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Industry structure and collusion with uniform yardstick competition: Theory and experiments
Accepted Manuscript
Industry Structure and Collusion with Uniform Yardstick Competition: Theory and Experiments Peter T. Dijkstra, Marco A. Haan, Machiel Mulder PII: DOI: Reference:
S0167-7187(16)30273-9 10.1016/j.ijindorg.2016.10.001 INDOR 2325
To appear in:
International Journal of Industrial Organization
Received date: Revised date: Accepted date:
19 December 2014 15 September 2016 3 October 2016
Please cite this article as: Peter T. Dijkstra, Marco A. Haan, Machiel Mulder, Industry Structure and Collusion with Uniform Yardstick Competition: Theory and Experiments, International Journal of Industrial Organization (2016), doi: 10.1016/j.ijindorg.2016.10.001
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Highlights • We study cartel stability in an industry subject to uniform yardstick regulation. • Theory shows that the number of symmetric firms does not affect collusion.
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• In an experiment we find that increasing the number of firms might foster collusion. • Increasing heterogeneity increases the competitive price but makes collusion harder.
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• Our experiment suggests that increasing heterogene-
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ity reduces prices.
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Industry Structure and Collusion with Uniform Yardstick Competition: Theory and Experiments✩ Peter T. Dijkstraa,∗, Marco A. Haana , Machiel Muldera,b,1 Department of Economics, Econometrics and Finance, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands. b Netherlands Authority for Consumers & Markets (ACM), PO Box 16326, 2500 BH The Hague, The Netherlands.
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a
Abstract
For an industry that is subject to uniform yardstick regulation, we study cartel stability and the impact of cartels on the regulated price. In a theoretical model, an increase in the number of symmetric firms may facilitate collusion. Our laboratory experiment suggests that this effect is even stronger than what theory predicts. Theory predicts that firm-size
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heterogeneity hinders collusion, but leads to higher regulated prices if firms do not collude. In a laboratory experiment we find that the first effect is stronger, implying that in a more heterogeneous industry regulated prices are lower. Keywords: Collusion, Industry Structure, Yardstick Competition, Experiment JEL: C73, C92, L13, L51
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1. Introduction
It is widely accepted that in unregulated markets, an
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increase in the number of firms hinders collusion: The
higher the number of firms, the higher the potential benefits of deviating from a collusive agreement (see e.g. Motta, 2004). The same holds for firm heterogeneity: the more firms differ, for example in terms of market share, the
We thank co-editor Martin Peitz, two anonymous referees, Jose
harder it is to collude. Smaller firms have stronger incen-
Luis Moraga-Gonz´alez and Bert Schoonbeek for helpful comments.
tives to deviate from collusive agreements (see e.g. Ivaldi
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✩
We are also indebted to participants of CRESSE 2014 (Corfu) and seminar participants at the University of Groningen (RUG) and the
et al., 2003). Experimental studies, such as Abbink and Brandts (2005) or Fonseca and Normann (2008), confirm
cial support of the Netherlands Authority for Consumers & Markets
these insights.
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Netherlands Authority for Consumers & Markets (ACM). Finan-
Yet, much less is known about the incentives to col-
for the contents of the paper. ∗ Corresponding author. Present address: Netherlands Authority for
lude in an industry that is subject to yardstick competition.
Consumers & Markets (ACM), PO Box 16326, 2500 BH The Hague,
With yardstick competition, prices for regional monopo-
The Netherlands. Email addresses: [email protected] (Peter T. Dijkstra),
lies are set by a regulator on the basis of actual costs of
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(ACM) is gratefully acknowledged. The authors are fully responsible
[email protected] (Marco A. Haan), [email protected] (Machiel Mulder) 1 Present address: Department of Economics, Econometrics and Finance, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands. Preprint submitted to International Journal of Industrial Organization
similar firms. This gives such firms an incentive to collude to keep their costs high (Tangerås, 2002). Many industries are subject to yardstick regulation. For example, in the US, the payment a hospital receives for a treatment under October 6, 2016
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Medicare, depends on the average cost of that treatment
do so in a theoretical framework. A theoretical model is
in other hospitals (Shleifer, 1985). But yardstick compe-
an important tool in helping to understand the fundamen-
tition is also used in industries as diverse as water supply
tal mechanisms that are at play, and the trade-offs that are
and sewerage2 , electricity networks3 , railways4 and bus
involved. However, repeated game models of the type we
transport services5 , to name but a few.
study do suffer from a multiplicity of equilibria, and remain silent on how and to what extent players in the real
petition are subject to consolidation. Regulators and an-
world may be able to coordinate on one of those equi-
titrust authorities then face an extra challenge: to decide
libria. Therefore, we also test our model in a laboratory
whether to allow mergers. To be able to do so, it is es-
experiment.
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More often than not, industries with yardstick com-
Ever since Shleifer (1985) it is well known that yard-
structure affects the incentives to collude. Insights from
stick competition is most effective with a discriminatory
unregulated markets do not necessarily carry over to these
yardstick. The price that one firm can charge then de-
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sential to have a thorough understanding of how industry
environments. This paper aims to contribute to that under-
pends on the cost levels of all other firms. In our analysis,
standing.
however, we focus on a uniform yardstick, where the price that one firm can charge depends on the weighted average
ple, the number of firms was about 350 in 1950. Due to
cost levels of all firms involved. Despite the superior theo-
mergers and acquisitions, it had decreased to 25 in 1998
retical properties of a discriminatory yardstick, a uniform
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In the Dutch energy-distribution industry, for exam-
(Arentsen and K¨unneke, 2003). In 2014, there were only
yardstick is more often used in practice.6 We use a simple model, loosely based on Shleifer (1985),
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8 companies left. The industry now consists of a few
where firms have to exert costly effort to lower their costs.7
large players, and some small ones (see Haffner et al., 2010). Over the years, firm-size heterogeneity has in-
6
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creased. These changes influence the effectiveness of yard-
One example being the Dutch energy-distribution industry dis-
cussed above (Haffner et al., 2010). 7 In Shleifer (1985), firms invest R(c) in a cost-reducing technol-
collude. Understanding the relationship between industry
ogy that lowers marginal cost by c. With downward sloping demand,
structure and collusion is thus essential for understanding
the monopolist cannot fully capture the social welfare generated by
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stick competition if they affect the incentives for firms to
lower costs. Hence it will underinvest from a welfare perspective.
the effectiveness of regulation through yardstick competi-
This is no longer true, however, if demand is inelastic. In that case,
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tion.
the monopolist does capture the entire surplus and hence there is no
In this paper, we study to what extent the number of
underinvestment. For experimental simplicity, we do assume inelastic
firms and firm-size heterogeneity in an industry subject to
demand. We therefore use a slightly different set-up. We effectively assume that the manager of the firm has to exert some costly effort to
uniform yardstick competition affects collusion. We first 2 3
lower costs. The discrepancy between the private and social optimum is then caused by the fact that the manager’s costly effort is not taken
Dassler, Parker and Saal (2006). Haffner, Delmer and Van Til (2010); Bl´azques-G´omez and
into account when maximizing social welfare. This allows us to gen-
Grifell-Tatj´e (2011); Jamasb and Pollitt (2007). 4 Mizutani, Kozumi and Matsushima (2009). 5 Dalen and Gomez-Lobo (2002).
erate the same theoretical prediction (that firms underinvest), but in a simpler set-up. Note that our interpretation closely follows Potters et
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3 symmetric firms are more collusive than industries with
marginal costs. Managers aim to maximize the sum of
2 such firms. We find some evidence that mergers that
profits and managerial benefits, and do so in a repeated
lead to symmetric firms indeed facilitate collusion. Rel-
game. Surprisingly, we find that with symmetric firms
ative to what theory predicts, we find that an increase in
the effect of the number of firms on cartel stability is am-
the number of firms makes it easier for experimental sub-
biguous. Hence, having more firms does not necessarily
jects to collude if they are symmetric, but harder if they
make a cartel harder to sustain, and may even make it eas-
are asymmetric. The theoretical effects of an increase in
ier. Moreover, we find that firm-size heterogeneity hinders
heterogeneity are ambiguous. On the one hand, such an
collusion.
increase raises prices if firms do not collude; on the other
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Managerial benefits are a concave function of a firm’s
hand, it makes collusion harder. Our experiment suggests
able to do so, we have to impose some additional struc-
that the net effect for consumers is positive. Summing
ture. In particular, we assume that managerial benefits are
up, we find that a heterogeneous industry structure is of
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We also test our model in a lab experiment. To be
quadratic in a firm’s marginal costs. The theoretical model
key importance for having relatively little collusion under
then predicts that the number of symmetric firms has no
yardstick competition. The remainder of this paper is organized as follows.
ambiguous. Mergers that lead to symmetric market shares
Section 2.1 provides more background on yardstick com-
facilitate collusion. Mergers that do not involve the small-
petition, while Section 2.2 discusses other experiments
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effect on cartel stability, while the effect of mergers is still
studying the impact of industry structure on the incidence
the competitive outcome for the smallest firm, making it
of collusion. Section 3 describes our theoretical model.
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est firm in the industry hinder collusion as they improve
Section 4 presents the experimental design, while Sec-
a collusive agreement. If two small firms merge, then such
tion 5 describes the results of our experiment. Section 6
a merger usually facilitates collusion.
concludes.
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more attractive for the manager of that firm to defect from
Our experimental implementation is loosely based on 2. Background
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Potters et al. (2004). To facilitate coordination we allow subjects to communicate in each round, before set-
2.1. Yardstick competition
ting their cost level. We look at five treatments that differ
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The last decades saw the liberalization of several net-
in the number of firms (2 or 3) and the extent of firm-size
work industries that used to be state-owned vertically-integrated
heterogeneity. In each treatment, we study how successful
monopolies. Examples include telecommunications, elec-
experimental subjects are in establishing collusion. To do
tricity, gas, water and sewerage. Parts of these industries
so, we look at three measures of collusion: the incidence
are natural monopolies characterized by subadditivity of
of full collusion, a collusion index, and the resulting price.
costs, which calls for regulatory supervision (Viscusi et
Our experimental findings suggest that industries with
al., 2005). A key component of such supervision is the regulation of tariffs. Historically, regulated tariffs were
al. (2004).
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United Kingdom (Dassler, Parker and Saal, 2006), elec-
However, such schemes give little incentive to increase
tricity networks in the Netherlands (Haffner, Delmer and
productive efficiency as lower costs directly translate to
Van Til, 2010), Spain (Bl´azques-G´omez and Grifell-Tatj´e,
lower revenues. Hence, the incentive power is low, as a
2011) and the United Kingdom (Jamasb and Pollitt, 2007),
change in costs hardly affect firms’ profits. Introducing
railways in Japan (Mizutani, Kozumi and Matsushima,
price-cap regulation solves this problem as any cost de-
2009) and bus transport services in Norway (Dalen and
crease then fully benefits the firm. But this method also
Gomez-Lobo, 2002). In several cases, the introduction
implies the risk of significant rents or losses for regulated
of yardstick regulation improved productive efficiency or
firms. To overcome both problems, Shleifer (1985) pro-
resulted in lower consumer prices.8 In other cases, the
posed to impose tariffs based on the actual costs of a group
experience appeared to be less successful.9
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either based on actual costs or an allowed rate of return.
What these experiences have taught us is that a num-
tion is called yardstick regulation. As tariffs are based
ber of conditions have to be met before yardstick regula-
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of similar (benchmark) firms. This type of tariff regula-
on the relative performance of a firm, yardstick regulation
tion can be effective. First, we need a sufficiently large
is generally seen as a powerful tool both for giving incen-
number of benchmark firms, using similar techniques op-
tives for productive efficiency as well as for rent extraction
erating within a similar environment. Second, revenues
(Tangerås, 2002; Burns, Jenkins, and Riechmann, 2005).
of regulated firms should be fully based on the yardstick. No other concerns should be taken into account, such as
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The incentive power of a yardstick partly depends on
the impact yardstick regulation may have on the risk of
determine tariffs. Essentially, there are two types of yard-
bankruptcy. Finally, firms should operate independently
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the exact manner in which costs of the benchmark firms
sticks. With a uniform yardstick every firm faces the same 8
cap on the tariffs it may charge, based on cost information
Mizutani et al. (2009), for instance, find that the efficiency of
Japanese railway companies, measured by the variable costs per unit of
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of all firms in the benchmark group. With a discrimina-
output, improved significantly after the introduction of yardstick regulation. Jamasb et al. (2004) report that the British model of incentive
costs of all other firms in the benchmark group. The latter
regulation in the electricity distribution industry has brought signif-
scheme gives a stronger incentive to improve efficiency.
icant price reductions to consumers. Haffner et al. (2010) find that
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tory yardstick, every firm faces a specific cap based on the
yardstick regulation of the Dutch electricity and gas networks resulted
For both schemes, numerous methods exist to determine
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in lower network tariffs without adversely affecting network invest-
the yardstick, including the (weighted) average costs, the
ments. 9 Dalen et al. (2002) for instance do not find an effect on the cost ef-
median costs, the 25th percentile of costs or just the best
ficiency of Norwegian bus companies, for which the authors blame the
practice of all benchmark firms (Yatchew, 2001). In the
bargaining power of the regulated firms to reduce the incentive power
Dutch regulation of electricity distribution networks, for
of the regulatory scheme. For the incentive regulation in the Spanish
instance, the yardstick is based on weighted average costs.
electricity-distribution industry, negative effects on consumer welfare
Experiences with yardstick regulation exist in several
were found by Bl´azques-G´omez and Grifell-Tatj´e (2011), which resulted from the fact that inefficient firms received financial compensa-
industries, including water supply and sewerage in the
tions afterwards in order to prevent bankruptcies.
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from each other. Any cooperation would reduce the effec-
with the number of firms when costs are private infor-
tiveness of yardstick competition (Jamasb, Nillesen and
mation. Abbink and Brandts (2008) find the same result
Pollitt, 2004).
with increasing marginal costs, and Fonseca and Normann (2008) with capacity constraints. Fonseca and Normann
with yardstick regulation under different industry struc-
(2012) also find more collusion with fewer firms. More-
tures. Tangerås (2002) observes that yardstick competi-
over, they find that the ability to communicate facilitates
tion is near useless if firms are able to collude. In his
collusion, the effect being strongest for industries with an
model, collusion takes place through joint manipulation
intermediate number of firms.
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In this paper we therefore assess the risk of collusion
Other experiments focus on asymmetries between firms.
of productivity reports that the regulator uses to determine
With quantity setting, Mason et al. (1992) find more col-
exerting effort to lower actual costs. In Tangerås (2002),
lusion if firms have equal rather than different marginal
collusion becomes less likely if the number of firms in-
costs. Phillips et al. (2011) confirm this result. With price
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tariffs. In our paper, firms that collude jointly refrain from
creases, which is in line with the literature on collusion in
competition, Fonseca and Normann (2008) find more col-
unregulated markets.
lusion if firms have identical capacity constraints, and Dugar and Mitra (2009) find more collusion if the range of pos-
depends on the heterogeneity of firms. The more firms
sible firm-specific marginal costs is smaller. Dugar and
differ in terms of market shares, the harder it is to reach
Mitra (2013) confirm the latter result in a slightly differ-
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In unregulated markets, the feasibility of collusion also
ent context. Argenton and M¨uller (2012) however, find
this is found for the US airline industry, where the extent
that collusion is unaffected if firms have different rather
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an agreement (see e.g. Motta, 2004). In empirical work
than identical cost structures. Still, overwhelmingly, ex-
size heterogeneity (Barla, 2000). In our paper, we study
periments on unregulated markets find that asymmetries
whether the same holds for industries with yardstick com-
do hinder collusion.
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of competition appears to be positively related to firm-
To the best of our knowledge, the only economic experiment that deals with yardstick competition is Potters
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petition.
2.2. Experiments on collusion
et al. (2004), on which our experiment is loosely based.
A number of economic experiments have analyzed the
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Potters et al. (2004) study the effect of yardstick design on
impact of industry structure on collusion in unregulated
collusion in a duopoly with symmetric firms. They com-
markets. Most studies focus on the effect of the number
pare a uniform yardstick to a discriminatory one. When
of firms. Huck et al. (2004) study homogeneous-product
firms behave non-cooperatively, the discriminatory yard-
quantity-setting oligopolies. They find some collusion in
stick yields lower prices and lower profits, making it more
markets with 2 firms, little collusion in markets with 3
prone to collusion. In their experiment, the authors indeed
firms, and no collusion in markets with 4 or 5 firms. Other
find higher cost levels with a discriminatory yardstick. Of studies look at homogeneous-product price-setting oligopolies.course, our research question is very different from that Abbink and Brandts (2005) find that collusion decreases 6
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addressed in Potters et al. (2004). Also, the experiment in
r(ci ). The firm’s problem is then to find the optimal mix of
that paper includes a stage in which firms set prices, while
marginal production cost and fixed cost, i.e. the efficient
we simply impose prices to equal the price cap imposed
scale. Our assumption implies that the optimal mix does
by the regulator. This simplifies the experiment. Anyhow,
not depend on the size of the firm, and hence that there are
Potters et al. (2004) find that subjects choose to set prices
constant returns to scale in the relevant interval.10 Hence,
equal to the price cap in about 99% of the cases.
when two firms merge, the optimal mix is unaffected. This allows us to isolate the effect of mergers on market perfor-
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3. The model
mance, without the confounding effect of already having lower average costs due to economies of scale. Our re-
3.1. Setup
sults should also be interpreted as such. In addition, our
There are n firms that play an infinitely repeated game.
assumption is consistent with the available evidence: ac-
Each firm acts as a local monopolist. For simplicity, we
cording to the empirical literature, economies of scale in
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assume that firm i faces demand that is completely inelas-
electricity distribution are already exhausted with some
tic and exogenously given by mass αi ∈ (0, 1). We norP malize total demand, so ni=1 αi = 1 and αi reflects firm i’s
20,000 customers, far below the size that is relevant on e.g. the Dutch market (see Yatchew, 2000, and the refer-
share of total demand. For ease of exposition, we will re-
ences therein).
fer to αi as the market share of firm i. In each round, firm
We assume that there is a unique cm that maximizes
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i decides on its constant marginal cost ci for that round. Firm i’s profits then equal πi (ci , c−i ) = (p − ci )αi , where p
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is determined by regulation. The vector c−i consists of the
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ated with a lack of maintenance. Such networks are prone to outages, compensation claims and political pressure, all
We assume that the manager of firm i maximizes
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ui (ci ) = πi (ci , c−i ) + r(ci )αi ,
be decreasing in ci for large enough ci . In electricity networks for example, high marginal costs are often associ-
cost levels chosen by other firms; these affect πi through the regulated price p.
managerial benefit, so ri0 (cm ) = 0. We thus allow r to
factors that do not exactly contribute to a quiet life for the manager.
(1)
The regulator uses a uniform weighted yardstick, where
with r a managerial benefit that is concave: r00 < 0 and
the price a firm is allowed to charge equals the weighted
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r(0) = 0. This specification is equivalent to Shleifer (1985),
average of all cost levels in the industry: n X p(ci , c−i ) = α jc j.
where managers invest in a cost-reducing technology. Crucially, we assume that total managerial benefit r(ci )αi is
(2)
j=1
proportional to market share. With equal marginal costs,
This implies that all firms get to charge the same price.
the total amount of managerial benefits in an industry is Total utility to manager i is thus given by P n then independent of the size distribution of firms: αi r(c) = X α + r(c )α . u (c , c ) = α c − c P i i −i j j i i i i r(c) αi = r(c). In a set-up similar to Laffont and Tij=1 role (1993), for example, a firm would earn αi (p − ci ) +
10
7
We thank an anonymous referee for pointing this out.
(3)
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If consumers value the product at v, total welfare equals X X W = (v − p) + p − αi ci + r(ci )αi (5)
3.2. Firm-size heterogeneity We are interested in how the extent of firm-size heterogeneity affects competition and collusion. We thus need
i
a measure of firm-size heterogeneity. For a fixed num-
where the first term is consumer surplus, and the other two
ber of firms, the most natural measure to look at is the P Herfindahl index H ≡ nj=1 α2j . With n equally-sized firms,
terms reflect total managerial utility. Maximizing with re-
firms differ more in size. To derive our analytical re-
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i
spect to ci a socially optimal cost level cW for all firms that
this equals 1/n, but it is easy to see that H increases as
is implicitly defined by r0 cW = 1.
sults, we will interpret an increase in firm heterogeneity
(6)
as the transfer of market share from a (weakly) smaller to
From (4), the social optimum is reached if all firms are
a (weakly) larger firm. Such an increase would necessar-
vanishingly small, so αi = 0 ∀i.
ily increase the Herfindahl index.
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In terms of comparative statics, we can now establish Theorem 1. In the competitive outcome, we have the fol-
3.3. Competition
lowing:
Consider the case where managers unilaterally maxi-
mize their utility. For ease of exposition, we will refer to
(a) An increase in firm-size heterogeneity implies an in-
ing (3) with respect to ci yields
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r0 (c∗i ) = 1 − αi .
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this as the competitive outcome. For given c−i , maximiz-
(4)
crease in the regulated price, provided that r0 is not too concave, i.e. provided r000 > 2r00 /α for all α’s that occur in the two situations we compare.
(b) With symmetric firms, an increase in the number of
costs level of other firms. Also note that c∗i is increasing
firms implies a decrease in the regulated price.
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This is a dominant strategy, as it does not depend on the in market share αi :11 the higher αi , the stronger the in-
(c) A merger leads to higher prices.
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fluence this firm has on the regulated price, and the more Proof. In Appendix.
attractive it is to choose a higher cost level. Finally, note
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that c∗i is strictly lower than the cost level that maximizes managerial
benefits12 .
Hence, if firms choose to compete, having more equally-
Lower costs increase profits, giv-
sized firms leads to lower prices, but a merger leads to
ing managers an incentive to be more efficient. 11
higher prices. Under an additional weak technical condition, we also have that a more heterogeneous market
A higher αi implies a lower equilibrium value of r0 (c∗i ), concavity
structure leads to higher prices. That technical condition
of r then implies a higher c∗i . 12 With a similar argument as in the previous footnote; maximizing 0
is satisfied for all functional forms that we consider in this
∗
managerial benefits requires setting r (c ) = 0, and is thus equivalent to having αi = 1 in (4).
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paper.13 For simplicity, we assume that it is always satis-
long-term losses due to a cartel breakdown. This implies
fied.
that we require
Note that mergers always increase prices. Hence, there
δ > δˆ i (αi ) ≡
is no merger paradox in our model; provided that firms
uiD − uiK uiD − u∗i
.
(8)
compete, a merger always makes their managers unam-
For collusion to be sustainable, we need that this condition
biguously better off. Admittedly, this may no longer be
is satisfied for all managers: o n δ > δˆ ≡ max δˆ i (α1 ) , . . . , δˆ i (αn ) .
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true if due to a merger, a cartel would no longer be stable. Still, with some potential abuse of terminology, in the re-
(9)
mainder of this paper we will talk about “the effects of a
We will focus on the cartel agreement in which all
merger” whenever we exogenously impose that two firms
managers set the same cost level ck that maximizes their
join forces.
total utility. Arguably, this is the most obvious and focal agreement. We will refer to it as the perfect symmetric
3.4. Collusion
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collusive agreement. Of course, if this would not yield a stable cartel, managers could still try to coordinate on a
ger strategies and look for the critical discount factor δˆ
different agreement, either symmetric or asymmetric. In
such that all managers have an incentive to stick to the car-
our theoretical analysis we rule out this option as we feel
tel agreement. With the usual arguments manager i does
that it would be much harder to coordinate on such an
not defect if and only if
alternative.14 Of course, in the experimental implementa-
(7)
tion of our model, firms are free to try to coordinate on whatever they can agree upon.
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uiK δu∗i > uiD + , 1−δ 1−δ
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Following e.g. Friedman (1971), we assume grim trig-
If firms coordinate on a common cost level ck , we im-
her utility in a cartel, and u∗i her utility in the competi-
mediately have p = ck , so all profits are zero and the car-
tive outcome. In other words, cartel stability requires that
tel simply maximizes joint managerial benefit r(c), which
the short-run benefits of defection are outweighed by the
implies ck = cm .
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In particular, in Corollary 1, we use the following functional
forms:
We now have:
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13
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with uiD the utility of manager i when she defects, uiK
(a) Suppose r(c) =
√ c − bc. We then have r0 (c) = 21 c−1/2 − b, so
Lemma 1. The perfect symmetric collusive agreement yields
r00 (c) = − 14 c−3/2 < 0 and r000 (c) = 38 c−5/2 > 0 > 2r00 /α, so the
a stable cartel if, for all i = 1, . . . , n, (1 − αi ) cm − c∗i + r(c∗i ) − r (cm ) ˆ i) ≡ P δ > δ(α . (1 − αi ) cm − nj,i α j c∗j
condition is satisfied.
(b) Suppose r(c) = bc − ac2 .In that case r000 (c) = 0 > 2r00 /α, so again the condition is satisfied.
(c) Suppose r(c) = bc − ec . In that case r00 = −ec and r000 = −ec , so
14
r000 > 2r00 /α if 1 < 2/α, which is true for any α ∈ (0, 1) .
(11)
Indeed, in our experiment we also observe that groups, if anything,
coordinate on the perfect symmetric collusive agreement, irrespective of the distribution of market shares.
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Proof. For a cartel to be stable, we need that (8) is satis-
This can be understood as follows. Defection util-
fied for all i. To evaluate the values for u∗i , uiK , and uiD , we
ity, cartel utility and competition utility all decrease in the
proceed as follows. First, using (3), we have n X uiK = α j cm − cm + r(cm ) αi = r cm αi .
number of firms.16 That implies from (8) that the net effect on cartel stability is ambiguous. In particular, it de(12)
pends on the curvature of r. Indeed, we can find families
j=1
of managerial benefit functions r such that an increase in
The optimal defection is simply to use your dominant strat-
Moreover
affect cartel stability:
= r c∗i − (1 − αi ) c∗i − cm αi .
(13)
n X u∗i = ui (c∗i , c∗−i ) = α j c∗j − c∗i + r(c∗i ) αi . j=1
Hence
r(c∗i )
cm
c∗i
r (cm )
Corollary 1.
(a) If r(c) =
√ c − bc, with b > 0, then a cartel is less
stable if the number of symmetric firms increases.
(14)
AN US
uiD
CR IP T
the number of firms either increases, decreases or does not
egy and set c∗i . That yields, again using (3),
(b) If r(c) = bc − ac2 , with a, b > 0, then cartel stability is unaffected by the number of symmetric firms.
3.5. Factors that facilitate collusion
M
− (1 − αi ) − − , P δˆ (αi ) = (c) If r(c) = bc − ec , with b > 2, then a cartel is more r c∗i − (1 − αi ) c∗i − cm − nj,i α j c∗j − c∗i + r c∗i stable if the number of symmetric firms increases. which simplifies to (11). Proof. In Appendix. To analyze how exogenous factors affect the stability
ED
We now study which factors facilitate or hinder collu-
of collusion with heterogeneous firms, we need to know
sion. Throughout the analysis, we assume that firm size
how these affect the manager with the highest critical dis-
PT
heterogeneity increases competitive prices, i.e. that the
count factor. We can establish:
condition in Theorem 1(a) is satisfied.15 We can now
16
CE
show
∂uiD ∗ m = r c∗i − (1 − αi ) c∗i − cm + r0 · c∗0 − (1 − αi ) c∗0 i + ci − c i αi ∂αi = r c∗i − (1 − αi ) c∗i − cm + c∗i − cm αi
Theorem 2. With symmetric firms, an increase in the num-
AC
ber of firms facilitates collusion if and only if r(c∗i ) is suf-
as r0 = 1 − αi . With uiD > 0, we have r c∗i − (1 − αi ) c∗i − cm > 0,
ficiently concave, more precisely if 00
r < −r
0
(c∗i )
r0 (c∗i ) (cm − c∗ ) − (r(cm ) − r(c∗ )) · (r(cm ) − r(c∗ )) (cm − c∗ )
hence c∗i > cm implies ∂uiD /αi > 0, which implies that uiD decreases with the number of firms. Moreover, from (12) and (14), ∂uiK = r (cm ) > 0 ∂αi ∂u∗i ∗0 0 ∗0 = p∗ − c + r c∗i + c∗i + αi c∗0 i − ci + r ci αi ∂αi = p∗ − c + r c∗i + c∗i αi > 0.
Proof. In Appendix. 15
For defection utility, note from (14)
Although note that we do not need to make this assumption for all
results that follow.
10
ACCEPTED MANUSCRIPT
Lemma 2. If the perfect symmetric collusive agreement is
Using this theorem, we can derive a number of scenar-
ˆ i ) is the highest stable, then the critical discount factor δ(α
ios in which a merger either always facilitates, or always
for the smallest firm.
hinders collusion:
Proof. In Appendix.
Theorem 4. A merger affects cartel stability in the following manner:
Lemma 2 immediately implies that a cartel is stable if and only if the smallest firm has no incentive to defect. Theorem 3. An increase in firm-size heterogeneity increases the critical discount factor and hence makes a cartel less
CR IP T
(a) If it does not affect the market share of the smallest firm, then a merger hinders collusion. (b) Suppose that we are in an environment where a car-
likely to be stable.
tel becomes weakly less stable if the number of symmetric firms increases. In that case, a merger that
Proof. In Appendix.
AN US
leads to symmetric firms facilitates collusion.
Comparing this result to Theorem 1, we thus have an
Proof. Part (a) is straightforward. From Lemma 2 the
neous industry structure implies a lower price – provided
smallest firm has the highest critical discount factor. If
that firms behave competitively. At the same time, how-
(one of the) smallest firm(s) is not involved in a merger,
ever, it makes collusion more likely.
its market share αi is not affected, so after the merger it is
M
interesting trade-off. On the one hand, a more homoge-
Now consider the effect of a merger. First, it is straight-
ED
forward to establish:
still the smallest firm. However, p∗ will increase, which from (11) implies that δˆ i (αi ) increases and collusion is less stable.
Lemma 3. Denote the pre-merger industry structure as A,
For part (b), note that, before the merger, there are
and the post-merger industry structure as B. The merger
PT
n > 2 firms, that differ in size. As a first step, redistribute
then facilitates collusion if and only if 1 − αB cm − c∗ αB + r(c∗ αB ) − r (cm ) < 1 − αB cm + αB c∗ αB − p∗B 1 − αA cm − c∗ αA + r(c∗ αA ) − r (cm ) , (15) 1 − αA cm + αA c∗ αA − p∗A
market shares between these firms in such a way that they
CE
are now equally sized. From Theorem 3, doing so facilitates collusion. As a second step, consider a change from
AC
n equally sized firms to n − 1 equally sized firms. By assumption, this also facilitates collusion. This establishes
with ακ the market share of the smallest firm in industry
the result.
structure κ, and p∗κ the competitive price in industry struc-
Result (a) is particularly strong: whenever the small-
ture κ, κ ∈ {A, B}.
est firm (or one of the smallest firms) is not part of the
Proof. Follows directly from Lemma 1 and Lemma 2.
merger, then the merger necessarily hinders collusion. The intuition is as follows. As the market share of the smallest firm is unaffected, the merger also has no effect on 11
ACCEPTED MANUSCRIPT
levels ci from the set {1, 2, . . . , 20}.19 Third, prices are de-
manager of that firm. However, it does affect her utility in
termined using (2). After each round, subjects learn their
the competitive outcome. From the proof of Theorem 1
profits and managerial benefits, and the cost levels each
a more concentrated industry implies higher competitive
subject has set.20 Although not very common in a car-
prices, and hence higher competitive utilities. Thus, de-
tel experiment, we feel that the unrestricted communica-
fection from a cartel becomes less costly, making it more
tion we allow for creates circumstances that are closest to
attractive to defect.
the real world. The Dutch regulatory framework for the
CR IP T
either the collusion utility or the defection utility of the
distribution-network operators, for example, does not for-
lead to a more homogeneous industry structure (such as
bid them to mutually exchange information on operational
the scenario described in (b) of the Theorem) facilitate
and strategic issues. The operators together have estab-
collusion, while mergers that lead to a less homogeneous
lished a joint organization (Netbeheer Nederland) which
industry structure (such as the scenario described in sce-
facilitates the cooperation among the operators besides
AN US
Broadly speaking, the results suggest that mergers that
nario (a)) hinder collusion – at least in an environment
representing the group of operators in the political debate
where having more equally sized firms makes it harder to
on energy policy in general and the design of regulation in
collude.17
particular. Without communication, it is also hard to sustain collusion in a cartel experiment with more than two
4. Experiment
M
players, see e.g. Haan et al. (2009).21
4.1. Design
ED
In our experiment, subjects play the game described
We use the following quadratic specification for the managerial benefit function: r(ci ) = bci − ac2i ,
above for at least 20 rounds. From round 20 onwards, the experiment ends with a probability of 20% in each round,
(16)
PT
where a, b ∈ R+ are parameters and ci ∈ [0, b/a]. We
to avoid possible end-game effects (see also Normann and
assume that a and b are such that all expressions we derive
CE
Wallace, 2012). We use fixed matching: every subject plays with the same group members in all rounds.
19
As saw in the theory section, both the competitive and the collu-
sive strategy are independent of the competitors’ actions. In that sense,
communicate using a chat screen. This chat is completely
our game boils down to a relatively straightforward prisoners’ dilemma
anonymous.18 . Second, subjects unilaterally choose cost
game with two strategies. Still, we choose not to represent it as such in
17
AC
Every round consists of three steps. First, subjects can
our experiment, as that would get rid of much of the regulatory context that we feel is crucial for our purpose. 20 Of course, in our set-up profit and managerial benefit are equally
If we are not in such an environment, the effect of a merger to
equality is ambiguous. It is hard to derive general conditions under
important in determining payoffs. This was also outlined in the experi-
which such a merger would or would not facilitate collusion. 18 We allowed subjects to chat for 150 seconds in round 1, for 120
mental instructions. Subjects indeed seemed to understand this as they correctly answered control questions on this topic. 21 Although illegal in practice, note that cartels cannot be prosecuted
seconds in round 3, for 60 seconds in rounds 3-10, and for 45 seconds in all remaining rounds.
or fined in this experiment.
12
ACCEPTED MANUSCRIPT
The 5 treatments allow us to evaluate the effect of the
are well defined. For this set-up, it is easy to drive that c∗i = m
b − 1 + αi ; 2a b . 2a
number of firms and firm-size heterogeneity on market
(17)
performance in terms of cost levels and regulated prices. They also allow us to evaluate the effect of a merger. In
We set a = 1/24 and b = 1. These choices assure that
our evaluation of the experimental results we do not in-
the competitive and collusive cost levels derived above are
terpret the critical discount factors as a strict prediction of
integers. We run 5 treatments that differ in the number of
the outcome of the experiment. Thus, we do not expect
firms and the extent of firm-size heterogeneity. For ease
that there will always be collusion whenever the δ we im-
of exposition, we normalize the total size of the market to
pose is larger than the δˆ we derived. Neither do we expect
12 when naming our treatments, which allows us to rep-
that there will never be collusion whenever the δ we im-
resent market sizes of all firms in all treatments as integer
pose is smaller than the δˆ we derived. Rather, we interpret
numbers.22 The first 3 treatments have 3 firms and are de-
a higher value of δˆ as making a cartel less likely to oc-
AN US
=
CR IP T
(18)
c
noted TrioXYZ, with X, Y and Z the size of firms 1, 2 and
cur. This is in line with other work (see e.g. Bigoni et al.,
3, respectively. The other treatments have 2 firms and are
2012).
From Corollary 1(b) we expect that, with symmetric
denoted DuoXY, with X and Y the sizes of firms 1 and 2.
firms, the number of firms does not affect collusion:
Table 1 provides information for each treatment. The
M
first panel gives the market share for each firm and the re-
sulting Herfindahl index. The second panel provides com-
Hypothesis 1 (Number of Firms). The amount of collu-
sion in Duo66 is the same as that in Trio444.
petitive cost levels and the resulting price. The third panel
ED
From Theorem 3 we expect that more heterogeneity
gives collusive cost levels and the critical discount factor
implies less collusion:
depend on industry structure; in all treatments, the perfect
Hypothesis 2 (Heterogeneity).
PT
δˆ of the cartel.23 From (10), collusive cost levels do not
symmetric collusive cost level equals 12.
CE
22
(a) There is more collusion in Duo66 than in Duo84;
In the actual experiment, we always normalize the size of the
(b) There is more collusion in Trio444 than in Trio633,
smallest firm to 1, which makes it easier to explain the experiment
and more collusion in Trio633 than in Trio642.
ample. 23 Strictly speaking, the critical discount factors we derive are only
From Theorem 4(b) we expect that mergers that lead
AC
to the subjects. See the instructions in the online appendix for an ex-
valid from round 20 onwards. Still, it is easy to see that this does not af-
to symmetric firms, facilitate collusion:
fect our qualitative results. If a cartel is stable from round 20 onwards,
Hypothesis 3 (Merger to Symmetry).
it is definitely stable in earlier rounds as well, since the incentives to defect become smaller as the first 19 rounds are played for sure. If a
(a) There is more collusion in Duo66 than in Trio633;
cartel is not stable from round 20 onwards, then subjects know it will break down in that round and hence, by backward induction, are not
(b) There is more collusion in Duo66 than in Trio642.
willing to form a cartel in an earlier round.
13
ACCEPTED MANUSCRIPT
Table 1: Overview of treatments.
Treatment
Market shares
Competition
Collusion
α2
α3
HHI
c∗1
c∗2
c∗3
p∗
c∗k = pk
δˆ
Trio444
33.3%
33.3%
33.3%
0.333
4
4
4
4.00
12
0.50
Trio633
50.0%
25.0%
25.0%
0.375
6
3
3
4.50
12
0.64
Trio642
50.0%
33.3%
16.7%
0.389
6
4
2
4.67
12
0.71
Duo66
50.0%
50.0%
0.500
6
6
Duo84
66.7%
33.3%
0.556
8
4
CR IP T
α1
6.00
12
0.50
6.67
12
1.00
Market shares give the markest shares that we impose for each experimental firm in that treatment, HHI is the resulting Hirschman Herfindahl Index. c∗i is the competitive cost level of firm i that our theory predicts, p∗ the resulting regulated price, c∗k and p∗k are the collusive cost levels and
AN US
prices predicted by our theory, δˆ the critical discount factor for collusion to be stable.
Finally, we look at mergers that lead to asymmetric firms. At least in this case, these hinder collusion:24
economics and business (59.3%). Every session consisted of one treatment and lasted between 80 and 115 minutes. Subjects signed in for sessions, while treatments were ran-
Hypothesis 4 (Merger to Asymmetry).
domly assigned to sessions. Every treatment with 2 subjects was played in two ses-
(b) There is less collusion in Duo84 than in Trio642.
sions while every treatment with 3 subjects was played in
ED
M
(a) There is less collusion in Duo84 than in Trio444;
three sessions. Between 14 and 18 subjects participated
Part (a) of this Hypothesis follows directly from Theo-
in a session, resulting in 16 to 17 groups per treatment.
rem 4(a). Part (b) does not follow directly from the results
PT
This is similar to other cartel experiments, see e.g. Bigoni
in the previous section, but can be shown to hold in our
et al. (2012), Dijkstra et al. (2014) and Hinloopen and
experimental implementation (see e.g. the last column of
Soetevent (2008). The experiment was programmed in z-
CE
Table 1).
Tree (Fischbacher, 2007). Printed instructions were provided and read aloud.25 On their computer, subjects first
AC
4.2. Implementation
had to answer a number of questions correctly to ensure
The experiment was conducted at the Groningen Ex-
understanding of the experiment. Participants were paid
perimental Economics Laboratory (GrEELab) at the Uni-
their cumulative earnings in euros. Since firm size differed
versity of Groningen in February and March 2013. A total
between treatments, exchange rates were varied such that
of 214 subjects participated, all students from the Uni-
participants would receive identical amounts with full col-
versity of Groningen (80.4%) or the Hanze University of Applied Sciences (19.6%), most of them in the fields of 24
25
Instructions for Trio633 are reproduced in the online appendix.
Instructions for other treatments are similar and available upon request.
Thanks to an anonymous referee for suggesting this hypothesis.
14
ACCEPTED MANUSCRIPT
lusion.26 Furthermore, they received an initial endowment
implies no difference between treatments we do a two-
of e4. Average earnings were e16.80 and ranged from
sided test. We also perform a two-sided test if the theoret-
e7.75 to e24.00.
ical result is ambiguous. This is may for example be the case with prices: when a treatment is expected to yield
5. Results
more collusion but also lower competitive prices, then the net effect on prices is ambiguous, so we do a two-sided
For comparison sake, we only include the first 20 rounds of each group in our analysis, as only these rounds are
tween treatments, we do a one-sided test.
First, we look at the incidence of full collusion. In
played by all groups. In this section, we introduce three
all treatments, the perfect symmetric collusive agreement
measures to evaluate the extent of collusion: the incidence
is for all firms to set a cost level of 12. For ease of expo-
of full collusion; a collusion index; and price. We explain
sition we refer to this as full collusion. The incidence of
AN US
these measures in more detail below. For each measure,
full collusion is thus defined as the percentage of markets
we give its development over time in each treatment, and
where all firms set a cost level of 12. Figure 1 shows how
compare averages between all treatments. In the next sec-
this measure develops over time, in all treatments.30 The
tion, we confront our hypotheses with the results of the experiment.
number of markets with full collusion is substantial. From the Figure, Trio444 seems to be the most collusive, while
M
As is common in experimental studies, we use non-
parametric tests for differences between treatments.27 The
ED
unit of observation is always the average value over the
first 20 rounds for each group.28 We use the Mann-Whitney U Test (MWU) to test for differences between two popu-
the least collusion is found in Trio642. Table 2 gives the average incidence of full collusion for all treatments, and reports on pairwise comparisons between treatments. The entries in the right-hand panel
PT
indicate whether the row treatment has an incidence of
lations, and the Jonckheere-Terpstra Test to test for an or-
collusion that is significantly higher (>) or significantly
dering of three populations.29 Whenever our hypothesis
lower (<) than the column treatment, or whether the dif-
CE
26
test. Whenever our hypothesis implies a difference be-
CR IP T
5.1. Three measures of collusion
ference is not significant (≈). We use this convention
This allows us to focus on the effect of asymmetries in industry
throughout the remainder of this paper. From the Table,
AC
structure without the possible interference of fairness issues. 27 Such tests do not impose a specific distribution on the data and are
we thus have for example that Trio444 leads to signifi-
less sensitive to outliers than parametric tests. Still, we also ran OLS
cantly more collusion than Trio642 and Duo84. The dif-
regression using panel data, which yielded the same qualitative results;
ference with Trio633 and Duo66 is not significant.
see the online appendix. 28 Arguably there may be learning and/or end-game effects present
One drawback of this measure is that it only considers
in the data. Still, our qualitative results do not change if we would
markets to be collusive if market participants succeed in
exclude first or last rounds from the analysis. Details are available from the authors upon request. 29 See Jonckheere (1954) or Terpstra (1952). The Kruskal-Wallis
30
test is equivalent but has an unordered alternative hypothesis.
The development of costs over time for each individual market are
given in the online appendix.
15
ACCEPTED MANUSCRIPT
4
8
12
16
Trio633 Duo84
20
Trio642
PT
ED
Trio444 Duo66
Round
M
0
AN US
0
CR IP T
Incidence of Full Collusion (%) 20 40 60 80
100
Figure 1: Incidence of full collusion (across all groups).
Table 2: Incidence of full collusion (across all rounds and groups).
AC
CE
Treatment
Average
Trio444
71.8%
Trio633
58.1%
Trio642
34.4%
Duo66
56.3%
Duo84
43.4%
Trio633
Trio642
Duo66
Duo84
≈o
>∗∗ o
≈t
>∗o
>+o
≈o
≈o
<∗o
≈o ≈o
Entries in the right-hand panel indicate whether the row treatment yields rates of full collusion that are significantly lower (<), significantly higher (>), or that do not differ significantly (≈) from the rates in the column treatment. Differences between treatments are tested using the MWU test for equality; subscript t denotes a two-sided test, o a one-sided test. + : significant at 10% level; ∗ at 5%; ∗∗ at 1%.
16
ACCEPTED MANUSCRIPT
achieving full collusion. Arguably, any market with prices
5.2. Test of Hypotheses
higher than those in the competitive outcome is collusive
In the previous subsection, we gave an overview of the
to at least some extent. Also, the closer prices are to the
experimental results. We now discuss to what extent those
fully collusive outcome, the more collusive that market
results confirm our hypotheses. Hypothesis 1 is concerned with the number of (sym-
relative premium that firms are able to achieve over and
metric) firms and argues that we expect no difference in
above the competitive outcome:
collusion between both treatments. From Table 5, the col-
collusion index ≡
price − p∗ , 12 − p∗
CR IP T
is. We therefore study the collusion index, which is the
lusion index is higher in Trio444 than in Duo66. The
(19)
other measures are not significantly different between the two treatments. In section 5.3 we further address the ques-
lows for the fact that the competitive price level differs
tion as to why subjects may find it easier to coordinate on
across industry structures (see Table 1). If the competi-
a collusive outcome with 3 rather than 2 players.
AN US
with p∗ the competitive price. Note that this measure al-
tive outcome is achieved, the collusion index equals 0. If
Hypothesis 2 is concerned with the effect of hetero-
the collusive outcome of 12 is achieved, it equals 1. The
geneity. For the case of 2 firms, Table 6 shows that the
higher the index, the more successful firms are in collud-
incidence of full collusion is higher in Duo66, but the av-
ing.
erage value of the collusion index is lower. Average prices
Figure 2 shows how the collusion index develops over
M
are also lower. However, none of these differences is sig-
time, in all treatments. Qualitatively, the picture looks
ED
very similar to that for the incidence of full collusion.
nificant. Thus, Hypothesis 2(a) is not confirmed. Table 7 looks at the results for three firms. Note that Trio444 is listed twice in this table, to facilitate all pair-
collusive treatment. Table 3 gives treatment averages and
wise comparisons. All results reported in Table 7 are con-
pairwise comparisons.
sistent with Hypothesis 2(b); for any measure, the extent
PT
Trio444 seems the most collusive, and Trio642 the least
of collusion is always higher if the industry structure is
that consumers ultimately pay. Despite a slightly higher
more homogeneous. Many of the pairwise comparisons
incidence of collusion, for example, an industry structure
are not significant, but results are highly significant in a
may still be preferable if it leads to lower competitive
Jonckheere-Terpstra test with the ordered alternative hy-
AC
CE
Of course, a regulator is most interested in the price
prices that more than outweigh the adverse effects of the
pothesis that Trio444 yields more collusion than Trio633
occasional cartel. For that reason, we also look at prices.
which in turn yields more collusion than Trio642. Note
Average prices over time are given in Figure 3. Consistent
that we argued in Section 3 that the effect of firm-size het-
with our earlier measures, average prices are high, and
erogeneity on price may be ambiguous, as our model pre-
come closest to the collusive outcome in Trio444. From
dicts that more heterogeneity makes collusion harder, but
Table 4, most pairwise comparisons are no longer signifi-
also raises the competitive price. Our experimental results
cant.
indicate that the effect on collusion is stronger, as prices 17
ACCEPTED MANUSCRIPT
Table 3: Collusion index (across all rounds and groups).
Average
Trio444
85.8%
Trio633
74.6%
Trio642
65.9%
Duo66
73.1%
Duo84
77.9%
Trio633
Trio642
Duo66
Duo84
≈o
>∗∗ o
>+t
>∗o
≈o
≈o
≈o
≈o
<+o ≈o
CR IP T
Treatment
Entries in the right-hand panel indicate whether the row treatment yields a collusion index that is significantly lower (<), significantly higher (>), or that does not differ significantly (≈) from the index in the column treatment. Differences between treatments are tested using the MWU test for
AN US
equality; subscript t denotes a two-sided test, o a one-sided test. + : significant at 10%; ∗ at 5%; ∗∗ at 1%.
M ED PT CE
0
AC
.2
Collusion Index (%) .4 .6 .8
1
Figure 2: Average collusion index per round (across all groups).
0
4
8
Round
Trio444 Duo66
12
Trio633 Duo84
18
16 Trio642
20
ACCEPTED MANUSCRIPT
13
Figure 3: Average price per round (across all groups).
4
8
12
16
Trio633 Duo84
20
Trio642
PT
ED
Trio444 Duo66
Round
M
0
AN US
7
8
9
Price 10
CR IP T
11
12
Collusive Price
Table 4: Price (across all rounds and groups).
AC
CE
Treatment
Average
Trio444
11.0
Trio633
10.0
Trio642
9.5
Duo66
10.3
Duo84
10.8
Trio633
Trio642
Duo66
Duo84
≈t
>∗t
≈o
≈t
≈t
≈t
≈t
≈t
<+t ≈t
Entries in the right-hand panel indicate whether the row treatment yields prices that are significantly lower (<), significantly higher (>), or that do not differ significantly (≈) from the column treatment. Differences between treatments are tested using the MWU test for equality; subscript t denotes a two-sided test, o a one-sided test. + : significant at 10%; ∗ : at 5%.
19
ACCEPTED MANUSCRIPT
Table 5: Comparison of treatments with symmetric firms (across all rounds and groups).
Trio444
Duo66
Incidence of full collusion
71.8%
≈o
56.3%
Collusion index
85.8%
>+o
73.1%
Price
11.0
≈t
10.3
CR IP T
Measure
Entries between values indicate whether the value to the left is significantly lower (<), higher (>), or does not differ significantly (≈) from the value to the right. Differences between treatments are tested using the MWU test for equality; subscript t denotes a two-sided test, o a one-sided
AN US
test. + : significant at the 10% level.
Table 6: Comparison of treatments with 2 firms (averages across all rounds and groups).
Measure
Duo66
Duo84
56.3%
≈o
43.4%
Collusion index
73.1%
≈o
77.9%
10.3
≈t
10.8
Price
M
Incidence of full collusion
ED
Entries between values indicate that the value to the left is sigificantly lower (<), higher (>), or does not differ significantly (≈) from the value to
CE
PT
the right. Differences between treatments are tested using the MWU test for equality; subscript t denotes a two-sided test, o a one-sided test.
Table 7: Comparison of treatments with 3 firms (averages across all rounds and groups).
Measure
Trio444
Trio633
Trio642
Trio444
≈o 58.1%
>+o 34.4% <∗∗ o 71.8%
Collusion index
85.8%
≈o 74.6%
≈o
Price
11.0
≈t 10.0
≈t
AC
Incidence of full collusion 71.8%
65.9% <∗∗ o 85.8% 9.5
<∗t 11.0
JT ∗∗o ∗∗o +t
Entries between values indicate whether the value to the left is significantly lower (<), significantly higher (>), or does not differ significantly (≈) from the value to the right. Differences between two treatments are tested using the MWU test for equality; differences between all treatments using the Jonckheere-Terpstra test (JT) for equality; subscript t denotes a two-sided test, o a one-sided test. + : significant at 10%; ∗ at 5%; 1%.
20
∗∗
at
ACCEPTED MANUSCRIPT
subjects. Coders coded the chat statements independently.
do go down with more heterogeneity. Summing up, in treatments with 2 firms we do not
They were unaware of the research questions addressed in
find evidence for an effect of the extent of firm-size het-
this study. For each group, the full chat in a given round
erogeneity on collusion. With 3 firms, however, we find
could only be assigned to one category. Out of all markets where subjects did chat, we re-
strong support for Hypothesis 2(b): more heterogeneity
port the fraction of markets where the relevant issue is
leads to less collusion.
at least discussed once.32 Table 10 provides these results.
tent to which a merger to symmetric firms facilitates col-
From the table, most groups that chat, discuss cost levels
lusion. We find that Duo66 is indeed more collusive than
at some point. In each treatment, some 90% of groups
Trio642, when measured by the incidence of full collu-
raised the possibility to coordinate on a cost level 12, with
sion. On the other two measures, Duo66 also scores higher,
a slightly lower percentage in Duo84.33 The more het-
but not significantly so. However, the differences between
erogeneous the market shares, the lower the percentage of
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Table 8 evaluates Hypothesis 3 and looks at the ex-
Duo66 and Trio633 are ambiguous and insignificant. Ta-
groups that actually agree to set cost levels of 12. Only
ble 9 evaluates the evidence concerning Hypothesis 4 that
few groups reach an alternative agreement, with the ex-
predicts that a merger to asymmetric firms hinders collu-
ception of Duo84.
Our results in Table 5 suggest that it might be easier
sures, a merger from Trio444 to Duo84 indeed hinders
to coordinate on a collusive outcome with 3 rather than 2
M
sion. Again, the evidence is mixed. On 2 out of 3 mea-
players. Table 11 takes a closer look at what might drive
to Duo84 only seems to facilitate collusion.
this result. The first thing to note is that in markets with 3
ED
collusion. However, if anything, a merger from Trio642
5.3. Chat Analysis
firms, there were only 5 instances in which subjects ex-
To quantify the agreements subjects make during the
PT
32
rather than one full observation. 33 In Duo84 the small firm earns the same with symmetric collusion
chats, we use the methodology of content analysis (see e.g. Cooper and K¨uhn, 2014, and Cason and Mui, forth-
CE
as it does when both firms compete. Yet, there are alternative collusive
coming). We employed two assistants to review the essence conversation31
We classify instances in which coders did not agree as one half
agreements where it earns more. The large firm also has an incentive
and classify them according to a
to make such agreements, as it implies a more stable cartel. The Table
classification scheme. We asked them to judge whether
suggests that this may be an issue, as we indeed see many alternative
AC
of each
agreements in this treatment. There were 49 alternative agreements out
an agreement was reached; whether this agreement was to
of 320 total rounds. In 25 agreements the large firm set a higher cost
“set all cost level 12” or a different agreement; whether no
level than the small firm, in 11 it set a lower cost level. 11 agreements
agreement was reached; or whether no proposal was made
implied setting different cost levels and rotating roles; 1 agreement set-
at all. Furthermore, we asked them to judge whether the
ting a symmetric cost level different from 12; and 1 agreement involved randomization of cost levels. However, these alternative agreements
possibility to all set cost level 12 was raised by any of the 31
were never long-lived, so they are more likely to reflect subjects trying different options to find the optimal solution, rather than that they
We define a conversation as the chat that takes place in a given
made a conscious effort to coordinate on an asymmetric equilibrium.
market in a given round.
21
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Table 8: Merger to symmetric firms (averages across all rounds and groups).
Measure
Trio642
Duo66
Trio633
Incidence of full collusion
34.4%
<∗o
56.3%
≈o
58.1%
Collusion index
65.9%
≈o
73.1%
≈o
74.6%
9.5
≈t
10.3
≈t
10.0
Price
Entries between values indicate whether the value to the left is significantly lower (<), higher (>), or does not differ significantly (≈) from the
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value to the right. Differences between treatments are tested using the MWU test for equality; subscript t denotes a two-sided test, o a one-sided test. ∗ : significant at 5% level.
Table 9: Merger to asymmetric duopoly (averages across all rounds and groups).
Measure
Trio642
Duo84
Trio444
34.4%
≈o
43.4%
<∗o
71.8%
Collusion index
65.9%
<+o
77.9%
<∗o
85.8%
9.5
<+t
10.8
≈t
11.0
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Incidence of full collusion
Price
Entries between values indicate whether the value to the left is significantly lower (<), higher (>), or does not differ significantly (≈) from the value to the right. Differences between treatments are tested using the MWU test for equality; subscript t denotes a two-sided test, subscript o a
M
one-sided test. + : significant at 10% level; ∗ at 5%.
the numbers in Table 11 reasonably well.35 The differ-
In markets with 2 firms, there were 25 such instances.34
ences cannot be explained by tacit agreements: in either
Hence, players in 2-player markets more readily agreed
treatment, there wasn’t a single instance in which all sub-
on alternative arrangements. Apparently, it is easier to
jects set costs equal to 12 without someone explicitly rais-
PT
ED
plicitly agreed on setting cost levels different from 12.
ing that possibility. Summing up, having 3 players makes
players need to agree.
it easier to agree on the focal equilibrium and less tempt-
CE
reach an alternative agreement when only 2 rather than 3
Moreover, groups that did figure out that it is a good
ing to try other scenarios. The most likely explanation is
idea to set cost level 12 took more time to do so in 2-firm
simply that 3 people know more than 2.
AC
markets than 3-firm markets. Arguably, it is more likely 35
that one player will come up with that possibility if we
Suppose for example that in each round each single player pro-
poses with probability 8% to set a common cost level of 12. In each
have 3 players, rather than if we have 2. This can explain
round, the probability that at least one subject makes this suggestion then equals 15.4% in a duopoly, and 22.1% in a triopoly. The ex-
34
In the 3-firm case, these alternative agreements all consisted of
pected number of rounds before this proposal is made is then 6.51 in a
setting different cost levels and rotating roles. In the 2-firm case, there
duopoly and 4.52 in a triopoly. These numbers are similar to those in
were 11 such rotating agreements, 2 agreements to set a symmetric
Table 11. We thank an anonymous referee for pointing this out.
cost level different from 12, 11 asymmetric agreements, and 1 agreement that only involved 1 player.
22
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6. Conclusion
the market structure pre-merger consists of equally sized firms. Otherwise such a merger even facilitates collusion.
In this paper we studied the effect of industry structure
Our findings also suggest that industries with 3 symmetric
on collusion among economic agents subject to uniform
firms are more collusive than industries with 2 such firms.
yardstick competition, an issue particularly relevant for
Summing up, our experimental findings tentatively sug-
the debate on the effectiveness of tariff regulation. We
gest the following. Relative to what theory predicts, an
did so both in a theoretical framework and in a laboratory
increase in the number of firms makes it easier for ex-
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experiment.
perimental subjects to collude if they are symmetric, but
In our theoretical model, we find that firm-size hetero-
harder if they are asymmetric. This may be understood as
geneity hinders collusion. More surprisingly, in a sym-
follows. An increase in their number makes it harder for
metric industry, the effect of the number of firms on cartel
subjects to coordinate. With asymmetry that is bad news,
stability is ambiguous, and an increase in the number of
as the lack of a focal outcome implies that it is now harder
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firms may even facilitate collusion. The theoretical effects
to reach any agreement. With symmetry, however, it is
of mergers on collusion are ambiguous. Mergers that do
good news, as the focal outcome becomes more salient
not involve the smallest firm in the industry hinder col-
and subjects are less tempted to try to explore alternative
lusion, as they improve the competitive outcome for the
agreements. We indeed find that 2-firm markets more of-
smallest firm, making it more attractive for the manager
ten reach alternative agreements, and take more time be-
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of that firm to defect from a collusive agreement. Mergers that lead to symmetric market shares facilitate collusion.
ED
Of course, no theoretical model can always predict
fore they start discussing the collusive outcome. Hence, firm-size heterogeneity in an industry subject to yardstick competition strongly affects collusion in tri-
what will happen in the real world. That is particularly
opoly markets. Our theory suggests that an increase in
true for models of collusion, as these assume that collu-
PT
heterogeneity increases the regulated price if firms do not
sive agreements are always reached instantaneously and
collude, but also makes collusion harder, rendering the net
coordination is not an issue. In the real world, coordi-
CE
effect ambiguous. Our experiment however suggests that
nation problems may be an important obstacle to estab-
the effect of collusion is stronger. Of course, it would be
lishing a cartel. Therefore, we also conducted an exper-
interesting to consider the effect on collusion of markets
AC
iment in which we allow for unrestricted communication
with even more firms (like e.g. Fonseca and Normann,
between agents.
2012).
In our experiment we find that in triopolies, size het-
Our conclusions are relevant for the debate on the op-
erogeneity indeed hinders collusion. We do not find ev-
timal industry structure. In a competitive market, a more
idence for this in duopolies. Also, we find weak evi-
homogeneous industry implies lower tariffs for network
dence that mergers that lead to symmetric market shares
users. When also taking the effects on collusion into ac-
indeed facilitate collusion, as theory predicts. Mergers to
count, our results suggest that more homogeneity may im-
asymmetry hinder collusion, as theory predicts, but only if 23
ACCEPTED MANUSCRIPT
ply higher tariffs. We address the uniform weighted yardstick in our experiment. Potters et al. (2004) compare this yardstick with a discriminatory yardstick in which the price a firm might charge depends on the cost levels of all other firms. They observe a higher tendency to collude with a discrimina-
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tory yardstick than with a uniform yardstick in a duopoly with symmetric firms. They refrained from investigating the effect of different yardsticks on collusion in different industry structures. We will address this question in future work.
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Another interesting extension is to include multiple dimensions of regulation. We considered price regulation,
but several industries also face quality regulation (Tangerås, 2009). This provides an interesting trade-off to firms, since higher quality is associated with higher costs while the in-
M
centive power tells them to reduce costs. In a theoretical model, Tangerås (2009) shows that larger firms need
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stronger incentives to improve quality and to reduce costs than smaller firms. It remains to be seen in which industry structures firms are able to collude on both dimensions
PT
and what effect this will have on social welfare.
CHECK OF HET NIET proofs moet zijn! https://www.elsevier.com/authors/author-
AC
CE
schemas/latex-instructions
24
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Table 10: Chat content.
Treatment
Discussed cost
Suggested
Agreement
Agreement
Other than
Never
all set 12
all set 12
made
all 12
all 12
5.9%
5.9%
100.0%
88.2%
79.4%
85.3%
Trio633
100.0%
96.9%
62.5%
68.8%
25.0%
6.3%
Trio642
97.1%
91.2%
52.9%
70.6%
29.4%
17.6%
Duo66
93.8%
87.5%
78.1%
78.1%
25.0%
0.0%
Duo84
100.0%
71.9%
53.1%
90.6%
62.5%
37.5%
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Trio444
ED
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Entries reflect percentage of groups to which the statement applies at some point during the first 20 rounds of the experiment.
Table 11: Characteristics Trio444 and Duo66.
Duo66
340
320
5
25
Average
3.71
7.06
Median
1
2.5
0
0
PT
Trio444
Total rounds
# rounds with agreement different from 12
AC
CE
Round first suggested to all set 12:
Instances of tacit collusion
“Round first suggested to all set 12” is the first round in which any subject in a market raised the possibility that all players should set their cost level equal to 12. When no subject ever raised that possibility, that observation is set equal to 21. Tacit collusion is defined in this context as all subjects setting cost level 12 without ever having discussed that possibility.
25
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Appendix A. Selected proofs
Note that this is indeed strictly positive for all ak > αi if and only if the term on the first line is always larger than
For part (a), using (4) and (2) the
Proof of Theorem 1.
the term on the second line; in other words, if
regulated price with competition is given by ∗
p =
n X j=1
α j c∗j = 1 − r0 (c∗j ) c∗j .
∂ 1 − r0 − r00 c∗ c∗0 > 0, ∂α
(A.1)
where we have dropped the arguments for ease of exposi-
to say, a large increase can always be written as the sum
tion. With 1 − r0 (c∗ (α)) = α, this simplifies to the require-
of many infinitesimally small increases. As noted, we
ment that
∂ αc∗0 − r00 c∗ c∗0 > 0. ∂α
will interpret an increase in firm size heterogeneity as the transfer of market share from a (weakly) smaller to a (weakly)
Hence we need
larger firm. Suppose that we transfer market share ∆α
c∗0 + αc∗00 −
∂ 00 ∗0 ∗ r c · c − r00 c∗0 c∗0 > 0. ∂α
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from firm i to firm k, with αi ≤ αk . This increases firm
size heterogeneity. The resulting change in regulated price then equals
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We look at an increase in firm heterogeneity. Needless
(A.2)
Note that 1 − r0 (c∗ (α)) = α. Differentiating through twice with respect to α, we obtain, respectively,
− +
ED
−
∗
M
∆ p∗ =
1 − r (c (αk + ∆α )) c∗ (αk + ∆α ) 1 − r0 (c∗ (αk )) c∗ (αk ) 1 − r0 c∗ (αi − ∆α ) c∗ (αi − ∆α ) 1 − r0 c∗ (αi ) c∗ (αi ) , 0
∂ (1 − r0 ) = r00 · c∗0 = −1 ∂α ∂ (r00 · c∗0 ) = r000 · c∗0 + r00 · c∗00 = 0. ∂α
c∗0 + αc∗00 − r00 c∗0 c∗0 > 0.
share α. Evaluating the effect on p∗ of a change in ∆α , we
PT
From (A.3), this simplifies to
∂∆ p∗ = 1 − r0 (c∗ (αk + ∆α )) c∗0 (αk + ∆α ) ∂∆α
CE
αc∗00 + 2c∗0 > 0
− r00 (c∗ (αk + ∆α ))c∗ (αk ) c∗0 (αk + ∆α ) − 1 − r0 c∗ (αi − ∆α ) c∗0 (αi − ∆α ) + r00 c∗ (αi − ∆α ) c∗ (αi ) c∗0 (αi − ∆α )
Note that this is equivalent to the requirement that αc∗ (α)is
AC
strictly convex in α. Using (A.4), we have ∗00
αc
For infinitessimally small ∆α , this implies ∂∆ p∗ = 1 − r0 (c∗ (αk )) c∗0 (αk ) ∂∆α ∆α →0
− r00 (c∗ (αk ))c∗ (αk ) c∗0 (αk ) − 1 − r0 c∗ (αi ) c∗0 (αi ) + r00 c∗ (αi ) c∗ (αi ) c∗0 (αi )
(A.4)
Using (A.4), condition (A.2) simplifies to
with c∗ (α) the equilibrium cost level of a firm with market
get
(A.3)
! −αr000 c∗0 αr000 ∗0 ∗0 + 2c = + 2c = 2 − 00 c . r00 r ∗0
With c∗0 > 0 and r00 < 0, we thus need the condition in (a). For part (b) note that with equal market shares, we have p∗ = c∗ (1/n) . With 1 − r0 (c (1/n)) = 1/n and r concave, this immediately implies the result. 26
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For part (c), denote the price before the merger as p∗ ,
(a) Suppose r(c) =
c∗ =
p∗0 − p∗ = (αi + αk ) c∗ (αi + αk ) − αi c∗ (αi ) − αk c∗ (αk ) .
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Now
r (cm ) − r (c∗ ) δˆ = 1 − 0 ∗ m . r (c ) (c − c∗ )
=1−
Consider this as a function of c∗ . For ease of exposition, r (c∗ ) .
∂δ = ∂c∗
c
r0
+ ∆r r00 ∆c r0 r0 ∆2c
− ∆r
r0
=
Note that
ED
∆c , ∆r > 0. Dropping arguments, we then have r0 ∆
r0 (r0 ∆c
− ∆r ) + r0 r0 ∆2c
PT
The denominator is obviously positive. With
∆r r00 ∆c
r0 (c∗ )
b . 1 + 2b − α
increasing in n. (b) Suppose r(c) = bc − ac2 , with a, b > 0. Note that r0 (c) = b−2ac and r00 (c) = −2a. Hence the function
.
is concave. To find c∗ , equate r0 (c) to 1 − α :
> 0
c∗ =
concavity implies that r0 (r0 ∆c − ∆r ) > 0. However, also due to concavity, we have ∆r r00 ∆c < 0. Hence the numer-
CE
AC
b+α−1 . 2a
To find cm , we simply set α = 1 :
ator is ambiguous. It is positive if and only if r00 <
1 2+b−2α
This critical discount factor is decreasing in α, hence
M
−
1
4b − 4 (1+b−α)2 r (cm ) − r (c∗ ) = 1 − δˆ = 1 − (1 − α) (cm − c∗ ) 1 (1 − α) 4b12 − 4(1+b−α) 2
which we can write as
and ∆r ≡
1 . 4b2
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has all firms setting c∗ ≡ c∗ (1/n) . From (11), we thus
−
.
! 1 1 1 1 r c =r − = = 2 2b 4b 4b 4b 1 2 + b − 2α 1 b = r c∗ = − 2 (1 + b − α) 4 (1 + b − α)2 4 (1 + b − α)2 m
αi = 1/n for all firms, whereas the competitive outcome
r (cm )
4 (1 + b − α)2
We also have
Proof of Theorem 2. First note that with symmetric firms,
r (cm ) − r (c∗ ) δˆ = 1 − , (1 − α) (cm − c∗ )
1
cm =
and αk c∗ (αi + αk ) > αk c∗ (αk ) . Hence p∗0 − p∗ > 0.
denote ∆c ≡
−b
To find cm , we simply set α = 1 :
With c∗ increasing, we have that αi c∗ (αi + αk ) > αi c∗ (αi )
c∗
1√ 2 c
concave. To find c∗ , equate r0 (c) to 1 − α to find
we thus have from (A.1)
cm
c − bc. Note that r0 (c) =
and r00 (c) = − 14 c−3/2 < 0. Hence the function is
the price after the merger as p∗0 . If firms i and k merge,
have
√
cm =
−r0 (r0 ∆c − ∆r ) , ∆r ∆c
b . 2a
We thus have b2 4a b2 − (1 − α)2 r(c∗ ) = . 4a
hence if r is sufficiently concave. In that case δ is increas-
r(cm ) =
ing in c∗ . With c∗ decreasing in the number of firms, it implies that δ is decreasing in the number of firms. Now Proof of Corollary 1.
First note that we look at equally
2
2
2
2a
2a
b −(1−α) b m ∗ 1 4a − ˆδ = 1− r (c ) − r (c ) = 1− 4a = . b b+α−1 (1 − α) (cm − c∗ ) 2 (1 − α) −
sized firms throughout, so for each firm α = 1/n. We consider the three families of functions in turn.
Hence, this is affected neither by α nor by n. 27
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(c) Suppose r(c) = bc−ec , with b > 2. Note that r0 (c) =
Proof of Lemma 2. Note that, from (11), we can write (1 − αi ) cm − c∗i + r(c∗i ) − r (cm ) ˆ i) ≡ δ(α , (A.5) (1 − αi ) cm + αi c∗i − p∗ P with p∗ ≡ j α j c∗j the competitive price. Take any two
b − ec and r00 (c) = −ec < 0. Hence the function is concave. To find c∗ , equate r0 (c) to 1 − α : c∗ = ln (b + α − 1) .
ˆ j ) − δˆ (αk ) , firms, j and k, with α j > αk . Consider δ(α
To find cm , we simply set α = 1 :
the difference between the critical discount factor of firm j and that of firm k. For the lemma to hold, we need this
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cm = ln b.
difference to be strictly negative. Using (11), we can write # Z αj " ˆ (α) ∂ δ 1 ˆ j ) − δˆ (αk ) = δ(α dα. (A.6) 2 αk ∂α
Note that the restriction b > 2 assures that both c∗ and cm are strictly positive. We thus have
Now
r (cm )
r (c∗ )
It is sufficient that ∂δˆ (α) /∂α < 0 for all relevant α. Denote the numerator of (A.5) as num, and the denominator
AN US
r cm = b ln b − b r c∗ = b ln (b + α − 1) − (b + α − 1)
as den. Hence ∂δˆ (α) /∂α < 0 if den · num0 − num · den0 < 0.
− (1 − α) (cm − c∗ ) b ln b − b − (b ln (b + α − 1) − (b + α − 1)) =1− (1 − α) (ln b − ln (b + α − 1)) b b ln b+α−1 + (α − 1) b 1 =1− =1− + b b 1 − α ln b+α−1 (1 − α) ln
If the perfect symmetric collusive agreement is stable, we necessarily have num < den and num, den > 0. Further-
b+α−1
so
ED
M
δˆ = 1 −
PT
∂δˆ b 1 =− + 2 2 ∂α b (1 − α) (b + α − 1) ln b+α−1 b
2
AC
(1 − α)
<
hence
b ln
b b+α−1
0 ∗0 m ∗ num0 = − cm − c∗i − (1 − αi ) c∗0 i + r · ci = − c − ci den0 = − cm − c∗i + αi c∗0 i .
fact that num < den, this necessarily implies that (A.7) is satisfied. This establishes the result.
1 2 b (b + α − 1) ln b+α−1
!2
more, note that
Hence num0 < 0, and den0 > num0 . Combined with the
CE
This is positive if
(A.7)
Proof of Theorem 3.
Consider a move from industry
structure A to industry structure B, where B is more heterogeneous. Without loss of generality, αiB = αiA − ∆ and
(b + α − 1) < (1 − α)2 .
αkB = αkA + ∆, for some ∆ > 0 and αiA ≤ αkA , whereas αAj = αBj for all j , i, k. Denote the smallest market share
Numerically, we can show that this is always the case.
of all other firms as α, so α = min j,i,k {α j }. From Lemma 2
Therefore this critical discount factor is increasing in α,
the smallest firm determines the critical discount factor.
hence decreasing in n.
We have three possibilities to consider: first, αiB > α; second, αiA < α and third αiB < α < αiA . 28
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In case I, αiB > α, the critical discount factor under industry structure κ ∈ {A, B} is given by 1 − α cm − c∗ (α) + r(c∗ (α)) − r (cm ) ˆ δˆ κ ≡ δ(α) , ≡ 1 − α cm + αc∗ (α) − p∗κ (A.8) with p∗κ the competitive equilibrium price in market structure κ. Note that c∗ (α) is not affected by the distribution
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of market shares over other firms. As market structure B is more heterogeneous than market structure A, we have p∗B > p∗A , hence δˆ B > δˆ A . In case II, αiA < α. Hence, the critical discount factor
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under industry structure κ is 1 − ακi cm − c∗ ακi + r(c∗ ακi ) − r (cm ) ˆ κ) = δˆ κ ≡ δ(α . i 1 − ακi cm + αi c∗ ακi − p∗κ (A.9)
From the proof of Lemma 2, for fixed p∗κ , δˆ i (αi ) is decreasing in αi . Hence, as αiB < αiA , for fixed p∗κ we have
M
δˆ B > δˆ A . The fact that p∗B > p∗A only strengthens this result.
ED
The proof for case III, where αiB < α < αiA follows from a combination of the above cases; first consider a
PT
decrease from αiA to α. From the analysis for case I, this ˆ Next consider a decrease from α leads to an increase in δ.
AC
ˆ increase in δ.
CE
to αiB . From the analysis for case II, this again leads to an
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Industry Structure and Collusion with Uniform Yardstick Competition: Theory and Experiments Peter T. Dijkstra, Marco A. Haan, Machiel Mulder PII: DOI: Reference:
S0167-7187(16)30273-9 10.1016/j.ijindorg.2016.10.001 INDOR 2325
To appear in:
International Journal of Industrial Organization
Received date: Revised date: Accepted date:
19 December 2014 15 September 2016 3 October 2016
Please cite this article as: Peter T. Dijkstra, Marco A. Haan, Machiel Mulder, Industry Structure and Collusion with Uniform Yardstick Competition: Theory and Experiments, International Journal of Industrial Organization (2016), doi: 10.1016/j.ijindorg.2016.10.001
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Highlights • We study cartel stability in an industry subject to uniform yardstick regulation. • Theory shows that the number of symmetric firms does not affect collusion.
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• In an experiment we find that increasing the number of firms might foster collusion. • Increasing heterogeneity increases the competitive price but makes collusion harder.
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• Our experiment suggests that increasing heterogene-
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ity reduces prices.
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Industry Structure and Collusion with Uniform Yardstick Competition: Theory and Experiments✩ Peter T. Dijkstraa,∗, Marco A. Haana , Machiel Muldera,b,1 Department of Economics, Econometrics and Finance, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands. b Netherlands Authority for Consumers & Markets (ACM), PO Box 16326, 2500 BH The Hague, The Netherlands.
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a
Abstract
For an industry that is subject to uniform yardstick regulation, we study cartel stability and the impact of cartels on the regulated price. In a theoretical model, an increase in the number of symmetric firms may facilitate collusion. Our laboratory experiment suggests that this effect is even stronger than what theory predicts. Theory predicts that firm-size
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heterogeneity hinders collusion, but leads to higher regulated prices if firms do not collude. In a laboratory experiment we find that the first effect is stronger, implying that in a more heterogeneous industry regulated prices are lower. Keywords: Collusion, Industry Structure, Yardstick Competition, Experiment JEL: C73, C92, L13, L51
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1. Introduction
It is widely accepted that in unregulated markets, an
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increase in the number of firms hinders collusion: The
higher the number of firms, the higher the potential benefits of deviating from a collusive agreement (see e.g. Motta, 2004). The same holds for firm heterogeneity: the more firms differ, for example in terms of market share, the
We thank co-editor Martin Peitz, two anonymous referees, Jose
harder it is to collude. Smaller firms have stronger incen-
Luis Moraga-Gonz´alez and Bert Schoonbeek for helpful comments.
tives to deviate from collusive agreements (see e.g. Ivaldi
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✩
We are also indebted to participants of CRESSE 2014 (Corfu) and seminar participants at the University of Groningen (RUG) and the
et al., 2003). Experimental studies, such as Abbink and Brandts (2005) or Fonseca and Normann (2008), confirm
cial support of the Netherlands Authority for Consumers & Markets
these insights.
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Netherlands Authority for Consumers & Markets (ACM). Finan-
Yet, much less is known about the incentives to col-
for the contents of the paper. ∗ Corresponding author. Present address: Netherlands Authority for
lude in an industry that is subject to yardstick competition.
Consumers & Markets (ACM), PO Box 16326, 2500 BH The Hague,
With yardstick competition, prices for regional monopo-
The Netherlands. Email addresses: [email protected] (Peter T. Dijkstra),
lies are set by a regulator on the basis of actual costs of
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(ACM) is gratefully acknowledged. The authors are fully responsible
[email protected] (Marco A. Haan), [email protected] (Machiel Mulder) 1 Present address: Department of Economics, Econometrics and Finance, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands. Preprint submitted to International Journal of Industrial Organization
similar firms. This gives such firms an incentive to collude to keep their costs high (Tangerås, 2002). Many industries are subject to yardstick regulation. For example, in the US, the payment a hospital receives for a treatment under October 6, 2016
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Medicare, depends on the average cost of that treatment
do so in a theoretical framework. A theoretical model is
in other hospitals (Shleifer, 1985). But yardstick compe-
an important tool in helping to understand the fundamen-
tition is also used in industries as diverse as water supply
tal mechanisms that are at play, and the trade-offs that are
and sewerage2 , electricity networks3 , railways4 and bus
involved. However, repeated game models of the type we
transport services5 , to name but a few.
study do suffer from a multiplicity of equilibria, and remain silent on how and to what extent players in the real
petition are subject to consolidation. Regulators and an-
world may be able to coordinate on one of those equi-
titrust authorities then face an extra challenge: to decide
libria. Therefore, we also test our model in a laboratory
whether to allow mergers. To be able to do so, it is es-
experiment.
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More often than not, industries with yardstick com-
Ever since Shleifer (1985) it is well known that yard-
structure affects the incentives to collude. Insights from
stick competition is most effective with a discriminatory
unregulated markets do not necessarily carry over to these
yardstick. The price that one firm can charge then de-
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sential to have a thorough understanding of how industry
environments. This paper aims to contribute to that under-
pends on the cost levels of all other firms. In our analysis,
standing.
however, we focus on a uniform yardstick, where the price that one firm can charge depends on the weighted average
ple, the number of firms was about 350 in 1950. Due to
cost levels of all firms involved. Despite the superior theo-
mergers and acquisitions, it had decreased to 25 in 1998
retical properties of a discriminatory yardstick, a uniform
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In the Dutch energy-distribution industry, for exam-
(Arentsen and K¨unneke, 2003). In 2014, there were only
yardstick is more often used in practice.6 We use a simple model, loosely based on Shleifer (1985),
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8 companies left. The industry now consists of a few
where firms have to exert costly effort to lower their costs.7
large players, and some small ones (see Haffner et al., 2010). Over the years, firm-size heterogeneity has in-
6
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creased. These changes influence the effectiveness of yard-
One example being the Dutch energy-distribution industry dis-
cussed above (Haffner et al., 2010). 7 In Shleifer (1985), firms invest R(c) in a cost-reducing technol-
collude. Understanding the relationship between industry
ogy that lowers marginal cost by c. With downward sloping demand,
structure and collusion is thus essential for understanding
the monopolist cannot fully capture the social welfare generated by
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stick competition if they affect the incentives for firms to
lower costs. Hence it will underinvest from a welfare perspective.
the effectiveness of regulation through yardstick competi-
This is no longer true, however, if demand is inelastic. In that case,
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tion.
the monopolist does capture the entire surplus and hence there is no
In this paper, we study to what extent the number of
underinvestment. For experimental simplicity, we do assume inelastic
firms and firm-size heterogeneity in an industry subject to
demand. We therefore use a slightly different set-up. We effectively assume that the manager of the firm has to exert some costly effort to
uniform yardstick competition affects collusion. We first 2 3
lower costs. The discrepancy between the private and social optimum is then caused by the fact that the manager’s costly effort is not taken
Dassler, Parker and Saal (2006). Haffner, Delmer and Van Til (2010); Bl´azques-G´omez and
into account when maximizing social welfare. This allows us to gen-
Grifell-Tatj´e (2011); Jamasb and Pollitt (2007). 4 Mizutani, Kozumi and Matsushima (2009). 5 Dalen and Gomez-Lobo (2002).
erate the same theoretical prediction (that firms underinvest), but in a simpler set-up. Note that our interpretation closely follows Potters et
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3 symmetric firms are more collusive than industries with
marginal costs. Managers aim to maximize the sum of
2 such firms. We find some evidence that mergers that
profits and managerial benefits, and do so in a repeated
lead to symmetric firms indeed facilitate collusion. Rel-
game. Surprisingly, we find that with symmetric firms
ative to what theory predicts, we find that an increase in
the effect of the number of firms on cartel stability is am-
the number of firms makes it easier for experimental sub-
biguous. Hence, having more firms does not necessarily
jects to collude if they are symmetric, but harder if they
make a cartel harder to sustain, and may even make it eas-
are asymmetric. The theoretical effects of an increase in
ier. Moreover, we find that firm-size heterogeneity hinders
heterogeneity are ambiguous. On the one hand, such an
collusion.
increase raises prices if firms do not collude; on the other
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Managerial benefits are a concave function of a firm’s
hand, it makes collusion harder. Our experiment suggests
able to do so, we have to impose some additional struc-
that the net effect for consumers is positive. Summing
ture. In particular, we assume that managerial benefits are
up, we find that a heterogeneous industry structure is of
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We also test our model in a lab experiment. To be
quadratic in a firm’s marginal costs. The theoretical model
key importance for having relatively little collusion under
then predicts that the number of symmetric firms has no
yardstick competition. The remainder of this paper is organized as follows.
ambiguous. Mergers that lead to symmetric market shares
Section 2.1 provides more background on yardstick com-
facilitate collusion. Mergers that do not involve the small-
petition, while Section 2.2 discusses other experiments
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effect on cartel stability, while the effect of mergers is still
studying the impact of industry structure on the incidence
the competitive outcome for the smallest firm, making it
of collusion. Section 3 describes our theoretical model.
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est firm in the industry hinder collusion as they improve
Section 4 presents the experimental design, while Sec-
a collusive agreement. If two small firms merge, then such
tion 5 describes the results of our experiment. Section 6
a merger usually facilitates collusion.
concludes.
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more attractive for the manager of that firm to defect from
Our experimental implementation is loosely based on 2. Background
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Potters et al. (2004). To facilitate coordination we allow subjects to communicate in each round, before set-
2.1. Yardstick competition
ting their cost level. We look at five treatments that differ
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The last decades saw the liberalization of several net-
in the number of firms (2 or 3) and the extent of firm-size
work industries that used to be state-owned vertically-integrated
heterogeneity. In each treatment, we study how successful
monopolies. Examples include telecommunications, elec-
experimental subjects are in establishing collusion. To do
tricity, gas, water and sewerage. Parts of these industries
so, we look at three measures of collusion: the incidence
are natural monopolies characterized by subadditivity of
of full collusion, a collusion index, and the resulting price.
costs, which calls for regulatory supervision (Viscusi et
Our experimental findings suggest that industries with
al., 2005). A key component of such supervision is the regulation of tariffs. Historically, regulated tariffs were
al. (2004).
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United Kingdom (Dassler, Parker and Saal, 2006), elec-
However, such schemes give little incentive to increase
tricity networks in the Netherlands (Haffner, Delmer and
productive efficiency as lower costs directly translate to
Van Til, 2010), Spain (Bl´azques-G´omez and Grifell-Tatj´e,
lower revenues. Hence, the incentive power is low, as a
2011) and the United Kingdom (Jamasb and Pollitt, 2007),
change in costs hardly affect firms’ profits. Introducing
railways in Japan (Mizutani, Kozumi and Matsushima,
price-cap regulation solves this problem as any cost de-
2009) and bus transport services in Norway (Dalen and
crease then fully benefits the firm. But this method also
Gomez-Lobo, 2002). In several cases, the introduction
implies the risk of significant rents or losses for regulated
of yardstick regulation improved productive efficiency or
firms. To overcome both problems, Shleifer (1985) pro-
resulted in lower consumer prices.8 In other cases, the
posed to impose tariffs based on the actual costs of a group
experience appeared to be less successful.9
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either based on actual costs or an allowed rate of return.
What these experiences have taught us is that a num-
tion is called yardstick regulation. As tariffs are based
ber of conditions have to be met before yardstick regula-
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of similar (benchmark) firms. This type of tariff regula-
on the relative performance of a firm, yardstick regulation
tion can be effective. First, we need a sufficiently large
is generally seen as a powerful tool both for giving incen-
number of benchmark firms, using similar techniques op-
tives for productive efficiency as well as for rent extraction
erating within a similar environment. Second, revenues
(Tangerås, 2002; Burns, Jenkins, and Riechmann, 2005).
of regulated firms should be fully based on the yardstick. No other concerns should be taken into account, such as
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The incentive power of a yardstick partly depends on
the impact yardstick regulation may have on the risk of
determine tariffs. Essentially, there are two types of yard-
bankruptcy. Finally, firms should operate independently
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the exact manner in which costs of the benchmark firms
sticks. With a uniform yardstick every firm faces the same 8
cap on the tariffs it may charge, based on cost information
Mizutani et al. (2009), for instance, find that the efficiency of
Japanese railway companies, measured by the variable costs per unit of
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of all firms in the benchmark group. With a discrimina-
output, improved significantly after the introduction of yardstick regulation. Jamasb et al. (2004) report that the British model of incentive
costs of all other firms in the benchmark group. The latter
regulation in the electricity distribution industry has brought signif-
scheme gives a stronger incentive to improve efficiency.
icant price reductions to consumers. Haffner et al. (2010) find that
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tory yardstick, every firm faces a specific cap based on the
yardstick regulation of the Dutch electricity and gas networks resulted
For both schemes, numerous methods exist to determine
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in lower network tariffs without adversely affecting network invest-
the yardstick, including the (weighted) average costs, the
ments. 9 Dalen et al. (2002) for instance do not find an effect on the cost ef-
median costs, the 25th percentile of costs or just the best
ficiency of Norwegian bus companies, for which the authors blame the
practice of all benchmark firms (Yatchew, 2001). In the
bargaining power of the regulated firms to reduce the incentive power
Dutch regulation of electricity distribution networks, for
of the regulatory scheme. For the incentive regulation in the Spanish
instance, the yardstick is based on weighted average costs.
electricity-distribution industry, negative effects on consumer welfare
Experiences with yardstick regulation exist in several
were found by Bl´azques-G´omez and Grifell-Tatj´e (2011), which resulted from the fact that inefficient firms received financial compensa-
industries, including water supply and sewerage in the
tions afterwards in order to prevent bankruptcies.
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from each other. Any cooperation would reduce the effec-
with the number of firms when costs are private infor-
tiveness of yardstick competition (Jamasb, Nillesen and
mation. Abbink and Brandts (2008) find the same result
Pollitt, 2004).
with increasing marginal costs, and Fonseca and Normann (2008) with capacity constraints. Fonseca and Normann
with yardstick regulation under different industry struc-
(2012) also find more collusion with fewer firms. More-
tures. Tangerås (2002) observes that yardstick competi-
over, they find that the ability to communicate facilitates
tion is near useless if firms are able to collude. In his
collusion, the effect being strongest for industries with an
model, collusion takes place through joint manipulation
intermediate number of firms.
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In this paper we therefore assess the risk of collusion
Other experiments focus on asymmetries between firms.
of productivity reports that the regulator uses to determine
With quantity setting, Mason et al. (1992) find more col-
exerting effort to lower actual costs. In Tangerås (2002),
lusion if firms have equal rather than different marginal
collusion becomes less likely if the number of firms in-
costs. Phillips et al. (2011) confirm this result. With price
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tariffs. In our paper, firms that collude jointly refrain from
creases, which is in line with the literature on collusion in
competition, Fonseca and Normann (2008) find more col-
unregulated markets.
lusion if firms have identical capacity constraints, and Dugar and Mitra (2009) find more collusion if the range of pos-
depends on the heterogeneity of firms. The more firms
sible firm-specific marginal costs is smaller. Dugar and
differ in terms of market shares, the harder it is to reach
Mitra (2013) confirm the latter result in a slightly differ-
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In unregulated markets, the feasibility of collusion also
ent context. Argenton and M¨uller (2012) however, find
this is found for the US airline industry, where the extent
that collusion is unaffected if firms have different rather
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an agreement (see e.g. Motta, 2004). In empirical work
than identical cost structures. Still, overwhelmingly, ex-
size heterogeneity (Barla, 2000). In our paper, we study
periments on unregulated markets find that asymmetries
whether the same holds for industries with yardstick com-
do hinder collusion.
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of competition appears to be positively related to firm-
To the best of our knowledge, the only economic experiment that deals with yardstick competition is Potters
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petition.
2.2. Experiments on collusion
et al. (2004), on which our experiment is loosely based.
A number of economic experiments have analyzed the
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Potters et al. (2004) study the effect of yardstick design on
impact of industry structure on collusion in unregulated
collusion in a duopoly with symmetric firms. They com-
markets. Most studies focus on the effect of the number
pare a uniform yardstick to a discriminatory one. When
of firms. Huck et al. (2004) study homogeneous-product
firms behave non-cooperatively, the discriminatory yard-
quantity-setting oligopolies. They find some collusion in
stick yields lower prices and lower profits, making it more
markets with 2 firms, little collusion in markets with 3
prone to collusion. In their experiment, the authors indeed
firms, and no collusion in markets with 4 or 5 firms. Other
find higher cost levels with a discriminatory yardstick. Of studies look at homogeneous-product price-setting oligopolies.course, our research question is very different from that Abbink and Brandts (2005) find that collusion decreases 6
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addressed in Potters et al. (2004). Also, the experiment in
r(ci ). The firm’s problem is then to find the optimal mix of
that paper includes a stage in which firms set prices, while
marginal production cost and fixed cost, i.e. the efficient
we simply impose prices to equal the price cap imposed
scale. Our assumption implies that the optimal mix does
by the regulator. This simplifies the experiment. Anyhow,
not depend on the size of the firm, and hence that there are
Potters et al. (2004) find that subjects choose to set prices
constant returns to scale in the relevant interval.10 Hence,
equal to the price cap in about 99% of the cases.
when two firms merge, the optimal mix is unaffected. This allows us to isolate the effect of mergers on market perfor-
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3. The model
mance, without the confounding effect of already having lower average costs due to economies of scale. Our re-
3.1. Setup
sults should also be interpreted as such. In addition, our
There are n firms that play an infinitely repeated game.
assumption is consistent with the available evidence: ac-
Each firm acts as a local monopolist. For simplicity, we
cording to the empirical literature, economies of scale in
AN US
assume that firm i faces demand that is completely inelas-
electricity distribution are already exhausted with some
tic and exogenously given by mass αi ∈ (0, 1). We norP malize total demand, so ni=1 αi = 1 and αi reflects firm i’s
20,000 customers, far below the size that is relevant on e.g. the Dutch market (see Yatchew, 2000, and the refer-
share of total demand. For ease of exposition, we will re-
ences therein).
fer to αi as the market share of firm i. In each round, firm
We assume that there is a unique cm that maximizes
M
i decides on its constant marginal cost ci for that round. Firm i’s profits then equal πi (ci , c−i ) = (p − ci )αi , where p
ED
is determined by regulation. The vector c−i consists of the
PT
ated with a lack of maintenance. Such networks are prone to outages, compensation claims and political pressure, all
We assume that the manager of firm i maximizes
CE
ui (ci ) = πi (ci , c−i ) + r(ci )αi ,
be decreasing in ci for large enough ci . In electricity networks for example, high marginal costs are often associ-
cost levels chosen by other firms; these affect πi through the regulated price p.
managerial benefit, so ri0 (cm ) = 0. We thus allow r to
factors that do not exactly contribute to a quiet life for the manager.
(1)
The regulator uses a uniform weighted yardstick, where
with r a managerial benefit that is concave: r00 < 0 and
the price a firm is allowed to charge equals the weighted
AC
r(0) = 0. This specification is equivalent to Shleifer (1985),
average of all cost levels in the industry: n X p(ci , c−i ) = α jc j.
where managers invest in a cost-reducing technology. Crucially, we assume that total managerial benefit r(ci )αi is
(2)
j=1
proportional to market share. With equal marginal costs,
This implies that all firms get to charge the same price.
the total amount of managerial benefits in an industry is Total utility to manager i is thus given by P n then independent of the size distribution of firms: αi r(c) = X α + r(c )α . u (c , c ) = α c − c P i i −i j j i i i i r(c) αi = r(c). In a set-up similar to Laffont and Tij=1 role (1993), for example, a firm would earn αi (p − ci ) +
10
7
We thank an anonymous referee for pointing this out.
(3)
ACCEPTED MANUSCRIPT
If consumers value the product at v, total welfare equals X X W = (v − p) + p − αi ci + r(ci )αi (5)
3.2. Firm-size heterogeneity We are interested in how the extent of firm-size heterogeneity affects competition and collusion. We thus need
i
a measure of firm-size heterogeneity. For a fixed num-
where the first term is consumer surplus, and the other two
ber of firms, the most natural measure to look at is the P Herfindahl index H ≡ nj=1 α2j . With n equally-sized firms,
terms reflect total managerial utility. Maximizing with re-
firms differ more in size. To derive our analytical re-
CR IP T
i
spect to ci a socially optimal cost level cW for all firms that
this equals 1/n, but it is easy to see that H increases as
is implicitly defined by r0 cW = 1.
sults, we will interpret an increase in firm heterogeneity
(6)
as the transfer of market share from a (weakly) smaller to
From (4), the social optimum is reached if all firms are
a (weakly) larger firm. Such an increase would necessar-
vanishingly small, so αi = 0 ∀i.
ily increase the Herfindahl index.
AN US
In terms of comparative statics, we can now establish Theorem 1. In the competitive outcome, we have the fol-
3.3. Competition
lowing:
Consider the case where managers unilaterally maxi-
mize their utility. For ease of exposition, we will refer to
(a) An increase in firm-size heterogeneity implies an in-
ing (3) with respect to ci yields
ED
r0 (c∗i ) = 1 − αi .
M
this as the competitive outcome. For given c−i , maximiz-
(4)
crease in the regulated price, provided that r0 is not too concave, i.e. provided r000 > 2r00 /α for all α’s that occur in the two situations we compare.
(b) With symmetric firms, an increase in the number of
costs level of other firms. Also note that c∗i is increasing
firms implies a decrease in the regulated price.
PT
This is a dominant strategy, as it does not depend on the in market share αi :11 the higher αi , the stronger the in-
(c) A merger leads to higher prices.
CE
fluence this firm has on the regulated price, and the more Proof. In Appendix.
attractive it is to choose a higher cost level. Finally, note
AC
that c∗i is strictly lower than the cost level that maximizes managerial
benefits12 .
Hence, if firms choose to compete, having more equally-
Lower costs increase profits, giv-
sized firms leads to lower prices, but a merger leads to
ing managers an incentive to be more efficient. 11
higher prices. Under an additional weak technical condition, we also have that a more heterogeneous market
A higher αi implies a lower equilibrium value of r0 (c∗i ), concavity
structure leads to higher prices. That technical condition
of r then implies a higher c∗i . 12 With a similar argument as in the previous footnote; maximizing 0
is satisfied for all functional forms that we consider in this
∗
managerial benefits requires setting r (c ) = 0, and is thus equivalent to having αi = 1 in (4).
8
ACCEPTED MANUSCRIPT
paper.13 For simplicity, we assume that it is always satis-
long-term losses due to a cartel breakdown. This implies
fied.
that we require
Note that mergers always increase prices. Hence, there
δ > δˆ i (αi ) ≡
is no merger paradox in our model; provided that firms
uiD − uiK uiD − u∗i
.
(8)
compete, a merger always makes their managers unam-
For collusion to be sustainable, we need that this condition
biguously better off. Admittedly, this may no longer be
is satisfied for all managers: o n δ > δˆ ≡ max δˆ i (α1 ) , . . . , δˆ i (αn ) .
CR IP T
true if due to a merger, a cartel would no longer be stable. Still, with some potential abuse of terminology, in the re-
(9)
mainder of this paper we will talk about “the effects of a
We will focus on the cartel agreement in which all
merger” whenever we exogenously impose that two firms
managers set the same cost level ck that maximizes their
join forces.
total utility. Arguably, this is the most obvious and focal agreement. We will refer to it as the perfect symmetric
3.4. Collusion
AN US
collusive agreement. Of course, if this would not yield a stable cartel, managers could still try to coordinate on a
ger strategies and look for the critical discount factor δˆ
different agreement, either symmetric or asymmetric. In
such that all managers have an incentive to stick to the car-
our theoretical analysis we rule out this option as we feel
tel agreement. With the usual arguments manager i does
that it would be much harder to coordinate on such an
not defect if and only if
alternative.14 Of course, in the experimental implementa-
(7)
tion of our model, firms are free to try to coordinate on whatever they can agree upon.
ED
uiK δu∗i > uiD + , 1−δ 1−δ
M
Following e.g. Friedman (1971), we assume grim trig-
If firms coordinate on a common cost level ck , we im-
her utility in a cartel, and u∗i her utility in the competi-
mediately have p = ck , so all profits are zero and the car-
tive outcome. In other words, cartel stability requires that
tel simply maximizes joint managerial benefit r(c), which
the short-run benefits of defection are outweighed by the
implies ck = cm .
(10)
In particular, in Corollary 1, we use the following functional
forms:
We now have:
AC
13
CE
PT
with uiD the utility of manager i when she defects, uiK
(a) Suppose r(c) =
√ c − bc. We then have r0 (c) = 21 c−1/2 − b, so
Lemma 1. The perfect symmetric collusive agreement yields
r00 (c) = − 14 c−3/2 < 0 and r000 (c) = 38 c−5/2 > 0 > 2r00 /α, so the
a stable cartel if, for all i = 1, . . . , n, (1 − αi ) cm − c∗i + r(c∗i ) − r (cm ) ˆ i) ≡ P δ > δ(α . (1 − αi ) cm − nj,i α j c∗j
condition is satisfied.
(b) Suppose r(c) = bc − ac2 .In that case r000 (c) = 0 > 2r00 /α, so again the condition is satisfied.
(c) Suppose r(c) = bc − ec . In that case r00 = −ec and r000 = −ec , so
14
r000 > 2r00 /α if 1 < 2/α, which is true for any α ∈ (0, 1) .
(11)
Indeed, in our experiment we also observe that groups, if anything,
coordinate on the perfect symmetric collusive agreement, irrespective of the distribution of market shares.
9
ACCEPTED MANUSCRIPT
Proof. For a cartel to be stable, we need that (8) is satis-
This can be understood as follows. Defection util-
fied for all i. To evaluate the values for u∗i , uiK , and uiD , we
ity, cartel utility and competition utility all decrease in the
proceed as follows. First, using (3), we have n X uiK = α j cm − cm + r(cm ) αi = r cm αi .
number of firms.16 That implies from (8) that the net effect on cartel stability is ambiguous. In particular, it de(12)
pends on the curvature of r. Indeed, we can find families
j=1
of managerial benefit functions r such that an increase in
The optimal defection is simply to use your dominant strat-
Moreover
affect cartel stability:
= r c∗i − (1 − αi ) c∗i − cm αi .
(13)
n X u∗i = ui (c∗i , c∗−i ) = α j c∗j − c∗i + r(c∗i ) αi . j=1
Hence
r(c∗i )
cm
c∗i
r (cm )
Corollary 1.
(a) If r(c) =
√ c − bc, with b > 0, then a cartel is less
stable if the number of symmetric firms increases.
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AN US
uiD
CR IP T
the number of firms either increases, decreases or does not
egy and set c∗i . That yields, again using (3),
(b) If r(c) = bc − ac2 , with a, b > 0, then cartel stability is unaffected by the number of symmetric firms.
3.5. Factors that facilitate collusion
M
− (1 − αi ) − − , P δˆ (αi ) = (c) If r(c) = bc − ec , with b > 2, then a cartel is more r c∗i − (1 − αi ) c∗i − cm − nj,i α j c∗j − c∗i + r c∗i stable if the number of symmetric firms increases. which simplifies to (11). Proof. In Appendix. To analyze how exogenous factors affect the stability
ED
We now study which factors facilitate or hinder collu-
of collusion with heterogeneous firms, we need to know
sion. Throughout the analysis, we assume that firm size
how these affect the manager with the highest critical dis-
PT
heterogeneity increases competitive prices, i.e. that the
count factor. We can establish:
condition in Theorem 1(a) is satisfied.15 We can now
16
CE
show
∂uiD ∗ m = r c∗i − (1 − αi ) c∗i − cm + r0 · c∗0 − (1 − αi ) c∗0 i + ci − c i αi ∂αi = r c∗i − (1 − αi ) c∗i − cm + c∗i − cm αi
Theorem 2. With symmetric firms, an increase in the num-
AC
ber of firms facilitates collusion if and only if r(c∗i ) is suf-
as r0 = 1 − αi . With uiD > 0, we have r c∗i − (1 − αi ) c∗i − cm > 0,
ficiently concave, more precisely if 00
r < −r
0
(c∗i )
r0 (c∗i ) (cm − c∗ ) − (r(cm ) − r(c∗ )) · (r(cm ) − r(c∗ )) (cm − c∗ )
hence c∗i > cm implies ∂uiD /αi > 0, which implies that uiD decreases with the number of firms. Moreover, from (12) and (14), ∂uiK = r (cm ) > 0 ∂αi ∂u∗i ∗0 0 ∗0 = p∗ − c + r c∗i + c∗i + αi c∗0 i − ci + r ci αi ∂αi = p∗ − c + r c∗i + c∗i αi > 0.
Proof. In Appendix. 15
For defection utility, note from (14)
Although note that we do not need to make this assumption for all
results that follow.
10
ACCEPTED MANUSCRIPT
Lemma 2. If the perfect symmetric collusive agreement is
Using this theorem, we can derive a number of scenar-
ˆ i ) is the highest stable, then the critical discount factor δ(α
ios in which a merger either always facilitates, or always
for the smallest firm.
hinders collusion:
Proof. In Appendix.
Theorem 4. A merger affects cartel stability in the following manner:
Lemma 2 immediately implies that a cartel is stable if and only if the smallest firm has no incentive to defect. Theorem 3. An increase in firm-size heterogeneity increases the critical discount factor and hence makes a cartel less
CR IP T
(a) If it does not affect the market share of the smallest firm, then a merger hinders collusion. (b) Suppose that we are in an environment where a car-
likely to be stable.
tel becomes weakly less stable if the number of symmetric firms increases. In that case, a merger that
Proof. In Appendix.
AN US
leads to symmetric firms facilitates collusion.
Comparing this result to Theorem 1, we thus have an
Proof. Part (a) is straightforward. From Lemma 2 the
neous industry structure implies a lower price – provided
smallest firm has the highest critical discount factor. If
that firms behave competitively. At the same time, how-
(one of the) smallest firm(s) is not involved in a merger,
ever, it makes collusion more likely.
its market share αi is not affected, so after the merger it is
M
interesting trade-off. On the one hand, a more homoge-
Now consider the effect of a merger. First, it is straight-
ED
forward to establish:
still the smallest firm. However, p∗ will increase, which from (11) implies that δˆ i (αi ) increases and collusion is less stable.
Lemma 3. Denote the pre-merger industry structure as A,
For part (b), note that, before the merger, there are
and the post-merger industry structure as B. The merger
PT
n > 2 firms, that differ in size. As a first step, redistribute
then facilitates collusion if and only if 1 − αB cm − c∗ αB + r(c∗ αB ) − r (cm ) < 1 − αB cm + αB c∗ αB − p∗B 1 − αA cm − c∗ αA + r(c∗ αA ) − r (cm ) , (15) 1 − αA cm + αA c∗ αA − p∗A
market shares between these firms in such a way that they
CE
are now equally sized. From Theorem 3, doing so facilitates collusion. As a second step, consider a change from
AC
n equally sized firms to n − 1 equally sized firms. By assumption, this also facilitates collusion. This establishes
with ακ the market share of the smallest firm in industry
the result.
structure κ, and p∗κ the competitive price in industry struc-
Result (a) is particularly strong: whenever the small-
ture κ, κ ∈ {A, B}.
est firm (or one of the smallest firms) is not part of the
Proof. Follows directly from Lemma 1 and Lemma 2.
merger, then the merger necessarily hinders collusion. The intuition is as follows. As the market share of the smallest firm is unaffected, the merger also has no effect on 11
ACCEPTED MANUSCRIPT
levels ci from the set {1, 2, . . . , 20}.19 Third, prices are de-
manager of that firm. However, it does affect her utility in
termined using (2). After each round, subjects learn their
the competitive outcome. From the proof of Theorem 1
profits and managerial benefits, and the cost levels each
a more concentrated industry implies higher competitive
subject has set.20 Although not very common in a car-
prices, and hence higher competitive utilities. Thus, de-
tel experiment, we feel that the unrestricted communica-
fection from a cartel becomes less costly, making it more
tion we allow for creates circumstances that are closest to
attractive to defect.
the real world. The Dutch regulatory framework for the
CR IP T
either the collusion utility or the defection utility of the
distribution-network operators, for example, does not for-
lead to a more homogeneous industry structure (such as
bid them to mutually exchange information on operational
the scenario described in (b) of the Theorem) facilitate
and strategic issues. The operators together have estab-
collusion, while mergers that lead to a less homogeneous
lished a joint organization (Netbeheer Nederland) which
industry structure (such as the scenario described in sce-
facilitates the cooperation among the operators besides
AN US
Broadly speaking, the results suggest that mergers that
nario (a)) hinder collusion – at least in an environment
representing the group of operators in the political debate
where having more equally sized firms makes it harder to
on energy policy in general and the design of regulation in
collude.17
particular. Without communication, it is also hard to sustain collusion in a cartel experiment with more than two
4. Experiment
M
players, see e.g. Haan et al. (2009).21
4.1. Design
ED
In our experiment, subjects play the game described
We use the following quadratic specification for the managerial benefit function: r(ci ) = bci − ac2i ,
above for at least 20 rounds. From round 20 onwards, the experiment ends with a probability of 20% in each round,
(16)
PT
where a, b ∈ R+ are parameters and ci ∈ [0, b/a]. We
to avoid possible end-game effects (see also Normann and
assume that a and b are such that all expressions we derive
CE
Wallace, 2012). We use fixed matching: every subject plays with the same group members in all rounds.
19
As saw in the theory section, both the competitive and the collu-
sive strategy are independent of the competitors’ actions. In that sense,
communicate using a chat screen. This chat is completely
our game boils down to a relatively straightforward prisoners’ dilemma
anonymous.18 . Second, subjects unilaterally choose cost
game with two strategies. Still, we choose not to represent it as such in
17
AC
Every round consists of three steps. First, subjects can
our experiment, as that would get rid of much of the regulatory context that we feel is crucial for our purpose. 20 Of course, in our set-up profit and managerial benefit are equally
If we are not in such an environment, the effect of a merger to
equality is ambiguous. It is hard to derive general conditions under
important in determining payoffs. This was also outlined in the experi-
which such a merger would or would not facilitate collusion. 18 We allowed subjects to chat for 150 seconds in round 1, for 120
mental instructions. Subjects indeed seemed to understand this as they correctly answered control questions on this topic. 21 Although illegal in practice, note that cartels cannot be prosecuted
seconds in round 3, for 60 seconds in rounds 3-10, and for 45 seconds in all remaining rounds.
or fined in this experiment.
12
ACCEPTED MANUSCRIPT
The 5 treatments allow us to evaluate the effect of the
are well defined. For this set-up, it is easy to drive that c∗i = m
b − 1 + αi ; 2a b . 2a
number of firms and firm-size heterogeneity on market
(17)
performance in terms of cost levels and regulated prices. They also allow us to evaluate the effect of a merger. In
We set a = 1/24 and b = 1. These choices assure that
our evaluation of the experimental results we do not in-
the competitive and collusive cost levels derived above are
terpret the critical discount factors as a strict prediction of
integers. We run 5 treatments that differ in the number of
the outcome of the experiment. Thus, we do not expect
firms and the extent of firm-size heterogeneity. For ease
that there will always be collusion whenever the δ we im-
of exposition, we normalize the total size of the market to
pose is larger than the δˆ we derived. Neither do we expect
12 when naming our treatments, which allows us to rep-
that there will never be collusion whenever the δ we im-
resent market sizes of all firms in all treatments as integer
pose is smaller than the δˆ we derived. Rather, we interpret
numbers.22 The first 3 treatments have 3 firms and are de-
a higher value of δˆ as making a cartel less likely to oc-
AN US
=
CR IP T
(18)
c
noted TrioXYZ, with X, Y and Z the size of firms 1, 2 and
cur. This is in line with other work (see e.g. Bigoni et al.,
3, respectively. The other treatments have 2 firms and are
2012).
From Corollary 1(b) we expect that, with symmetric
denoted DuoXY, with X and Y the sizes of firms 1 and 2.
firms, the number of firms does not affect collusion:
Table 1 provides information for each treatment. The
M
first panel gives the market share for each firm and the re-
sulting Herfindahl index. The second panel provides com-
Hypothesis 1 (Number of Firms). The amount of collu-
sion in Duo66 is the same as that in Trio444.
petitive cost levels and the resulting price. The third panel
ED
From Theorem 3 we expect that more heterogeneity
gives collusive cost levels and the critical discount factor
implies less collusion:
depend on industry structure; in all treatments, the perfect
Hypothesis 2 (Heterogeneity).
PT
δˆ of the cartel.23 From (10), collusive cost levels do not
symmetric collusive cost level equals 12.
CE
22
(a) There is more collusion in Duo66 than in Duo84;
In the actual experiment, we always normalize the size of the
(b) There is more collusion in Trio444 than in Trio633,
smallest firm to 1, which makes it easier to explain the experiment
and more collusion in Trio633 than in Trio642.
ample. 23 Strictly speaking, the critical discount factors we derive are only
From Theorem 4(b) we expect that mergers that lead
AC
to the subjects. See the instructions in the online appendix for an ex-
valid from round 20 onwards. Still, it is easy to see that this does not af-
to symmetric firms, facilitate collusion:
fect our qualitative results. If a cartel is stable from round 20 onwards,
Hypothesis 3 (Merger to Symmetry).
it is definitely stable in earlier rounds as well, since the incentives to defect become smaller as the first 19 rounds are played for sure. If a
(a) There is more collusion in Duo66 than in Trio633;
cartel is not stable from round 20 onwards, then subjects know it will break down in that round and hence, by backward induction, are not
(b) There is more collusion in Duo66 than in Trio642.
willing to form a cartel in an earlier round.
13
ACCEPTED MANUSCRIPT
Table 1: Overview of treatments.
Treatment
Market shares
Competition
Collusion
α2
α3
HHI
c∗1
c∗2
c∗3
p∗
c∗k = pk
δˆ
Trio444
33.3%
33.3%
33.3%
0.333
4
4
4
4.00
12
0.50
Trio633
50.0%
25.0%
25.0%
0.375
6
3
3
4.50
12
0.64
Trio642
50.0%
33.3%
16.7%
0.389
6
4
2
4.67
12
0.71
Duo66
50.0%
50.0%
0.500
6
6
Duo84
66.7%
33.3%
0.556
8
4
CR IP T
α1
6.00
12
0.50
6.67
12
1.00
Market shares give the markest shares that we impose for each experimental firm in that treatment, HHI is the resulting Hirschman Herfindahl Index. c∗i is the competitive cost level of firm i that our theory predicts, p∗ the resulting regulated price, c∗k and p∗k are the collusive cost levels and
AN US
prices predicted by our theory, δˆ the critical discount factor for collusion to be stable.
Finally, we look at mergers that lead to asymmetric firms. At least in this case, these hinder collusion:24
economics and business (59.3%). Every session consisted of one treatment and lasted between 80 and 115 minutes. Subjects signed in for sessions, while treatments were ran-
Hypothesis 4 (Merger to Asymmetry).
domly assigned to sessions. Every treatment with 2 subjects was played in two ses-
(b) There is less collusion in Duo84 than in Trio642.
sions while every treatment with 3 subjects was played in
ED
M
(a) There is less collusion in Duo84 than in Trio444;
three sessions. Between 14 and 18 subjects participated
Part (a) of this Hypothesis follows directly from Theo-
in a session, resulting in 16 to 17 groups per treatment.
rem 4(a). Part (b) does not follow directly from the results
PT
This is similar to other cartel experiments, see e.g. Bigoni
in the previous section, but can be shown to hold in our
et al. (2012), Dijkstra et al. (2014) and Hinloopen and
experimental implementation (see e.g. the last column of
Soetevent (2008). The experiment was programmed in z-
CE
Table 1).
Tree (Fischbacher, 2007). Printed instructions were provided and read aloud.25 On their computer, subjects first
AC
4.2. Implementation
had to answer a number of questions correctly to ensure
The experiment was conducted at the Groningen Ex-
understanding of the experiment. Participants were paid
perimental Economics Laboratory (GrEELab) at the Uni-
their cumulative earnings in euros. Since firm size differed
versity of Groningen in February and March 2013. A total
between treatments, exchange rates were varied such that
of 214 subjects participated, all students from the Uni-
participants would receive identical amounts with full col-
versity of Groningen (80.4%) or the Hanze University of Applied Sciences (19.6%), most of them in the fields of 24
25
Instructions for Trio633 are reproduced in the online appendix.
Instructions for other treatments are similar and available upon request.
Thanks to an anonymous referee for suggesting this hypothesis.
14
ACCEPTED MANUSCRIPT
lusion.26 Furthermore, they received an initial endowment
implies no difference between treatments we do a two-
of e4. Average earnings were e16.80 and ranged from
sided test. We also perform a two-sided test if the theoret-
e7.75 to e24.00.
ical result is ambiguous. This is may for example be the case with prices: when a treatment is expected to yield
5. Results
more collusion but also lower competitive prices, then the net effect on prices is ambiguous, so we do a two-sided
For comparison sake, we only include the first 20 rounds of each group in our analysis, as only these rounds are
tween treatments, we do a one-sided test.
First, we look at the incidence of full collusion. In
played by all groups. In this section, we introduce three
all treatments, the perfect symmetric collusive agreement
measures to evaluate the extent of collusion: the incidence
is for all firms to set a cost level of 12. For ease of expo-
of full collusion; a collusion index; and price. We explain
sition we refer to this as full collusion. The incidence of
AN US
these measures in more detail below. For each measure,
full collusion is thus defined as the percentage of markets
we give its development over time in each treatment, and
where all firms set a cost level of 12. Figure 1 shows how
compare averages between all treatments. In the next sec-
this measure develops over time, in all treatments.30 The
tion, we confront our hypotheses with the results of the experiment.
number of markets with full collusion is substantial. From the Figure, Trio444 seems to be the most collusive, while
M
As is common in experimental studies, we use non-
parametric tests for differences between treatments.27 The
ED
unit of observation is always the average value over the
first 20 rounds for each group.28 We use the Mann-Whitney U Test (MWU) to test for differences between two popu-
the least collusion is found in Trio642. Table 2 gives the average incidence of full collusion for all treatments, and reports on pairwise comparisons between treatments. The entries in the right-hand panel
PT
indicate whether the row treatment has an incidence of
lations, and the Jonckheere-Terpstra Test to test for an or-
collusion that is significantly higher (>) or significantly
dering of three populations.29 Whenever our hypothesis
lower (<) than the column treatment, or whether the dif-
CE
26
test. Whenever our hypothesis implies a difference be-
CR IP T
5.1. Three measures of collusion
ference is not significant (≈). We use this convention
This allows us to focus on the effect of asymmetries in industry
throughout the remainder of this paper. From the Table,
AC
structure without the possible interference of fairness issues. 27 Such tests do not impose a specific distribution on the data and are
we thus have for example that Trio444 leads to signifi-
less sensitive to outliers than parametric tests. Still, we also ran OLS
cantly more collusion than Trio642 and Duo84. The dif-
regression using panel data, which yielded the same qualitative results;
ference with Trio633 and Duo66 is not significant.
see the online appendix. 28 Arguably there may be learning and/or end-game effects present
One drawback of this measure is that it only considers
in the data. Still, our qualitative results do not change if we would
markets to be collusive if market participants succeed in
exclude first or last rounds from the analysis. Details are available from the authors upon request. 29 See Jonckheere (1954) or Terpstra (1952). The Kruskal-Wallis
30
test is equivalent but has an unordered alternative hypothesis.
The development of costs over time for each individual market are
given in the online appendix.
15
ACCEPTED MANUSCRIPT
4
8
12
16
Trio633 Duo84
20
Trio642
PT
ED
Trio444 Duo66
Round
M
0
AN US
0
CR IP T
Incidence of Full Collusion (%) 20 40 60 80
100
Figure 1: Incidence of full collusion (across all groups).
Table 2: Incidence of full collusion (across all rounds and groups).
AC
CE
Treatment
Average
Trio444
71.8%
Trio633
58.1%
Trio642
34.4%
Duo66
56.3%
Duo84
43.4%
Trio633
Trio642
Duo66
Duo84
≈o
>∗∗ o
≈t
>∗o
>+o
≈o
≈o
<∗o
≈o ≈o
Entries in the right-hand panel indicate whether the row treatment yields rates of full collusion that are significantly lower (<), significantly higher (>), or that do not differ significantly (≈) from the rates in the column treatment. Differences between treatments are tested using the MWU test for equality; subscript t denotes a two-sided test, o a one-sided test. + : significant at 10% level; ∗ at 5%; ∗∗ at 1%.
16
ACCEPTED MANUSCRIPT
achieving full collusion. Arguably, any market with prices
5.2. Test of Hypotheses
higher than those in the competitive outcome is collusive
In the previous subsection, we gave an overview of the
to at least some extent. Also, the closer prices are to the
experimental results. We now discuss to what extent those
fully collusive outcome, the more collusive that market
results confirm our hypotheses. Hypothesis 1 is concerned with the number of (sym-
relative premium that firms are able to achieve over and
metric) firms and argues that we expect no difference in
above the competitive outcome:
collusion between both treatments. From Table 5, the col-
collusion index ≡
price − p∗ , 12 − p∗
CR IP T
is. We therefore study the collusion index, which is the
lusion index is higher in Trio444 than in Duo66. The
(19)
other measures are not significantly different between the two treatments. In section 5.3 we further address the ques-
lows for the fact that the competitive price level differs
tion as to why subjects may find it easier to coordinate on
across industry structures (see Table 1). If the competi-
a collusive outcome with 3 rather than 2 players.
AN US
with p∗ the competitive price. Note that this measure al-
tive outcome is achieved, the collusion index equals 0. If
Hypothesis 2 is concerned with the effect of hetero-
the collusive outcome of 12 is achieved, it equals 1. The
geneity. For the case of 2 firms, Table 6 shows that the
higher the index, the more successful firms are in collud-
incidence of full collusion is higher in Duo66, but the av-
ing.
erage value of the collusion index is lower. Average prices
Figure 2 shows how the collusion index develops over
M
are also lower. However, none of these differences is sig-
time, in all treatments. Qualitatively, the picture looks
ED
very similar to that for the incidence of full collusion.
nificant. Thus, Hypothesis 2(a) is not confirmed. Table 7 looks at the results for three firms. Note that Trio444 is listed twice in this table, to facilitate all pair-
collusive treatment. Table 3 gives treatment averages and
wise comparisons. All results reported in Table 7 are con-
pairwise comparisons.
sistent with Hypothesis 2(b); for any measure, the extent
PT
Trio444 seems the most collusive, and Trio642 the least
of collusion is always higher if the industry structure is
that consumers ultimately pay. Despite a slightly higher
more homogeneous. Many of the pairwise comparisons
incidence of collusion, for example, an industry structure
are not significant, but results are highly significant in a
may still be preferable if it leads to lower competitive
Jonckheere-Terpstra test with the ordered alternative hy-
AC
CE
Of course, a regulator is most interested in the price
prices that more than outweigh the adverse effects of the
pothesis that Trio444 yields more collusion than Trio633
occasional cartel. For that reason, we also look at prices.
which in turn yields more collusion than Trio642. Note
Average prices over time are given in Figure 3. Consistent
that we argued in Section 3 that the effect of firm-size het-
with our earlier measures, average prices are high, and
erogeneity on price may be ambiguous, as our model pre-
come closest to the collusive outcome in Trio444. From
dicts that more heterogeneity makes collusion harder, but
Table 4, most pairwise comparisons are no longer signifi-
also raises the competitive price. Our experimental results
cant.
indicate that the effect on collusion is stronger, as prices 17
ACCEPTED MANUSCRIPT
Table 3: Collusion index (across all rounds and groups).
Average
Trio444
85.8%
Trio633
74.6%
Trio642
65.9%
Duo66
73.1%
Duo84
77.9%
Trio633
Trio642
Duo66
Duo84
≈o
>∗∗ o
>+t
>∗o
≈o
≈o
≈o
≈o
<+o ≈o
CR IP T
Treatment
Entries in the right-hand panel indicate whether the row treatment yields a collusion index that is significantly lower (<), significantly higher (>), or that does not differ significantly (≈) from the index in the column treatment. Differences between treatments are tested using the MWU test for
AN US
equality; subscript t denotes a two-sided test, o a one-sided test. + : significant at 10%; ∗ at 5%; ∗∗ at 1%.
M ED PT CE
0
AC
.2
Collusion Index (%) .4 .6 .8
1
Figure 2: Average collusion index per round (across all groups).
0
4
8
Round
Trio444 Duo66
12
Trio633 Duo84
18
16 Trio642
20
ACCEPTED MANUSCRIPT
13
Figure 3: Average price per round (across all groups).
4
8
12
16
Trio633 Duo84
20
Trio642
PT
ED
Trio444 Duo66
Round
M
0
AN US
7
8
9
Price 10
CR IP T
11
12
Collusive Price
Table 4: Price (across all rounds and groups).
AC
CE
Treatment
Average
Trio444
11.0
Trio633
10.0
Trio642
9.5
Duo66
10.3
Duo84
10.8
Trio633
Trio642
Duo66
Duo84
≈t
>∗t
≈o
≈t
≈t
≈t
≈t
≈t
<+t ≈t
Entries in the right-hand panel indicate whether the row treatment yields prices that are significantly lower (<), significantly higher (>), or that do not differ significantly (≈) from the column treatment. Differences between treatments are tested using the MWU test for equality; subscript t denotes a two-sided test, o a one-sided test. + : significant at 10%; ∗ : at 5%.
19
ACCEPTED MANUSCRIPT
Table 5: Comparison of treatments with symmetric firms (across all rounds and groups).
Trio444
Duo66
Incidence of full collusion
71.8%
≈o
56.3%
Collusion index
85.8%
>+o
73.1%
Price
11.0
≈t
10.3
CR IP T
Measure
Entries between values indicate whether the value to the left is significantly lower (<), higher (>), or does not differ significantly (≈) from the value to the right. Differences between treatments are tested using the MWU test for equality; subscript t denotes a two-sided test, o a one-sided
AN US
test. + : significant at the 10% level.
Table 6: Comparison of treatments with 2 firms (averages across all rounds and groups).
Measure
Duo66
Duo84
56.3%
≈o
43.4%
Collusion index
73.1%
≈o
77.9%
10.3
≈t
10.8
Price
M
Incidence of full collusion
ED
Entries between values indicate that the value to the left is sigificantly lower (<), higher (>), or does not differ significantly (≈) from the value to
CE
PT
the right. Differences between treatments are tested using the MWU test for equality; subscript t denotes a two-sided test, o a one-sided test.
Table 7: Comparison of treatments with 3 firms (averages across all rounds and groups).
Measure
Trio444
Trio633
Trio642
Trio444
≈o 58.1%
>+o 34.4% <∗∗ o 71.8%
Collusion index
85.8%
≈o 74.6%
≈o
Price
11.0
≈t 10.0
≈t
AC
Incidence of full collusion 71.8%
65.9% <∗∗ o 85.8% 9.5
<∗t 11.0
JT ∗∗o ∗∗o +t
Entries between values indicate whether the value to the left is significantly lower (<), significantly higher (>), or does not differ significantly (≈) from the value to the right. Differences between two treatments are tested using the MWU test for equality; differences between all treatments using the Jonckheere-Terpstra test (JT) for equality; subscript t denotes a two-sided test, o a one-sided test. + : significant at 10%; ∗ at 5%; 1%.
20
∗∗
at
ACCEPTED MANUSCRIPT
subjects. Coders coded the chat statements independently.
do go down with more heterogeneity. Summing up, in treatments with 2 firms we do not
They were unaware of the research questions addressed in
find evidence for an effect of the extent of firm-size het-
this study. For each group, the full chat in a given round
erogeneity on collusion. With 3 firms, however, we find
could only be assigned to one category. Out of all markets where subjects did chat, we re-
strong support for Hypothesis 2(b): more heterogeneity
port the fraction of markets where the relevant issue is
leads to less collusion.
at least discussed once.32 Table 10 provides these results.
tent to which a merger to symmetric firms facilitates col-
From the table, most groups that chat, discuss cost levels
lusion. We find that Duo66 is indeed more collusive than
at some point. In each treatment, some 90% of groups
Trio642, when measured by the incidence of full collu-
raised the possibility to coordinate on a cost level 12, with
sion. On the other two measures, Duo66 also scores higher,
a slightly lower percentage in Duo84.33 The more het-
but not significantly so. However, the differences between
erogeneous the market shares, the lower the percentage of
AN US
CR IP T
Table 8 evaluates Hypothesis 3 and looks at the ex-
Duo66 and Trio633 are ambiguous and insignificant. Ta-
groups that actually agree to set cost levels of 12. Only
ble 9 evaluates the evidence concerning Hypothesis 4 that
few groups reach an alternative agreement, with the ex-
predicts that a merger to asymmetric firms hinders collu-
ception of Duo84.
Our results in Table 5 suggest that it might be easier
sures, a merger from Trio444 to Duo84 indeed hinders
to coordinate on a collusive outcome with 3 rather than 2
M
sion. Again, the evidence is mixed. On 2 out of 3 mea-
players. Table 11 takes a closer look at what might drive
to Duo84 only seems to facilitate collusion.
this result. The first thing to note is that in markets with 3
ED
collusion. However, if anything, a merger from Trio642
5.3. Chat Analysis
firms, there were only 5 instances in which subjects ex-
To quantify the agreements subjects make during the
PT
32
rather than one full observation. 33 In Duo84 the small firm earns the same with symmetric collusion
chats, we use the methodology of content analysis (see e.g. Cooper and K¨uhn, 2014, and Cason and Mui, forth-
CE
as it does when both firms compete. Yet, there are alternative collusive
coming). We employed two assistants to review the essence conversation31
We classify instances in which coders did not agree as one half
agreements where it earns more. The large firm also has an incentive
and classify them according to a
to make such agreements, as it implies a more stable cartel. The Table
classification scheme. We asked them to judge whether
suggests that this may be an issue, as we indeed see many alternative
AC
of each
agreements in this treatment. There were 49 alternative agreements out
an agreement was reached; whether this agreement was to
of 320 total rounds. In 25 agreements the large firm set a higher cost
“set all cost level 12” or a different agreement; whether no
level than the small firm, in 11 it set a lower cost level. 11 agreements
agreement was reached; or whether no proposal was made
implied setting different cost levels and rotating roles; 1 agreement set-
at all. Furthermore, we asked them to judge whether the
ting a symmetric cost level different from 12; and 1 agreement involved randomization of cost levels. However, these alternative agreements
possibility to all set cost level 12 was raised by any of the 31
were never long-lived, so they are more likely to reflect subjects trying different options to find the optimal solution, rather than that they
We define a conversation as the chat that takes place in a given
made a conscious effort to coordinate on an asymmetric equilibrium.
market in a given round.
21
ACCEPTED MANUSCRIPT
Table 8: Merger to symmetric firms (averages across all rounds and groups).
Measure
Trio642
Duo66
Trio633
Incidence of full collusion
34.4%
<∗o
56.3%
≈o
58.1%
Collusion index
65.9%
≈o
73.1%
≈o
74.6%
9.5
≈t
10.3
≈t
10.0
Price
Entries between values indicate whether the value to the left is significantly lower (<), higher (>), or does not differ significantly (≈) from the
CR IP T
value to the right. Differences between treatments are tested using the MWU test for equality; subscript t denotes a two-sided test, o a one-sided test. ∗ : significant at 5% level.
Table 9: Merger to asymmetric duopoly (averages across all rounds and groups).
Measure
Trio642
Duo84
Trio444
34.4%
≈o
43.4%
<∗o
71.8%
Collusion index
65.9%
<+o
77.9%
<∗o
85.8%
9.5
<+t
10.8
≈t
11.0
AN US
Incidence of full collusion
Price
Entries between values indicate whether the value to the left is significantly lower (<), higher (>), or does not differ significantly (≈) from the value to the right. Differences between treatments are tested using the MWU test for equality; subscript t denotes a two-sided test, subscript o a
M
one-sided test. + : significant at 10% level; ∗ at 5%.
the numbers in Table 11 reasonably well.35 The differ-
In markets with 2 firms, there were 25 such instances.34
ences cannot be explained by tacit agreements: in either
Hence, players in 2-player markets more readily agreed
treatment, there wasn’t a single instance in which all sub-
on alternative arrangements. Apparently, it is easier to
jects set costs equal to 12 without someone explicitly rais-
PT
ED
plicitly agreed on setting cost levels different from 12.
ing that possibility. Summing up, having 3 players makes
players need to agree.
it easier to agree on the focal equilibrium and less tempt-
CE
reach an alternative agreement when only 2 rather than 3
Moreover, groups that did figure out that it is a good
ing to try other scenarios. The most likely explanation is
idea to set cost level 12 took more time to do so in 2-firm
simply that 3 people know more than 2.
AC
markets than 3-firm markets. Arguably, it is more likely 35
that one player will come up with that possibility if we
Suppose for example that in each round each single player pro-
poses with probability 8% to set a common cost level of 12. In each
have 3 players, rather than if we have 2. This can explain
round, the probability that at least one subject makes this suggestion then equals 15.4% in a duopoly, and 22.1% in a triopoly. The ex-
34
In the 3-firm case, these alternative agreements all consisted of
pected number of rounds before this proposal is made is then 6.51 in a
setting different cost levels and rotating roles. In the 2-firm case, there
duopoly and 4.52 in a triopoly. These numbers are similar to those in
were 11 such rotating agreements, 2 agreements to set a symmetric
Table 11. We thank an anonymous referee for pointing this out.
cost level different from 12, 11 asymmetric agreements, and 1 agreement that only involved 1 player.
22
ACCEPTED MANUSCRIPT
6. Conclusion
the market structure pre-merger consists of equally sized firms. Otherwise such a merger even facilitates collusion.
In this paper we studied the effect of industry structure
Our findings also suggest that industries with 3 symmetric
on collusion among economic agents subject to uniform
firms are more collusive than industries with 2 such firms.
yardstick competition, an issue particularly relevant for
Summing up, our experimental findings tentatively sug-
the debate on the effectiveness of tariff regulation. We
gest the following. Relative to what theory predicts, an
did so both in a theoretical framework and in a laboratory
increase in the number of firms makes it easier for ex-
CR IP T
experiment.
perimental subjects to collude if they are symmetric, but
In our theoretical model, we find that firm-size hetero-
harder if they are asymmetric. This may be understood as
geneity hinders collusion. More surprisingly, in a sym-
follows. An increase in their number makes it harder for
metric industry, the effect of the number of firms on cartel
subjects to coordinate. With asymmetry that is bad news,
stability is ambiguous, and an increase in the number of
as the lack of a focal outcome implies that it is now harder
AN US
firms may even facilitate collusion. The theoretical effects
to reach any agreement. With symmetry, however, it is
of mergers on collusion are ambiguous. Mergers that do
good news, as the focal outcome becomes more salient
not involve the smallest firm in the industry hinder col-
and subjects are less tempted to try to explore alternative
lusion, as they improve the competitive outcome for the
agreements. We indeed find that 2-firm markets more of-
smallest firm, making it more attractive for the manager
ten reach alternative agreements, and take more time be-
M
of that firm to defect from a collusive agreement. Mergers that lead to symmetric market shares facilitate collusion.
ED
Of course, no theoretical model can always predict
fore they start discussing the collusive outcome. Hence, firm-size heterogeneity in an industry subject to yardstick competition strongly affects collusion in tri-
what will happen in the real world. That is particularly
opoly markets. Our theory suggests that an increase in
true for models of collusion, as these assume that collu-
PT
heterogeneity increases the regulated price if firms do not
sive agreements are always reached instantaneously and
collude, but also makes collusion harder, rendering the net
coordination is not an issue. In the real world, coordi-
CE
effect ambiguous. Our experiment however suggests that
nation problems may be an important obstacle to estab-
the effect of collusion is stronger. Of course, it would be
lishing a cartel. Therefore, we also conducted an exper-
interesting to consider the effect on collusion of markets
AC
iment in which we allow for unrestricted communication
with even more firms (like e.g. Fonseca and Normann,
between agents.
2012).
In our experiment we find that in triopolies, size het-
Our conclusions are relevant for the debate on the op-
erogeneity indeed hinders collusion. We do not find ev-
timal industry structure. In a competitive market, a more
idence for this in duopolies. Also, we find weak evi-
homogeneous industry implies lower tariffs for network
dence that mergers that lead to symmetric market shares
users. When also taking the effects on collusion into ac-
indeed facilitate collusion, as theory predicts. Mergers to
count, our results suggest that more homogeneity may im-
asymmetry hinder collusion, as theory predicts, but only if 23
ACCEPTED MANUSCRIPT
ply higher tariffs. We address the uniform weighted yardstick in our experiment. Potters et al. (2004) compare this yardstick with a discriminatory yardstick in which the price a firm might charge depends on the cost levels of all other firms. They observe a higher tendency to collude with a discrimina-
CR IP T
tory yardstick than with a uniform yardstick in a duopoly with symmetric firms. They refrained from investigating the effect of different yardsticks on collusion in different industry structures. We will address this question in future work.
AN US
Another interesting extension is to include multiple dimensions of regulation. We considered price regulation,
but several industries also face quality regulation (Tangerås, 2009). This provides an interesting trade-off to firms, since higher quality is associated with higher costs while the in-
M
centive power tells them to reduce costs. In a theoretical model, Tangerås (2009) shows that larger firms need
ED
stronger incentives to improve quality and to reduce costs than smaller firms. It remains to be seen in which industry structures firms are able to collude on both dimensions
PT
and what effect this will have on social welfare.
CHECK OF HET NIET proofs moet zijn! https://www.elsevier.com/authors/author-
AC
CE
schemas/latex-instructions
24
ACCEPTED MANUSCRIPT
Table 10: Chat content.
Treatment
Discussed cost
Suggested
Agreement
Agreement
Other than
Never
all set 12
all set 12
made
all 12
all 12
5.9%
5.9%
100.0%
88.2%
79.4%
85.3%
Trio633
100.0%
96.9%
62.5%
68.8%
25.0%
6.3%
Trio642
97.1%
91.2%
52.9%
70.6%
29.4%
17.6%
Duo66
93.8%
87.5%
78.1%
78.1%
25.0%
0.0%
Duo84
100.0%
71.9%
53.1%
90.6%
62.5%
37.5%
CR IP T
Trio444
ED
M
AN US
Entries reflect percentage of groups to which the statement applies at some point during the first 20 rounds of the experiment.
Table 11: Characteristics Trio444 and Duo66.
Duo66
340
320
5
25
Average
3.71
7.06
Median
1
2.5
0
0
PT
Trio444
Total rounds
# rounds with agreement different from 12
AC
CE
Round first suggested to all set 12:
Instances of tacit collusion
“Round first suggested to all set 12” is the first round in which any subject in a market raised the possibility that all players should set their cost level equal to 12. When no subject ever raised that possibility, that observation is set equal to 21. Tacit collusion is defined in this context as all subjects setting cost level 12 without ever having discussed that possibility.
25
ACCEPTED MANUSCRIPT
Appendix A. Selected proofs
Note that this is indeed strictly positive for all ak > αi if and only if the term on the first line is always larger than
For part (a), using (4) and (2) the
Proof of Theorem 1.
the term on the second line; in other words, if
regulated price with competition is given by ∗
p =
n X j=1
α j c∗j = 1 − r0 (c∗j ) c∗j .
∂ 1 − r0 − r00 c∗ c∗0 > 0, ∂α
(A.1)
where we have dropped the arguments for ease of exposi-
to say, a large increase can always be written as the sum
tion. With 1 − r0 (c∗ (α)) = α, this simplifies to the require-
of many infinitesimally small increases. As noted, we
ment that
∂ αc∗0 − r00 c∗ c∗0 > 0. ∂α
will interpret an increase in firm size heterogeneity as the transfer of market share from a (weakly) smaller to a (weakly)
Hence we need
larger firm. Suppose that we transfer market share ∆α
c∗0 + αc∗00 −
∂ 00 ∗0 ∗ r c · c − r00 c∗0 c∗0 > 0. ∂α
AN US
from firm i to firm k, with αi ≤ αk . This increases firm
size heterogeneity. The resulting change in regulated price then equals
CR IP T
We look at an increase in firm heterogeneity. Needless
(A.2)
Note that 1 − r0 (c∗ (α)) = α. Differentiating through twice with respect to α, we obtain, respectively,
− +
ED
−
∗
M
∆ p∗ =
1 − r (c (αk + ∆α )) c∗ (αk + ∆α ) 1 − r0 (c∗ (αk )) c∗ (αk ) 1 − r0 c∗ (αi − ∆α ) c∗ (αi − ∆α ) 1 − r0 c∗ (αi ) c∗ (αi ) , 0
∂ (1 − r0 ) = r00 · c∗0 = −1 ∂α ∂ (r00 · c∗0 ) = r000 · c∗0 + r00 · c∗00 = 0. ∂α
c∗0 + αc∗00 − r00 c∗0 c∗0 > 0.
share α. Evaluating the effect on p∗ of a change in ∆α , we
PT
From (A.3), this simplifies to
∂∆ p∗ = 1 − r0 (c∗ (αk + ∆α )) c∗0 (αk + ∆α ) ∂∆α
CE
αc∗00 + 2c∗0 > 0
− r00 (c∗ (αk + ∆α ))c∗ (αk ) c∗0 (αk + ∆α ) − 1 − r0 c∗ (αi − ∆α ) c∗0 (αi − ∆α ) + r00 c∗ (αi − ∆α ) c∗ (αi ) c∗0 (αi − ∆α )
Note that this is equivalent to the requirement that αc∗ (α)is
AC
strictly convex in α. Using (A.4), we have ∗00
αc
For infinitessimally small ∆α , this implies ∂∆ p∗ = 1 − r0 (c∗ (αk )) c∗0 (αk ) ∂∆α ∆α →0
− r00 (c∗ (αk ))c∗ (αk ) c∗0 (αk ) − 1 − r0 c∗ (αi ) c∗0 (αi ) + r00 c∗ (αi ) c∗ (αi ) c∗0 (αi )
(A.4)
Using (A.4), condition (A.2) simplifies to
with c∗ (α) the equilibrium cost level of a firm with market
get
(A.3)
! −αr000 c∗0 αr000 ∗0 ∗0 + 2c = + 2c = 2 − 00 c . r00 r ∗0
With c∗0 > 0 and r00 < 0, we thus need the condition in (a). For part (b) note that with equal market shares, we have p∗ = c∗ (1/n) . With 1 − r0 (c (1/n)) = 1/n and r concave, this immediately implies the result. 26
ACCEPTED MANUSCRIPT
For part (c), denote the price before the merger as p∗ ,
(a) Suppose r(c) =
c∗ =
p∗0 − p∗ = (αi + αk ) c∗ (αi + αk ) − αi c∗ (αi ) − αk c∗ (αk ) .
AN US
Now
r (cm ) − r (c∗ ) δˆ = 1 − 0 ∗ m . r (c ) (c − c∗ )
=1−
Consider this as a function of c∗ . For ease of exposition, r (c∗ ) .
∂δ = ∂c∗
c
r0
+ ∆r r00 ∆c r0 r0 ∆2c
− ∆r
r0
=
Note that
ED
∆c , ∆r > 0. Dropping arguments, we then have r0 ∆
r0 (r0 ∆c
− ∆r ) + r0 r0 ∆2c
PT
The denominator is obviously positive. With
∆r r00 ∆c
r0 (c∗ )
b . 1 + 2b − α
increasing in n. (b) Suppose r(c) = bc − ac2 , with a, b > 0. Note that r0 (c) = b−2ac and r00 (c) = −2a. Hence the function
.
is concave. To find c∗ , equate r0 (c) to 1 − α :
> 0
c∗ =
concavity implies that r0 (r0 ∆c − ∆r ) > 0. However, also due to concavity, we have ∆r r00 ∆c < 0. Hence the numer-
CE
AC
b+α−1 . 2a
To find cm , we simply set α = 1 :
ator is ambiguous. It is positive if and only if r00 <
1 2+b−2α
This critical discount factor is decreasing in α, hence
M
−
1
4b − 4 (1+b−α)2 r (cm ) − r (c∗ ) = 1 − δˆ = 1 − (1 − α) (cm − c∗ ) 1 (1 − α) 4b12 − 4(1+b−α) 2
which we can write as
and ∆r ≡
1 . 4b2
CR IP T
has all firms setting c∗ ≡ c∗ (1/n) . From (11), we thus
−
.
! 1 1 1 1 r c =r − = = 2 2b 4b 4b 4b 1 2 + b − 2α 1 b = r c∗ = − 2 (1 + b − α) 4 (1 + b − α)2 4 (1 + b − α)2 m
αi = 1/n for all firms, whereas the competitive outcome
r (cm )
4 (1 + b − α)2
We also have
Proof of Theorem 2. First note that with symmetric firms,
r (cm ) − r (c∗ ) δˆ = 1 − , (1 − α) (cm − c∗ )
1
cm =
and αk c∗ (αi + αk ) > αk c∗ (αk ) . Hence p∗0 − p∗ > 0.
denote ∆c ≡
−b
To find cm , we simply set α = 1 :
With c∗ increasing, we have that αi c∗ (αi + αk ) > αi c∗ (αi )
c∗
1√ 2 c
concave. To find c∗ , equate r0 (c) to 1 − α to find
we thus have from (A.1)
cm
c − bc. Note that r0 (c) =
and r00 (c) = − 14 c−3/2 < 0. Hence the function is
the price after the merger as p∗0 . If firms i and k merge,
have
√
cm =
−r0 (r0 ∆c − ∆r ) , ∆r ∆c
b . 2a
We thus have b2 4a b2 − (1 − α)2 r(c∗ ) = . 4a
hence if r is sufficiently concave. In that case δ is increas-
r(cm ) =
ing in c∗ . With c∗ decreasing in the number of firms, it implies that δ is decreasing in the number of firms. Now Proof of Corollary 1.
First note that we look at equally
2
2
2
2a
2a
b −(1−α) b m ∗ 1 4a − ˆδ = 1− r (c ) − r (c ) = 1− 4a = . b b+α−1 (1 − α) (cm − c∗ ) 2 (1 − α) −
sized firms throughout, so for each firm α = 1/n. We consider the three families of functions in turn.
Hence, this is affected neither by α nor by n. 27
ACCEPTED MANUSCRIPT
(c) Suppose r(c) = bc−ec , with b > 2. Note that r0 (c) =
Proof of Lemma 2. Note that, from (11), we can write (1 − αi ) cm − c∗i + r(c∗i ) − r (cm ) ˆ i) ≡ δ(α , (A.5) (1 − αi ) cm + αi c∗i − p∗ P with p∗ ≡ j α j c∗j the competitive price. Take any two
b − ec and r00 (c) = −ec < 0. Hence the function is concave. To find c∗ , equate r0 (c) to 1 − α : c∗ = ln (b + α − 1) .
ˆ j ) − δˆ (αk ) , firms, j and k, with α j > αk . Consider δ(α
To find cm , we simply set α = 1 :
the difference between the critical discount factor of firm j and that of firm k. For the lemma to hold, we need this
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cm = ln b.
difference to be strictly negative. Using (11), we can write # Z αj " ˆ (α) ∂ δ 1 ˆ j ) − δˆ (αk ) = δ(α dα. (A.6) 2 αk ∂α
Note that the restriction b > 2 assures that both c∗ and cm are strictly positive. We thus have
Now
r (cm )
r (c∗ )
It is sufficient that ∂δˆ (α) /∂α < 0 for all relevant α. Denote the numerator of (A.5) as num, and the denominator
AN US
r cm = b ln b − b r c∗ = b ln (b + α − 1) − (b + α − 1)
as den. Hence ∂δˆ (α) /∂α < 0 if den · num0 − num · den0 < 0.
− (1 − α) (cm − c∗ ) b ln b − b − (b ln (b + α − 1) − (b + α − 1)) =1− (1 − α) (ln b − ln (b + α − 1)) b b ln b+α−1 + (α − 1) b 1 =1− =1− + b b 1 − α ln b+α−1 (1 − α) ln
If the perfect symmetric collusive agreement is stable, we necessarily have num < den and num, den > 0. Further-
b+α−1
so
ED
M
δˆ = 1 −
PT
∂δˆ b 1 =− + 2 2 ∂α b (1 − α) (b + α − 1) ln b+α−1 b
2
AC
(1 − α)
<
hence
b ln
b b+α−1
0 ∗0 m ∗ num0 = − cm − c∗i − (1 − αi ) c∗0 i + r · ci = − c − ci den0 = − cm − c∗i + αi c∗0 i .
fact that num < den, this necessarily implies that (A.7) is satisfied. This establishes the result.
1 2 b (b + α − 1) ln b+α−1
!2
more, note that
Hence num0 < 0, and den0 > num0 . Combined with the
CE
This is positive if
(A.7)
Proof of Theorem 3.
Consider a move from industry
structure A to industry structure B, where B is more heterogeneous. Without loss of generality, αiB = αiA − ∆ and
(b + α − 1) < (1 − α)2 .
αkB = αkA + ∆, for some ∆ > 0 and αiA ≤ αkA , whereas αAj = αBj for all j , i, k. Denote the smallest market share
Numerically, we can show that this is always the case.
of all other firms as α, so α = min j,i,k {α j }. From Lemma 2
Therefore this critical discount factor is increasing in α,
the smallest firm determines the critical discount factor.
hence decreasing in n.
We have three possibilities to consider: first, αiB > α; second, αiA < α and third αiB < α < αiA . 28
ACCEPTED MANUSCRIPT
In case I, αiB > α, the critical discount factor under industry structure κ ∈ {A, B} is given by 1 − α cm − c∗ (α) + r(c∗ (α)) − r (cm ) ˆ δˆ κ ≡ δ(α) , ≡ 1 − α cm + αc∗ (α) − p∗κ (A.8) with p∗κ the competitive equilibrium price in market structure κ. Note that c∗ (α) is not affected by the distribution
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of market shares over other firms. As market structure B is more heterogeneous than market structure A, we have p∗B > p∗A , hence δˆ B > δˆ A . In case II, αiA < α. Hence, the critical discount factor
AN US
under industry structure κ is 1 − ακi cm − c∗ ακi + r(c∗ ακi ) − r (cm ) ˆ κ) = δˆ κ ≡ δ(α . i 1 − ακi cm + αi c∗ ακi − p∗κ (A.9)
From the proof of Lemma 2, for fixed p∗κ , δˆ i (αi ) is decreasing in αi . Hence, as αiB < αiA , for fixed p∗κ we have
M
δˆ B > δˆ A . The fact that p∗B > p∗A only strengthens this result.
ED
The proof for case III, where αiB < α < αiA follows from a combination of the above cases; first consider a
PT
decrease from αiA to α. From the analysis for case I, this ˆ Next consider a decrease from α leads to an increase in δ.
AC
ˆ increase in δ.
CE
to αiB . From the analysis for case II, this again leads to an
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