Markov Perfect Industry Dynamics with Many Firms∗
Gabriel Y. Weintraub
Stanford University
C. Lanier Benkard
Stanford University
and NBER
Benjamin Van Roy
Stanford University
PRELIMINARY FIRST DRAFT (TYPOS MAY REMAIN)
August 2005
Abstract
We propose an approximation method for analyzing Ericson and Pakes (1995)-style dynamic models
of imperfect competition. We derive a simple computational algorithm for computing an “oblivious”
equilibrium, in which each firm is assumed to make decisions based only on its own state and knowledge
of the long run average industry state, but where firms ignore current information about rivals’ states.
We prove that, if the distribution of firms obeys a certain “light-tail” condition, then as the market size
becomes large oblivious equilibrium closely approximates a Markov perfect equilibrium. We derive
a performance bound that can be used to assess how well the approximation performs in any given
applied problem. We apply these methods to a dynamic investment game and find that the approximation
typically works well for markets with hundreds of firms, and in some cases works well even for markets
with only tens of firms.
JEL classification numbers: C63, C73, L11, L13
∗
We have had very helpful conversations with Herman Bennett, José Blanchet, Uli Doraszelski, Liran Einav, Hugo Hopenhayn,
Cristóbal Huneeus, Ken Judd, and Jon Levin, and well as seminar participants at SITE, UCLA, and University of Chile. This
research was supported by General Motors, the National Science Foundation, the Office of Naval Research, and the Lillie fund.
Correspondence: [email protected]; [email protected]; [email protected].
1
Contents
1
Introduction
1
2
A Dynamic Industry Model of Imperfect Competition
2.1 Model and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
5
6
7
3
Oblivious Equilibrium
9
3.1 Oblivious Strategies and Entry Rate Functions . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Oblivious Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 The Invariant Industry Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4
Asymptotic Results
12
4.1 Uniform Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Asymptotic Markov Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 A Light-Tail Sufficient Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5
Algorithm and Performance Bounds
18
5.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.2 Performance Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6
Computational Experiments
6.1 The Computational Model . . . . . . . . . . . . . . . . . . . .
6.2 A Tighter Bound . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Numerical Results: Behavior of the Bound . . . . . . . . . . . .
6.4 Closeness to Markov Perfect Equilibrium . . . . . . . . . . . .
6.5 Example: Changes in Market Structure as Market Size Increases
7
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Conclusions and Future Research
20
20
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25
26
A Appendix: Proofs and Mathematical Arguments
A.1 Proofs and Mathematical Arguments for Section 2
A.2 Proofs and Mathematical Arguments for Section 3
A.3 Proofs and Mathematical Arguments for Section 4
A.4 Proofs and Mathematical Arguments for Section 5
A Appendix: Tables and Figures
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31
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2
1
Introduction
Many problems in applied economics are dynamic in nature. Entry and exit, collusion between firms,
mergers, advertising, investment in R&D or capacity, network effects, durable goods, consumer learning,
learning by doing, and transaction or adjustment costs are inherently dynamic. A model useful for analysis
of policy or environmental changes must account for these dynamic phenomena. Additionally, empirical
evidence suggests that there is significant heterogeneity between firms in the same industry (Davis and
Haltiwanger (1992) and Dunne, Roberts, and Samuelson (1988)). This heterogeneity is expressed, for
example, in simultaneous entry and exit in narrowly defined sectors of the economy.
Ericson and Pakes (1995) introduced an approach to modeling dynamics in an industry with heterogeneous firms. In their model, entry and investment, together with idiosyncratic shocks, produce heterogeneity
among firms. The literature on such models — which we will refer to as Ericson and Pakes (EP)-type models — computes Markov perfect equilibria (MPE) using dynamic programming algorithms (e.g., Pakes and
McGuire (1994)). One reason for the success of the EP framework is that it is easily extended to many
important dynamic problems in economics. See, for example, Benkard (2004), Berry and Pakes (1993),
Besanko and Doraszelski (2004), Doraszelski and Markovich (2003), Fershtman and Pakes (2000), Goettler, Parlour, and Rajan (2004), Gowrisankaran (1999), Jenkins, Liu, Matzkin, and McFadden (2004), Judd,
Schmedders, and Yeltekin (2002), Langohr (2003), Markovich (2003), de Roos (2002), Song (2003), as well
as Pakes (2000) for a survey. A major shortcoming, however, is the computational complexity involved in
solving for the MPE. Several papers have proposed methods to alleviate the curse of dimensionality that
arises in these models (Pakes and McGuire (2001) and Doraszelski and Judd (2003)). However, even with
these improvements it remains difficult to compute equilibria for markets with more than a handful of firms
and/or more than 10-20 points per firm in the state space.1 We believe that this has been a major obstacle for
the EP model to become a more widely used computational and empirical framework as initially intended.
In this paper we propose an approximation approach that dramatically reduces the computational complexity of an EP-type model. Consider an industry with a large number of firms. In every period some
firms receive positive shocks and some receive negative shocks. However, when there are a large number
of firms changes in individual firms’ states average out at the industry level such that the industry state, a
vector containing the number of firms at each state, may not change very much from period to period. A
key assumption for this to be true is that shocks are idiosyncratic. If the industry state does not change
much between periods, firms can make close to optimal decisions knowing only their own state and the long
1
Doraszelski and Judd (2003) report results with up to 14 firms and 9 state points per firm.
1
run average industry state. We call oblivious strategies, strategies that only consider each firm’s own state
and the average industry state, and oblivious equilibrium, an equilibrium where every firm uses oblivious
strategies. The advantage of analyzing oblivious equilibrium is that solving for an oblivious equilibrium is
easy: it is a one-dimensional dynamic programming problem.
Based on the logic above, we first derive an asymptotic result that provides sufficient conditions for
an oblivious equilibrium to well approximate equilibrium behavior in an EP-type model as the market size
grows. For a fixed market size, an oblivious equilibrium constitutes an −equilibrium in the sense that a
firm cannot improve its expected discounted profit by more than by unilaterally deviating and using a
strategy that keeps track of the industry state. We provide conditions under which converges to zero as
the market size grows. It may seem that this would be true provided that the average number of firms in the
industry grows to infinity as the market size grows. However, even if there are many firms in the market
asymptotically, if one firm is extremely large relative to the others, capturing a significant portion of the
market (e.g., Microsoft in the software industry), then the approximation may fail. Instead, we show that a
sufficient condition for an oblivious equilibrium to provide a good approximation asymptotically is that the
industry states generated by the sequence of oblivious equilibria are “light-tailed” in a precise mathematical
way. The condition can be easily checked in particular applications. For example, if the demand system
is logit with a common parameter specification and the spot market equilibrium is Nash in prices then the
condition holds if the average firm size in the industry is finite for all market sizes.2
We next provide a simple algorithm for computing oblivious equilibrium that works as follows. Conditional on a value for the long run average industry state, it is straightforward to solve for the optimal
oblivious strategy that a firm should follow who believes that his competitors will always be in that state
(this is a one-dimensional dynamic programming problem). Supposing that every firm in the industry follows this oblivious strategy, it is then straightforward to compute the long run average industry state. The
algorithm iterates these steps until convergence, such that the long run average state and oblivious strategy
obtained are consistent with each other.
The algorithm is computationally light and easy to program, typically requiring just 100-200 lines of
code. This is a considerable advantage over existing algorithms, which require a large investment in computer programming to be adapted to new problems. Another feature of the algorithm that differentiates it
from the previous literature is that it places no restrictions on how many firms may be present in the market
at one time and no restrictions on how large each firm may grow (i.e., it places no restrictions on the number
2
Note that the light-tail condition does not rule out the approximation working for industries that have infinite sized firms when
the market size is large, as long as these firms are observed with small enough probability.
2
of possible states per firm). Instead these are determined endogenously in equilibrium and computed alongside the equilibrium. Moreover, unlike previous algorithms, the computational burden of the algorithm is
fairly insensitive to the expected number of firms in the industry. It is easy to compute oblivious equilibria
even for industries with many thousands of firms.
The asymptotic results provide conditions under which the approximation works well for large enough
markets, but in practice one would also like to know exactly how well the approximation worked for a particular applied problem. For that purpose we derive performance bounds on the approximation error (an
expression for above) that can be computed via simulation. In particular, it is not necessary to know the
Markov perfect equilibrium (which would be computationally intractable) in order to compute the performance bounds. To our knowledge, these are the first bounds of this type in this literature. Using these
numerical bounds we find that an oblivious equilibrium typically yields a good approximation of the MPE
in industries with a couple hundred firms, and sometimes yields a good approximation even with just tens of
firms. These results suggest that oblivious equilibrium may be useful in many industry studies, particularly
those that consider whole industries and not just the top few largest firms.
The concept of oblivious equilibrium is closely related to Hopenhayn (1992). Hopenhayn assumes that
the industry hosts an infinite number of firms, each of which garners an infinitessimal fraction of the market.
His model is tractable because it assumes the industry state is constant every period, implicitly assuming a
law of large numbers holds. An oblivious equilibrium is inspired by the same intuition. The main difference
is that our goal is to analyze industries with a finite (possibly small) number of firms where a law of large
numbers does not hold exactly. Also, our EP-type model is a generalization of Hopenhayn’s model because
the transitional dynamics resulting from firms’ investment policies are generated by equilibrium behavior
that is explicitly modeled. Hopenhayn abstracts from this aspect of the model and instead assumes that
firms’ state trajectories follow exogenous Markov processes that are independent across firms.
Note that there is also a nice behavioral interpretation of oblivious strategies. If firms cannot increase
profits very much by using a strategy that keeps track of the complete industry state, and observing the
industry state is costly, then they may be better off using an oblivious strategy.
Our light-tail condition is similar to Sutton (1991)’s exogenous sunk costs model, which suggests conditions that lead to a diffuse industry structure when market size is large. Though our goals are different, our
asymptotic results are also similar in spirit to Novshek and Sonnenschein (1978) who, in a static setting, provide conditions under which Cournot-Nash equilibrium converges to Walras competitive equilibrium when
there is free entry. Also related is Vives (2002), which analyzes the impact of ignoring strategic behavior and
incomplete information as market size grows in a Cournot market. Though a detailed proof is beyond the
3
scope of this paper, our results also imply that a Hopenhayn-style model that assumes an infinite number of
infinitesimal firms, yields a good approximation of the behavior of an industry with a large but finite number
of firms, if the light-tail condition is satisfied.
Finally, to show the power of our results in practice, we apply our methods to a dynamic investment
model in which firms can invest (or advertise) to improve the quality of their product over time. An important question in industrial organization is to understand what features of an industry determine the industry’s
structure. Of particular interest (Sutton 1991) is whether a market becomes fragmented, or remains concentrated, as the market grows in size. Sutton (1991) sought to describe factors that determine industry structure
in large markets and that are robust across a wide class of models. In our model we find that, by changing
one parameter that determines the extent of vertical product differentiation a tiny amount, we are able to
generate two very different industry structures as the market size increases. In the first case the industry
becomes diffuse, with increasing numbers of firms but a stable distribution of firm sizes. In the second,
as the market size increases firms grow larger and larger without a proportional increase in the number of
firms. These results contradict Sutton (1991) in one sense, because market structure is not robust to a small
change in a single model parameter. An implication of the results is that different market structures might
be observed in the same industry across markets that are the same size and have very similar characteristics.
However, the results may also be seen as broadly consistent with Sutton (1991), since the extent of vertical
product differentiation impacts the returns to investment. Consistent with Sutton (1991), industries with
higher returns to investment tend to be concentrated.
The paper is organized as follows. In Section 2 we outline the dynamic industry model. In Section 3 we
introduce the concept of oblivious strategies and oblivious equilibrium. In Section 4 we provide conditions
under which oblivious strategies approximate MPE strategies asymptotically as the market size grows. In
Section 5 we provide an algorithm and performance bounds for an oblivious equilibrium. In Section 6 we
perform computational experiments. Finally, Section 7 contains some conclusions and discussion for future
research.
2
A Dynamic Industry Model of Imperfect Competition
In this section we formulate a general model of an industry in which firms compete in a single-good market.
The general model captures many of the dynamic phenomena mentioned above, and is easily extended to
cover many more. Our model is close in spirit to that of Ericson and Pakes (1995), but with some differences.
4
Most notably, we generalize the entry and exit processes in Ericson and Pakes (1995) so as to make them
more realistic when there are a large number of firms. Additionally, our approximation method cannot
handle aggregate industry shocks so our model includes only idiosyncratic shocks.
2.1
Model and Notation
The industry evolves over discrete time periods and an infinite horizon. We index time periods with nonnegative integers t ∈ N (N = {0, 1, 2, . . .}). All random variables are defined on a probability space
(Ω, F, P) equipped with a filtration {Ft : t ≥ 0}. We adopt a convention of indexing by t variables that are
Ft -measurable.
At each time t ∈ N, there are nt incumbent firms. Each firm that enters the industry is assigned a unique
positive integer-valued index. The set of indices of incumbent firms at time t is denoted by St . Firms are
heterogeneous, and this heterogeneity is reflected in a set of firm-specific state variables. We will refer to
firms’ states as their quality level. However, more generally firms’ states might reflect the quality of their
products, their capacities, their capital stocks, or any other aspect of the firms that affects their profits. At
time t, the quality level of firm i ∈ St is denoted by xit ∈ N. For each state x ∈ N, we define the industry
state, st (x) ∈ N, to be the number of incumbent firms at time t with state x. The set of possible industry
P
n
o
states is denoted by S = s ∈ N∞ ∞
s(x)
<
∞
. For each i ∈ St , we define s−i,t ∈ S to be the state
x=0
P
of the competitors of firm i. Similarly, let n−i,t = x∈N s−i,t (x) refer to the number of competitors of firm
i. It is also useful to define f−i,t (x) as the fraction of firms i’s competitors at quality level x in time t, such
that, f−i,t =
s−i,t
n−i,t .
In each period, each incumbent firm earns profits on a spot market. A firm’s expected profits depend on
its quality level, xit , its competitors’ state, s−i,t , and the market size, m, and are denoted by πm (xit , s−i,t ).
With some abuse of notation, we define πm (xit , f−i,t , n−i,t ) ≡ πm (xit , n−i,t · f−i,t ).3 Firms have common
discount factor β.
In each period, each incumbent firm i ∈ St observes a positive real-valued sell-off value φit that is
private information to the firm. If the sell-off value exceeds the value of continuing in the industry then the
firm may choose to exit, in which case it earns the sell-off value and then ceases operations permanently.
If the firm instead decides to remain in the industry, then it can invest to improve its quality level. If a
3
n
For reasons
that will o become clear later, we extend the domain of πm (x, ·) to the real numbers, S
P
∞
s ∈ <∞
s(x)
<∞ .
+
x=0
5
=
firm invests ιit , then the firm’s state at time t + 1 is given by,
xi,t+1 = xit + w(ιit , ζi,t+1 ),
where ζi,t+1 reflects the uncertain impact of an investment. Uncertainty may arise, for example, due to the
risk associated with a research and development endeavor or a marketing campaign. We denote the unit cost
of investment by d.
Finally, in each period new firms can enter the industry by paying a setup cost κ. Entrants do not earn
profits in the period that they enter. Instead, they appear in the following period at state xe ∈ N.
2.2
Assumptions
We make the following assumptions on the profit function:
Assumption 2.1.
1. For all m, s, πm (x, s) is increasing in x.
2. For all m, x, f and n2 > n1 , πm (x, f, n1 ) ≥ πm (x, f, n2 ).
3. For all m, x, f and n, πm (x, f , n) ≥ πm (x, f, n), where f = (1, 0, 0, 0, ...).
4. There exists a scalar Y such that for all x, s, 0 < πm (x, s) ≤ mY .
The assumptions are natural. Assumption 2.1.1 ensures that increases in quality lead to increases in
profit. Assumption 2.1.2 states that, for a fixed distribution over firm sizes, as the total number of firms
increases, the profit garnered by any particular firm does not increase. Assumption 2.1.3 ensures that for a
fixed number of competitors, a firm makes the largest profits if all its competitors are at the lowest quality
level. Assumption 2.1.4 ensures that profits are positive and that they scale at most linearly with market size
— Y should be thought of as the income of a representative consumer.
We also make assumptions on the distributions of the private shocks,
Assumption 2.2.
1. φit is i.i.d. across firms and time and has a well defined density function with support <+ and finite
expectation.
2. The random variables {ζit |t ≥ 0, i ≥ 1} are i.i.d. and independent of {φit |t ≥ 0, i ≥ 1}.
3. w(ι, ζi,t+1 ) is nondecreasing in ι, and P[w(ι, ζi,t+1 ) > 0] > 0, if ι > 0.
4. There exists a positive constant w ∈ N such that |w(ι, ζ)| ≤ w, for all (ι, ζ), and such that P[w(ι, ζi,t+1 ) =
k] is continuous in ι, for all k ∈ {−w, ..., w}.
6
Assumptions 2.2.1 and 2.2.2 imply that investment and exit outcomes are idiosyncratic conditional on
the state. Assumption 2.2.3 implies that investment is productive. Assumption 2.2.4 places a finite bound
on how much progress can be made or lost in a single period through investment.
Entry is determined as follows,
Assumption 2.3.
1. The number of entrants in period t is a Poisson random variable with mean λ(st ).
2. κ > β · φ̄, where φ̄ is defined as the expected discounted value of entering the market, investing zero
each period, and then exiting at an optimal stopping time.
Below, we will require that λ(st ) satisfies a zero expected profits condition at each state. We use the
Poisson distribution because for our asymptotic results we require an entry process that is robust to changes
in the market size. The Poisson distribution well-approximates a binomial distribution when there are a large
number of potential entrants each of which has a small probability of entering the industry (see appendix for
a derivation). Modelling the number of entrants as a Poisson random variable also simplifies the mathematical analysis and leads to more elegant results. Assumption 2.3.2 ensures that the sell-off value by itself is
not sufficient reason to enter the industry.
Finally, in each period, we assume that events occur in the following order:
1. Incumbent firms make their exit and investment decisions.
2. The number of entering firms is determined by the outcome of a Poisson random variable with mean
λ(st ). Each entrant pays an entry cost κ.
3. Firms compete in the spot market, receiving profits πm (xit , s−i,t ).
4. Exiting firms exit and receive their sell-off values.
5. Investment outcomes are determined, new entrants enter, and the industry takes on a new state st+1 .
2.3
Equilibrium
To analyze equilibrium behavior we focus on pure strategy Markov perfect equilibrium (MPE), in the sense
of Maskin and Tirole (1988). We further assume that equilibrium is symmetric, such that every firm uses
the same stationary exit/investment strategy. In particular, there is a function ι such that at each time t, each
incumbent firm i ∈ St invests an amount ιit = ι(xit , s−i,t ). Similarly, each firm follows an exit strategy
that follows a cutoff rule: there is a real-valued function ρ such that an incumbent firm i ∈ St exits at time t
if and only if φit ≥ ρ(xit , s−i,t ). In the appendix we show that there always exists an optimal exit strategy
7
of this form. Let M denote the set of investment and exit strategies such that an element µ ∈ M is a pair
of functions µ = (ι, ρ), where ι : N × S → <+ is an investment strategy and ρ : N × S → <+ is an
exit strategy. Similarly, we denote the set of entry rate functions by Λ, where an element of Λ is a function
λ : S → <+ .
We define the value function V (x, s|µ0 , µ, λ) to be the expected discounted value of profits for a firm
at state x when the industry is in state s, and given that all rival firms follow policy µ ∈ M, the entry rate
function is λ ∈ Λ and the firm itself follows strategy µ0 ∈ M.
"
V (x, s|µ0 , µ, λ) = Eµ0 ,µ,λ
τi
X
#
β k−t (πm (xik , s−i,k ) − dιik ) + β τi −t φi,τi xit = x, s−i,t = s ,
k=t
where i is taken to be the index of a firm at quality level x at time t and τi is a random variable representing
the time at which firm i exits the industry. In an abuse of notation, we will use the shorthand, V (x, s|µ, λ) ≡
V (x, s|µ, µ, λ), to refer to the expected discounted value of profits when firm i follows the same policy µ as
its rivals.
An equlibrium to our model comprises an investment/exit strategy µ = (ι, ρ) ∈ M, and an entry rate
function λ ∈ Λ that satisfy the following conditions:
1. Incumbent firm strategies represent a MPE:
sup V (x, s|µ0 , µ, λ) = V (x, s|µ, λ)
(2.1)
µ0 ∈M
∀x ∈ N, ∀s ∈ S.
2. At each state, either entrants have zero expected profits or the entry rate is zero (or both):
P
s∈S
λ(s) (βEµ,λ [V (xe , st+1 |µ, λ)|st = s] − κ) = 0
βEµ,λ [V (xe , st+1 |µ, λ)|st = s] − κ ≤ 0
∀s ∈ S
λ(s) ≥ 0
∀s ∈ S.
Doraszelski and Satterthwaite (2003) show existence of equilibrium in pure strategies in a closely related
model. We do not provide an existence proof here because it is long and cumbersome and would replicate
this previous work. With respect to uniqueness, in general we presume that our model may have many
equilibria.4
If the number of firms and the number of states per firm are not both very small, the state space of our
model will be enormous. Therefore, computation of equilibrium policies is subject to a curse of dimen4
Doraszelski and Satterthwaite (2003) also provide an example of multiple equilibria in their closely related model.
8
sionality, and exact computation of equilibrium would be prohibitive in many applications. Several papers
have proposed methods to alleviate this curse of dimensionality (Pakes and McGuire (1994), Pakes and
McGuire (2001) and Doraszelski and Judd (2003)). However, even with these enhanced methods current
algorithms can only handle industries with a small number of firms and around 10-20 state points per firm.
This difficulty motivates our alternative approach.
3
Oblivious Equilibrium
In this section we consider restricting firms’ strategies so that each firm’s decisions depend only on the
firm’s own quality level. We call such restricted strategies oblivious since they involve decisions made
without full knowledge of the circumstances — in particular, the state of the industry. It may seem that
oblivious strategies and entry rate functions should lead to industry behavior very different from what would
be observed in an equilibrium of the kind defined above. However, as we will establish later in the paper,
when the number of firms is large (and under some additional conditions), oblivious strategies and entry rate
functions can closely approximate a MPE. Additionally, the computation of oblivious strategies bypasses
the curse of dimensionality because it effectively restricts the state space considered by each firm to a single
dimension — its quality level xit .
3.1
Oblivious Strategies and Entry Rate Functions
Let M̃ ⊂ M and Λ̃ ⊂ Λ denote the set of oblivious strategies and the set of oblivious entry rate functions.
Since each strategy µ = (ι, ρ) ∈ M̃ generates decisions ι(x, s) and ρ(x, s) that do not depend on s, with
some abuse of notation, we will often drop the second argument and write ι(x) and ρ(x). Similarly, for
an entry rate function λ ∈ Λ̃, we will denote by λ the real-valued entry rate which persists for all industry
states.
3.2
Oblivious Equilibrium
Note that if all firms use a common strategy µ ∈ M̃, the quality level of each evolves as an independent transient Markov chain. Let the transition sub-probabilities of this transient Markov chain be denoted by Pµ (x, y). Then, the expected time that an entering firm spends at a quality level x is given by
P∞
P∞ P
k e
k e
k=0 Pµ (x , x), and the expected lifespan of a firm is
k=0
x∈N Pµ (x , x). Denote the expected state
of the industry at time t by s̃t (x) = E[st (x)], ∀x. The following result offers a nice expression for the
9
long-run expected state when dynamics are governed by oblivious strategies and entry rate functions.
Lemma 3.1. If firms make decisions according to an oblivious strategy µ ∈ M̃ and enter according to an
oblivious entry rate function λ ∈ Λ̃, and the expected time that a firm spends in the industry is finite, then
lim s̃t (x) = λ
(3.1)
t→∞
∞
X
Pµk (xe , x),
k=0
for all x ∈ N.
We omit the proof, which is straightforward. To abbreviate notation, we let s̃µ,λ (x) = limt→∞ s̃t (x) for
µ ∈ M̃, λ ∈ Λ̃, and x ∈ N. For an oblivious strategy µ ∈ M̃ and an oblivious entry rate function λ ∈ Λ̃
we define an oblivious value function
"
Ṽ (x|µ0 , µ, λ) = Eµ0
τi
X
#
β k−t (πm (xik , s̃µ,λ ) − dιik ) + β τi −t φi,τi xit = x .
k=t
This value function should be interpreted as the expected discounted value of profits perceived by a firm
under the assumption that its competitors’ state will be equal to s̃µ,λ in all future periods (and when all rival
firms follow oblivious strategy µ, the entry rate is λ and firm i follows oblivious strategy µ0 ). Note that
only the own firm’s strategy µ0 enters the expectation operator because neither the profit function nor the
strategy µ0 depends on the industry state. The oblivious value function depends on rival strategies, µ, and
the entry rate, λ, only through the expected industry state s̃µ,λ . Similarly to above, we abuse notation by
using Ṽ (x|µ, λ) ≡ Ṽ (x|µ, µ, λ) to refer to the oblivious value function when firm i follows the same policy
µ as its rivals.
We now define a new solution concept: an oblivious equilibrium consists of a strategy µ ∈ M̃ and an
entry rate function λ ∈ Λ̃ that satisfy the following conditions:
1. Firm strategies optimize an oblivious value function assuming state s̃µ,λ is self-generated:
(3.2)
sup Ṽ (x|µ0 , µ, λ) = Ṽ (x|µ, λ),
∀x ∈ N.
µ0 ∈M̃
2. Either the oblivious expected value of entry is zero or the entry rate is zero (or both):
λ β Ṽ (xe |µ, λ) − κ = 0
β Ṽ (xe |µ, λ) − κ ≤ 0
λ ≥ 0.
10
It is straightforward to show that an oblivious equilibrium always exists using a fixed point argument.
Furthermore, if the entry cost is not prohibitively high then an oblivious equilibrium with a positive entry
rate always exits. We omit a proof of this for brevity and because our approach is computational. In Section
5 we introduce an algorithm to compute oblivious equilibrium. Once an equilibrium is found numerically,
it follows that an equilibrium exists. With respect to uniqueness, we have been unable to find multiple
oblivious equilibria in any of our examples but, similarly to the case for MPE, we have no reason to believe
that in general there is a unique oblivious equilibrium.
Note that we assume that, even if firms are using oblivious strategies, the sequence of single-period
profits received are the ones associated with Nash, and not oblivious, static behavior. For example, if singleperiod profits are derived from a game in which firms compete in prices, we assume firms price according to
Nash equilibrium strategies, and are not oblivious with respect to pricing behavior. We make this assumption
because we are interested in the dynamic behavior of the industry, and we believe that assuming oblivious
pricing behavior, which would be cumbersome, would not change our conclusions in that respect.
We will later explore situations where the number of firms is large and an oblivious equilibrium approximates a true equilibrium of the model in a sense that we will define precisely. First, we analyze the long-run
behavior of an industry where strategies and the entry rate function are oblivious.
3.3
The Invariant Industry Distribution
In Lemma 3.1, we characterized the long-run expected industry state. Our next result characterizes the
long-run distribution. The symbol ⇒ denotes weak convergence as t → ∞.
Lemma 3.2. Suppose firms make decisions according to an oblivious strategy µ ∈ M̃ and enter according
to an oblivious entry rate function λ ∈ Λ̃, and the expected time that a firm spends in the industry is finite.
Then, {st : t ≥ 0} is an irreducible, aperiodic and positive recurrent Markov chain that admits an invariant
distribution with the product form of Poisson random variables. Moreover, let {Zx : x ∈ N} be a sequence
of independent Poisson random variables with means {s̃µ,λ (x) : x ∈ N}. Then, st (x) ⇒ Zx , ∀x. Also,
P
nt ⇒ Z, where Z is a Poisson random variable with mean x∈N s̃µ,λ (x).
To conclude this subsection we state an important result for later use. First note that
st (x) =
X
1{xit = x} =
nt
X
1{x(j)t = x},
j=1
i∈St
where {x(j)t : j = 1, ..., nt } is a random permutation of {xit : i ∈ St }. That is, we randomly pick a firm
11
from St and assign to it the index j = 1; from the remaining firms we randomly pick another firm and assign
to it the index j = 2, and so on.
Lemma 3.3. Suppose firms make decisions according to an oblivious strategy µ ∈ M̃ and enter according
to an oblivious entry rate function λ ∈ Λ̃, and the expected time that a firm spends in the industry is
finite. Let {Yn : n ∈ N} be a sequence of independent random variables, each distributed according to
P
s̃µ,λ (·)/ x∈N s̃µ,λ (x). Then, for all n ∈ N,
x(1)t , ..., x(nt )t nt = n ⇒ (Y1 , ..., Yn ).
The proof of both lemmas can be found in the appendix. Assuming a Poisson entry process is key
for proving the result. Essentially, the result ensures that if we sample a firm randomly from those firms
currently in the industry and the industry state is distributed according to the invariant distribution, the
firm’s state will be distributed according to the normalized expected industry state. Further, each subsequent
time we sample without replacement we get an independent sample from the same distribution. For brevity,
when we consider sampling a random firm from among those currently in the industry and the industry state
is distributed according to the invariant distribution, we will abbreviate by saying that we are sampling a
firm from the industry’s invariant distribution.
Note that it is straightforward to show that the expected time a firm spends inside the industry is finite for
any oblivious strategy µ ∈ M̃ that comprises an oblivious equilibrium. This follows from the fact that the
sell-off value has support in <+ and the continuation value from every state is bounded above by
mY
1−β
+ φ.
Therefore, the probability of exiting is bounded below by a strictly positive constant. Then we use the fact
that the expected value of a geometric random variable is finite. This implies that the previous lemmas apply
when firms use oblivious equilibrium strategies.
4
Asymptotic Results
In this section we consider a sequence of industries indexed by the market size m. We index functions
and random variables associated with market m with a superscript (m). To help clarify the notation, from
this point onward we denote oblivious equilibrium using (µ̃, λ̃), so that (µ̃(m) , λ̃(m) ) denotes an oblivious
equilibrium for industry m. Let V (m) and Ṽ (m) represent the value function and oblivious value function
respectively when the market size is m. To further shorten notation we denote the expected industry state
associated with (µ̃(m) , λ̃(m) ) as s̃(m) ≡ s̃µ̃(m) ,λ̃(m) .
12
(m)
The random variable st
denotes the industry state at time t when every firm uses strategy µ̃(m) and the
(m)
entry rate is λ̃(m) . We denote the invariant distribution of {st
: t ≥ 0} by q (m) . In order to simplify our
(m)
analysis, we assume that the initial industry state for industry m, s0 , is sampled from q (m) . This makes
(m)
st
(m)
a stationary process: st
is distributed according to q (m) for all t ≥ 0. Note that this assumption
simplifies the exposition and analysis but does not impact any of the asymptotic results.
(m)
It will be helpful to decompose st
(m)
as follows, st
(m) (m)
nt ,
= ft
(m)
where ft
is the fraction of firms
(m)
(m)
is the total number of firms, respectively. Similarly, let f˜(m) ≡ E[ft ] denote the
P
(m)
expected fraction of firms in each state and ñ(m) ≡ E[nt ] = x∈N s̃(m) (x) denote the expected number
in each state and nt
of firms. Using Lemma 3.3, it is easy to check that f˜(m) =
s̃(m)
.
ñ(m)
The main objective of this section is to establish conditions under which an oblivious equilibrium yields
a good approximation of the behavior of the dynamic industry model as market size grows.
4.1
Uniform Law of Large Numbers
The following theorem (uniform law of large numbers) says that when the number of firms is large, the
(m)
industry state becomes approximately constant (st
≈ s̃(m) ).
Theorem 4.1. For all t > 0, if limm→∞ ñ(m) = ∞, then
#
s(m) (x) s̃(m) (x) lim P sup t (m) −
> =0.
m→∞
ñ
ñ(m)
x∈N
"
The theorem can be proved by invoking a uniform law of large numbers (Vidyasagar (2002)) and using
Lemma 3.3. It suggests that when the expected number of firms is large, using an oblivious strategy might
be close to optimal. However, for this to be the case it turns out that additional conditions are required. To
make this rigorous we introduce some additional concepts.
4.2
Asymptotic Markov Equilibrium
Consider an oblivious equilibrium, (µ̃(m) , λ̃(m) ), in market m. We say that this is an “−Markov” equilibrium if, on average over the long run distribution of states (q (m) ), no firm can improve its expected
discounted profits by more than by unilaterally deviating from the oblivious equilibrium strategy µ̃(m) and
instead following an optimal non-oblivious strategy:
13
Definition 4.1. (µ̃(m) , λ̃(m) ) is an −Markov equilibrium if:
"
E
#
sup V
µ0 ∈M
(m)
(m)
(x, st |µ0 , µ̃(m) , λ̃(m) )
−V
(m)
(m)
(x, st |µ̃(m) , λ̃(m) )
∀x ∈ N .
< ,
Similarly, a sequence of oblivious equilibria, (µ̃(m) , λ̃(m) ), has the “asymptotic Markov equilibrium”
(AME) property if it represents a sequence of −Markov equilibria in which tends to zero (point-wise for
all states x ∈ N).
Definition 4.2. A sequence (µ̃(m) , λ̃(m) ) ∈ M × Λ has the asymptotic Markov equilibrium (AME) property
if for all x ∈ N,
"
lim E
m→∞
#
sup V
µ0 ∈M
(m)
(m)
(x, st |µ0 , µ̃(m) , λ̃(m) )
−V
(m)
(m)
(x, st |µ̃(m) , λ̃(m) )
=0.
Below we establish conditions under which a sequence of oblivious equilibria has the AME property. Note that when standard solution algorithms (e.g., Pakes and McGuire (1994)) are used a point-wise
−equilibrium is obtained; that is, a firm can not improve its value function more than starting from any
state (x, s). The AME property instead considers the benefit of deviating to an optimal policy starting from
state x, averaged over the invariant distribution of industry states. It would not be possible to obtain our
results point-wise. This is because in an oblivious equilibrium firms may be making poor decisions in states
that are far from the expected state. Offsetting this effect is the fact that these states have very low probability
of occurrence, so they have a small impact on expected discounted profits.5
If a sequence of oblivious equilibria has the AME property then, asymptotically as m grows, we have
that, supµ0 ∈M V (m) (x, s|µ0 , µ̃(m) , λ̃(m) ) ≈ V (m) (x, s|µ̃(m) , λ̃(m) ), for states s in the recurrent class. This
implies that, asymptotically, µ̃(m) is a MPE strategy for games restricted to the recurrent class. Similarly,
since β Ṽ (m) (xe , s) = κ for all m, asymptotically it must be that βV (m) (xe , s|µ̃(m) , λ̃(m) ) ≈ κ for states s in
the recurrent class. Hence, asymptotically, λ̃(m) satisfies the zero profit condition for states in the recurrent
class, implying that (µ̃(m) , λ̃(m) ) represents a MPE of the game. In summary, if an oblivious equilibrium is
an −Markov equilibrium and is small, MPE strategies and entry rates in the recurrent class should be well
approximated by oblivious ones. We will later check, using computational experiments, that this is actually
the case.6
5
Note that Pakes and McGuire (2001) use a similar concept as a stopping rule in their stochastic algorithm. They compare the
difference in the value function from two consecutive iterates, weighting states according to their empirical distribution of visits in
the long run.
6
One could try to formalize this heuristic argument by following a similar line of reasoning as Fudenberg and Levine (1986) or
Altman, Pourtallier, Haurie, and Moresino (2000). They provide conditions under which, if a sequence of restricted games, Gm ,
14
4.3
A Light-Tail Sufficient Condition
In this subsection, we derive a sufficient condition that ensures that the sequence of oblivious equilibria
has the AME property. Theorem 4.1 says that if the expected number of firms grows to infinity, then the
industry state converges to a constant state. This suggests that a sequence of oblivious equilibria should
have the AME property. However, this is not necessarily the case. Even when there are many firms in the
overall market, it may be that there are one or more very large firms whose actions (particularly exit) affect
the profits of the remaining firms in important ways. In that case a strategy that does not keep track of the
large firms’ states is unlikely to perform well. Thus, having a large number of firms is a necessary but not
sufficient condition for a sequence of oblivious equilibria to have the AME property. It turns out that we also
need to impose a “light-tail” condition over the sequence of oblivious equilibria industry states that rules out
very large (in a relative sense) firms occurring with high probability.
We now make this statement more precise, but first we require some additional assumptions. We begin
with some technical conditions on the profit function.
Assumption 4.1.
1. Let Fz = {f ∈ ∆ : f (x) = 0, ∀x > z}, where ∆ is the infinite dimensional unit simplex. Let n(m)
be a sequence, such that, limm n(m)
m = 0. Then, for all z and all x ≥ z:
lim πm (x, f, n(m)) = ∞ , ∀f ∈ Fz .
m→∞
2. There exists constants B, C, α > 0, with αw β < 1, such that, for all x, m, and n,
πm (x, f , n) ≤ πm (0, f , n) (B + Cαx ). Additionally, for all m and n, πm (0, f , n) > 0.
Assumption 4.1.1 is natural. It states that if the number of firms is sufficiently small compared to the
market size then one period profits for the largest firm become arbitrarily large when the market size is
large. Assumption 4.1.2 states that a firm with all its competitors at the lowest state will make strictly
positive profits, and states that the relative advantage of being at higher quality levels cannot grow faster
than exponentially (note that α could be greater than one). In the appendix (at the end of subsection A.3) we
argue that this assumption is essentially without loss of generality as it is a necessary condition for the lighttail assumption below. Note also that these requirements are satisfied by many profit functions of practical
interest.7
converges to a game of interest, G, in an appropriate way, then any convergent sequence of m −equilibria of Gm with m → 0
converges to an equilibrium of G. In our case, we do not have a well defined limit game. If m = ∞, profits are infinite and we
would have infinite entry. This makes it hard to use their line of thought.
7
For example, if demand is given by a logit model and the static equilibrium is Nash in prices then the assumptions are satisfied.
A linear-quadratic Cournot model satisfies the first assumption and violates the second but it is straightforward to find an alternative
assumption that guarantees the same results.
15
Let,
(4.1)
∂ ln π (y, f, n) m
g(k) = sup .
∂fk
m,y,f,n
g(k) is the maximum percentage change on any firm’s single-period profits that a small change in the fraction
of firms at state k can inflict (more on this below).8 We assume that g is a strictly increasing function.
Assumption 4.2. g is defined in equation (4.1) and is a strictly increasing function.
The fact that g is strictly increasing is natural considering that by Assumption 2.1.1 firms in larger states
should have larger market shares. Hence, changes in the fraction of firms in higher states should have a
larger impact on profits.
We also require that the profit function satisfy a continuity condition. Let k f k1,g =
P
x |f (x)|g(x)
be
a 1-weighted norm.
Assumption 4.3. For all > 0, there exists δ > 0 such that for all f, fˆ, n, n̂ satisfying
n
k f − fˆ k1,g < δ and − 1 < δ ,
n̂
we have
π (y, f, n) − π (y, fˆ, n̂) m
m
sup < , ∀m .
ˆ
πm (y, f , n̂)
y∈N
Note that because we use a weighted norm this assumption allows changes in the fraction of firms at
larger states to have a greater effect on profits, as we would expect.9
Finally, we introduce our light-tail condition.
Assumption 4.4. There exists γ > 0 such that
h
i
sup E g 1+γ (x̃(m) ) < ∞ ,
m
where x̃(m) is a nonnegative discrete random variable distributed according to f˜(m) .
The tail of x̃(m) must decay fast enough to make E[g 1+γ (x̃(m) )] uniformly bounded over m (see explanation below). Note that using Lemma 3.3 it is easy to check that f˜(m) =
(m)
P s̃ (m)
.
(y)
y s̃
Therefore, x̃(m)
represents the state of a random sampled firm from industry m’s invariant distribution.
8
Note that in principle f lives in the infinite dimensional unit simplex. To avoid problems when evaluating its partial derivatives,
we assume that πm (y, f, n) is well defined for values of f in an open set that strictly contains the infinite dimensional simplex.
9
It is also possible to show that this assumption follows from differentiability of the profit function in f .
16
The following proposition says that, under the assumptions above, the number of firms grows with the
market size:
Proposition 4.1. Under Assumptions 2.1.2, 4.1.1, 4.2, 4.3, and 4.4,
ñ(m) = Ω(m).10
The proof can be found in the appendix. The intuition behind the proposition is simple. If the number
of firms were to grow slower than the market size, profits would blow up and the zero profit condition at the
entry state would not be met. The proposition is used to prove that the sequence of oblivious equilibrium
has the AME property when the sequence of industry states is light-tailed, one of our main results:
Theorem 4.2. Under Assumptions 2.1, 4.1, 4.2, 4.3 and 4.4, the sequence of oblivious equilibria has the
AME property.
A proof can be found in the appendix. The key assumption of the theorem is the light-tail condition.
Note that
∂ ln πm (y,f,n)
∂fk
is the semi-elasticity of one period profits with respect to the fraction of firms in
state k. The light tail condition requires that (up to the arbitrarily small γ) the supremum of this semielasticity over m, y, f, n has a uniformly bounded expectation. The expectation is taken with respect to
the distribution of x̃(m) , which is a random draw from industry m’s invariant distribution (i.e., a random
draw from the distribution of firms in industry m). Thus, the light-tail condition requires that states where
a change in the fraction of firms has a large impact on profits must have a small probability of occurrence
under the invariant distribution.
While the condition is quite intuitive, it may also be instructive to consider an example. If demand is
given by the logit model with Cobb-Douglas utility and the static equilibrium is Nash in prices, then the
1+γ
condition requires that supm E[ x̃(m)
] < ∞. This condition amounts to requiring that the distribution
of firms have a finite first (plus γ) moment. This condition would be satisfied by most distributions (e.g.,
lognormal, exponential, etc.) but would not be satisfied for a Pareto with parameter one.11 Note that the light
tail condition can also be verified by computing oblivious equilibria for different values of m (see examples
below).
(m)
10
I.e., there exists a finite constant C > 0, such that, C ≤ lim inf m ñ m . Note that with an additional regularity condition it is
also straightforward to show that n(m) = O(m) (i.e., the number of firms cannot grow faster than rate m).
11
A more detailed discussion can be found in the appendix.
17
5
Algorithm and Performance Bounds
Under the light-tail condition an oblivious equilibrium should yield a good approximation of an industry
with a large market size. Motivated by this fact, in this section we propose an algorithm to compute an
oblivious equilibrium. We then show that an oblivious equilibrium is an −Markov equilibrium and derive
an expression for that can be easily computed via simulation. This expression is very useful as it can
be used to evaluate the oblivious equilibrium as an approximation for MPE. Note that in this section we
consider an industry with a fixed market size, so to simplify notation we suppress the index m.
5.1
Algorithm
The following algorithm can be used to compute an oblivious equilibrium as defined in Subsection 3.2:
1. Begin with an arbitrary entry rate λ0 . Let i = 0.
2. Find µi such that:
sup Ṽ (x|µ0 , µi , λi ) = Ṽ (x|µi , λi ), ∀x.
µ0 ∈M̃
To do this:
(a) Begin with strategy µi0 . Let j = 0.
(b) Compute expected state s̃µij ,λi using equation (3.1). This involves inverting a matrix, because
P∞
k e
−1 e
k=0 Pµ (x , x) = (I − Pµ ) (x , x), where I is the identity matrix. Note that the inverse
always exists if the expected time a firm spends inside the industry is finite.
(c) Solve for supµ0 ∈M̃ Ṽ (x|µ0 , µij , λi ), by solving the corresponding Bellman’s equation. Call the
solution µij+1 . Note that at each iteration (i, j) there will be a state x̄ij (determined endogenously) such that ι(x) = 0, ∀x > x̄ij . This allows the state space to be reduced to [0, x̄ij ] when
solving the Bellman equation.
(d) If k µij − µij+1 k∞ < 1 , where 1 is a small predefined precision chosen by the researcher, stop
and let µi = µij . If not, let µij+1 = η j µij + (1 − η j )µij+1 , where η j is a predefined sequence
in the interval [0, 1] that converges to 1 that is used to smooth the algorithm. Let j = j + 1 and
go back to step (b).
3. If |β Ṽ (xe |µi , λi ) − κ| < 2 , where 2 is a small predefined precision chosen by the researcher, then
stop. If not, let λi+1 = λi + h(β Ṽ (xe |µi , λi ) − κ), where h is an increasing function chosen by the
researcher such that h(0) = 0. Let i = i + 1 and go back to (2).
18
Note that the algorithm is easy to program and computationally light. The Bellman equation solution
involves a one dimensional dynamic program that in practice would have no more than a couple of hundred
states (the exact number is determined endogenously). We programmed the algorithm in Matlab and on
average in our example models it required a couple of minutes on a Pentium IV laptop (run time is mostly
independent of the number of firms in the resulting equilibrium).
5.2
Performance Bounds
The next theorem provides two different bounds on the performance of an oblivious equilibrium strategy,
(µ̃, λ̃). As before, let q be the invariant distribution of {st : t > 0} when every firm uses strategy µ̃ and the
entry rate is λ̃ and let s be a random vector distributed according to q. s̃ is the expected state in equilibrium
(E(s)). Finally, let ax (y) be the discounted expected total number of visits to state y for a firm starting at
state x that uses strategy µ̃ .
Theorem 5.1.
1.
E[ sup V (x, s|µ0 , µ̃, λ̃) − V (x, s|µ̃, λ̃)] ≤
µ0 ∈M
2
E[∆|πm |(s)] , ∀x ∈ N ,
1−β
where ∆|πm |(s) = maxy∈N |πm (y, s) − πm (y, s̃)|.
2. For all x ∈ N,
E[ sup V (x, s|µ0 , µ̃, λ̃)−V (x, s|µ̃, λ̃)] ≤
µ0 ∈M
X
1
E[∆πm (s)]+
ax (y) (πm (y, s̃) − E [πm (y, s)]) ,
1−β
y∈N
where ∆πm (s) = maxy∈N (πm (y, s) − πm (y, s̃)).
The derivation of these bounds can be found in the appendix. Note that these bounds are very general.
They depend on the idiosyncratic nature of the shocks but do not require any of the previous assumptions on
the profit function, the entry process or the specific structure of the model. The first bound is simpler so we
will use it to build the intuition behind the bounds. According to the first bound, if one period profits when
the state is sampled from the invariant distribution are close to one period profits evaluated at the expected
state, then an oblivious equilibrium will be an −Markov equilibrium with a small . This is because, in that
case, firms cannot improve their expected discounted profits very much by unilaterally deviating from the
oblivious strategy and keeping track of the industry state. We will come back to this point later.
19
The second bound is more involved. We provide it because it is much tighter (and we use it for the
computational results presented in the next subsection). Note that the second bound is a point-wise, rather
than uniform, bound (different for each x). It is possible to derive even tighter bounds that take advantage
of the specific structure of the model that we use for the computations, and we will provide one example of
this below.
Importantly, both bounds can be easily computed via simulation from the oblivious equilibrium. Computing the bounds involves computing an expectation over the industry state s. Once the oblivious equilibrium has been computed, the industry state has a known distribution, namely, the product form of Poisson
random variables with mean s̃ (by Lemma 3.2). In particular, note that the bounds are not a function of the
true MPE or even of the optimal non-oblivious best response policy, both of which we do not know because
their computation is subject to a curse of dimensionality.
6
Computational Experiments
In this section we conduct some computational experiments to evaluate how oblivious equilibrium performs
in practice. We begin with the model to be analyzed. The model is based on that of Pakes and McGuire
(1994).
6.1
The Computational Model
S INGLE - PERIOD
PROFIT FUNCTION .
We consider an industry with differentiated products, where each
firm’s state variable represents the quality of its product. There are m consumers in the market (definition
of the market size). In period t, consumer j receives utility uijt from consuming the good produced by firm
i given by:
uijt = θ1 ln(xit + 1) + θ2 ln(Y − pit ) + νijt , i ∈ St , j = 1, ..., m,
where Y is the consumer’s income and pit is the price of the good produced by firm i. νijt are i.i.d. random
variables distributed Gumbel that represent unobserved characteristics for each consumer-good pair. There
is also an outside good that provides consumers zero utility. We assume consumers buy at most one product
each period and that they choose the product that maximizes utility. Under these assumptions our demand
system is a classical logit model.
Let n(xit , pi ) = exp(θ1 ln(xit + 1) + θ2 ln(Y − pi )). Then, the expected market share of each firm is
20
given by:
σ(xit , s−i,t , p) =
n(x , p )
P it i
, ∀i ∈ St .
1 + j∈St n(xjt , pj )
We assume that firms set prices in the spot market. If there is a constant marginal cost c, the Nash equilibrium
of the pricing game satisfies the first-order conditions,
Y − pi + θ2 (pi − c)(σ(xit , s−i,t , p) − 1) = 0 , ∀i ∈ St .
There is a unique Nash equilibrium in pure strategies, denoted p∗i (Caplin and Nalebuff (1991)). Expected
profits are given by:
πm (xit , s−i,t ) = mσ(xit , s−i,t , p∗ )(p∗i − c) , ∀i ∈ St .
S ELL - OFF PRICE . φit are i.i.d. exponential random variables with mean K.
T RANSITION DYNAMICS . A firm’s investment is successful with probability
aι
1+aι ,
in which case the quality
of its product increases by one level. The firm’s product depreciates one level with probability δ, independently each period. Note that our model differs from Pakes and McGuire (1994) here because the depreciation shocks in our model are idiosyncratic. Combining the investment and depreciation processes, it follows
that the transition probabilities for a firm in state x that invests ι are given by:
h
P xit+1
6.2
i
= y xit = x, ι =


(1−δ)aι



1+aι








if y = x + 1
(1−δ)+δaι
1+aι
δ
1+aι
if y = x
if y = x − 1 .
A Tighter Bound
In this section we derive a tighter version of the performance bound in Theorem 5.1.2 that takes advantage
of the fact that in the model above a firm can change its state at most in one quality level each period. The
derivation is similar to that of the bounds above so it is omitted.
Corollary 6.1. For all x,
1 E[∆A πm (s)] + β a−x E[∆Ac πm (s)] +
1−β
X
+
ax (y) (πm (y, s̃) − E [πm (y, s)]) .
E[ sup V (x, s|µ0 , µ̃, λ̃) − V (x, s|µ̃, λ̃)] ≤
µ0 ∈M
(6.1)
y∈N
where ∆A πm (s) = maxy∈A (πm (y, s) − πm (y, s̃)), A = {0, ..., a − 1} and x ∈ A.
21
In general, E [πm (y, s)]−πm (y, s̃)] ≥ 0 and is increasing in y. Therefore, E [maxy∈A (πm (y, s) − πm (y, s̃))]
increases as the set A increases, making the first term in the bound from Theorem 5.1.2 large. In the bound
presented in the corollary, the term that involves maxy∈Ac is the larger one because it involves the maximum
over the larger states. However, this is offset by its multiplication by β a−x . The term β a−x appears because
it takes at least a − x periods to go from state x to some state in Ac . We use this bound in the numerical
experiments. In each case we choose the set A to minimize the bound.12
6.3
Numerical Results: Behavior of the Bound
In this subsection we present numerical results for the performance bound under different model specifications. In our computational experiments we found that most important parameter for the performance
bounds was θ1 , which determines the importance that consumers place on product quality. If θ1 is small,
the degree of vertical differentiation between products is small. This reduces the impact of changes in the
industry state on profits, making the MPE policies less sensitive to the industry state. Additionally, when θ1
is small it turns out that the invariant distribution s̃ is very “light-tailed”. Oblivious policies work well in this
case, and the performance bound is small. If θ1 is large, we get the reverse implications and the performance
bound is larger.
Following Pakes and McGuire (1994) we fix a = 3 and δ = 0.7. Additionally, we fix marginal cost at
c = 0.5, income at Y = 1, and θ2 = 0.5. The discount factor is β = 0.95. The entry cost is κ = 35 and
the entry state is xe = 10. The average sell-off value is K = 10. For these parameters we find the oblivious
equilibrium and we compute the performance bound for different values of θ1 and the investment cost d.
We consider two pairs (θ1 , d): (0.1, 0.1) and (0.5, 0.5). The former (“Low”) is a situation where the level
of vertical differentiation is low and it is inexpensive to invest to improve quality. The latter (“High”) is the
opposite. As a point of comparison, if a firm increases its state from x = 10 to x = 20, its single-period
profits increase by 7% and 40% respectively in the two cases (holding rivals constant). For each set of
parameters we compute the performance bound for different market sizes. As the market size increases, the
expected number of firms increases and the performance bound decreases.
In Figure 1 we present the percentage performance bound as a function of the expected number of firms
for the two levels of vertical differentiation (the two curves are obtained by varying the market size). The
12
Note also that in the bound in Theorem 5.1.2 we take maxy∈N in the first term. The theorem still holds if we only take
maxy∈{0,...,xmax } , where xmax = max(x̃max , x∗max ). x̃max and x∗max are the largest states a firm can reach when using strategies
µ̃ and the optimal non-oblivious strategy respectively. In our experiments we compute µ̃ so we know x̃max . However, we do not
know x∗max because the computation of the optimal non-oblivious strategy is subject to the curse of dimensionality. Instead,
we assume that xmax = C x̃max , where C is a constant that we fix between two and five. Similarly, in Corollary 6.1, Ac =
{a, ..., xmax }. The selection of C is arbitrary, however, its particular choice does not have much effect on the bound in Corollary
6.1. This is another advantage of using the latter bound.
22
bounds are expressed as a percentage of the value function and we estimate the bounds using simulation.13
For the low vertical differentiation case it takes around 60 firms to bring the bound down to 2%, and 250
firms to bring it below 1%. For the high case it takes around 250 firms to bring the bound to 5% and 1000
firms to bring it near 2%.
When the level of vertical differentiation is high, the number of firms required to have a good approximation is large, requiring hundreds and even thousands of firms. The approximation would be better if the
state s were always close to its mean, s̃. One aspect of the model that interferes with this is the Poisson
entry process. Because of this entry process the number of firms in the industry is Poisson with mean ñ (the
expected number of firms). The coefficient of variation of the number of firms (i.e., the standard deviation
over the mean) is
√1 ,
ñ
which converges to zero but at a very slow rate. Hence, the entry process leads to a
large amount of variability in the number of firms even for large market sizes. This causes problems for the
oblivious equilibrium approximation, which relies on the industry state staying close to its mean. We chose
to model the entry process this way because it facilitated the asymptotic arguments in Section 4. However,
since the expressions for the performance bounds are still correct without the assumption there is no need to
maintain it in our computations. As an alternative we tried assuming a fixed number of entrants each period
(as in Ericson and Pakes (1995)). This entry process implies a smaller variability in the number of firms.
Note that to avoid non-existence problems we allowed for a fractional number of entrants. For example, if
the zero profit condition requires 2.5 entrants then every period the number of entrants is either 2 or 3 with
probability 0.5.
Figure 2 presents the results with the new entry process. In the case of low vertical differentiation, the
performance bound is less than 2% with just 30 firms and it is around 1% with 100 firms. When the level
of vertical differentiation is high the performance bound is around 5% when there are 120 firms and around
2% for 700 firms.
Going one step further, we tried shutting down entry and exit altogether and considered an industry with
a fixed number of firms (see Figure 3). For the low case the performance bound is less than 0.5% with just
5 firms while for the high case it is 5% for 5 firms, less than 3% with 40 firms, and less than 1% with 400
firms.
13
The value function is estimated with a relative precision of 0.5% and a confidence level of 99%. The bound is estimated with
a relative precision of 10% and a confidence level of 99%. Note that the percentage performance bound depends on the state x so
we consider the percentage bound evaluated at the entry state.
23
6.4
Closeness to Markov Perfect Equilibrium
Having gained some insight into what features of the model lead to low values of the percentage bounds,
the question arises as to what value of the percentage bounds is required to obtain a good approximation.
To answer this question we compare long-run statistics for the same industry primitives under oblivious
equilibrium and MPE strategies. A major constraint on this excercise is that it requires the ability to actually
compute the MPE. With the current algorithms we are able to compute MPE for industries with a maximum
of five to ten firms — to keep computation manageable we use four firms here. We therefore limit our
analysis to the case of a fixed number of firms (no entry and exit), because only for that case were the
performance bounds small under oblivious strategies (with only four firms). We use the same parameter
specifications as in the previous subsection. Because of computational constraints in computing the MPE,
we also impose a maximum state that a firm can reach of xmax = 15, (at which point investment is assumed
to have no further effect).14
Recall that under oblivious equilibrium strategies, the industry state is described by an ergodic Markov
chain (Lemma 3.2). This is also true under MPE strategies (Ericson and Pakes (1995)). Therefore, both
systems have a well defined invariant distribution that describes their long-run behavior. We compare different expected quantities under the oblivious equilibrium and the MPE invariant distributions. The quantities
compared are: average investment, average producer surplus, average consumer surplus, average share of
the largest firm (C1), and average share of the largest two firms (C2). Table 1 reports these statistics for a
wide range of parameters. We also report the maximum value (across all states) and weighted average of
the performance bound, as well as the maximum and weighted average actual benefit from deviating and
keeping track of the industry state (the latter quantity should always be smaller than the performance bound).
From the computational experiments we conclude the following (see Table 1):
1. When the bound is less than 1% the long-run quantities estimated under oblivious equilibrium and
MPE strategies are very close.
2. Performance of the approximation depends on the shape of the average industry state. When the bound
is between 1-20% and the average industry state is symmetric (see Figure 4), the long-run quantities
estimated under oblivious equilibrium and MPE strategies are still quite close. When the bound is
above 1% and the average industry state is very skewed (see Figure 5), the long-run quantities can be
quite different on a percentage basis (5% to 20%). However, as shown in Table 2, this is at least partly
due to the fact that there is very little investment in these industries.
14
The code used to compute MPE was generously provided by Uli Doraszelski.
24
3. The performance bound is not tight. For a wide range of parameters the performance bound is as
much as 10 to 20 times larger than the actual benefit from deviating.
6.5
Example: Changes in Market Structure as Market Size Increases
In this subsection we analyze the structure of the industry as market size grows under different parameter
specifications. We note that the intention of this subsection is to show the potential of our methods and to
provoke interest in future research in this area, and not to make robust and conclusive statements.
An important question in industrial organization is to understand what features of an industry are most
important to determining the industry’s structure. Of particular interest (Sutton 1991) is whether a market
becomes fragmented, or remains concentrated, as the market grows in size. Sutton (1991) and Shaked and
Sutton (1987) suggest that the presence of “endogenous” sunk costs imposes a lower bound to concentration.
Our model is a dynamic model of endogenous sunk costs because, while entrants pay the same cost of
entry regardless of market size (an “exogenous” sunk cost), in larger markets they may need to invest large
amounts to become as large as the incumbent firms (an “endogenous” sunk cost). In what follows we show
that under slightly different parameterizations of our model we get two extremely different outcomes: a
fragmented and a very concentrated industry.
We use the same parameter specifications as in Subsection 6.3 with the only difference that now the
utility of the consumers is given by:
utij
= θ1 ln
xit
t
+ 1 + θ2 ln(Y − pti ) + νij
.
7
This specification makes it easier to obtain the two different situations. We fix d = 0.5 and we compute the
oblivious equilibrium for θ1 = 0.9 and θ1 = 1.2. In Figures 6 and 7 we present the average industry states
obtained for different market sizes in the two cases. For θ1 = 0.9, as the market size grows the industry
becomes fragmented. The number of firms grows but firms do not grow in size, so the market share for
every firm converges to zero. When θ1 = 1.2 the entry rate decreases and firms grow larger as market size
grows. In this second case, the industry state is heavy-tailed.15
These results have a clear intution: when θ1 > 1 consumers care more about quality and this increases
the returns to investment. What results is a quality race between firms, so there are a small number of firms
in the industry that become ever larger with the market size. When θ1 < 1 returns to investment are lower
15
To show the results more clearly, we forced the highest quality level to be 80. This leads the industry state to have a mass
of firms near 80 for the larger market sizes. Without this arbitrary cutoff firms would have grown even larger and it would have
become impossible to see the lower portion of the firm size distribution on the graphs.
25
and investment is not worthwhile above a certain point regardless of the market size.
These results contradict Sutton (1991) in one sense because Sutton (1991) sought predictions about
market structure that are robust across a wide class of models. Here, different market structures result from
the same model through a tiny change in a single parameter. An implication of this result is that the same
industry might be observed with very different market structures across markets that are the same size but
have small differences in consumer tastes.
Our results remain broadly consistent with Sutton’s predictions because the parameter that determines
market structure pertains directly to the returns to investment. Consistent with the predictions of Sutton
(1991), industries with higher returns to investment tend to be concentrated. However, our results show that
this relationship can be much more fragile than it first appears.
7
Conclusions and Future Research
The goal of this paper has been to increase the set of applied economic problems that can be addressed using
Ericson and Pakes (1995)-style dynamic models of imperfect competition. Due to the curse of dimensionality, existing methods have limited application of these models to industries with a small number of firms
and a small number of states per firm. As an alternative, we proposed a method for approximating MPE
behavior using an oblivious equilibrium, where firms make decisions only based on their own state and the
long run average industry state.
We began by showing that the approximation works well asymptotically, where asymptotics were taken
in the market size. A sufficient condition for an oblivious equilibrium to well approximate a MPE asymptotically is that the sequence of industry states generated by the oblivious equilibria is “light-tailed” (as
described by Assumption 4.4). This condition is also sufficient to establish that a model of the type introduced in Hopenhayn (1992) yields a good approximation of a finite industry. We also introduced a simple
algorithm to compute an oblivious equilibrium. A nice feature of the algorithm is that there is no need to
place a’ priori restrictions on the number of firms in the industry or the set of states that a firm can reach.
To facilitate using oblivious equilibrium in practice, we derived performance bounds that indicate how
good the approximation is in any particular problem under study. These performance bounds are quite
general and thus can be used in a wide class of models. We believe them to be the first bounds of this type
in this literature. In an example model we showed that oblivious equilibrium yields a good approximation
of MPE behavior for industries with a couple hundred firms, and sometimes even with just tens of firms.
26
An interesting and useful generalization would be to introduce aggregate shocks into the model. For
example, suppose there were aggregate shocks to profits that followed an ergodic Markov process. It is not
possible to generalize the asymptotic results of Theorem 4.2 to handle these shocks because even if there
were an infinite number of firms the industry state would not be constant. Thus, the approximation error
would never go to zero. However, it is straightforward to extend the performance bounds in Theorem 5.1 to
this case, so the approximation method may still be useful in practice (of course, the approximation is likely
to be less precise for a model with aggregate shocks).
Finally, our results also show the potential of state space reduction approximations in general in this
context. We have considered very simple strategies that are functions only of a firm’s own state and the
long run average industry state. It may be possible to obtain better approximations using additional information such as the total number of firms in the industry and/or the average state across firms. Solving for
equilibria of this type would be more difficult than solving for oblivious equilibria, but is still likely to be
computationally feasible. However, the performance bounds would be more difficult to obtain for this case
because, even if the shocks to individual firms remained idiosyncratic, firms would have outcomes that were
correlated through their strategies. This will be a matter of further research.
27
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30
A
Appendix: Proofs and Mathematical Arguments
A.1
Proofs and Mathematical Arguments for Section 2
The Poisson Entry Model
Suppose there are K potential entrants at a given state. Each potential entrant enters if the setup cost κ
is less than the present value of future cash flows upon entry. Let vK (i) be the present value of future cash
flows for each entering firm if i of the K firms enter simultaneously. vK (i) is nonincreasing in i. One can
then pose the problem faced by potential entrants as a game in which each entrant employs a mixed strategy
and enters with some probability pK . If we assume that every potential entrant employs the same strategy,
the condition for a mixed strategy Nash equilibrium when κ ∈ [vK (K), vK (1)] can be written as
K−1
X
(A.1)
i=0
K −1
i
piK (1 − pK )K−1−i vK (i + 1) − κ = 0,
which is solved by a unique pK ∈ [0, 1]. If κ < vK (K), the equilibrium is a pure strategy with pK = 1,
whereas if η > vK (1), the equilibrium is given by pK = 0. The following result, which we state without
proof, establishes that our Poisson entry model can be viewed as a limiting case as the number of potential
entrants K grows large.
Lemma A.1. Let the following conditions hold:
1. vK (i) is non-increasing in i, ∀K and non-increasing in K, ∀i;
2. there exists a positive constant M such that |vK (i)| < M, ∀K, ∀i;
3. there exists K, such that ∀K > K, vK (i) changes sign in (0, K);
4. there exists a function v(i) such that limK→∞ maxi≤i |vK (i) − v(i)| = 0, for all i ∈ N.
Then,
1. for each K > K, Equation (A.1) has a unique solution p∗K ∈ (0, 1);
2. the limit λ = limK→∞ Kp∗K exists, and if YK is a binomial random variable with parameters (K, p∗K )
and Z is a Poisson random variable with mean λ, then YK ⇒ Z.
The result states that if the number of potential entrants grows to infinity then the entry process converges
to a Poisson random variable. Hence, Poisson entry can be understood as the result of a large population of
potential entrants, each one entering the industry with a very small probability.
31
Bellman’s Equation
We define a dynamic programming operator:
(Tµ,λ V )(x, s) = πm (x, s)
+E max φit , sup (−dι + βEµ,λ [V (xi,t+1 , s−i,t+1 )|xit = x, s−i,t = s, ιit = ι]) ,
(A.2)
ι≥0
for all x ∈ N and s ∈ S.
0
µ
To simplify the notation in this section we will let Vµ,λ
≡ V (·|µ0 , µ, λ).
Lemma A.2. For all µ ∈ M and λ ∈ Λ, there exists µ∗ ∈ M, such that:
∗
0
∗
µ
µ
µ
Vµ,λ
= sup Vµ,λ
= Tµ,λ Vµ,λ
.
µ0 ∈M
∗
µ
is the unique fixed point of Tµ,λ within the class of bounded functions.
Further, Vµ,λ
∗
∗
µ
µ
(x, s) ≤
, is bounded by 0 ≤ Vµ,λ
Proof. First note that that the value function Vµ,λ
mY
1−α +φ, ∀x
∈ N, s ∈ S.
Therefore, it will never be optimal to invest more than a finite quantity, ιm . Hence we can assume without
loss of generality that 0 ≤ ι(x, s) ≤ ιm , ∀x ∈ N, ∀s ∈ S. Additionally, βφ ≤ ρ(x, s) ≤
mY
1−β
+ φ,
∀x ∈ N, ∀s ∈ S. Therefore the action space for each state is compact.
For a given state (x, s), expected one period profits including investment and sell-off value can be written
as:
πm (x, s) − dι(x, s)P[φ < ρ(x, s)] + P[φ ≥ ρ(x, s)]E[φ | φ ≥ ρ(x, s)] .
Note that πm (x, s) < mY , investment is bounded by the previous argument and φ has finite expectation.
Therefore, expected one period profits including investment and the sell-off value are uniformly bounded
for all states (x, s). The result follows by Propositions 1.2.2 and 3.1.7 in Bertsekas (2001).
A.2
Proofs and Mathematical Arguments for Section 3
Lemma 3.2
Proof. If every firm uses a strategy µ ∈ M̃ and entry is according to an entry rate function λ ∈ Λ̃, then
X ≡ {st : t ≥ 0} is clearly an irreducible Markov chain. All states reach the state ∅ = {0, 0, ...} with
positive probability and all states can be reached from ∅ as well. Now, take any state s ∈ S. There is a
32
positive probability that there is no entry, no exit and no firms’ state transitions. Therefore, for all s ∈ S,
P[st+1 = sst = s] > 0, so X is aperiodic. Moreover X is positive recurrent because the expected time to
come back from state ∅ to itself is finite (see Kleinrock (1975)).
Now, let us write:
(A.3)
st (x) =
Wτ
t X
X
1[Xi,t−τ = x] ,
τ =0 i=1
where Wτ are i.i.d. Poisson random variables with mean λ, the first sum is taken over all periods previous
to (and including) t, the second sum is taken over the firms that entered the industry in each period, and for
each τ , Xi,t−τ are random variables that represent the state of firm i after t − τ periods inside the industry
when using strategy µ. Since firms use oblivious strategy µ ∈ M̃ their state evolutions are independent, so
P τ
1[Xi,t−τ = x] are i.i.d. It follows that W
i=1 1[Xi,t−τ = x] is a filtered Poisson random variable, so it is a
Poisson random variable. Thus st (x), as a sum of independent Poisson random variables, is also Poisson.
Given that the expected time a firm spends inside the industry is finite, using characteristic functions it is
straightforward to show that st (x) ⇒ Zx , ∀x ∈ N. To show that {Zx : x ∈ N} is a sequence of independent
random variables note that by the filtering property of Poisson random variables (Durrett (1996)), for all t,
{st (x) : x ∈ N} is a sequence of independent random variables.
By summing over x ∈ N, we can show that nt ⇒ Z.
Lemma 3.3
Proof. The proof is rather involved, but relies on a simple idea. For a Poisson process, conditional on
n arrivals on an interval [0, T ], the unordered arrival times have the same distribution as n i.i.d. uniform
random variables in [0, T ].
Let us condition on nt = n. {x(j)t : j = 1, ..., n} are the random variables that represent the state
of each of the n firms in the industry when they are sampled randomly. The expected time a firm spends
inside the industry is finite, so the time a firm spends inside the industry is finite with probability one. Recall
that a firm can grow at most w states each period. Therefore, ∀ > 0, there exists a state z, such that for
all j ∈ 1, ..., n, ∀t, P[x(j)t > z] < . That is, the probability of observing a firm in a very large state is
negligible, because firms “do not have time” to get that far. Hence,

P
n
[

{x(j)t > z} | nt = n < , ∀t .
j=1
33
Therefore, the sequence of random vectors (x(1)t , ..., x(nt )t | nt = n) is tight. Theorem 9.1 in Durrett (1996)
states that Zn ⇒ Z if and only if for all sets A with P[Z ∈ ∂A] = 0, limn→∞ P [Zn ∈ A] = P [Z ∈ A],
where ∂A denotes the boundary of the set A. Using this result and the tightness previously established, it is
enough to show that for all n, for all (z1 , ..., zn ):
(A.4)
h
lim P x(j)t
t→∞
where p(·) is the pmf s̃µ,λ (·)/
n
i Y
= zj , j = 1, ..., nnt = n =
p(zj ) ,
j=1
P
x∈N s̃µ,λ (x).
Let T̃j be the entry date of firm (j) and Tj = t − T̃j be its
age. Then we can write:
h
i
P x(j)t = zj , j = 1, ..., nnt = n =
X
P[x(j)t = zj , j = 1, ..., n | T1 = t1 , ..., Tn = tn , nt = n] ·
0≤t1 <∞,...,
0≤tn <∞
P [T1 = t1 , ..., Tn = tn | nt = n]
n
X Y
=
P x(j)t = zj | Tj = tj ·
0≤t1 <∞,...,
0≤tn <∞
j=1
P [T1 = t1 , ..., Tn = tn | nt = n] ,
(A.5)
The last equation follows because the evolution of firms is independent across firms. Note that if any tj has
h
i
a value greater that t, then P T1 = t1 , ..., Tn = tn nt = n = 0. We can write
h
P x(j)t
(A.6)
P x(j)t = zj , Tj = tj
P [Tj = tj ]
P Tj = tj , Xj,tj = zj
=
P [Tj = tj ]
P[Tj = tj ]P[Xj, tj = zj ]
=
P [Tj = tj ]
= P[Xj,tj = zj ] ,
i
= zj T j = t j =
where, similarly to above, Xj,tj is a random variable that represents a firm’s state after tj periods conditional
on having stayed in the industry. Note that for all k, {Xj,k : j ≥ 1} is i.i.d. The second to last equation
follows because the evolution of a firm is independent of its entry time.
Now we show that
lim P [T1 = t1 , ..., Tn = tn | nt = n] =
t→∞
n
Y
u[tj ] ,
j=1
for some probability mass function u. This equation can be derived directly. However, it is easier to invoke
34
the relationship between nt and a Poisson process and show that the equation holds using known results for
Poisson processes.
Similarly to equation (A.3), we can write:
nt =
Wτ
t X
X
1[Ai,t−τ = 1] ,
τ =0 i=1
where, for each τ , Ai,t−τ are i.i.d. Bernoulli random variables that equal one if the firm is still in the
P τ
industry after t − τ periods when using strategy µ. Since Ai,t−τ are i.i.d., nt,τ ≡ Z
i=1 1[Ai,t−τ = 1] is a
filtered Poisson random variable, and is therefore Poisson. Call its mean αt,τ . It follows that nt is a sum of
P
independent Poisson random variables, so it is Poisson with mean tτ =0 αt,τ .
Consider {N (t) : t ≥ 0}, a homogeneous Poisson process on the real line with rate 1. Note that N (t)
P
and nt are equivalent in the sense that we can construct nt using the process N (s) : 0 ≤ s ≤ tτ =0 αt,τ
in the following way: for each 0 ≤ τ ≤ t, consider the events in the interval [αt,τ −1 , αt,τ −1 + αt,τ ], where
αt,−1 = 0. Assign those events to firms that are still in the industry at time t and entered at period τ ,
nt,τ . Note that: i) N (αt,τ ) and nt,τ are both Poisson with mean αt,τ ; ii) N (t) has independent increments
P
and nt,τ are independent, for all τ ; iii) N ( tτ =0 αt,τ ) and nt are both sum of independent Poisson random
P
variables with mean tτ =0 αt,τ .
P
Suppose N ( tτ =0 αt,τ ) = n. Conditional on that event, the unordered arrival times of N (t) have the
P
same distribution as n i.i.d. uniform random variables in [0, tτ =0 αt,τ ] (Durrett (1996)). By the equivalency
argument described above, conditional on nt = n, the unordered arrival times of the n firms are i.i.d. discrete
random variables with pmf:
αt,τ
vt (τ ) = Pt
,0 ≤ τ ≤ t .
j=0 αt,j
Recall that αt,τ is the expected number of firms that entered at time τ and are still inside the industry at time
t. Since the entry rate is the same every period and every firm uses the same strategy which is independent
of every other firm, αt,τ = α̃k , where α̃k is the expected number of firms that entered the industry at time
s, for any s, and are still inside the industry at time s + k. This suggests making a change of variable and
defining:
α̃k
ut (k) = Pt
j=0 α̃j
,0 ≤ k ≤ t .
ut (k) is the probability a random sampled firm from the industry at time t entered k periods ago, conditional
35
on nt = n. Taking the limit as t tends to infinity, we get that:
α̃k
lim ut (k) ≡ u(k) = P∞
j=0 α̃j
t→∞
provided that limt→∞ E[nt ] =
P∞
j=0 α̃j
, 0≤k<∞ ,
< ∞, which is true by assumption. u(k) is the probability a ran-
dom sampled firm from the industry at time t 0 entered k periods before the sampling period. Therefore
n
Y
lim P [T1 = t1 , ..., Tn = tn | nt = n] =
t→∞
u[tj ] .
j=1
Replacing the previous equation and equation (A.6) in equation (A.5) we obtain:
n
Y
X
lim P x(j)t = zj , j = 1, ..., n | nt = n =
P[Xj,t = zj ]u(t)
t→∞
j=1 0≤t<∞
where the interchange between the infinite sum and the limit follows by dominated convergence. The sum
yields the pmf p(·). The previous equation proves that Y1 , ..., Yn are i.i.d. with pmf p(·) which is independent
of n.
Consider t 0. More formally, suppose that s0 is sampled from the invariant distribution, which is well
defined by Lemma 3.2. In this case, st is a stationary process; st is distributed according to the invariant
distribution for all t ≥ 0.


nt
X
s̃µ,λ (x) = E[st (x)] = E 
1[x(j)t = x] .
j=1
Conditioning on nt , and considering that we already proved that {x(j)t : j = 1, ..., n} are i.i.d. with pmf
P
p(·) which is independent of n, we get that p(·) = s̃µ,λ (·)/ x∈N s̃µ,λ (x).
A.3
Proofs and Mathematical Arguments for Section 4
Proposition 4.1
Proof. First we argue that for all δ > 0, there exists z such that, for all f˜(m) , there exists fˆ(m) ∈ Fz (Fz
defined in Assumption 4.1.1) such that k f˜(m) − fˆ(m) k1,g < δ, for all m.
By Assumption 4.4,
sup
m
X
g 1+γ (x)f˜(m) (x) < ∞ .
x∈N
36
Therefore, the sequence of random variables g(x̃(m) ) is uniformly integrable (Billingsley (1995)), hence,
for all η > 0, there exists z such that
sup
X
g(x)f˜(m) (x) < η ,
m x>z
where we used the fact that g is strictly increasing. Given this, for all δ > 0, we can choose z such that there
is a sequence fˆ(m) ∈ Fz that satisfies k f˜(m) − fˆ(m) k1,g < δ, for all m. Choose < 1. Now, by Assumption
4.3 and the previous argument, find δ (and z and a sequence fˆ(m) ∈ Fz ), such that,
πm (x, f˜(m) , ñ(m) ) ≥ πm (x, fˆ(m) , ñ(m) )(1 − ) , ∀x, ∀m .
Assume lim inf m
ñ(m)
m
= 0. Hence, there is a subsequence, such that, limk
ñ(mk )
mk
= 0. If this is the case,
by Assumption 4.1.1 and the previous inequality, limk πmk (x, f˜(mk ) , ñ(mk ) ) = ∞, ∀x ≥ z. We will show
that, under these conditions, the zero profit condition at the entry state cannot be met for market sizes mk
with k big enough.
If a firm invests ι > 0, there is a strictly positive probability, p(ι), increasing in ι, that the firm will grow
at least one quality level. Let T be the time to get from state xe to state z. If a firm invests ιt ≥ ι > 0, every
period t, by a geometric trials argument, we conclude that E[T ] < ∞. Therefore, there exists an investment
strategy for which the expected time and cost to go from xe to z are uniformly bounded above over all
m. Once a firm gets to z, it receives πmk (z, f˜(mk ) , ñ(mk ) ), which becomes arbitrarily large as k grows.
This should be discounted by E[β T ] ≥ β E[T ] > 0, ∀k. Hence, there is a strategy, for which, expected
discounted profits at the entry state become arbitrarily large as k grows. This is a contradiction because the
zero profit condition at the entry state would not be met, so ñ(mk ) cannot be the expected number of firms
under oblivious equilibrium strategies for k big enough.
Theorem 4.2
Proof. We first discuss the main intuition behind the theorem. A key step in our analysis is to relate the
difference in value functions to differences in one period profits. One can show that
(m)
E[ sup V (m) (x, st
µ0 ∈M
(m)
|µ0 , µ̃(m) , λ̃(m) ) − V (m) (x, st
|µ̃(m) , λ̃(m) )]
(m)
can be bounded above by an infinite sum of terms of the form E[|πm (xt , st
37
) − πm (xt , s̃(m) )|]. Given
i
h
(m)
this, a crucial step on proving the theorem is to show that E πm (x, sk ) − πm (x, s̃(m) ) → 0. In turn,
h
i
(m)
the key step to showing this is to show that P k f
− f˜(m) k1,g > δ converges to zero. Recall that,
k
by Assumption 4.3, if two states are close under the k · k1,g norm, then profits are close. To show that
h
i
(m)
P k fk − f˜(m) k1,g > δ → 0 we use a weak law of large numbers for triangular arrays and we need
Assumption 4.4 (the light tail assumption). The uniformly bounded expectation in the assumption plays a
similar role to the expectation assumption in a standard strong law of large numbers. The γ > 0 appears
because we are considering not only one random variable but a sequence. It is used to put a uniform bound
on their tails.
(m)
Similarly to the result in Subsection 4.1 one can show that k fk
− f˜(m) k∞ converges to zero in
probability if ñ(m) → ∞ independent of the tail of the distributions f˜(m) . However, this is not sufficient for
(m)
the sequence of oblivious equilibria to have the AME property. We need fk
− f˜(m) to converge to zero
in probability but with the appropriate norm, namely, the k · k1,g norm. The difference in the fraction of
firms in each state under the k · k∞ norm could be small, but the difference in profits could be huge because
what impacts profits is the “sum” of all the differences. Also a small difference in the fraction of firms
in a very high state could have a significant impact on profits. The k · k1,g norm takes this into account.
(m)
If fk
(m)
− f˜(m) is close to zero in the appropriate norm, then πm (x, sk ) will be close to πm (x, s̃(m) )
and therefore supµ0 ∈M V (m) (·|µ0 , µ̃(m) , λ̃(m) ) will be close to V (m) (·|µ̃(m) , λ̃(m) ). We now formalize this
argument.
Let us write,
V (m) (x, s|µ∗ , µ̃(m) , λ̃(m) ) − Ṽ (m) (x|µ̃(m) , λ̃(m) )
+
Ṽ (m) (x|µ̃(m) , λ̃(m) ) − V (m) (x, s|µ̃(m) , λ̃(m) ) ,
V (m) (x, s|µ∗(m) , µ̃(m) , λ̃(m) ) − V (m) (x, s|µ̃(m) , λ̃(m) ) =
(A.7)
where µ∗(m) is defined as the optimal (non-oblivious) best response to (µ̃(m) , λ̃(m) ) for a firm that is keeping
track of the industry state, such that
V (m) (x, s|µ∗(m) , µ̃(m) , λ̃(m) ) = sup V (m) (x, s|µ0 , µ̃(m) , λ̃(m) ).
µ0 ∈M
We will consider the first difference above. The treatment of the second one is analogous.
The equilibrium oblivious value function in market m can be written as,
(A.8)
"
Ṽ (m) (x, s|µ̃(m) , λ̃(m) ) = Eµ̃(m) ,µ̃(m) ,λ̃(m)
τi
X
β k−t πm (xik , s̃(m) ) − dιik
k=t
38
#
+ β τi −t φiτi xit = x, sit = s .
Note that the latter two subindices µ̃(m) , λ̃(m) on the expectation and conditioning in s−i
t = s are redundant
because neither the one period profit function nor the strategy µ̃(m) depends on the competitors’ state s−i
t .
However, we write the dependence explicitly to make the following argument clear. The oblivious value
function does not depend on the actual outcomes for rivals’ states (it is a function only of the expected
industry state). Therefore, the strategy µ̃(m) maximizes the oblivious value function among all strategies M
(not just oblivious strategies M̃). In particular, the oblivious equilibrium strategy µ̃(m) must be at least as
good as strategy µ∗(m) for the oblivious value function. Therefore:
(A.9)
"
Ṽ (m) (x, s|µ̃(m) , λ̃(m) ) ≥ Eµ∗ ,µ̃(m) ,λ̃(m)
τi
X
β k−t πm (xik , s̃(m) ) − dιik
#
+ β τi −t φiτi xit = x, sit = s .
k=t
Let ∆m (k, i) ≡ πm (xik , s−i,k ) − πm (xik , s̃(m) ). Then, negating equation (A.9) and adding
V (m) (x, s|µ∗(m) , µ̃(m) , λ̃(m) ) to both sides gives,
(A.10)
V (m) (x, s|µ∗(m) , µ̃(m) , λ̃(m) ) − Ṽ (m) (x, s|µ̃(m) , λ̃(m) )
"
τi
X
"
k=t
∞
X
≤ Eµ∗(m) ,µ̃(m) ,λ̃(m)
≤ Eµ∗(m) ,µ̃(m) ,λ̃(m)
#
β k−t ∆m (k, i)xit = x, s−i,t = s
β
k−t
#
∆m (k, i)xit = x, s−i,t = s
k=t
=
∞
X
h
i
β k−t Eµ∗(m) ,µ̃(m) ,λ̃(m) ∆m (k, i)xit = x, s−i,t = s .
k=t
The last equation follows by monotone convergence. Suppose s is distributed according to the invariant
distribution, q (m) . If s−i,t is distributed according to q (m) , then the marginal distribution of s−i,k is also distributed q (m) for all k > t. Recall that q (m) is the invariant distribution of the industry state when every firm
(m)
uses the strategy µ̃(m) and the entry rate is λ̃(m) . Let sk
(m)
are q (m) for all k ≥ t. Let Qk
be random vectors whose marginal distributions
be the joint distribution of (s−i,t , ..., s−i,k ). Taking expectations we have:
∞
h
i X
(m)
(m)
E V (m) (x, st |µ∗(m) , µ̃(m) , λ̃(m) ) − Ṽ (m) (x, st |µ̃(m) , λ̃(m) ) ≤
β k−t Eµ∗
k=t
where we suppress index i to simplify notation.
Note that
∆m (k) ≤ ∆nm (k) + ∆fm (k) ,
39
(m)
h
(m)
,Qk
i
∆m (k)xt = x ,
where
(m) (m)
(m)
∆nm (k) = πm (xk , fk , nk ) − πm (xk , fk , ñ(m) ) ,
(m)
∆fm (k) = πm (xk , fk , ñ(m) ) − πm (xk , f˜(m) , ñ(m) ) .
Therefore,
h
i
(m)
(m)
(A.11) E V (m) (x, st |µ∗(m) , µ̃(m) , λ̃(m) ) − Ṽ (m) (x, st |µ̃(m) , λ̃(m) ) ≤
∞
X
∞
h
i X
h
i
β k−t Eµ∗(m) ,Q(m) ∆nm (k)xt = x +
β k−t Eµ∗(m) ,Q(m) ∆fm (k)xt = x .
k
k=t
k=t
k
Let us analyze the first sum in the previous equation.
∞
X
β k−t Eµ∗(m) ,Q(m)
h
k
k=t
(A.12)
#
#
"
" (m)
∞
n
i
X
∆nm (k)xt = x =
β k−t Eµ∗(m) ,Q(m) ∆nm (k)1 k(m) − 1 ≤ δ xt = x
ñ
k
k=t
"
"
#
#
∞
n(m)
X
k
n
k−t
β Eµ∗(m) ,Q(m) ∆m (k)1 (m) − 1 > δ xt = x ,
+
ñ
k
k=t
where 1[] is an indicator function. By Assumption 4.3, if δ is chosen appropriately, the first sum in equation
(A.12) is bounded by:
(A.13) ∞
X
∞
h
i
X
(m)
β k−t Eµ∗(m) ,Q(m) πm (xk , fk , ñ(m) )xt = x ≤ πm (0, f , ñ(m) )
β k (B + Cαx+kw ) ,
k=t
k
k=0
where the inequality follows by Assumptions 2.1.1, 2.1.3, 4.1.2 and because a firm can grow at most a finite
number of states in one time step. Note that by Assumptions 2.1.1 and 2.1.4, and by Proposition 4.1,
(A.14)
πm (0, f , ñ(m) ) ≤
O(m)
= O(1) .
Ω(m)
The last inequality follows because the amount of money to split is O(m) and if a firm is in the smallest
k
P
state, it can get at most O(m) divided by the number of firms, which is Ω(m).
αβ w converges
because, by assumption, αβ w < 1. By the previous argument, πm (0, f , ñ(m) ) is uniformly bounded over
all m. Therefore, the first sum in equation (A.12) can be made arbitrarily small, for all m, by choosing small enough. The second sum converges to zero as m → ∞ by observing that ∆nm (k) = O(m) and the
following lemma.
40
Lemma A.3. For all δ > 0, there exists D > 0 such that:
"
#
n(m)
t
P (m) − 1 > δ < O(e−Dm ).
ñ
(A.15)
(m)
Proof. nt
is a Poisson random variable with mean ñ(m) . Let Zi be i.i.d. Poisson random variables with
mean 1. Then we can write:
(m)
nt
=
(m)
ñ
X
Zi .
i=1
There is a rounding error because ñ(m) might not be an integer. However, we will take the limit as ñ(m) →∞
so the error is negligible. The result follows because the Poisson distribution has exponential moments, by
Chebyshev’s inequality and Proposition 4.1.
Therefore, we have shown that the first sum in equation (A.11) converges to zero as m → ∞. Let us
analyze the second sum similarly.
∞
X
β
k−t
Eµ∗(m) ,Q(m)
k
k=t
h
∆fm (k)xt
∞
h
h
i
i
i
X
(m)
=x =
β k−t Eµ∗(m) ,Q(m) ∆fm (k)1 k fk − f˜(m) k1,g ≤ δ xt = x
(A.16)
+
k=t
∞
X
k
h
h
i
i
(m)
β k−t Eµ∗(m) ,Q(m) ∆fm (k)1 k fk − f˜(m) k1,g > δ xt = x
k
k=t
The first term in equation (A.16) can be made arbitrarily small, for all m, by choosing δ appropriately and using a similar argument to the one used for the first sum in equation (A.12) above. Using the exact same arguh
i
(m)
ment we can also show that the second sum in equation (A.16) is bounded by F P k fk − f˜(m) k1,g > δ ,
for a constant F > 0. Therefore, it is enough to show that this probability converges to zero. We prove
this in the proposition below. Hence, the sum in equation (A.16) converges to zero, and hence both sums
in equation (A.11) converge to zero. This proves that the expectation of the first term in equation (A.7)
converges to zero. The argument for the second term is analogous. Note that the convergence in the theorem
is not uniform over all x, because in several parts of the proof the bounds we have used actually depend on
x (for example, in equation (A.13)).
As above, {xj (m) t : j = 1, ..., n(m) } are the random variables that represent the state of each of the n
firms in the industry when they are sampled randomly.
41
Proposition A.1. Under Assumptions 2.1.1, 4.1.1, 4.2, 4.3 and 4.4, for all δ > 0,

X lim P 
m→∞
Pn(m)
t
j=1
g(x)1[xj (m) t = x]
(m)
− g(x)P[x̃(m)
nt
x∈N

= x] > δ  = 0 .
Proof. To simplify notation, let us define the event

(m)
Ψt
X =
Pn(m)
t
j=1
g(x)1[xj (m) t = x]
(m)
nt
x∈N
− g(x)P[x̃(m)

= x] > δ  .
Then,
i h
i X h
i h
i
h
i
h
(m)
(m) (m)
(m)
(m)
(m) (m)
P Ψt nt = n P nt = n .
P Ψt
= P Ψt nt ≤ n P nt ≤ n +
(A.17)
n>n
Let > 0. We first prove that there exists n such that, for all m,
h
i (m) (m)
P Ψt nt = n < , ∀n > n .
2
(A.18)
(m)
By Lemma 3.3, conditional on nt
= n, 1[xj (m) t = x] are i.i.d. random variables whose distribution is
independent of n. Therefore, conditional on
(m)
nt ,
(m)
mean of an i.i.d. process. Conditional on nt
Pn(m)
t
j=1
g(x)1[xj (m) t =x]
(m)
nt
can be understood as an empirical
, xj (m) t are i.i.d. random variables with the same distribution
as the random variable x̃(m) . Therefore,
h
"
#
P
i
X nj=1 g(x)1[yj(m) = x]
(m)
(m)
Ψm nt = n =
− g(x)P[x̃
= x] > δ ,
n
x∈N
(m)
where yj
are i.i.d. random variables with the same distribution as the random variable x̃(m) . Finally, by
the previous discussion, equation (A.18) follows by proposition A.3 that we prove next using the light-tail
(m)
is a Poisson random variable with mean ñ(m) and by proposition 4.1,
h
i
(m)
→ ∞. Therefore, there exists m, such that, P nt ≤ n < 2 , ∀m > m. By the previous arguments
condition. On the other hand, nt
ñ(m)
and equation (A.17):
(A.19)
h
i (m)
P Ψt
< + = , ∀m > m .
2 2
42
To prove Proposition A.3, we introduce a slightly modified version of the weak law of large numbers for
triangular arrays (Theorem 1.5.5. in Durrett (1996)). Let {Yjm : j = 1, 2, ...} be a sequence of i.i.d. random
m,n
variables for all m. Let Y j
= Yjm 1[|Yjm | < n].
Proposition A.2. Suppose that
1. limn→∞ supm n Pr[|Y1m | > n] = 0.
m,n 2
2. limn→∞ supm n1 E[ Y 1
] = 0.
Then, for all > 0,


X
1 n m
m,n
lim sup Pr 
Yj − E[Y 1 ] >  = 0 .
n→∞ m
n j=1
Proposition A.3. Let X m , X1m , X2m , ... be i.i.d. integer valued random variables for each m = 1, 2, .... Let
m = g(k)1(X m = k), for some positive strictly increasing function g(k). Suppose sup E[g 1+γ (X m )] <
lk,i
m
i
P
n
m , be the empirical mean of lm . Let Elm = E[g(k)1(X m =
∞, for some γ > 0. Let Ên lkm = n1 i=1 lk,i
k
k,i
k)] = g(k) Pr[X m = k]. Then,
lim sup Pr[
n→∞ m
Proof. Note that
P
k
X
|Ên lkm − Elkm | > ] = 0 .
k
|Ên lkm − Elkm | > implies that, for all k ,
X
|Ên lkm − Elkm | +
X
Ên lkm +
k>k
k≤k
X
Elkm > ,
k>k
which implies

X

k≤k



X


X
|Ên lkm − Elkm | >
∨
Ên lkm +
Elkm >
.

2
2
k>k
k>k
Therefore,
(A.20)
"
Pr
#
X

|Ên lkm − Elkm | > ≤ Pr 
k

X
k≤k

|Ên lkm − Elkm | >  + Pr 
2

X
k>k
Ên lkm >
−
2
Now,

Pr 
X
k≤k

X
m
m
m
m

|Ên lk − Elk | >
≤
Pr |Ên lk − Elk | >
.
2
2k
k≤k
43
X
k>k
Elkm  .
Since each of the functions {lkm : k = 1, ..., k } is bounded, it is easy to check, using Proposition A.2, that
each term in the sum of the right hand side converges to zero uniformly over all m. Since the sum involves
a finite number of terms we get that, for all k ,

lim sup Pr 
(A.21)
n→∞ m

X
|Ên lkm − Elkm | >  = 0 .
k≤k
Let us analyze the second term in equation (A.20). Note that:

(A.22)
Pr 

X
k>k
Ên lkm >
−
2
X


Elkm  = Pr 
k>k
X
Ên lkm −
k>k
X
Elkm >
k>k
−2
2
X
Elkm  .
k>k
Define the following function:
h(X) ≡ g(X)1[X > k ]
= g(X)1[g(X) > g(k )] ,
since g is strictly increasing. Since supm E[g 1+γ (X m )] < ∞, the sequence of random variables g(X m ) is
uniformly integrable. Therefore, we can choose k big enough such that g(k ) is sufficiently big so that:
E[h(X m )] <
(A.23)
, ∀m.
8
Note that

X
k>k
Elkm = E 

X
g(k)1[X m = k]
k>k
m
= E [g(X )1[X m > k ]]
= E[h(X m )] ,
44
where the first equation follows by the monotone convergence theorem. Similarly,
X
n
X 1X
g(k)1[Xim = k]
n
Ên lkm =
k>k
=
=
i=1
k>k
n X
X
1
n
1
n
g(k)1[Xim = k]
i=1 k>k
n
X
h(Xim ) .
i=1
By the previous arguments and equations (A.22) and (A.23) we have that:


Pr 
(A.24)
X
Ên lkm
k>k
> −
2
X
Elkm 
"
n
1X
≤ Pr
h(Xim ) − E[h(X m )] >
n
4
#
.
i=1
k>k
Now,
"
n
1X
h(Xim ) − E[h(X m )] >
Pr
n
4
#
i=1
(A.25)
" n
1 X
m
m n ≤ Pr h(Xi ) − E[h(X ) ] >
n
i=1
h
i
+ Pr |E[h(X m )n ] − E[h(X m )]| >
8
8
#
,
where h(X m )n = h(X m )1[|h(X m )| < n]. The second term converges to zero uniformly over all m by
the uniform integrability of the sequence of random variables g(X m ). The first term converges to zero
uniformly over all m by Proposition A.2. Let us check that its conditions are satisfied. The first one is
satisfied by Markov’s inequality and the fact that supm E[g 1+γ (X m )] < ∞:
sup n Pr [|h(X m )| > n] = sup n Pr |h1+γ (X m )| > n1+γ
m
m
1
E[h1+γ (X m )]
γ
m n
1
≤ sup γ E[g 1+γ (X m )] ,
m n
≤ sup
whcih converges to zero as n → ∞. The second condition can be checked using an analogous argument
to the one used to prove the Weak Law of Large Numbers (Theorem 1.5.6) in Durrett (1996). Therefore,
the probability in equation (A.24) converges to zero uniformly over all m. This fact together with equations
(A.20) and (A.21) prove the result.
Light-Tail Condition for Logit Profit Function
45
Consider a profit function where the demand system is logit and static equilibrium is Nash in prices. A
detailed description of the model is provided in Subsection 6.1. For simplicity, suppose that θ1 = 1. To
compute
∂ ln πm (y,f,n)
∂fk
we only consider the first order effect, namely, the direct effect of the change of fk in
πm . We ignore the second order effect through prices. Numerical experiments suggest that when number of
firms is large the latter effect is negligible. In that case it is easy to show that:
∂ ln πm (y, f, n)
∂fk
m,y,f,n
sup
< Ck ,
for a constant C > 0. The light-tail condition follows. Similarly, we can also check directly that Assumption
4.3 is satisfied.
Discussion of Assumption 4.1.2
Assumption 4.1.2 is used to prove Theorem 4.2. In this subsection we argue that the assumption is
essentially a necessary condition for the light-tail assumption to be satisfied (Assumption 4.4), and hence is
made without loss of generality in conjunction with the light tail assumption.
Assumption 4.1.2 in the proof of Theorem 4.2 is used to guarantee that
∞
X
i
h
(m)
β k−t EQ(m) πm (x + (k − t)w, fk , ñ(m) )xt = x
k
k=t
remains uniformly bounded over all market sizes when the light-tail condition is satisfied and, hence, when
(m)
ñ(m) = Ω(m) (see equation A.13). Note that if fk
is replaced by f˜(m) , then, it is easy to see that when
the light-tail condition is satisfied the sum must be uniformly bounded over all market sizes. If not, the
zero profit condition at the entry state in the oblivious equilibrium would be violated for m big enough.
(m)
However, with very small (but positive) probability, fk
can be arbitrarily far from f˜(m) . Therefore, we
(m)
need to ensure that the sum remains uniformly bounded for any value of fk
sup
∞
X
. Specifically, we need
β k−t πm (x + (k − t)w, f, ñ(m) ) < ∞ ,
m,f k=t
if ñ(m) = Ω(m). Assumption 4.1.2 ensures that this is the case.
The assumption is a necessary condition for the light-tail condition. If profits grow just faster than
assumed in 4.1.2 the above sum would be unbounded. Therefore, if 4.1.2 were not satisfied, it would not
be enough to have ñ(m) growing at rate m to keep discounted profits uniformly bounded. One can check
that the zero profit condition would then require f˜(m) to grow with m. But, under the light-tail condition,
46
f˜(m) converges. Therefore, if Assumption 4.1.2 were not satisfied then the light-tail condition could not be
satisfied either.
A.4
Proofs and Mathematical Arguments for Section 5
Theorem 5.1
Proof. We derive the second bound, beginning with the following proposition. Let µ∗ be the optimal (nonoblivious) best response to an oblivious equilibrium (µ̃, λ̃) for a firm that is keeping track of the industry
state.
Proposition A.4.
E[V (x, s|µ∗ , µ̃, λ̃) − Ṽ (x, s|µ̃, λ̃)] ≤
1
E[∆πm (s)] , ∀x ∈ N .
1−β
Proof. By a similar argument to the one at the beginning of the proof of Theorem 4.2, we have that
(A.26)
"
V (x, s|µ∗ , µ̃, λ̃) − Ṽ (x, s|µ̃, λ̃) ≤ Eµ∗ ,µ̃,λ̃
τi
X
#
β k−t (πm (xik , s−i,k ) − πm (xik , s̃)) xit = x, s−i,t = s .
k=t
The equation can be rewritten as:
V (x, s|µ∗ , µ̃, λ̃) − Ṽ (x, s|µ̃, λ̃) ≤
∞
X
k=t
(A.27)
β k−t
X
Pµ∗ ,µ̃,λ̃ [xik = y, s−i,k = s0 | xit = x, s−it = s] ·
y∈N
s0 ∈S
πm (y, s0 ) − πm (y, s̃) ,
where Pµ∗ ,µ̃,λ̃ [xik = y, s−i,k = s0 | xit = x, s−i,t = s] is the probability firm i, currently in state x with
competitors in state s, will be in state y and s0 , respectively, k − t periods from now.
We can write:
Pµ∗ ,µ̃,λ̃ [xik = y, s−i,k = s0 | xit = x, s−i,t = s] = Pµ∗ ,µ̃,λ̃ [xik = y | s−i,k = s0 , xit = x, s−i,t = s] ·
Pµ∗ ,µ̃,λ̃ [s−i,k = s0 | xit = x, s−i,t = s]
= Pµ∗ ,µ̃,λ̃ [xik = y | s−i,k = s0 , xit = x, s−i,t = s] ·
Pµ̃,λ̃ [s−i,k = s0 | s−i,t = s].
47
The last equation follows because rival firms use strategy µ̃, which only depends on their own state, and the
entry rate is λ̃ independent of the industry state. Substituting into equation (A.27) gives:
V (x, s|µ∗ , µ̃, λ̃) − Ṽ (x, s|µ̃, λ̃) ≤
∞
X
β k−t
≤
(A.28)
y∈N
∞
X
Pµ̃,λ̃ [s−i,k = s0 | s−i,t = s]
s0 ∈S
k=t
X
X
Pµ∗ ,µ̃,λ̃ [xik = y | s−i,k = s0 , xit = x, s−i,t = s] πm (y, s0 ) − πm (y, s̃)
β k−t
k=t
X
Pµ̃,λ̃ [s−i,k = s0 | s−i,t = s] max πm (y, s0 ) − πm (y, s̃) ,
y∈N
s0 ∈S
Recall that q(s) is the invariant distribution of {st : t ≥ 0}, where st is the industry state at time t
when every firms uses strategy µ̃ and the entry rate is λ̃. Note that {s−i,t : t ≥ 0} is the same process as
{st : t ≥ 0}. Hence for any k ≥ t:
q(s0 ) =
(A.29)
X
q(s)Pµ̃,λ̃ [s−i,k = s0 | s−i,t = s]
s∈S
Multiplying equations (A.28) by q(s) and summing over all s ∈ S we obtain:
X
∞
X
X
X
q(s) V (x, s|µ , µ̃, λ̃) − Ṽ (x, s|µ̃, λ̃)
≤
q(s)
β k−t
Pµ̃,λ̃ [s−i,k = s0 | s−i,t = s] ·
∗
s∈S
s∈S
s0 ∈S
k=t
0
max πm (y, s ) − πm (y, s̃)
y∈N
=
∞
X
β k−t
XX
q(s)Pµ̃,λ̃ [s−i,k = s0 | s−i,t = s] ·
s0 ∈S s∈S
0
k=t
max πm (y, s ) − πm (y, s̃)
y∈N
=
(A.30)
∞
X
β k−t
k=t
X
s0 ∈S
q(s0 ) max πm (y, s0 ) − πm (y, s̃)
y∈N
The second equation follows by Fubini and the last one by equation (A.29). The previous argument is valid
for any x ∈ N, therefore:
E[V (x, s|µ∗ , µ̃, λ̃) − Ṽ (x, s|µ̃, λ̃)] ≤
where s is a random vector distributed according to q.
48
1
E[∆πm (s)] , ∀x ∈ N ,
1−β
Returning to the derivation of the bound, we have that:
(A.31)
E[V (x, s|µ∗ , µ̃, λ̃) − V (x, s|µ̃, λ̃)] = E[V (x, s|µ∗ , µ̃, λ̃) − Ṽ (x, s|µ̃, λ̃)] + E[Ṽ (x, s|µ̃, λ̃) − V (x, s|µ̃, λ̃)]
The first term is bounded by the previous proposition. Let us analyze the second term:
"
Ṽ (x, s|µ̃, λ̃) − V (x, s|µ̃, λ̃) = Eµ̃,λ̃
"
− Eµ̃,λ̃
τi
X
k=t
τi
X
i
β k−t πm (xik , s̃) − dιk
β
k−t
πm (xik , s−i,k ) −
#
+ β τi −t φiτi xit = x, s−i,t = s
dιik
+
β τi −t φiτi xit
#
= x, s−i,t = s
k=t
"
= Eµ̃,λ̃
τi
X
#
β k−t (πm (xik , s̃) − πm (xik , s−i,k )) xit = x, s−i,t = s
k=t
=
∞
X
β k−t
(A.32)
Pµ̃,λ̃ [s−i,t = s0 | s−i,t = s] ·
s0 ∈S
k=t
X
X
Pµ̃ [xik = y | xit = x] πm (y, s̃) − πm (y, s0 ) .
y∈N
The last equation follows because under oblivious strategies firms’ trajectories are independent. Multiplying
each term by q(s), summing over all s ∈ S and interchanging sums in the right hand side using Fubini we
obtain:
(A.33) E[Ṽ (x, s|µ̃, λ̃)−V (x, s|µ̃, λ̃)] =
∞
X
β k−t
k=t
X
P µ̃ [xik = y | xit = x] (πm (y, s̃) − E [πm (y, s)]) ,
y∈N
where s is a random vector distributed according to q. Finally, interchanging the sums
(A.34)
E[Ṽ (x, s|µ̃, λ̃) − V (x, s|µ̃, λ̃)] =
X
ax (y) (πm (y, s̃) − E [πm (y, s)]) .
y∈N
The second bound follows by equations (A.31), (A.34) and the proposition. The first one follows by a similar
argument, but with the difference that we take maxy∈N in equation (A.32) and we take absolute value of the
difference of one period profits in equations (A.28) and (A.32).
49
A
Appendix: Tables and Figures
Table 1: Comparison of MPE and OE strategies (4 firms, no entry and exit)
Parameters
Long Run Statistics (% Diff)
Perf Bound (% Diff)
Actual (% Diff)
Prod
Cons
Max
Weighted
Max
Weighted
θ1
d
Inv.
Surp
Surp
C1
C2
Diff
Avg
Diff
Avg
0.10
0.10 −0.26 −0.01 −0.02
0.03
0.03
0.14
0.13
0.08
0.07
0.15
0.27
3.54
0.14
0.2
1.22
0.46
0.36
0.35
0.1
0.1
0.20
0.35
4.18
0.29
0.42
1.93
1.03
0.81
0.77
−0.09
−0.05
0.30
0.30 −0.13
0.06
0.08
0.08
0.16
1.67
1.22
0.04
0.01
0.30
0.55
9.28
0.93
1.31
5.10
2.45
1.96
1.85
0.26
0.25
0.40
0.80 21.02
2.10
2.93 11.58
4.12
3.01
2.92
0.30
0.29
0.50
1.00 18.62
3.30
4.33 15.69
5.94
6.29
5.86
0.32
0.30
0.50
0.50 −0.11
0.20
0.28
0.18
0.50
6.64
3.61
0.21
0.06
0.70
0.70 −2.21
0.40
0.15
1.08
2.09
18.85
8.35
1.60
0.67
0.85
0.70 −2.19
0.23 −0.28
1.37
2.10
30.80
9.64
1.80
0.20
1.20
1.00 −8.41
1.59
0.36
2.73
8.41 131.30
48.50
23.70
7.90
Long run statistics and value functions simulated with a relative precision of 1.0% and a confidence level of
99%. Performance bound simulated with a relative precision of at most 10% and a confidence level of 99%.
50
Table 2: Comparison of MPE and OE Investment (4 firms, no entry and exit)
Parameters
Investment
θ1
d
MPE
OE
% Diff
0.10
0.10
0.752
0.754 −0.26
0.15
0.27
0.192
0.185
3.54
0.20
0.35
0.261
0.250
4.18
0.30
0.30
0.754
0.755 −0.13
0.30
0.55
0.238
0.216
9.28
0.40
0.80
0.168
0.133
21.02
0.50
1.00
0.195
0.158
18.62
0.50
0.50
0.741
0.742 −0.11
0.70
0.70
0.694
0.709 −2.21
0.85
0.70
0.748
0.765 −2.19
1.20
1.00
0.553
0.599 −8.41
Investment simulated with a relative precision of
1.0% and a confidence level of 99%.
Figure 1: Percentage performance bound for Poisson entry process for different market sizes.
20
Low
High
18
16
% Bound
14
12
10
8
6
4
2
0
30
60
120
250
500
1000
Expected Number of Firms
51
1800
2600
Figure 2: Percentage performance bound for deterministic entry process for different market sizes.
12
Low
High
10
% Bound
8
6
4
2
0
30
60
120
250
500
1000
1800
Expected Number of Firms
Figure 3: Percentage performance bound for fixed number of firms.
5
Low
High
% Bound
4
3
2
1
0
5
10
20
40
80
120
190
Number of Firms
52
380
500
750
1000
Figure 4: Average industry state for θ1 = 0.5 and d = 0.5.
Expected Number of Firms
0.12
0.1
0.08
0.06
0.04
0.02
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Quality Level
Figure 5: Average industry state for θ1 = 0.4 and d = 0.8.
Expected Number of Firms
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
Quality Level
53
10
11
12
13
14
15
Figure 6: Average industry state for different market sizes for θ1 = 0.9.
100
M=100
90
M=400
Expected Number of Firms
80
M=1600
70
60
50
40
30
20
10
0
1
21
Quality Level
Figure 7: Average industry state for different market sizes for θ1 = 1.2.
8
M=100
Expected Number of Firms
7
M=400
M=1600
6
5
4
3
2
1
0
1
21
41
Quality Level
54
61