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IB31
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working paper
department
of
economics
and the Emergence of
Market Dominance
Innovation
Susan Athey
Armin Schmutzler
No. 99-18
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
Innovation and the Emergence of
Market Dominance
Susan Athey
Armin Schmutzler
No. 99-18
October 1999
MASSACHUSEHS
INSTITUTE OF
TECHNOLOGY
50
MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
Innovation and the Emergence of Market Dominance
Susan Athey and Armin Schmutzler
This Draft: October, 1999.
analyzes a model of oligopolistic competition with ongoing
Abstract: This paper
investment.
It
incorporates the following models as special cases: incremental invest-
ment, patent races, learning-by-doing, and network externalities.
cumstances under which a firm with low costs or high quality
will
We
investigate cir-
extend
through further cost-reducing or quality-improving investments. In
studied oligopoly games, such investments are strategic substitutes.
its initial
lead
many commonly-
We
derive a
new
comparative statics result that applies to games with strategic substitutes, and we use
the result to derive conditions under which leading firms invest more than lagging firms.
We
show that the conditions
models.
We
are satisfied in a variety of commonly-studied oligopoly
also highlight plausible countervailing effects
First, leading firms
may
find
it
more
from two distinct sources.
costly than others to achieve the
same increment
many models of patent races, where
firms make research investments in an attempt to find a new technology that delivers
a given level of cost or quality. Second, countervailing effects may arise in dynamic
games with more than two firms, when firms are sufficiently patient.
to their state. This force
JEL
Classification
is
particularly salient in
Numbers: Lll, L13, L41, O30.
Keywords: Oligopoly games, strategic substitutes, innovation, investment, increasing
dominance, market concentration.
*AfSliations:
Susan Athey, Department of Economics, Massachusetts Institute of Technology,
MA 02142, USA; and National Bureau of Ek;onomic
Armin Schmutzler, SozialOkonomisches Seminar, Universitat
50 Memorial Drive, E52-252C, Cambridge,
Research, e-mail: athey@mit.edu.
Zurich, Bliimlisalpstr. 10, 8006 Zurich, Switzerland, e-maU: arminsch@sozoec.unizh.ch.
We
Rabah Amir, Zava Aydemir, Kyle Bagwell, Abhijit Banerjee, Men-Andri
Thomas Gehrig, Bob Gibbons, Aija Leiponen, Paul Milgrom, Wally Mullin,
Andreas Polk, Mike Riordan, and seminar audiences at MIT, Freiburg, Torino (EARIE) and
Zurich for helpful discussions. Thomas Borek provided excellent research assistance. Athey
are grateful to
Benz, Luis Cabral,
thanks the
NSF
(SBR-9631760)
for fmancial support.
Introduction
1
This paper studies the dynamics of market structure
in
a setting where firms have the
opportunity to innovate, for example by investing in cost reduction. Our goal
is
to understand
conditions under which firms with an initial advantage increase or decrease their investments
and market shares over time.
We focus on changes in concentration that
choices about activities such as investment in research
arise
through firms'
and development and advertising.
We
examine conditions under which weak increasing dominance emerges, whereby leading firms
invest
more
into improving their state.
We
also discuss
under which conditions this implies
strong increasing dominance, whereby the higher investment levels of leading firms result in
increasing market shares.
The
empirical evidence about the dynamics of concentration illustrates that a variety of
phenomena are
possible. Increasing
dominance of the market leader
most obvious examples concern network
is
not uncommon.
effects or learning-by-doing. ^ Increasing
The
dominance
has also been documented as arising through a process of cost-reducing or demand-enhancing
investments in retail markets (Bagwell et
(Sutton (1991), Bagwell and
Ramey
al (1997))
(1994)).^
On
and advertising-intensive industries
the other hand, increasing dominance
does not always arise in such settings.^
To provide
competition
insight into these patterns,
among two
in each period,
^
Network
more
oligopolists.
In our model, firms
may make
investments
where we use the term "investment" very broadly. For example, firms may
make investments
negUgible
or
and we develop a general theoretical model of
in process innovations that lead to cost reduction, so that the current
have been widely cited as the reason why VHS managed to rapidly extend comparatively
advantages over Betamax in the market for videocassette recorders; more recently, policy-
effects
initial
makers have focused on the role of network effects in computer operating systems and apphcations. Seminal
models of network effects include Katz and Shapiro (1986) and Farrell and Saloner (1986). Learning-bydoing effects have been analyzed in the context of aircraft production and other manufacturing processes;
see Cabral and RiordEui (1994) for further examples.
^In another example, the empirical literatine concerning the product cycle emphasizes that
when a
mature stage of market development is reached, a shake-out of firms arises. Such a shake-out
often arises because some firms manage to reduce production costs faster than others (Klepper 1996).
relatively
^Gruber's (1994) account of the semiconductor industry shows that the dynamic patterns within genermemory chips vary from period to period and from product to product. IBM lost its early lead
ations of
mainframe computer market to market entrants in the 1960's and 1970's (see Sutton 1998, ch.l5), in
network effects. The aircraft industry is another example for a technology-intensive industry
characterized by technological leapfropping between a small number of firms, resulting in leadership reversals
and significant shifts in market shares.
in the
spite of strong
state of the firm depends on past investments. Investments can result in small incremental
improvements
relative to the earlier state, or else in
n&D-investment
refers to
Our
first result
firms are myopic.
We
show that
more when the following conditions are met:
product market payoffs are convex
(ii)
level that
maximizes
establishes a general set of conditions under
when
(i)
(i)-(iii)
is
all
Indeed,
much
games with
functions must be "exchangeable."
differences
among
firms are
we impose an
about strategic trade
Within the
That
is,
and Milgrom and Roberts
(1990)),
it
in favor of conditions (i)-(ii).
equilibrium demand. Clearly, the higher
many
firm's
variables.
may
it
Our
result represents
an
also
be applied to other games
can be
vised to consider questions
games, we show that forces arising naturally in
and consider decomposing a firm's product market
demand, and a
games
policy.
work
to lowering cost. In
competing
they must satisfy a certain kind of sjrmmetry:
summarized by the state
class of investment
oligopoly models
*
should be noted that the result
alternative assumption: the firms' profit
with strategic substitutes. For example, we show that
its
adjustment costs do
strategic substitutes potentially have
extension of the comparative statics literature, and
and
(iii)
of the existing literature restricts attention to two-player
In this paper,
for just this reason.^
all
It
and
choices are mutually reinforcing and thus comparative statics on equilibria can be
easily obtained, multi-player
effects. "*
which weak increasing dom-
not immediate. In contrast to games with
strategic complementarities (Topkis (1979), Vives (1990),
where
a single period.
investments are strategic substitutes,
in the state variable,
imply weak increasing dominance
profits in
firms with higher state variables will
not rise too quickly with the level of the state variable.
that
(as in stochastic
a learning-by-doing or network externality setting, "investment"
a choice of output that exceeds the
inance emerges
invest
races). In
major breakthroughs
Consider the example of cost reduction,
profits into the
product of a firm's markup
a firm's demand, the higher are the returns
models, an opponent's investment decreases a firm's equiUbrirun
own investment
Observe that an increase
is
many
increases
in player I's action
its
equiUbrium demand. Thus,
it is
perhaps
has direct negative effects on aU opponents; but the resulting
decrease in player 2's action might lead to an even larger increase in player 3's action. Thus, the effect of an
increase in player I's state variable has ambiguous effects on player 3.
^This is the approach taken by Novshek (1985) in an analysis of Com-not ohgopoly. Amir (1996) and
Davis (1999) use the tools of games with strategic complementarities to analyze Cournot ohgopoly games
with more than two firms, but maintain the assumption that each firm cares only about the siun of opponent
output.
not surprising that a wide variety of commonly- used models satisfy conditions
including: Bertrand
and Cournot competition with
differentiated
and
(i)
(ii),
goods and hnear demand;
horizontal competition on the line or on the circle with quadratic transportation costs; and
some
to
vertical quality differentiation models.
some
cases where firms undertake
We
also
show that our
results
more than one type of investment
can be extended
at the
same
time,
such as quality-improving and cost-reducing investments.^
However, we have already noted that even
in innovative industries,
not always follow patterns of increasing dominance.
favor of
dominance?
What
market dynamics do
serves to mitigate the forces in
We only briefly discuss the potential for competing effects that arise due
to product market competition, instead focusing on the properties of the technology with
which state variables can be adjusted upwards
(effect (iii)).
Two
polar examples serve to
highhght the scenarios under which adjustment costs do or do not compete with increasing
dominance
effects.
First, consider
a stylized incremental investment model, where the cost
of achieving a given increment to the state variable
is
the same for
of the state variables. Contrast this with a second example,
a manner that
compete
closer to
is
for the
what has been modeled
all firms,
and independent
where innovation takes place
in the literature
on patent
races:'^
in
firms
next technological innovation, and each firm requires a similar investment
to achieve a given level of the state variable (or a given probability of achieving that level
of the state variable) in the next period.
per unit of investment
is
In such models, the magnitude of improvement
larger for laggards than for leaders,
than laggards. However, the
latter effect
and so leaders may
must be large to overcome
forces
(i)
invest less
and
(ii)
from
above.
In a second set of results,
concern for the future
of the state variable
aflfects
is
dynamics reinforce the
we relax the assumption that
our results about dominance.
deterministic
resiilts
firms are myopic,
We
find that
and consider how
when the
and firms commit to an investment plan
evolution
in advance,
from the myopic-firm model.^ However, when we allow firms to
^Cabral (1999) proposes an alternative theory of increasing dominance that does not rely on the scale
He identifies a different force: leaders have an incentive to imdertake R&D investment
with retinrns that are correlated with the laggards, while laggards desire an opportmiity to leapfrog, which
can be accomplished using independent investments.
effects described above.
^See Reinganmn 1985, Vickers 1986, Beath et
(1979)
and Lee and Wilde
* Indeed,
al.
1987, as well as the earlier patent race papers
by Lom-y
(1980).
the hteratm-e highhghts the fact that the results are sensitive to the specification of the model.
^This highhghts an advantage to our
lattice- theoretic
approach: the theorem applies without modification
condition each period's investment on the current level of the state variable, then a variety of
competing
effects
most robust
can
in scenarios
where firms are
require advance planning
we show
Finally,
Thus, we show that our results about increasing dominance are
arise.
and are
fairly short-sighted, or
difficult to
modify once
where investment strategies
in place.
^°
that our framework can be used to organize and extend the results
of specific models from the existing literature, including incremental investment games,
learning- by-doing models,
and patent
races. ^^
Thus, our model provides a lens through
which to analyze the extent to which increasing dominance should be expected
in
a partic-
ular industry.
We
main
proceed as follows.
In Section
results for this model.
and applications of the
2,
we introduce the model. Section
In Section 4,
results
from Section
3 contains the
we
present examples of the general framework
3.
Section 5 examines poUcy implications and
conclusions.
The Model
2
This section specifies a model of dynamic oligopoly that
is
general enough to incorporate
incremental investment races, patent races, learning-by-doing and network models as special
cases.
Our
interpretations will vary across models; Section 4
makes the interpretations more
precise in the context of the specific applications.
when the
choice set for each player
is
an
infinite series rather
than a real number.
on understanding general properties in a dynamic context, our paper is similar to Budd et al.
framework to identify factors favoring and preventing increasing
dominance. Their paper considers a continuous-time game between two firms who make investments in
changing the state variable, and each firm's profits depends on the state variable. They fink the slope of the
profit functions to investments, but they do not attempt to give conditions on oUgopoly models that lead to
^°In its focus
(1993)
who
also apply a relatively abstract
increasing dominance.
^^
Consider some examples from the existing hteratiue. Flaherty's (1980) incremental investment model
homogenous goods.
allows for cost-reducing investment between two rounds of Cournot competition with
Dominance emerges. A similar effect is present in the product life cycle model of Klepper (1996), although
the scale effect is dominated by others in some stages of the life cycle. In Bagwell et al. (1997), customers
learn retail prices slowly and, by using low prices today, firms signal that they have low costs and will also
have low prices in the future. This is credible if low-cost firms expect high future demand, which is the scale
effect described earUer. Cabral and Riordan (1994) examine a duopoly model with random product sales,
where a lucky firm that manages to sell a lot in early stages proceeds along its learning cin-ve more rapidly,
allowing it to set prices lower than the competitor, thereby increasing the chances of future sales and thus
the chances of futiue leairning.
T
There are
t,
firm
i
periods,
t
=
1,...,T
though
in a
=
and /
1, ..,n.
> Yj
firms,
where
=
3^^
componentwise order:
Since the goal of this paper
assume that Y^
3^i,
few apphcations we will have
will order vectors using the standard,
i
oo),
characterized by a state variable Y^ 6
is
typically E,
for
<
(T
implies that firm
is
i
3^^
=
In each period
I,..., I.
a partially ordered
is
E". Throughout the paper, we
for x, x'
G E", x >
state variable might represent the extent of previous cost reductions
j.
As
from some
number
by a
of product variants offered
firm; or,
a combination of
x'-
we
such, the
initial level;
a demand-enhancing parameter, such as product quality or a cumulated advertising
or the
>
x' if Xi
to analyze the dynamics of market share,
has greater market share than firm
i
set,
level,
demand and
cost
parameters.
Let
Y'
=
(y/, ...,y/)
ey =
The
x^J^^.
initial state of
the market,
exogenously given. Given Y^~^, each firm chooses an action variable
We
the increment y| in the state variable.
a*
—
(a\,
...,
G
a\)
A=
XiAi.
The
In the simplest applications, yl
turns,
we allow
y*
=
(2/1, •••, ?//)
=
=
a-.
Now
when
a-
y/-^
+
i
€ ^i, which
(1)
random
we
variable with distribution
the distribution of the change in the
let 7r*(a',y*, Y'~-^)
it
first case, a* is
depend only on the state
Instead,
we
variables,
denote product market profit for firm
this case,
i.
For example, a'
product market profits
and we do not formally model the product market
let 7r*(a*,y*,
Y'~^) represent the reduced-form profit to firm
from a product market equilibrium. Since
many examples, such
wish to
represents output choice in a learning-by-
might represent the vector of cost-reducing investments. In
^^In
We
the vector of firm choices
In the second case, investments precede product market competition.
competition.
re-
represents an Pl&D-effort.
market competition. For example,
doing game. Then,
Let
develops according to
to be the reahzation of a
may improve
is
influences
an "investment."
consider representing the profits from product market competition.
in product
-^YP),
yl
handle two distinct cases in a unified notation. In the
resulting
{Y^°,
However, to incorporate stochastic investment
if (y'|a', Y*~^).^^ In this case, investment
state variable, as
will refer to this choice as
state vector of firm
F/
a*
Y° =
in this case, investments affect
as incremental investment games, the distribution of y* in period
on y/~^ and a*, not on the other firms' state variables and investments. In contrast,
investments have externalities across firms.
t
i
product
depends only
in patent race models,
market competition only through the evolution of the state variable,
for
each
i
there must
exist a function 7r*(Y) such that
n^{a\y\Y'-')
We
refer to the function
tt*
when
= r{Y'-' +
y').
(2)
deriving general results about our model, and
we analyze
the case where (2) holds in greater depth in Section 3.3 and in the context of applications.
We
write the expected product market profit to firm
En'{a.\ Y'-')
We
i
as follows:
= f 7r'(aS y*, Y'-')dH{y'
\
a*,
Y'"^).
further allow for an investment cost function, denoted A;'(a-,y/~^) for firm
assumed to
Thus, the expected
payoff" to firm
Y-''"^) is
i
To keep the
will
exposition concise,
though our main
results
non-decreasing in
in period
W{a\ Y*-^) =
we
which
is
satisfy:
k\al,
entiable,
i,
t is
do not
rely
that
all
k'{al F/"^).
ni,,y.(a,
Y) =
^ff
Y),
(a,
(4)
of the relevant functions are differ-
on that assumption. Throughout the paper,
use subscripts to note partial derivatives of IT and
^n'(a, Y),
(3)
written
E7i'{a\ Y'-')
we assume
a^.
(E7r0a.,y.
En\
as follows:
= ^^Tr^a, Y), and
n^.(a,
Y) =
likewise for the
other variables.
We will
assume throughout the paper that an equilibrium
the assumption that the equilibritmi
is satisfied: for
is
each i,j and each Y,
"conditionally unique," that
if
Yi
=
on firms k
playing equihbrium actions a*_^j{Y),
j^ i,j
variables, there
is
then a*(Y)
is,
Further,
we maintain
the following condition
^ Yj and there exist two equiUbria, a*(Y), a*(Y)
= a*(Y) and a*(Y) = a*(Y). In words, conditional
such that a*_ij{Y)
a*_ij{Y),
exists.^"'
if
firms
a unique equilibrium. For the case where
^*This assumption might be restrictive in some cases.
z
tt'
and j have
different state
represents a reduced- form
The hteratvre on existence of equihbrium
in
Cournot
quantity games might be instructive here (Novshek (1985), Amir (1996), Davis (1999)), as this work concerns
games with strategic substitutes; but, each of these papers considers models where firms care only about the
smn
of opponent outputs.
from competition
profit function
in the
product market, we impose the conditional uniqueness
assumption on the product market equilibrium as
The assumption
that there
of conditional uniqueness plays an important role, as
Observe that
further below.
this
assumption
5.7.1.2), if
each firm's objective function
In order to introduce our final assumption,
poses two elements of a vector. Formally,
Xi
=
Xi.
Then we
|n^._„.
|
>
exchangeable
if
for
=
conditions hold: f\x.)
Using
x=
if
define a
Tjfc(x),
/*
set of I functions,
all
map
Tj^
=
E"
:
globally concave.
for i,j
,a,-
we must
then Xj
we can
The
/^x) =
—
XiXi
:
i,j,k G {1,..,/} such that
f^{Tij{x.))-
this definition,
we
a;
is
(see, for
=
1,...,/,
verify that for all
^ R",
=
Xk, Xk
that trans-
Xj,
and
for all
have:
Definition 1 Consider a
are
that
is
In contrast, to guarantee uniqueness in the /-player game,
^ j,k,
will discuss
considerably weaker than an assumption
is
a sufficient condition for conditional uniqueness
i
we
a unique equilibrium of the /-player game. Using familiar arguments
is
example, Tirole 1988, section
i y^ j.
well.
>
i
^
M.
j
for i=l,..,I. The functions
^
k
^
i,
the following two
f'{Tjk{x)).
state the assumption:
(EXCH)
firms' profit functions are exchangeable.
This assumption will be maintained throughout the paper. Exchangeability reqmres a kind
of
symmetry
variables of
in the identities of firms:
its
if
firm j was in firm
actions
holds
and
all
We
i
cares only about the actions
It
z's
implies that firm
z's
profits are the
same as firm
situation. It further implies that firm i's profits are
state variables of two opponents are exchanged. Intuitively,
difTerences
among
firms are
summarized
wiU provide exphcit examples that
however, the assumption
differentiation,
for all firms.
and
state
opponents, but not about the match between an opponent's identity and
actions /state variables.
be
each firm
and
What
is
would
unchanged
if
the
when exchangeabiUty
in the state variables.
into this firamework in Section 4; in general,
consistent with models of Cournot oUgopoly, vertical product
differentiated product
is
fit
j's profits
models where the cross-price
ruled out by this assumption?
are horizontally differentiated on a Hotelling
line,
effects are identical
Consider a simple example.
and each firm can
Firms
invest to decrease its
Then, firm
marginal cost.
lowers
its
i's
profit will
depend on whether a near neighbor or a distant firm
>
2 firms, exchangeability will not hold in a Hotelling
marginal cost. Hence,
framework.
For 1
=
for /
2 firms, exchangeability will only hold
when
firms are restricted to
locate symmetrically around the midpoint of the interval.^'*
Exchangeability can also be understood in relation to the concept of anonymity, as used
game theory and
in cooperative
among a group
of agents.
A
This literature considers allocations
social choice theory.
welfare function w{x.)
is
anonymous
if
an allocation x yields
the same welfare as Tjfe(x); and a social choice function (mapping agent state variables, or
preferences, to allocations)
is
anonymous
if
permuting the state variables of two agents leads
to a permutation of the allocations (see, for example, Moulin (1988)).
the exchangeability assumption in this context.
that in terms of firm
is
opposing firms j
^
i
first
consider stating
requirement of exchangeability
is
are anonymous; the second requirement
that the vector of profit functions, which can be viewed as a single social welfare function
mapping
3
z's profits,
The
Now
state variables
Dominance
In this section,
and actions to "allocations" of
utility, is
anonjonous.
results
we study conditions under which
firms with higher state variables
make
higher
We begin
by introducing an abstract theorem that applies to games with strategic substitutes. We then
investments in equilibrium. In such cases,
we speak
of weak increasing dominance.
apply the theorem to our model under the assumption that firms are myopic. For the case
where investments take place prior to product market competition, we further explore the
conditions on the oligopoly model that lead to both
weak and strong increasing dominance,
so that leading firms increase their market share over time.
where firms are
^''Of course,
Finally,
we
consider the case
far-sighted.
we could
consider each firm's horizontal location as a state variable, but this approach will
not be useful for our purposes.
Ranking Equilibrium Actions
3.1
Games with
in
Strategic Sub-
stitutes
As we
will see in
more
many
detail below,
strategic substitutes, in the sense that firm
firm j's investment. In this section,
with strategic substitutes. To do
Consider a
element
game between
=
Let A^
Xi.
Xj
so,
given by u'
We
:
Pi!
e
X
decreasing in the level of
Assume that
A",
i's
is
is
=
and
games
notation.
strategy space by Xi, with typical
N
a product set in M^,
<
oo (we
and
as well as static ones),^^
an exogenous "state variable,"
{9i, ..,6j)
statics result for
more abstract
slightly
we can analyze dynamic games
=
is
games are games with
Oi
G
0j,
where 0,
let
is
a
XiQi. Let the players' utility functions be
-^ R.
will typically
we
(x, 0)). Further,
we introduce a
/ players. Denote player
XiXi. For each player, there
product set in E"^, and 6
incentive to invest
i's
we introduce a new comparative
for all i,j.
allow this generality so that
X=
oligopolistic investment
assume that the
utility functions are
exchangeable (as functions of
require that:
(UNQ)
For each 6, there exists a conditionally unique equihbrium,
where the term "conditionally unique" was defined
in Section 2.
Before proceeding, we introduce an important definition (Topkis, 1978).
X,y
Definition 2 Let
be partially ordered sets.
increasing differences in
fix"",
If
y =
Xiyi
differences in
If
/
:
R^
To
a product
is
[yi, yj)
^E
is
smooth,
we
begin,
we do
added generahty
is
set,
all
it
y^)
i
x^ >
- /(x^ y^) >
/
3^
— M
>
is
x^,
function f
y^ >
/(x^, y^)
-
:
X
x
y
—^
M.
satisfies
y^,
/(rr^ y^).
supermodular
in
y
and only
if
if it satisfies increasing
^ j.
has increasing differences
state a condition that appUes
choice sets (such as
^^ Everything
for
{x; y) if for all
A
if
when the
in this section
can be rephrased
for the case
>
0.
players have multi-dimensional
when they play a dynamic game and choose a
not required for this paper.
/^^
where Xi
series of investments): for
is
an arbitrary
lattice,
but the
and
all
x_i,
all
u^{xi,yi-i\6)
Condition
(OSPM)
supermodular
(OSPM)
in Xj.
requires that each player's payoffs are supermodular in her
vector. If the choice set
is
the player's choice vector
Games with
is
choice
multidimensional, this assumption requires that each component of
is
complementary with the other components of the choice
strategic substitutes can
vector.
be usefully contrasted against games with strategic
Games with strategic complementarities
complementarities.^^
own
that each u^ satisfies increasing differences in {xi;xj) for
all
are defined by the requirement
i
^
j.
The
following result
is
due to Topkis (1979).
Lemma
3 Suppose
that:
(OSPM)
(i)
holds;
(ii)
the players' actions are strategic comple-
ments, and (Hi) v} has increasing differences in {xi\6j) for
highest equilibrium. Then,
The
6" > 6^
implies x.*{6")
>
and suppose
this
change directly
For each
is
fairly straightforward.
affects only firm j,
mental returns to investing in the sense defined by
6, let:K*{6) be the
x*(0^).
intuition behind this comparative statics result
increasing 9j,
all j.
Lemma
3.
If
Consider
by increasing her
incre-
opponents' actions were
held fixed, firm j would want to increase her action. However, such a change would lead
opponents to desire increases
in
games of
Now
in their actions. Since
strategic complementarity, the
consider a
game with
such increases are mutually reinforcing
equihbrium action vector must go up.
strategic substitutes, defined
—Xj)
satisfies increasing differences in (xj;
for j
^
i;
that
action decreases the incremental return to a player's
ences in
{xi]
— x_j).
own
now we suppose
—6-j)
for all j
Theorem 4
any increase
action:
^
i.
any opponent's
has increasing
differ-
such that u^ has increasing
is,
u^ has increasing differences in
Then, we have:
Suppose there are only two players, that
functions are exchangeable.
Suppose
that:
^^See Bulow, Geanakoplos, and Klemperer (1985)
this distinction in oligopoly
9i
u''
in
u'
that increases in any opponent's state
variable decrease the incremental return to acting, that
{xi]
is,
by the requirement that each
For such games, we consider parameters
differences in {xi]6i), as before, but
all
(i)
(OSPM)
(UNQ)
holds;
holds,
(ii)
and
the players' actions are
and Fudenberg and Tirole (1984)
games.
10
that the payoff
for
more discussion of
strategic substitutes,
Then
>
di
=
i
1,2.
In the
=
—X2,
61
=
=
61, 62
=
for
first
vector, (^1,^2)
-9h)- As {9h, -9l)
{9l,
each parameter vector
is
>
^
—9j) for j
i.
Lemma
Now, we compare two
Ol
=
—
w'(xi, —£2;^!, —62)
3.
For a given,
alternative param-
Lemma
=
In the second vector, 9i
^2-
9i\
{9h,—9l), whereas for the second vector,
-9h), and since (UNQ) implies that the
{9l,
unique.
=
—^2) andlet {i*(x,^)
the conditions of
satisfies
= 9h >
vector, 9i
first
Hence, for the
9h-
(^1,^2)
rium
Xi, £2
[xi]
x*{d).
Then the modified game
eter vectors.
—
>
x*(^) be the equihbrium of this game.
9, let
92
=
and
(Hi) u' has increasing differences in {xi;6i)
implies that x*(9)
9j
Proof. Define Xi
for
and
3 implies that x\{9h,
and hence x'{{9h,9l) > x\{9l,9h)- By exchangeabiUty xI{9h,9l)
=
equilib-
—9l) > £i(^l, —9h)
xI{9l,9h) and hence
the result follows.
The proof proceeds by observing
for
one player,
it
is
that,
by re-ordering the action
possible to convert the two-player
game with
set
and
state variable
strategic substitutes to
a game with strategic complementarities.^^ Then, the comparative statics result
v^dth strategic complementarities,
Lemma
under two scenarios: one where the
player's,
and one where the
the
player's state variable
first
first
3,
for
games
can be used to compare the equihbriimi choices
player's state variable
roles of the players are reversed.
and increasing the second
is
higher than the second
Lemma 3 implies that decreasing
firm's state variable decreases the
equilibrium choice for firm one, and increases the equihbrium choice for firm two. Finally,
we
exploit exchangeability: the equihbrium in the second case
the
first case,
merely the equihbrium of
with the roles of the players reversed. Since reversing the state variables of the
players leads to an equihbrium with a lower choice for player 1
2,
is
we conclude
and a higher choice
for player
that the player with the higher state variable chooses a higher action.
Now consider extending this result to more than two players.
In a geime with
many
players, even
if
the conditions of
we vxxn into a difficulty.
Here,
Theorem 4
hold,
an increase
does not necessarily lead to an increase in Xi and a decrease in the choices of
^^Vives (1990) and
Amir
-"^^
all
in 9i
opponents.
(1994) use a similar approach to analyze Cournot oligopoly using the tools of
supermodular games.
^*To see the role of the condition (UNQ), suppose that actions are chosen from {0, 1}. If (UNQ) fails,
both (0,1) and (1,0) can be equihbria for a given set of parameters. In fact, both might be equihbria for
both parameter vectors {9h,^l) and {9i,6h)\ this is fully consistent with our assumption that actions are
strategic substitutes. Clearly, however, the equilibrium (0,1)
the conclusion of the theorem.
11
when the parameter
vector
is
(9h,0l) violates
Consider the intuition. Clearly the direct
effect of 6^
supports the specified changes in the
when
choice vectors, and one set of indirect effects does as well:
all
player
i
increases her action,
opponents have a lower incremental return to their actions. The problem
may
set of indirect effects
dominate: when player j decreases her action, player k has an
Whether player k
incentive to increase his.
that a second
is
more
is
sensitive to a
change
in player z's action
or player j's action depends on the functional form.
Given that comparative
and more than two
players, the existing literature often
assumptions. For example,
firm's profit
statics results are quitesubtle in
it is
common
have been exploited
game
effectively
imposes a number of simplifying
assume that a
(as in
a Cournot model with perfect
becomes a two- player game.
These properties
proof of Selten (1970), and in the comparative statics
in the existence
discussion of Dixit (1986).
strategic substitutes
to consider two-firm models, or to
depends only on the sum of opponent actions
substitutes), so that the
games with
However, these assumptions are
restrictive, for
example ruling
out models with imperfect substitutes.
Thus motivated, we now introduce a new
result for
Although we cannot generalize the comparative
games with
statics results of
provide sufficient conditions for weak increasing dominance.
The
strategic substitutes.
Lemma
3,
we can
assumption
critical
for
still
our
result is exchangeability.
Theorem
Then
6i
>
5 Suppose the assumptions of Theorem 4 hold, except there are /
6j implies that
Proof. Without
=
Ti2{0).
that xli2(^)
2.
By
—
x*{6)
>
2 players.
6i
>
x*{G).
loss of generahty, consider players 1
and
2,
and suppose
62-
Let
our exchangeability assumption, there exists an equilibrium x*(0) such
^-i2(^)- -^i^ ^-12
=
^-i2(^) ^^'^ consider the
game between
players 1
Let x**(0) and x**(^) be equilibria of the two-player game, and observe that
imphes that each of these equilibria
assumption, players
2.
>
3,
..,
is
unique given the parameters.
(UNQ)
our exchangeabiUty
/ are not affected by the reversal of the state vairiables of firms
Thus, the equilibrium of the two- player game
are not constrained:
By
x**(0)
=
{xl{0), xl{6))
and
is
also
an equilibrium when players
likewise for 0.
and
1
3,
and
..,
/
But Theorem 4 imphes that
xriO) > xriO). m
The
intuition behind the result can
be
easily related.
librium choices for two vectors of state variables:
12
Our
goal
is
to
compare the equi-
the original vector, and a vector with
the
first
affected
two elements transposed. The key insight
is
when we
players.
variables of the
reverse the roles of the
first
first
two
that players three and higher are not
Further, transposing the state
two players should merely transpose their equilibrium
we can proceed by holding
fixed the actions of players three
and higher
values for the original vector of state variables, and analyze the
players.
But then, the
Theorem 4
logic of
Thus,
choices.
at the equilibrium
game between
first
two
applies: decreasing player I's state variable
and
the
increasing player 2's state variable decreases the equilibrium choice of player 1 and increases
the equilibrium choice of player
Thus, we see the
critical role
2.
played by exchangeability:
to hold fixed the behavior of players three
Without
this assumption,
asymmetries
Theorem
we could
and
higher,
it
gives us just
enough structure
and focus on the two-player game.
find a counter-example.
Such an example might exploit
in the extent to
which one player cares about the choices of the others.
5 can potentially
be applied
in a
wide variety of oligopoly problems, as discussed
below, as well as in a variety of other economic contexts; Section 4.4 considers a model of
strategic trade, while the conclusion discusses a potential application to tournaments.
It is instructive
to
compare the approach pursued here with more standard approaches
that might be used to reach the conclusion
alternative set of sufficient conditions:
weakly increasing
satisfies
in 6i
and
x*j{6) is
6i
(a) 6i
=
=^ x*{9) > x*{0).
=» ^*(^) = ^j(^)) and
>
6j
6j
weakly decreasing in
di for
j
^^
i.
Consider an
(b) x*(9) is
a particular game
If
the requisite regularity conditions, condition (b) could be verified using the implicit
This approach would
function theorem.
differ
from ours
require additional (dominant-diagonal) conditions
the conditions of
increase of
some
Theorem
x*{0), j
j^
5,
i,
strategic substitutes. Indeed,
in
two respects.
on second
(b) excludes situations
derivatives.
where an increase
we
when
there are three or
(b).
we
Second, unlike
in 6i leaxls to
an
more
players,
it is
possible to construct
fails.
Thus, by imposing
are able to include a range of economic behaviors
Further, as discussed in Section
essential uniqueness
would
are able to dispense with smoothness requirements, dominant diagonal
conditions on second derivatives, and
excluded by
it
even though such situations are quite plausible in games with
examples where the conditions of Theorem 5 hold but condition (b)
exchangeability,
First,
is
2,
even our maintained assumption of
weak compared to the global dominant diagonal condition.
13
Weak
3.2
Using the results of the
in
Dominance
Increasing
we now give conditions
last subsection,
our investment game,
when
Myopic Firms
for
weak increasing dominance
for
firms are myopic.
Proposition 6 Suppose firms are myopic. Suppose
that for all
i
Then, weak increasing dominance holds: for a given
^
i
the equilibrium investment of firm
j, if firm
greater than that of firm
is
i
i
^ j,
all
a and
all
Y,
has a higher state variable,
j,
that
Yi
is,
>
Yj implies
a*(Y)>a;(Y).
The Proposition
is
a direct application of Theorem
can be understood as follows.
First, in
The
5.
conditions of the Proposition
terms of adjustment costs, the incremental cost of
investment must not increase too rapidly as the
own
state variable increases.
conditions concern expected product market profits. For future reference,
conditions that would be required to satisfy
(^^l.,a,
<
0'
(WID)
(^^l.,y. >
0'
in the
^^d
The remaining
we summarize the
absence of adjustment costs:
{En^)^^^^^
<
(WID-P)
0.
In words, (WTD-P) requires that in terms of expected product market profits, the investments
of the two firms are strategic substitutes; higher levels of a firm's
own
state variable increase
the marginal returns to investment; and higher levels of the opponent's state variable decrease
the marginal returns to investment. In Section
4,
we
will further analyze these conditions in
the context of oligopoly models.
Of
course, not
aU applications
will
be characterized by weak increasing dominance.
In-
deed, the following simple corollary gives sufficient conditions for "weak decreasing domi-
nance," meaning that leaders invest less than laggards:
Corollary 7 Suppose firms are myopic. Suppose
K,aj <
Then, Yi
>
Yj implies
In words,
and
if
a*(Y)
<
0,
ni^.y.
<
0,
that for all
and
U^^^y^
i
>
^j
and
all
(a,Y),
0.
a*(Y).
a higher state variable for firm
i
decreases firm
i's
investment incentives
increases firm j's investment incentives, the leading firm will invest less
myopic.
14
if
the firms are
Investments that Precede Product Market Competition
3.3
we
In this section,
further explore the special case where the firms invest prior to prod-
uct market competition, and thus the profits in the product market can be represented as
reduced- form functions of the state variables, as in
under which (WID-P)
the subscripts on
tt
is satisfied.
and
tt'
When
(2) holds,
we have the
is
to analyze the conditions
following (where, as usual,
7^V.,,.(a^y^Y'-l)
=
F^.,y,(Y'-^
^V„..(a',ySY'-^)
=
7r;,,^^.(a^y^r-^)=r^,,^^.(Y*-^+/)
that (5) describes the interactions
a'
=
+ /)
y*,
and
(5)
(2) holds, £;7r*(a', Y*~-^)
=
7r'(Y*~^-|-
among investments and between investments and
In this case, the following condition wiU be sufficient to guarantee that
state variables.
(WID-P)
goal
denote partial derivatives):
When investment returns are deterministic,
a*), so
Our
(2).
holds:
7rV.,y.
>
and
<
7r^._y.
(WID-0)
0.
Next, we analyze (WID-0) in terms of a more primitive (but
still
reduced-form) oligopoly
model. Bagwell and Staiger (1994) argued that a wide range of ohgopoly models have the
feature that investments in cost-reduction are strategic substitutes.
might hold
for firm
i
variable
is
in general,
and m'(Y')
first
are
is
first
-
markup
for
gyt Qyt
many
D'(Y*) m'(Y*), where D'(Y')
•
is
the
result
demand
product market equilibrium when the state
in the
+
Qyt Qyt
applications,
market share and the markup:
of a competitor's improvement
+ "^ QytQyt + ^ dYldYj
^'
own
on a
firm's
improvements in the opponent's state variable
|^ <
and |pr
firm's
imderstand the second term, note that the positive
effect
state,
<
own mark-up has
and hence the higher the
liigher one's
up has
why such a
two terms on the right-hand side are typically negative. To understand the
term, observe that in
bad
the
=
see
Y'. Observe that:
""YuYi
Here, the
suppose that ^'(Y*)
To
0.
effect
greater effects on profits the
own demand (f^fpr < 0). To
of a firm's state on its own mark-
greater effect on profits the lower the competitor's state
15
Then, the negative
and hence the higher one's
demand (f^|^ <
0). Similarly,
d^D'
_ dD'dm'
- ^QYl dYl + ""
^
{dY!f
^i
^YuY.
^,
,
Recall our initial (defining) assumption that
|^ >
d^m'
^^'
{dYlf
0; further,
many
since in
a higher F/ corresponds to factors such as lower costs or higher quality,
as well.
If so,
then
f^f^ >
from reduced marginal costs
improved quality)
is
enhanced by the positive
reduction on demand. So long as neither mark-up nor
own
Lemma
result follows directly
8 Suppose that D^ {Y^,
dD'/dYl >
0,
dm^/dYl >
|^ >
demand
is
effect of cost-
extremely concave in the
dominates and (WID-0) holds. Similarly,
state variable, this effect
The next
we expect
the positive effect on the mark-up that stems
0. Intuitively,
(or
applications,
0,
from these arguments:
...,Yj)
and
m'' (Y^, ...,Yj)
dD'/dYj <
and dm'/dYj
are linear functions,
<
0.
and
that
Then Condition WID-0
holds.
By
hold
Proposition
if
6,
the conditions of
Lemma
assumption
the following
is
much more
Lemma, which shows
special than required. Nevertheless,
that
many
Lemma
In the following models, condition
WID-0
firm
i 's
and demand
is
linear (Dixit, 1979), where
is
constant, goods are
Y} represents either marginal
cost,
quality level, or the difference between the latter two parameters.
Models of horizontal competition on the
line
(d'Aspremont
(Salop, 1979) vjith quadratic transpoHation costs, where
(c)
can be applied in
holds:^^
Bertrand or Cournot competition where each firm 's marginal cost
differentiated,
Obviously, the
(WID-0).
in
9
it
will
familiar oligopoly models satisfy the properties
summarized
(b)
weak increasing dominance
adjustment costs do not increase too rapidly with state variables.
linearity
(a)
8 imply that
The Shaked/Sutton (1982) model of
Yl
is
et ai,
1979) or on the circle
as in (a).
vertical quality differentiation with potentially dif-
ferent marginal costs, where Y} represents firm
i 's
marginal cnst}^
^^Bagwell and Staiger (1994) establish that several of these examples satisfy strategic substitutability.
However, they do not discuss convexity.
^°In this case, condition
WID-0
does not necessarily hold
16
when V/
represents firm
f's
quahty
level.
The Shaked/Sutton (1982) model of
(d)
assumed
to he covered,
where the market
vertical quality differentiation
the firms have identical marginal costs,
and Y^ represents firm
is
i 's
quality level.
A
sketch of the proof can be found in the Appendix. In the
Lemma
follows from direct application of
quality parameter as the state variable
Lemma
of
demand and
Proposition
particular,
Lemma
6, it
9
the mark-up, in particular for a low-quality firm that
but
it
we must
verify the assumptions that profits are exchangeable
requires that
we
horizontally differentiated.
we
the
Exchangeability always holds in models
restrict attention to only
Even
and that there
two firms
for the two-firm case,
in
when the
model
(b),
(a),
exists
(c),
and
where firms are
firms are located on a line,
require the additional assumption that the locations are equidistant from the midpoint of
line.
for Yi
is
apply
in order to
remains to check that the models satisfy our maintained assumptions. In
a conditionally unique equilibrium.
(d),
its rival.^^
whether (WID-0) holds,
useful in determining
is
case of vertical differentiation with the
vertical differentiation: increasing quality does
reduces vertical differentiation by moving closer to
Although
three cases, the result
not quite as straightforward, because the conditions
model of
8 do not hold for every
not necessary increase
is
The
8.
first
The
y^Yj,iy^
sufficiently
If
conditional uniqueness requirement
is
satisfied if ^^^^ai^i
In terms of our primitives, this will hold
j.
if
,a
~ ^ai,a^i
,ai
¥"
the adjustment cost function
convex in investment.
investment
is
deterministic and yj
=
a-,
we conclude that product market competition
creates strong forces in favor of the result that leading firms have stronger incentives to
increase their state variable than lagging firms.
Proposition 10 Suppose
that firm investments precede product
investment returns are deterministic, and that
competition
8 are
is
y\
Finally,
assume
that
Try. y.
>
a\.
Suppose further that product market
Lemma
described by one of the models of
satisfied.
=
fc^.
y
market competition, that
9,
for
or that the conditions of
all (a,
Lemma
Y). Then weak increasing
dominance holds for myopic firms.
^^Indeed,
Ronnen
(1991) considers the Shaked/Sutton
model when the market
is
not covered, and finds
conditions whereby vertical investments are strategic complements; similarly, the conditions do not necessarily
hold
if
See also EUickson (1999) for an example
complements depending on the parameter values.
the firms have different marginal cost parameters.
where quahty investments may be
strategic substitutes or
17
In Section
we
4,
will consider specific
models that
We
further explore stochastic investment.
into our general framework,
fit
will also highlight
and
circumstances under which
countervailing effects are likely to dominate over the forces identified in this section. Before
proceeding, however,
we consider conditions under which leading
firms will not only choose
higher investments, but will also increase their market shares over time.
Weak
is,
—
Strong Increasing Dominance
3.3.1
-
increasing dominance does not necessarily imply strong increasing dominance, that
that leading firms will increase their market share over time.
ments need not necessarily imply
relation
faster
Clearly, stronger invest-
growth of the state variable
if
there
between investment and investment success. ^^ However, even
if
is
a stochastic
the state variable
of the leading firm grows faster than the others, additional conditions are required for
increasing
We
dominance to imply strong increasing dominance.
continue to focus on the case where investment
is
7r'(Y)
=
D'(Y)
straightforward to show that under these assumptions a firm that increases
than
its
competitors can increase
its
market share
=
deterministic (y'
ment takes place before product market competition, and
faster
weak
if
and only
if
•
its
a*), invest-
m^(Y).
It is
state variable
(letting
D
be total
demand) :^^
DZ^-D^ZZ^>0.
Olk
Olk
To
(8)
k=lj=l
fc=l
interpret this condition, consider a special case:
demand
functions are linear in states
symmetric, so that there exist two constants, Ki and K21 such that for
all i,j,
I
=
Ki and dD'/dYj
K2. In this case,
(8)
becomes 1// > {<)D'/D
for
and
dD^ /dYi
—
I
EEfp^
fc=ii=i
>
(<)0-
'^
Since the leading firm by definition has a market share greater than 1//, strong increasing
dominance holds
if
and only
if
X] X^
fc=ij=i
market demand
if
the interpretation
dominance
if
^<
every firm can increase
is
straightforward.
and only
simultaneous increase in
if
0.
As
its
state variable
this last
term describes the growth of total
''
Weak
by the same marginal amount,
increasing dominance implies strong increasing
market demand remains unaffected or shrinks as a result of a
all
state variables.
Also, consider the discussion of leaxning- by-doing below.
^^The claim follows by taking the sum of derivatives of firm
18
z's
profit over
i
=
1, ...,!
and rearranging.
Of
course, these are cases
demand-creating
Other things being equal, we should therefore expect strong
effects.
more
creasing dominance to be
than
for cost-reducing
where the investments only have demand-steahng rather than
typical, say, for advertising investments in a
investments in products with
fairly elastic
in-
mature industry
demand.
Far-Sighted Firms
3.4
Now suppose
that firms discount the future at rate 6
>
0.
We wish
to generalize Proposition
6 to the case where firms look forward to the future, maximizing present discounted expected
profits.
We
assume that the actions of firms are observable
simplify matters,
To
will
assume that the evolution of the state variable
we show that we
begin,
attention to the
In an
we
OPSNE,
benchmark case of "open- loop" pure strategy Nash
in the
it
may
OPSNE
when considered
when we
equilibria
restrict
(OPSNE).
at date 0.
While
in general, such a restriction
OPSNE
is
also a
can
Nash equihbrium
player can condition his actions on the observed history of past play,
not be subgame-perfect (see Fudenberg and Tirole (1991), pp. 130-133).^^
As
omits the strategic effects that might arise when firms attempt to manipulate
the future investments of opponents,
firms are fairly impatient
OPSNE may
be required
deterministic.
the investment plans of the firms must best
later;
the deterministic context: every
less so in
game where each
though
the
it is
However, to
each firm makes a deterministic investment plan at the beginning of the
responses to one another
severe,
is
are able to provide powerful conclusions
game, and this plan cannot be modified
be
in every period.
and must
provide a good
if
research
first
it
serves as a useful point of comparison. Further,
fix their
if
investment plans several periods in advance,
approximation of behavior. Such advance planning might
and development requires
large capital expenditures or specialized
technology, such as laboratories.
Milgrom and Roberts (1990) showed that an advantage of the
to
games
is
lattice-theoretic
approach
that the theory of supermodular games can be appUed to problems with a wide
variety of choice sets, including problems
where the agent chooses an
infinite
sequence of
^^To see this, suppose that a^ is an open-loop NE. Then, consider the unrestricted game, where players
can condition their investment in period t on thf history of play up to period t. Suppose that player i
chooses a strategy where his action at every step depends only on t and Y", but not on any other aspect of
the history. Suppose in particular, player i's plan is to use a'-'^. Then, the same conditions that guarantee
that a*-' is an OPSNE imply that a.^'^ is a best response to a^''-' for j ^ i. This argument breaks down if the
game
is
stochastic or firms use
mixed
strategies.
19
actions.
They apply the theory of superraodular games
dynamic games. The following Proposition uses a
to analyze
OPSNE of infinite- horizon
similar approach, applying
Theorem
5 to
the infinite-horizon problem.
Proposition 11 Suppose
(WW)
satisfies
=
T <
oo periods and are far-sighted, and that
Suppose further that the evolution of the state variable
holds.
Y^
that firms live for
Y^'^
+ aj,
(£;7r%^_^^
and that for
>
0,
7^ j,
i
{Ett'),.^,.
unique
If there is a conditionally
all
<
OPSNE,
0,
and
and
all (a,
is
deterministic and
Y),
(£^7r%^_^,
<
denoted a*(Y°), Y^'
(WID-Dl)
0.
>
Yj' implies that
for
all t,
ar(Y°) > 4* (Y°).
The proof
is
given in the Appendix. Proposition 11 imposes several conditions beyond
those required in Proposition
6.
The
functional restriction on the evolution of the state
variable simplifies the problem, allowing us to consider directly the effect of today's action
all
future periods. Condition
(WID-Dl)
is
on
required to guarantee that the following additional
conditions hold: the actions of a given firm in two different periods are complementary in
increasing the profit in
all
future periods (requiring the additional assumption
Ily, y.
>
0);
and, across any pair of periods, the actions of the two firms are strategic substitutes (requiring
the additional assumptions H^.
not affect firm
product market
z's
adjustment
y.
<
and
Ely.
y <
0).
Since firm j's actions
costs, the conditions reduce to restrictions
and
states
do
on the expected
profits.
Consider the special case where investment precedes product market competition. Then,
the restriction analyzed in the last subsection, (WID-0), impUes both
Our
restriction to
OPSNE
in Proposition 11
is
(WID) and (WID-Dl).
potentially severe.
While
it
illustrates
that some aspects of dynamic competition reinforce our results about increasing dominance,
the result ignores the incentives of firms to adjiost their investment strategies over time in an
attempt to manipulate the investment response of opposing firms. Unfortunately, when we
enlarge the strategy space of firms in the dynamic
conditions, a variety of
competing
of sufficient conditions for
effects
game
to allow
them
to respond to current
can emerge. The following result highUghts a set
weak increasing dominance
20
in a
dynamic game between two
firms.
Proposition 12 Suppose
the assumptions of Proposition 11 hold. In addition,
in each period, each firm's investment is chosen
period, the opponent
from a compact subset o/E, and
(1) the conditions of
twice continuously differentiable,
that in each
and for
i
^j
Lemma
8 hold,
and for
all (a,
or,
more
generally, (2)
allt, a'(Y'~-^) is continuously differentiable,
a^(Y')
is
dY!
W
Y),
(^^%,,y,>Oi (^^%,,a,>0! (^^l,,a,>0; (^^1, <0;
and for
that
equilibrium action induces a unique, interior best-response investment.
assume that either
Finally,
is
's
assume
(WID-D2)
and
nondecreasing in {—Yl,Yj), and "^r;7<^j(Y*)
is
nonincreasing in Yj.
(WID-A)
// there
is
a conditionally unique Markov-perfect equilibrium, where strategies in period
denoted a'*{Y), then F/"^
>
F/"^ implies that af (Y*-i)
>
t
are
af{Y'-^).
This result provides sufficient conditions for weak increasing dominance in a Markovperfect equilibrium,
in period
t.
whereby each
firm's investment in period
Under the assimiptions
equilibrium wdth the
OPSNE,
of Proposition 12,
t
depends on the state variable
we can compare the Markov-perfect
finding that the additional strategic effects reinforce the ten-
dency towards weak increasing dominance. However, when the assumptions are relaxed, the
additional strategic effects
weak
increasing
may
serve as mitigating factors. If firms are sufficiently patient,
dominance may be overturned.
Consider the role of each assumption. First, we maintain the assumption that
order to avoid technical issues concerning differentiability of the value function.
can be extended to an
infinite horizon,
but the proof
is
The
oo in
results
more cumbersome. DifferentiabiUty
simphfies the analysis by allowing us to apply the envelope theorem
interactions between investments across firms
T<
when analyzing
the
and over time.
Second, we have imposed additional conditions on partial derivatives of the product
market payoffs (WID-D2) and the pohcy functions (WID-A). They play a role because,
to determine whether investments are strategic substitutes,
investments are substitutes in affecting tomorrow's
the conditions of
Lemmas
8 and
9.
profit.
One consequence
21
we must
These
verify that today's
restrictions are
of condition
(WID-A)
is
impUed by
that firm j's
period-t investment decreases the marginal return to firm
effect
on firm
j's period-i
+
investment.
1
and the
policy functions are linear,
When
z's
period-i investment, through
its
per-period payoffs are quadratic, optimal
latter effect
zero.
is
However, in general, the third
derivatives of the value function can potentially generate strategic interaction effects that
work against weak increasing dominance.
Now
quence of this assumption, exploited
be monotone: by
Lemma
when
3,
vestment increases and firm
With more than
ther,
The
consider the role of the assumption that there are only two firms.
even
if
in
firm
the proof,
is
conse-
that equilibrium policy functions will
has a higher state variable, firm
i
first
z's
equilibrium
equihbrium investment decreases. Formally, ^a^j{Y^)
j's
<
in0.
2 firms, this conclusions no longer holds, as discussed in Section 4.1. Fur-
equilibrium policy vectors are monotone, competing effects can
similar reasons). Consider the interaction
The
variable in the final period.
arise (for
still
between firm Vs state variable and firm
Y^)
cross-partial derivative gytg^y n'(a'^(Y-^),
j's state
includes the
following terms:
nka.(a^(Y^), Y^)^ar(Y^)
Notice that
if
g^a'^{Y'^)
substitutability
is
0,
between firm
the sign of H^^^. Thus,
two firms
<
the
i's
+
first
term
is
positive, creating
a force opposing strategic
investment and firm j's investment.
when more than two
complicated by the
Y^)^aJ(Y^)^ar(Y^).
nL^,„,(a^(Y^),
We
have not specified
firms are present, strategic interax:tion between
effects of the
two
firms' investments
on the investments of
other firms.
In summary,
we
find that
model remain present; and
in advance,
when
if
firms are forward-looking, the basic forces
from our
static
the firms are forced to commit to strategic investment plans
our results about weak increasing dominance are reinforced. Thus, our results
about weak increasing dominance are perhaps most salient when firms are impatient, or when
investment plans are inherently long-term. However,
if
firms are far-sighted
their investments in response to the evolution of the state variable,
be
qualified.
Competing
effects
curvature of the profit function
^^Of coures, increasing dominance
not
satisfied.
may
is
arise
when
there are
and
if
they adjust
then our results must
more than two
firms, or
when the
very sensitive to the level of the state variables. ^^
may
arise
even when the
sufficient conditions of
Proposition 12 are
For instance, in a model of Cabral (1999), increasing dominance arises despite the fact that
expected payoffs are concave in the state variable for leaders and convex for laggards.
22
Before proceeding,
we mention a
comment about our extension
final
our result about decreasing dominance, Corollary
7,
to far-sighted firms:
cannot be extended to allow
for far-
sighted firms, even under the additional restrictive assumptions outlined above. For example,
applying the logic of the proof of Proposition 11 would require that the intertemporal profit
function satisfies increasing differences in (a-;-a^), and in {a];Yj) for
condition generally requires Ily
<
y.
0,
this violates the
The
first
Hy y >
0.
investment in terms of
today's profits, then firm j's investment increases the returns to firm
But
7^ j.
whereas the second condition requires
Intuitively, if firm j's state variable increases the returns to firm z's
future.
i
z's
investment in the
requirement that across any pair of periods, the investments of
the two firms are strategic substitutes. Thus,
if
firms are far-sighted, another approach
is
required to provide conditions under which lagging firms "catch up" to leading firms.
Examples
4
In the following,
we apply the framework
Incremental Investment
4.1
to several specific examples of investment games.
Games
Several authors (e.g. Flaherty 1980) have considered incremental investment games. ^^
now
introduce a generalized model of incremental investments, allowing for more than two
firms, general functional
We
We
forms
for
product market competition, and stochastic investment. ^^
show that increasing dominance holds quite generally
competing
We
effects
from adjustment costs are not too
interpret a' as
large,
in this context, so long as
and the firms are not too
any
patient.
an investment that takes place prior to product market competition,
such as a cost- reducing or demand-enhancing investment.
the cumulated cost reduction from
some common
The
state variable
V/ represents
reference level c^ or else the quality level.
Hence, the profit from product market competition can be written as in
(2). If
investments
are deterministic. Proposition 10 applies directly: so long as the marginal cost of adjustment
^^The product life cycle model of Klepper (1996) also incorporates incremental investments; however, that
model has a richer structure, as firms take three different kinds of iavestment decisions.
^^Flaherty (1980) considers
more than two
firms,
but does not derive general results about weak increasing
dominance.
23
does not increase too fast with a firm's
sighted, increasing
dominance should
Two extreme examples
own
and firms are
state variable,
sufficiently short-
hold.
of adjustment costs can be used to highlight scenarios under
which the conditions on adjustment costs are
be
likely to
satisfied.
suppose that
First,
adjustment costs are entirely independent of the state variable. At the other extreme, there
exists
a strictly increasing and convex function
k^ (Y^).
such that
k^
A;'(a-,y/~-')
=
k'^{Yl~^
+ a') =
In words, the adjustment costs depend only on the target level of the state variable,
not on the
initial state.
max {y/~\
...,
^/~^})
if
This type of adjustment cost
the firm invests in a radically
or organizational form, so that earlier expertise
firm's state variable, the
cheaper
it is
is
of
likely to arise (at least for
is
diff'erent
little use.
technology, product variant
In this case, the lower
is
a
to attain a given increase in the state variable, a-;
thus, the adjustment cost function creates a force that could potentially upset
Now
Y^ >
A
consider the case of stochastic investments.
(WID).
model of incremental investment
motivates several assumptions about the investment technology. First, assume that the distribution of yl depends only
on firm Vs investment. Further, assume that higher
levels of
investment lead to a First-Order Stochastic Dominance (FOSD) improvement in the distribution of
yl,
and that investment returns are distributed independently across
Formally, there exist marginal distribution functions H^{yl a-),
|
//(y>*, Y*-^)
= Yl H\y\
\
a^),
and H\y\
\
a*) is
i
=
1,
..,
/,
firms.
^^
such that:
nonincreasing in
a\.
(9)
i
Lemmas
8 and 9 provide conditions under which,
product market
firms;
if
and (WID-0) are
profits,
following result, proved in the Appendix,
sufficient to
imply the desired conditions on expected
(WID-P). For simpUcity,we
firms are far-sighted, the
are deterministic, the
The
payoff functions satisfy condition (WID-0).
indicates that (9)
when investments
restrict attention to
game can be analyzed
the case of myopic
using the approach of Proposition
12.
Proposition 13 In
that
(WID-0) and
the incremental investment
(9) hold,
and
that
(£J7r')a._y;
>
model with stochastic investments, suppose
k'^. y.
for
all {ai,Yi).
Then weak increasing
dominance holds for myopic firms.
^*In fact, independence
is
tion of the restrictions that
not necessary; see Athey (1999) and Athey and Schmutzler (1995) for a descripwould be required on the joint distribution of investment returns to preserve oui
results.
24
Our
results
Bagwell and
about incremental investment games can be applied to a variety of industries.
Ramey
(1994) argue that in the retail sector, leading firms
made a variety of cost-
reducing investments as they gained in market share. For example, Wal-Mart invested in
own
its
and a satellite-based communication
distribution system, including trucks, warehouses,
system, as well as advanced information technology systems to
manage
its
inventory. Sutton
(1991) provides a variety of other examples.
Multi-dimensional investments
4.1.1
In this section,
we show that our
about incremental investment games extend to the
results
case where firms invest in both cost reduction
5 applies to
and quality improvement. Recall that Theorem
games where the firms make more than one type of investment, provided these
investments are complementary; thus. Propositions 6 and 11 can be easily extended to that
case. It
remains to be verified that the relevant restrictions are satisfied in the incremental
investment games.
Cost-reducing and demand-enhancing investments for monopolists have been shown to
be complementary under
ohgopoly setting. Formally,
Y-
let
=
(I^/c^/q) ^^^ ^^^ ^i
cost functions are additively separable for the
and
further,
if
for
77,
now, we generalize that result to an
fairly general conditions;^^
dY-^dY-^
^
IT)^ ^ IK.
two investments and
nv') >
~
'
and
one firm gains an
this advantage, investing
To
is
more
further understand
Lemma 8 to
straightforward to
in
initial
sufficiently small,
—g— F(Y') <
(WID-0')
0,
well.
An implication is that,
advantage in either cost or quality,
it
starting
wiU retain
both cost-reduction and quahty improvement.
when (WID-0') wiU be
satisfied,
observe that
we can extend
effect:
it
show that condition (WID-0') holds provided demand and mark-up can
linear functions £>'
(V/c
..,
Y/^; Y^q,
..,
Y}q) and
m'
[Y^q^
••'
^/c! ^iQ)
•'
^q)'
Ramey (1994) and Athey and Schmutzler (1995). The driving force behind this result
a low-cost firm will have higher per-miit profit and hence values an increase in demand more
^^See Bagwell and
a scale
is
allow for both cost-reducing and quahty-improving investments. In particular,
be written as
is
A;^. y.
dY-^dY^
ITj^ JK
'
then we can apply Propositions 6 and 11 to this game as
if
(^iC'^ig)- ^^ ^^^ adjustment
k G {C, Q},
^'
from symmetry,
—
than a high-cost firm.
25
and
dD'jdYi^
further,
These conditions are
When
ment
>
0,
satisfied in
firms are myopic,
models
we can
(ii)
and dm^/dV^^ <
dD'/dYj^ <
0,
(a)
and
(b) of
further extend the
Athey and Schmutzler
returns, as in
each type of investment,
for
dm'/dY^^ >
Lemma
for
77
=
9.
model to allow
(1995). For example,
for stochastic invest-
suppose that
(i)
investments in cost-
(iii)
reduction do not affect the distribution of quality improvements (and vice versa).
more
holds
(9)
the returns to demand-enhancing innovation are distributed
independently of the returns to cost-reducing innovation, ^° and
leading firms will invest
C, Q.
in
Then,
both cost-reducing and demand-enhancing innovation.
Patent race models
4.2
we apply our framework
In this section,
Although our
to models of patent races.
results
are ambiguous, they can be used to organize the competing effects that arise in the existing
(Reinganum 1985, Vickers 1986, Beath
literature
As
in the case of
common
incremental investment, Yl
et al. 1987).
is
we modify our assumptions about
the investment technology. In each period the firms invest in
If
firm
state variable,
maxfce{i^..,_/}
states
receives the patent,
i
where that
level is
it
remain the same (F/
where dP'/da\
>
Denoting the
=
Y^~^).
The
and dP'/da\ <
/c-th unit
will achieve
R&D
in
an attempt to attain
an exogenously fixed
an improvement over the best present
Yl~^ ?^ In each period, one firm
for
[^'
state:
level of the
Y^
=Y >
the patent; for the other firms, the
i
obtains a patent
is
P'
(a^),
^ iP
vector as e^, firm
En\B^X-') = iZ
will receive
probability that firm
k
R&D investment.
a\ is
the cumulated cost reduction from some
reference level c or the quality level. However,
a patent.
Suppose
i's
expected product market profit
(y'"' +eA: (y'
- Yl''))] P'
is
written:
(a')
fc=i
^°More generally, if the investments are
and Schmutzler (1995).
^
As
in
many
of the patent race papers,
as a result of a patent
'^It
is
possible to
is
map
correlated, the investments
we
satisfy
a condition found
in
Athey
therefore assiune that the level of the state that can be achieved
given exogenously in each period.
these assimiptions into conditions on the joint distribution function i/(y*|a*, Y'~^),
P is more intuitive for this problem. More generally, we might also consider
more than one firm can obtain a patent and the size of the prize may not be fixed (because firms
but the notation in terms of
races where
must
can pursue different R&D-projects).
26
Finally,
(
assume that previous investment success has no
=
t^yt-i ^'
Of
0).
course,
if
we assume
leaders have a lower marginal cost of
effect
on marginal
R&D
costs
instead (in the spirit of Rogerson (1982)) that
R&D,
the case for
weak increasing dominance would
be strengthened.
If
there are only two firms, then
as required
the larger
(WID-P).'^'^ Intuitively, the lower the level of the opponent's state variable,
by
is
the incremental effect of a patent, and thus the greater the profit increase
to the firm from taking the patent
away from the opponent. Next, consider whether firm
investments are strategic substitutes. Suppose that profits are close to zero
not attain a patent. Then,
the probability that firm
i
<
(£^7r')^ ^,
if
attains a patent
g^g^i P^ (a')
is less
is
the firm does
sufficiently negative.
sensitive to firm
j invests more. However, even under this assumption,
if
(WID-P)
In words,
Vs investment when firm
fails in
general, since firms
with higher current states will see lower returns to investing:
(^-)«,.,
The
intuition
that
it
is
=
Z K (y- +
simple: the better a firm
- rr))]
e. (y'
aheady
is,
^P' (a')
<
0.
the smaller the increase of the state
receives as a consequence of a successful patent,
and hence the smaller the resulting
profit increase.
In conclusion,
we cannot make an unambiguous
dominance, since the own-state
works against the opponent-state
effect
can be shown that increasing dominance
higher state on investment incentives
own
state
on investment
Without
lated.
there must exist
and
a2
'''^In
^
=
loss of generality,
still
holds
if
effect.
o-l
it
the negative effect of a competitor's
see this, suppose that increasing
dominance
suppose that the violation concerns firms
Yh > Yl and an >
However,
substantial relative to the negative effect of a firm's
is
To
incentives.
case for either increasing or decreasing
such that in equiUbrium, Vi
=
Y//,
12
1
and
= Y^,
2.
ai
vio-
is
Then,
=
a^,,
an- Suppose that adjustment costs are sufficiently convex in actions such that each
the
(y*-^
case
of
/>2
firms,
+ e, (r - Yl-'))
it
even
is
'-^
is
possible
positive.
27
that
iv^-i
>
0;
since
for
k
^
i,
j,
firm's objective function is globally
concave
in the investment,
and investment choices are
Hold fixed the actions and states of firms 3 and higher at
interior.
al;^2(^)) ^ind suppress
and
these in the notation. Then, the first-order conditions for players 1
2
and exchangeability
imply that
=
n^, (ol, an; Yh, Yl)
By
concavity, H^^ (a^,
a^,;
Yl, Yfj)
n^^ (a^,
aL-,
>
0.
H^^ (a^,
Hence,
Yl, Yh)
- U^
a^,;
Y^, Yh)
=
0.
Yh, Yl)
>
0.
"
"^
(a^, a^;
(10)
But,
n^, (aL, aL, Yl, Yh)
-U^
(cl, an;
+ Ul
Yh, Yl)
= U^
{aL, aL, Yl,
Yh)
-U^
{aL, an, Yl,
Yh)
-K^ {aL, an, Yh, Yh)
+ K^ i^L, an; Yh, Yh) -U^ {aL, an, Yh, Yl)
<
Then,
aL)
is
if Il\^ ^^ is
(an
-
{aL, an; Yl,
aL) (^-infUl^^,^^
+
{Yh - Yl) (^- iniUl^y^
+ supU^^y^^
not too negative, adjustment costs are sufficiently high such that {an
not too large relative to {Yh
— Yl), and sup^ y ^iiY2 <
increasing dominance holds. In words,
are not strong strategic substitutes,
from a higher
Yh)
level of
a firm's
own
i'^^a.v
weak increasing dominance
and the decrease
state
is
n^^y^
,
(10) fails
follows
if
—
and weak
the investments
in the marginal return to investment
small relative to the increase in the marginal
return to investment from a higher level of opponent states. ^^
Note, however, that even
if
weak
increasing dominance holds, leapfrogging
sheer luck, a laggard might overtake the leader even though he invests
is
possible.
less. If this
By
happens,
he wiU have higher investment incentives once he takes the lead.
A
4.2.1
Now
Potential Extension: Incremental versus Radical Innovation
consider whether
weak
increasing dominance can be expected in a
model where firms
can choose to invest in both incremental improvements and more radical iimovation. For
^^
Existing literature on patent races has taken notice of the fact that product market competition might
matter
whether or not increasing dominance arises. For instance, Ln Vickers (1986), weak increasing
arises for Bertrand competition, but not for Cournot competition.
However, the conditions
here have to our knowledge not been derived elsewhere.
for
dominance
identified
28
example, suppose that the state variable for each firm
level of cost reduction),
is
one-dimensional
the current
(i.e.
but that firms can choose two action variables, denoted {aic,aip)
,
where Uic as an incremental investment, and of aip as a radical investment which can poa major jump
tentially lead to
state variable: yi
=
in the state variable.
with dyi/daic
yi {aic, aip)
These action variables both increase the
and dyi/daip >
>
Further, assume that
0.^^
these variables are substitutes in improving the state: d'^yi/daicdaip
<
0,
would be true
as
implementing the radical investment makes incremental cost improvements
suming that investments take place before competition, product market
given by
tt'
(Y
-I-
In the one-dimensional case
mental investments
firms invest
general.
To
more
As-
profits are therefore
{ac, ap)).
y
natural conjecture
obsolete."^^
if
if
is
WID-0
we saw
holds for
that weak increasing dominance
tt',
but
is
likely for incre-
investments. Thus, a
less likely for radical
that leading firms invest more in incremental investment, while lagging
in radical innovation.
While
this behavior
see this, observe that deriving such a result from
might obtain,
Theorem
need not in
it
5 would require the
following conditions.
(a)
Kca,.
(c) Ui^^^y^
The second
of Vi [dic
all
1
o-ip)
1
<
0;
>
(6) n^,^,„^.^
and
condition in (b) will
ciiC
increases
yi,
ir^^^^y^
fail
<
<
and
0;
n^,^,,^,^
and ajp increases
see why, observe that
y^,
but
Ily. y.
<
own investments
in the
if
<
0.
by the definition
WID-0
holds. Thus,
in qualitatively similar ways.
investment precedes product market competition, then
interact with
H^,^,^.
To
opponent actions interact with a firm's own actions
itively, if
(11)
0;
> Oand
(d) IV^^^^y^
in general.
>
same way. Thus, an
all
Intu-
opponent investments
increase in firm j's investment in
incremental innovation decreases the returns to both incremental and radical iimovation for
firm
jF
''^More generally,
we could
consider combining the model of incremental investment with a patent race
model, as described above.
^^We
ignore comphcations arising because of stochastic
R&D here.
model where firms can choose between two different types
of iimovative investments. In this model, if the frrms make identical choices about investment, the outcomes
are perfectly correlated, while otherwise they are independent. In this setting, it turns out that leaders want
to imitate laggards, whereas laggards want to differentiate themselves from leaders.
^^In a related paper, Cabral (1999) analyzes a
29
Learning- by-doing
4.3
In learning-by-doing models (e.g. Cabral and Riordan (1994), Sutton (1998, ch. 14)), costs
are
monotone decreasing functions
force
of previous output levels. It
well understood that this
is
works towards increasing dominance, although countervailing
firms have exhausted
most of the opportunities
how
our approach to show
dominates.
for learning. In
effects arise
once leading
the following, we shall apply
the nature of product market competition influences which force
^^
Denote the output
reference level
c.
level as a\.
Y^
is
common
the cumulated cost reduction from some
Hence, we write
\r=l
where r
is
a strictly increasing and concave function. In this case,
deterministic,
we
let
tt*
if
investment returns are
(a^ Y*~^) represent the product market profit of a firm
cumulated cost reduction
is
given by Y^~^ and the output level by
a\.
i
when
its
Further, the state
variable develops according to
Hence,
t-i
satisfies
d^yl/dal
>
and d'^y\/dYl~^da\ <
Finally,
0.
we
set
A;*
=
0,
as
all costs
and
benefits
of increasing output are borne through the product market profit.
Consider the example of Cournot competition with inverse market
so that IT (Y, a)
= p( Ylk=i ^k] —c + Yi
model.
is
First,
it
ai.
We now
standard to assume that H^
^.
<
''^It is
an increase
in
output
is
more valuable
for
function p{-),
argue that (WID-0) holds for this
in
a Cournot model, in order to
guarantee existence of equilibrium (see Novshek 1985). Further, H^.
effects:
demand
a low cost firm.
y.
=
1
>
Finally, H^.
0,
due to scale
y
=
0.
Hence,
simple to reinterpret the follow'.ng as a model of network or standardization effects along the hnes
and Saloner (1986), Katz and Shapiro (1986), where firms that manage to acquire many customers
an early phase of the market benefit if customers' wiUingness to pay grows with the number of other
customers adhering to the same standard or using the same network. Here Y^ is a measure of perceived
quality rather than total cost reduction.
of Farrell
in
30
when
firms are myopic, leading firms will always have stronger incentives to increase output
and thus increase
That
their state variable.
is,
weak increasing dominance holds without
game has a unique
additional assumptions, provided the Cournot
equilibrium.
For forward-looking firms, however, the nature of product market competition plays a
decisive role in determining
whether increasing dominance
two-period framework.
The
long-run profit of firm
LR^ (Y°, a^)
=r
(a\ Y°)
where
7r'(Y^)
if
da]dY^^
da\dY^^
d'LR _
+ y^^A al),Y'_, + yi,(Y°„ a^J)
dH^
d'^LR
da}da]
\dY°
{av^^f
d^¥
^
_
~
d''¥
da}da]
d^l^'
"^
As
long as
tt'
satisfies
However, q^qyo V]
competing
increasing
effect
course,
^ dY^ ) "
dy] dy]
dY^^dY^ da] da}
tt^
^
'
'
"
have the correct sign under
to the remaining terms, there are two countervailing effects.
is
tt*
works against the product market
effects
arise
with forward-looking firms,
the
it is
first
expression. For
This
weak
therefore important that the
dominate.
most apphcations
For example,
effect in
As
are correct.
negative, reflecting the slow-down in learning for a better firm.
will
mental investment, and investments
models.
^^ j:
condition (WID-0), the signs of the second derivatives of
dominance to
product market
i
weak
dy} \
have already shown that the mixed derivatives of
very general conditions.
all
5,
dajdYP dy}
J da}
dy} (
dY^dY^' da} \
,
Applying Theorem
the following conditions hold for
da\dY^
da]dY^
Of
n^ (y,°
can be written as
the Cournot profit for cost structure (Y^).
is
increasing dominance will obtain
We
+
i
For simphcity, consider a
arises.
it
combine some elements of learning-by-doing,
in
more
incre-
radical innovations similar to the patent-ra€e
has been widely asserted that in the market for
memory
chips,
learning-by-doing effects play an important role. However, increasing dominance only arises
in
some parts
of this market. For example, the market for
DRAM chips does not exhibit in-
creasing dominance (Gruber, 1994). Within each product generation (4K, 16K, 64K, 256K,
31
1MB
chips), concentration levels did not increase (at least, not substantially);
The
each product generation, leadership patterns often changed.
prising, given that across generations, learning-by
Dynamic competition
across generations
latter finding
doing effects are
more akin
is
much
is
not sur-
less significant.
to patent competition,
potentially have higher investment incentives. In contrast, the
and across
where laggards
EPROM market shows stable
leadership patterns, even across product generations; Gruber (1994) argues that learning-by
doing
4.4
effects
tend not to be exhausted within one product generation in this market.
Strategic Trade Policy
Our model can
also
be apphed to analyze strategic trade
the consequences of
setting with one
increasing
its
R&D
home
own
typically strategic substitutes
R&D
if
home
generally,
and Brander,
country, since
Staiger, 1994).
home
firm's
R&D
1983).
R&D,
same time reduce the
In a
thereby
foreign firm's
investments are
Such an argument can be used
presumably desirable only
market share. However, following the
increasing dominance plays a role, temporary subsidies
goal unless the
More
(BagweU and
home
subsidies, subsidies that are
lead to lasting increases in the
paper,
of questions concerns
firm and one foreign firm, a government that subsidizes
firm's innovation incentives, will at the
temporary
One set
subsidies by governments (Spencer
innovation incentives, to the advantage of the
to support
policy.
may
if
they
logic of this
not serve the long-term
firm actually surpasses the foreign firm.
we can use the
results in this
paper to formally analyze the dynamic
evolution of other trade policies, such as export subsidies, even in the absence of innovation.
Rather than apply our framework to product market competition between the
instead analyze the
game between
firms,
we
the governments of two countries, where the governments
choose to subsidize their export industries (and these choices might evolve over time). For
simplicity
we consider the simple case
(first
analyzed by Brander and Spencer (1985)) where
the firms, after receiving their subsidies, play a Cournot game;
incorporate learning-by-doing.
Our goal
is
to use
Theorem
more
generally,
we might
5 to analyze conditions under
which a leading country sees a greater return to using subsidies than a lagging country. In
that case, lagging countries
may be
better off
and demand
export subsidies are prohibited.
i
=
is linear:
q
Formally, suppose two firms from coimtries
are Cournot competitors,
if
32
1,2 serve a third country's market. Firms
= a — p. Each
firm chooses output levels
Qi
SO as to maLximize firm profits plus
country
i.
Ziq^,
where
Zi is tlie
per-unit export subsidy chosen by
Countries choose subsidies so as to maximize their firm's profits in the ensuing
oligopoly game. Simple derivations (see, for example, Fershtman
show
that,
when each
,
{z^,Zj)
11
{a
=
firm
z's
constant marginal cost
+ q/ (1 + z,) + cj/ (1 + Zj) -
- 2q/
—
(1
+
z,)
is
,
ni^•^^
.
^^
=
(1+2,)'
(1
o
(1
do not
differ
is
negative
too
if
5 leads us to conclude that
and only
when
home
Cj
+ Z,r
much from each
>
p
(1
+ 2,)
if
{a
—
0.
firm's cost:
12ci
(1
+ Z,Y
sign of the last expression depends on parameter values.
costs
the
+ 2,)'(l + ^i)
not be monotonic in the
8c,
(1
=
(l
may
,
^
+ Z,f
and m.
,
'"''
<
+ 2,)'
a
"^''^'
subsidies, the sign
when
higher
is
higher:
However, the incentive to subsidize
The
930))
+ c,-/ (1 + zj))
Hence, subsidies are strategic substitutes, and the incentive to subsidize
opponent's cost
p.
denoted q,
is
3q) {a
and Judd (1987,
VlCi) (1
(1
+ ZiY
For approximately identical
+ 2) + Cj + 8cj <
0,
which holds
other and from a. Applying the logic of
if
Theorem
countries are not very different, the leading country has
a larger incentive to subsidize exports, undermining the standard infant industry protection
argument.
One would therefore expect a leading country to expand its lead in the early stages
of such a strategic subsidy game.
When
firms are very different, however.
Theorem 5 cannot
be applied; either leading or lagging countries might have higher investment incentives,
depending on the parameters. For example,
have
5
little
if
the cost of the leading firm
is
very low,
it
will
incentive to subsidize.
Conclusions
This paper analyzes ohgopolistic firms that can engage in demand-enhancing or cost-reducing
activities,
providing conditions under which a firm
likely to increase
an
initial
advantage
might lead to such a development are scale
effects
from product market competition. Whether there axe strong countervaihng
effects
over competitors.
resulting
The main
is
forces that
depends on the nature of the investment process.
33
When
"investment"
is
just a by-product of
large output, such as in learning-by-doing or network models, the product-market effects are
When
reinforced by the investment process.
investment, countervailing forces arise
a systematic
product market
effect
lagging firms find
if
When
leading firms to improve their states.
is
firms can improve their states via incremental
investment
somewhat
Finally,
effect.
competing
effects
will often
dominate the
about games with strategic substitutes represent a contribution
could be applied to a variety of other games with strategic substitutes.
we have argued
that most of the
commonly studied
in /-player
opponent choices. This
oligopoly models have forces
to a two-player game.
suffices.
in
In this paper,
sufficient to generate
is
statics results.
firms
is
imposed
in order to effectively
we have shown
that a
Namely, we require exchangeability of the firm
assumption
commit
in
For example,
it
sum
of
ohgopoly games that one firm cares only about the
restriction
result
In the present
functional form assumptions, can be used to extend the existing literature.
commonly assumed
The
Thus, our approach, which does not rely on specific
that favor strategic substitutability.
is
far-
'
to the literature that uses lattice-theoretic tools to analyze strategic behavior.
context,
than
can arise when firms are sufficiently
^
generally, our result
less costly
of the patent race type, there
working against increasing dominance, which
sighted.
More
is
it
many
However, our approach
reduce an /-player
much more
general assumption
profit functions.
We
interesting empirical predictions
still
game
show that
this
and comparative
has limitations in dynamic games, unless the
advance to investment plans and do not adjust them in response to changes
opponent investments.
We
expect that the techniques developed in this paper can be fruitfully applied in a
variety of other problems in industrial organization. Section 4.4
is
suggestive of applications
to delegation games. In an example of an application outside of the oligopoly context, the
players could be workers in a firm engaged in repeated tournaments for promotions, where
human
capital investments are possible in each period.
applied to understand under which circumstances
ahead of others
in early periods of the
In conclusion,
we observe that our
public policy questions.
game
(e.g.
The methods
promotion
of this paper could be
policies)
workers
who
get
are likely to increase their lead.
results provide a lens
through which to analyze several
In particular, several anti-trust pohcies that might serve to limit
increasing concentration, such as prohibitions against predatory pricing
34
and mergers, can
have ambiguous welfare
effects.
Even
if it
can be established that a particular policy works
dominance, often the investment required for a firm to achieve
for or against increasing
market dominance are beneficial to consumers.
Consider
Even
the case of mergers.
first
an industry
if
is
characterized by increasing
dominance, some symmetry-increasing mergers (such as mergers among lagging firms)
slow the evolution of market dominance. However, even mergers
some welfare
benefits: for example, in
among
may
leading firms have
an incremental investment game, increases in market
share are the direct result of larger improvements in cost or quality.
Similar logic apphes to predatory pricing.
when
Broadly speaking, predatory pricing occurs
firms set low prices in order to induce market exit, reduce a competitor's output, or
prevent entry.
The
rule of Areeda- Turner (1975) considers
predatory. In contrast,
Baumol
competitor
after the exit of a
is
(1979) proposes a rule whereby a firm that increases
may be
satisfied,
game, a firm drives out
its
Our approach
guilty of predatory pricing.
a view suggested elsewhere (Bagwell
condition
any prices below marginal costs as
1997, Cabral
et al.
even when welfare
price
consistent with
and Hiordan 1994): the Baumol
not harmed. In an incremental investment
is
competitor by sufficiently lowering
firm raises prices after a competitor exits, the price might end
were
is
its
its cost.
Even
if
the leading
up lower than when the firms
fairly similar in size.
Appendix
6
Proof of Lemma
in Section 3.3.
In
the argument relies on the decomposition of profits outlined
9:
For parts
all
the cases under consideration,
(a)-(c)
,
demand and markups
are linear functions of
the state variables. Hence the desired properties follow,
(a)
Consider a duopoly with inverse
Pi
for
output levels
gi
and
q^
=
ai
-
demand
Pqi
- 792;
and parameters
/?
functions
=
P2
>7>
- 791 - Pq2;
"2
0; ai,
02
>
0.
For the case of price competition, straightforward calculations show that
_ (2/3^-7^)(a,-Q)-/?7(a,-c,) '^^ _
4/32-72
{213^
- j^)
{g,
- a) - P^j (aj -
(4/32-72)(/32-72)
Clearly, these expressions are linear in marginal costs, increasing in c,
that
Lemma
cj)
and decreasing
8 can be applied for cumulated cost reduction as the state variable.
35
in Cj, so
The same
argu-
ment works
parameters
for the quality
(WID-0) and (WID-0')
conditions
and the
Qj,
cti,
difference
For the case of quantity competition, the argument
_
- q) - 7 (ttj -72
2/3 (qj
'z
Cj)
iVi
€
_
_
4/?-'=
(b)
producing a good
20^
{aj-a) - Pj {aj -
Wi)
.
covered,
Then, assuming that
Di
=
-
—
{Vi
-^^
in equilibrium
-
Ci
demand
{Vi
—\ mi =
-
Ci
Suppose customers
vt.
for
customer
w
are
-
sis
+
Vj
Cj)
—.
6
Wi\
— q
ov vi
Ci^vt
with firms located at
line,
both firms have unit demand and the entire interval
Cj)
i
bt\Wj
Hence, no matter whether
—+
Vj
—
-;
Cj)
Suppose transportation costs
markup and demand can be written
turns out that the
it
Hence,
_y
4^2
which consumers have^^illingness to pay
for
cost.
the same, using
'
are distributed uniformly on the unit interval.
is
is
Consider the standard Hotelling model of competition on the
[0,1]
t{w —
between quality and
(the latter condition defined in Section 4.1.1) hold.
is
used as a state variable, the equilibrium markup and
are both linear functions of these state variables.
when
Similar reasoning shows that
demand and mark-up
firms are located on a circle,
are linear
functions of marginal costs (see Eswaran and Gallini 1996 for details).
(c),(d) In this model,^^ firms sell
a
valuation
where a
for Yi
for quality.
This taste parameter
>
2a. In this case,
c
— Q where
=
c
products of different qualities
is
it is
Vi.
Customers
distributed uniformly across the interval
is
straightforward to show that the conditions of
some
differ in their
reference cost level.
Now
consider Yi
=
Vi, in
Lemma
8
[o;,a],
still
hold
the boundary case
that costs are identical. Profits can be written as
^i
_
(I
[vj
- Vi)
\l{viThus
TTy. y.
=
wherever this
0,
is
is [2
a—
a\
dy
for firm
Y^ > Y^
can show that
that
(OSPM)
•^^To
be
holds,
11: Let a*
implies a^
=
>
is
-af
for Vi
i,
i
than
Yj) {2
;
Vj^
Vj J
show that
for Yi
>
Yj,
any incremental
This follows because the value of
for firm j it is
a - af -
Ej
<
>
for firm j.
whereas
{a\,...,aj)
=
{Y^
- Yj){a-2af
(a^,...,ajj
a^, the result follows.
and the game has
precise, the following
for Vi
defined. It suffices to
max {a-2afdy,{dy-Yi +
Proof of Proposition
- 2af
Vj) [2a
investment dy increases profits more for firm
such an investment
[a
and a
=
{ai,aj). Clearly,
To apply Theorem
strategic substitutes
5, it suffices to
and increasing differences
if
we
show
in (aj^YP)
a slightly modified version of the original Shaked and Sutton model (Tirole
1989, eh. 7.5.1).
36
and
i
in (a^, —Yj^)
To
.
see that these conditions hold, note that the long-run profit function of firm
can be written as
+(5*-itf (a^
(OSPM)
requires:
d LB}
+ a'-^) +
...
i
strategic substitutes,
J
da\da'
1-
...
40
i
To guarantee
Y° + a^ +
ct-lrrt
we
require:
=
6'-'K^^,^+6'nv„Y,
=
<5'"^n^.,y.
,
sT-l-ni
+ - + s'~"^Y„Y,<o^oTte{i,...,T}.
Rt-ni
,
,
J
d'^LR}
The
= 6^-^Ul
Y-
+ S'U'y.^Y. +
+(5*nt.
+
y.
...
...
+ 6^ "^nV,_y. <
fori
6
<
fori
G {1,...,T} and s<t.
+ (5^~^n'y
y,
{1, ...,r}
and
s
>
t.
increasing differences conditions require the following:
fl2 T Tji
-
S'-'K„y,
+ S^'^kv, +
...
+ <5^-^nV.,y, >
^tt^ =
<5'-^n|,.y.
+
+
...
+ <5^-inv.y. <oforie{i,...,r}
-Q^gyo
for i
G
{1, ...,T}
a2r pi
<5*n^.y
All of these conditions are clearly implied
Proof of Proposition
12: Let F''*(Y') be the value of the firm in period
be the equilibrium policy vector in period
Step
1:
Show
by (WID) and (WID-Dl).
t,
and
let
a*(Y')
t.
that condition (1) in the Proposition impUes (2),
i.e.,
the assumptions of
Lemma
8
imply linear (and thus differentiable) policy functions in each period, (WID- A) and (WID-D2). Let
where
a.^ is the
by IT (a, Y)
Nash equilibrium
+ 6V"'{a + Y). Then
game where player i's payoffs
!P(V) maps quadratic functions into
of the auxiliary static
if 11* is
functions, as the equilibrium strategies
quadratic,
a^
will
be
linear.
Since ^'^"^(Y-^"^)
is
are given
quadratic
quadratic, the value
function in each period will therefore be quadratic, and the policy functions linear and conti .luously
differentiable.
'"'Here
affect
and
(WID-A) and (WID-D2)
in the following,
ouj results,
mixed
therefore hold.
partials differ according to
we drop the arguments.
37
where they are evaluated; as
this does not
Step
2:
Show
that condition (2) of the Proposition implies that the value function
differentiable for each
is
Since each firm has a unique best response, actions are chosen from a compact
is
continuously differentiable in Y^~^
theorem implies that
if
y
yi,t-i^Y'-^)
is
continuously
t.
differentiable in Y*~-^,
^yi,t-i(Y«-i)
''
is
,
and
11'
set, a*(Y'~-^)
twice continuously differentiable, the envelope
is
continuously differentiable, then
= maxff(a„a*(Y*-i),Y'-^)
+<5F^'*((a„a*(Y*-i))+Y^-i)
and
=
nV.(a*(Y*-^),Y'-^)+5T4f(a*(Y*-^)+Y*-i)
dYl
The
derivative ^yt-i. y^'^~^ 0^'^~^)
sumed
is
obtained analogously.
Since the equilibrium policies are as-
to be continuously differentiable, the derivatives of the value function are continuous as
well.
Thus, since y'^~i(Y'^~^)
period
Step
3:
is
is
continuously differentiable, by induction the value function in each
continuously differentiable.
Establish conditions required for weak increasing dominance in the final period, T.
Now, assume
for simplicity that the value function
not true,
we can redo the
and policy function
following analysis with differences rather than
differentiable
(if
derivatives).
Consider the properties required for increasing dominance.
this
is
in each period are twice
As
(WID-Dl), and (WID-D2) by assumption, analogous conditions hold replacing
11' satisfies
(WID),
with H'
+ 6V^''^,
11'
if:
^y;!y,
Consider
first
Differentiating
whether y'-^-i
<
0> V^-,Y,
satisfies
0>
and vp^^
>
(WID-V)
0.
(WID-V). Observe that F''^-i(Y^-i)
and using the envelope theorem
can be written as
>
as described above, for
i
^ j,
= n'(a^(Y^-^), Y^"^).
r_f
T_i n'(a-^(Y-^~-^), Y-^~^]
follows:^-^
j
I
j
'>
3
+ni,..,^«T(Y'-->)^af (Y-) + ni,,,„,-|3raJ(Y'--)^aJ(Y'--')
^^Here and in the following, we drop the arguments {a^ {Y'^) ^Y"^) to simplify the exposition.
38
>
Note that --fnaj(y^-i)
^pf^af (y^-^) <
by
guarantee that ^^rJlyT-i ^K^'^C^'^''^),^'^'^)
sufficient to
WID-A
D2, and
and
Lemma
<
for
i
Thus, the following are
3.
^
j
:
WID, WID-Dl, WID-
(observing that by exchangeability, the signs of the derivatives of aj can be
inferred)
^^^;|^^n'(a^(Y^-i), Y^-i)
Similarly,
This
guarantee that
Step
same
positive under the
is
f _i^ n^(a-^(Y-^~-^), Y-^~-^)
The assumptions
holds for
t
=
set assumptions; finally, these
show
to
that
WID- V
of the Proposition guarantee that
WID-V
T. Suppose that
of n'(a*(Y^-i), Y'-^)
Step
if
5:
WID-V
holds in period
Athey
will
Then, following similar arguments to
t,
the derivatives
hold for
i
-
1.
By
induction,
WID-V
WID-V
holds for
3.
all t.
imply that in each period
t,
this result,
we use
the following
lemma
(see, for
example,
(1999)):
Lemma
14 Let /
X
:
y —
x
M, and
>
let
G^{»;z) and G^{»;w) be probability distributions.
If f{x,y) satisfies increasing differences in (a;,y),
in z
WID-V
weak increasing dominance.
To prove
13:
and that
all t,
can be analyzed following the approach of Step
the Proposition together with
applies to guarantee
Proof of Proposition
i, it
t.
holds for
continuously diff'erentiable for each
+ (5y^''(a*(Y'-i)+Y*-^)
The assumptions of
Theorem 5
is
sufficient to
t.
WID-A
holds for arbitrary
assumptions are also
can be verified in a similar way.
positive, as
holds for each
above, and since Step 2 established that V^'^
Thus,
is
'
Use induction
4:
given by:
is
and w,
then
respectively,
J f{x,y)dG^{x;z)dGy{y;w)
^
Under (WID-0),
y*) Y[j
dH^{yj
\
J f{x,y)dG^{x;z)
<
0.
>
Since
14 implies that {En^)
.
and
is
(WID-0)
<
0.
<
^0^
0.
^ <
^
i,
and
i 7^ j,
(E-k')
7
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