The Comparative Statics of Collusion Models∗
Kai-Uwe Kühn†
University of Michigan and CEPR
Michael S. Rimler
Xavier University
This Version: August 2007
Abstract
We develop and illustrate a methodology for obtaining robust comparative statics results for collusion models in markets with differentiated goods by analyzing the homogeneous goods limit of these
models. This analysis reveals that the impact of parameter changes on the incentives to deviate from
collusion and the punishment profits are often of different order of magnitude yielding comparative
statics results that are robust to the functional form of the demand system. We demonstrate with
numerical calculations that these limiting results predict the global comparative statics at any degree
of product differentiation. We use this methodology to demonstrate the non-robustness of comparative statics obtained from Nash reversion equilibria and to develop new results in the comparative
statics of collusion.
∗
Acknowledgements: We are very grateful to Joe Harrington, Daisuke Nakajima, Volker Nocke, Yusufcan Masatlioglu, Pierre Regibeau, Tom Ross, Steve Salant, and Ennio Stacchetti as well as seminar participants at the University of
British Columbia and the 2006 Duke-Northwestern-Texas IO theory conference at Northwestern University for extremely
helpful comments and discussions. All errors are, of course, our own.
†
Corresponding author: Kai-Uwe Kühn, Department of Economics, University of Michigan, 258 Lorch Hall, 611 Tappan
St., Ann Arbor, MI 48109. Tel.: (734) 763 5317. Fax.: (734) 764 2769. e-mail: [email protected]
1
1
Introduction
Competition and merger policy often draw heavily on the conclusions of collusion theory to determine
what environments increase the likelihood and severity of collusion. Such assessments have become
even more important with the increased attention given recently to the assessment of coordinated
effects of mergers both in Europe and the US.1 For policy purposes, there is a need to determine specific
characteristics of a market that make it particularly at risk from collusive behavior. For this reason, the
literature on collusion has been driven by a strong policy interest in answering questions like: Should
collusion be seen as more important a threat in homogeneous goods markets than in differentiated
goods markets? Is cross-ownership between competitors a reason to be more worried about collusion?
What other factors should be considered as important to assess whether collusion may be an important
phenomenon in a specific market?
Unfortunately, it is often extremely difficult to characterize the whole set of collusive equilibria. For
this reason researchers have often relied on particular functional forms for demand and cost functions
as well as specific restrictions on the equilibrium strategies considered. This literature has often found
that results critically depend on such assumptions. This raises the new issue from a policy perspective
whether any of the results are robust enough to guide the formulation of policy rules.
In this paper, we suggest a method that allows us to obtain comparative statics results for a very
general class of models. This method allows us to make claims about how robust these results are both
to assumptions on functional form and on the equilibrium strategies allowed. The idea is to analyze
the comparative statics of collusion in a model with product differentiation close to the limit of perfect
homogeneity. We show that often the effects of a parameter change on the incentives to deviate from
collusion and on the severity of punishment are of different order of magnitude. This allows us to
conclude which effect should generally be expected to dominate. We confirm with numerical methods
that, for the usual product differentiation models used, the limiting analysis predicts the comparative
statics for the whole range of our product differentiation parameter. There is therefore a clear sense
in which models with unambiguous comparative statics in the limit yield more robust insights than
those that have limit comparative statics that depend on functional form. We also show that generally
symmetric optimal punishment equilibria are more likely to generate robust results than Nash reversion
equilbiria.
This paper is not the first to provide robust comparative statics results without restrictions on the
equilbrium strategies used. For example, Abreu, Pearce, and Stacchetti (1990) provide such results with
respect to the discount factor and Kandori (1992) with respect to uncertainty in imperfect monitoring
models for very general settings. However, these results refer to changes in the whole equilibrium value
set and show that equilibrium value sets are nested as the parameter is changed. Such results can make
no predictions about the comparative statics of collusive profits and prices in response to parameter
changes. In order to generate these results, which are of particular policy interest, one has to augment
the collusive model by specifying a method to select an equilibrium from the equilibrium set. In the
spirit of Harrington (1991), we make the assumption that the equilibrium is selected by some bargaining
procedure. Under some weak general assumptions on the bargaining procedure and profit functions we
show that the qualitative properties of comparative statics on values and prices obtained from the
1
See Kühn (2007) for references to the recent policy debate on coordinated effects in mergers in Europe and the US.
2
study of symmetric optimal punishment equilibria carry over to the comparative statics for equilibria
selected through bargaining over the unrestricted equilibrium set when the model is symmetric. This
result justifies the use of symmetric optimal punishment equilibria in symmetric models of collusion. In
contrast, the use of Nash reversion equilibria cannot generally be justified because they can generate
results at odds with the comparative statics of the unrestricted equilibrium set. We then show how the
techniques developed in that analysis can be applied to more general comparative statics exercises.
In Section 2 we develop a general model of collusion in markets with product differentiation that has
a well-defined homogeneous goods limit. We specify the properties of the bargaining rule and discuss
its role in the analysis in more detail.
In Section 3, we demonstrate the basic analysis for the comparative statics of collusive equilibria
at the homogeneous goods limit for the case of changes in a product differentiation parameter. We
show why the comparative statics for symmetric optimal punishment equilibria is, in contrast to Nash
reversion equilibria, robust to assumptions about the functional form of demand. We then show that
the qualitative comparative statics result for the symmetric optimal punishment equilbiria must also
hold for the unrestricted equilibrium value set given our assumptions on bargaining.
The analysis of the impact of product differentiation on collusion is of some interest in the industrial
organization literature in its own right. Part of the general folklore of competition policy holds that
collusion is considered easier in homogeneous goods markets than in markets with differentiated goods.
The degree of product “heterogeneity” appears, for example, on the European Commission checklist
used to determine whether coordinated effects of mergers are likely to arise (see Kühn 2001). But it is
not obvious that this claim of practitioners is supported by collusion theory. A change in the degree
of product differentiation will effect, on one hand, the incentive to deviate from collusion and, on the
other hand the severity of the punishments. First, an increase in the degree of product differentiation
must reduce the incentives to deviate from a collusive price because any given price cut wins over less
market share. This is an effect that works in the opposite direction of the policy view. However, there
should also be some effect on the punishment side. As goods are more differentiated, firms have more
market power and therefore will be more reluctant to participate in severe punishments. This suggests
a general trade-off between the impact of product differentiation on the incentives to deviate from
collusive agreements and its impact on the intensity of punishments that can be imposed.
In the theoretical literature, results about the impact of product differentiation on collusion are
ambiguous. Deneckere (1983) shows in the context of a duopoly model with linear demand that there
may be theoretical support for the prevailing policy view: He demonstrates that, starting from perfect
homogeneity, the minimal discount factor that can support collusion at monopoly prices is initially
increasing for the best Nash reversion equilibrium. However, Ross (1992) shows that this conclusion
is not robust to the demand system assumed and Lambertini and Sasaki (1999) demonstrate that the
critical discount factor for sustaining the monopoly price is monotonically decreasing in the degree of
product differentiation in the best symmetric optimal punishment equilibrium of Deneckere’s model.
Our analysis explains why the differences between these studies arise and show that robust results on
this issue are available under optimal punishment schemes.
In Section 4, we give three examples of how the framework developed in section 3 can be extended
to obtain general robust comparative statics results in other parameters. This analysis generates new
results in the comparative statics of collusion. The first example analyzes the consolidation of differ3
entiated products in the hands of less firms in a symmetric version of the product line model of Kühn
(2004). Standard reasoning would suggest that there may be countervailing effects from such consolidation: while firms have less incentive to deviate from a collusive price when they earn a higher proportion
of varieties in the market, they also have less incentives to use severe punishments. The analysis shows
that the former effect is of larger order of magnitude. More fragmentation hinders collusive activity.2
The second example concerns another issue for which the literature has traditionally pointed to countervailing effects (see Malueg 1992): the existence of cross-ownership shares in competing firms. We show
that cross-ownership shares robustly facilitate collusion since, for optimal punishment equilibria, there
exists no trade-off for low degrees of product differentiation. The incentives to deviate from collusion
are reduced and the punishment profits are increased at the same time. Furthermore, numerical analysis shows that the impact of small cross-shareholdings on the collusive price can be substantial. This
suggests that cross-shareholdings should be an important factor in assessing the coordinated effects of
mergers. From a policy point of view, it is also important to note that the use of equilibria generated
from Nash reversion would have severly underestimated the collusive impact of such cross ownership.
The third comparative static we analyze concerns an increase in the marginal cost for all firms. Antitrust
investigations of gasoline retailing are often triggered when input price shocks lead to price increases
that are more than one to one. We show that collusion is indeed facilitated when marginal costs rise,
but the analysis does not support that evidence of prices rising more than one-to-one is evidence of
collusion. Furthermore, it can be concluded from the analysis that the impact of increases in marginal
cost on the ability to collude are of small magnitude.
These examples show that our method of analyzing incentive constraints at the homogeneous goods
limit is a powerful instrument to derive new results in the comparative statics of collusion and distinguish
between results that policy makers can consider robust and those that they should not. Furthermore,
they show that these methods can also yield insight about the relative order of magnitude of different
factors that might impact the likelihood to collude. Section 6 concludes the paper.
2
A General Model of Collusion in Differentiated Products Markets
2.1
The Model
We analyze a general oligopoly model with differentiated goods in which the degree of product differentiation is well defined and captured by a one dimensional parameter φ ∈ [0, 1]. At φ = 0 products are
homogeneous. Product differentiation increases in φ. The model is sufficiently general to encompass
all of the popular single parameter symmetric product differentiation models in the literature. These
include the linear demand system of Shubik (1980), CES preferences as in Spence (1976) or Dixit and
Stiglitz (1977), the symmetric nested logit model with an outside option (see Anderson, De Palma, and
Thisse 1992), and the Salop (1979) circular road model after appropriate normalization.3
The demand system is symmetric. Demand for product i is given by the demand function D(pi , p−i , φ, n),
2
This is not a trivial result. For example, in a homogeneous goods market with capacity constraints, Kühn (2005)
shows that fragmentation of a given amount of capacity among more firms facilitates collusion.
3
We show explicitly in Appendix B that CES preferences, the nested logit model, and a Hotelling duopoly model satisfy
all the assumptions.
4
where pi is own price, p−i is the vector of prices charged for other products, and n is the number of products in the market.4 Demand is strictly decreasing and strictly log-concave in pi , and
for which the demands for all products
are strictly
limpi →∞ D(pi , p−i , φ, n) = 0. For every
¯ price vector
¯ 2
¯ pP
¯
¯ ∂D(pi ,p−i ,φ,n) ¯
¯ ∂ ln D(pi ,p−i ,φ,n) ¯
∂D(pi ,p−i ,φ,n)
>
positive, demand is differentiable, ¯
>
0,
and
¯
¯
¯ >
2
j6=i
∂pi
∂pj
∂pi
2
P
∂ ln D(pi ,p−i ,φ,n)
> 0 for all φ ∈ (0, 1). The latter assumptions guarantee that goods are imperj6=i
∂pi ∂pj
fect substitutes, prices are strategic complements, and cross-price effects dominate own price effects
in demand. To obtain well defined comparative statics results for Nash equilibrium prices we also assume that the elasticity of demand at any Nash equilbirium decreases with product differentiation and
2
2
D(p,p,φ,n)
D(p,p,φ,n)
> 0 and ∂ ln ∂p
< 0.
increases in the number of firms: ∂ ln ∂p
i ∂φ
i ∂n
To eliminate market size effects, we adopt a normalization for the demand functions that guarantees
that the degree of product differentiation does not affect monopoly profits.5 At equal prices pj = p for all
j = 1, ..., n, D(pi , p−i , φ, n) = n1 D̄(p), where D̄(p) is independent of φ and n. D̄(p) can be interpreted as
the aggregate demand function and has elasticity ε(p).Given symmetry in demand and cost function, this
assumption guarantees that the competitive and the industry profit maximizing allocations do not vary
with the degree of product differentiation or the number of firms in the market. The monopoly price pm
is independent of φ and n. The product differentiation parameter only affects the intensity of preferences
of customers for their preferred brands. Consumers become less price sensitive if there is more product
differentiation, so that larger price cuts are needed to attract the same number of customers. This is
−i ,φ,n)
> 0 (< 0) if pi < minj6=i pj (pi > maxj6=i pj ). For given
captured by the assumption ∂D(pi ,p
∂φ
prices a firm charging the lowest price will lose demand to the firms with higher prices when product
differentiation increases.
To have a consistent model that has the homogeneous goods model as a proper limit for φ → 0,
we also need the demand functions for individual goods to converge pointwise to the residual demand
function of the homogeneous goods model. At prices strictly above any rival price, demand must
converge to zero. At a price strictly below the price of rivals, a firm must face the aggregate demand
function in the limit. If several firms charge the lowest price, the demand is shared equally between the
,p−i ,φ,n)
= −∞. Demand converges to zero if
firms. More formally, if pi ≥ minj6=i pj , then limφ→0 ∂D(pi ∂p
i
1
the inequality is strict and to m D̄(p) otherwise, where m is the number of firms charging minj6=i pj . If
,p−i ,φ,n)
pi
pi < pj for all j 6= i, then limφ→0 D(pi , p−i , φ, n) = D̄(pi ) and limφ→0 ∂D(pi ∂p
D(pi ,p−i ,φ,n) = ε(p) >
i
−∞. In addition, we make the following assumption on the limiting behavior of the demand function:
Assumption 1: Let p(φ) map the degree of product differentiation into a price vector.
−i (φ),φ,n)
< 0,
limφ→0 D(pi (φ), p−i (φ), φ, n) = D̄(limφ→0 pi (φ)), then limφ→0 ∂D(pi (φ),p
∂φ
If
Note that along the path of p(φ), a firm must be strictly undercutting all other firms in the neighborhood of φ = 0 in order to obtain all of the demand in the limit. Assumption 1 then says that any
firm that strictly undercuts all other firms in the homogeneous goods limit will strictly lose demand
4
The symmetry assumption does not exclude the possibility of different degrees of substitution between different goods.
For example, symmetric spatial models are covered by our analysis. In that case, the second element in the price vector
of the demand function refers to the left neighbor, the third to the right neighbor, etc. As long as these patterns are
symmetric as in the Salop circular road model such a model will fit our assumptions.
5
In the language of Shaked and Sutton (1990) we adopt a normalization that eliminates any market expansion effect of
product differentiation.
5
when some product differentiation is introduced even if it undercuts by an arbitrarily small amount.6
Every product can be produced at constant marginal cost c > 0. The payoffs in period t from
product i are given by π it (pit , p−it , φ) = (pit − c)D(pit , p−it , φ, n). Future profits are discounted at rate
δ ∈ (0, 1). There are n firms. A firm i produces only variety i. Firms play an infinite horizon game.
In each time period, a publicly observable signal σ with continuous distribution on [0, 1] is realized.7
Then all firms simultaneously set their price pi and receive the profits obtained in that period. Let P
be the (one-dimensional) set of non-negative prices. A history of the game at time t is then defined
as Ht ∈ P n×t−1 × [0, 1]t . A pure strategy pei for firm i, is a sequence of functions {pit }∞
t=0 , where
pit : P n×t−1 × [0, 1]t → P is a mapping from t-period histories to a price pi . Denote the set of all such
strategies by P.
p∗ , φ) denote an average
We analyze the set of subgame perfect equilibria of this game. Let vi∗ (e
e ∗ and v∗ (e
p∗ , φ) ∈ Rn a vector of such payoffs.8
equilibrium payoff for some equilibrium strategy profile p
∗
The set V (φ) contains all such vectors and is called the “equilibrium value set”. Associated with the
equilibrium value set is a set P∗ (φ) ⊂ Rn of all price vectors that can be sustained in some period of
some equilibrium of the game. Given our assumptions, a Nash equilibrium of the one shot game exists,
so that V∗ (φ) and P∗ (φ) are not empty:
Proposition 1 For every (φ, n), there exists a unique Nash equilibrium of the one shot game with
price vector pN (φ, n). It is symmetric. Nash equilibrium prices increase in φ, decrease in n, and
limφ→0 pN
i (φ, n) = c for all i.
Proof. See Appendix B.
The proposition follows directly from the assumptions by standard arguments. Besides establishing
the existence of a subgame perfect equilibrium in our game, it also establishes some properties of the
Nash equilibrium prices that are useful for the later analysis: In the one-shot Nash equilibrium, an
increase in product differentiation must increase prices, while an increase in the number of competitors
decreases prices. The Nash equilibrium prices converge to marginal cost as φ → 0. The last property
follows directly from the fact that the residual demand functions converge to the homogeneous goods
limit. Note, that proposition 1 does not include a convergence result for large numbers of firms. The
reason is that we allow for models of true monopolistic competition for which price remains above
marginal cost for n → ∞.9
6
This appears like a very natural property to require and it is satisfied for all examples in Appendix B.
The public signal is a purely technical device that allows firms to correlate their strategies and thereby serves to
convexify the equilibrium value set. This assumption is not necessary at all, when the most severe punishment can be
implemented through a low price in the first period of punishments and a return to thew Pareto frontier of the equilibrium
value set after wards. The assumption does nothing more than convexify the equilibrium value set, which permits the
application of the Abreu, Pearce, Stacchetti (1990) algorithm for computing equilibrium value sets in infinitely repeated
games.
8
“Average payoffs” here refer to the present value of profits in the equilibrium, multiplied by 1 − δ. This convention is
standard in the theory of repeated games and is convenient to keep mathematical expressions simple.
9
Of the one-parameter product differentiation models cited earlier, only the Salop circular road model has a competitive
limit. The CES, the nested logit, and the linear demand models are examples that converge to monopolistic competition.
There are many utility functions that do give rise to competitive limits. Kühn and Vives (1998) give examples in the
Spence-Dixit-Stiglitz framework. However, there is no widely used single parameter model in this class other than the
CES formulation.
7
6
2.2
The Comparative Statics of Collusion
Rigorous work on the comparative statics of collusion models has focused on the change in the whole
equilibrium value set in response to a parameter change. (The discount factor in Abreu, Pearce, Stacchetti 1990 and the degree of uncertainty in Kandori 1992.) Such studies can claim that collusion “gets
easier” with an increase in a parameter when the new equilibrium value set nests the old equilibrium set
after the parameter change. However, such comparative statics results on the equilibrium value set do
not allow for a prediction for the comparative statics of profits and prices. This problem for comparative
statics analysis arises because the collusive model is not closed. It is silent about the way that firms
coordinate on the collusive equilibrium. To close the collusion model we assume that firms select from
the set of incentive compatible strategy profiles by bargaining in the spirit of Harrington (1991). Given
a fixed bargaining rule, the comparative statics of equilibrium values in any parameter are well defined.
We represent a bargaining rule by a function B(V∗ (φ)) : V → Pn , where V is the set of compact,
convex subsets of Rn . The bargaining rule maps from equilibrium value sets to a unique equilibrium
strategy profile.10 We make the following assumption on B:
Assumption 2: For any set value set V , and any π ∈ V , B(V ) satisfies the following properties:
(a) It selects a strategy profile that generates a Pareto optimal outcome in V
(b) If V is symmetric, it selects a strategy profile with a symmetric outcome in V
We call the equilibrium selected by the bargaining rule the “best collusive equilibrium”. The Nash
bargaining solution over the equilibrium value set and all of the frequently considered bargaining solutions fall into this class.11 For any symmetric collusion model, all bargaining rules that satisfy assumption 2 select the same outcome because there is a unique symmetric outcome on the Pareto frontier of
the equilibrium value set.
To derive the comparative statics of prices, this selection procedure still poses potential problems.
The reason is that, in general, strategies supporting the best collusive equilibrium need not be stationary
on the equilibrium path. This means that the collusive prices charged can change over time in the best
collusive equilibrium. To derive the change of such non-stationary strategies due to changes in parameters of the model appears an impossible task either analytically or numerically. For this reason we
specialize our selection rule further, by requiring firms to only select the best outcome from the subset
Vs (φ) of those equilibria in V∗ (φ) in which the same price is set in every period on the equilibrium path.
Note that these are equilibria that may be supported by the most severe punishments from the full equilibrium set. Our comparative statics analysis will focus on B(Vs (φ)), which we call the “best stationary
collusive equilibrium”. Even in asymmetric models, this equilibrium can easily be computed using the
data generated from the Abreu-Pearce-Stacchetti algorithm for computing equilibrium value sets. In
symmetric models we also show that this equilibrium coincides with the best collusive equilibrium (i.e.
B(Vs (φ)) = B(V∗ (φ))) under fairly weak conditions on payoff functions. Furthermore, the numerical
10
We may allow this mapping to depend on the static Nash equilibrium profits π N (φ) to model a threat point. We
suppress this potential dependence to save on notation.
11
There can, of course, be several equilibria that support the outcome selected by the standard bargaining rules. We
do not elaborate on this issue since the strategy profile supporting the selected proves to be unique in a large class of
symmetric models.
7
results for the comparative statics of the valuations in the best stationary collusive equilibrium and the
best collusive equilibrium are virtually indistinguishable for the asymmetric games we analyze in this
paper.
Most of the literature has imposed much more severe restrictions on the set of equilibria considerd
than the stationary requirement for the best stationary collusive equilibrium. The reason is, that it is
typically very difficult, if not impossible, to derive comparative statics results analytically otherwise.
The restrictions on equilibrium strategies imposed generate restricted equilibrium value and price sets,
Vr (φ) and Pr (φ) that are strict subsets of Vs (φ) and Ps (φ). For example, the set of “symmetric optimal
punishment equilibria” is generated by the restriction that only symmetric strategies are used on and
off the equilibrium path.12 Abreu (1988) has shown that the best symmetric optimal punishment
equilibrium is typically less collusive than the best collusive equilibrium. This result arises because
symmetric optimal punishment equilibria exclude the possibility of asymmetric continuation equilibria
after deviations. Such asymmetric continuation equilibria typically allow for more severe punishments.
Much of the literature has been even more restrictive by focusing on symmetric subgame perfect
equilibria that are supported by threats to switch to one shot Nash behavior forever (“Nash reversion
equilibrium”). This imposes an even more severe restriction on the equilibrium value set and the
set of supportable prices. Since it limits the punishments relative to symmetric optimal punishment
equilibria, the best Nash reversion equilibrium will again be less collusive than the best symmetric
optimal punishment equilibrium.
Studying such restricted equilibrium value sets will therefore not allow us to identify the correct
comparative statics of the best collusive equilibrium. However, the modelling shortcuts could still give
valuable insights if the qualitative comparative statics of the best collusive equilibrium can be expected
to be the same as that for the best equilibrium in restricted set. The robustness analysis in this paper
will allow us to answer precisely under what circumstances we can expect our shortcuts to give valid
comparative statics results in this sense.
3
The Impact of Product Differentiation on Collusion
In this section, we introduce our method of obtaining robust comparative statics results in collusion
models using the example of the comparative statics in the degree of product differentiation. We study
both the robustness of results with respect to assumptions on the functional form of demand as well as
to the restrictions on the strategies permitted for different classes of equilibria.
We start the analysis with a global comparison of the best stationary collusive equilibrium, the
best symmetric optimal punishment equilibrium, and the best Nash reversion equilibrium. Since it is
analytically impossible to derive global comparative statics in the general model, we then study the
comparative statics in product differentiation at the homogeneous goods limit. As a benchmark case,
we show that the limit results for the comparative statics of the best symmetric optimal punishment
equilibrium are robust to functional form. This analysis immediately reveals why the comparative statics
12
For asymmetric games, researchers have sometimes applied quasi-symmetric equilibria for the same reason (see Compte,
Jenny and Rey 2002).
8
of the best Nash reversion equilibrium depends on functional form assumptions. We demonstrate this
sensitivity to functional form with three simple examples. In contrast, we show that the qualitative
comparative statics of best symmetric optimal punishment equilibrium coincide with those of the best
stationary collusive equilibrium and (under relatively weak conditions on profit functions) the best
stationary collusive equilibrium.
The section concludes with a robust global comparative statics result for product differentiation
that holds at the limit of a large number of firms. This result is derived for the best Nash reversion
equilibrium, but carries over to the more general analysis. We, therefore, generate further insight about
the circumstances under which modelling shortcuts will obtain valid qualitative comparative statics
results for the general model.
3.1
Some Preliminary Results
For symmetric models our bargaining rule implies that strategies are symmetric and stationary on
the equilibrium path for the best Nash reversion equilibrium, the best symmetric optimal punishment
equilibirum, and the best stationary collusive equilibrium. In each of these equilibria, firms charge the
highest sustainable price below the monopoly price pc ∈ (pN (φ), pm ] in every period. In order for pc to
be supported in an equilibrium, the net present value of obtaining a share n1 of the industry profits at
pc forever must exceed the benefits obtained from deviating from collusion in one period and reverting
to the worst feasible equilibrium in the set of permitted equilibrium strategies. Hence, in all cases the
incentive compatibility constraint on pc for given (φ, δ, n) can be written as
π(pc , pc , n) ≥ (1 − δ)π(p∗ (pc , φ), pc , φ, n) + δv(φ, δ, n),
(1)
where v(φ, δ, n) is the worst possible average continuation value in any equilibrium given the permitted
strategies and p∗ (pc , φ) is the short run best response price when all other firms set pc . The essential
difference for the different equilibrium concepts arise from differences in the continuation value of the
worst continuation equilibrium. In the best Nash reversion equilibrium this value is given by the static
Nash equilibrium profits, i.e. v(φ, δ, n) = π(pN (φ), pN (φ)). In the best symmetric optimal punishment
equilibrium and the best stationary collusive equilibrium, v(φ, δ, n) is determined by an incentive compatibility condition. Differences in the comparative statics of the best equilibria under the different
restrictions on equilibrium strategies can only arise because the comparative statics of v(φ, δ, n) in φ
differ.
We are primarily interested in deriving the comparative statics of profits and price. However, in
the literature authors have also frequently asked a question that is slightly easier to analyze: What is
the lowest discount factor δ(φ, n) at which pc = pm ? We demonstrate a close relationship between the
comparative static of the critical discount factor δ(φ, n) and the comparative static of the price in the
best equilibrium in this section. Using the superscript N for the best Nash reversion equilibrium, O for
the best optimal symmetric punishment equilibrium, and S for best stationary collusive equilibrium,
we can immediately state some simple results on the comparison of these equilibria:
Lemma 1 For any φ ∈ (0, 1), the set of symmetric equilibria generated from Nash punishments is a
strict subset of the set generated from optimal symmetric punishments. In the limit as φ → 0, the two
sets coincide. In particular, for all φ ∈ (0, 1)
9
(a) v N (φ, δ, n) > v O (φ, δ, n) ≥ v S (φ, δ, n) ≥ 0.
(b) pcN (φ, δ, n) ≤ pcO (φ, δ, n) ≤ pcS (φ, δ, n) ≤ pm
(c) 1 > δ N (φ, n) ≥ δ O (φ, n) ≥ δ S (φ, n) > 0
(d) limφ→0 vN (φ, δ, n) = limφ→0 v O (φ, δ, n) = limφ→0 v S (φ, δ, n) = 0,
limφ→0 pcN (φ, δ, n) = limφ→0 pcO (φ, δ, n) = limφ→0 pcS (φ, δ, n) = pc (0, δ, n),
and limφ→0 δ N (φ, n) = δ O (φ, n) = δ S (φ, n) =
n−1
n
Proof. See Appendix A.
The intuition for this result is simple. Punishments for a deviation are strictly harsher in optimal
symmetric punishment equilibria than in Nash reversion equilibria. The reason is that the initial
punishment price is allowed to fall below the Nash price. Punishments in the unrestricted equilibrium set
can be even harsher because the punishments of symmetric optimal punishment equilibria are available,
but even harsher asymmetric punishments can be used. Since the best stationary collusive equilibrium
is supported by punishments from the unrestricted set, this observation directly implies that prices are
highest in the best stationary collusive equilibrium and lowest in the best Nash reversion equilibrium.
The opposite ranking follows for the critical discount factors. However, note that continuation profits
are always bounded from below by zero since a firm can always guarantee such profits by setting price
equal to marginal cost. The lowest equilibrium vlaues and equilibrium prices must then coincide in the
homogeneous goods limit.
It is important to note that some of the qualitative features of the different equilibria also coincide.
These are summarized in proposition 2:
Proposition 2 (a) For any (φ; δ, n) with φ > 0, the highest sustainable collusive price strictly exceeds
the Nash equilibrium price, i.e. pc (φ; δ, n) > pN (φ, n).
(b) If δ ≥
n−1
n ,
then limφ→0 pc (φ; δ, n) = pm . If δ <
n−1
n ,
then limφ→0 pc (φ; δ, n) = c.
Proof. See Appendix A.
Proposition 2 has some strong implications, which are independent of the equilibrium set studied.
First, some collusion is always sustainable in models with product differentiation in contrast to the
homogeneous goods limit model in which collusion is either sustainable at any price or not at all. In
product differentiation models, a less favorable environment for collusion will be accommodated by a
lower maximal collusive price pc . Second, behavior of the firms sharply divides for discount factors
above and below the critical discount factor of the homogeneous goods limit model.
Proposition 2 also imposes restrictions on the qualitative comparative statics of collusion that are
independent of the equilibrium set studied. For δ < n−1
n , it implies that, close to the homogeneous goods
limit, the price in the best equilibrium must be increasing faster in φ than the Nash equilibrium price.
The only difference between the different equilibrium concepts considered is that, in the equilibrium
10
with higher prices for a given φ, the price rises faster above the Nash equilibrium price around the
homogeneous goods limit. For δ ≥ n−1
n , proposition 2 also gives us a clean classification of the potential
comparative statics for the most collusive price. If δ ≥ n−1
n and δ(φ, n) ≤ δ globally, the most collusive
price is always pm , independent of φ. On the other hand, if there exists some interval (φL , φH ) on
which δ(φ, n) > δ it must be the case that pc (φ; δ, n) is decreasing in φ on (that) interval since
limφ→0 pc (φ; δ, n) = pm and pc (φ; δ, n) < pm for φ ∈ (φL , φH ) by definition of δ(φ, n). Hence, if δ(φ, n)
increases on some range, pc (φ; δ, n) must decrease on some range as well. Note, that these results imply
that pc (φ; δ, n) can only be non-monotonic in φ if δ(φ; n) is as well.
To determine the comparative statics in φ, define ξ as the the degree of slackness in the incentive
compatibility constraint (1):
ξ(pc , δ, φ, n) ≡ π(pc , pc , n) − (1 − δ)π(p∗ (pc , φ), pc , φ, n) − δv(φ, δ, n)
(2)
Let δ(φ, n, pc ) be the lowest discount factor such that pc can be sustained so that δ(φ, n) = δ(φ, n, pm ).
At δ(φ, n, pc ), we have ξ = 0. The comparative statics of pc (φ, δ, n) and of δ(φ, n, pc ) are completely
determined by:
∂v(φ, δ, n)
dπ(p∗ (pc , φ), pc , φ)
−δ·
(3)
ξ φ (pc , δ, φ, n) = −(1 − δ)
dφ
∂φ
Expression (3) represents the marginal trade-off between the deviation and the punishment effect induced by a small change in φ. The first term is positive since the profit from an optimal deviation
decreases as products become more differentiated. The second term is generally negative because profits in the worst equilibrium will rise when product differentiation is increased. The reason is that it
becomes less feasible to punish harshly because each firm has more market power. Collusion is facilitated by a small increase in product differentiation at (pc , δ, φ, n) if and only if ξ φ (pc , δ, φ, n) > 0. In
this case, either the highest sustainable price increases when product differentiation is slightly increased
or the lowest feasible discount factor for sustaining a collusive price pc decreases:
Lemma 2 The highest sustainable collusive price pc (φ, δ, n) increases in φ at (pc , δ, φ, n) and the lowest feasible discount factor for collusion at pc , δ(φ, n, pc ) decreases in φ at (pc , δ, φ, n) if and only if
ξ φ (pc , δ, φ, n) > 0.
Proof. See Appendix A.
The result follows directly because in models with product differentiation an increase in the collusive
price must tighten the incentive constraint. Furthermore, it is always the case that increasing the
discount factor relaxes the incentive constraint.
3.2
Robust Comparative Statics
Generally it is not possible to push the analysis beyond propositions 2 and Lemma 2 because it is not
possible to sign ξ φ for the whole domain. Instead we analyze (3) at the limit of completely homogeneous
goods. This allows us to identify circumstances in which the impact of a small increase of product
differentiation in the neighborhood of φ = 0 is of a greater order of magnitude for the first term in
11
(3) then for the second term. When this is the case we call comparative statics “robust” because the
limiting comparative statics results are independent of the functional form of demand. Furthermore,
when the effects on the incentive to deviate and the punishment are of a different order of magnitude
close to homogeneity, it would be extremely difficult for the punishment effect ever to dominate. We
confirm this intuition through numerical analysis for the most popular product differentiation models
in the literature.13
The comparative statics of the best symmetric optimal punishment equilibrium are “robust” because
punishments are so severe in the short run that the impact of a change in product differentiation on the
lowest equilibrium value is of second order at φ = 0. To see this, note that in any optimal punishment
equilibrium the worst equilibrium value equals the profits from optimally deviating from the punishment,
i.e. v(φ, δ, n) = π(p∗ (pL , φ, n), pL , φ, n), where pL is the price charged in the first punishment period.
Optimal punishments allow punishment price pL to drop significantly below Nash equilibrium prices.
Any short run optimal deviation p∗ (pL , φ, n) from such punishments leads to an upward deviation to
a price below the Nash equilibrium price. Thus a deviation under optimal symmetric punishments
generates a smaller margin and a smaller output than Nash reversion punishments. As goods become
more and more homogeneous, both the price-cost margin earned from a deviation, p∗ − c, as well as
the overall demand when deviating, converge to 0. Even though an increase in product differentiation
increases price cost margins and increases demand at given prices, this effect means that the impact of
product differentiation on punishment profits are of second order in φ.
More formally, the impact of an increase in product differentiation on the punishment profits can
then be written as:
dv(φ, δ, n)
dφ
∂π(p∗ (pL , φ, n), pL , φ, n) ∂pL
+ π φ (p∗ (pL , φ, n), pL , φ, n)
∂pj
∂φ
µ
¶
−Dpj ∂pL
∗
= D(p (pL , φ, n), pL , φ, n)
Dpi
∂φ
∗
∗
+Dφ (p (pL , φ, n), pL , φ, n) [p (pL , φ, n) − c] ,
=
(4)
In the homoegenous goods limit p∗ (pL , φ, n) converges to c and pL converges to a price strictly below
c. This means that Dφ > 0 in the limit. Hence, the deviating firm gains demand from an increase in
product differentiation. But the price cost margin is zero, so there is no first order profit effect from
L
this gain. Similarly, ∂p
∂φ > 0 even in the homogeneous goods limit, because of the demand enhancing
effect of product differentiation. But since p∗ (pL , φ, n) > pL in the limit, total demand from such a
deviation converges
to zero so that this effect is also of second order. In contrast, Assumption 1 assures
dπ(p∗ (pc ,φ),pc ,φ)
< 0. An undercutting firm gains strictly less consumers from optimally
that limφ→0
dφ
undercutting when product differentiation is increased. It thus follows that the effect on deviation
profits is of a higher order of magnitude at the homogeneous goods limit than the effect on punishment
profits. Hence, we obtain the result:
Proposition 3 For each n, there exists φ̂ ∈ (0, 1) such that δ O (φ, n) is strictly decreasing for all
c
m for all φ ∈ (0, φ̂). If δ < n−1 , pc (φ, δ, n) is increasing on
φ ∈ (0, φ̂). If δ ≥ n−1
n then p (φ, δ, n) = p
n
e ≤ φ̂ such that pc (φ, δ, n) is strictly increasing on (0, φ).
e
(0, φ̂) and there exists φ
13
Note that “robustness” in this sense does not mean that it is never possible to find a functional form for which the
comparative statics differ nor that such cases are of measure zero.
12
Proof. See Appendix B.
Given the comparative statics of δ O (φ, n) the results on the comparative statics of pc follow directly
from proposition 2. In all of our numerical examples δ O (φ, n) is monotonically increasing globally so
that the qualitative comparative statics of proposition 3 hold for the whole range φ ∈ (0, 1).
It follows from this analysis that product differentiation robustly facilitates collusion relative to the
perfectly homogenous case. This conclusion is in contrast to the Nash reversion case. The reason for
this difference can be seen directly from equation (4). This equation applies to the case of the best
Nash reversion equilbrium with the interpretation that pL is set at pN and therefore p∗ (pL , φ, n) = pL .
Note that the term in the last line is always zero because Dφ = 0 at equal prices. As under optimal
N
1
punishments limφ→0 ∂p
∂φ > 0. But demand equals n D̄(pL ) at Nash prices. There is therefore a first
order effect on punishment profits because, in the limit as φ → 0, the price increase is applied to n1 th of
the total demand. The impact of product differentiation on the incentives to deviate from a collusive
price remain unchanged and, hence, of first order.
The countervailing effects on the deviation profits and the punishment profits are therefore of the
same order of magnitude for the best Nash punishment equilibrium. This implies that under Nash
reversion there is a general tradeoff between the degree to which an increase in product differentiation
decreases the demand obtained at an optimal deviation, and the degree to which an increase in product
differentiation increases the static Nash equilibrium price explaining why the result in Deneckere (1983)
violates our robust result for symmetric optimal punishment equilibria.
We now show that the direction in which this trade-off is resolved for the best Nash reversion equilibrium depends critically on the functional form and the parameters assumed. We focus on demand
systems that have not previously been analyzed in the literature: Spence-Dixit-Stiglitz CES preferences
and the symmetric nested logit model. Again we use limiting analysis to derive the qualitative comparative statics for each specific functional form. The proofs are provided in Appendix B. We have
confirmed numerically that the qualitative results extend to the whole range of product differentiation
parameters. We start with the example of CES preferences:
m
)
Example 1: With CES preferences δ(φ, n) is strictly increasing at φ = 0 if and only if D(p
D(c) ln(n) <
1. This inequality holds for all κ if n ≤ 15. For n ≥ 16, there exists κ(n) ∈ (0, ∞), which is strictly
increasing in n, such that for all κ ∈ (0, κ(n)], δ(φ, n) is strictly decreasing.
With CES preferences it is very difficult to obtain anything else but the result that product differentiation makes collusion more difficult. Indeed, Figure 1 below shows that for n = 2, collusion is always
more difficult under any finite degree of product differentiation than under homogeneous products. Over
a wide range of product differentiation parameters δ(φ, 2) remains increasing in φ. Later it declines and
converges to .5.
[Insert Figure 1]
By itself this result would be very misleading. For the nested logit demand it gets much easier to
obtain the result that collusion is facilitated by product differentiation:
13
m
)
Example 2:With nested logit preferences δ(φ, n) is increasing at φ = 0 if and only if D(p
D(c) ln(n) <
1. In particular, if n = 2 then this condition always holds. If n ≥ 3, then there exists A(n) > −∞,
such that δ(φ, n) is increasing at φ = 0 if and only if A ∈ (−∞, A(n)). A(n) is strictly decreasing in
n.
Note that the limiting condition for the qualitative comparative statics in the SDS and the nested
logit model are the same. However, in the nested logit model the demand reduction due to monopoly
pricing tends to be smaller, which makes it much less likely that collusion becomes easier for differentiated products than for undifferentiated products. However, these models and the linear model of
Deneckere (1983) coincide in the result that collusion is hindered by product differentiation in the two
firm case. Again this is not a robust result. For a simple Hotelling model we obtain:
Example 3:Consider a two firm Hotelling model with firms located at the endpoints of a line and
quadratic transportation costs. Then an increase of product differentiation measured by transport costs
around the homogeneous goods limit facilitates collusion.
We have thus shown that restricting analysis to the best Nash reversion equilibrium can generate
results that contradict those of the best optimal punishment equilbirium, but that these results are highly
non-robust in the sense that they depend on functional form and even specific parameter values.14
3.3
Comparative Statics of the Best Collusive Equilibrium
We have shown that the best symmetric optimal punishment equilbirium has robust comparative statics
in the degree of product differentiation. But does this comparative static result necessarily capture
the qualitative comparative statics in the best collusive equilibrium? Our analysis in the previous
subsection immediately gives the answer to this question. Since the punishment payoffs in the best
symmetric optimal punishment equilibrium are of second order they should be of second order for the
full equilibrium set as well. The reason is that any punishment value available from symmetric strategies
will be available in the full equilibrium set.
To see this for the best stationary collusive equilibrium, simply note that the worst continuation
equilibrium v S (φ) has a value not exceeding v O (φ). Since both converge to zero profits as φ → 0, it
follows that vS (φ) cannot grow faster in φ than vO (φ) close to φ = 0. Hence, punishment profits will
be at most of second order as in the case of symmetric optimal punishment equilibria:
Proposition 4 For each n, there exists φ̂ ∈ (0, 1) such that δ S (φ, n) is strictly decreasing for all
φ ∈ (0, φ̂).
Proof. See Appendix A.
14
We have not reported the numerical results for the comparative statics in prices because they do not add much insight.
For example, for SDS preferences, prices dramatically fall below the monopoly price for slightly differentiated products
and then increase as implied by proposition 2. The monopoly price will only be reached for very high degrees of product
differentiation.
14
Obviously, the argument on comparative statics of price is exactly the same as in the case of the
best symmetric optimal punishment equilibrium. It should only be added that pS (φ, n) increases faster
than pO (φ, n) close to φ = 0, since pS (φ, n) ≥ pO (φ, n).
We now show that under weak assumptions on the demand function this result carries over to the
Best Collusive Equilibrium. To do so, we make two assumptions:
Assumption 2: π(p, φ, n) is concave in p.
Assumption 3: π i (p∗i (p−i ), p−i , φ, n) is strictly convex in p−i .
Given these assumptions it is the straightforward to prove:
Proposition 5 Under assumptions 2 and 3, the Best Stationary Collusive Equilibrium is the Best
Collusive Equlibrium.
Proof. See Appendix A.
The proof proceeds by showing that under the assumptions above it is always easier to sustain a
symmetric value pair with a symmetric price charged forever than a non-stationary price path. The
concavity assumption ensures that for every asymmetric price vector charged in the first period of a
Best Collusive Equilibrium there exists a uniform price below the average price in that price vector, that
would yield exactly the same profits if repeated forever. It is then shown that a stationary price can
be sustained as a subgame perfect equilibrium given the value of the lowest equilibrium, if the greatest
one period profits from deviating from this price are no larger than the average profits from an optimal
one period deviation across all firms in the hypothesized Best Collusive Equilibrium. The convexity
condition then guarantees that this inequality is strict for the average price, showing that the original
equilibrium cannot be the Best Collusive Equilibrium.
It should be noted that the qualitative equivalence of the comparative statics for Best Collusive
Equilibrium and the Best Symmetric Optimal Punishment Equilibrium arise because punishment profits
are of the same order close to the homogeneous goods limit. This comes from the property that maximal
punishments for both concepts rely on maximal frontloading of the punishments. This is the feature
that drives the robustness of the results. In the rest of the paper we discuss how these insights can
be carried over to more general comparative statics problems in the context of differentiated products
models.
3.4
Are there Robust Comparative Statics for the Best Nash Reversion Equilibrium?
Are there ever circumstances under which the best Nash punishment equilbirium has robust comparative
statics in the sense that it both is robust to functional form and captures the qualitative comparative
statics of the best collusive equilibrium? We now show that one can use the best Nash punishment
equilibrium to generate global comparative statics results in φ when the number of competing firms in
the market is large enough. In the limit of an arbitrarily large number of firms the punishment profits
15
again become arbitrarily small relative to the deviation profits yielding unambiguous comparative statics
in φ. Note that this result must carry over to the comparative statics of the best collusive equilibrium
by the same arguments we used previously in this section.
For the large number case, we need to distinguish between models in which behavior converges to
perfect competition as numbers get large and those models that have a monopolistically competitive
limit. In the latter, Nash equilibrium prices will not converge to marginal cost as the number of
firms in the industry becomes arbitrarily large. The conditions under which robust results for the
monopolistically competitive case are obtained are slightly stricter. However, all of the standard product
differentiation models with a monopolistically competitive limit have this property:
¯
¯
¯
,p−i ,φ,n) ¯
Proposition 6 Assume that for every φ ∈ (0, 1) there exists Z(φ) < ∞ such that ¯ ∂D(pi ∂p
¯ < Z(φ)
i
∗
m
for all n, p. Furthermore, suppose the model has a competitive limit or that limn→∞ Dφ (p (p , φ), pm , φ) =
−∞. Then, for every φ ∈ (0, 1), there exists n̄ < ∞ such that for all n > n̄,
(a) δ N (φ, n) is strictly decreasing in φ and
(b) pcN (φ, δ, n) is either strictly increasing in φ or constant at pm .
Proof. See Appendix A.
The intuition for this result is fairly simple. First consider
the effect on the deviation profits from
∗ c
c
the collusive price when φ is slightly increased, i.e. dπ(p (pdφ,φ),p ,φ) in (3). For any reduction in price a
firm wins less consumers and the incentive to undercut is reduced. Even as n gets large there must be a
strict reduction in the demand the firm can gain when product differentiation is increased. This effect
is therefore of first order. However, the weight on the profits from deviation in total profits is given by
(1 − δ). As n goes up the critical discount factor converges to 1 at rate 1/n. Hence, the total effect of
an increase of product differentiation on the incentive to deviate from collusion is negative and of order
1/n.
Now consider the impact on the continuation profits after deviation, the second line in (3). If the
model has a competitive limit, price will converge to marginal cost, whatever the value of the product
differentiation parameter. Essentially, product space becomes so tightly packed that for any degree of
product differentiation there is still an arbitrarily close competitor available. At the same time, output
in the market goes to zero at rate 1/n. Hence, in the limit, as n becomes arbitrarily large, the impact
of an increase on product differentiation on punishments goes to zero at a rate strictly faster than 1/n.
Hence, for large numbers of firms the effect on the incentives to deviate from collusion dominates and
an increase in product differentiation unambiguously facilitates collusion.
In contrast, for monopolistically competitive models the effect on punishments is only of order 1/n.
However, in that case we assume that limn→∞ Dφ (p∗ (pm , φ), pm , φ) = −∞, a property of all models
with a monopolistically competitive limit that we know of. Since the marginal effect of φ on the Nash
equilibrium price remains finite, it again follows that collusion is facilitated by an increase in product
differentiation when the number of firms is large enough.
Proposition 6 generalizes an observation in Mollegard and Overgaard (2002) that, in the linear
model, the results of Deneckere (1983) are reversed for n > 5. Our proposition reveals the general
16
underlying reason why such a result would robustly come about for large enough n: for large n the
impact of φ on deviation profits and punishment profits is of a different order of magnitude. This result
thus confirms the insight of the limiting analysis for φ → 0 that we should consider an increase in
product differentiation to robustly facilitate collusion.15
4
General Implications for Comparative Statics in Symmetric Models
In the previous section, we introduced a general technique for deriving robust comparative statics for
collusion models that rely on the analysis of the model around the homogeneous goods limit or the
limit for an arbitrarily large number of firms. In this section we demonstrate that this technique can be
applied with minor modifications to the comparative statics in any parameter θ. Again we analyze the
sign of the marginal change of the slack in the incentive constraint, ξ, in the parameter θ as the degree of
product differentiation goes to zero, i.e. limφ→0 ξ θ . The qualitative comparative statics can be obtained
directly from the homogeneous goods limiting model when limφ→0 ξ θ 6= 0. When limφ→0 ξ θ = 0 the
homogeneous goods limit model has degenerate comparative statics in price and the critical discount
factor. In that case, an analysis of limφ→0 ξ θφ still provides the desired comparative statics.
One complication arises for the comparative statics of price. If δ exceeds the critical discount factor
we have shown that the price in the best collusive equilibrium is the monopoly price for all φ. This
will be independent of the value of other parameters. Interesting comparative statics can therefore only
be obtained for δ lower than that critical discount factor. However at the limit as φ → 0 price cost
margins under collusion, under optimal punishments and under Nash punishments all converge to zero.
As a consequence it is practically impossible to determine the relative order of magnitude of the effects
on deviation and punishments profits from a change in a parameter θ at the homogeneous goods limit.
To get insight into the comparative statics of price we fix a collusive price pc and consider the critical
discount factor δ(φ, θ, pc ) that can sustain this price. Then we look at the comparative statics at pc
when taking the limit in φ, adjusting the discount factor along δ(φ, θ, pc ). This leads to some limitation
for the analysis. While δ(φ, θ, pc ) < δ(φ, θ) for all φ ∈ (0, 1), we have limφ→0 δ(φ, θ, pc ) = δ(0, θ). This
means that we may not capture the comparative statics in price at the homogeneous goods limit when
δ is small. However, we believe that this analysis remains in the spirit of the analysis presented in
the previous section. Furthermore, our numerical analysis confirms in each case that the analytical
comparative statics results obtained in this way also hold for the small δ case.
n−1
n ,
We discuss three examples in this section. First, we give an example in which the effects on profits
from deviation and profits in the punishment equilibrium are of different order of magnitude in φ.
Second, we present an example in which robust comparative statics can be obtained from the analysis
of the best Nash punishment equilbirium. However, we also show that quantitatively the comparative
statics result for the best Nash reversion equilibrium greatly underestimate the potential effect of small
cross-share holdings on collusion. Third, we show that our method can generate new results even when
there is no impact of the parameter change in the homogeneous goods limit model. These examples
demonstrate how our technique of deriving the comparative statics at the homogeneous goods model
15
Note that many of the arguments that have been made in antitrust about product differentiation making collusion
harder are not invalidated by this argument. For example, it is often argued that product differentiation makes it harder
to monitor a cartel or increases the complexity of cartel agreements. For critique of such arguments see Kühn (2007).
17
can be used to derive a variety of new and robust results that would be difficult to obtain otherwise.
4.1
The Impact of Consolidating Varieties
In this example we consider a variant of our basic model in which each of k < n firms owns nk varieties.
This is a symmetric version of the model of Kühn (2004). We ask the question whether collusion is
facilitated or hindered if less firms own the same number of total varieties. As Kühn (2004, 2006) has
stressed, this is very different from decreasing the number of varieties. The latter would unambiguously
lead to facilitated collusion. But redistributing a given number of varieties (symmetrically) among less
firms has countervailing effects. On one hand, it becomes easier to collude because the short run benefits
of cheating on a collusive agreement are reduced. On the other hand, owning a greater proportion of
the varieties makes punishments less severe. A similar tradeoff exists when a fixed amount of capacity
is distributed among fewer firms. Kühn (2006) has shown in the latter model that fragmentation of
capacity facilitates collusion.
It is easy with our methodology to show that the result of Kühn (2006) for capacity fragmentation
does not hold for the fragmentation of varieties. The reason is that, as in the base model, the effects
on the profits in the punishment equilibrium are of smaller order of magnitude than the impact on the
incentives to deviate from the collusive price. Collusion becomes easier with fewer firms selling the same
varieties. To see this formally, note that for any k the incentive compatibility condition for deviation
from a stationary collusive price pc is given by:
n
n
n
π(pck , pck , n) ≥ (1 − δ) π k (p∗ (pck ), pck , φ, n) + δ v(φ, k)
(5)
k
k
k
where π k (p∗ (pc ), pc ) refers to the profit function for one variety in which the first nk prices are set at
p∗ (pc ) and v(φ, k) is the average value per variety in the worst equilibrium. Let k > k. Then an increase
of the number of firms from k to k has a marginal effect on the incentive condition for given pc and
given δ of:
h
i
ξ k = (1 − δ) π k (p∗ (pck ), pck , φ, n) − π k (p∗ (pck ), pck , φ, n) + δ[v(φ, k) − v(φ, k)]
(6)
ck
ck
ck
ck
It is immediately clear that π k (p∗ (p ), p , φ, n) − π k (p∗ (p ), p , φ, n) < 0 for all φ ∈ (0, 1). The best
response profit per variety of an optimally undercutting firm is larger when it is smaller in the market.
Suppose that δ ≥ k−1
k . Then full collusion can be sustained close to the homogeneous goods limit when
ck
ck
ck
the number of firms is small. Note that This means that π k (p∗ (p ), pck , φ, n)−π k (p∗ (p (φ)), p (φ), φ, n)
k̄] ck
ck
converges to a number no larger than [k−
n (p − c)D̄(p ) < 0 as φ → 0. By the arguments made in the
previous section limφ→0 v(φ, k) = 0 for all k, so that limφ→0 [v(φ, k) − v(φ, k)] = 0. Hence, the impact on
the incentives to deviate from the collusive price dominates and ξ k < 0. It follows that δ(k, φ) increases
> δ ≥ k−1
in k. For φ close to zero, the highest collusive price pck (φ) strictly falls in k if k̄−1
k or if
k̄
δ<
k−1
k
and δ = δ(k, φ, pck (φ)):
Proposition 7 For any k̄ > k there exists φ̂(pc ) > 0, such that for all φ ∈ (0, φ̂(pc ))
(a) δ(k̄, φ, pc ) > δ(k, φ, pc ).
18
(b) pck̄ (φ, δ) ≤ pck (φ, δ) = pm , where the inequality is strict if
k−1
k .
k−1
k
≤δ<
k̄−1
k̄
or if δ = δ(kφ, pck ) <
Proof. See Appendix A.
As in the basic product differentiation analysis, the comparative statics is robustly determined
because close to perfect homogeneity the impact of the number of firms on the incentives to deviate is
of higher order of magnitude than the impact on the punishment profits. The dominating effect is that
with less firms in the market the market share gain from optimally undercutting is smaller.
4.2
Symmetric Cross Ownership Shares
In this section we study the impact of increasing symmetric cross-shareholdings between all firms in
the market on the ability to collude. The current literature on the subject (see Malueg 1992) suggests
that there are countervailing effects from such shareholdings: The incentives to deviate from collusion
are reduced, but punishments are also weakened because the profits in a one shot Nash equilibrium
increase in cross-shareholdings. Malueg (1992) therefore finds that cross-ownership has ambiguous
effects on collusion. We show in this subsection that in way defined in our paper, the best Nash reversion
equilibrium generates the robust result that collusion is facilitated by increases in cross-ownership. This
implies that it is robust for the other equilibrium concepts considered in this paper as well. However,
even though the best Nash reversion equilibrium produces robust qualitative comparative statics, it
remains misleading on the quantitative size of these effects.
To remain in the framework of symmetric models, we analyze a special case of cross-ownership in
which ownership shares are symmetric and each firm i controls the production of only one good, good
i. We assume that each firm owns a share α in each rival j 6= i, with 0 ≤ α < n1 . The share of firm i
in its own product i is given by 1 − (n − 1)α. Let α̂ ≡ (n − 1)α. Then total profits of firm i, Πi (p, α̂),
can be written as the weighted average of the profits from variety i and the average profits from all
other varieties j 6= i, with (1 − α̂) and α̂ as the respective weights. Notice that, in the limit, as α → n1 ,
α̂ → n−1
n and the profits of all varieties have equal weights in the one period profit function of firm i.
We can write the incentive condition determining the lowest δ at which collusion at pc can be
sustained under the threat of Nash reversion as:
Π(pc , pc ) = (1 − δ)Πi (p∗ (pc , α̂, φ), pc , φ) + δΠ(pN (α̂, φ), pN (α̂, φ))
(7)
The comparative statics of the critical discount factor and the collusive price is determined by sign of:
ξ α̂ = (1 − δ)[π(p∗ (pc , α̂, φ), pc , φ) − π(pc , p∗ (pc , α̂, φ), pc , φ)]
dpN
,
−δπ pj (pN (α̂, φ), pN (α̂, φ))
dα̂
(8)
where we differentiated the right hand side of (7) with respect to α̂ using the envelope theorem. The
first term in (8) represents the fact that a symmetric increase in cross-share holdings will reduce the
weight of own product profits in the overall profit calculation of the firm and increase the weight of the
19
average profits of others. Hence, there is less of an incentive to deviate because less profits are won
by undercutting. On the other hand, the Nash equilibrium price is increased, reducing the intensity of
punishments that can be imposed for higher levels of cross share holdings. This effect makes collusion
more difficult to achieve. This apparent tradeoff has been at the heart of a small literature on the effect
of cross-share holdings on collusion (see Malueg 1992).
However, using limiting results we can obtain the relative order of magnitude of the two effects at
the homogeneous goods limit. Note that for any α̂ < n−1
n a firm gains from slightly undercutting at
the homogeneous goods limit and winning the whole market. This means that the term in brackets
in (8) converges to (pc − c)D̄(pc ) > 0. By exactly the same undercutting incentive it should also be
clear that the one shot Nash equilbirium at the homogeneous goods limit has pN (0, α̂) = c for all
α̂ < n−1
n . Nash punishment profits therefore converge to zero for every α̂. This means that the effect
of cross-shareholdings on punishment profits must be of second order in α̂ at the homogeneous goods
limit.16
This insight directly leads to the two main comparative statics results for collusion under Nash
reversion:
Proposition 8 There exists φ̂(pc ) > 0 such that for all φ ∈ (0, φ̂(pc )),
(a) δ N (α̂, φ, pc ) is strictly decreasing in α̂.
(b) pc (α̂, φ, δ) is strictly increasing at (φ, δ N (α̂, φ, pc )) if δ N (α̂, φ, pc ) < δ(α̂, φ) and constant at pm
otherwise.
Proof. See Appendix A.
Close to perfect homogeneity only the incentive to deviate is of first order. This incentive is zero
when all firms own n1 of the share in every other firm, i.e. when α̂ = n−1
n . As α̂ increases the gain
from deviation becomes smaller and smaller so that the critical discount factor for collusion at the
monopoly price increases and converges smoothly to 0 as α̂ → n−1
n . The comparative statics in price is
trivial when full collusion is possible in the homogeneous goods limit. Then it must remain possible for
higher α̂. More interesting is the case in which collusion cannot be sustained for perfect homogeneity
for some level of cross-shareholdings. As in the pure product differentiation model, the highest collusive
price then collapses to marginal cost at the homogeneous goods limit. However, slightly away from the
homogeneous good limit, the highest sustainable price strictly exceeds the Nash equilibrium price if
the discount factor is high enough. Then an increase in α̂ will strictly increase the highest sustainable
collusive price and will increase it all the way to the fully collusive price at some α̂ < n−1
n . This is the
α̂, where cross-ownership just compensates for the low discount factor.
Although the analysis of the best Nash reversion equilbrium generates a robust comparative statics
result in this case, it still misses a crucial feature of the structure of the collusion problem that leads to
a severe underestimation of the quantitative effects of cross-shareholdings on collusion. The intuition
for this problem can most easily be conveyed in the homogeneous goods limiting model. We show that
16
This is a strong version of the general insight that small share holdings only have small impact on static equilibrium
in the market. Any cross-shareholding below α̂ = n−1
is small in the homogeneous goods model.
n
20
in the limiting model an increase in cross-shareholdings not only reduces the gain of deviating from the
equilibrium, but also makes the punishments more severe. The main insight from looking at symmetric
optimal punishment equilibria is that punishments will not be limited to a zero continuation profit.
Even if the firm deviates from a low punishment price, it can never escape the losses from its ownership
in rivals that price below marginal cost at the beginning of punishments in the homogeneous goods limit
model. But then punishments can be increased, by increasing the stakes in rivals. To see this write the
incentive constraint for an optimal symmetric punishment as:
v = (1 − δ)π i (pL ) + δπ i (p̂) ≥ (1 − δ)α̂nπ i (pL ) + δv,
(9)
where p̂ is the common price set in the first period of the punishment equilibrium and p̂ is the price
charged thereafter. The expression α̂nπ i (pL ) is the profit a firm makes when deviating upwards from
the punishment price pL . Without cross shareholdings it would be zero. With cross-shareholdings the
firm still participates in the losses other firms make in the first punishment period. We can simplify
that incentive constraint to:
(10)
δπ i (p̂) ≥ (α̂n − 1 + δ)π i (pL )
To simplify the exposition assume that limp→0 D̄(p) = ∞. Then p̂ = pm . Furthermore, if δ > 1 − α̂n,
then the right hand side is negative and a further decrease in pL is incentive compatible. But then
punishments are made more severe and the incentive constraint on the best collusive price can also be
relaxed. Hence δ O < 1 − α̂n, and (10) strictly binds. Substituting from (13) into (11) and rearranging
terms yields:
1
1
− α̂[n −
] < δN .
δO = 1 −
n(1 − α̂)
1 − α̂
1
. Note that the critical discount factor
where the last inequality follows from the fact that δ N = 1− n(1−α̂)
under symmetric optimal punishments is strictly below that of Nash reversion. Indeed, the difference is
substantial. At a 5% cross shareholding and 2 firms in the market δ O = 0.426 32 while δ N = 0.473 68,
i.e. the critical discount factor under Nash reversion is 10% higher. At a 10% cross-ownership we have
δ O = 0.355 55 and δ N = 0.444 44. This shows that even small cross-shareholdings can have large effects
on the critical discount factor in the best optimal symmetric punishment equilibrium.
The homogeneous goods model gives little sense for the order of magnitude of price effects that
one might expect from corss-shareholdings. This is the reason one might want to calibrate a product
differentiation model like ours. Since the basic logic of punishment prices below marginal costs will
hold even for models with some product differentiation we should expect the quantitive effects on price
to by significantly greater in the best symmetric optimal punishmentequilibrium than in the best Nash
reversion equilibrium. This fact is shown in Figure 2:
[Insert Figure 2]
The figure contrasts the price effects of going from a 5% cross-shareholding to a 10% cross shareholding in a duopoly for three different assumptions on the equilibirium played and for the whole range
of levels of product differentiation. The discount factor is at a level such that in all cases the solution
collapses to perfect competition at the homogeneous goods limit. The lowest two lines correspond to
the impact of cross-shareholdings on Nash equilibria. As is well known, such cross-shareholdings have
21
virtually no impact on the static Nash equilibrium. For the best Nash reversion equilibrium prices are
higher but the marginal effect of an increase in cross shareholdings from 5% to 10% are still very small.
However, under optimal symmetric punishments price effects of such small cross-shareholding changes
can have an impact of up to 5% in the price level. This analysis thus suggests that, for example, in
mergers that raise coordinated effects concerns small cross-shareholdings should be of significant importance for remedies while this would not be the case in unilateral effects cases. It should be noted how
important it is to analyze optimal punishment equilibria in order to get a proper insight into the order
of magnitude of this effect.
4.3
The Impact of Marginal Cost on Collusion
In some markets, in particular gasoline retailing, increases in input prices often lead to increased margins.
In public debate this is often attributed to increased input prices facilitating downstream collusion.
However, our basic homogeneous goods collusive model does not conform to that prediction. The
incentive condition to sustain some collusive price pC > c is given by:
1
(pC − c) D(pC ) ≥ (1 − δ)(pC − c)D(pC )
(11)
n
This means that the incentive compatibility constraint depends neither on pc nor on c. This resultdepends on the very special property of the homogemeous goods model that deviation profits are proportional to collusive profits. This is not the case as soon as we introduce some product differentiation.
We now show that generally an increase in marginal costs does facilitate collusion in the sense that
the critical discount factor at which full collusion at the monopoly price is feasible falls. However, it is
unlikely for price cost margins to increase as a result of increased marginal costs.
In all our earlier examples the monopoly price was independent of the parameter for which we
analyzed the comparative statics of collusion. When we study the comparative statics with respect to c,
the monopoly price must increase when c increases. We therefore interpret δ(c, φ) as the lowest discount
factor for which the monopoly price pm (c) can be sustained for a given φ. We are interested in the
comparative statics of δ(c, φ) in c. The incentive compatibility constraint for any δ and pc is given by:
1
(12)
(pc (c) − c) D̄(pc (c))
n
∗ c
∗ c
c
≥ (1 − δ)[p (p (c), φ) − c]Di (p (p (c), φ), p (c), φ) + δv(c, φ)
where the constraint is binding if pc (c) < pm (c) or if δ = δ(c, φ) and pc (c, φ) = pm (c). Note that for
any given c, p∗ (pm (c), φ) → pm (c) for φ → 0 and δ(c, φ) → n−1
n for φ → 0. We now want to show that
for small degrees of product differentiation collusion is strictly facilitated in the sense that δ(c, φ) is
strictly decreasing in c when φ is close to zero. We discuss the implications for the comparative statics
in price below. As in previous analyses we consider the marginal change in the tightness of the incentive
constraint as c changes, by looking at:
1
(13)
ξ c = − D̄(pm (c)) + (1 − δ(c, φ))D(p∗ (pm (c), φ), pm (c), φ)
n
∂Di ∂pm
−(1 − δ(c, φ))(p∗ (pm (c), φ) − c)
∂pj ∂c
∂v(c, φ)
−δ
∂c
22
The first line captures the fundamental effect that dominates the comparative statics in the model. An
increase in c will decrease the profits from collusion, which tightens the constraint. This is represented
by the first term. On the other hand, the profits from deviation are reduced. These two terms cancel as
∗ m
m
m
φ → 0 since limφ→0 δ(c, φ) = n−1
n and limφ→0 Di (p (p (c), φ), p (c), φ) = D̄(p (c)). Slightly away from
zero this term becomes strictly positive: There is a direct effect that demand at the optimal deviation
declines when product differentiation is introduced. Close to φ = 0, this effect is exactly compensated
by the decline in the critical discount factor that arises when products become more differentiated. The
effect that remains is that the best response price declines, leading to a relative increase in demand,
making the first term positive. A countervailing effect arises from the second line. It is generated because
the monopoly price increases in c. Since all other firms raise the monopoly price, a best response to
collusion becomes more profitable at given costs. Hence, the incentive constraint is tightened. However,
since limφ→0 Dpj (p∗ (pm , φ), pm , φ) = 0, the effects in the first and second line are of the same order.
We show in the proof of proposition 9 in Appendix A that the first effect dominates the second effect
and the incentive constraint is strictly relaxed. The third line captures the impact on the continuation
value in the worst continuation equilibrium, ∂v(c,φ)
∂c . This effect is positive but converges to zero at the
homogeneous goods limit. To see this note that there are two things affecting this continuation profit.
First, costs go up, reducing the profits from punishments. However, the punishment price that is set
will also increase, increasing punishment profits. It can be shown that the net effect is positive.
To prove these results at the homogeneous goods limits we need two more assumptions that impose
some more weak regularity conditions on demand at the homogeneous goods limit in the spirit of
Assumption 1:
Assumption 4: Let p(φ) map the degree of product differentiation into a price vector.
If limφ→0 D(pi (φ), p−i (φ), φ, n) = D̄(limφ→0 pi (φ)), then
h
i
P
∂D(pi (φ),p−i (φ),φ,n)
−i (φ),φ,n)
(a) limφ→0 ∂D(pi (φ),p
+
= D̄0 (limφ→0 pi (φ))
j6=i
∂pi
∂pj
(b) limφ→0
S
∂ 2 D(pi (φ),p−i (φ),φ,n)/∂pi ∂φ− j6=i ∂ 2 D(pi (φ),p−i (φ),φ,n)/∂pj ∂φ
∂D(pi (φ),p−i (φ),φ,n)/∂pi
>0
Assumption 4 (a) asserts that at the limit the demand from the undercutting firms must increase
with the slope of aggregate demand when all prices are increased by the same amount. Assumption 4
(b) says that an increase in product differentiation at the homogeneous goods limit strictly increases
the elasticity of demand of a strictly undercutting firm and more so than the cross-elasticity of demand
is increased. Essentially this says that some product differentiation will make some customers stop
purchasing because their ideal product is not available. These properties are satisfied in our examples
in Appendix A. With these assumptions in place we can prove:
Proposition 9 There exists φ̂ > 0 such that for all φ ∈ (0, φ̂) the critical discount factor δ(c, φ̂) is
strictly decreasing in c.
Proof. See Appendix A.
We omit a statement and proof of a related result on the comparative statics of price, because it
is relatively complicated. It is analogous to the proof of comparative statics in price of the first two
23
examples and works along the same lines as the proof of the proposition on the critical discount factor.
Intuitively, the proposition about the discount factor suggests that at a price close to the collusive price,
the collusive price should rise at least as fast as the monopoly price if the monopoly price increases less
than one to one in costs (this is the case when demand is log-concave). Simulations on a wide range of
parameter values for the nested logit model confirms that this is the case over a wide range of parameters.
When demand is not log-concave as in the CES case, prices always rise more than one-to-one in the
cost parameter, but less than under full monopoly.
This conclusion is in contrast to a model of transitory cost shocks. In a version of the Rotemberg
and Saloner model with cost instead of demand uncertainty Rimler (2005) shows that price rises more
than one to one in transitory cost shocks. In that model the anticipation of lower costs in the future
(and with it higher profitability) makes collusion easier in a high cost state. Our analysis combined
with this result suggests that one to one increases in costs are only likely to be related to collusion if
they arise from cost shocks that are perceived to be transitory.
This example also demonstrates that our method allows us to assess the magnitude of different
comparative statics effects on collusion. The impact of increases in marginal cost on collusion turn
out to be very small. This is unsurprising since there is no effect in the homogeneous goods limit. A
similar result has been obtained by Schultz (2005) for the impact of consumer market transparency
on collusion. In his model there is no impact of consumer market transparency in the limit model as
product differentiation goes to zero. Schultz’s result is derived for a simple Hotelling model but could
be greatly generalized to other demand structures using our methods. For policy purposes such results
give important insights to the type of variables an antitrust authority should look at: the effect of
consumer market transparency or marginal cost changes on collusion is small relative to the impact
of, for example, cross-ownership. This is thus another example of how the analysis of a change in the
incentive constraints close to homogeneity can lead to a rich set of new results that allow us to assess
both the robustness and order of magnitude of different variables on the ability to collude.
5
Conclusions
In this paper we have presented a framework to study the robustness and order of magnitude of results
in collusion theory. On the basis of a general product differentiation model we have been able to contrast
the robustness of symmetric optimal punishments relative to equilibria obtained under Nash reversion
in symmetric models. We have also identified why symmetric optimal punishments can be legitimately
used as a modelling shortcut in symmetric models. We have used these results to prove new results
on the impact of product differentiation, the fragmentation of asset ownership, cross-shareholdings,
and cost changes on the ability to collude. We have also shown in related work that the methods
developed here are useful for exploring the comparative statics of collusion in asymmetric models (see
Kühn and Rimler, 2007). Generally, this work illustrates how a combination of theoretical tools using
the behavior at the homogeneous goods limit and numerical methods can be applied to obtain insight
into the robustness and order of magnitude of the comparative statics of collusion.
24
6
Appendix A: Proofs
Proof of Proposition 1: (Existence and Comparative Statics of the Nash Equilibrium)
Proof: The assumptions in the model section guarantee that demand is log-supermodular in prices
and that best responses are contraction mappings that implies that there exists a unique Nash equilib2
2
i ,p−i ,φ,n)
i ,p−i ,φ,n)
> 0 and ∂ ln D(p
< 0 guarantees that
rium in prices. Furthermore, assuming ∂ ln D(p
∂pi ∂φ
∂pi ∂n
equilibrium prices are increasing in φ and decreasing in n. Finally, note that
¢
¡
D(pN , pN )
lim pN (φ) − c = − lim
φ→0
φ→0 Dpi (pN , pN )
= −
1
D̄(pN )
lim
=0
n φ→0 Dpi (pN , pN )
since limφ→0 Dpi (pN , pN ) = −∞. QED.
Lemma A1: For any function p(φ) with p(φ) ∈ [pN (φ), pm ] for all φ, we have:
a) lim p∗ (p(φ), φ) =
b) lim −
φ→0
φ→0
(p∗ (p(φ), φ), φ)
Dpi
D(p∗ (p(φ), φ), φ)
=
lim p(φ)
φ→0
1
D̄0 (p(φ))
≥ lim −
,
φ→0 p(φ) − c
φ→0
D̄(p(φ))
lim
where (b) holds with equality if and only if limφ→0 p(φ) = pm .
Proof: Let p∗ = limφ→0 p∗ (p(φ), φ) and p = limφ→0 p(φ). Then p∗ > p is impossible since the firm
would make zero profits and it could make strictly positive profits and face the market demand function
by setting pi < p. We now show that p∗ < p is also impossible. Suppose that would be the case. Then
D̄(p)
D̄0 (p)
p∗ − c < p − c ≤ − D̄
0 (p) , where the last inequality follows from the fact that D̄(p) is decreasing in p, due
to log-concavity of demand. Hence, by quasi-concavity of the limit industry profit function in p, the
firm could, for φ close to zero, increase price and increase profits. Since the model is smooth for every
φ the elasticity has to converge to the inverse of p−c
p . QED.
Proof of Lemma 1: We begin by proving that the highest sustainable collusive price under Nash reversion always exceeds the Nash price for φ ∈ (0, 1). Consider the incentive condition on pc given by (1).
Differentiating π(pc , pc , n) − (1 − δ)π(p∗ (pc , φ, n), pc , φ, n) with respect to pc and evaluating this at pc =
N N
pN (φ, n) yields δ ∂π(p∂p,pj ,n) > 0, which implies that the incentive constraint is strictly relaxed. Hence,
pc > pN and π(pNc , pNc , φ, n) > π(pN (φ, n), pN (φ, n), φ, n) > 0. Since the Nash punishment strategy is available when considering the set of fully optimal punishments, v O (φ, δ, pc , n) ≤ v N (φ, δ, pc , n)
and π(pNc , pNc , n) can be sustained as average profits under optimal punishment equilibria. Furthermore, by the existence of a public signal any v ∈ [v N (φ, δ, pc , n), π(pNc , pNc , n)] can be supported as an
average payoff in some equilibrium belonging to the set of optimal symmetric punishment equilibria.
Now choose pL = pN (φ, n) − ε so that π(pN (φ, n), pN (φ, n), φ, n) − π(p∗ (pL , φ, n), pL , φ, n) = η(ε) and
η(ε)
η(ε)
< 1 for all ε > 0, γ(ε)
is increasing
π(pN (φ, n), pN (φ, n), φ, n) − π(pL , pL , φ, n) = γ(ε), where clearly γ(ε)
η(ε)
in ε in the neighborhood of ε = 0, limε→0 γ(ε)
= 1, and limε→0 η(ε) = limε→0 γ(ε) = 0. Then there
exists ε̂ > 0 such that, for all ε ∈ (0, ε̂), η(ε) − (1 − δ)γ(ε) > 0. Then one can choose ε ∈ (0, ε̂), such
25
that v̂(ε) = π(pN (φ, n), pN (φ, n), φ, n) + 1δ [η(ε) − (1 − δ)γ(ε)] and v̂ ∈ (v N (φ, δ, pc , n), π(pNc , pNc , n)).
By construction this satisfies (1 − δ)π(pL , pL , n) + δv̂ = π(p∗ (pL , φ, n), pL , φ, n). Hence, pL < pN (φ, n)
is sustainable as a punishment equilibrium and v O (φ, δ, pc , n) < v N (φ, δ, pc , n). Proving property (a)
since the strategy supporting v O (φ, δ, pc , n) is also available for the best stationary collusive equilibrium.
This result immediately implies property (b). To see this note that at pN c the incentive constraint for
the most profitable collusive price is binding under Nash punishments unless pNc = pm . Hence, it is
slack under symmetric optimal punishments at pNc and pOc > pNc unless pNc = pm . To see property
(c), first note that at δ = 1 the constraint is strictly slack at pc = pm . If δ = 0, it is violated. Property (c) follows by noting that δ(φ, n) is strictly increasing in v. Property (d) almost directly follows
from the previous results: Average profits are bounded from below by 0 since a firm can always deviate to a strategy of marginal cost pricing in every period. Since vN (φ, δ, pc , n) → 0 as φ → 0, the
same holds for v O (φ, δ, pc , n) by property 1. Note that at the limit the same incentive condition has
to be satisfied for pc whether Nash reversion punishments are used or symmetric optimal punishments.
Hence limφ→0 pNc = limφ→0 pOc . Finally, note that by Lemma A1, limφ→0 p∗ (pm , φ, n) = pm . Hence,
limφ→0 π(p∗ (pNc (φ), φ, n), φ, n) = (pm − c)D̄(pm ). Consider a strategy where the undercutting firm sets
p∗ (pm , φ, n) − ε for every φ. By Lemma B1, limφ→0 p∗ (pm , φ, n) − ε = pm − ε. Since pm − ε < pm , the
firm would serve the whole market at pm − ε in the limit. Hence, in the limit, the firm would make a
profit of (pm − ε − c)D(pm − ε). Since the firm can use such a strategy for every ε > 0, in the limit,
profits cannot fall below (pm − c)D(pm ), the maximal attainable profits in the market. Hence,
lim δ(φ, n) =
φ→0
=
limφ→0 π(p∗ (pNc (φ), φ, n), φ, n) − limφ→0 π(pNc , pN c , n)
limφ→0 π(p∗ (pN c (φ), φ, n), φ, n) − limφ→0 v(φ, δ, pm , n)
n−1 m
m
n (p − c)D̄(p )
(pm − c)D̄(pm )
=
n−1
n
where we have used the fact that limφ→0 v(φ, δ, pm , n) = 0 independently of the punishment strategies
allowed. This completes the proof.QED.
Lemma A2: Let p > c, then limφ→0 D(p∗ (p, φ), p, φ) = D̄(p).
Proof: Suppose limφ→0 D(p∗ (p, φ), p, φ) = λD̄(p) < D̄(p). Now consider the alternative strategy
p∗ (p, φ) − ε. Since p∗ (p, φ) → p it follows from our assumptions that limφ→0 D(p∗ (p, φ) − ε, p, φ) =
D̄(p − ε). Clearly for ε small enough (p − ε − c)D̄(p − ε) > (p − c)λD̄(p). Hence, for some φ̂ > 0 this
strategy give strictly higher profits for all φ ∈ (0, φ̂), contradicting that p∗ (p, φ) is a best response on
this range. Hence, limφ→0 D(p∗ (p, φ), p, φ) = D̄(p). QED
Proof of Proposition 2: The fact that pc > pN in part (a) follows directly from Lemma 1. Now
∗ m
m
∗ Nc
Nc
suppose δ ≥ n−1
n . By Lemma A1, limφ→0 p (p , φ, n) = p and by Lemma A2 limφ→0 D(p (p (φ), φ), p (φ), φ) =
D̄(pNc (0)) where pNc (0) ≡ limφ→0 pNc (φ). Then the right hand side of (1) converges to
(1 − δ) lim π(p∗ (pNc (φ), φ, n), pNc (φ), φ, n)
φ→0
≤
1 Nc
(p (0) − c)D̄(pNc (0)) = π(pNc (0), pNc (0), n)
n
and the incentive compatibility constraint is satisfied. It follows that pm can be sustained and therefore
pN c (φ) (and by Lemma 1 also pOc ) must converge to pm . Now assume for contradiction that p(0) =
∗ Oc
limφ→0 pOc > c if δ < n−1
n . Consider a strategy where the undercutting firm sets p (p (φ), φ, n) − ε for
26
every φ. By Lemma A1, limφ→0 p∗ (pOc (φ), φ, n) − ε = p(0) − ε. Since p(0) − ε < p(0), the firm would
serve the whole market at p(0) − ε in the limit. Hence, in the limit, the firm would make a profit of
(p(0) − ε − c)D̄(p(0) − ε). Since the firm can use such a strategy for every ε > 0, in the limit, average
profits from deviation have to exceed (1 − δ)(p(0) − c)D(p(0)) > n1 (p(0) − c)D(p(0)). Hence, the firm
would have a strict incentive to deviate close to φ = 0. It follows that limφ→0 pOc = limφ→0 pNc = c,
where the first equality follows from Lemma 1. QED.
We now derive a Lemma that allows us to characterize the payoffs from symmetric optimal punishment equilibria. To do this let pL be the price set at the start of a punishment period and let v̂ be the
continuation value after the first punishment period, i.e. v O (φ, δ, pc , n) = (1 − δ)π(pL , pL , n) + δv̂. Then
we have:
Lemma A3: The incentive constraint for optimal punishments is always binding, i.e. v O (φ, δ, pc , n) =
π(p∗ (pL , φ, n), pL , φ, n). Either pL = 0 and v̂ < π(pc , pc , n), or the optimal punishment price solves
π(p∗ (pL , φ, n), pL , φ, n) = (1 − δ)π(pL , pL , n) + δπ(pc , pc , n)
(14)
In this case pL (φ, δ, pc , n) is increasing in φ and n and decreasing in δ and pc .
Proof of Lemma A3: Suppose that the incentive constraints on the worst punishment equilibrium
were slack, i.e.vO (φ, δ, pc , n) = (1−δ)π(pL , pL , n)+δv̂ > π(p∗ (pL , φ, n), pL , φ, n). Then π(pc , pc , n) ≥ v̂ >
v O since π(pL , pL , n) ≤ π(p∗ (pL , φ, n), pL , φ, n). By convexity of the value set v̂ could be strictly reduced
and the constraint would still be slack and the constraint on the highest sustainable price would be relaxed. This contradicts the definition of v O (φ, δ, pc , n). Hence, v O (φ, δ, pc , n) = π(p∗ (pL , φ, n), pL , φ, n).
Now suppose pL > 0 and v̂ < π(pc , pc , n). Then pL could be lowered slightly and v̂ increased to leave
v O (φ, δ, pc , n) unchanged. But then the incentive constraint is slack since π(p∗ (pL , φ, n), pL , φ, n) strictly
increases in pL . It follows that either pL = 0 and v̂ < π(pc , pc , n) or pL > 0 and v̂ = π(pc , pc , n). In the
first case pL does not vary with small changes in φ, δ, or pc . In the second case pL (φ, δ, pc , n) solves
dπ(p∗ (pL ,φ,n),pL ,φ,n)
L ,pL ,n)
−
> 0.
(14). First note, that at any optimal punishment price pL , (1 − δ) dπ(pdp
dpL
L
Otherwise, pL could be slightly reduced and the incentive constraint be relaxed. By totally differentidpL
∗
L
ating (??) we find that dp
dφ has the same sign as π φ (p (pL , φ, n), pL , φ, n) > 0, dδ has the same sign as
c
c ,n)
dπ(p ,p
L
−[π(pc , pc , n) − π(pL , pL , φ)] < 0, dp
dpc has the same sign as −δ
dpc
1
∗ (p , φ, n), p , φ, n) + π (p∗ (p , φ, n), p , φ, n) > 0. QED.
π(p
n
L
L
L
L
n
and
dpL
dn
has the same sign as
Proof of Lemma 2: The comparative statics of collusion can be calculated from (1) by totally
differentiating. This yields
∂pc (φ, δ, n)
=
∂φ
and
ξ φ (φ, pc (φ, δ, n), δ, n)
dπ(pc ,pc ,n)
dpc
− ξ pc (φ, pc (φ, δ, n), δ, n)
ξ φ (φ, pm , δ(φ, n), n)
∂δ(φ, n)
=−
∂φ
ξ δ (φ, pm , δ(φ, n), n)
c
c
Now note that dπ(pdp,pc ,n) − ξ pc (φ, pc (φ, δ, n), δ, n) < 0. Otherwise the incentive constraint could be
relaxed by increasing pc , contradicting the definition of pc . Furthermore,
ξ δ (φ, pm , δ(φ, n), n) = −(π(p∗ (pm , φ, n), pm , φ, n) − v(φ, pm , δ(φ, n), n)) + δ
27
∂v(φ, pm , δ(φ, n), n))
.
∂δ
In the case of Nash punishments
punishment equilibria,
∂v(φ,pm ,δ(φ,n),n))
∂δ
= 0 and ξ δ < 0. For the case of symmetric optimal
dπ(p∗ (pL , φ, n), pL (φ, δ, pm , n), φ, n) ∂pL
∂v(φ, pm , δ(φ, n), n))
=
≤ 0,
∂δ
dpL
∂δ
where the last inequality follows from Lemma A3. Hence,
ξ φ < 0.QED.
O
∂pc (φ,δ,n)
∂φ
> 0 and
∂δ(φ,n)
∂φ
< 0 if and only if
m
,n)
Proof of Proposition 3: We show that limφ→0 ∂v (φ,δ,p
= 0, implying that limφ→0 ξ φ < 0.
∂φ
O
Then by continuity the proposition follows. By Lemma B2 we have v (φ, δ, pm , n) = π(p∗ (pL , φ, n), pL , φ, n).
By Lemma 2, limφ→0 π(p∗ (pL , φ, n), pL , φ, n) = 0, and hence, from (14),
1
n−1
lim π(pL , pL , n) +
π(pm , pm , n) = 0
n φ→0
n
It follows limφ→0 pL < c. Since π(p∗ (pL , φ, n), pL , φ, n) ≥ 0, it follows that p∗ (pL , φ, n) ≥ 0. Hence,
limφ→0 D(p∗ (pL , φ, n), pL , φ, n) = 0, and, consequently, limφ→0 p∗ (pL , φ, n) = c. Now note that:
∂v O (φ, δ, pm , n)
∂φ
∂pL
+ π φ (p∗ (pL , φ, n), pL , φ, n)
∂φ
∂pL
+ π φ (p∗ (pL , φ, n), pL , φ, n)
[p∗ (pL , φ, n) − c] [Dpj (p∗ (pL , φ, n), pL , φ, n)
∂φ
D(p∗ (pL , φ, n), pL , φ, n)
∂pL
Dpj
+ π φ (p∗ (pL , φ, n), pL , φ, n)
−
∗
Dpi (p (pL , φ, n), pL , φ, n)
∂φ
∂pL
+ π φ (p∗ (pL , φ, n), pL , φ, n)
D(p∗ (pL , φ, n), pL , φ, n)
∂φ
⎡
⎤
∗ (p , φ, n), p , φ, n)
D(p
L
L
⎣
+ 1⎦ [p∗ (pL , φ, n) − c] Dφ (p∗ , pL , φ, n)
dπ(pL ,pL ,n)
dπ(p∗ (pL ,φ,n),pL ,φ,n)
(1 − δ)
−
dpL
dpL
= π pj (p∗ (pL , φ, n), pL , φ, n)
=
=
≤
=
∗
,φ,n),pL ,φ,n)
L ,pL ,n)
L ,pL ,n)
Now 0 ≤ limφ→0 (1−δ) dπ(pdp
− dπ(p (pL dp
= n1 limφ→0 dπ(pdp
< ∞. Furthermore we have
L
L
L
∗
0 < limφ→0 Dφ (p (pL , φ, n), pL , φ, n) = Dφ (c, pL , φ, n) < ∞. But since demand and price cost margin
O
c
,n)
= 0. It follows that δ(φ, n) is decreasing at φ = 0, so that pc must
converge to zero limφ→0 ∂v (φ,δ,p
∂φ
be constant at pm in the neighborhood of φ = 0. The result for δ < n−1
n follows directly from Lemma
1 and proposition 2. QED.
Proof of Proposition 4: Since for all φ ∈ (0, 1), v O (φ, δ, pc , n) ≥ vS (φ, δ, pc , n) ≥ 0 and
O
c ,n)
limφ→0 vO (φ, δ, pc , n) = limφ→0 v S (φ, δ, pc , n) = 0 (by Lemma 1) it follows directly that limφ→0 ∂v (φ,δ,p
≥
∂φ
limφ→0
∂vO (φ,δ,pc ,n)
.
∂φ
The proposition then follows from the proof of proposition 3. QED.
Proof of Proposition 5: Since the game is symmetric the bargaining solution over the equilibrium
value set selects a symmetric value vector on the Pareto frontier of the equilibrium value set. We will
now show that under assumptions 2 and 3 strategies in this equilbirium must be symmetric on the
equilibrium path. Suppose at the best collusive equilibrium firms charge a price vector p in period 1
28
and have an average continuation profit of v̂i , i = 1, ...n. For contradiction assume that firms do not
charge the same price and have different continuation profits. For every i we have
(1 − δ)π i (p,φ) + δv̂i ≥ (1 − δ)π i (p∗ (p−i ), p−i , φ) + δv
where by symmetry of the equilibrium value set the punishment value for each i must be the same. It
follows that
1X
1X
1X
(1 − δ)
π i (p,φ) + δ
v̂i ≥ (1 − δ)
π i (p∗ (p−i ), p−i , φ) + δv,
n
n
n
i
i
i
P
P
where (1 − δ)π i (p,φ)
+ δv̂i = (1 − δ) n1 i π i (p,φ) +
δ n1 i v̂i by symmetry
of the selected equilibrium
1 P
1 P
1 P
value. Let p̄ = n i pi .PBy strict concavity of n i π i (p,φ) inPp, n i π i (p,φ) < π i (p̄, p̄). Furthermore, by convexity of n1 i π i (p∗ (p−i ), p−i , φ) in p−i we have n1 i π i (p∗ (p−i ), p−i , φ) ≥ π i (p∗ (p̄), p̄, φ).
Furthermore, by convexity of the equilibrium value set, there exists an equilibrium
with a continuation
1 P
value in which each player obtains an average continuation value of v̄ = n i v̂i . But then a price p̄ for
every player in period 1 with a continuation value of v̄ satisfies the incentive constraint for all players.
Furthermore, the value of the equilibrium is higher, contradicting the assumption that the previous
equilibrium was the best collusive equilibrium. Hence, the best collusive equilibrium is symmetric on
the equilibrium path. But then the equilibrium must also be stationary. Suppose otherwise. Then either
π i (pc , pc ) > v̄ or π i (pc , pc ) < v̄. Consider the first case. Since pc is sustainable, it can be played forever
and will satisfy incentive compatibility. Hence, there would be a better equilibrium contradicting that
we are at the best collusive equilibrium. Consider π i (pc , pc ) < v̄. If v̄ is available as a continuation
equilibrium outcome it can also be played at the beginning. But v̄ > (1 − δ)π i (pc , pc ) + δv̄, contradicting
that we are at the best collusive equilibrium. Hence, the equilibrium is stationary on the equilibrium
path. QED.
N
(φ,n)
< 0, which occurs
Proof of Proposition 6: Part (a): We show that for sufficiently large n, ∂δ ∂φ
by Lemma 1 if and only if nξ φ (pm , δ(φ, n), φ, n) < 0. To show that the limit of this expression is strictly
negative, we take the limits of some of the component parts of this expression. First, we show that
limn→∞ n(1 − δ N (φ, n)) > 0. We have:
lim n
n→∞
(pm − c)D(pm ) − limn→∞ (pN − c)D̄(pN )
π(pm , pm , n) − π(pN , pN , n)
=
π(p∗ (pm ), pm , n) − π(pN , pN , n)
limn→∞ (p∗ (pm ) − c)D(p∗ (pm ), pm , n)
Clearly, the sign of limn→∞ (p∗ (p, n) − c) D(p∗ (pm ), pm , n) is bounded from above by (pm − c)D̄(pm ).
Since (pm − c) D̄(pm )−limn→∞ (pN (n)−c)D(pN (n)) > 0 by quasi-concavity of (p−c)D(p) in p it follows
that limn→∞ n(1 − δ N (φ, n)) > 0.
Second, we show that limn→∞
dπ(p∗ (pm ,φ),pm ,φ)
dφ
< 0. By the envelope theorem
dπ(p∗ (pm ,φ),pm ,φ)
dπ(p∗ (pm ,φ),pm ,φ)
dφ
=
(p∗ (pm , n) − c) Dφ (p∗ (pm , φ), pm , φ). Then limn→∞
< 0 immediately follows from our
dφ
assumptions on demand. Indeed, the expression must go to −∞ when Dφ (p∗ (pm , φ), pm , φ) → −∞ for
Dp (pN ,pN ,n)
N
∂v
N
n → ∞. Finally, we can write nδ N (φ, n) ∂φ
as nδ N (φ, n)D(pN , pN , n) −Djpi (pN ,pN ,n) dp
dφ . Since δ (φ, n) ≤
µ S
¶
N
N
=i Dpj (p (n),p (n),n)
1, D(pN (n)) ≤ D(c), and − j6D
≤ 1 by our assumptions on demand, this expression
N
N
pi (p (n),p (n),n)
is positive and bounded from above by D(c) limn→∞
dpN
dφ .
29
By Lemma A3,
dpN
dφ
is bounded and converges
to zero for n → ∞ if the model has a competitive limit, i.e. if limn→∞ εi (p, n, φ) = ∞. Hence, if the
∂v
vanishes as n → ∞, while the first term in
model has a competitive limit, the term nδ N (φ, n) ∂φ
nξ φ converges to a strictly negative number. If the model has a monopolistically competitive limit,
∗
∂v
nδ N (φ, n) ∂φ
is bounded from above in the limit. But if limn→∞ dπ(p (p,φ),p,φ)
= −∞ as assumed in the
dφ
proposition, nξ φ is still strictly negative in the limit. This completes the proof.
Part (b): The proof of part (b) is similar to the proof of part (a). First, we show that limn→∞ n(1 −
∗
By the envelope theorem dπ(p (p,φ),p,φ)
= (p∗ (pc , φ) − c) Dφ (p∗ (pc , φ), pc , φ). Since
dφ
∗ c
c
δ) dπ(p (pdφ,φ),p ,φ) < 0.
¯
¯
¯ ∂D(pi ,p−i ,φ) ¯
¯ < Z(φ)
¯
∂pi
then
< ∞, it follows that limn→∞ n(pN − c) = limn→∞
lim n
n→∞
dπ(p∗ (p, φ), p, φ)
dφ
=
≤
<
D̄(pN (φ,n))
Dpi (pN (φ,n)
>
D̄(pN (φ,n))
.
Z(φ)
But
lim n (p∗ (pc , n) − c) Dφ (p∗ (pc , φ), pc , φ)
¡
¢
lim n pN (φ, n) − c Dφ (p∗ (pc , φ), pc , φ)
n→∞
n→∞
D̄(pN (φ, n))
Z(φ)
lim Dφ (p∗ (pc , φ), pc , φ)
n→∞
The first inequality follows because limn→∞ Dφ is strictly negative by assumption. The second inequality
∂v
directly follows by the limiting assumption in the slope of demand. The limiting argument for nδ ∂φ
is
essentially the same as in proposition 3. The result then follows.QED.
Proof of Proposition 7: Fix pck and let δ(k, φ, pck ) be the lowest discount faactor at which pck can
just be sustained. Note that the incentive constraint is just binding when there are fewer firms. We show
that the incentive constraint for the market with the strictly larger number of firms is strictly slack for φ
close to zero. This implies that δ(k, φ, pck ) < δ(k̄, φ, pck ) and pck (φ, δ(k, φ, pck )) > pck̄ (φ, δ(k, φ, pck )).To
show the result it is sufficient to show that
h
i
(15)
ξ k (φ) = (1 − δ(k, φ, pck )) π k (p∗k (pck , φ), pck , φ, n) − π k (p∗k̄ (pck , φ)), pck , φ, n)
+δ(k, φ, pck )[v(φ, k) − v(φ, k)]
is strictly positive for φ close enough to zero. Since limφ→0 (p∗k (p, φ)−c) = p−c > 0 and limφ→0 v(φ, k) =
0 for all k,
¸
´∙k
k̄
1 ³ ck̄
p −c
−
D̄(c) < 0.
(16)
lim ξ k (φ) <
φ→0
k
n n
It follows directly that δ(k, φ, pck ) < δ(k̄, φ, pck ). Furthermore, if ξ increases in pc at pck , then pck is not
the highest sustainable collusive price. Hence, if pck is the highest sustainable collusive price at φ the
highest collusive price for k̄ at the same φ and the same δ must be strictly lower, i.e. pck (φ, δ(k, φ, pck )) >
pck̄ (φ, δ(k, φ, pck )) QED.
We now derive a result for the comparative statics of the Nash equilibrium when there are crossshareholdings that we use for the proof of proposition 9. The short run best response is determined
from the first order condition:
D(p∗ (p, α, φ), p, φ)+(p∗ (p, α, φ) − c) Dpi (p∗ (p, α, φ), p, φ)+
30
1
α̂
(p−c)Dpj (p, p∗ (p, α, φ), p, φ) = 0
1 − α̂ n − 1
(17)
At a Nash equilibrium (17) is satisfied for all firms. By an argument analogous to that in proposition
1 this equilibrium exists and is unique. Nash equilibrium prices have the following limiting properties
when goods become perfectly homogeneous:
Lemma A4: Nash equilibrium prices converge to marginal cost for every α̂ ∈ [0, n−1
n ). Furthermore,
Nash equilibrium prices are constant at φ = 0, i.e. limφ→0
dpN
dα̂
= 0.
Proof of Lemma A4: The first order condition for Nash equilibrium prices can be written as:
pN (α̂, φ) − c
pN (α̂, φ)
∙
¸
D(pN (α, φ), pN (α, φ))
α̂
1 Dpj −1
= −
1+
Dpi (pN (α, φ), pN (α, φ), φ)pN (α, φ)
1 − α̂ n − 1 Dpi
∙
¸−1
α̂
1 Dpj
1
1+
=
N
εi (p (α, φ), φ)
1 − α̂ n − 1 Dpi
From the proof of proposition B1 we can use the argument that limφ→0
h
limφ→0 1 +
α̂ Dpj
1−α̂ Dpi
i
= 1−
α̂
1
1−α̂ n−1
> 0 for all α̂ <
n−1
n .
Dpj
Dpi
= −1. Hence,
By our assumptions on demand,
limφ→0 εi (pN (α, φ), φ) → −∞. Hence, limφ→0 pN (α̂, φ) = c for all α̂. It follows that limφ→0
0.
∂pN (α̂,φ,n)
∂ α̂
=
Proof of Proposition 8:
We first show the limit result on δ(α̂, φ, pc ) as φ → 0.
Π(pc , pc ) − Π(pN , pN , α̂, φ)
φ→0 Π(p∗ (pc , α̂, φ), α̂, φ) − Π(pN , pN , α̂, φ)
1
π(pc , pc )
=
=
c
c
(1 − α̂)nπ(p , p )
(1 − α̂) n
lim 1 − δ(α̂, φ, pc ) =
φ→0
lim
Then
1
lim [π(p∗ (pc , α̂, φ), pc , φ) − π(pc , p∗ (pc , α̂, φ), pc , φ)]
(1 − α̂)n φ→0
µ
∙
¶
¸
1
dpN
N
N
+ 1−
lim π pj (p (α̂, φ), p (α̂, φ))
(1 − α̂)n φ→0
dα̂
lim ξ α̂ = −
φ→0
The term on the first line in brackets converges to (pc − c)D̄(pc ) > 0.
N
In the second term limφ→0 π pj (pN (α̂, φ), pN (α̂, φ)) = n1 D̄(c) and limφ→0 dp
dα̂ = 0. Hence, limφ→0 ξ α̂ =
1
(pc − c)D̄(pc ) < 0. The comparative statics results on δ and pc follow directly. QED.
− (1−α̂)n
Proof of Proposition 9: First note that limφ→0 ξ c = 0. To find out the sign of ξ c close to zero, we
analyze limφ→0 ξ cφ , where we are changing the discount factor δ, so that pm (c) remains just sustainable
31
as φ → 0. We have:
ξ cφ = (1 − δ)Dφ (p∗ (pm (c), φ), pm (c), φ) −
∂δ ∗ (c, φ)
D(p∗ (pm (c)), pm (c), φ)
∂φ
(18)
∂δ ∗ (c, φ) ∗
∂pm (c)
(p − c)Dpj
∂φ
∂c
∗
∂p
∂ 2 D ∂pm (c)
+(1 − δ)Dp1
− (1 − δ)(p∗ − c)
∂φ
∂pj ∂φ ∂c
¸
∙ ∗
2
∂p
∂ D ∂p∗ ∂pm (c)
∗
Dp + (p − c)
−(1 − δ)
∂φ j
∂pj ∂pi ∂φ
∂c
−δv cφ
+
Suppose that for a given c the incentive constraint is just binding for δ ∗ (c, φ). If limφ→0 ξ cφ > 0
or limφ→0 [ξ cφ /Dφ (p∗ (pm (c), c, φ), pm , φ)] < 0 then it follows that for small φ the incentive constraint
is strictly relaxed if marginal costs rises to c + ε and δ ∗ (c + ε, φ) < δ ∗ (c, φ). Since we know that
limφ→0 Dφ (p∗ (pm (c), φ), pm (c), φ) < 0, but cannot rule out that this expression converges to −∞,we
ξ
proceed by analyzing limφ→0 Dcφφ .
First, it is immediate from (12) that
(1 − δ ∗ (c, φ))(p∗ − c)Dφ + δv φ (c , φ)
∂δ ∗ (c, φ)
=
∂φ
(p∗ − c)D − v(c , φ)
From our analysis in section 3 it follows that limφ→0 v(c, φ) = limφ→0 vφ (c, φ) = 0, so that
lim
φ→0
∂δ ∗ (c,φ)
∂φ
Dφ
=
1−δ
.
D̄
It follows that the first term is zero for φ close to zero. Similarly, the second line converges to zero, since
∂δ ∗ (c,φ)
∂φ
and, by assumption 4, limφ→0 Dpj = 0. It is also easy to see that the last term
limφ→0 Dφ = 1−δ
D̄
coming from the continuation value under optimal punishments must converges to zero: Since vcφ = v φc
and limφ→0 v φ = 0 for all φ, it follows that limφ→0 v φc = limφ→0 (vφc /Dφ ) = 0.
To deal with the remaining two lines in ξ cφ , first note that
∙ ∗
¸
− [(p∗ − c)Dpi φ + Dφ ]
∂p
1 dpm (c)
/Dφ = lim
lim
lim
φ→0 ∂φ
φ→0 Dpi
dc φ→0
Dφ
Furthermore, the elasticity of D(p∗ , pj , φ) in pi is given by εi = −
∂εi
∂φ
εi
D(pi , pj , φ) =
We have:
D
Dp φ − Dφ = − [(p∗ − c)Dpi φ + Dφ ]
Dpi i
∂εi
∂φ
> 0 and the fact that [(p∗ − c)Dpi φ + Dφ ] can therefore go no slower
h ∗
i
−[(p∗ −c)Dpi φ +Dφ ]
∂p
it follows that limφ→0
<
0
and
thus
lim
/D
> 0. If
φ→0 ∂φ
φ
Dφ
From our assumption limφ→0
to −∞ than Dφ
Dpi pi
D .
32
limφ→0
h
∂p∗
∂φ /Dφ
i
< ∞, the fourth line in (18) converges to zero as φ → 0, since limφ→0 Dpj =
h ∗
i
/D
limφ→0 Dpj pi = 0. In the case limφ→0 ∂p
φ = ∞ it is easy to show that all our results go through
∂φ
∗
by analyzing limφ→0 (ξ cφ / [(p − c)Dpi φ + Dφ ]) instead.
To derive the sign of line 3 in (18), note that
(p∗ − c)Dpj φ =
where εj =
Dpj pj
D .
∂εj
∂φ
Dpj
pi
D(pi , pj , φ) −
Dφ
εi pj
Dpi
It follows that
∗
lim
φ→0
m
2
∂ D ∂p (c)
∗
(1 − δ)Dp1 ∂p
∂φ − (1 − δ)(p − c) ∂pj ∂φ ∂c
⎡
Dφ
= (1 − δ) ⎣ lim
pi ∂εj 1
pj ∂φ εi D(pi , pj , φ)
i 1
− ∂ε
∂φ εi D(pi , pj , φ) +
Dφ
φ→0
φ→0
⎡³
⎡
³
= (1 − δ) lim − ⎣
= −(1 − δ) ⎣ lim
φ→0
∂εi 1
∂φ εi
∂εi 1
∂φ εi
−
−
pi ∂εj 1
pj ∂φ εi
D(pi , pj , φ)
Dφ
´⎤
∂εj 1
∂φ εi
Dφ
´
m
−
Dpj
Dpi
⎤
Dpj ∂pm (c)
⎦
+
Dpi
∂c
∂p (c)
⎦ D̄(pm
< 0.
i )
∂c
where the last inequality follows from the fact that limφ→0
³
Dφ
∂εi 1
∂φ εi
−
∂εj 1
∂φ εi
´
⎤
m
⎦ ∂p (c)
∂c
> 0 by assumption 4 and
∂εi 1
1
∗
i 1
since ∂ε
∂φ εi = − D [(p − c)Dpi φ + Dφ ], so that ∂φ εi is of the same order in φ as Dφ . It follows that, for
φ close to zero, the incentive constraint becomes strictly slack when c is increased. Hence, there exists
φ̂ > 0 such that δ ∗ (c, φ) is strictly decreasing in c for all φ ∈ (0, φ̂). QED.
7
Appendix B: Examples for the General Model
In this appendix we show that some commonly used specifications of product differentiation models fall
in the class of our general formulation and prove the comparative statics claims in for these examples
in Section 3.2. We use the following limit result to facilitate the calculation of these examples:
Lemma B1: At the homogeneous goods limit the largest discount factor that supports the monopoly
price in the best Nash reversion equilibrium is decreasing in φ if and only if
lim −nξ φ = (pm − c) lim Dφ (p∗ (pm , φ), φ) +
φ→0
φ→0
where limφ→0 Dφ (p∗ (pm , φ), φ) < 0 and limφ→0
dpN
dφ
> 0.
33
dpN
n−1
D(c) lim
< 0.
φ→0 dφ
n
Proof of Lemma B1: From (3) we have:
¾
½
∂v(φ, δ, pm , n)
dπ(p∗ (pc , φ), pc , φ)
+δ·
lim nξ φ (p , φ, δ(φ, n), n) = ξ φ (p , δ, φ, n) = lim n (1 − δ)
φ→0
φ→0
dφ
∂φ
dπ(p∗ (pc , φ), pc , φ)
= n lim (1 − δ(φ, n)) lim
φ→0
φ→0
dφ
X ∂D(pN , pN , n) dpN
+n lim δ(φ, n) · lim (pN − c)
φ→0
φ→0
∂pj
dφ
j6=i
¾
½
∂D(p∗ (pm , φ), pm , φ)
= lim (p∗ (pm , φ) − c)
φ→0
∂φ
⎧
⎫
⎨ −D(pN , pN , φ) X ∂D(pN , pN , n) dpN ⎬
+ (n − 1) lim
φ→0 ⎩ D1 (pN , pN , φ)
∂pj
dφ ⎭
m
c
j6=i
m
m
∗
m
= (p − c) lim Dφ (p (p , φ), p , φ)
φ→0
⎧ P
⎫
∂D(pN ,pN ,n) ⎬
⎨
−
dpN
n−1
j6=i
∂pj
+
D̄(c) lim
lim
φ→0 ⎩
n
D1 (pN , pN , φ) ⎭ φ→0 dφ
where the first equality follows from the envelope theorem and the fact that Dφ = 0 at equal prices.
N
The second equality follows from the fact that limφ→0 δ(φ, n) = n−1
n and by substituting for p − c
from the first order condition of the firm’s maximization problem. The last equality follows because
N
limφ→0 p∗ (pm , φ) = pm and because limφ→0 D(pN , pN , φ) = limφ→0 D̄(pn ) = D̄(c)
n . The proposition then
follows by showing that
⎧ P
⎫
N N
⎨ − j6=i ∂D(p ,p ,n) ⎬
∂pj
lim
=1
φ→0 ⎩
Dpi (pN , pN , φ) ⎭
To see this note that
∂D(pN ,pN ,n)
∂pj
+
Hence,
∂D(pN ,pN ,n) N
p
∂pj
n
D(pN , pN , n)
∂D(pN ,pN ,n) N
p
∂pj
D(pN ,pN ,n)
P
j6=i
N
D̄(p )
⎡
⎣1 +
∂D(pN ,pN ,n)
∂pj
N
p
P
= ε(pN )
⎤
∂D(pN ,pN ,n)
∂pj
⎦
N
(p , pN , φ)
j6=i
D1
S
But limφ→0
= ∞ and hence, limφ→0
of the expression to satisfy ε(c) in the limit. QED.
j6=i
D1
∂D(pN ,pN ,n)
∂pj
(pN ,pN ,φ)
= ε(pN )
= −1 in order for the left hand side
Example 1: Spence-Dixit-Stiglitz preferences with Constant Elasticity of Substitution
The first example is the widely used CES version of the Spence-Dixit-Stiglitz model. There is a
³
´ 1−κ
n1−κ 1 P 1−φκ 1−φκ
representative consumer, whose utility function is given by U (x)= 1−κ n xi
+ m, where x
is the vector of consumption of the n differentiated products, m is expenditure on outside goods, and
34
κ ∈ [0, 1] and φ ∈ [0, 1] are parameters. The direct demand function for product i generated from this
utility function is given by:
1−φ
1 −1
(19)
D(pi , p−i , φ, n) = pi κ (nQi ) 1−φκ
n
µ
¶−1
P ³ pi ´ 1−φκ
φκ
where Qi =
. We now show that this demand function has the desired properties.
j pj
First, note that
lim Qi =
φ→0
⎧
⎨ 1
⎩
1
m
0
if
pi < minj6=i pj
if pi = minj6=i pj and m − 1 other firms charge pi
if
pi > minj6=i pj
(20)
It follows immediately from (19) and (20) that demand is independent of the product differentiation
parameter if all products are offered at the same price. Furthermore, total demand at equal prices is
1
given by p− κ , i.e. “aggregate demand” is independent of the product differentiation parameter and
the number of firms. If a firm has the strictly lowest price its demand converges to the aggregate
demand as products become homogeneous, as required. Furthermore, demand converges to zero in the
homogeneous goods limit if a firm charges a price strictly above the lowest price other firms charge.
We now verify that the assumptions on the slope of the demand functions made in our paper are
also satisfied:
1
pi
∂Di (pi , p−i , φ)
=
εi (pi , p−i , φ)) = −
∂pi
Di (pi , p−i , φ)
κ
½
¾
1−φ
1+
[1 − Qi ]
φ
(21)
Unless firm i has all the demand, i.e. Qi = 1, the elasticity of a product i’s residual demand is greater
than that of the aggregate demand function. If all varieties set the same price, the term in curly brackets
n−1
becomes 1 + 1−φ
φ
n and the elasticity converges to ∞ as φ → 0 as required. If firm ı́’s price is higher
than the lowest price Qi → 0 as φ → 0 and again demand becomes infinitely elastic. However, when
1
i
i
pi < minj6=i pj , limφ→0 1−Q
= − limφ→0 ∂Q
φ
∂φ = 0 and limφ→0 εi → κ , the market elasticity of demand.
To see the latter point, note that
∂Qi 1
∂φ Qi
X ∂ µ pi ¶
= −Qi
∂φ pj
1−φκ
φκ
(22)
j6=i
= Qi
X µ pi ¶
j6=i
pj
1−φκ
φκ
1
ln
φ2 κ
µ
pi
pj
¶
(23)
³ ´ 1−φκ
φκ
1
For ppji < 1, ppji
→ 0 if φ → 0. Hence, we confirm that in the limit residual demand has
φ2 κ
the same properties as the residual demand function of the homogeneous goods model. From (21) we
c
and
can immediately determine the monopoly price and the static Nash equilibrium price as pm = 1−κ
c
N
k
l
p (φ, n) =
respectively.
1−φ n−1 −1
(1−κ 1+
φ
n
)
Furthermore, we can show that demand has the properties assumed in the paper. First, we have:
35
∂Di (pi , pj , φ) 1
∂ ln Di
1 − φ ∂Qi 1
1−κ
=
ln (nQi ) +
=−
2
∂φ
Di
∂φ
(1 − φκ)
(1 − φκ) ∂φ Qi
(24)
When pi = pj for all j, the first term in (24) is zero since ln (nQi ) = 0, and the second term in (24)
1
i
is also zero, since, from (22), ∂Q
∂φ = 0 at pi =³ pj ´for all j. For pi < minj6=i pj , Qi > n and hence
pi
i
< 0. Hence, demand goes down as products get
− ln (nQi ) < 0. Furthermore ∂Q
∂φ < 0 since ln pj
more differentiated for a given price ratio at which firm i has the lower price. The sign is reversed for
pi > maxj6=i pj . Note that (24) also implies that our limiting condition is satisfied:
h
´i
³
1
p
1
Lemma B2: For SDS preferences limφ→0 Dφ (p∗ (p, φ), p, φ) = − (p)− κ 1−κ
ln
(n)
+
−
κ
p−c
κ
Proof: By Lemma A1, limφ→0 εi (p∗ (p, φ), φ) =
1
lim εi (p (p, φ), φ) = lim
φ→0
φ→0 κ
∗
p
p−c .
Hence,
½
µ
¶
¾
1
p
1+
− 1 (1 − Qi ) =
φ
p−c
rearranging terms solving the last equation for limφ→0 φ1 (1 − Qi ), yields
1
lim (1 − Qi ) = κ lim
φ→0 φ
φ→0
½
∙
¸¾
p
1
1
p
1
−
=
− ,
1−φ p−c κ
p−c κ
p
1
i
It follows, by L’Hôpital’s rule, that limφ→0 − dQ
dφ = p−c − κ . Taking limits in (24) above and using
Lemma A1 then yields:
¶¾
½
µ
1
p
∗
−
lim Dφ (p (p, φ), p, φ) = D̄(p) − (1 − κ) ln (n) −
φ→0
p−c κ
Lemma B3:For SDS preferences limφ→0
Proof: Since pN (ρ, n) =
lim
ρ→0
dpN (φ,n)
dφ
c
l−1 ,
k
n−1
(1−κ 1+ 1−φ
)
φ
n
∂pN
⎡
n
= κ n−1
c
we have:
⎢
= c lim ⎣κ h
ρ→0
∂φ
1+
1 n−1
φ2 n
1−φ n−1
φ
n
⎤
n
⎥
i2 ⎦ = cκ
n−1
−κ
Result Example 1: By Lemmas B2 and B3 and proposition B1 above, the sign of the comparative
statics under CES preferences is the same sign as:
1−κ
ln (n) + D(c)c
κ
= −pm D(pm )(1 − κ) ln (n) + D(c)c
∙
¸
D(pm )
= D(c)c −
ln(n) + 1
D(c)
36
− (pm − c) D(pm )
(25)
1
1
1
Now note that D(pm ) = D(c)(1−κ) κ . The term (1−κ) κ is strictly decreasing in κ and limκ→0 (1−κ) κ =
1
e . Hence, if ln(n) < e the expression in brackets in the last line of is strictly positive for all κ. Since
15 < ee < 16, the first part of the proposition follows. If n ≥ 16 there then exists some κ > 0 such that
1
the expression is strictly negative. Furthermore, for κ → 1, (1 − κ) κ → 0, so that there must exist some
1
κ(n) > 0 beyond which the sign switches. Clearly, by monotonicity of (1 − κ) κ in κ, the higher n the
higher κ can be for δ to decrease in φ.
Example 2: A Nested Logit Model
We now show analyze a simple nested choice model in which consumers first have to decide whether
to purchase or not and then decide, which product to purchase. Conditional on having made the decision
to purchase, the probability of choosing brand i is given by:
1
exp {−pi /φκ}
=P
j exp {(−pj ) /φκ}
j exp {−(pj − pi )/φκ}
Qi = P
(26)
This can be interpreted as the market share of product i. The expected value to the customer from
purchasing in the market is then by standard reasoning given by (up to a constant):
AP C
"
#
1X
= φκ ln
exp {(−pi ) /φκ}
n
i
⎡
⎤
X
1
= −pi + φκ ln ⎣
exp{−(pj − pi )/φκ}⎦
n
(27)
j
= −pi + φκ ln[
1
] = −pi − φκ ln [nQi ]
nQi
The choice between purchasing and not purchasing is again given by a simple logit choice:
Q̂(Qi ) =
1
exp{AP C /κ}
=
exp{A/κ} + exp{AP C /κ}
exp{(pi + A)/κ} (nQi )φ + 1
(28)
where A > c is a constant and κ is a parameter determining the aggregate demand elasticity. Demand
for variety i is then given by:
D(pi , p−i , φ, n) = Q̂Qi =
1
1
(n Qi )
n exp{(pi + A)/κ} (nQi )φ + 1
(29)
When all prices are the same, Qi = n1 , and the demand function reduces to:
D(p, p, φ, n) =
1
1
n exp{(p + A)/κ} + 1
(30)
1
which is independent of φ. Furthermore, for pi < pj , demand converges to D̄(pi ) = exp{(pi +A)/κ}+1
as
φ → 0, i.e. as goods become homogeneous the lowest price firm faces total demand. Conversely for
pi > pj , demand converges to zero. To check the other properties of demand first consider:
37
Dpi (pi , pj )
D(pi , pj , φ)
∂Qi 1
dQ̂ 1
+
dpi Q̂
∂pi Qi
¸
∙
1
1
(1 − Qi ) ,
= − (1 − Q̂)Qi +
κ
φκ
=
which is a weighted average of the aggregate semi-elasticity
1
. We also have that:
monopolistically competitive firm φκ
1
κ (1
(31)
− Q̂) and the semi-elasticity of a
p − pi
(1 − Qi )D(pi , p−i , φ)
(32)
φ2 κ
⎛
⎡
⎞
⎤
X
1
1
φ p − pi
+ ⎣ ln ⎝
exp {−(p − pi )/φ}⎠ −
(1 − Qi )⎦ (1 − Q̂)D(pi , p, φ)
κ
n
κ φ2
j
Dφ (pi , pj , φ) = −
At pi = p this collapses to 0 for any φ.
Lemma B4: For the nested logit model limφ→0 Dφ (p∗ (p, φ), p, φ) = −ε(p) 1p ln (n) D(p).
Proof:
1
Note that aggregate demand is given by D(p) = exp{(p+A)/κ}+1
and that limφ→0 Q̂(p∗ (pm ), pm , φ) =
∗
m
m
D(p). By Lemma A2 it follows that limφ→0 Qi (p (p , φ), p , φ) = 1, so that
lim −
φ→0
Dpi (p∗ (pm , φ), pm , φ)
1
= [1 − D̄(pm )]
∗
m
m
D(p (p , φ), p , φ)
κ
By (31) this implies that:
1
∂Qi (p∗ (pm , φ), pm , φ)
[1 − Qi (p∗ (pm , φ), pm , φ)] = lim
= 0.
φ→0 φ
φ→0
∂φ
− lim
Then:
½
¾
dQi 1
dQi
1
− Q̂(1 − Q̂)
− Q̂Qi (1 − Q̂) ln (nQi )
lim Dφ (p (p , φ), p , φ) = lim Q̂
φ→0
φ→0
dφ
dφ Qi
κ
∗
m
m
1 − limφ→0 Q̂
1
ln (n) D̄(pm ) = − [1 − D̄(pm )] ln(n)D̄(pm )
κ
κ
1
= −ε(pm ) m ln (n) D(pm ),
p
= −
where the first equality in the second line comes from limφ→0
limφ→0 Qi (p∗ (pm , φ), pm , φ) = 1.
Lemma B5: For the nested logit model, limφ→0
38
dpN (φ,n)
dφ
=
∂Qi (p∗ (pm ,φ),pm ,φ)
∂φ
n
n−1
= 0 and the fact that
Proof: From (31):
dpN (φ, n)
φ→0
dφ
lim
pN (φ, n) − c
φ→∞
φ
1
n
i=
= lim h
1
1
1
n−1
φ→∞ φ
n−1
φ n + κ (1 − Q̂) n
=
lim
QED.
Result Example 2:
The precise expression for limφ→0 ξ φ given in the example directly follows from Lemmas B1, B4,
and B5. To derive the numerical cutoffs note that:
D(pm )
=
D(c)
exp
½
exp
© c+A ª
κ
1
1−Q̂m ( c+A
)
κ
+1
+
c+A
κ
¾
(33)
+1
By our arguments before, this expression converges to 1 when c+A
κ converges to −∞. Hence, if ln(n) > 1,
D(pm )
then, by continuity, there must exist A > −∞ such that D(c) ln(n) > 1. Clearly ln(2) < 1. However,
ln(3) = 1. 098 6 > 1. Hence, for all n ≥ 3, there exists A(n) such that δ(φ, n) is decreasing at φ = 0 for
all A ∈ (−∞, A(n)). The remainder of the result is implied by (33) increasing in c+A
κ , which is easily
c+A
m
confirmed knowing that Q̂ is decreasing in κ .
Example 3: A normalized Hotelling Duopoly
In the standard Hotelling model with quadratic transport costs, the monopoly price changes as
the product differentiation parameter changes. We here describe a simple modification of a Hotelling
duopoly model for which all the assumptions of our model hold. There are two firms each located at
the end of a line of length 1. A buyer x is located a distance x away from the left end of the line. The
utility from buying good 1 at the left end of the line is v + φ4 − φx2 − p1 . The utility of buying from
firm 2 at the right end of the line is v + φ4 − φ(1 − x)2 − p2 . Note that our only modification from the
standard Hotelling model is that the reservation price changes in φ. We assume that φ < (v −c)/3. This
condition is sufficient for a monopolist with two outlets to cover the market in equilibrium. Clearly, if
the market is covered, the monopoly price is pm = v, and monopoly profit (v − c) 12 . The elasticity of
demand for a given variety at equal prices is φp . Hence, the Nash equilibrium between two competing
firms is pN = φ + c, which converges to marginal cost as φ → 0. Nash equilibrium profits are given by
φ 12 .
The condition φ < (v − c)/3 also guarantees that the optimal deviation from a common price pm = v
is to the highest price at which the deviating firm just obtains the full market, i.e. p∗ (v) = v − φ. Hence,
∗ m
m
deviation profits are given by (v − φ − c). It follows that dΠ(p (pdφ),p ,φ) = −1 < 0, a property that we
have generally assumed in the paper.
39
Result Example 3: From the exposition of the Hotelling model in Appendix A it follows that
dΠ(pN , pN , φ)
dΠ(p∗ (pm , φ), pm , φ)
+ δ(φ)
φ→0
dφ
dφ
1
1
= lim − (1 − δ(φ)) + δ(φ) = − < 0
φ→0
2
4
lim (1 − δ(φ))
where the first equality comes from the profit expressions derived in Appendix A and the last inequality
derives from the fact that limφ→0 δ(φ) = 12 . It follows that a small amount of product differentiation
strictly reduces the critical discount factor at which collusion at the monopoly price is feasible.
40
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42
Critical Delta
Kappa = 0.50, SDS, Symmetric Nash Reversion
0.6
0.5
Delta
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
0.2
0.25
Phi
Figure 1:
43
0.3
0.35
0.4
0.45
0.5
Optimal Collusive Price (Delta = 0.35)
(SDS Preferences, Kappa=0.5, N=2, Monopoly Price = 20)
25
20
Nash Reversion, 5% Share
Nash Reversion, 10% Share
OptSymmPun, 5% Share
OptSymmPun, 10% Share
One-Shot, 5% Share
One-Shot, 10% Share
Price
15
10
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Product Differentiation (Phi)
Figure 2:
44
0.8
0.9
1