Price Fixing and Non-Price Competition
Marco A. Haan∗
Bastiaan M. Overvest∗∗
February 15, 2007
Abstract
We study a model in which firms can collude on prices, but not
on other aspects of their product. Consumers search sequentially
for a product and have firm-specific matching values. Firms exert
consumer-specific effort if they are visited. We find that collusive
prices may be lower than competitive prices, as low prices commit
firms not to compete too aggressively in the non-price dimension. Nevertheless, price fixing always hurts consumers and is unambiguously
bad for welfare.
JEL Classification Numbers: D43, D83, L41.
Keywords: price fixing, search, partial collusion.
1
Introduction
The typical industrial organization model of collusion considers the case that
firms are able to fully collude on every single aspect of their product. In
practice, however, firms are often only able to collude on a limited number
∗
Department of Economics, University of Groningen, PO Box 800, 9700 AV Groningen,
The Netherlands, e-mail: [email protected].
∗∗
Department of Economics, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands, e-mail: [email protected]. This paper benefited greatly from
comments by participants of the ASSET 2006 meeting, the EARIE 2006 conference, the
2006 NAKE Day, Ben Heijdra, and José Luis Moraga-Gonzalez.
1
of dimensions. Shops that have agreed on a high price for their merchandize
will be inclined to exert more effort to secure a sale from a customer that
has entered their shop. Construction firms that collude on prices are still
able to unilaterally decide on the amount of effort they put in proposing a
plan to suit the tastes of the customer they are facing. In these examples,
price-fixing firms conspire against the consumer, but still compete against
each other in the quality dimension.
A natural question that arises under such circumstances is to what extent
price fixing still helps firms in increasing their joint profits. Indeed, this
question was already posed by Stigler (1968, pg. 149):
”When a uniform price is agreed upon, or agreed to by, an industry, some or all of the other terms of the sale are left unregulated.
[...I]n the absence of free entry [...] the question arises: Will
any monopoly profit achieved by suppressing price competition
be eliminated by non-price competition?”
In this paper, we aim to answer this very question. We also study the welfare
effects of such partial collusion. One may argue that a high fixed price gives
firms an incentive to provide higher quality, which may ultimately benefit
consumers. We analyze whether this can be the case.
To study these issues, we use a model in which consumers search for a
product, and firms exert consumer-specific effort to try to secure a sale to
a consumer that has entered their store. Each consumer has firm-specific
matching values, which reflect the extent to which this consumer likes the
products that a firm has on offer. These matching values are unobservable
to firms, and only observable to a consumer after it has visited a firm. Firms
may form a price-fixing cartel, but are not able to collude on effort.
As an example, consider a home-owner who wishes her bathroom to be
restyled.1 She may ask a specialized firm to propose a design, tailored to her
specific needs and desires. The attractiveness of the offer will in part depend
on how well the design matches the taste of the home-owner, something that
is unobservable for the firm. It will also depend on the amount of effort that
1
This example is due to Wolinsky (2005).
2
the firm exerts. Higher effort implies higher quality. Upon learning the price
and details of the offer, the home-owner is free to contract with the current
firm, to contact another firm, or to quit being active in this market altogether.
In the last two cases, the firm incurs a loss, because the costs of preparing the
offer is buyer-specific and cannot be recouped. The general features of this
market (a buyer searches for potential sellers and sellers exert buyer-specific
effort) are shared by many real-world markets. They are exactly the types
of markets that we have in mind. We study price fixing in such markets.
Most surprisingly, we find that collusion may yield prices that are lower
than the competitive price. The intuition is that, by setting a low price, the
cartel effectively discourages competition in costly effort. If collusion does
increase prices, then it also increases equilibrium effort, and hence the quality
of the product that firms provide. This suggest that collusion may be good
news for consumers as it either lowers price, or raises quality. Alas, we show
that that is not the case. Also in our set-up, collusion lowers both consumer
and social welfare.
Admittedly, our paper is not the first one that studies a case in which
firms can collude in one dimension, but have to compete in some other.
We already mentioned Stigler (1968), who studies a perfectly competitive
industry with persuasive advertising, where firms can either collude on prices
or on advertising levels. He shows that, in the context of his model, any
benefits from price fixing are competed away provided that there are constant
returns to scale.
Jehiel (1992) and Friedman and Thisse (1993) consider collusion in a
duopoly model with horizontal product differentiation á la Hotelling. They
find that if firms first choose their locations non-cooperatively, and then
collude in prices, the equilibrium features minimum product differentiation.
Fershtman and Gandal (1994) and Brod (1999) analyze how collusion in
the product market affects profits in a model of R&D along the lines of
d’Aspremont and Jacquemin (1988). They show that if firms decide noncooperatively on R&D investments, and then expect to collude in the product
market, profits may be even lower than if they act non-cooperatively in both
stages. This depends on the size of the technological spillovers.
3
What these papers have in common is that firms first choose some investment before they compete on the product market. Current investments
then become a bargaining chip in future cartel negotiations. In this paper,
we are not interested in such bargaining. In our model, firms first choose collusive prices, and only then decide on some quality aspects of their product.
This set-up captures more naturally the issue that we are interested in: how
does a market operate where, after having fixed prices, firms are still able to
compete on other dimensions.
Also, we are not the first to study collusion in a model with search costs.
Bae (2003) also shows that collusive prices may be lower than equilibrium
prices, but through a different mechanism. He introduces consumer heterogeneity in the Diamond (1971) model and shows that in the noncooperative
equilibrium, too many consumers drop out of the market and do not search
at all. Firms are better off if they can coordinate on a lower price. Campbell
et al. (2005) show that, in a dynamic version of Stahl’s (1989) search model,
a decrease in search costs may increase collusive prices as such a decrease
also makes it easier for firms to observe defections from the cartel price.
The remainder of this paper is structured as follows. Section 2 introduces
our model. In section 3 we solve the model for the case that firms compete
in all dimensions, while section 4 studies the equilibrium with price fixing.
In section 5 we examine the welfare effects. Section 6 concludes.
2
The Model
In this section, we give the set-up of our model. The benchmark model that
we study is based on Anderson and Renault (1999) and Wolinsky (2005).
We consider a search model with a finite number of firms in which firms
have to exert buyer-specific effort, and buyers enjoy a firm-specific matching
value, that is a priori unknown to both the firm and the buyer. Anderson and
Renault (1999) also consider a search model with matching values and a finite
number of firms, but do not include buyer-specific effort. Wolinsky (2005)
considers a search model with matching values and buyer-specific effort, but
uses an infinite number of firms. Needless to say, neither of these papers
4
considers the possibility of collusion between firms.
More specifically, our model is as follows:
Buyers
We consider one buyer that has unit demand (e.g. a bathroom). It is immediate to generalize the analysis to an arbitrary number of symmetric buyers.
The buyer may approach a firm at cost s > 0. These costs include the costs
of finding a firm and the costs of transferring detailed information about her
demand. If the buyer approaches a firm i, she learns the price pi of the good,
and her total valuation for the product of the firm, which equals v + ei + ui .
Here, v > 0 is the intrinsic utility the buyer derives from obtaining the product, ei is the buyer-specific effort exerted by the firm, and ui is the matching
value between the buyer and the firm. We assume that v is sufficiently large
so as to guarantee that the buyer will always obtain the good. The matching
value ui is a random draw from a cumulative density function F , which has
support [0, 1]. For convenience, we also suppose that F is twice differentiable
and has positive marginal probability on its domain. The value ui is only observable to the buyer, and only after she has visited firm i. To guarantee that
R1
the buyer has a positive reservation value, we assume that s < 0 udF (u).
After any visit to a firm, the buyer may grant the project to a firm from
which she has received an offer, or search for another firm. Her payoff if she
buys from firm i after k ≥ 1 visits is given by U (ei , ui , pi , k) = v + ei + ui −
pi − ks. A buyer is allowed to come back to a previously visited firm at zero
costs. It is assumed that the firm commits to a price once it is announced.
We assume sequential search, which implies that the buyer does not have
to commit in advance to the number of searches that she will make. The
optimal strategy for the buyer is thus some stopping rule that indicates that
she should stop searching once she has found an offer that gives her net utility
of at least x̂, where x̂ is to be determined endogenously.
5
Firms
There are n firms. For ease of exposition, we assume that the marginal costs
of production are equal to zero for all firms. If a firm is not contacted by
the buyer, it obtains zero profits. If a firm i is approached by the buyer, it
announces its price pi and exerts effort ei to formulate a plan. This effort
may also include e.g. time to inform the consumer regarding all aspects of
the product.
The cost of effort is given by the function c(ei ). This function satisfies
the standard regularity assumptions c(0) = 0, c0 (ei ) > 0 for ei > 0, c0 (0) = 0
∂ c0 (e)
and c00 (e) ≥ 0. We also make the technical assumption that ∂e
> 0.
c00 (e)
Effectively, this condition requires that the marginal costs of effort are not
too convex.2 For e.g. a quadratic cost function, it is always satisfied.
Effort is sunk. Hence the profit of the firm conditional on being approached by the buyer is pi − c(ei ) if the buyer buys from this firm and
−c(ei ) if she does not. For simplicity, we assume that firms are not able
to charge diagnosis costs, i.e. firms cannot charge buyers a price solely for
formulating a plan. In a slightly different set-up, Wolinsky (2005) does allow for this possibility, but he finds that firms charge zero diagnosis costs in
equilibrium because of Bertrand competition in diagnosis fees.
Timing
We consider two scenarios as to how firms compete on this market. First,
they may act purely non-cooperatively. Second, they may form a price-fixing
cartel. In that case, the cartel announces a collusive price that maximizes
expected joint profit conditional on every firm adhering to the agreement.
Each firm that is visited by the buyer still exerts effort in a non-cooperative
fashion. This may be the case, for example, since prices are observable, but
effort levels are not. For any cartel, it will be next to impossible to monitor
the effort that individual firms exert in trying to close a deal with the buyer.
2
Denote marginal costs of effort as m(e) ≡ c0 (e). The condition then requires that
(m m0 − mm00 )/m0 m0 > 0, or m00 < m0 m0 /m. With m0 ≥ 0 and m > 0, this implies that
m00 should not be too high, hence that marginal costs should not be too convex.
0
6
For simplicity, we assume that firms are patient enough for the cartel to be
stable. Calculating the exact value of the discount rate δ for which this is
satisfied would greatly complicate the analysis without adding much insight.
In the non-cooperative benchmark, the timing in our model is as follows.
First, the buyer decides whether to search. If so, she picks a firm at random.
Upon being contacted, the firm announces price pi and exerts effort level ei .
After having observed this offer, the buyer may decide to search more firms.
After she has decided to search no longer, or after she has visited all firms,
the buyer decides whether and where to obtain the product.
In the price-fixing model, the timing is as follows. First, the firms collectively decide which uniform price p to set. Second, the buyer decides whether
to search. If so, she picks a firm at random. Upon being contacted, the firm
sets effort level ei . After having observed this offer, the buyer may decide
to search more firms. After she has decided to search no longer, or after she
has visited all firms, the buyer decides whether and where to buy.
3
The competitive benchmark
In this section, we derive the Nash equilibrium for the case that firms noncooperatively set both prices and efforts. We focus on the symmetric pure
strategy Nash equilibrium that consists of price pN and effort eN . The analysis here largely follows Wolinsky (2005).
The search-and-stop rule
Suppose that the buyer has not yet approached all n firms and that her
current best option yields utility v +ei +ui −pi ≥ 0. In the Nash equilibrium,
a visit to a new firm j will yield v + eN + uj − pN . The consumer will prefer
to buy from j if
uj > pN − pi − eN + ei + ui ≡ x.
R1
Therefore, the expected benefit of one more search is x (u − x)f (u)du Such
a search is worthwhile if and only if these benefits exceed the costs on one
more search, s. We thus have that the buyer is exactly indifferent between
7
searching and stopping if x ≥ x̂, with x̂ implicitly defined by
Z
1
(u − x̂)f (u)du = s.
(1)
x̂
Hence, the buyer stops searching as soon as the current best offer is such
that x > x̂, where x̂ is the unique solution to g(x̂) = s. In the event that the
buyer approached all n firms, it is clearly optimal for her to return to the
firm that offered her the best offer.
Competitive effort and price
Consider the decision of firm i. Suppose that all other firms charge pN and
set effort level eN . We will derive the best rep ly of firm i, which in turn will
allow us to derive the equilibrium values pN and eN .
The probability that the buyer samples firm i is given by
P r{firm i is sampled} =
1 F (x̂) F (x̂)2
F (x̂)n−1
+
+
+ ... +
,
n
n
n
n
which simplifies to
P r{firm i is sampled} =
1 1 − F (x̂)n
·
.
n 1 − F (x̂)
as F (x̂) ∈ (0, 1).
The probability that the buyer stops at firm i (i.e. makes an immediate
purchase), given that firm i is sampled, is equal to P r(x > x̂), or
P r(ei − eN + pN − pi + ui > x̂) = P r(ui − ∆ > x̂) = 1 − F (x̂ + ∆),
where ∆ ≡ pi − pN − ei + eN . Conditionally on firm i being sampled, the
buyer returns to firm i after having visited all firms if maxj6=i {uj } < ui − ∆,
R x̂+∆
which occurs with probability 0
F (u − ∆)n−1 f (u)du.
Combining these elements, the expected profits of firm i, conditionally on
8
being sampled, are given by
·
1 − F (x̂)
π(∆) = pi 1 − F (x̂ + ∆) + n
1 − F (x̂)n
Z
¸
x̂+∆
F (u − ∆)
n−1
f (u)du − c(ei ).
0
(2)
Following Anderson and Renault (1999), we assume that the profit function
is concave. Straightforward calculations show that
Proposition 1 The unique symmetric equilibrium has
pN =
1−F (x̂)n
1−F (x̂)
f (x̂) − n
c0 (eN ) =
1
R x̂
0
F (u)n−1 f 0 (u)du
1 − F (x̂)
.
1 − F (x̂)n
,
(3)
(4)
The buyer searches until she encounters a firm i with pN −pi −eN +ei +ui > x̂,
with x̂ given by (1), or until she has visited all firms. In the latter case, she
buys from the firm that offers her the highest net utility.
Proof. The expressions follow directly from solving the first-order conditions and imposing symmetry. Existence and uniqueness then follows from
the concavity of the profit function and convexity of the feasible set.
The equilibrium price pN coincides with Anderson and Renault’s (1999)
result. This should not come as a surprise. Given that effort is set optimally,
firms in our model face the exact same decision problem regarding price as
the firms in their model do. The comparative statics are as follows:
Proposition 2 If 1 − F (x) is log-concave on [0, 1], then price pN is (i) increasing in search costs s and (ii) decreasing in the number of firms n. Furthermore, effort eN is (iii) increasing in s and (iv) decreasing function in
n.
Proof. Part (i) and (ii) follow directly from Anderson and Renault
(1999). For (iii), first note from (1) that x̂ is decreasing in³ s. Hence,
´ to
1−F
(x̂)
show that eN is decreasing in s, we need to establish that ∂∂x̂ 1−F (x̂)n < 0.
9
This derivative equals
∂
∂ x̂
µ
1 − F (x̂)
1 − F (x̂)n
¶
=
f (x̂) [nF (x̂)n − F (x̂) − (n − 1)F (x̂)n+1 ]
.
F (x̂)(1 − F (x̂)n )2
Its sign is strictly negative if nF (x̂)n−1 − (n − 1)F (x̂)n < 1, which clearly
holds for n = 2. Now suppose it holds for some n > 1. Then it will also hold
for n + 1 if nF (x̂)n − (n − 1)F (x̂)n+1 > (n + 1)F (x̂)n+1 − nF (x̂)n+2 . This
inequality can be reduced to F (x̂)2 − 2F (x̂) + 1 > 0, which is satisfied for
all F (x̂) ∈ (0, 1). Hence, by induction, the result follows. Finally, to derive
result (iv), note that
∂
∂n
µ
1 − F (x̂)
1 − F (x̂)n
¶
=
(1 − F (x̂))F (x̂)n ln(F (x̂))
.
(1 − F (x̂)n )2
which is strictly negative, as ln(F (x̂)) < 0.
Proposition 2 shows that the equilibrium properties of our model are
intuitive. For well-behaved distribution functions3 , equilibrium price and
effort are decreasing in the number of firms. As competition intensifies,
firms lower their price to attract the buyer. At the same time, firms exert
less effort as the probability that the buyer eventually buys at a given firm
decreases. As search costs increase, firm effectively have more market power
once they are visited by a consumer. This allows them to set a higher price.
At the same time, firms exert more effort as the probability that the buyer
eventually buys at a given firm increases.
4
The price-fixing outcome
Under price-fixing, each cartel member has agreed to post a fixed price, but
is free to compete as vigorously in effort as it pleases. This type of collusive
behavior (colluding in prices, but competing in quality), makes sense if prices
are freely observable for all firms, but effort is not. In our bathroom example,
the consumer will be able to communicate the price that some other firm has
3
Logconcavity of 1 − F is equivalent to an increasing hazard rate, a property that holds
for most common distribution functions.
10
charged, but will not be able to communicate how much effort that firm
has put into the design. Also, a cartel will prefer a unidimensional collusive
scheme if agreements about quality require laborious product descriptions.
These descriptions are not only costly to write, but they may also be used
by the antitrust authority as evidence of collusive behavior.
One example of such a price-fixing industry is the American cigarette
industry in the 1920s and 1930s. There, according to Fershtman and Gandal
(1994), the three major firms colluded on price but competed on advertising
to garner more sales.
As an aside, one can also imagine a cartel that makes a quality-fixing
agreement, but competes in prices. Consider, for example, the Sugar Institute. This cartel of American sugar-refining firms, that operated between
1927 and 1936, did not make price agreements, but instead forced its members to obey extensive quality regulations, see Genesove and Mullin (2001).
A quality-fixing cartel in our model would simply exert zero effort because
raising effort only increases the costs for the winning firm, without increasing
the probability of a sale.
We look for the profit-maximizing cartel price p∗ . Given that it is finite
(we will confirm this below), our assumption that the market is covered
implies that the probability that the buyer will ultimately obtain the good
is equal to 1. The expected total cost for the cartel is the expected sum of
P
i
effort, which is n−1
i=0 c(e)F (x̂) . Hence, the problem of the cartel is to set a
price p∗ as to maximize
Π = p∗ − c(e) ·
1 − F (x̂)n
.
1 − F (x̂)
Effort e is implicitly defined by the first-order condition for individual firm
optimality. Thus, the cartel explicitly takes the effect of the fixed cartel price
on the firms’ equilibrium effort into account. Under the assumption that all
other firms set e∗ , we will derive the best reply of one firm, which allows us
to derive e∗ . It will be convenient to work with a short-hand expression of
per-firm profit:
π(∆) = p∗ H(∆) − c(ei ).
11
where
1 − F (x̂)
H(∆) ≡ 1 − F (x̂ + ∆) + n
1 − F (x̂)n
Z
x̂+∆
F (u − ∆)n−1 f (u)du.
0
is the probability that the buyer obtains the good from firm i, conditional
on firm i being sampled by the buyer. Now, per-firm effort can be implicitly
written as
c0 (e) = p∗ hei |∆=0 .
(5)
where hei |∆=0 denotes the partial derivative of H(∆) with respect to ei ,
evaluated in ∆ = 0. We evaluate this marginal probability of sale in ∆ = 0
as in equilibrium all firms charge the same (collusive) price and provide the
same level of effort. For notational convenience, we will simply write he .
Equation (5) implicitly defines effort as a function of price. Totally differentiating, we find
Lemma 1
e0 (p) =
he
00
c (e)
> 0.
(6)
Proof. Recall that c00 (ei ) > 0. Therefore, it is sufficient to show that the
marginal probability of sale he > 0. We have
h(e) = f (x̂) +
h
i
R x̂
n [1 − F (x̂)] (n − 1) 0 F (u)n−2 f (u)2 du − F (x̂)n−1 f (x̂)
1 − F (x̂)n
.
This expression is strictly positive if
f (x̂) −
nf (x̂)F (x̂)n−1 (1 − F (x̂))
> 0.
1 − F (x̂)n
Clearly, this is true for n = 1. Given that it holds for n, it invariantly holds
for n + 1 if
1 − F (x̂)n+1 − (n + 1)(1 − F (x̂))F (x̂)n > 1 − F (x̂)n − n(1 − F (x̂))F (x̂)n−1 .
This reduces to n(1 − F (x̂)) > 0 which is satisfied for all F (x̂) ∈ (0, 1).
12
Lemma 1 is intuitive; as a firm increases its effort, it increases the probability that it trades with the buyer. Therefore, as price increases, individual
firms try to win the contract by exerting more effort. A bigger trophy results
in more effort. We can now show:
Proposition 3 With price fixing, the optimal cartel price is
p∗ =
c00 (e∗ ) 1 − F (x̂)
.
h2e 1 − F (x̂)n
The effort exerted by each firm in the cartel is given by
c00 (e∗ ) 1 − F (x̂)
.
c (e ) =
he 1 − F (x̂)n
0
∗
2
Proof. First, we show that Π is strictly concave. Note that − ∂∂pΠ2 has the
sign of c00 (e)e0 (p) + c0 (e)e00 (p). As e00 (p) = 0 and e0 (p) > 0 (by Lemma 1),
strict concavity of Π follows. Therefore, the sufficient condition for the unique
optimal price p∗ is ∂Π
= 0, or
∂p
∂Π
1 − F (x̂)n
= 1 − c0 (e) · e0 (p) ·
.
∂p
1 − F (x̂)
This derivative can be simplified by inserting the expressions for c0 (e) and
e0 (p) in (5) and Lemma 1. Rewriting gives p∗ .
The level of effort exerted by each firm in the cartel is found by substituting the expression for p∗ into equation (5). Given that the regularity
0
∂ c (e)
> 0 is satisfied, a unique positive solution to this equation
condition ∂e
c00 (e)
exists.
The cartel sets a fixed price p∗ , taking the opportunistic behavior of its
own members into account. Agreeing to post a very high collusive price may
not be in the cartel’s best interest. When firms have a strong incentive to
compete along the effort dimension, the cartel’s profit may be competed away
through vigorous competition in effort. Therefore, it is sometimes optimal
for the colluding firms to set a relatively low price, as the following result
shows.
13
Corollary 1 The optimal price-fixing cartel price p∗ is lower than the Nash
equilibrium price pN if and only if
c00 (e∗ ) < he .
Proof. The Nash price pN is strictly higher than the collusive price p∗ if and
only if
1−F (x̂)n
f (x̂)
1−F (x̂)
−n
1
R x̂
This is equivalent to
−
0
F (u)n−1 f 0 (u)du
>
c00 (e∗ ) 1 − F (x̂)
.
h2e 1 − F (x̂)n
H
c00 (e∗ ) 1 − F (x̂)
>
.
hp
h2e 1 − F (x̂)n
Note that hp = −he : a unit decrease in effort has the same effect on the
probability that the firm is selected by the buyer as a unit increase in price.
This allows us to simplify the above inequality to
c00 (e∗ ) <
1 − F (x̂)n
· H · he .
1 − F (x̂)
(7)
If all firms charge the same price and provide the same level of effort, we
1−F (x̂)
have that H = 1−F
, and therefore a collusive firm charges a lower price
(x̂)n
(and exerts less effort) than a non-cooperative firm if
c00 (e∗ ) < he .
Thus, if the costs of effort respond sluggishly to an increase in effort (i.e.
c (e) is sufficiently small), the cartel chooses to set a low price in order to
prevent a costly war in effort between the firms. For instance, if the cost
00
of effort function is given by c(e) = ce2 and the matching value is drawn
from the uniform distribution, a price-fixing cartel agrees on a price below
the competitive price if c < 1/2.4
4
Note that concavity of Π, and hence an equilibrium in pure strategies, requires c > 1/4.
14
With price fixing, we have the same comparative statics as in the competitive benchmark:
Proposition 4 If 1 − F (x) is log-concave on [0, 1], then price p∗ is (i) increasing in search costs s and (ii) decreasing in the number of firms n. Furthermore, effort e∗ is (iii) increasing in s and (iv) decreasing function in
n.
Proof. First, we proof part (iii). We show that effort decreases in x̂, which
implies the desired result as x̂ decreases in s. Note that e∗ is uniquely deter00 (e)
(x̂)n
c00 (e)
mined by cc0 (e)
= he K(x̂), where K(x̂) ≡ 1−F
As
decreases in e, e∗
1−F (x̂)
c0 (e)
decreases in x̂ if he K(x̂) increases in x̂.
The derivative of he K(x̂) with respect to x̂ is f 0 (x̂)(K(x̂) − nF (x̂)n−1 ) +
f (x̂)K 0 (x̂) where K 0 denotes the derivative of K. Using induction on n, it
is straightforward to show that K(x̂) − nF (x̂)n−1 is positive. Furthermore,
f (x̂)2
. Hence, we have
log-concavity implies f 0 (x̂) > − 1−F
(x̂)
∂he K(x̂)
f (x̂)2
> −
(K(x̂) − nF (x̂)n−1 ) + f (x̂)K 0 (x̂)
∂ x̂
1 − F (x̂)
(n − 1)F (x̂)(1 − F (x̂)n−1 )
=
(1 − F (x̂))2
> 0.
0
∗
For part (i), note that we can write p∗ as p∗ = c h(ee . As e∗ is increasing
in s, it is sufficient to have he decreasing in s, and thus to have he increasing
in x̂. It can be shown that this is true if and only if
Z
0
x̂
n
f (x̂)(1 − F (x̂) ) + nf (x̂)
F (u)n−1 f 0 (u)du ≥ 0,
0
which requires
f 0 (x̂) ≥
−nf (x̂)
R x̂
F (u)n−1 f 0 (u)du
.
1 − F (x̂)n
0
2
(8)
f (x̂)
Again, log-concavity implies f 0 (x̂) > − 1−F
. Hence, (8) is always satisfied
(x̂)
15
if
−nf (x̂)
R x̂
F (u)n−1 f 0 (u)du
f (x̂)2
≥
−
.
1 − F (x̂)n
1 − F (x̂)
0
Rewriting this yields
1 − F (x̂)n
f (x̂) − n
1 − F (x̂)
Z
x̂
F (u)n−1 f 0 (u)du ≥ 0.
0
But the left-hand side is exactly the denominator of pN in (3). As the Nash
equilibrium price pN is well-defined, the condition is always satisfied. This
proves the result.
Along the same lines, for (iv) and (ii) to hold, we need to show that he is
increasing in n. First of all, note that
Z
F
n+1
Z
x̂
(x̂)
F
n−2
2
(u)f (u)du > F (x̂)
0
and
Z
x̂
n
F n−1 (u)f 2 (u)du
0
Z
x̂
F
n−2
x̂
2
(u)f (u)du <
0
F n−1 (u)f 2 (u)du.
0
This implies
¡
1−F
n+1
¢
(x̂)
Z
Z
x̂
F
n−2 2
x̂
n
f < (1 − F (x̂))
0
F n−1 f 2 ,
0
where we have shortened the integrals for brevity. We thus have
¡
¢
1 − F n+1 (x̂)
Z
Z
x̂
F
n−2 2
x̂
n
f < (1 − F (x̂))
0
F n−1 f 2 +F (x̂)n−1 f (x̂) (1 − F (x̂)) ,
0
which can be written
·Z x̂
¸
¢
n−2 2
n−1
1−F
(x̂)
F
f − F (x̂) f (x̂)
0
·Z x̂
¸
n
n−1 2
n
< (1 − F (x̂))
F
f − F (x̂) f (x̂) .
¡
n+1
0
16
This immediately implies
R x̂
1 − F n+1 (x̂) 0 F n−2 f 2 − F (x̂)n−1 f (x̂)
n+1
<
,
R
x̂ n−1 2
n f (x̂)
1 − F n (x̂)
n−1
F
f
−
F
(x̂)
0
so
(n + 1)
hR
x̂
0
i
F n−1 f 2 − F (x̂)n f (x̂)
1−
F n+1 (x̂)
>
(n − 1)
hR
x̂
0
i
F n−2 f 2 − F (x̂)n−1 f (x̂)
1 − F n (x̂)
,
hence
hR
x̂
0
i
F n−1 f 2 − F (x̂)n f (x̂)
>
1 − F n+1 (x̂)
hR
i
x̂ n−2 2
n−1
(n − 1) n (1 − F (x̂)) 0 F
f du − F (x̂) f (x̂)
f (x̂) +
f (x̂) +
(n + 1) n (1 − F (x̂))
1 − F n (x̂)
,
which exactly implies he (n + 1) > he (n). Hence, he is increasing in n, which
proves the theorem.
Also with price fixing, the comparative statics are intuitive. For wellbehaved distribution functions, equilibrium price and effort are decreasing
in the number of firms. As the number of firms increases, the competitive
pressure increases. This implies that the cartel has to set a lower price to
avoid a costly war in effort. As a result, equilibrium effort levels also decrease.
As search costs increase, firms effectively have more market power, once they
are visited by a consumer. This allows the cartel to set a higher price, which
results in higher equilibrium effort.
5
Social Welfare
In this section, we study the welfare effects of price fixing in our model.
Our measure of social welfare W is the sum of expected buyer’s utility and
expected industry profit. Formally,
W = E[v + e + u − k(c(e) + s)].
17
Using the facts that s ≡
E[k] equals
1−F (x̂)n
,
1−F (x̂)
R1
x̂
(u − x̂)f (u)du and the expected number of visits
welfare boils down to
1 − F (x̂)n
W =v+e−
· c(e) + x̂ [1 − F (x̂)n ] +
1 − F (x̂)
Z
x̂
udF (u)n .
0
Since the third and fourth terms of this expression are independent of effort,
a social planner who can dictate effort would set effort eW such that
c0 (eW ) =
1 − F (x̂)
= c0 (eN ).
n
1 − F (x̂)
Hence, the socially optimal level of effort equals the level of effort under Nash
play. This is a notable result in itself. Wolinsky (2005) finds that, as buyers
do not fully bear the costs of search themselves, they visit too many firms
in equilibrium. This leads to an inefficient level of effort, as compared to
the first-best social welfare optimum. Our result shows that, if the social
planner is unable to meddle with the buyer’s search strategy, social welfare
under non-cooperative play is optimal in a second-best sense.
The finding that in the competitive mode effort is set (second-best) optimally implies that a price-fixing industry always reduces social welfare, as
c0 (e∗ ) =
c00 (e∗ ) 1 − F (x̂)
1 − F (x̂)
6=
= c0 (eW ).
n
hei 1 − F (x̂)
1 − F (x̂)n
Thus, effort is distorted away from the optimal competitive level. Moreover,
the search strategy of the buyer is invariant under both modes of competition
and therefore price-fixing firms necessarily yield lower welfare levels than noncooperative firms.
Return to the question of Stigler (1968) that we quoted in the introduction, we clearly have that firms are strictly better off with price fixing than
they are when competing in prices: the cartel can still choose to fix prices
at the competitive level pN . The fact that it chooses not to do so5 implies
that firms are strictly better off under price fixing. It is also clear that firms
5
Except, that is, in the knife-edge case that c00 (e∗ ) = he .
18
could do even better under full collusion, i.e. if they were also able to collude on effort. Thus, in the words of Stigler, the monopoly profit achieved
by suppressing price competition will be lowered, but not be eliminated by
non-price competition.
The welfare effects of price fixing in our model can thus be summarized
as follows:
Proposition 5 Regardless of its effect on price, price fixing is bad for welfare
and bad for consumers. Firms are better off if they can fix prices.
6
Concluding Remarks
In the real world, fully collusive industries in which all firms can collude
on every single aspect of their product, are hard if not impossible to find.
We explored the effects of a price-fixing agreement in a model with vertical
product differentiation and sequential search. In a model that builds on
Anderson and Renault (1999) and Wolinsky (2005), we showed that if firms
collude in prices but not in effort, they may prefer to set prices below the
competitive price to avoid a costly war in effort. Still, social welfare is always
lower in the case of price fixing. Firms are strictly better off with price fixing.
Our model has important implications for antitrust policy. To find evidence of collusion, competition authorities tend to look for industries where
prices are too high. Our analysis suggests that in industries where firms
can collude on prices but not on other aspects of their product, prices may
actually be lower than competitive prices.
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