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Federal Reserve Bank of Chicago
Competition in Large Markets
Jeffrey R. Campbell
REVISED December, 2006
WP 2005-16
Competition in Large Markets
Jeffrey R. Campbell∗
December, 2006
Abstract
This paper evaluates the simplifying assumption that producers compete in a large
market without substantial strategic interactions using nonparametric regressions of
producers’ choices on market size. With such atomistic competition, increasing the
number of consumers leaves the distributions of producers’ prices and other choices
unchanged. In many models featuring non-trivial strategic considerations, producers’
prices fall as their numbers increase. I examine observations of restaurants’ sales, seating capacities, exit decisions, and prices from 222 U.S. cities. Given factor prices and
demographic variables, increasing a city’s size increases restaurants’ average sales and
decreases their exit rate and prices. These results suggest that strategic considerations
lie at the heart of restaurant pricing and turnover.
JEL Classification: L11, L81
Keywords: Atomistic Competition, Market Size, Free Entry, Exit, Nonparametric Test.
∗
Federal Reserve Bank of Chicago and NBER. I am grateful to Gadi Barlevy, Meredith Crowley, Hugo
Hopenhayn, and Ruilin Zhou for their helpful comments. The National Science Foundation supported my
research on this topic through grant SBR-0137048 to the NBER. Please address correspondence to Jeffrey
Campbell, Economic Research, Federal Reserve Bank of Chicago, 230 South LaSalle St., Chicago, IL 60604.
e-mail: jcampbell@frbchi.org
1
Introduction
Observations of producers’ actions from firm registries or national statistical agencies typically lack an accompanying description of their strategic environments. This unfortunate
fact tempts one to assume that producers compete anonymously in a large market, but casual observation nearly always suggests some scope for strategic interaction between firms.
This paper quantifies this informal suspicion using nonparametric regressions of producers’
choices on market size. The data come from 222 U.S. cities’ restaurant industries and are
reported in the 1992 Census of Retail Trade. Under the null hypothesis of atomistic competition, market size has no impact on these decisions. This result is familiar from highly stylized
models of Chamberlinian monopolistic and perfect competition, and this paper proves it in a
very general framework without substantial restrictions on the market demand system, producers’ cost functions, or the variables over which they compete. It requires nonparametric
regressions to display no influence of market size on producers’ choices given a sufficiently
rich set of control variables. In fact, restaurants in larger cities have lower prices, exit less
frequently, and have greater sales revenues. Even if one finds atomistic competition implausible ex ante, the theory and regressions together quantify how the strategic environment
influences producers’ observable choices. Such quantification is essential for extending the
domain of strategically-oriented empirical industrial organization to large markets.
The analysis rests on a nonparametric free-entry model. Potential producers make entry
choices and then compete across a possibly large number of variables; such as price and
advertising. A producer’s profit depends only on the distribution of its rivals’ actions and
not on any particular rival’s choices. This allows the transformation of a free-entry equilibrium for a given market size into one for a market twice as large with double the number of
producers and the same distribution of their observable actions. Of course, the distribution
of producers’ actions could differ across large and small markets even without substantial
strategic interaction if the production technology and consumer tastes systematically change
with market size. I eliminate dependence of an individual’s demand and producers’ costs on
1
market size in the model by assumption; and the regressions control for differences in production possibilities and consumer tastes across U.S. cities with factor prices and demographic
measures.
Previous contributions to international trade and industrial organization have recognized
the importance of oligopolistic competition in large markets, but this recognition has taxed
the desire to work with analytically tractable models. Standard Chamberlinian monopolistic
competition cannot capture the idea that increasing market size makes competition “fiercer”
by increasing the number of producers, but true oligopoly models with strategic interaction
raise difficult issues of dynamic game theory that are not necessarily central to a particular
author’s problem. This difficulty has led some authors to use a model with a continuum of
producers and goods due to Ottaviano, Tabuchi, and Thisse (2002), in which the elasticity
of any given good’s demand decreases with the number of goods offered even though no two
producers compete head-to-head. Campbell and Hopenhayn (2005) show that such a model
predicts firm size to increase with market size, because the product of the falling markup
with firms’ average sales must equal the constant fixed cost of entry. Their empirical results
confirm this relationship for a large number of U.S. retail trade industries. Asplund and
Nocke (2006) and Nocke (2006)find that an otherwise-standard model of industry dynamics
with this specification predicts that producer entry and exit rates increase with market size.
Asplund and Nocke’s observations of Swedish hairdressers and Syverson’s (2004) observations
of U.S. concrete industries support this conclusion. I find the opposite to be true for U.S.
restaurant industries. Apparently, the dynamic aspects of oligopolistic interaction that this
specification omits are unimportant for the industries examined by Asplund and Nocke and
Syverson, but they substantially lower restaurants’ exit rates in larger markets.
The approach to evaluating competition in large markets I advocate in this paper has
one limit worth noting. A model in which oligopolists successfully collude and keep markups
at their monopoly level but do not deter entry will replicate the scale invariance of atomistic competition. That is, the test has no power to reject the null in favor of the specific
2
alternative of collusion with free entry. The empirical results of this paper as well as those
of Campbell and Hopenhayn (2005) and Yeap (2005) indicate that this lack of power is not
a practical problem for work with U.S. data.
The remainder of this paper proceeds as follows. The next section sets the stage for the
analysis with an empirical examination of how restauranteurs’ decisions vary with market
size. Section 3 then provides a structural interpretation of these nonparametric results using
the general model of atomistic competition. Section 4 relates this paper’s results to those
from the relevant literature, and Section 5 offers some concluding remarks.
2
Competition among Restaurants
To motivate this paper’s analysis, consider the U.S. Restaurant industry. The U.S. Census
questions the population of restaurants about their sales, cuisine, and pricing decisions every
five years when creating the Economic Census. These observations allow researchers to
address fundamental questions about the process of business formation, growth, and exit;
but they contain only little information about the potential for strategic interactions. This
is particularly the case for restaurants in cities, who have a great scope for differentiating
themselves by location and cuisine.
The hypothesis that the firms in this data set compete atomistically can greatly simplify
its analysis, because each firm’s actions can be cast as the outcome of a single-agent decision
problem. This simplification could come at a high price if strategic interaction is a first-order
feature of competition, so I desire a simple procedure that can evaluate it before proceeding
with a more complicated analysis.
Campbell and Hopenhayn (2005) use a symmetric model of oligopolists with constant
marginal cost to build such a procedure. They note that oligopolists’ average sales must rise
with market size if their markups fall with additional entry, because they must recover the
same fixed cost with a lower markup by selling more. Hence, modelling an industry as a
3
collection of oligopolies seems promising if we see average sales rising with market size. The
two shortcomings of their procedure are its reliance on a stylized model of competition and
its exclusive focus on producers’ average sales. This paper constructs a very general model
of the null hypothesis which implies that all observable producer decisions are invariant to
market size. The following description of how U.S. restauranteurs’ actions vary with market
size provides this theoretical analysis with a concrete empirical context.
2.1
Data
For this paper, I use observations from the 1992 Census of Retail Trade for the same sample of
MSAs examined by Campbell and Hopenhayn (2005). The volume RC92-S-4, “Miscellaneous
Subjects”, reports the number of restaurants operating at any time during 1992 and at the
end of that year. These observations immediately yield one measure of the annual exit rate.
This volume also reports restaurants’ average seating capacities for each MSA, the sales of all
restaurants and of those operating at the end of the year, and the fraction of restaurants with
typical meal prices greater than or equal to $5.00. Although the Census records information
about each restaurant’s cuisine, this information is not disclosed publicly by MSA.1
From these observations, we construct four variables of interest. The first summarizes
firms’ pricing decisions. Denote the fraction of restaurants charging a typical meal price of
$5.00 or more with S($5.00), and consider its logistic transformation
L($5.00) ≡ ln(S($5.00)/(1 − S($5.00))
This is the logarithm of the ratio of “high priced” restaurants’ share of the population to
that of their “low priced” counterparts. Figure 1 plots this variable against the demeaned
logarithm of MSA population. The observations corresponding to the smallest and largest
MSA’s (Enid, OK and Atlanta, GA) are labelled, as are the observations with extreme values
1
It would be desirable to examine more recent observations. Unfortunately, the Census has not published
MSA level observations of these variables from the two most recent Economic Censuses.
4
of the log relative market share. The median value of S($5.00) across the sample’s MSA’s is
0.67. The Census reports that only 13 percent of restaurants in Rocky Mount, NC charge
$5.00 or more for a meal, and it reports that 96 percent of restaurants charge $5.00 or more in
both Longview-Marshall TX and Jackson, MS. Aside from these three outliers, the minimum
and maximum values of S($5.00) are 0.32 and 0.92. The correlation between the log relative
market share and MSA population equals 0.09.
The second variable of interest measures one aspect of industry dynamics, the exit rate.
I constructed this by dividing the number of firms operating at some time of the year but
not at the end of the year by the total number of firms to operate in that year. The plot of
this against MSA log population in Figure 2 shows a negative correlation. The exit rate for
Enid, OK is very close to the maximum observed, 19 percent, while that for Atlanta, GA is
close to the median across all MSA’s, 10.3 percent. The correlation between these variables
equals −0.11.
The other two variables of interest both measure average restaurant size, restaurants’
average revenue and average seating capacity. This average revenue variable differs differs
from that used by Campbell and Hopenhayn only because it excludes restaurants not operating at the end of the year. Figures 3 and 4 plot these variables against MSA population.
The strong positive association between MSA population and sales revenue documented by
Campbell and Hopenhayn is evident in Figure 3. Figure 4 reveals little correlation between
MSA population and average seating capacity.
2.2
Regression Results
Let Yi denote the value of one of these four measures of restaurateurs’ actions for MSA i,
and use Si and Wi to represent that MSA’s population and a vector of control variables
that includes relevant factor prices and consumer demographics. The factor prices account
for larger cities’ higher cost of commercial space and wages and lower cost of advertising
per consumer exposure. The demographic variables control for differences in preferences
5
associated with income, race, and education that could shift the the nature of producers’
products and thereby indirectly influence their observable decisions. These control variables
are identical to those used in Campbell and Hopenhayn). The regression of Yi on ln Si and
Wi is
Yi = m(ln Si , Wi ) + Ui .
Here, m(·) is not restricted to a particular functional form.2
The curse of dimensionality makes the estimation of m(ln S, W ) infeasible. However, it
is still possible to test the hypothesis that its dependence on ln S is trivial using estimates
of the regression function’s density-weighted average derivatives. These are
(1)
δS
δW
∂m (S, W )
≡ E
f (ln S, W ) /E [f (ln S, W )]
∂ ln S
∂m (S, W )
≡ E
f (ln S, W ) /E [f (ln S, W )] ,
∂W
where f (ln S, W ) is the joint density function of ln S and W across markets and expectations
are taken with respect to the same joint density function. Powell, Stock, and Stoker (1989)
provide a simple instrumental variables estimator of δS and δW which converges to the true
√
parameter values at the parametric rate of N . If market size does not directly impact
producers’ decisions, then δS = 0.
For the four measures of restaurateurs’ actions, Table 1 reports the estimated values
of δS and δW along with consistent estimates of their asymptotic standard errors. Before
estimation, the elements of W were scaled by the standard deviation of ln S, which is 0.86 in
this sample. Powell et al.’s (1989) estimator requires a first-stage nonparametric estimation
of ∂f (ln S, W )/∂ ln S and ∂f (ln S, W )/∂W . The estimates reported here are based on the
tenth-order bias-reducing kernel of Bierens (1987) and use a bandwidth equal to 2. To
increase the precision of the estimates’ reports, all entries in the table and in the text have
been multiplied by 100.
2
In the case where Yi = ln(Si ($5.00)/(1 − Si ($5.00)), this specification for the regression function is
equivalent to assuming that Si ($5.00) = em(ln Si ,Wi )+Ui /(1 + em(ln Si ,Wi )+Ui ).
6
The estimate of δS for the regression of L($5.00) equals −12.90 and is statistically significant at the 5 percent level. Thus, restaurants in larger markets charge lower prices given
factor costs. To gauge the magnitude of this coefficient, consider an MSA with S($5.00) at
the median level of 0.67. Set all of the elements of W equal to their means and consider
increasing S by one standard deviation. If we assume that ∂m(ln S, W )/∂ ln S is constant,
then such an increase in ln S decreases S($5.00) to 0.65. The hypothesis that increasing market size lowers prices permeates empirical industrial organization, but to date only Syverson
(Forthcoming) has verified that this is so for a particular industry. This finding that typical
meal prices fall with market size complements his results.
The coefficients on two of the factor costs, commercial rent and the retail wage, are
positive. They are both statistically significant at the 10 percent level, so the regression
confirms the basic intuition that prices rise with factor costs. The third factor price, the cost
of purchasing 1, 000 advertising exposures in a Sunday newspaper, has a negative coefficient
which is statistically significant at the 10 percent level. Perhaps high advertising costs allow
producers to segment the market more effectively, thereby raising prices. Certainly, the effect
of advertising costs on restaurant prices merits further investigation.
The estimate of δS for the exit rate is also negative, −0.77, and statistically significant
at the 5 percent level. This implies that doubling S decreases restaurants’ exit rate by
0.53 percentage points. As Campbell and Hopenhayn (2005) document, an increase in ln S
strongly raises restaurants’ average revenue. The estimated coefficient is 4.68, and it is statistically significant at the one-percent level.3 The final dependent variable is the logarithm
of average seats per restaurant. The estimated coefficient is positive, 2.07, but its standard
error equals 1.99. Hence, these observations are uninformative about whether the increase in
average revenue per restaurant arises from increased capacity utilization or increased average
3
This estimate differs greatly from that reported by Campbell and Hopenhayn (2005) for the iden-
tical regression with nearly the same sample.
The discrepancy between the two reflects an error in
Campbell and Hopenhayn’s calculations. An erratum to that paper available at http://www.nber.org/˜
jrc/marketsizematters corrects that error.
7
capacity. Nevertheless, the estimates in Table 1 clearly indicate that important decisions of
restauranteurs vary systematically with market size.
The estimates in Table 1 depend on the particular measure of market size (population)
and the bandwidth choice. Table 2 examines the robustness of the estimates of δS to these
choices. Its first column reproduces the estimates from the first row of Table 1, and its next
two columns report alternative estimates based on measuring market size with geographic
population density and the number of housing units. Using either of these alternatives brings
the estimate of δS for the regression of L($5.00) closer to zero. It equals −8.91 and has a
p-value of 12 percent with population density, and it equals −11.93 with a p-value of 5.4
percent using housing units. All other inferences are invariant to changing the measure
of market size. The final two columns of Table 2 report estimates based on changing the
bandwidth h from its baseline value of 2 to either 1 or 3. Changing the bandwidth moves the
estimated standard errors in the opposite direction. Otherwise, the estimates are unaffected.
The only inference to change relative to the baseline specification is in the regression of
L($5.00). When h = 1, the p-value for δS equals 10.3 percent.
I have undertaken three other checks of these estimates’ robustness worth mentioning
here. First, I have estimated all of the regression equations using ordinary least squares. The
estimated coefficients are similar to the nonparametric estimates of δS . The only notable
change in inference regards the coefficient in the regression of L($5.00). Its estimate drops to
−9.79, and its p-value rises to 6.6 percent. Second, the Census reports the share of restaurants
charging less than $7.00 per meal. When I regress L($7.00) on ln S and W , I find no effect of
market size on prices. Apparently, the reduction of restaurant prices occurs at the market’s
“low end”. Nothing in principle prevents estimating δS using the original values of S($5.00)
as a dependent variable. When I do so, the p-value for δS rises to 5.9 percent. Finally, I
have also constructed analogues of Tables 1 and 2 for a sister industry, Refreshment Places.
In that industry, market size has no measurable effect on typical meal prices, but its effects
on the other three dependent variables are the same as with Restaurants.
8
3
A General Model of Atomistic Competition
The results of the previous section clearly conflict with very basic models of atomistic competition. Consider for example Dixit and Stiglitz’s (1977) model of Chamberlinian monopolistic
competition. Its free-entry condition implies that each firm’s sales equals the product of the
exogenous fixed cost with consumers’ constant elasticity of demand. Doubling the number
of consumers leaves producers’ average sales unchanged. In this section, I show that the
abstraction from strategic interaction is the sole source of the conflict between such simple
models and the data. Their other simplifying features are not to blame.
To do so, I develop the cross-market predictions of atomistic competition in a very general
model with no parametric restrictions. So that the analysis is as broadly applicable as
possible, I do not present specific conditions to guarantee the existence and uniqueness of a
free-entry equilibrium. Instead, the analysis begins with the assumption that an equilibrium
exists for a particular market size, and it then constructs an equilibrium with the same
observable distribution of producers’ actions for a larger market.
To make following the general model easier, this section begins with a specific example of
atomistic competition. It then proceeds to the general model, referring back to the specific
example to explain its moving parts.
3.1
A Specific Example
Consider a market for restaurant meals of heterogeneous quality. Production takes place in
two stages, entry and competition. In the entry stage, a large number of potential restaurateurs simultaneously decide whether to pay a sunk cost of i to enter the market or to
remain inactive at zero cost. After the restaurateurs commit to their entry decisions, each
restaurant receives a random endowment of quality, which can equal either the high value
qH with probability w or the low value qL with the complementary probability.
The competitive stage consists of two periods, early and late. All entrants can operate
9
with zero fixed costs in the early period, but continuing to the late period requires paying a
continuation cost i0 . Exit allows a restaurateur to avoid this cost. In both periods, consumers
randomly match with restaurants. The market is populated by S identical consumers, and
equal numbers of them match with each restaurant. Restaurateurs simultaneously post their
prices, and consumers decide on their purchases. A consumer matched with a restaurant
charging a price p for a meal of quality q purchases d(p/q) meals. This demand function
is strictly decreasing and concave. Restaurants’ variable cost functions are identical and
feature a constant marginal cost of production, m.
A free entry equilibrium consists of a number of entrants, N , quality-contingent pricing
decisions for each of the two periods, and quality contingent exit decisions such that each
active restaurateur maximizes profit, entry earns a non-negative return, and no inactive
potential entrant regrets staying out of the market. It is straightforward to show that this
model has a unique free-entry equilibrium. First, consider the restaurants’ pricing decisions,
which satisfy the usual inverse-elasticity rule.
p d0 (p/q)
p−m
=
p
q d(p/q)
Because d(·) is concave, there is a unique price that satisfies this for each quality level. The
optimal price increases with the restaurant’s quality.
The assumption of a constant marginal cost implies that a restaurant earns a constant
profit per customer. Denote these with πL and πH for the low and high quality restaurants. Restaurateurs’ exit decisions depend on these profits, the number of entrants, and the
cost of continuation. Denote the number of active restaurants in the late period with N 0 .
Restaurateurs’ optimal continuation decisions imply that
N if i0 ≤ (S/N ) × πL ,
πL
S i0 if (S/N ) × πL < i0 ≤ (S/wN ) × πL ,
0
N =
wN if (S/wN ) × πL < i0 ≤ (S/wN )πH ,
S πH if (S/wN )π < i0 .
H
i0
10
In the first case all restaurants can profitably produce during the late period. In the second
case, low-quality restaurants exit until their continuation value equals zero. In the third
case, all low-quality restaurants exit, but all high-quality restaurants continue. In the final
case, the continuation cost is high enough so that high-quality restaurants exit until their
continuation value equals zero. The equilibrium exit decisions allow the definition of low
and high quality restaurants’ values at the beginning of the competitive stage, VL (S/N ) and
VH (S/N ). These are both strictly increasing in S/N , so there exists a unique value of N
that equates the ex-ante value of a new entrant with the entry cost.
Before proceeding to the general model, it is worth highlighting the scale invariance
of this free-entry equilibrium. Because the ex-ante value of an entrant depends only on
S/N , increasing the number of consumers raises the number of entrants proportionately.
Restaurants’ optimal prices depend on neither S nor N , while increasing both S and N
raises N 0 by the same proportion and leaves the exit rate, 1 − N 0 /N , unchanged. Hence,
increasing the number of consumers in the market leaves the distributions of all observable
producer decisions unchanged.
This specific example is far too stylized for empirical work, but suppose for the moment
that it generated the MSA-level observations of restauranteurs’ decisions used in Section
2. If restaurateurs’ marginal costs and consumers’ demand curves depend on a vector of
market-specific variables like the factor prices and demographics in W , then regressions of
restaurants’ exit rate and of the fraction of restaurants with “high” prices on this vector and
ln S would detect no dependence of these market-level summaries of producer actions on
market size. In this sense, the specific example yields a testable prediction for cross-market
comparisons of producer actions. The fact that the results in Section 2 refute this prediction
implies that this very simple model could not have generated the data in hand. The analysis
of the general model demonstrates that the conflict arises from the assumption of atomistic
competition rather than one of the example’s other simplifying assumptions.
11
3.2
The General Model
Like the specific example, the general model consists of two stages, entry and comptition.
In the first stage, a large number of potential entrants simultaneously make their entry
decisions. At the same time, entering producers make their product choices. That of a
particular entrant is x, and this lies in the set of all possible choices, X ⊂ Rk , where k < ∞.
The number of producers that made choice x is F (x) ∈ N, which I call the industry’s entry
profile. The example did not make restaurateurs’ product choices explicit, but this can be
remedied by assuming that they choose product addresses in R and that all consumers match
in equal numbers with all offered products .
In the second stage, producers compete to sell their products to the market’s S consumers.
Producers simultaneously choose actions, a ∈ A ⊂ Rl , where l < ∞. Producers’ profits
depend on these choices and on realization of a vector of aggregate shocks, Z, which occurs
before producers choose actions. An action profile is a function A (x; Z, F ) → A. If F (x0 ) >
0, then A (x0 ; Z, F ) gives the action of a producer that chose x0 at entry. In the example,
a represents a restaurants’ early and late prices and its continuation probability and Z
determines restaurants’ qualities.4
For simplicity, we assume that if two or more entrants chose x, they both choose the
same post entry action.5 The total revenues of a producer at x0 that chooses the action a0
when all other producers’ use the action profile A (x; Z, F ) and the entry profile is F (x) are
4
The specific example relies on idiosyncratic shocks to entrants’ qualities. To use the general model’s
aggregate shocks to represent idiosyncratic shocks, assume that Z is a uniformly distributed location on
the unit-circumference circle and that a restaurant has high quality if the clockwise distance between x/N
(interpreted as a location on this circle) and Z is less than w. A potential entrant is indifferent across all
locations on [0, N ) if entrants uniformly distribute themselves on this interval, so such a uniform distribution
is an equilibrium outcome that generates the same distribution of high and low quality as in the example.
5
As in the specific example, an element of a can represent a mixed strategy over a discrete and finite
set of actions; and the revenues and costs specified below can be reinterpreted as expected values. Hence
this assumption allows for mixed strategies. However, it does remove asymmetric Nash equilibria from
consideration.
12
S × r (a0 , x0 ; A, Z, F ). Here, S denotes the number of consumers and r (·) is the producer’s
average revenue per consumer, which does not directly depend on S. That producer’s costs
are c (a0 , x0 ; A, Z, F, S).
The expected post entry profit to a producer choosing x0 at entry when it and its competitors follow the action profile A (x; Z, F ) are
π (x0 ; A, F, S) ≡ E [S × r (A (x0 ; Z, F ) , x0 ; A, Z, F ) − c (A (x0 ; Z, F ) , x0 ; A, Z, F, S)] .
Here, the expectation is taken with respect to the distribution of Z. This expectation exists
under the assumption that that r(·) and c(·) are uniformly bounded functions of a and Z.
For the example, denote the prices charged by a restaurant and the probability that it
produces in the late period with a1 , a2 , and a3 . The revenue and cost functions in the case
where a single restaurant occupies an address are
a2
a1
× d(a1 /q) + a3 × 0 × d(a2 /q), and
N
N
S
S
0
c(·) = i + m × d(a1 /q) + a3 × i + m × 0 d(a2 /q) .
N
N
r(·) =
In these expressions, the restaurant’s quality q is the function of Z and x described in
Footnote 4.
Define a strategy profile to be an action profile A (x; Z, F ) paired with an entry profile
F (x) and denote it with (A, F ). With this notation in place, the definition of a free-entry
equilibrium may proceed.6
Definition
A strategy profile (A? , F ? ) is a free-entry equilibrium for a market with S
consumers if it satisfies the following conditions.
6
Conventional notation for a dynamic game takes a set of players with names, a strategy space, and payoff
functions as primitives. The application of that approach to this model would specify the set of players as
an unbounded set of potential entrants with names in Rk , the strategy space as X × {A(x; Z, F ) ∈ A}, and
the payoffs as profit defined above. Because F (x) and Z directly index all subgames, working directly with
the strategy profile as defined here simplifies the model’s exposition.
13
(a) Take any entry profile F (x). If F (x0 ) > 0, then for all a ∈ A and all possible realizations of Z,
S × r (a, x0 ; A, Z, F ) − c (a, x0 ; A, Z, F, S) ≤
S × r (A? (x0 ; Z, F ) , x0 ; A, Z, F ) − c (A? (x0 ; Z, F ) , x0 ; A, Z, F, S) .
(b) For all x0 ∈ X , π (x0 ; A, F ? + I {x = x0 } , S) ≤ 0.
(c) If F ? (x0 ) > 0; then π (x0 ; A? , F ? , S) ≥ 0, and for all x00 ∈ X
π (x0 ; A? , F ? , S) ≥ π (x00 ; A? , F ? + I {x = x00 } − I {x = x0 } , S)
Condition (a) of this definition ensures that the action profile A? (x; Z, F ) forms a Nash
equilibrium for all subgames following the entry stage. Condition (b) requires that no further
entry is profitable, and condition (c) states that each active producer’s entry decision and
choice of x is optimal given all other potential entrants’ decisions. Together, the definition’s
three conditions are equivalent to requiring the strategy profile (A? , F ? ) to correspond to a
subgame perfect Nash equilibrium with pure strategies in the entry stage.
3.3
Atomistic Competition
At this level of generality, the framework encompasses many models. To specialize it and
thereby derive the implications of atomistic competition, we impose the following two conditions. The first condition allows for only trivial strategic interactions between producers
when no two of them occupy the same location in X , and the second ensures that no such
“local oligopolies” will arise in a free-entry equilibrium. Henceforth, I assume that X is a
Borel measurable set with positive measure, denote the set of its Borel measurable subsets
with M, and use µ (M ) to denote the Borel measure of M ∈ M.
Assumption A1 (Atomistic Competition) Let (A, F ) be a strategy profile with F (x) ≤
1 and define M = {x|F (x) = 1}. If F (x) is Borel-measurable, µ(M ) > 0, A(x; Z, F 0 ) is
14
Borel-measurable given any shock realization Z and Borel-measurable entry profile F 0 , and
F (x0 ) = 1, then the revenues of the producer at x0 choosing the action a0 satisfy
S × r (a0 , x0 ; A, Z, F ) = S × ρ (a0 , x0 ; G (A, Z, F ) , Z, NF ) ,
where NF ≡ µ(M ) is the mass of producers operating and
Z
1
0
G (A, Z, F ) (a ) ≡
I {A (x; Z, F ) ≤ a0 } × F (x)dµ (x) .
NF X
Two aspects of Assumption A1 capture the idea that producers compete atomistically.
First, a producer’s revenues only depend on its own choices, aggregate shocks, the mass
of competing producers, and the empirical distribution of their actions. Second, any one
producer has measure zero when computing this distribution, so changing a single producer’s
conduct alters no other producer’s revenue. The example revenue function above satisfies
Assumption A1, because each producer’s profit depends on its rivals actions only through
S/N and S/N 0 . A finite-horizon version of Hopenhayn’s (1992) model of perfect competition
also satisfies Assumption A1. In any particular industry, the number of producers is obviously
countable and not continuous. Models of atomistic competition are of empirical interest
because their predictions might fit the data well in spite of the false simplifying assumption
of a continuum of producers.
Assumption A2 (Product Differentiation) If F (x0 ) ≥ 2 and A satisfies condition (a)
of the definition of a free-entry equilibrium, then π (x0 ; A, F, S) < 0.
Assumption A2 states that competition between producers of identical products is tough
enough to guarantee that no more than one producer will occupy any location in X . Thus,
the observed market structure will not contain any “local” oligopolies; and competition is
“global” in the sense of Anderson and de Palma (2000). The specific example satisfies this
assumption. Any model in which firms’ producing exactly the same product act as Bertrand
competitors will satisfy Assumption A2.7
7
A model with price-taking producers of a homogeneous good, such as Hopenhayn’s (1992), could accom-
modate this assumption by defining a trivial product placement choice x on the real line and assuming that
15
3.4
Intrinsic Scale Effects
Thus far, the model’s specification does not rule out direct effects of the scale of the market,
measured with either S or NF , on producers’ revenues or costs. For example, the product
space might be limited so that entry cannot continue indefinitely. The market shares of
producers with particular choices of x might be more or less sensitive to the size of the
market, or directly raising S could systematically reduce costs and so encourage entry and
production. For all of these reasons, the distribution of producers’ decisions across large and
small markets could differ. The following three conditions eliminate them as a theoretical
possibility.
Assumption S1 (Invariance of Market Shares) The per consumer revenue function
ρ (·) is homogeneous of degree -1 in NF .
This assumption states that doubling the number of producers while holding the distribution
of their actions fixed cuts each producer’s revenue in half. In the example, it follows from
the uniform random matching of consumers with firms. This assumption is closely related
to the independence of irrelevant alternatives: Adding a producer to a market does not
change the relative market shares of any two incumbents. The quadratic demand system
of Ottaviano, Tabuchi, and Thisse (2002) generally violates Assumption S1, because each
consumer’s elasticity of demand depends on the number of varieties for sale. This and
similar models hard-wire markups which decline with market size. These arise naturally
in true oligopoly models, so excluding them from the definition of atomistic competition is
appropriate if one wishes the definition to distinguish substantially between strategic and
anonymous competition.8
the cost of entry at a given “location” increases steeply with the number of entrants there.
8
Whether or not one agrees with this distinction, the result of Asplund and Nocke (2006) implies that a
model which violates only Assumption S1 cannot reproduce the exit rate’s decline with market size documented in Section 2.
16
Assumption S2 (No Productive Spillovers) For any entry choice x0 ∈ X ,action a0 ∈
A, and any two strategy profiles (A, F ), (A? , F ? ) and market sizes S and S ? , if
c (a0 , x0 ; A, Z, F, S) < c (a0 , x0 ; A? , Z, F ? , S ? )
then
S × r (a0 , x0 ; A, Z, F ) < S ? × r (a0 , x0 ; A? , Z, F ? )
Assumption S2 implies that is impossible to hold a producer’s choices of x and a fixed,
change its competitive environment, and lower that producer’s costs without simultaneously
lowering its revenues. Any model in which producers’ costs depend only on their own actions
(such as a quantity setting game with no productive spillovers) satisfies this assumption. If
the market faces an upward sloping supply curve for some input, as in some versions of
Hopenhayn’s (1992) model, then this assumption would be violated. The simple affine
technology of the example obviously satisfies Assumption S2.
Assumption S3 (Distinct Observationally-Equivalent Strategy Profiles) For any
market size S and strategy profile (A, F ) such that F (x) ≤ 1 for all x ∈ X , there exists
a continuous, one to one, and onto function g : X → X such that if we define F T (x) ≡
F (g −1 (x)), and AT (x; Z, F T ) = A(g −1 (x), Z, F ) then
(a) ∀x ∈ X , F (x) + F T (x) ≤ 1;
(b) if F (x0 ) > 0, then
S × r (a0 , x0 ; A, Z, F ) = S × r a0 , g(x0 ); AT , Z, F T
and
c (a0 , x0 ; A, Z, F, S) = c a0 , g(x0 ); AT , Z, F T , S ;
(c) ∀M ∈ M, µ (g −1 (M )) = µ (M ).
17
In many models of competition with product differentiation, it is possible to rearrange
producers’ locations in X , hold their actions fixed, and leave their payoffs unchanged. Consider two examples of such a rearrangement, moving all producers a short distance to the
right in Salop’s (1979) circle model and changing the particular products chosen by entrants
in Dixit and Stiglitz’s (1977) model of Chamberlinian monopolistic competition. In both
cases, the rearrangement leaves the game played after product placement unaltered. In Assumption S3, conditions (a) and (b) require such a rearrangement to be possible for any given
strategy profile. In this section’s simple example, X is an infinite set of all possible products
without any spatial structure or asymmetries in demand or cost, so such a rearrangement is
possible. Condition (c) requires g (x) to be measure preserving, so that the rearrangement
does not alter the mass of producers. Overall, Assumption S3 asserts that no location in X
has payoff-relevant characteristics that are unique.
3.5
Equilibrium
The following two conditions ensure that potential entrants’ expectations about post-entry
competition can be well-defined and that a free-entry equilibrium exists.
Assumption E1 (Existence of Nash Equilibrium) For any market size S, there exists
a strategy profile A (x, Z, F ) that satisfies condition (a) of the definition of a free-entry equilibrium.
Assumption E2 (Existence of a Measurable Free Entry Equilibrium) There exists
a market size S0 > 0 with a corresponding free-entry equilibrium (A0 , F0 ) such that
(a) F0 (x) and A0 (x; Z, F0 ) are Borel measurable functions of x for any Z, and
(b) A0 (x0 ; Z, F ) = A0 (x0 ; Z, F0 ) if F (x0 ) = F0 (x0 ) = 1 and F (x) = F0 (x) almost everywhere.
18
The goal of this section is to show that no well-defined model of atomistic competition
can reproduce the dependence of firms’ choices on market size. A model that violates E1 can
make no equilibrium prediction for some market size, and a model that violates part (a) of E2
makes no prediction for any market size. In this specific sense, a model that violates either of
these assumptions is not well defined. Part (b) of Assumption E2 eliminates the possibility
that a positive measure of firms respond to deviations from the equilibrium by a single
(measure zero) firm. That is, there is no producer whose actions act as a pure coordination
device for the others. Both E1 and E2 are clearly true for this section’s introductory example.
Accordingly, I view both assumptions as regularity conditions that any model must satisfy
before it could be taken seriously as an explanation for the data.
3.6
Market Size and Producers’ Actions
The specific example and many other models of monopolistic competition satisfy all of the
above assumptions. Together, they place sufficient structure on the model to imply the
following observational implication.
Proposition
If S=2j × S0 , where j is a non-negative integer and S0 , A0 , and F0 are
defined in Assumption E2, then there exists a free entry equilibrium (Aj , Fj ) such that
G (Aj , Z, Fj ) = G (A0 , Z, F0 )
where G (A, Z, F ) is as defined in Assumption A1.
The proposition says that for every measurable equilibrium with a given market size
there exists a corresponding equilibrium for a market with twice as many consumers with
identical distributions of producers’ actions. Unless there exists more than one free-entry
equilibrium and the equilibrium selection rule systematically depends on S, there can be no
observable relationship between market size and the distribution of any producer choice. The
appendix presents the proposition’s proof. Here, I only outline the argument. Consider the
19
free-entry equilibrium (A0 , F0 ) for S0 . We know from Assumption S3 that there is a different
but observationally equivalent strategy profile, AT0 , F0T . Now consider a market with S1 =
2×S0 and entry profile, F0 +F0T . If all producers duplicate the actions they take in the smaller
market, then the empirical c.d.f. of producers actions remains unchanged, so Assumptions
A1, S1, and S2 imply that each producer’s profit maximizing action remains unchanged.
That is, the action profile that duplicates producers’ actions is a Nash equilibrium profile
for this larger market size and entry profile. Each producer’s profits remain unchanged, and
the profits from producing in an unoccupied location in X are identical to their value in the
original free entry equilibrium with the smaller market size, so conditions (b) and (c) of the
definition are satisfied. Each producer’s actions equal those from the original equilibrium,
so their empirical distributions are also unchanged as the proposition asserts.9
The proposition illustrates that the invariance of producers’ decisions to the market’s
size in the specific example extends well beyond its particular assumptions. All of the
assumptions excepting A1 are regularity conditions, so I interpret the fact that market size
does influence restauranteurs’ observable decisions as a rejection of atomistic competition.
3.7
Extensions
The general model is restrictive in two ways that are worth noting. First, it has no role for
actions that are taken prior to the realization of Z that do not directly differentiate firms’
products, such as investments that increase the likelihood of having a high quality restaurant.
Adding such pre competition actions to the general model increases its notational burden
but does not alter its scale invariance. Second, the use of the Borel integral to form the
c.d.f. of producers’ actions in Assumption A1 restricts product placement decisions to be
9
The proposition’s focus on doubling market size can easily be changed if the assertion that g(x) is
measure preserving in Assumption S3 is replaced with the assumption that for any t > 1, there exists a
gt (x) satisfying the assumption’s other conditions and which satisfies µ(gt−1 (M )) = t × µ(M ). With this, a
parallel argument establishes that a free-entry equilibrium that replicates producers’ decisions exists for any
market size greater than S0 .
20
continuous choices. Scale invariance requires some continuous product placement decision
to differentiate firms’ products, but it does not require all product placement decisions to be
continuous. Extending the general model to allow for discrete dimensions of firms’ productplacement decisions is straightforward.
3.8
Atomistic and Monopolistic Competition
Before concluding, it is helpful to clarify the relationship between what I have labelled
“atomistic competition” with the large theoretical literature on monopolistic competition.
For some authors, “monopolistic competition” refers to all imperfect competition among
a large number of producers. Models that prominently feature strategic interaction, such
as Salop’s (1979) model of spatial competition, then go by the label of “Hotelling-style”
monopolistic competition. Models with only trivial strategic considerations, such as Spence’s
(1976) are called “Chamberlin-style”.
Hart (1985) andWolinsky (1986) propose a more exclusive definition of “monopolistic
competition” based on four criteria.
(1) there are many firms producing differentiated commodities; (2) each firm is
negligible in the sense that it can ignore its impact on, and hence reactions from,
other firms; (3) each firm faces a downward sloping demand curve and hence the
equilibrium price exceeds marginal cost; (4) free entry results in zero-profit of
operating firms (or, at least, of marginal firms).(Wolinsky, 1986, page 493)
These clearly correspond to what others call Chamberlin-style monopolistic competition.
Hart and Wolinsky’s first two criteria correspond to Assumptions A2 and A1, and the fourth
criterion is implicit in the definition of a free-entry equilibrium. The definition of atomistic
competition does not require firms to face downward sloping demand curves, but it clearly
allows for that possibility. Hence, models of monopolistic competition (in the sense of Hart
and Wolinsky) can usually be written without economically substantial changes to satisfy the
21
assumptions this paper places on atomistic competition. However, the definition of atomistic
competition is broad enough to also encompass models without market power.
4
Related Literature
Structure-conduct-performance studies gave rise to many examinations of competitive outcomes’ dependence on market size. One strand of this literature uses the empirical relationship between market size and the number of competitors to infer how adding competition
lowers markups. If doubling market size leads to a less than proportional increase in the
number of producers, either per-consumer profits fall with entry or incumbents raise entrants’ fixed costs. Bresnahan and Reiss (1990) apply this approach to concentrated retail
automobile markets in isolated towns. Berry and Waldfogel (2006) examine the influence of
market size on the number of competitors in a slightly broader sample of MSAs than that
used in this paper, and they find that the number of restaurants increases less than proportionally with MSA population.10 The proof of this paper’s proposition makes it clear that
the number of producers in atomistically competitive markets is proportional to the number
of consumers, so Berry and Waldfogel’s finding reinforces this paper’s empirical conclusion.11
This paper’s proposition does not stress the relationship between market size and the
number of firms under atomistic competition, because a finding that doubling market size
less than doubles the number of firms could arise solely from measurement error in market
size. Measurement error could make the rejection of this paper’s exclusion restrictions less
likely when they are false, but it does not lead directly to their rejection when they are
true. In this sense, a test of atomistic competition based on the relationship between noisily
10
11
See the third and fourth columns of their Table 3.
Berry and Waldfogel’s finding also manifests itself in the observations used in the present paper. The
estimate of δs from a nonparametric regression of the number of restaurants’ logarithm on population’s
logarithm and the other control variables listed in Table 1 using Campbell and Hopenhayn’s (2005) sample
of MSAs equals 0.93, and this is significantly different from one.
22
measured market size and measures of producer actions is conservative.12
This paper derives testable predictions of a free-entry model without the use of parametric
assumptions. In this respect, Sutton’s (1991) analysis of models with endogenous sunk costs
precedes it. He considers a model of competition in which entrants compete with sunk
investments in product quality. The firm with the greatest investment earns a guaranteed
minimum market share, regardless of the number of other producers. Sutton shows that
these features together imply a nonparametric upper bound on the number of entrants, and
he demonstrates that cross-country data from several advertising-intensive food-processing
industries satisfy this bound. As the number of consumers grows, the number of entrants
remains bounded from above. In this sense, industries that satisfy his model’s assumptions
are natural oligopolies. As noted above in Subsection 3.7, it is notationally burdensome
but straightforward to add pre-entry investments in quality to this paper’s model. This
extension leaves the model’s nonparametric testable implications unaltered. In particular,
the number of producers grows linearly with market size. The contrast between that result
and Sutton’s highlights the role of endogenous sunk costs in his results: They are necessary
but not sufficient for an industry to be a natural oligopoly. Hence, the simple observation
that an industry’s producers incur endogenous sunk costs does not imply that its firms
are oligopolists. However, tests of the exclusion restrictions from atomistic competition do
provide information about the nature of competition.
The analysis of the exit of restaurants places this paper in another vast literature which
examines the rate of producer turnover and the reallocation of resources between firms.
These papers have focused on differences in firm growth and survival across the life cycle (as
in Dunne, Roberts, and Samuelson (1988)) and on the interaction of resource reallocation
with the business cycle (as in Davis, Haltiwanger, and Schuh (1996), Campbell (1998), and
12
Bresnahan and Reiss (1990) can measure market size accurately because they carefully chose their
sample towns. This strategy becomes infeasible when considering competition in large markets in which the
definition of the market and industry are themselves somewhat subjective, so prudence requires accounting
for possible measurement error.
23
Campbell and Lapham (2004)). Analysis of how the pace of resource reallocation varies
with local market conditions, similar to that in this paper, is much scarcer. Syverson (2004)
shows that ready-mixed concrete producers serving geographically concentrated markets
have higher average productivity and less productivity dispersion than their counterparts in
more sparsely populated areas, and he interprets this as the result of more intense selection
in highly competitive markets. As noted in the introduction, Asplund and Nocke (2006)
create a model of such selection by incorporating markups which depend on the number of
producers into an otherwise standard model of industry dynamics with Chamberlinian monopolistic competition. They confirm Syverson’s finding by showing that Swedish hairstylists
are younger in larger markets. This paper finds the opposite relationship between market
size and exit for U.S. restaurants. Together, this paper’s theoretical results and those of Asplund and Nocke imply that dynamic aspects of strategic interaction substantially influence
the rate of restaurant turnover.
5
Conclusion
Researchers’ prior beliefs about the usefulness of different modelling approaches influence
their investigations of industries. The relative simplicity of atomistic competition models
makes them a tempting first choice for the empirical study of competition in large markets. However, those who believe that strategic interaction permeates all producers’ choices
have chosen to focus instead on industries with few competitors and relatively well-defined
strategic environments. This paper’s results are of use to both sorts of researchers. For
those who regularly abstract from strategic interaction, the nonparametric test of atomistic
competition can be used to subject this assumption to empirical scrutiny before proceeding
to a more involved investigation.13 For those unwilling to part from a strategic focus, the
regressions indicate the dimensions of the data along which strategic interaction manifests
13
For example, Abbring and Campbell (2006) apply this papers’ results to test the assumption of atomistic
competition in their structural model of new Texas bars’ growth and exit decisions.
24
itself quantitatively. This is an essential first step to extending the strategic analysis of
competition to large markets.
The application of this paper’s theoretical analysis to observations of U.S. restaurants’
prices, exit rates, sales, and seating capacity indicates that atomistic competition cannot
explain how restauranteurs’ key choices depend on market size. A particularly important
aspect of this is that exit rates fall with market size. In related work, Yeap (2005) documents that this increase in average size reflects only the decisions of firms owning two
or more restaurants. Taken together these findings indicate that better understanding of
competition among restaurants in large markets requires confronting restaurateurs’ strategic behavior. Toivanen and Waterson (2005) take an important step in this direction by
empirically modelling entry decisions into well defined duopoly fast-food markets. Extending such an analysis to large samples of restaurants without high-quality information about
market definitions and strategic interactions is the subject of my current research with Jaap
Abbring.
Proof of the Proposition
Clearly, the proposition is true for j = 0. We now wish to show that it is true for j = 1.
The proposition can then be demonstrated recursively for greater values of j.
Let g (x) be the function assumed to exist in Assumption S3. Define the entry profile
F1 (x) = F0 (x) + F0T (x), where the latter entry profile is defined as in the statement of
Assumption S3. From Assumption A2 and the definition of a free-entry equilibrium, we know
that F0 (x) ∈ {0, 1}. Therefore, condition (a) of Assumption S3 ensures that F1 (x) ∈ {0, 1}.
We know from Assumption E1 that there exists an action profile A (x; Z, F ) that satisfies
condition (a) of a free-entry equilibrium’s definition for S1 = 2 × S0 . We now wish to use this
and A0 (x; Z, F ) to construct an action profile that forms a candidate free-entry equilibrium
when paired with F1 . For any entry profile F (x) such that either
25
(i) F (x0 ) ≥ 2 for some x0 ∈ X or
(ii) {x|F (x) 6= F1 (x)} is either not measurable or has positive measure,
define A1 (x; Z, F ) = A (x; Z, F ) .
For any entry profile F (x) ∈ {0, 1} for which F (x) = F1 (x) almost everywhere, there
exists two measurable sets Cp and Cm with µ (Cp ) = µ (Cm ) = 0 and F (x) = F1 (x) +
I {x ∈ Cp } − I {x ∈ Cm } . Define F0 (Cp ) (x) = F0 (x) + I {x ∈ Cp }. If F (x) = 1, then either
F0 (Cp ) (x) = 1 or F0T (x) = 1. Therefore, we can define the action profile for these values of
x with
A1 (x; Z, F ) =
A0 (x; Z, F0 (Cp ))
if F0 (Cp ) (x) = 1,
A0 (g −1 (x) ; Z, F0 (Cp )) otherwise.
Because the composition of Borel measurable functions is itself Borel measurable, A1 (x; Z, F )
is a Borel measurable function of x.
The next step is to show that (A1 , F1 ) is a free-entry equilibrium. To do so, consider the
definition’s three conditions in turn.
Condition (a)
Note that by construction A1 (x; Z, F ) satisfies the inequality in condition (a) of a free-entry
equilibrium’s definition if it satisfies either (i) or (ii) above. Suppose that F (x) ∈ {0, 1} and
F (x) = F1 (x) almost everywhere. This implies
Z
1
I {A1 (x; Z, F ) ≤ a0 } F (x) dµ (x)
G (A1 , Z, F ) (a ) ≡
NF X
Z
1 1
=
I {A0 (x; Z, F0 (Cp )) ≤ a0 } F0 (Cp ) (x) dµ (x)
2 NF0 (Cp ) X
Z
1 1
+
I A0 g −1 (x) ; Z, F0 (Cp ) ≤ a0 F0T (x) dµ (x)
2 NF0T X
0
= G (A0 , Z, F0 (Cp )) (a0 ) .
The first equality holds because F1 and F0 (Cp ) + F00 (Cp ) differ by a set of measure zero, and
the last equality follows from Proposition 1 in Chapter 15 of Royden (1988).
26
With this and Assumptions A1 and S1, we can conclude that if F0 (Cp ) (x) = 1, then for
all a0 ∈ A, S1 × ρ (a0 , x0 ; G (A1 , Z, F ) , Z, NF ) = S0 × ρ a0 ; , x0 ; , G (A0 , Z, F0 (Cp )) , Z, NF0 (Cp )
In turn, this and Assumption S2 imply that
S1 × ρ (a0 , x0 ; G (A1 , Z, F ) , Z, NF ) − c (a0 , x0 ; A1 , Z, F, S1 )
= S0 × ρ a0 ; , x0 ; , G (A0 , Z, F0 (Cp )) , Z, NF0 (Cp ) − c (a0 , x0 ; A0 , Z, F0 (Cp ) , S0 )
The action A0 (x0 ; Z, F0 (Cp )) = A1 (x0 ; Z, F ) maximizes the right-hand side, so it must also
maximize its left-hand side.
Alternatively, if F00 (Cp ) (x) = 1, then we can construct a parallel argument to show
that A1 (x0 , Z, F ) maximizes the firm’s profit. Thus A1 (x, Z, F ) satisfies condition (a) of a
free-entry equilibrium’s definition.
Condition (b)
Next, consider condition (b) of the definition. Extending the notation above, denote F1 (x)+
I {x = x0 } with F1 ({x0 }) (x), If F1 (x0 ) = 1, then the definition of A1 and Assumptions
A2 and E1 imply that π (x0 ; A1 , F1 ({x0 }) , S) ≤ 0. Next, note that if F1 (x0 ) = 0, then we
know from above that G (A1 , Z, F1 ({x0 })) (a) = G (A0 , Z, F0 ({x0 })) (a) , and that NF1 ({x0 }) =
2 × NF0 ({x0 }) . Therefore, Assumptions A1, S1, and S2 and the definition of a free-entry
equilibrium imply that π (x0 ; A1 , F1 ({x0 }) , S) ≤ 0 in this case as well. Hence, condition (b)
of the definition is satisfied.
Condition (c)
Finally, consider condition (c) of a free-entry equilibrium’s definition. Because
G (A1 , Z, F1 ) (a) = G (A0 , Z, F0 ) (a)
and NF1 = 2 × NF0 , so if F0 (x0 ) = 1 then
π (x0 ; A1 , F1 , S1 ) = π (x0 ; A0 , F0 , S0 ) ≥ 0.
27
Furthermore, conditions (b) and (c) of Assumption S3 imply that this inequality also applies
if F0T (x0 ) = 1. Therefore, the first inequality in condition (c) of the definition holds good.
The second inequality in this condition holds trivially from Assumption A2 and the
definition of A1 if F1 (x00 ) = 1. Suppose instead that F1 (x00 ) = 0 and F1 (x0 ) = 1. We know
that F1 (x) + I {x = x00 } − I {x = x0 } = F1 (x) + I {x = x00 } almost everywhere. From this
and the fact that we have already verified condition (b) of an equilibrium’s definition, we
conclude that
π (x00 ; A1 , F1 + I {x = x00 } − I {x = x0 } , S1 ) ≤ 0.
Thus, the second inequality of condition (c) holds and (A1 , F1 ) is a free-entry equilibrium.
28
References
Abbring, J. H. and J. R. Campbell (2006). A firm’s first year. Technical report, Federal
Reserve Bank of Chicago. 24
Anderson, S. P. and A. de Palma (2000). From local to global competition. European
Economic Review 44 (3), 423–448. 15
Asplund, M. and V. Nocke (2006, April). Firm turnover in imperfectly competitive markets.
Review of Economic Studies 73 (2), 295–327. 2, 16, 24
Berry, S. and J. Waldfogel (2006). Product quality and market size. Working Paper, Yale
University and University of Pennsylvania. 22
Bierens, H. J. (1987). Kernel estimators of regression functions. In T. Bewley (Ed.), Advances
in Econometrics. Fifth World Congress, Volume I. Cambridge University Press. 6
Bresnahan, T. F. and P. C. Reiss (1990). Entry in monopoly markets. Review of Economic
Studies 57 (4), 531–553. 22, 23
Campbell, J. R. (1998, April). Entry, exit, embodied technology, and business cycles. Review
of Economic Dynamics 1 (2), 371–408. 23
Campbell, J. R. and H. A. Hopenhayn (2005). Market size matters. Journal of Industrial
Economics 53 (1), 1–25. 2, 3, 4, 5, 6, 7, 22
Campbell, J. R. and B. Lapham (2004). Real exchange rate fluctuations and the dynamics
of retail trade industries on the U.S.-Canada border. American Economic Review 94 (4),
1194–1206. 24
Davis, S. J., J. C. Haltiwanger, and S. Schuh (1996). Job Creation and Job Destruction.
Cambridge, MA: MIT Press. 23
29
Dixit, A. K. and J. E. Stiglitz (1977). Monopolistic competition and optimum product
diversity. American Economic Review 67 (3), 297–308. 9, 18
Dunne, T., M. J. Roberts, and L. Samuelson (1988). Patterns of firm entry and exit in U.S.
manufacturing industries. RAND Journal of Economics 19 (4), 495–515. 23
Hart, O. (1985). Monopolistic competition in the spirit of Chamberlin: A general model.
Review of Economic Studies 52 (4), 529–546. 21
Hopenhayn, H. A. (1992). Entry, exit, and firm dynamics in long run equilibrium. Econometrica 60 (5), 1127–1150. 15, 17
Nocke, V. (2006). A gap for me: Entrepreneurs and entry. Journal of the European Economic
Association 4 (5), 929–956. 2
Ottaviano, G., T. Tabuchi, and J.-F. Thisse (2002, May). Agglomeration and trade revisited.
International Economic Review 43 (2), 409–435. 2, 16
Powell, J. L., J. H. Stock, and T. M. Stoker (1989). Semiparametric estimation of index
coefficients. Econometrica 57 (6), 1403–1430. 6
Royden, H. L. (1988). Real Analysis. Macmillan Publishing Company, New York. 26
Salop, S. C. (1979). Monopolistic competition with outside goods. Bell Journal of Economics 10, 141–156. 18
Spence, M. (1976). Product selection, fixed costs, and monopolistic competition. Review of
Economic Studies 43 (2), 217–235. 21
Sutton, J. (1991). Sunk Costs and Market Structure. MIT Press. 23
Syverson, C. (2004). Market structure and productivity: A Concrete example. Journal of
Political Economy 112 (6), 1181–1222. 2, 24
30
Syverson, C. (Forthcoming). Prices, spatial competition, and heterogeneous producers: An
empirical test. Journal of Industrial Economics. 7
Toivanen, O. and M. Waterson (2005). Market structure and entry: Where’s the Beef?
RAND Journal of Economics 36 (3), 680–699. 25
Wolinsky, A. (1986). True monopolistic competition as a result of imperfect information.
Quarterly Journal of Economics 101 (3), 493–512. 21
Yeap, C. (2005). Competition and market structure in the food services industry: Changes
in firm size when market size expands. University of Minnesota. 3, 25
31
Table 1: Nonparametric Regression Estimates(i,ii,iii)
Log Average
Revenue Seats per Restaurant
4.68???
2.07
(1.76)
(1.99)
L($5.00)
-12.90??
(6.13)
Exit Rate
-0.77??
(0.32)
Commercial Rent
11.16?
(6.29)
-0.29
(0.30)
1.66
(1.53)
-2.23
(1.89)
Retail Wage
17.37??
(7.12)
0.69??
(0.35)
-0.49
(1.56)
-2.56
(2.18)
Advertising Cost
-9.00?
(5.47)
-0.30
(0.27)
-1.43
(1.65)
-0.35
(1.95)
Income
-1.67
(7.03)
-0.55?
(0.33)
6.17???
(1.63)
4.90??
(2.39)
Percent Black
20.89???
(5.91)
0.62??
(0.26)
0.90
(1.31)
-3.95?
(2.16)
Percent College
19.70???
(6.22)
-0.09
(0.33)
8.48???
(1.28)
2.50
(1.83)
-2.41
(5.08)
-0.57??
(0.28)
-1.16
(1.58)
1.38
(2.00)
Population
Vehicle Ownership
Notes: (i) The table reports estimates of density-weighted average derivatives from the
regressions of the indicated variables on the regressors listed in the first column. Asymptotic
standard errors appear below each estimate in parentheses. The superscripts ?, ??, and ? ? ?
indicate statistical significance at the 10, 5, and 1 percent levels (ii) In the table, L($5.00)
refers to the logistic transformation of the fraction of restaurants in an MSA with typical
meal prices greater than or equal to $5.00. (iii) All estimates have been multiplied by 100.
See the text for further details.
32
33
Market Size Measures
Bandwidth Choices
Population Density Housing Units h = 1
h=3
?
-8.91
-11.93
-12.67
-12.58 ??
(5.74)
(6.19)
(7.76)
(5.72)
-0.89???
-0.75 ??
-0.94 ??
-0.73 ??
(0.29)
(0.32)
(0.39)
(0.30)
??
???
???
4.36
5.03
5.76
4.57???
(1.77)
(1.78)
(1.99)
(1.71)
2.69
2.45
2.83
1.97
(2.00)
(2.05)
(2.55)
(1.89)
Notes: (i) The table’s entries are estimated density-weighted average derivatives of the indicated variable with respect to the
logarithm of the indicated measure of market size. Heteroskedasticity-consistent standard errors appear in parentheses. The
superscripts ?, ??, and ? ? ? indicate statistical significance at the 10%, 5%, and 1% levels. (ii) In the table, L($5.00) refers to
the logistic transformation of the fraction of restaurants in an MSA with typical meal prices greater than or equal to $5.00. (iii)
All estimates have been multiplied by 100. See the text for further details.
Log of Average Seats per Restaurant
Log of Restaurants’ Average Revenue
Exit Rate
L($5.00)(ii)
Population
-12.90 ??
(6.13)
-0.77 ??
(0.32)
4.68???
(1.76)
2.07
(1.99)
Table 2: Alternative Regression Estimates(i,iii)
Figure 1: Logistic Transformation of the Share of Establishments with High Meal Prices(i)
Note: The figure plots the logistic transofrmation of the share of establishments with typical
meal prices exceeding $5.00 against the demeaned logarithm of MSA population.
34
Figure 2: Restaurants’ Annual Exit Rate in Percentage Points
35
Figure 3: Logarithm of Restaurants’ Average Revenue
36
Figure 4: Logarithm of Average Seats per Restaurant
37
Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
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Working Paper Series (continued)
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8
@FedFRASER