ(t, a)da. The average markups of firms in
the economy, both weighted and unweighted by awareness, are defined as
Z ∞
1
p̄(t) ≡ M e (t)
(1 − fˆ(t, 0))φ(t, a)p(a)da
(57)
0
Z ∞
u
1
p̄ (t) ≡ M (t)
φ(t, a)p(a)da
(58)
0
1.015
1.01
U (t)
W (t)
P (t)
1
1.01
0.99
1.005
0.98
0.97
0
5
10
15
t
20
25
30
1
1.002
1.6
1
1.4
0.998
1.2
0.996
0.994
5
10
15
t
20
25
5
10
15
t
20
30
30
Πr (t)
r̃(t)
0.8
0
5
10
15
t
20
1.015
1.03
Ωr (t)
1.025
25
1
H(t)
0
0
25
30
p̄(t)
p̄u (t)
1.01
1.02
1.005
1.015
1
1.01
0
5
10
15
t
20
25
30
0.995
0
5
10
15
t
20
Figure 16: Transition Dynamics for Off-Equilibrium Age Distribution
Examining equation 44, the price index P (t) is decreasing since the firms’ pricing decisions
are dependent only on industry age in equation 43. Therefore, the additional mass of young
varieties in φ(0, t) lowers this cost of living index, even though p̄(t) and p̄u (t) are higher than
their stationary value. From the decreasing price index, as long as nominal income is weakly
increasing, examining equations 47 and 50 shows that the addition of new varieties improves
welfare, as typically the case in models with CES preferences.
Real firm profits, Πr (t), increase slightly due to the presence of higher markups for the
new varieties. As consumers prefer variety, they purchase new products as they become aware
of the industry, diverting their income from older, more competitive industries to those that
are less competitive with higher markups. Even though H(t) drops, the real income Ωr (t)
increases because the consumer owns the stream of dividends from all firms,
Even in a case with relatively inelastic labor supply, total labor (or output), H(t), decreases due to the changes in the composition of the consumer’s bundle. The quantity
32
25
30
consumed in younger industries is lower due to two reasons: 1) the industries are less competitive and hence prices are higher, and 2) the industries are less sorted, so consumers are
unlikely to have good matches. For these reasons, with an elasticity of labor supply of .75,
equilibrium labor H(t) is dropping.
Figure 17 shows the transition dynamics for valuations and aggregate Price to Earnings
ratio, P E(t). The valuation with selection, Vtp , provides the aggregate value of only the
public firms, where firms have a constant hazard of entering the public markets, to be
captured in aggregate stock market earnings and valuation data.39
1.04
1.015
1.05
1.03
1.01
1.04
1.02
1.03
1.005
1.01
1.02
1
1
1.01
0.995
0.99
1
0.99
0.98
0.99
0.985
0.97
V (t)
V p (t)
0.96
0.95
0.94
0
5
10
15
t
20
25
0.98
Π(t)
Πp (t)
0.98
0.975
30
0.97
0
5
10
15
t
20
25
PE(t)
PEp (t)
0.97
0.96
30
0.95
0
5
10
15
t
20
25
Figure 17: Transition Dynamics for Aggregation Valuation, Earnings, P/E
While real profits are increasing, interest rates are very high as the new varieties enter
consumer’s bundles. Hence, valuations discount future profits heavily after the shock, lowering nominal value of the firms. When selecting for public firms, the stock market begins
to rise rapidly as the new firms, with higher markups, enter the index of public valuations.
At peak, the large 16.7% increase in additional industries results in nearly a 3% drop in
the cost of living index, 0.5% drop in labor or physical output, 0.5% increase in welfare, 2%
increase in real income, and 1.2% increase in a naive calculation of average firm markup.
Since firms price using the consumer’s discount factor, the nominal value of the stock market
can drop by 6% initially if there is selection for older firms in the index of public markets,
and the price-to-earnings ratio can swing by approximately 10% during the transition.
In Figure 18, a misspecified estimate of TFP is used, assuming that the econometrician
can calculate a quality adjusted cost of living index by sampling firms and their demand
functions. The estimate Z 1 (t) based on H(t) is described in equation 54. The estimates of
Z 2 (t) and Z 3 (t) are based on equation 53, and built, respectively, using a correctly adjusted
price index, and a price index without weighting for awareness.
In order to determine the importance of θ on consumer utility and the price index,
Figure 19 shows the stationary utility and price index relative to the calibrated θ = .15 as the
baseline. While θ is not endogenously chosen in this model, it is a reasonable assumption that
it is positively related to aggregate expenditures on advertising and marketing expenditures.
A properly weighted average markup in this model is related to P (t) through a monotone
transformation, so the negative correlation between P (t) and θ suggests a negative correlation
39
The yearly hazard of entering the public markets, 0.14 in this example, is calibrated to the average amount of time before firms IPO. From the SEC IPO Taskforce,
http://www.sec.gov/info/smallbus/acsec/ipotaskforceslides.pdf, the average time to IPO is
between 4.8 and 9.4 in the US, and much longer in Europe.
33
30
1.08
Z 1 (t)
1.06
Z 2 (t)
Z 3 (t)
1.04
1.02
1
0.98
0.96
0.94
0
5
10
15
t
20
25
30
Figure 18: Misspecified TFP Calculations
1.6
U∞ (θ)
P∞ (θ)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.05
0.1
0.15
θ
0.2
0.25
Figure 19: Comparative statics on P (t) and U (t) over θ
34
between proxies for awareness investment per capita and markups.40 Using data on 15 OECD
countries, the correlation between the specified advertising intensity measure and average
markup is −0.14.
5.7
Model Predictions and Panel Data
The following summarizes fixed effect regression with 333 industries, as discussed in Appendix
F.3. The data is a subset of that used in Figure 2, except that the industry code is treated as
a panel identifier rather than averaging across all industries for a particular age. As before,
the “age” in these regressions is the relative age, where the peak is at age = 1 and birth is
at age=0.
The goal of these regressions is to summarize co-movements in the data, and the magnitudes of the coefficients are not interpreted. All regressions are done controlling for year
effects. Full results are reported in Table 4.
Panel Data Regression
TFP is increasing in age
Model Explanation
Customer sorting increases average quality
Price is decreasing in age
Increasing awareness increases the intensity
of competition41
Concentration is increasing in age
Decreasing profits due to increasing competition can create exit42
Markup is increasing in concentration (controlling for age)
Oligopolistic pricing with differentiated
products
Markup is decreasing in age (controlling for
concentration)
Increasing awareness increases the intensity
of competition43
Table 1: Comparing Panel Data Regressions and the Model
40
The advertising intensity measure is given in Van der Wurff, Bakker, and Picard (2008), and average
markups are given in Martins, Scarpetta, and Pilat (1996). This data analysis is preliminary. There may
also be major endogeneity issues in the choice of advertising intensity as a proxy since higher investment in
advertising may be chosen to overcome idiosyncratic impediments to awareness expansion in a country.
43
While research such as Greenwood, Hercowitz, and Krusell (1997) show that price indices of investment
goods fall over time, Hobijn (2001) cautions that price decreases may actually be capturing the erosion of
market power for older vintages rather than increasing productivity. This model provides a partial explanation for within industry market power erosion.
43
This version of the model does not contain any firm exit. Technical Appendix D details a version of the
model with fixed costs of operation, which can lead to exit.
43
Markups decreasing in age when controlling for concentration and year effects may be a new result, and
appears robust to several calculations of markups.
35
6
6.1
Related Literature
Motivations for the Product Awareness Process
In this model, product awareness captures several potentially independent effects. The following components, discussed in the literature, would be part of awareness:
Knowledge of product or firm existence: This is a key goal of models with informative advertising in customer markets, as discussed in Bagwell (2007) and Dinlersoz and
Yorukoglu (2012). Sethi (1983) and other papers in the marketing literature examine how
advertising can influence or build a customer base for the product, which I interpret as
spreading awareness of the product to new consumers. Product awareness may also occur due to word-of-mouth diffusion, independent of the advertising decisions of the firm, as
surveyed in Mahajan, Muller, and Bass (1990).44 The most direct comparison within the discrete choice literature is Goeree (2008), which estimates a static discrete choice BLP model
of the personal computer industry where advertising effects the probability of a consumer
being aware of a product.
Access to purchase the good: In the case of consumer products, this may include
ensuring that a distributor exists within a particular geography, or that a relationship exists
between an importer and an exporter. An example of a model which investigates this margin
and relates it to advertising expenditures is Arkolakis (2010). Drozd and Nosal (2012) has
a similar notion of investment in market capital, which acts as a capacity constraint on
production. Other approaches include modeling the slow process of building relationships
between importers and exporters, as in Eaton, Eslava, Krizan, Kugler, and Tybout (2011),
or by examining networks, as in Chaney (2011).
Quality of the product and match value: Consumer preferences in this model include
components for both a product quality common to all consumers and an idiosyncratic match
quality. If these are unknown to the consumer, evaluating both sources can be a slow
and costly process, especially when the match quality dominates the average quality or
the product is an experience good. Gourio and Rudanko (2011) models this process as a
two-sided search and matching friction between consumers and producers, and connects the
friction to advertising expenditures.45 Grossman and Shapiro (1984) models firms using
informative advertising to communicate general qualities of a differentiated product. At
the micro-level, many papers have modeled a dynamic demand choice for consumers with
idiosyncratic preferences for products, often in a learning environment such as Ackerberg
(2003), Crawford and Shum (2005), Bergemann and Välimäki (2006) and De los Santos,
Hortaçsu, and Wildenbeest (2012).46
44
In the baseline law of motion used, awareness is an uncontrolled process unrelated to any sort of awareness
contagion. However, the Markov transition matrix described in Section 3.2 is a model intrinsic and easily
extended to include these elements. Some of these extensions are discussed Technical Appendix G.
45
At the aggregate level, Bai, Rios-Rull, and Storesletten (2011) explores shocks to the shopping preferences
of consumers in a matching model to explain business cycles.
46
In particular, the decreasing prices over time in this model have similarities to Bergemann and Välimäki
(2006), albeit due to different forces. The expansion of informed consumers over time in that paper is
analogous in the expansion of consumer awareness, and the “niche” vs. “mass market” dichotomy is captured
36
Knowledge of prices for each good: A related literature includes models of price dispersion driven through search frictions. For example, Butters (1977), Burdett and Judd
(1983), and Burdett and Coles (1997) are interested in explaining price dispersion of a homogeneous good, often motivated by Diamond (1971). This model assumes that within
their information set, consumers have perfect information of the prices and costlessly switch
between firms. As the goal of this model is to explore the expansion of awareness for new
products and industries rather than price competition among homogeneous goods, it also
assumes that consumers aware of a product are also aware of its price at any point in time.
Finally, producers cannot price discriminate, unlike Butters (1977) and others where the
dissemination of price is the important equilibrium choice.
6.2
Other Approaches to Explaining the Industry Life Cycle
Technological explanations for the industry life cycle and the eventual decline of firms goes
back to Gort and Klepper (1982) and continues with papers such as Klepper and Graddy
(1990) and Dinlersoz and MacDonald (2009). For the most part, these explanations are
based on technological cycles of product introduction and refinement. Similarly, Chari and
Hopenhayn (1991) explains the peak and decline of industries through vintages of human
capital. In contrast, my model attempts to explain elements of the life cycle in the absence
of any technological change or vintages.
Industry selection driven through stochastic productivity or imperfect signals on profitability is the core mechanism of papers such as Jovanovic (1982), Hopenhayn (1992), Ericson and Pakes (1995), Melitz (2003), and Luttmer (2007). Part of the rationale for models
such as Luttmer (2007) is to explain deviations from Gibrat’s Law of constant growth rates
independent of size, as discussed in Sutton (1997). In these models, growth rates have size
dependence conditional on survival. Since young firms exit on average more than older
firms, survivors tend to have higher growth rates and tend to be smaller. Another approach
to understanding industry equilibrium and non-Gibrat growth is taken in Rossi-Hansberg
and Wright (2007), where heterogeneity in industry-specific human capital leads to mean
reversion of growth rates. In this model, growth rates are instead declining in firm age due
to limitations in demand from the consumer preferences, which is consistent with meanreversion.
Another reason to understand firm and industry growth is to explain the discrepancy
between firms’ stock market valuations and book values, as captured in Tobin’s Q and
discussed in Hall (2001). Papers such as Hall (2004) have cast doubt on the role of capital
adjustment costs, while Abel and Eberly (1994) and others explain deviations in Tobin’s
Q with costless adjustment. The general consensus is that sources of intangible capital are
necessary to explain growth rates and deviations of Tobin’s Q from unity. For example,
Arrow (1962), Stokey (1988), and Klenow (1998) use a learning-by-doing mechanism to
explain growth of productivity. When learning-by-doing is taken literally as an increase in
physical productivity, it can sometimes contradict the micro-evidence of a lack of systematic
TFP differences between young and old firms, as described Section 2.2. However, if the
mechanism is interpreted more broadly to include expansion of demand, then there may be
a more direct comparison to the expansion of awareness in my model.
A related explanation for intangible capital, as in Prescott and Visscher (1980) and
in the σ parameter in my model.
37
Atkeson and Kehoe (2005), is to model the creation of organization specific knowledge to
explain how firm growth occurs without systematic differences in labor or capital productivity
between firms. In this model, the accumulation of organizational capital over a plant’s life
cycle would be similar to the accumulation of product awareness, acting as a shifter of
demand rather than an organization specific technology change.
More directly, my model fits into the literature on customer capital, such as Phelps
and Winter (1970), Hall (2008), and Gourio and Rudanko (2011). Customer capital is often
explained by switching costs, as described in Klemperer (1995), or through habits in consumer
preferences. Recent work by Ravn, Schmitt-Grohe, and Uribe (2006) and Nakamura and
Steinsson (2011) explore the creation of habits on a good-by-good basis. In contrast to these
approaches, this model assumes no intrinsic habits exist in the preferences and consumers
can costlessly switch between goods they are aware of. Customer turnover decreases over
time due to customer sorting as they build awareness of many firms.47
7
Conclusion
This paper contributes a novel mechanism to understand firm growth and the industry life
cycle as the expansion of new product awareness among consumers. This information friction
is modeled as a stochastically evolving set of firms that each consumer is aware of, where
consumers can only purchase products from within this information set.
The expansion of awareness causes two countervailing forces on firm profitability. First,
an increase in the size of the average information set increases the intensity of competition
among firms, decreasing market power. Second, as consumers gain access to more goods,
they choose the product that best matches their needs. Since consumers tend to demand
more of goods they strongly prefer, the sorting of consumers into their preferred products
increases the average quantity demanded for all goods in that industry.48 If an industry
has a low degree of product differentiation, then decreasing market power can dominate
the increasing demand, leading to non-monotone firm profits and valuations. On the other
hand, if an industry has highly differentiated products, then the sorting of consumers into
their preferred products strongly offsets any intensifying competition and leads to monotone
profits and valuations.
While stylized, the mechanism is consistent with the rough macroeconomic evidence on
industry evolution as well as evidence from specific industry studies. In particular, at the
aggregate level, the model can explain why industries start small, and how the aggregate
capitalization of an industry can drop over time, even if productivity and output remain
high. At the micro level, this paper explains how entrants can have higher productivity and
lower prices than incumbents, yet grow slowly.
47
This explanation is consistent with consumer learning models such as such as Crawford and Shum (2005)
or Ackerberg (2003), where consumers dynamically choose whether to sample new goods.
48
Consequently, this model urges caution in interpreting aggregate data in the context of a representative
consumer model with monopolistically competitive firms. If common data sources, such as revenue, profits,
and labor inputs, are used to estimate industry productivity or quality, then the the sorting of consumers
into their preferred products is captured in estimates of quality growth (or, equivalently, TFP growth), even
though the fundamental quality and productivity of each firm is constant.
38
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43
Appendix A
A.1
Markov Process Details
Fully Differentiated Firms
The possible states for awareness A are 2I , with cardinality
N I X
N
N≡ 2 =
k
k=0
Define the lexicographic ordering matrix as

0 0 ...
 1 0 ...

L=
 0 1 ...
. . .
1 1 ...
(A.1)

0
0

0


1 N×N
(A.2)
Additionally, define the following vectors and matrices:
1. I is the diagonal matrix of size N × N
2. ~1 is the vector of 1’s of size N
3. eι is a vector of 0’s with a 1 at the ι’th position
ι Index
1
2
3
4
5
...
N
Set Ajt
{∅}
{1}
{2}
{1, 2}
{1, 3}
Row
of L
0
0
0
1 0 0
0 1 0
1 1 0
1 0 1
{1, 2 . . . N }
0
0
0
0
0
...
...
...
...
...
0
0
0
0
0
1 1 1 1 ... 1
Table 2: Lexicographic Ordering of States
The unconditional distribution of awareness in the economy is the vector ft ∈ RN , using
this ordering of the states. With equations 2 and 3, the Markov transition matrix C of
transition probabilities from ι to ι′ can be written as
Cιι′ =


1 − θ +
θ
e L ~1T
N ι
θ
N


0
ι′ = ι
(eι′ − eι )L = 1
o.w.
(A.3)
Using standard properties of Markov chains, the unconditional distribution of awareness in
the economy is
ft = e1 C t
44
A.2
Ergodic Distribution
Proof of Proposition 1. The solution for the unconditional distribution in the population
follows from standard properties of Markov Chains with a continuum of mass 1 of agents.
The ergodic distribution is the solution, f∞ , to the following equation:
f∞ = f∞ C
(A.4)
From standard Markov theory, C is ordered such that there is a single absorbing state
at ft (I). More formally, because C is constructed to be an upper triangular matrix, its
eigenvalues are along the diagonal. Since there is only a single unit eigenvalue in the location
CNN , there is only one ergodic set. Hence
there is a unique eigenvector solving equation A.4,
up to a scale. Since the vector f∞ = 0 0 . . . 1 fulfills equation A.4 and sums to 1, it is
the unique ergodic distribution.
Appendix B
B.1
Consumer Demand and Price Derivation
Lagrangian and First Order Necessary Conditions
To set up the Lagrangian for the consumer’s optimization problem in equation 8, define
λj > 0 as the Lagrange multiplier on the budget constraint and µimj ≥ 0 as the Lagrange
multipliers ensuring weak positivity on every choice of yimj .
Z
L=
1
0
qi′ m eσξi′ mj yi′ mj
i′ ∈Am
Z
− λj
X
1
X
0 i′ ∈A
m
!ρ !1/ρ
!
pi′ m yi′ mj dm − Ω
+
Z
1
X
µi′ mj yi′ mj dm
(B.1)
0 i′ ∈A
m
The first order condition with respect to yimj is
Sj
X
eσξi′ mj qi′ m yi′ mj
i′ ∈Am
!ρ−1
qim eσξimj = λj pim − µimj
λj > 0 µimj ≥ 0, ∀ i, m
(B.2)
(B.3)
with the following definition
Sj ≡
Z
1
0
X
qi′ m eσξi′ mj yi′ mj
i′ ∈Am
!ρ
dm
!1/ρ−1
(B.4)
Solving for Lagrange Multiply on Budget Constraint As discussed in section 3.3,
since this paper concentrates on partial equilibrium for a particular industry m, assume that
there is only one product in every m′ 6= m, qm′ = 1, and ξm′ j = 0 for all consumers. Define
45
κ≡
1
,
1−ρ
the elasticity of substitution between industries.49
To solve for λj and the price index, follow standard CES algebra and look at the first order
condition in equation B.2 for industry m′ 6= m where µm′ = 0. The i index is dropped since
there is only 1 good per category, and the j index is dropped since all j are identical outside
of industry m.
ρ−1
Sj ym
′ j = λj p m′
(B.5)
Take the ratio of two industries m′ and m̃ outside of the industry of interest m
ρ−1
ym̃
ρ−1
y m′
Rearrange and define κ ≡
pm̃
p m′
=
(B.6)
1
1−ρ
ym̃ = ym′ pκm′ p−κ
m̃
(B.7)
Multiply both sides by pm̃ and integrate for all m̃ 6= m
Z
Z
κ
pm̃ ym̃ dm̃ = ym′ pm′
[0,1]\{m}
[0,1]\{m}
1−κ
pm̃
dm̃
(B.8)
Recognize that industry m is infinitesimal, so the integrals are identical with or without
industry m. Therefore, the left hand side of the integral is the endowment. Define the price
index
1
Z 1
1−κ
1−κ
(B.9)
P ≡
dm̃
pm̃
0
Reorganize and define real income as Ω̄ ≡
Ω
P
pm′ −κ
P
Ω
ym′ = p−κ
m′ P 1−κ =
ρ−1
ym
=
′
pm′ ρ−1
Ω̄
P
Ω̄
(B.10)
(B.11)
Substitute equation B.11 into equation B.5 to find
pm′ ρ−1
Ω̄
P
=
λj
p ′
Sj m
(B.12)
Rearrange to get an expression for the Lagrange multiplier in terms the aggregate price index
real income
−κ
λj
= Ω̄P κ ∀ j
(B.13)
Sj
49
Useful expressions:
κ≡
ρ=
1
ρ
1
1−ρ
κ−1
κ
− 1 = −κ
=1−
−ρκ = 1 − κ =
(1 − κ)(ρ − 1) = ρ =
46
κ−1
κ
1
κ
ρ
ρ−1
Solution to Consumer’s Problem
Proof of Proposition 2. For the industry of interest, m, this section will prove that measure
1 of consumers will choose 1 product for each industry. Drop the m index for simplicity. The
proof strategy is to assume a single product is chosen, derive its intensive demand, and use
the non-negativity on the Lagrange multipliers for the other products to determine a set of
inequalities necessary for this choice to hold.
Assume that the chosen product has index i, so yij > 0 and yi′ j = 0 ∀i′ 6= i. The Lagrange
multipliers on the non-negativity constraints are: µij = 0 and µi′ j ≥ 0 ∀i′ 6= i.
Use the first order necessary conditions in equation B.2 to find the first order condition for
the chosen product i
ρ−1 σξij
Sj eσξij qi yij
qi e
= λj p i
(B.14)
Reorganize
eσξij qi yij
Reorganize and take everything to the
1
ρ−1
ρ−1
=
λj pi −σξij
e
Sj qi
(B.15)
= −κ power to solve for demand
yi (ξij ) =
Use the substitution from equation B.13
−κ −κ
λj
Sj
pi
qi
eσ(κ−1)ξij q1i
κ−1 σ(κ−1)ξij κ
P Ω̄
yi (ξij ) = p−κ
e
i qi
The inequality constraints for all other products i′ 6= i gives
ρ−1
Sj eσξij qi yij
qi′ eσξi′ j ≤ λj pi′ , ∀ i′ 6= i
(B.16)
(B.17)
(B.18)
Substitute from equation B.15
λj pi −σξij
e
qi′ eσξi′ j
Sj qi
≤
λj
p′
Sj i
(B.19)
Divide by qi′ and take logs
log
p i′
qi ′
− log( pqii ) ≥ σ(ξi′ j − ξij )
(B.20)
Finally, show that choosing multiple products only happens for measure 0 consumers. Without loss of generality, assume that there are only 2 products in the industry and that both
y1 > 0 and y2 > 0. The first order conditions are then
ρ−1 σξij
Sj eσξij qi yij + eσξi′ j qi′ yi′ j
qi e
= λj p i
(B.21)
ρ−1
σξi′ j
σξi′ j
σξij
q i′ yi′ j
= λ j p i′
(B.22)
q i′ e
Sj e qi yij + e
Take the ratio and then logs
ξi′ j − ξij = σ −1 (log
47
p i′
qi ′
− log( pqii ))
(B.23)
(B.24)
Hence, for a given set of prices and a given distribution of ξ, there may be an infinite
number of agents with the particular combination of these (ξi′ j , ξij ) given a and p vectors.
However, the solution is an affine subset of the ξ1 , ξ2 space. Given Assumption 1, the measure
of this affine subset is 0. The conclusion is that the set of agents who purchase multiple
products is measure 0 if prices are positive.
To summarize, for all but measure 0 of consumers, the optimal choice is
p
(B.25)
log qi′′ − log( pqii ) > σ(ξi′ j − ξij ), ∀ i′ ∈ Aj \ i
i
κ−1 σ(κ−1)ξij κ
P Ω̄,
=⇒ yi (ξij ) = p−κ
e
i qi
yi′ = 0 ∀ i′ 6= i
(B.26)
∂yi
i
i
Note that since κ > 1, ∂p
< 0, ∂y
> 0, ∂y
> 0. Also, if real income Ω̄, is kept constant,
∂qi
∂ξi
i
∂yi
then ∂P > 0, reflecting substitution away from other goods to this industry.
Appendix C
C.1
Customers and Total Demand Derivation
Number of Customers
Proof of Proposition 3. Define the number of customers as the measure of consumers choosing product i from equation B.25:
Z
p
xit (p) = 1{log qi′′ − log( pqii ) > σ(ξi′ j − ξij )|∀ i′ ∈ Aj \ i} dΨ(Aj , ξ1j , ξ2j , . . . ξN j ) (C.1)
i
Using the independence of A and ξj , the integral can be split using Fubini’s Theorem over
all awareness states that contain product i
Z
X
p
xit (p) =
(C.2)
ft (A) 1{log qi′′ − log( pqii ) > σ(ξi′ j − ξij )|∀ i′ ∈ A \ i}dG(ξj )
i
A|i
This this can be reorganized as
Z
X
p
xit (p) =
ft (A) 1{− σ1 log( pqii ) − σ1 (− log qi′′ ) > ξi′ j − ξij |∀ i′ ∈ A \ i}dG(ξj )
i
(C.3)
A|i
This form is of the same structure as many discrete choice problems. Given a particular
functional form of g(ξj ), this can be solved numerically or analytically. Under the assumption
that g(ξj ) is iid Gumbel, this can be solved in closed form following Anderson, De Palma,
and Thisse (1992).
xit (p) =

X
ft (A)
A|i
This is simplified further to
xit (p) =
X
A|i

e
P
i′ ∈A
e
− σ1
p −1/σ
i
ft (A)
P
48
qi
i′ ∈A

pi
)
qi

p i′ 
log
qi ′
− σ1 log(
p′ i
qi ′

−1/σ 
(C.4)
(C.5)
C.2
Total Demand
Proof of Proposition 4. Since intensive demand is a function of ξj , to find the total demand
for product i, the firm will sum up demand with ξij conditional on product i being chosen.
Define the total demand for product i given a price vector p as
Z
X
p
yit (p) =
ft (A) yi (ξij )1{log qi′′ − log( pqii ) > σ(ξi′ j − ξij )|∀ i′ ∈ A \ i}dG(ξj ) (C.6)
i
A|i
As before, this expression can be simplified using the independence between the A distribution and the ξ distribution. Use equation B.26 and take out constants
Z
X
p
−κ κ−1 κ
σ(κ−1)ξij
yit (p) = pi qi P Ω̄
ft (A) e
1{log qi′′ − log( pqii ) > σ(ξi′ j − ξij )|∀ i′ ∈ A \ i}dG(ξj )
i
A|i
(C.7)
For a particular A, find the total demand from agents conditional on having that awareness
Z
p
−κ κ−1 κ
yi (p, A) = pi qi P Ω̄ eσ(κ−1)ξij 1{log qi′′ /σ − log( pqii )/σ + ξij > ξi′ j |∀ i′ ∈ A \ i}dG(ξj )
i
(C.8)
For arbitrary g(ξ) this expression could be calculated numerically. For iid Gumbel distributions with pdf g(ξi ), the integral is solved in two parts: first use Fubini’s Theorem and the
independence of the ξj distributions to solve for the inner non-ξi variables, defined as ξ−ij ,
and then integrate with respect to the the ξi variable. This is the standard technique in the
derivation of the Logit probabilities.50
yi (p, A)
≡
−κ κ−1 κ
pi qi P Ω̄
Z
∞
e
σ(κ−1)ξij
−∞
Z
1{log
p i′
q i′
/σ −
log( pqii )/σ
′
+ ξij > ξi′ j |∀ i ∈ A \ i}dG−i (ξ−ij ) dGi (ξij )
(C.9)
The inner integral is the cdf of the joint distribution of ξ−ij . Use the cdf of the Gumbel along
each dimension other than i
p ′
Z ∞
p
− log( q i )/σ−log( q i )/σ+ξij
Y
′
i
i
=
eσ(κ−1)ξij
e−e
dGi (ξij )
(C.10)
−∞
i′ ∈A\i
Substitute in the pdf of the Gumbel
=
50
Z
∞
e
−∞
σ(κ−1)ξij
Y
e
−e
p ′
p
− log( q i )/σ−log( q i )/σ+ξij
i
i′
e−ξij e−e
i′ ∈A\i
The pdf and cdf of the univariate Gumbel are:
f (x) = e−x e−e
F (x) = e−e
49
−x
−x
−ξij
dξij
(C.11)
=
Z
∞
−∞
pi
qi
/σ − log( pqii )/σ = 0 allows taking the sum for all i′ , including i
!
X − log pi′ /σ−log( pi )/σ+ξij
qi ′
qi
eσ(κ−1)ξij exp −
e
e−ξij dξij
(C.12)
Recognize that log
i′ ∈A
Simplify by factoring the exponential
!
Z ∞
1/σ X −1/σ
p i′
−ξij pi
dξij
exp (−(1 − σ(κ − 1))ξij ) exp −e
=
qi
q′
i
−∞
(C.13)
i∈A
R∞
−x
If B > 0 and A > 0, then −∞ e−Ax e−Be dx = B −A Γ(A) where Γ(·) is the Gamma function.
The Gamma function occurs in this expression since E eσξ 6= 0 for the Gumbel distribution.
This distortion could be removed by a σ specific normalization in the average quality. Use
this expression,
!σ(κ−1)−1
σ(κ−1)−1
X p ′ −1/σ
σ
pi
i
= Γ(1 − σ(κ − 1)) qi
(C.14)
q′
i
i∈A
1
for finiteness of the integral. This ensures that
Here there is a requirement that σ < κ−1
the variance of the idiosyncratic preferences is not so large that the total demand explodes
as the demand from agents with large ξij are summed.
To find the total demand for product i given price vectors p, integrate over the distribution
of A states in the economy.
First define the constant
π̄ ≡ P κ Ω̄ Γ(1 − σ(κ − 1))
(C.15)
Then, from equation C.14 and equation C.7, the total demand is the sum of all awareness
states that contain product i

!σ(κ−1)−1 
X p ′ −1/σ
π̄ pi −1−1/σ X 
i

yit (p) =
(C.16)
ft (A)
qi
qi ′
qi
i∈A
A|i


−1/σ
pi

qi
π̄ X 


yit (p) =
(C.17)
ft (A) 1−σ(κ−1) 
pi
−1/σ


P
p′
A|i
i∈A
i
qi ′
As a test of the specification, assume that there is a single, monopolistic firm. Substituting
for the two possible awareness states, yi (pi ) = (1−ft (∅))π̄qiκ−1 p−κ
i . This is the same functional
form as the standard demand in a monopolistically competitive economy with an evolving
consumer population size. The dispersion comes into the Γ(1 − σ(κ − 1)) constant. If σ = 0,
then Γ(1) = 1, hence this returns the standard demand in a monopolistically competitive
economy with a growing number of consumers. Therefore, this specification nests models
with heterogeneous productivity in monopolistic competition. Note, however, that if σ > 0,
interpreting or estimating 1/ρ directly as the markup over marginal cost is distorted due
to the heterogeneity of consumers. Also note that the dispersion of the preferences σ only
appears in this formula through the Γ(·) distortion, confirming the intuition that there are
no sorting effects for monopolists, and the quality is simply the unconditional average.
50
C.3
Average Quality
Proof of Proposition 5. The average quality q̄it , is the average of the multiplicative quality
terms, weighted by the number of units consumed,51
Z
1 X
p
σξij
q̄it =
ft (A) qi e yi (ξij )1{log qi′′ − log( pqii ) > σ(ξi′ j − ξij )|∀ i′ ∈ A \ i}dG(ξj )
i
yit
A|i
(C.18)
Use the same methods as in Section C.2 to simplify and factor the integral
X
q̄it yit
=
ft (A)
−κ
pi
κ
P Ω̄
A|i
qi
Z
∞
−∞
exp (−(1 − σκ)ξij ) exp −e−ξij
1/σ X pi
qi
i∈A
p i′
qi ′
−1/σ
!
dξij
(C.19)
Use the same approach to calculate the integral,
= Γ(1 − σκ)
X
ft (A)
A|i
σκ−1
σ
X p ′ −1/σ
pi
qi
i
qi ′
i∈A
!σκ−1
(C.20)
Substitute in for yit from equation C.17 and simplifying
q̄it =
C.4
P
pi Γ(1 − σκ)
Γ(1 − σ(κ − 1) P
ft (A)
A|i
A|i
[ft (A)
P
P
i∈A
i∈A
p i′
qi ′
p i′
qi ′
−1/σ σκ−1
!
−1/σ σ(κ−1)−1
!
(C.21)
Symmetric Equilibrium
Notation To simplify notation, define the following vectors as functions of n: Jnκ K is a
κ
vector ofκ length
N + 1 where the first element is 0. For example, if N = 3 then Jn K =
κ
0 1 2 3 .
Proof of Proposition 6. Integration over the information sets requires summation over the
set of consumers who are aware of n firms, of which i is one. Given N firms, this mass is:
n ˆ
f (n). Simplify equation 15,
N t


N
−1−1/σ
X
pi

yit (pi , p̄) = π̄
(C.22)
 Nn fˆt (n) 1−σ(κ−1) 
−1/σ
−1/σ
n=1
+ (n − 1)p̄
pi
Simplification gives equation 26. Substitution of p̄ = p gives equation 27
51
An alternative approach would be to weight by the number of customers.
51
Proof of Proposition 7. Assume a symmetric price p̄, and use equation 26 with equation 22
to form the optimization problem
n
σ(κ−1)−1 o
K
(C.23)
p̄ = arg max (p − w) Nπ̄ p−1−1/σ fˆt · Jn p−1/σ + (n − 1)p̄−1/σ
p≥0
σ(κ−1)−1
Define the vector valued function g(p, p̄) ≡ Jn p−1/σ + (n − 1)p̄−1/σ
K. Assume existence and concavity as discussed in Technical Appendix E.1, and take the first order condition
with respect to p
∂g(p, p̄)
0 = (−p + w + wσ)fˆt · g(p, p̄) + p(p − w)σ fˆt ·
∂p
(C.24)
At a symmetric equilibrium p̄ = p, substitute and simplify
g(p, p) = p1/σ−κ+1 Jnσ(k−1) K
∂g(p, p)
p1/σ−κ Jnσ(k−1)−1 K
= 1−σ(k−1)
σ
∂p
(C.25)
(C.26)
ˆ
σ(κ−1)−1
K
0 = −p + w + wσ + (p − w)(1 − σ(κ − 1)) ftfˆ·Jn
·Jnσ(κ−1) K
t
(C.27)
Finally, solve for the price
ˆ
pt =
σ(κ−1)−1
K
1 + σ − (1 − σ(κ − 1)) ftfˆ·Jn
·Jnσ(κ−1) K
t
1 − (1 − σ(κ −
σ(κ−1)−1 K
ˆ
1)) ftfˆ·Jn
σ(κ−1) K
·Jn
t
w
(C.28)
For symmetric firms, recall the transformation in Proposition 6 from ft to fˆt conditional
on i being in the set as the sum of Nn fˆt (n). Substitute this into equation C.3 with symmetric
p and a,
σκ−1 PN
p −1/σ
n ˆ
n=1 N ft (n) n a
pΓ(1 − σκ)
q̄it =
(C.29)
σ(κ−1)−1 Γ(1 − σ(κ − 1) PN
p −1/σ
n ˆ
n=1 N ft (n) n a
Simplifying
i
PN h ˆ
σκ
f
(n)n
t
n=1
Γ(1 − σκ)
i
h
=a
P
Γ(1 − σ(κ − 1) N fˆ (n)nσ(κ−1)
t
n=1
(C.30)
Finally, define using the dot product and vector notation
q̄t =
Γ(1 − σκ)
fˆt · Jnσκ K
Γ(1 − σ(κ − 1)) fˆt · Jnσ(κ−1) K
(C.31)
For the other results,
ˆ
σ(κ−1)−1
K
1. Note that ftfˆ·Jn
< 1 and by Assumption 2, σ(κ − 1) < 1. Therefore, pt is
σ(κ−1) K
t ·Jn
decreasing in σ.
2.
fˆt ·Jnσ(κ−1)−1 K
fˆt ·Jnσ(κ−1) K
is weakly decreasing in N and hence pt is weakly decreasing in N .
52
Appendix D
D.1
Aggregation and Asymptotics
Pricing Limits
Proof of Proposition 8. The t = 1 price, i.e., after consumers have either 1 or 0 firms in their
information set
n
σ(κ−1)−1 o
(D.1)
p =arg max (p̃ − w)C p̃−1−1/σ p̃−1/σ
p̃≥0
arg max (p̃ − w)Cp−κ
(D.2)
p̃≥0
=
w
ρ
(D.3)
For the N asymptotics,
fˆt ·Jnσ(κ−1)−1 K
σ(κ−1) K
t→∞ fˆt ·Jn
lim
=
1
N
(D.4)
Use equation 29, substitute in equation D.4 and take limits
p∞ =
w(N − 1 + σ(N − 1 + k))
N − 1 + σ(k − 1)
(D.5)
Proof of Proposition 9. As a 2nd order approximation, E [f (X)] ≈ f (E [X]) + f
Using this,
α(α − 1)
E [X]α−2 V [X]
2
√
V[X]
Using the coefficient of variation, CV [X] ≡ E[X]
E [X α ] ≈ E [X]α +
= E [X]
α
α(α − 1)
1+
CV [X]2
2
′′ (E[X])
2
V [X].
(D.6)
(D.7)
For the zero-removed Poisson distribution with parameter λ are,
λ
1 − e−λ
1
1 − (1 + λ)e−λ
CV [n̂]2 =
λ
(D.8)
E [n̂] =
(D.9)
Hence, an approximation for fractional moments is
α −λ
1
E [n̂α ] ≈ 1−eλ−λ
1
−
(1
+
λ)e
1 + α(α−1)
2
λ
(D.10)
And the ratio,
E [n̂α−1 ]
−λ 1 +
−1
1
−
e
≈
λ
E [n̂α ]
1+
53
α(α−1) 1
2
λ
α(α+1) 1
2
λ
1 − (1 + λ)e−λ
(1 − (1 + λ)e−λ )
(D.11)
Note that for large λ, i.e. for large t in a Poisson counting process,
E [n̂α−1 ]
≈ λ−1 ,
α
E [n̂ ]
for large λ
(D.12)
Using equation D.12 with equation 29, for large t, the prices are
pt ≈
1 + σ − (1 − σ(κ − 1))(θt)−1
w
1 − (1 − σ(κ − 1))(θt)−1
It can be shown that the markup converges towards 1 + σ at the rate of 1/t as is consistent
with the more general results in Gabaix, Laibson, Li, Li, Resnick, and de Vries (2012).
Figure 20 shows examples of these approximations for large N .
1.8
Exact
1st Order
2nd Order
1.7
Et [n̂.5 ]
1.6
1.5
1.4
1.3
1.2
1.1
1
0
2
4
6
8
10
12
t
14
16
18
20
Figure 20: Approximations to Et [n̂.5]
Proof of Proposition 10. The proof will solve for a modified price index using a composite
good, derive labor supply and demand, and then solve for the equilibrium income.
Price Index Using the same reorganization of industries as in equation 40, the budget
constraint is
Z
Z ∞
∞
X
ˆ
ˆ
φ(t, a) 1 − f (a, 0)
fa (n) p(a)y(a, ξ)g(n) dξda
(D.13)
Ω(t) =
0
n=1
For the Gumbel distribution, the order statistic for the maximum of n draws is
g(n) (ξ) = ne−ξ e−ne
−ξ
(D.14)
From this, the following integral is calculated using
Z
eσ(κ−1)ξ g(n) (ξ)dξ = Γ(1 − σ(κ − 1))nσ(κ−1)
54
R∞
−∞
−x
e−Ax e−Be dx = B −A Γ(A)
(D.15)
The price index and Lagrange multiplier are time dependent through the distribution
φ(t, a). To find the price index, use age heterogeneity in industries with the same approach
of equation B.5. take the ratio of the first order condition in equation B.2, using the optimal
choice of one purchase per industry.
Rewrite equation B.2 with age heterogeneity and use the result of purchasing from 1 firm
per industry
S(t)y(a, ξ)ρ−1 = e−ρσξ λ(t)p(a)
(D.16)
Choose two industries with potentially different ages, take the ratio, multiply by p(ã), and
reorganize
˜ = y(a, ξ)eσ(1−κ)ξ p(a)κ p(ã)1−κ eσ(κ−1)ξ̃
p(ã)y(ã, ξ)
(D.17)
Integrate by weighting by the age of each industry, and use the order statistic for each
industry within a cohort
Z
e
σ(1−κ)ξ
∞
0
Z
∞
X
ˆ
ˆ
φ(t, ã) 1 − f (ã, 0)
f (ã, n) p(ã)y(ã, ξ)g(n) (ξ)dξda =
(D.18)
n=1
y(a, ξ)p(a)
κ
Z
∞
p(ã)
1−κ
0
φ(t, ã) 1 − fˆ(ã, 0)
∞
X
fˆ(ã, n)
n=1
Z
eσ(κ−1)ξ g(n) (ξ)dξda
(D.19)
The left hand side is the budget constraint. Define the price index as
P (t)
1−κ
=
Z
∞
p(ã)
1−κ
0
Z
∞
X
ˆ
ˆ
f (ã, n) eσ(κ−1)ξ g(n) (ξ)dξda
φ(t, ã) 1 − f (ã, 0)
(D.20)
n=1
Use equation D.15 to simplify this expression
P (t) =
Γ(1 − σ(κ − 1))
Z
∞
0
1/(1−κ)
σ(κ−1)
1−κ
ˆ
φ(t, a) 1 − f (a, 0) Ea n̂
p(a) da
(D.21)
Composite Good To solve for Y (t), simplify equation 40,
ρ
Y (t) =
Z
∞
0
Z
∞
X
fˆa (n)
φ(t, a) 1 − fˆ(a, 0)
eσξ y(a, ξ)
n=1
Substitute from equation B.26 with time varying price index and simplify
Z
Z
ρ
σξ
κ−1
1−κ ρ
e y(a, ξ)g(n) (ξ) dξ = P (t) p(a) Ω̄
eσ(κ−1)ξ g(n) (ξ)dξ
55
ρ
g(n) (ξ)dξda
(D.22)
(D.23)
Use equation D.15
= P (t)κ−1 p(a)1−κ Ω̄(t)ρ Γ(1 − σ(κ − 1))nσ(κ−1)
(D.24)
Substitute for the price index P (t) to find the relationship
Y (t) = PΩ(t)
(t)
(D.25)
This expression is a standard result from CES preferences and monopolistic competition.
From these aggregation results, for a given P (t), w(t), and Ω(t), rewrite the period utility
with γ > 1 and ν > 0 as
o
n
η
1−γ
1+1/ν
1
Y
(t)
−
H
U (t) = max
1−γ
1+1/ν
0<C, 0<H<1
s.t. Y (t)P (t) ≤= w(t)H(t) + Π(t) = Ω(t)
Aggregate Profits Define a constant and integrate with the same use of D.15,
Z ∞
Π̄(t) ≡
Γ(1 − σ(κ − 1))(1 − fˆ(a, 0)(p(a) − 1)p(a)−κ Ea n̂σ(κ−1) φ(t, a)da
(D.26)
(D.27)
0
−1/(γν+1)
Ω(t) = η ν P (t)ν(1−γ) (1 − P (t)κ−1 Π̄(t))
Z ∞
(1 − fˆ(a, 0))Γ(1 − σ(κ − 1))P (t)κ−1 Ω(t)p(a)−κ Ea n̂σ(κ−1) φ(t, a)da
H(t) =
(D.28)
(D.29)
0
Π(t) = Π̄(t)P (t)κ−1 Ω(t)
(D.30)
Hence, the budget constraint in equation D.26 is
Ω(t) = H(t) + Ω(t)P (t)κ−1 P¯i(t)
(D.31)
Labor Demand As each firm has unit productivity, the labor demand is the total physical
output
Z ∞
Z
∞
X
ˆ
ˆ
H(t) ≡
φ(t, a) 1 − f (a, 0)
fa (n) y(a, ξ)g(n) (ξ)dξda
(D.32)
0
n=1
Use equation B.26
= P (t)
κ−1
Ω(t)
Z
∞
0
Z
∞
X
−κ
φ(t, a) 1 − fˆ(a, 0) p(a)
fˆa (n) eσ(κ−1) g(n) (ξ)dξda (D.33)
n=1
From equation D.15, define
Z ∞
H̄(t) =≡
Γ(1 − σ(κ − 1))(1 − fˆ(a, 0)p(a)−κ Ea n̂σ(κ−1) φ(t, a)da
(D.34)
0
Then,
H(t) = P (t)κ−1 Ω(t)H̄(t)
(D.35)
And labor demand can be related to aggregate profits through
H(t) =
H̄(t)
Π(t)
Π̄(t)
(D.36)
56
Labor Supply Choose w(t) as the numeraire. Assume that the endogenous labor supply,
H(t), is interior and take the first order conditions from equation 41
ηH(t)1/ν = λ(t)w(t)
Y (t)−γ = λ(t)P (t)
(D.37)
(D.38)
Take the ratio, use w(t) = 1, and Y (t) = Ω(t)/P (t), and reorganize
H(t) = η −ν Ω(t)−γν P (t)ν(γ−1)
(D.39)
Use the budget constraint in equation D.31
Ω(t)(1 − P (t)κ−1 Π̄(t)) = η −ν Ω(t)−γν P (t)ν(γ−1)
(D.40)
Solve for Ω(t)
−1/(γν+1)
Ω(t) = η ν P (t)ν(1−γ) (1 − P (t)κ−1 Π̄(t))
(D.41)
Proof of Proposition 11. If the econometrician assumes that firms are monopolistically competitive, then the price is a constant markup over marginal cost. If w is chosen to be the
1
renumeraire and a aggregate shock to productivity of Z(t) is assumed, then p(t) = ρZ(t)
gardless of age.
P (t) =
=
Z
p(t)
1−κ
φ(t, a)da
1
M (t)1/(1−κ)
ρZ(t)
1/(1−κ)
(D.42)
(D.43)
If P (t), or a misspecified P (t) is observed directly, then
Z(t) = ρP (t)M (t)1/(κ−1)
If H(t) is observed, then from the first order condition
−1
P (t)−1 = ηH(t)1/ν Y (t)γ
(D.44)
(D.45)
Use the misspecified price index in equation D.42 and that Y (t) = (Z(t)H(t))1/ρ
ρZ(t)M (t)1/(κ−1) = ηH(t)1/ν Z(t)γ/rho H(t)γ/rho
(D.46)
Solve for Z(t) given H(t) and M (t)
1
Z(t) =
γ
1
H(t) ν + ρ M (t) 1−κ η
ρ
57
ρ
!− γ−ρ
(D.47)
Appendix E
E.1
Alternative Laws of Motion
Contagion and Word-of-Mouth Diffusion
Following a version of Mahajan, Muller, and Bass (1990), the continuous time law of motion
will include contagion from consumers with awareness of a product in the industry to those
unaware of any product. This law will follow the continuous time Markov chain, and include
the simplification that no diffusion occurs between users of products.
The transition rate for n(t) ≡ |Aj (t))| is the limit as ∆ → 0 of
ˆ
Pr {n(t + ∆) = 1 |n(t) = 0 } = θx + θd (1 − f (t, 0)) ∆ + o(∆)
(E.1)
Pr {n(t + ∆) = n(t) + 1 |n(t) } = θ∆ + o(∆)
(E.2)
where o(∆) is using order notation. With this Markov chain, the transition rates from n̂ = n
for n ≥ 1 are identical to those in 3.7, and maintains the Poisson counting process for large
N
For consumers entering the market, following the terminology of Mahajan, Muller, and
Bass (1990), θx > 0 represents the spread of awareness through external influences, while
θd > 0 parametrizes the spread of awareness through diffusion from existing customers. From
an initial state of no awareness, this process yields the following process for fˆ(t, 0)
dfˆ(t,0)
ˆ(t, 0) 1 − θx − θd (1 − fˆ(t, 0))
=
f
(E.3)
dt
This is a first-order non-linear ODE. With the initial condition fˆ(0, 0) = 1, the solution is
1 − fˆ(t, 0) =
θx + θd
θd + θx e(θd +θx )t
(E.4)
(E.5)
This process will show an inflection point as the diffusion from existing customers begins to
overtake the external influence of awareness growth. See Figure 21.
58
1
1.4
fˆ(t, 0)
0.8
p(t)
1.3
0.6
1.2
0.4
1.1
0.2
0
0
5
10
15
t
20
25
1
30
1
1
0.8
0.8
0.6
0.6
0.4
0.4
Y (t)
0.2
0
0
5
10
15
t
20
25
0
5
10
0
30
0
5
10
0.96
0.94
V (t)
0
5
10
15
t
20
25
30
Figure 21: Model with Bass-Style Contagion
59
25
30
Π(t)
1
0.9
20
0.2
0.98
0.92
15
t
15
t
20
25
30
Appendix F
F.1
Empirical Evidence
Description and Notes on the Data
The primary data sources are:
• Productivity, output, and price index data comes from the NBER-CES Manufacturing
Industry Database, Bartelsman, Becker, and Gray (2000) with data at yearly intervals
from 1958 to 2009.
http://www.nber.org/nberces/nbprod96.htm
• Data on concentration and the number of firms comes from the Census. It is by 4 digit
SIC code from 1947 to 1992, sampled every 5 years.
http://www.census.gov/epcd/www/concentration92-47.xls The sampling of every
5 years is the main reason for the jagged concentration and number of firms in the
industry lifecycle in Figure 2.
• BEA NIPA, http://www.bea.gov/iTable/iTable.cfm?ReqID=9&step=1, is used for
GDP and aggregate price deflators.
• Valuations come from http://wrds-web.wharton.upenn.edu, Wharton Research Data
Services, Center for Research in Security Prices (CRSP). The data is monthly from 1925
to 2012 by 4 digit SIC code. I find the market cap for each firm in December, then
aggregate across all firms by 4 digit SIC code.
Some notes on the calculations and data series:
• While the census database has tried to keep things consistent, interpretation or adding
of new SIC codes can cause reclassification of firms over time. To try to filter out
results which are likely due to this effect, industries where the market capitalization
shift between years of > 100% have been removed.
• For productivity, use the 5 factor TFP index.
• For concentration, use the number of firms and the market share of the top 20 firms,
though the results appear robust to Herfindahl and other concentration indices.
• Birth year is defined as when an SIC first shows up in the either the CRSP or productivity database.52 The distribution of peak ages and years is described in Figure 22.
The discrete jumps at certain years are partially due to the removal of suspicious data
as discussed above. A histogram of the data to see the density of observations relative
to the normalized maximum value is given in Figure 23.
• The peak of the industry is calculated from the real market capitalization. One concern
is that a correlation due to aggregate stock market movements may cause significant
52
Due to the lag between SIC code created and when firms are classified within them, an alternative was
tested as a robustness check where the “birth” date when the industry hit various proportions of its maximum
output. The features of the data appeared to be robust to these different definitions.
60
auto-correlation of industry peaks. This does not appear to be the case, as can be seen
in Figure 22.53
• I calculate markups as proportional to the inverse share of value-added in wages, in
the same sense as Hall (1988). This is a crude approximation to markup, but the
results were found to be robust to other definitions of markup including the data from
Nekarda and Ramey (2011). The summary statistics of the price-cost markup are given
in Table 5. The drop to averaging is 213 industries was due to censoring industries
with negative minimum markups. The unconditional average is similar.
After transformation and cleaning, there is data for 439 unique SIC codes for the panel data
tests, and 333 SIC with concentration data.
F.2
Calibration
The parameters for the experiments are discussed in Table 3. The calibration of σ comes
from the asymptotic markup in a state of full awareness, and κ comes from the maximum
markup, which corresponds to monopolistic pricing and minimum awareness, where Table 5
has descriptive statistics of this data. The value of θ is calibrated from the annual growth
rate of employment for industries less than half of their peak age.54
Variable
σ
Value
0.2
1/ρ − 1
.4
θ
.15
δ
.123
β
ν
γ
[.9, .99]
.75
[1, 5]
Description
Minimum industry markup = σ from Proposition 8. Calculated as the
average minimum markup from NBER-CES Manufacturing Industry
Database.
Maximum industry markup = 1/ρ − 1 from Proposition 8. Calculated
as the average maximum markup from NBER-CES Manufacturing
Industry Database. See Proposition 8.
From the annual growth rate of employment in industries of age ≤ .5
of the normalized peak age. This is 18% in the NBER-CES Manufacturing Industry Database
1 − e−δ = .116, for the exponential parameter of the firm age distribution where 11.6% is the firm entry/death rate per annum from the
US Small Business Administration. See Luttmer (2007).
Targeting different annual interest rates, and aggregate P/E ratios
From Chetty, Guren, Manoli, and Weber (2011)
Broad range of elasticity of intertemporal substitution
Table 3: Parameter Calibration
53
As a robustness check, I also tested the results with peak market cap relative to aggregate stock market
valuation, which should remove those shifts. The results were even more stark, with a substantially lower
drop in relative market cap after the peak.
54
These are crude estimates intended to provide baseline values for these simple examples. The degree of
differentiation, σ, is heterogeneous among industries, and its calibration is only a bound since it assumes
both full awareness and a large number of firms.
61
(1)
Price
-0.00271
[-0.0555,0.0500]
age
age2
(3)
Output
-0.0297
[-0.113,0.0536]
(4)
Markup
-0.0245
[-0.0510,0.00212]
-0.00194
0.00474
-0.00470
0.00117
[-0.00976,0.00588] [-0.00179,0.0113] [-0.0187,0.00928] [-0.00317,0.00551]
Concentration
N
(2)
TFP
-0.0318
[-0.0718,0.00818]
-0.0413
[-0.113,0.0307]
1474
0.0877
[0.0348,0.141]
1973
-0.0756
[-0.171,0.0193]
1222
0.0837
[0.0485,0.119]
1973
95% confidence intervals in brackets
Table 4: Results from Panel Data Regression
F.3
Panel Data Regressions
As everything is rescaled, these are summary statistics and I give only an interpretation of
the sign and not the magnitude. The goal is not to estimate the parameters, but rather to
look at patterns in the data without assuming a particular model.
All regressions controls for year effects. The typical issue that comes from simultaneously
estimating age and year effects is avoided, as the stretching of data ensures they are no longer
collinear. These regressions are naive in the sense they do not control for endogeneity of the
concentration variable.
Note that TFP is decreasing in age. This is partially due to controlling for year effects.
It appears that TFP growth is due in a large part to a economy wide trend, and hence
spillovers may be occurring across industries.
Table 5: Summary Statistics for Min and Max Markup by Industry
Variable Mean Std. Dev. Min. Max.
min pcm
0.193
0.077
.018
.501
max pcm 0.391
0.099
.15
.87
N
208
62
.04
0
.01
.02
Density
.03
.04
.03
.02
Density
.01
0
0
20
40
60
80
1940
1960
Age at Peak
1980
2000
2020
Year of Peak
0
0
1
5
Density
Density
2
10
3
4
15
Figure 22: Histogram of Peak Age and Year
.2
.4
.6
.8
real_price normalized by its maximum value’
1
0
.2
1
.2
.4
.6
.8
con20 normalized by its maximum value’
1
0
0
1
1
Density
Density
2
2
3
4
3
0
.4
.6
.8
rel_mc normalized by its maximum value’
0
.2
.4
.6
tfp5 normalized by its maximum value’
.8
1
Figure 23: Histogram of TFP, Price, Concentration, Market Cap Relative to Maximum Value
63