1. society
2. welfare

Welfare Changes in the Cournot Setting With
An Empirical Application to the
Telecommunications Industry
Pedro M. Ferreira⇤
[email protected]
Heinz College, Carnegie Mellon University
October 8, 2014
Abstract
This paper characterizes the welfare efficiency of the Cournot equilibrium and provides bounds for the loss in consumer surplus, producer
surplus and welfare when the number of firms in the market changes.
I only assume that demand is decreasing in price and costs increasing
in the quantity produced as long as Cournot equilibrium exists. I show
how price, demand and average cost, before and after the number of
firms in the market changes, are enough to compute these bounds. I
apply these bounds to the Portuguese wireline market and conclude
that the welfare loss carried by Portugal Telecom’s monopoly in 2005
reduced significantly when the company was split in 2007.
JEL Classification: D43, L13
⇤
I would like to thank Professors Lowell Taylor, Rahul Telang, Ramayya Krishnan, John
Chuang and Marvin Sirbu for comments to previous versions of this paper. I would also
like to thank participants in the Heinz College Faculty Seminar Series (Carnegie Mellon
University, Pittsburgh, PA, September 22, 2009), in the 2009 Informs Annual Meeting
(San Diego, CA, October 14, 2009) and in the 1st Portuguese Workshop on the Economics
of ICTs (Porto, Portugal, March 11, 2010), for suggestions of improvement. This work
has been supported by the CMU-Portugal Program under grant number NGN56. All
remaining errors are my own.
1
Keywords: Cournot Equilibrium, Welfare Loss, Telecommunications Industry
2
1
Introduction
Research on measuring the welfare inefficiency of oligopolies captured significant attention during many decades now. Most of this research was prompted
by the early result from (Harberger 1954), who found that the welfare loss
associated with monopolies in US manufacturing could be surprisingly low.
Other researchers using the same methodology found similar results in other
sectors and countries. Among other concerns, these studies were criticized because they failed to measure costs well and then assumed constant marginal
costs. In more recent years, a number of theoretical studies determined
bounds for the welfare loss associated with Cournot competition ((Johari
and Tsitsiklis 2005), (Guo and Yang 2005)). Yet, most of these studies assume increasing convex costs. However, firms in a number of economically
relevant sectors exhibit economies of scale, such as capital intensive industries, to which the prior literature does not apply. Also in recent years, a
parallel stream of research looked at the e↵ect of the curvature of demand
on welfare loss but assuming again constant marginal costs ((Anderson and
Renault 2003), (Corchon 2008)).
This paper develops bounds for welfare loss allowing both for flexible
costs curves and for exploring the e↵ect of the curvature of demand. I study
how welfare under Cournot competition changes when the number of firms
in the market changes. I also provide bounds for the efficiency of Cournot
equilibrium. These bounds rely on data observed at equilibrium, and in particular data that are typically publicly available, and thus they can be widely
applied in practice. With more market data one may try to estimate demand
more precisely and, perhaps compute welfare loss directly. However, using
only observational data to regress demand served on price can only provide
approximate estimates for the price elasticity of demand due to the typical
endogeneity between these covariates that is hard to adjust for. The bounds
for welfare loss developed in this paper, or any other bounds for welfare
loss developed in the literature for that matter, are most interesting in cases
where little data are available. Bounds can be used to obtain approximate
estimates that can later be used as anchors to run sensitivity analyses to
learn more about potential losses.
The bounds for welfare loss developed in this paper rely only on price,
average cost and demand served, before and after the number of firms in the
3
market changes, and on the curvature of demand. The intuition for these
bounds is the following. Consider a demand function locally concave around
the market configuration before the number of firms in the market changes.
Firm entry decreases prices and moves equilibrium along a relatively inelastic
portion of the demand curve. As a result, total output is unlikely to change
much but consumers are likely to become better o↵. If, instead, demand is
locally convex then firm entry still reduces price but moves the equilibrium
along a relatively elastic portion of the demand curve. This results in a large
increase in total output but most of the surplus gain will go to firms.
Bounds for the change in welfare loss associated with the movements
described above can then be obtained by constraining the global degree of
concavity and convexity of the demand function. Intuitively, such a constraint ensures that the curvature of demand at the market equilibrium after
the number of firms in the market changes is relatively similar to the one at
the market equilibrium before this change. Otherwise, a significant change in
the curvature of the demand away from the equilibrium before the number of
firms in the market changed could break apart the arguments described in the
previous paragraph. Bounding the curvature of demand can be accomplished
resorting to the concept of ⇢-concavity as (Anderson and Renault 2003) did.
However, they assumed constant marginal costs and no fixed costs. These are
unnecessary restrictions that I lift in this paper. One only needs equilibria
to exist to study its welfare properties. In this regard, ?? showed that with
⇢-concave demand equilibrium exists and is unique as long as marginal costs
do not decrease at rate higher than demand does. As such, my results allow
for (not too) concave costs and thus may apply to capital intensive industries.
A main contribution of this paper is that my results allow both for exploring the e↵ect of the curvature of demand on welfare as well as for costs
to be concave. I allow average cost to change with production and show that
bounds for the welfare loss can be computed using only observed information
about the market at the equilibrium. In the later part of this paper I provide
an empirical application to the telecommunications sector and show ways to
bound the curvature of demand in a pragmatic setting. I also discuss why
applying Cournot competition to capital intensive industries is reasonable.
4
2
Literature Review
Capital intensive industries are likely to exhibit economies of scale. In this
industries there is typically room for only a few firms who benefit from
decreasing average costs to drive competition away. Firms with sufficient
capital for upfront investment who tap the market first are likely to enjoy
an advantage. Natural oligopolies emerge and regulation is often used to
preclude firms from abusing from market power. Otherwise, firms can raise
equilibrium prices well above average cost hurting consumers. Unfortunately,
most empirical studies looking at the welfare efficiency of oligopolies assume
constant marginal costs. See, for example, (Harberger 1954) and follow-up
studies such as ((Schwartzman 1960), (Worcester 1973), (Siegfried and Tiemann 1974), (Worcester 1975), (Gisser 1982), (Gisser 1986), (He↵ord and
Round 1978), (Jenny and Weber 1983) and (Dickson and Yu 1989). This assumption precludes them from considering how costs change when increased
competition expands output, which a↵ects profits, and possibly demand,
and thus changes welfare. (Stigler 1956), (Bergson 1973), (Tullock 1967) and
(Posner 1975) o↵er a number of additional arguments in favor of dismissing
this approach to measure welfare loss in oligopolies.
A number of theoretical studies that look at welfare efficiency do not
restrict marginal costs to be constant. However, all of them assume increasing convex costs and often require similar firms. For example, (Johari and
Tsitsiklis 2005) showed that if n similar firms with increasing convex costs
compete a-la Cournot to meet affine decreasing demand then welfare loss
is at most 1/(2n + 1). (Guo and Yang 2005) showed that with increasing
convex costs, decreasing demand and concave revenues, the welfare
Pn inefficiency of the Cournot equilibrium is given by (1 + 2✓ s1 )/(1 + 2✓ i=1 s2i ),
where si is the market share of the ith firm. Firms are ordered so that
s1 is the largest market share. Parameter is the maximum of (x) for
x > 0 with (x) implicitly defined by p(x) + s1 xp0 (x) = p( (x)x), where
p represents the price Rfunction. ✓ is the maximum of ✓(x) for x > 0 with
x
✓(x) = 1/2x2 p0 (x)/( 0 p(w)dw xp(x)). The bounds provided in these
papers for the welfare efficiency of a Cournot equilibrium are interesting because they do not require full knowledge of the cost functions. However,
the first paper requires firms to be similar. The second paper relaxes this
assumption as well as allows for a more general demand curve. Still, the
expressions obtained, including implicitly defined functions, are hard to un5
derstand from an intuitive point of view.
Other papers impose more restrictive assumptions on the shape of the
cost functions and instead explore the curvature of demand to study the
welfare efficiency of the Cournot equilibrium. For example, (Anderson and
Renault 2003) considered n similar firms with constant marginal cost and
no fixed costs facing a ⇢-concave demand with ⇢ > 1. They showed that
the Percentage Welfare Loss (PWL) of the Cournot equilibrium is given by
1 (1+(1/(n+⇢))(1+⇢/n) 1/⇢ . (Corchon 2008) showed that PWL decreases
in n, is quasi-concave in ⇢ and goes to 0 when ⇢ goes to either infinity or
-1. He also computed bounds for the PWL and extended them to the case
of firms with di↵erent constant marginal costs to show that assuming similar
constant magical costs underestimates PWL.
This paper o↵ers bounds for the welfare inefficiency of the Cournot equilibrium allowing both for a very flexible definition for costs and for exploring
the e↵ect of the curvature of demand thus bringing together under more
general results the advantages of the prior literature that only studied these
issues separately. I will also resort to the potencial ⇢-concavity of demand
to study welfare efficiency. However, I will not require costs to be convex as
long as Cournot equilibrium exists. More specifically, I use the result in ??
who generalized findings in (Novshek 1985) to show that if Ci00 (qi ) < P 0 (Q)
and P 0 (Q) + QP 00 (Q)  0 then Cournot equilibrium exists and is unique. Ci
represents the cost function of firm i and qi represents the quantity produced
at the Cournot equilibrium by this firm. P and Q represent the price function and the total quantity produced, respectively. Intuitively, the former
condition ensures that profits are concave. The latter condition implies that
reactions curves have negative slope.
3
Definitions and Assumptions
Consider n firms that o↵er a homogeneous product and compete a-la Cournot.
Let pn represent the price they charge at equilibrium and let p¯  +1 represent a large enough price such that pn  p¯. Consider the following two
assumptions, which will be assumed true throughout this paper: A1) de+
2
mand is given by D : <+
¯ and D(p) = 0
0 ! <0 in C with D(p) > 0 for p  p
6
+
0
for p > p¯; A2) firm j has variable marginal
cost cj : <+
0 ! <0 in C and
Rq
fixed cost Fj 0. Also let c¯j (q) = ( 0 cj (˜
q )d˜
q + F )/q represent the average
cost faced by firm j.
A non-negative function f defined in a convex domain R is ⇢-concave,
with ⇢ 6= 0, if f ((1
)x + y) [(1
)f ⇢ (x) + f ⇢ (y)]1/⇢ for 0   1 and
for all x, y 2 R. The traditional definition of concavity is obtained for ⇢ = 1.
This traditional definition, however, does not capture the fact that concave
functions may have di↵erent degrees of curvature. Parameter ⇢ in the definition of ⇢-concavity is introduced to do so. A function f is ⇢-concavity (with
⇢ > 0) if the straight line connecting any two of its points lies below f ⇢ (for
⇢ < 0 it must lie below f ⇢ ). This means that f ⇢ is concave (or convex
when ⇢ < 0). This non-linear transformation of the vertical axis from f to
f ⇢ allows for associating ⇢ with the degree of curvature of f . In words, correcting for ⇢ the function ”loses” its concavity and becomes ”only” concave.
Furthermore, a function with a higher ⇢ is ”more concave” and a ⇢-concave
function is also ⇢0 -concave for ⇢0 < ⇢. Figure 1 illustrates these propositions.
⇢-convexity is defined by reversing the inequality.
<< Figure 1 about here >>
For any positive monotone function f defined in a convex domain there exist ⇢1 and ⇢2 in the extended real line such that f is ⇢1 -concave and ⇢2 -convex
(see (Anderson and Renault 2003) for a proof) and ⇢1  ⇢2 . Therefore, in
+
my case, there exist ⇢d and ⇢+
d , with ⇢d  ⇢d , such that D is ⇢d -concave
and ⇢+
d -convex. As Figure (?) illustrates, this means, for example, that
demand can be, at the same time, convex and ⇢-concave, for some ⇢ < 1.
This allows for bounding demand from below and above, which I will use
extensively in this paper to establish my results. Figure 3 shows an example of these bounds in practice1 . Finally, a function is ⇢-linear if it is both
⇢-concave and ⇢-convex. Also, ⇢-concavity is defined for ⇢ = 1, 0, +1
using continuity arguments. For example, f is 0-concave, or log-concave, i↵
ln[f ((1
)x + y)] (1
)ln[f (x)] + ln[f (y)]2 . Notably, ⇢ = 1 is the
Because x2 is 1-convex then x2
y 2 (1 + ⇢(2/1)(x y)f 0 (y)/f (y))1/⇢ . Make y = 1
2
to obtain x
2x 1. Similarly, because x2 is 1/3-concave then x2  f (y)(1 + 1/3(x
0
1/3
y)f (y)/f (y)) . Make again y = 1 to obtain x2  (1 + 2(x 1)/3)3 .
2
The results in the paper do not explicitly consider the case ⇢ = 0 for sake of space.
All results extend trivially to this case using the definition of log concavity.
1
7
weakest case of ⇢-concavity, requiring only the function to be quasi-concave.
<< Figure 2 about here >>
<< Figure 3 about here >>
In line with Anderson and Renault (2003), all analyses in this paper assume ⇢d > 1. In this case (1 ⇢d )P 0 (D)+P 00 (D)D < 0, where P represents
the inverse of D3 . (Novshek 1985) showed how log-concavity of demand, that
is P 0 (D) + P 00 (D)D < 0, yields existence and uniqueness of Cournot equilibrium with increasing convex costs. Gaudet and Salant (1991) showed how
this result can be extended to allow for concave costs as long as c0j > P 0 .
Essentially, the first and second order conditions for Cournot equilibrium for
firm j with n firms in the market are P (Dn ) + Djn P 0 (Dn ) cj (Djn ) = 0 and
[P 0 (Dn ) + Djn P 00 (Dn )] + [P 0 (Dn ) c0j (Din )] < 0, respectively, where Djn represents the demand served by firm j and Dn the total demand served. The
second order conditions are satisfied with log concave demand and c0j > P 0
because both terms in brackets are negative (see (Vives, 1999) for a detailed
proof). To accommodate 1 < ⇢d < 0 rewrite the second order conditions as
[(1 ⇢d )P 0 (Dn ) + Djn P 00 (Dn )] + [(1 + ⇢d )P 0 (Dn ) c0j (Djn )] < 0 and therefore
for ⇢d -concave demand with ⇢d > 1 I need a stronger condition, namely
c0j > (1 + ⇢d )P 0 (Dn ), which still allows for concave costs4 .
Throughout the paper, and similarly to Anderson and Renault (2003), I
will assume that the first order conditions characterize the Cournot equilibrium. The first order condition for firm j using only the demand function is5
Dn /D0n = [¯
cj (Djn )
pn ]/[snj (1
c¯0j (Djn )D0n )]
(1)
When D is ⇢d -concave then D⇢d /⇢d is concave. This implies (1 ⇢d )D02 + DD00 < 0.
Substituting D0 = 1/P 0 and D00 = P 00 /P 03 yields the required condition.
4
For a similar argument see (Ewerhart 2013) who assuming (↵, )-biconcave demand
establishes new existence and uniqueness results (in my case ↵ = 1 and = ⇢d ).
5
The profit of firm j is given by (P (Q) c¯j (qj ))qj , where qj represents the quantity
produced by firm j. Setting the derivative to zero yields (P 0 (Dn ) c¯0j (Djn ))Djn + P (Dn )
c¯j (Djn ) = 0. Substituting D0 = 1/P 0 readily yields Expression 1.
3
8
where snj = Djn /Dn represents the share of the market served by firm j
at equilibrium and D0n is an abbreviation for D0 (pn ). Finally, note that in
this setting I also have pn  pn˜ for n
˜ n.
4
Changes in Surplus and Welfare
The ⇢d -concavity and the ⇢+
d -convexity of demand can be used to compute bounds for the loss in consumer surplus, producer surplus and welfare when the number of firms in the market changes. Let CS n , P S n
and W n represent the consumer surplus, producer surplus, and welfare at
Cournot equilibrium with n firms in the market. Let nb
na and define
P W Lna ,nb = (W nb W na )/W nb as the welfare loss, in percentage terms,
when the number of firms in the market decreases from nb to na . Similarly,
define P CSLna ,nb = (CS nb CS na )/CS nb for the case of consumer surplus
and define P P SLna ,nb = (P S nb P S na ) /P S nb for the case of producer surplus. I start with a fundamental Lemma that will be used later in the paper.
All proofs of all Lemmas and Results in this section are provided in Appendix
8.
Lemma 1: CS n 
D 2 (y)
(1
(1+⇢d )D 0 (y)
+ ⇢d (pn
0
(y) 1+1/⇢
d , y
y) DD(y)
)
pn .
P
n
0n
Define f n (⇢) = (1 + ⇢) nj=1 snj 2 (1 c¯0j (D
j )D ). Summing Expression
P
n
n
1 over all firms and substituting P S n =
c¯j (Djn ))Djn leads to
j=1 (p
2
f n (⇢) = (1 + ⇢)P S n D0n /Dn . This expression is still valid when some
firms decide not to produce at the Cournot equilibrium because such a firm
enjoys no profit and its market share is zero. This expression shows that
f n (⇢) can be compute using data observed at the equilibrium on the demand
served, how demand changes locally, price and average cost. It is now also
easy to see that f n (⇢) > 0 for all ⇢ > 1. The next Lemma relates producer
and consumer surplus at the Cournot equilibrium:
Lemma 2: P S n /CS n
f n (⇢d ).
Lemma 2 shows that f n (⇢) represents the ratio between producer surplus
and consumer surplus at the Cournot equilibrium with n firms in the market
and ⇢-linear demand. Note that the definition of f n (⇢) includes P S n explic9
itly. The intuition behind the other factors in this definition is illustrated in
Figure 4. Demand lies below a ⇢d -linear function that is tangent to it at the
Cournot equilibrium with n firms because demand is ⇢d -concave. Therefore,
consumer surplus is less that what it would be if demand was given by this
⇢d -linear function and pn the price charged. In this case, consumer surplus
2
would be given by Dn /[(1 + ⇢d )D0n ], which are the remaining factors in
the definition of f n (⇢)6 . Therefore, studying the behavior of f n (⇢) is enough
to understand how welfare at equilibrium splits between consumers and producers, which can provide regulators with valuable insight about the fairness
of the Cournot allocation.
<< Figure 4 about here >>
My first result pertains to consumer surplus loss at the Cournot equilbrium. Let g n,˜n (⇢) = (1 + ⇢(pn pn˜ )D0˜n /Dn˜ ) (1+1/⇢) for n > n
˜ . Note
n,˜
n
that g (⇢) can be computed using data observed at the equilibrium on the
demand served, how demand changes locally and price. Note also that the ⇢concavity of demand implies (1 + ⇢(x y)D0 (y)/D(y))1/⇢ > D(x)/D(y) > 0,
for y x. Making y = pn˜ and x = pn shows that g n,˜n (⇢) > 0 for any ⇢ > 1.
Result 1: P CSLnb ,na
1
1+⇢+
d
g nb ,na (⇢+
d ),
1+⇢d
n b > na .
Result 1 shows that g n,˜n (⇢) represents the ratio of consumer surplus at
Cournot equilibrium with n
˜ and n firms. The intuition behind the definition
n,˜
n
of g (⇢) is illustrated in Figure 5. Demand lies above a ⇢+
d -linear function
that is tangent to it at price pn because demand is ⇢+
-convex.
Therefore,
d
consumer surplus is more than what it would be if demand was given by this
+
n
⇢+
d -linear function with price p . The consumer surplus with this ⇢d -linear
demand function and price pn can be related to the consumer surplus if den
˜
mand was still given by a ⇢+
d -linear demand curve but shifted to yield D at
+
n
˜
n,˜
n +
price p . This relationship is given by g (⇢d ). The term (1 + ⇢d )/(1 + ⇢d )
in Result 1 arises from adjusting the latter consumer surplus with n
˜ firms
+
n
˜
in the market and price p from a ⇢d -linear demand curve to a ⇢d -linear
demand curve that is tangent to demand at price pn˜ . Finally, the consumer
surplus with such a ⇢d -linear demand curve, as seen above in the definition of
6
The reader familiar with computing consumer surplus with affine demand (⇢d = 1)
2
will recognize this expression since in this case consumer surplus is given by Dn /(2D0n ).
10
f n (⇢), is more than the consumer surplus with the true demand curve at the
Cournot equilibrium with n
˜ firms in the market because demand is ⇢d -linear.
Studying the behavior of g n,˜n (⇢) is enough to understand the percentage loss
in consumer surplus when the number of firms in the market changes. This is
often the fundamental information sought by regulators to learn how changes
in the number of firms in the market may a↵ect consumers. Result 2 pertains
to producer surplus loss at the Cournot equilibrium:
<< Figure 5 about here >>
Result 2: P P SLnb ,na
1
na (⇢+ )
1+⇢+
d f
d
g nb ,na (⇢+
d ),
1+⇢d f nb (⇢d )
n b > na .
Finally, I can bound the welfare loss at the Cournot equilibrium when
the number of firms in the market changes:
Result 3: P W Lnb ,na
1
na (⇢+ )
1+⇢+
d 1+f
d
g nb ,na (⇢+
d ),
1+⇢d 1+f nb (⇢d )
n b > na .
Result 3 shows how welfare can still increase when the number of firms
in the market reduces from nb to na . Let mnj = 1 c¯j (Djn )/pn represent the percent margin of firm j at equilibrium. Substituting
P Expression 1 into the definition of f n (⇢) yields f n (⇢) = [Rn /CS n ] nj=1 mnj snj ,
where Rn represents the sum of the revenues of the n firms in the market at the Cournot equilirium. This expression is the weighted average
of the percent margins of the firms in the market at the Cournot equilibrium, the weights being their market shares, scaled by the ratio of revenues
to consumer surplus.
Therefore, with ⇢-linearPdemand, P W Lnb ,na > 0 if
P
n
a
b
[1 + [Rna /CS na ] j=1
mnj a snj a ]/[1 + [Rnb /CS nb ] nj=1
mnj b snj b ] > (1/g nb ,na (⇢)).
In words, welfare increases if this scaled version of the weighed average of
the percent margins counters the loss in consumer surplus.
Consider now that one knows P P SLnb ,na . In this case, the following results hold:
Result 4: P CSLnb ,na
↵ if 1
(1
1/f na (⇢ )
P P SLnb ,na ) 1/f nb (⇢+d )
d
Result 5: P W Lnb ,na
↵ if 1
1+1/f na (⇢ )
P P SLnb ,na ) 1+1/f nb (⇢+d )
(1
d
↵, nb > na .
↵, nb > na .
Result 5 is particularly relevant for a decision maker because it defines an
11
area within the (⇢d , ⇢+
1 < ⇢d  ⇢+
d ) space, with
d , in which one knows that
welfare loss is at least ↵. Let = (1 ↵)/(1 P P SLnb ,na ). Figure 6 shows
the several cases that arise for the shape of this area, which depends on how
na
compares to 1 and to f nb (⇢+
d )/f (⇢d ). Similar results can be obtained for
P W Lnb ,na  ↵ with = (1 ↵)/(1 P P SLnb ,na ). Figure 7 shows the several
cases that arise. Combining Figures 6 and 7 provides a good understanding
of the regions where one is sure about where welfare loss lies as a function of
⇢d and ⇢+
d.
<< Figure 6 about here >>
<< Figure 7 about here >>
So far, I have been assuming that the demand curve does not change
while the number of firms in the market does. This, however, may be a
strong simplifying assumption. A number of factors, some potentially unknown to us, may lead demand to change. The fact that the number of firms
in the market changes may, just by itself, a↵ect the consumers’ willingness
to pay. It is hard to empirically separate the e↵ect of the change in the
number of firms in the market from the e↵ect of unobserved changes in other
covariates that may a↵ect demand (for example, changes in marketing) in
particular with observational data alone. However, it might still be useful to
know what happened to welfare even though one may not be able to isolate
the e↵ect due to the change in the number of firms. Results 4 and 5 are
powerful in this respect because they apply even when the demand curve
changes. To see this note that the proofs of these results rely on bounding the ratio between consumer and producer surplus before and after the
change in the number of firms in the market. Each of these bounds can be
obtained using a di↵erent demand curve. For example, in the proof of Result 4, I have used CS na /CS nb = [P S na /P S nb ][CS na /P S na ]/[CS nb /P S nb ].
Bounding [CS na /P S na ], using Lemma 2, is obtained using the demand curve
when there were na firms in the market. Likewise, bounding [CS nb /P S nb ],
using also Lemma 2, is obtained using the demand curve when there were nb
firms in the market. Nothing in the proof of this result requires these two
demand curves to be the same. The connection between them is provided
by [P S na /P S nb ], which is assumed to be known in this result. Similar arguments apply to Result 5, which also holds even when the demand curve
12
changes while the number of firms in the market does.
Finally, assume ⇢-linear demand and constant average costs. In this case,
I obtain:
P CSLnb ,na (⇢) = 1
P W Lnb ,na (⇢) = 1
(
na nb + ⇢ 1+ ⇢1
)
nb na + ⇢
nb na + 1 + ⇢ na nb + ⇢ (1+ ⇢1 )
(
)
na nb + 1 + ⇢ nb na + ⇢
b 1 (1
Note that P CSLnb ,na (⇢) = 1 ⇧n=n
P CSLn,n+1 (⇢)) and likewise for
na
nb ,na
PWL
(⇢). Therefore the change in consumer surplus and in welfare when
one firm shuts down provides already very useful information. Figures 8 and 9
show P CSLn+1,n and P W Ln+1,n for n = {1, 2, 3, 4, 5} and ⇢ 2 [ 1, 10]. Both
P CSLn+1,n and P W Ln+1,n decrease as the number of firms grows, for any
⇢ > 1. That is, the e↵ect of losing one firm on consumer surplus and welfare
decreases as more firms are in the market. P W Ln+1,n is quasi-concave in ⇢
and P CSLn+1,n is increasing and concave in ⇢. I have lim⇢! 1 P W Ln+1,n =
lim⇢!1 P W Ln+1,n = limn!1 P W Ln+1,n = 0. In this case of ⇢-linear demand
with constant average costs, the maximum welfare loss from losing one firm
is roughly 20%, obtained with a monopolist and ⇢ ⇡ 0.42. Note also that
the welfare efficiency of the Cournot equilibrium found in (Anderson and Renault 2003) for the case of ⇢-linear demand with similar constant marginal
costs and no fixed costs is obtained by making na = n in expression 2 and
allowing nb ! 1. I also have lim⇢! 1 P CSLn+1,n = limn!1 P CSLn+1,n = 0
but now lim⇢!1 P CSLn+1,n = 1/(1 + n). These results show that consumer
surplus and welfare loss are small when the Cournot equilibrium is around
an elastic portion of the demand curve. When the Cournot equilibrium is
around an inelastic portion of the demand curve, welfare loss is also small
but consumer loss can be significant. The latter is inversely proportional to
the number of firms in the market. Therefore, markets with inelastic demand
and few firms render the most concern for regulatory agencies when mergers
need to be considered. Typical sectors with impact on everyone’s daily life
include electricity and gasoline.
<< Figure 8 about here >>
13
<< Figure 9 about here >>
5
Efficiency of the Cournot Equilibrium
The ⇢d -concavity and the ⇢+
d -convexity of demand can also one used to compute bounds for the efficiency of the Cournot equilibrium with n firms. Define efficient allocation as one that maximizes welfare. Let w CS n , w P S n
and w W n represent the consumer surplus, the producer surplus and the welfare, respectively, at the efficient allocation with n firms. Define w P W Ln =
(w W n W n )/w W n as the percentage welfare loss carried by the Cournot equilibrium with n firms in the market. Similarly, define w P CSLn = (w CS n
CS n )/w CS n and w P P SLn = (w P S n P S n )/w P S n for the cases of consumer
and producer surplus, respectively.
Let wP
Djn represent the demand met by firm j at such an allocation. Make
w n
D = nj=1 w Djn and w snj = w Dj /w Dn . Let w pn represent the price charged
0
at the efficient allocation and define w D n = D0 (w pn ) for sake of abbreviation.
Let qi represent the quantity produced by firm i. The welfare with n firms
is expressed as
n
W =
Z
Pn
i=1 qi
P (q)dq
0
n
X
c¯(qi )qi
i=1
The first order conditions to maximize welfare yield
w n
p
c¯j (w Djn ) = c¯0j (w Djn )w Djn
(2)
for all j = 1, ...n. This expression shows that at the efficient allocation
profit is positive as long as average cost is increasing. Consider the following
lemma:
Lemma 3:
w
CS n 
D 2 (y)
(1
(1+⇢d )D 0 (y)
+ ⇢d ( w p n
14
0
(y) 1+1/⇢
d , y
y) DD(y)
)
w n
p .
The next Lemma relates consumer surplus
producer surplus at the efPn and
w n
w n 2 0 w n w 0n
ficient allocation. Let f (⇢) = (1+⇢) j=1 ( sj ) P
c¯j ( Dj ) D . Summing
Expression 2 over all firms and substituting w P S n = nj=1 (w pn c¯j (w Djn )w Djn
0
leads to w f n (⇢) = (1 + ⇢)w P S nw D n /w Dn 2 and therefore it is easy to see
that the sign of w f n (⇢) is the same as that of w P S n for ⇢ > 1. This expression also holds when some firms do not produce at the efficient allocation
because such a firm enjoys zero profit and has zero market share.
Lemma 4:
w
P S n /w CS n
w n
f (⇢d ), with w P S n > 0.
The next result relates consumer surplus at the Cournot allocation and
0
at the efficient allocation. Let w g n (⇢) = (1 + ⇢(w pn pn )D n /Dn ) (1+1/⇢) . As
before in the case of g n,˜n (⇢), I have w g n (⇢) > 0, for all ⇢ > 1. The next
Result pertains to consumer surplus loss:
Result 6:
w
P CSLn
1
1+⇢+
d w n
g (⇢+
d ).
1+⇢d
Lemma 4 shows that w f n (⇢) represents the ratio between the producer
surplus and the consumer surplus at the efficient allocation with n firms and
a ⇢-linear demand. Studying the behavior of w f n is thus enough to understand how welfare splits between consumers and producers at the efficient
allocation. This can provide valuable insight for policy making. In addition,
Result 6 shows that w g n (⇢) measures the loss in consumer surplus, in percentage terms, carried by the Cournot equilibrium. Studying the behavior of
w n
g (⇢) is thus enough to learn about how changes in the number of firms in
the market a↵ect consumers. Again, this can also provide valuable insight for
regulators. The next two results pertain to producer surplus and welfare loss:
Result 7:
w
P P SLn
Result 8:
w
P W Ln
1
1
n +
1+⇢+
d f (⇢d ) w n
g (⇢+
w
d ),
1+⇢d f n (⇢d )
with w P S n > 0.
n +
1+⇢+
d 1+f (⇢d ) w n
g (⇢+
d ),
1+⇢d 1+w f n (⇢+
d)
with w P S n > 0.
Result 8 cannot be used when w P S n = 0. An example is when c0j (q) < 0
for all q 0. In such a case the following result should be used instead:
Result 9:
w
P W Ln
1
1+⇢d+ w n +
g (⇢d )(1
1+⇢d
15
w
n
+ f n (⇢+
d )), with P S = 0.
Note that w P S n = 0 arises also when one is interested in learning about
the welfare loss of the Cournot equilibrium relative to the market configuration that maximizes welfare and allows firms to exactly break even. In this
case, the FOC in Expression 2 does not hold but the FOC in Expression
1 still does. This allows for using Result 9 because its proof relies only on
bounding the ratio of consumer and producer surplus at the Cournot equilibrium and the ratio of consumer surplus at the Cournot equilibrium and at
the allocation that maximizes welfare keeping the firms’ profit non-negative
and the latter relies only on the price at that such allocation.
Assume now ⇢-linear demand and constant average cost. In this case, I
obtain
w
w
P CSLn = 1
P W Ln = 1
(1 +
(1 +
⇢
)
n
(1+1/⇢)
1+⇢
⇢
)(1 + )
n
n
(1+1/⇢)
which recovers the result obtained in (Anderson and Renault 2003). Figures 10 and 11 show w P CSLn and w P W Ln for n = {1, 2, 3, 4, 5} and ⇢ 2
[ 1, 10]. As before for the cases of P CSLn+1,n and P W Ln+1,n , both w P CSLn
and w P W Ln decrease as the number of firms grows, for any ⇢ > 1. That
is, consumer surplus loss and welfare loss are greater the fewer firms in the
market. w P W Ln is quasi-concave in ⇢ and w P CSLn is increasing in ⇢. I
have lim⇢! 1 w P W Ln = lim⇢!1 w P W Ln = limn!1 w P W Ln = 0. In this
case of ⇢-linear demand and constant average costs, the maximum welfare
loss of the Cournot equilibrium is roughly 26.4% obtained with a monopolist
and ⇢ ⇡ 0. I also have lim⇢! 1 w P CSLn = limn!1 w P CSLn = 0 but now
lim⇢!1 w P CSLn = 1 for any n 1. As before for the cases of P CSLn+1,n
and P W Ln+1,n , consumer surplus loss and welfare loss are small when the
Cournot equilibrium is around an elastic portion of the demand curve. However, severe consumer surplus loss may arise when the Cournot equilibrium is
around an inelastic portion of the demand curve irrespectively of the number
of firms in the market. Once again, markets with inelastic demand render
the most concern for regulatory agencies when mergers are to be considered.
16
<< Figure 10 about here >>
<< Figure 11 about here >>
6
An Example of Application
This section illustrates how the bounds obtained in this paper can be used
in practice. I study a split that occurred in the Portuguese wireline market
– namely in fixed broadband access – at the end of 2007 but other interesting examples in the scope of other capital intensive industries can be easily
developed using the same approach. The main contribution of this paper is
to provide a way to obtain approximations for the change in welfare when
the number of firms in the market changes when little data are available.
The bounds for such change rely on very little data, namely just the demand
served, the price charged and the average cost before and after the number of
firms in the market changes. These data are often publicly available. Hence,
these bounds can be widely applied in practice.
When more data are available one may be able to estimate demand more
precisely. For example, one may fit a ⇢-linear demand curve to the data and
use the obtained estimate of ⇢ with the results provided in this paper to
learn about changes in welfare. However, such an approach can only yield
approximate results given the endogeneity between demand and price that
one is usually unable to correct for using only observational data. Below I
start by o↵ering an approximation for the change in welfare associated with
the above mentioned split using very little data. Then I use more data for
comparison purposes keeping in mind the caveat above.
All data used in this section are drawn from the quarterly accounting reports of Portugal Telecom (PT) and ZON, unless otherwise indicated. I use
two main sources of data. The first dataset includes, for each firm separately,
revenues and the number of Revenue Generating Units (RGUs) served per
quarter. An RGU is a standard concept used in the scope of the telecommunications industry and corresponds to a pair (costumer, service subscribed).
Operating revenues include payments obtained to provide voice and data services in retail and wholesale markets. The second dataset includes operating
17
costs (opex) and capital expenditures (capex). For each firm separately, and
on a quarterly basis, operating costs include wages and salaries, direct costs
and commercial costs. Capex includes investments in infrastructure and
maintenance costs. It is spread over the useful life of the assets purchased
and capitalized on a quarterly basis.
Before delving into the analysis note that using Cournot to describe competition in this market seems appropriate. As discussed in (Cabral 2000)
Cournot and Bertrand models rely on similar assumptions but predict very
di↵erent behavior. In industries where firms need to decide on both capacity
and price the crucial aspect to choose a model is the relative timing of these
decisions. If price is easier to adjust than capacity then the best model is one
in which firms first decide on capacity and later choose prices. This is what
typically happens in capital intensive industries such as telecommunications.
The Cournot model is known to be the one that corresponds to this setup.
As (Cabral 2000) discusses most real world industries are indeed closer to
the case in which capacity is difficult to adjust though in recent years digitization has been allowing some firms, such as software developers, to adjust
capacity rather quickly for example by simply shipping electronic copies of
their applications.
In 1Q2005, PT dominated the Portuguese wireline market. The company’s market share was roughly 90% according to ANACOM 2006 (the
Portuguese Telecommunications regulator). PT served 6.3 million RGUs at
an average price of 30.50 Euros/month. PT’s average cost per RGU served
was 15.46 Euros/month. PT’s wireline business in 2005 included telephony,
DSL and cable. PT-Multimedia, a branch within PT, supervised the cable operation. In early 2006, Sonae.com, a large retailer store in Portugal
with a small telecommunications operation, made a hostile take-over bid
for PT. This transaction was approved by the Portuguese Competition Authority with remedies, namely that PT-Multimedia should be structurally
separated. PT shareholders rejected Sonae.com’s o↵er but PT still spun o↵
PT-Multimedia in late 2007 into a new company called ZON. This resulted in
the continued steady growth of the cable market in Portugal in competition
with PT’s DSL.
In 4Q2009, ZON was serving 2.8 million RGUs at an average price of
19.33 Euros/month. ZON’s average cost per RGU served was 13.46 Eu18
ros/month. Also in 4Q2009, PT was still serving roughly 6.3 million RGUs
but now at an average price of 19.96 Euros/month. ZON’s average cost per
household served was 12.19 Euros/month. During 2009, ZON and PT combined accounted for more than 85% market share in the wireline business in
Portugal. PT’s and ZON’s average costs in 2009 were not very di↵erent from
each other because ZON’s backbone infrastructure was not remarkably different from PT’s. In fact, ZON and PT shared infrastructure for sometime
after the split until they were technically able to separate their backbone
networks. Also, PT’s and ZON’s average prices were not very di↵erent from
each other due to the fierce competition between these two firms in the Portuguese wireline market. Figure 12 depicts this market dynamics.
<< Figure 12 about here >>
The split of PT-Multimedia from PT and the subsequent growth of ZON
must have contributed to increase welfare in the Portuguese wireline market.
The bounds obtained in this paper can shed more light on this issue. A first
approach to compute changes in welfare is to note that aggregate profits were
94.89 million Euros in 1Q2005 and 62.00 million Euros in 4Q2009. Therefore, P P SL = 0.53. For now, assume that the demand curve was ⇢-linear
and did not change from 1Q2005 to 4Q2009. I will relax these assumptions
later. Result 2 allows for estimating ⇢ if one observes D0na and D0nb . A first
approximation is to make D0 na = D0 nb = (D4Q2009 D1Q2005 )/(p4Q2009
p1Q2005 ) = 0.26. However, with a linear approximation to demand, Result 2 yields, without surprise, ⇢ˆ = 1. With a little bit more data, one
can estimate D0na = (D2Q2005 D1Q2005 )/(p2Q2005 p1Q2005 ) = 0.20 and
D0nb = (D4Q2009 D3Q2009 )/(p4Q2009 p3Q2009 ) = 0.21. Note that I reverse
time, calling year 2005 the ”after” period and year 2009 the ”before” period,
to ensure that na < nb . Plugging these estimates for D0na and D0nb into
ˆ = 52.0%
Result 2 yields ⇢ˆ = 1.65. Using Result 1 and Result 3 yields P CSL
ˆ L = 21.5%. This means that in 1Q2005 only 48% of the consumer
and P W
surplus in 4Q2009 was realized. Likewise, in 1Q2005 only 78.5% of the welfare
in 4Q2009 was realized. This estimates are consistent with previous findings
(Pereira and Ribeiro, 2010). I note that a little bit more data before and
after the number of firms in the market changes is likely to be available as
these situations, such as splits and mergers, are likely to be highly scrutinized.
With more data on demand one can try to fit a ⇢-linear functional form
19
to them to estimate ⇢. However, as discussed before, this approach does
not necessarily yield better results due to the potential bias associated with
endogeneity. For example the technology and consequently the quality of the
broadband services provided by PT and ZON may have changed from 2005
to 2009. This is unobserved in my setting. However, such a change could be
related to price and to demand through mechanisms other than price, which
would render OLS estimates inconsistent. It is however unlikely that the
technology changed because PT’s network was already ready to o↵er DLS
and cable service to the whole country in early 2005 and upgrades to fiber
took place only in 2010. In fact regressing Ln(¯
c(qt )) = ↵+ qt + 1 t+ 2 t2 +✏t ,
or Ln(¯
c(qt )) = ↵ + 1 qt + 2 qt ⇤ t + 1 t + 2 t2 + ✏t , using quarterly data on
these firms’ average cost per RGU yields statistically non-significant coefficients for the time trend, or for its interaction with output. It is however
important to note that for as much as these regressions partial out the effect of changes in demand it is still true that demand and time are highly
correlated during the period of analysis and thus this provides only limited
evidence that changes in the technologies used by PT and ZON must have
been small or did not have a relevant impact on average costs. Figure 13
plots all the quarterly data available to me, namely, average cost, average
price and number of RGUs served.
<< Figure 13 about here >>
Nevertheless, I show results using these data for sake of comparison. I
estimate D(pt ) = (↵ pt ) + 1 t + 2 t2 + ✏t . This approach allows the size
of the market to change over time. I obtain ↵
ˆ = 53.83 (p < 0.001), ˆ = 0.61
(p < 0.001), ˆ1 = 0.23 (p < 0.001) and ˆ2 = 0.01 (p < 0.001). Therefore
⇢ˆ = 1/0.61 = 1.63, which is remarkably close to the estimate of ⇢ used above,
which may indicate that assuming ⇢-linear demand is reasonable in this market. In fact, the R2 for this model is above 0.99. Moreover, these parameters
yield a market size of roughly 10.5 million RGUs, which makes sense for Portugal. With these estimates I get D0na = 0.17 and D0nb = 0.15. Substiˆ = 48.1% and P W
ˆ L = 15.6%.
tuting into Result 1 and Result 3 yields P CSL
Note, however, that this approach, fixing the size of the market, the curvature of demand and the elasticity of demand, does not necessarily ensure
P P SL = 0.53. In fact, with these data I have P P SL = 0.91. An alternative way to use these data is to set P P SL = 0.53, use the estimates
for ⇢, D0na and D0nb obtained in the previous paragraph and adjust the size
20
of the market accordingly. This approach seems more reasonable given that
one observe P P LS with certainty. With these constraints, and for an estiˆ
mated market size of roughly 9.7 million RGUs, I obtain P CSL
= 51.5%
ˆ
and P W L = 21.4%.
One way to reduce the potential correlation between the error term and
price in the regression above is to control for additional covariates that may
have a↵ected demand during the period of analysis. For this purpose, I
estimate D(pt ) = (↵ pt ) + 1 t + 2 t2 + Xt + ✏t , where Xt includes quarterly data on the economic climate index, gross domestic product, available
income, unemployment rate and population in Portugal. Running this regression still yields ⇢ˆ = 1.63 (p < 0.001). Furthermore, none of the new
covariates are statistically significant in this regression (note however, that
some of them are highly collinear). A potentially better way to control for
time varying unobserved e↵ects is to add semester dummies. Therefore, I
estimate D(pt ) = (↵ pt ) + 1 t + 2 t2 + dbt/2c + ✏t , where dbt/2c indicate
the semester dummies. In this case, all semester dummies are statistically
ˆ
significant (p < 0.01) and I obtain ⇢ˆ = 1.32, which yields P CSL
= 48.3%
ˆ
and P W L = 19.2%.
Another approach using the same data is to allow for the curvature of demand to change over time. To this end, I estimate D(pt ) = (↵ pt ) 1 + 2 t +
2
ˆ = 57.70 (p < 0.001), ˆ1 = 0.75 (p < 0.001),
1 t + 2 t + ✏t . I obtain ↵
ˆ2 = 0.01 (p < 0.001), ˆ1 = 0.33 (p < 0.001) and ˆ2 = 0.01 (p < 0.1).
The R2 for this model is also greater than 0.99. This regression yields different estimates for ⇢ in 1Q2005 and 4Q2009, which can be used as ⇢d and
+
0na
⇢+
and
d . I have ⇢d = 1.45 and ⇢d = 1.73. Using the estimates above for D
0nb
D , which yield a market size similar to the ones reported above, Results
ˆ
1, 2 and 3 yield 0.87 < P PˆSL < 0.29, 46.9% < P CSL
< 54.8% and
ˆ
10.5% < P W L < 30.2%. The bounds for P P SL include the true value. Yet,
we can do better by setting P P SL = 0.53 and using Results 4 and 5, which
ˆ < 56.7% and 17.5% < P W
ˆ L < 27.1%.
yield 46.4% < P CSL
The functional form used above to estimate demand for allows for the
market size and the curvature of demand to change over time. Despite
this flexibility it would still be more interesting to allow di↵erent and independent demand curves before and after the split. To this end, I estimate
D(pt ) = (↵ pt ) + 1 t + 2 t2 + ✏t separately for the ”before” and the ”after”
21
period (using data between 1Q2005 and 3Q2007 and then between 4Q2007
and 4Q2009). For the ”after” period I obtain ↵
ˆ = 66.80 (p < 0.01), ˆ = 0.52
(p < 0.001), ˆ1 = 0.02 and ˆ2 = 0.002 but the latter two parameters
are not statistically significant. For the ”before” period I obtain ↵
ˆ = 49.50
(p < 0.001), ˆ = 0.72 (p < 0.05), ˆ1 = 0.48 and ˆ2 = 0.02 but the latter
two parameters are also not statistically significant. The estimates for ⇢ are
1.92 and 1.39 in the ”after” and the ”before” period, respectively. Moreover, the parameters on the time trends are no longer statistically significant
when the data are split into these periods, which provides some evidence that
this data split captures well the changes in demand that may have occurred
around the time at which PT spun o↵ PT-Multimedia. Using the same estimates for D0na and D0nb as before (which yield a market size similar to the
ˆ = 60.6% and
ones reported above) and setting P P LS = 0.53 yields P CSL
ˆ
P W L = 30.5%.
It is still possible that the curvature of demand may have changed over
time within both the ”before” and the ”after” periods. To capture this I also
estimate D(pt ) = (↵ pt )( 1 + 2 t) + t+✏t separately as above. For the ”after”
period I obtain ↵
ˆ = 70.11 (p < 0.05), ˆ1 = 0.50 (p < 0.001), ˆ2 = 0.00 and
ˆ = 0.04 but the latter two parameters are not statistically significant. For
the ”before” period I obtain ↵
ˆ = 55.72 (p < 0.02), ˆ1 = 0.56 (p < 0.09),
ˆ2 = 0.01 (p < 0.01) and ˆ = 0.26 but the latter parameter is not statistically significant. In this case, I have ⇢1Q2005 = 1.98 and ⇢4Q2009 = 1.49. Using
the same estimates for D0na and D0nb as before (which again yield a market
size similar to the ones reported above) and setting P P SL = 0.53 yields
ˆ = 59.7% and P W
ˆ L = 28.8%.
P CSL
All results presented before show consistent evidence that the split of PT
increased both welfare and consumer surplus in the Portuguese wireline market. The estimated magnitude of the e↵ect depends on the estimates for ⇢d ,
0na
⇢+
and D0nb . Using the estimates above for the latter two variables
d, D
and setting P P SL = 0.53 I can use Results 4 and 5 to draw Figure 14,
which shows how the bounds for P W L change with ⇢d and ⇢+
d . For example, this figure shows that 0.3 P W L 0.2 for ⇢d = ⇢+
=
1.63
and that
d
+
0.3 P W L 0.1 for ⇢d 2 [1.39, 1.49] and ⇢d 2 [1.73, 1.98]. This figure allows a policy maker obtaining di↵erent estimates for ⇢d and ⇢+
d from di↵erent
policy studies to analyze how welfare may have increased with the split of PT.
22
<< Figure 14 about here >>
7
Conclusions
This paper provides bounds for the loss in consumer surplus, producer surplus and welfare when the number of firms in the market changes. These
bounds apply to capital intensive industries by allowing costs to be concave
as long as Cournot equilibrium exists. This is a departure from the previous literature that required convex costs and, often, constant marginal costs
and similar firms, to provide such bounds. In addition, this paper uses the
concept of ⇢-concavity to bound demand, which allows for exploring how its
curvature a↵ects these losses. Therefore, this paper combines under more
general results analyses that the previous literature considered only separately.
The bounds developed in this paper rely on very little data observed at
the Cournot equilibrium. Estimates for the demand served, the price charged
and the average costs experienced by firms before and after the change in
the number of firms in the market are enough to compute them. This information is typically publicly available and thus these bounds can be widely
used in practice. They are, however, mostly useful when only little data are
available. With more data, one may be able to roughly estimate demand and
cost curves, which may allow for computing losses directly. However, such
an approach is also only approximate given the typical endogeneity concerns
that would preclude one from estimating these functions precisely with only
observational data. The bounds developed in these paper provide preliminary indications for how consumers may be a↵ected when the number of
firms in the market changes and thus provide anchors around which policy
analysts can run sensitivity analyses to conclude how results may change
with the uncertainty around costs and around the curvature of demand.
As a special case, I look at how mergers a↵ect losses by studying consumer
surplus loss and welfare loss when one firm shuts down. While welfare losses
are not necessarily large when demand is both relatively elastic or relatively
inelastic around the Cournot equilibrium, the loss in consumer surplus due
to a merger can be significant in the latter case and, in particular, when only
23
a few firms remain in the market. Hence, anti-trust authorities should keep
monitoring industries with ”very concave” demand and only a few firms, such
as telecommunications, electricity and gasoline, to name a few examples with
significant impact on everyone’s daily life. I also present an empirical example to show how the bounds developed in this paper can be used in practice.
I look at the e↵ect of the split of ZON-Multimedia from Portugal Telecom on
the Portuguese wireline business – namely, fixed broadband access. This split
took place in late 2007. Using data between 2005 and 2009 I provide a series
of consistent results showing that welfare and consumer surplus increased as
a result of this split triggered by the Portuguese Competition Authority.
Nonetheless, this paper does not come with limitations. Far and foremost, the bounds developed in this paper provide only preliminary estimates
for losses. Yet, they apply in contexts where only little data are available and
thus provide guidance when it is perhaps most needed. In addition, it is hard
to empirically separate the e↵ect of the change in the number of firms from
confounding e↵ects that may have a↵ected demand concurrently. A number
of results in this paper allow for such changes to take place and for comparing
welfare at two moments in time with di↵erent number of firms and di↵erent
demand curves but ultimately one needs additional assumptions to better
isolate the e↵ect of changing the number of firms in the market on welfare.
Examples of concurrent changes that may take place include the rollout of
new technologies or even maintenance of existing ones. In the example explored in this paper I provide empirical evidence that such changes must have
been small at best but this is not always necessarily the case. Finally, the
bounds o↵ered in this paper apply only in the case of a single product but,
in practice, and in particular in modern telecommunication markets, several
companies launch series of heterogeneous products that hit the market at
the same time, which renders the question of welfare efficiency significantly
harder and potentially a subject for future research.
8
Appendix on Proofs
This appendix includes proofs of all Lemmas and Results included in sections
4 and 5 as well as the analysis of the behvior of P CSLn+1,n , P W Ln+1,n ,
w
P CSLn and w P W Ln .
24
Lemma 1: CS n 
D 2 (y)
(1
(1+⇢d )D 0 (y)
0
(y) 1+1/⇢
d , y
y) DD(y)
)
+ ⇢d (pn
Proof7 : by definition I have CS n =
R +1
pn
pn .
D(x)dx. The ⇢d -concavity of
demand implies D (x)  D (y) + (x y)⇢d D⇢d 1 (y)D0 (y). Rearranging
terms leads to D(x)  D(y)[1 + ⇢d (x y)D0 (y)/D(y)]1⇢d .Therefore, CS n 
R +1
D(y)[1 + ⇢d (x y)D0 (y)/D(y)]1/⇢d dx with y
pn . When ⇢d > 0 the
pn
new integrand has an intercept z that satisfies 1 + ⇢d (z y)D0 (y)/D(y) = 0.
When ⇢d < 0 this new integrand goes to zero as x goes to infinity so define
z = +1 in this case. The Lemma is obtained from evaluating the integral
between pn and z. The reverse inequality can be obtained substituting ⇢d
for ⇢+
d . Therefore, Lemma 1 can also be stated with equality using ⇢ is lieu
+
of ⇢d when demand is ⇢-linear.
⇢d
⇢d
Lemma 2: P S n /CS n
f n (⇢d ).
Proof: make y = pn in Lemma 1 and use the definition of f n (⇢d ).
The reverse inequality can be obtained substituting ⇢d by ⇢+
d and thus
P S n /CS n = f n (⇢) when demand in ⇢-linear.
Result 1: P CSLnb ,na
1
1+⇢+
d
g nb ,na (⇢+
d ),
1+⇢d
n b > na .
R +1
R +1
Proof: make CS na /CS nb = pna D(x)dx/ pnb D(x)dx. Using both the
⇢+
d -convexity and the ⇢d -concavity of demand yields
na
CS

CS nb
0
1+
1
⇢
d
1+
1
⇢+
d
D 2 (y1 )
(1
(1+⇢d )D 0 (y1 )
+ ⇢d (pna
(y1 )
y1 ) DD(y
)
1)
D 2 (y2 )
(1
0
(1+⇢+
d )D (y2 )
nb
⇢+
d (p
0 (y )
2
y2 ) DD(y
)
2)
+
(3)
with y1 pna and y2 pnb . The result comes from making y1 = y2 = pna
(note that pna
pnb ) and using the definition of g nb ,na (⇢+
d ). Reversing the
roles of the ⇢d -concavity and of the ⇢+
-convexity
of
demand
allow for showd
+
ing the reverse inequality swapping ⇢d by ⇢d . Consequently, P SCLnb ,na =
7
This proof is based in the seminal result provided in (Anderson and Renault 2003).
25
1
g nb ,na (⇢) when demand is ⇢-linear.
Result 2: P P SLnb ,na
na (⇢+ )
1+⇢+
d f
d
g nb ,na (⇢+
d ),
1+⇢d f nb (⇢d )
1
n b > na .
Proof: make P S na /P S nb = [P S na /CS na ][CS na /CS nb ][CS nb /P S nb ]. Use
Lemma 2 twice, with n = na and n = nb , to bound P S na /CS na and
CS nb /P S nb , respectively. Use Result 1 to bound CS na /CS nb . The reverse
nb ,na
inequality can be obtained swapping ⇢+
=
d and ⇢d . Consequently, P P SL
n
a
f (⇢) nb ,na
1 f nb (⇢) g
(⇢) when demand is ⇢-linear.
Result 3: P W Lnb ,na
1
na (⇢+ )
1+⇢+
d 1+f
d
g nb ,na (⇢+
d ),
1+⇢d 1+f nb (⇢d )
n b > na .
Proof: make W na /W nb = [P S na /CS na +1]/[P S nb /CS nb +1][CS na /CS nb ].
Use Lemma 2 twice, with n = na and n = nb , to bound P S na /CS na
and P S nb /CS nb , respectively. Use Result 1 to bound CS na /CS nb . The
reverse inequality can be obtained swapping ⇢+
Consequently,
d and ⇢d .
1+f na (⇢) nb ,na
nb ,na
PWL
= 1 1+f nb (⇢) g
(⇢) when demand is ⇢-linear.
Result 4: P CSLnb ,na
↵ if 1
1/f na (⇢ )
P P SLnb ,na ) 1/f nb (⇢+d )
(1
d
↵, nb > na .
Proof: make CS na /CS nb = [P S na /P S nb ][CS na /P S na ]/[CS nb /P S nb ]. Use
Lemma 2 twice, with n = na and n = nb , to bound CS na /P S na and
CS nb /P S nb , respectively. Substitute P S na /P S nb by 1 P P SLnb ,na and the
Result follows. A similar result, with the inequality reversed, can be obtained
swapping ⇢d and ⇢+
d.
Result 5: P W Lnb ,na
↵ if 1
1+1/f na (⇢ )
P P SLnb ,na ) 1+1/f nb (⇢+d )
(1
d
↵, nb > na .
Proof: make W na /W nb = [P S na /P S nb ][1+CS na /P S na ]/[1+CS nb /P S nb ].
Use Lemma 2 twice, with n = na and n = nb , to bound CS na /P S na and
CS nb /P S nb , respectively. Substitute P S na /P S nb by 1 P P SLnb ,na and the
Result follows. A similar result, with the inequality reversed, can be obtained
swapping ⇢d and ⇢+
d.
Lemma 3:
w
CS n 
D 2 (y)
(1
(1+⇢d )D 0 (y)
Proof: by definition w CS n =
R +1
w pn
+ ⇢d ( w p n
0
(y) 1+1/⇢
d , y
y) DD(y)
)
w n
p .
D(x)dx. The result follows from us-
26
ing the ⇢d -concavity of demand as in the proof of Lemma 1. The reverse
inequality can be obtained substituting ⇢d for ⇢+
d . Therefore, Lemma 3 can
also be stated with equality using ⇢ is lieu of ⇢+
d when demand is ⇢-linear.
Lemma 4:
w
P S n /w CS n
w n
f (⇢d ), with w P S n > 0.
Proof: make y = w pn in Lemma 3 and use the definition of w f n (⇢d ). Similar results can be easily obtained by reversing the inequality when w P S n < 0.
The reverse inequality can be obtained substituting ⇢d by ⇢+
d and thus
w
P S n /w CS n = w f n (⇢) when demand in ⇢-linear.
Result 6:
w
P CSLn
1
1+⇢+
d w n
g (⇢+
d ).
1+⇢d
R +1
R +1
Proof: make CS n /w CS n = pn D(x)dx/ w pn D(x)dx. Using both the
⇢+
d -convexity and the ⇢d -concavity of demand this yields
CS
D 2 (y1 )
(1
(1+⇢d )D 0 (y1 )
n
w CS n

D 2 (y2 )
(1
0
(1+⇢+
d )D (y2 )
+ ⇢d (pn
+
w n
⇢+
d( p
0
(y1 )
y1 ) DD(y
)
1)
1+
0 (y )
2
y2 ) DD(y
)
2)
1
⇢
d
1+
1
⇢+
d
(4)
with y1 pn and y2 w pn . The result comes from making y1 = y2 = pn
(note that w pn  pn ) and using the definition of w g n (⇢+
d ). Reversing the roles
of the ⇢d -convexity and of the ⇢d -concavity of demand allows for showing
w
n
w n
the reverse inequality swapping ⇢+
g (⇢)
d by ⇢d and thus P CSL = 1
when demand is ⇢-linear.
Result 7:
w
P P SLn
1
n +
1+⇢+
d f (⇢d ) w n
g (⇢+
d ),
1+⇢d w f n (⇢d )
with w P S n > 0.
Proof: make P S n /w P S n = [P S n /CS n ][CS n /w CS n ][w CS n /w P S n ]. Use
Lemma 2, Result 6 and Lemma 4 to bound the terms in the first, second
and third brackets, respectively, and the result follows. Similar results can
be easily obtained by reversing the inequality when w P S n < 0. Similar arguments can be used to show that a lower bound holds swapping ⇢+
d by ⇢d
f n (⇢) w
w
n
and thus P P SL = 1 w f n (⇢) g(⇢) when demand is ⇢-linear.
Result 8:
w
P W Ln
1
n +
1+⇢+
d 1+f (⇢d ) w n
g (⇢+
w
d ),
1+⇢d 1+ f n (⇢+
d)
27
with w P S n > 0.
Proof: make W n /w W n = [P S n /CS n + 1]/[w P S n /w CS n + 1][CS n /w CS n ].
Use Lemma 2, Result 6 and Lemma 4 to bound the first, second and third
terms in brackets, respectively, and the result follows. The reverseninequality
1+f (⇢) w n
w
n
can be obtained swapping ⇢d by ⇢+
g (⇢)
d and thus P W L = 1
1+w f n (⇢)
when demand is ⇢-linear.
Result 9:
w
P W Ln
1
1+⇢d+ w n
1+⇢d
n +
w
n
g (⇢+
d )(1 + f (⇢d )), with P S = 0.
Proof: In this case, w W n /W n = [CS(n)/w CS n ][1 + P S n /CS n ]. Use Result 6 and Lemma Lemma 2 to bound the first and second terms in brackets,
respectively, and the result follows. Similar arguments allow to show a lower
bound swapping ⇢+
d by ⇢d .
Properties of P CSLn+1,n , P W Ln+1,n , w P CSLn and w P W Ln
These properties are derived assuming ⇢-linear demand and constant average cost at c˜. Note that in this case the Lerner rule yields nD0n /Dn =
(˜
c pn ) 1 . Also, the FOC for welfare maximization, w pn c¯j = c¯0j (w Djn )w Djn
yields w pn = c˜.
Computing w P CSLn : Substituting w pn = c˜ into the definition of w g n
and using the Lerner rule readily provides w g n (⇢) = (1 + ⇢/n) (1+1⇢) .Therefore, w P CSLn = 1 (1 + ⇢/n) (1+1/⇢) .
2
Computing w P W Ln : Note that in this case f n (⇢) = (1+⇢)P S n D0n /Dn =
2
(1+⇢)(Dn (pn c˜))D0n /Dn . Using the Lerner rule yields f n (⇢) = (1+⇢)/n.
Furthermore, w f n (⇢) = 0. Therefore, w P W Ln (⇢) = 1 (1 + f n (⇢))/(1 +
w n
f (⇢))w g n (⇢) = 1 (1 + (1 + ⇢)/n)(1 + ⇢/n) (1+1/⇢) .
w
Computing P CSLnb ,na : Note that in this case
P S n = 0. Therefore,
R
w Dn
w
W n = w CS n . Furthermore, in this case, w W n = 0
P (q)dq w Dn c˜. Note
R
D(˜
c)
that w Dn = D(w pn ) = D(˜
c) and therefore, w W n = 0
P (q)dq D(˜
c)˜
c and
w
n
thus W does not depend on n.
Write P CSLnb ,na = 1
1 (1 w P CSLna (⇢))(1
(CS na /w CS na )(w CS nb /CS nb )(w CS na /w CS nb ) =
w
P CSLnb (⇢)) 1 once I have that w CS na /w CS nb =
28
w
W na /w W nb = 1. Substituting the expressions for w P CSLna and w P CSLnb
readily provides P CSLnb ,na (⇢) = 1 ((na )/(nb )(nb + ⇢)/(na + ⇢))1+1/⇢ .
Computing P W Lnb ,na : Using the same approach as above P W Lnb ,na =
1 (1 w P W Lna (⇢))(1 w P W Lnb (⇢)) 1 . Substituting yields P W Lnb ,na (⇢) =
1 (nb /na )(na + 1 + ⇢)/(nb + 1 + ⇢)((na /nb )((nb + ⇢)/(na + ⇢)))1+1/⇢ .
Properties of w P CSLn :
@ w P CSLn
@n
=
1+⇢
(1
(n+⇢)2
@ w P CSLn
@⇢
=
n(1+⇢/n) 1/⇢
S(⇢).
⇢2 (n+⇢)2
With S(⇢) =
Consider n
+ ⇢/n)
= log(n/(n
lim⇢!1
@ w P CSLn
@⇢
= 0.
lim⇢!
S(⇢) = (n
lim⇢!1 S(⇢) =
Then S 0 (⇢) =
lim⇢!
1
w
P CSLn is decreasing in n.
2. Then:
@ w P CSLn
@⇢
1
< 0.
⇢(1 + ⇢) + (n + ⇢)log(1 + ⇢/n).
1
lim⇢!
1/⇢
1)log(1
1/n) < 0.
1.
2⇢ + log(1 + ⇢/n).
S 0 (⇢) = 2 + log(1
lim⇢!1 S 0 (⇢) =
1)) > 0.
1/n) > 0.
1.
S 00 (⇢) = 2 + 1/(n + ⇢), which is always negative for ⇢ > 1. Therefore:
S (⇢) is positive, then zero and then negative. The maximum of S(⇢) is obtained when n = ⇢/(e2⇢ 1). This maximum is ⇢(1+⇢)+2⇢2 /(1 e 2⇢ ). This
maximum is negative, then zero when ⇢ = 0 and positive thereafter. Coinciw CSLn
dently, S(0) = 0 and therefore, S(⇢) < 0 for all ⇢ > 1. Hence, @ P@⇢
>0
w
n
for all ⇢ > 1 and P CSL (⇢) is increasing in ⇢. Figure 15 illustrates these
0
29
w
n
CSL
arguments. For n = 1, @ P@⇢
= (1 + ⇢) (1+1/⇢) /⇢2 (⇢ log(1 + ⇢)) > 0 for
⇢ > 1. The analysis for P CSLn+1,n is similar and available upon request.
<< Figure 15 about here >>
Properties of w P W Ln :
@ w P W Ln
@n
=
@ w P W Ln
@⇢
=
1+⇢
(1
n(n+⇢)2
+ ⇢/n)
1/⇢
< 0.
w
P W Ln is decreasing in n.
(1+⇢/n) 1/⇢
S(⇢).
⇢2 (n+⇢)2
Let S(⇢) =
⇢(1 + n + 2⇢) + (n(n + 1) + ⇢(⇢ + 1) + 2n⇢)log(1 + ⇢/n).
Consider n
2. Then:
1
@ w P W Ln
@⇢
=
lim⇢!1
@ w P W Ln
@⇢
= 0.
lim⇢!
S(⇢) = (n
lim⇢!
1
(1 + nlog(1
1/n))/n > 0.
1)(1 + nlog(1
1/n)) < 0.
lim⇢!1 S(⇢) = +1.
Then S 0 (⇢) =
lim⇢!
1
3⇢ + (1 + 2n + 2⇢)log(1 + ⇢/n).
S 0 (⇢) = 3 + (2n
1)log(1
1/n) > 0.
lim⇢!1 S 0 (⇢) = +1.
Then S 00 (⇢) = 1/(n + ⇢)
lim⇢!
1
S 00 (⇢) = (2
1 + 2log(1 + ⇢/n).
n + 2(n
1)log(1
1/n))/(n
1) < 0.
lim⇢!1 S 00 (⇢) = +1.
Then S 000 (⇢) = (2(n+⇢) 1)/(n+⇢)0 , which is always positive for ⇢ > 1.
Therefore: S 00 (⇢) is negative, then zero then positive. The minimum of S 0 (⇢)
30
is obtained when 2log(1 + ⇢/n) = 1 1/(n + ⇢). This minimum is always negative (because, for example, S 0 (1) = 3 + (3 + 2n)log(1 1/n) <
0.Consequently, S(⇢) increases, then decreases and then increases again. The
maximum of S(⇢) is obtained when ⇢ = 0 and is also zero. It is readily shown that this minimum is always negative. Therefore, S(⇢) is negative, then zero, then negative and zero again and finally positive. Hence,
@ w P W Ln
is first positive and then negative. Consequently, w P W Ln (⇢) is
@⇢
quasi-concave in ⇢. Figure 16 illustrates these arguments. For n = 1,
@ w P W Ln
= (1 + ⇢) (1+1/⇢) /⇢2 ((2 + ⇢)log(1 + ⇢) 2⇢)). The last bracket
@n
is negative and then positive. Therefore, again, w P W Ln (⇢) is quasi-concave
in ⇢. The analysis for P W Ln+1,n is similar and available upon request.
<< Figure 16 about here >>
31
9
Appendix with Figures
ρ=10%
ρ=0.2&
ρ=20%
ρ=0.4&
ρ=40%
ρ=30%
ρ=50%
ρ=0.6&
ρ=0.8&
ρ=1.0&
Figure 1: Examples of ⇢-linear functions with ⇢ > 0.
32
(x2)1/3(
(x2)1&
Figure 2: A function (f (x) = x2 ) that is both convex and 1/3-concave.
References
Anderson, S. and R. Renault (2003). Efficiency and surplus bounds in
cournot competition. Journal of Economic Theory 113, 253–264.
Bergson, A. (1973). On monopoly welfare losses. The American Economic
Review 53 (5), 853–870.
Cabral, L. M. (2000). Introduction to industrial organization. Cambridge,
Mass.: MIT Press.
Corchon, L. (2008). Welfare losses under cournot competition. International Journal of Industrial Organization 26 (5), 1102–1131.
Dickson, V. and W. Yu (1989). Welfare losses in canadian manufacturing
under alternative oligopoly regimes. International Journal of Industrial
Organization 7, 257–267.
Gisser, M. (1982). Welfare implications of oligopoly in u.s. food manufacturing. American Journal of Agricultural Economics 64 (4), 616–624.
Gisser, M. (1986). Price leadership and welfare losses in u.s. manufacturing.
The American Economic Review 76 (4), 756–767.
33
f(x)≤(1+2(x+1)/3)3'
f(x)=x2'
f(x)≥2x+1'
Figure 3: Using the convexity and 1/3-concavity of x2 to bound it.
34
p"
CS#with#ρd''linear#demand#
going#through#(Dn,pn)##
CSn#
pn#
ρd''linear#
func0on#
Dn#
D"
Figure 4: Consumer surplus with n firms in the market, CS n , is less than the
consumer surplus with ⇢d -linear demand going though (Dn , pn ). The latter
2
is given by Dn /((1 + ⇢d )D0n ).
35
p"
CSna$
CSnb$
CS$with$ρd+*linear$demand$
going$through$(Dna,pna)$$
pna$
CS$with$ρd+*linear$demand$
going$through$(Dnb,pnb)$$
pnb$
ρd+*linear$
func2on$
Dna$
Dnb$
D"
Figure 5: Consumer surplus at Cournot equilibrium with nb firms is greater
nb
nb
than that with a ⇢+
d -linear demand going though (D , p ). The consumer
surplus with this ⇢d -linear demand going through (Dna , pna ) is related to the
nb ,na +
latter by (1 + ⇢+
(⇢d ). The latter is greater than the consumer
d )/(1 + ⇢d )g
surplus at the Cournot equilibrium with na firms in the market as depicted
in the previous figure.
36
+
Figure 6: Area in the (⇢d , ⇢+
⇢d > 1 in which
d ) space with ⇢d
nb ,na
na
PWL
↵ (in the figure hna = f (⇢d )/(1 + ⇢d ) and hnb = f nb (⇢+
d )/(1 +
+
⇢d ) for sake of simplicity).
37
+
Figure 7: Area in the (⇢d , ⇢+
⇢d > 1 in which
d ) space with ⇢d
nb ,na
na
PWL
 ↵ (in the figure hna = f (⇢d )/(1 + ⇢d ) and hnb = f nb (⇢+
d )/(1 +
+
⇢d ) for sake of simplicity).
38
n=1$
n=2$
n=3$
n=4$
n=5$
Figure 8: Consumer surplus loss for the case of ⇢-linear demand, constant
average costs, as a function of ⇢ when one firm shuts down and n firms remain
in the market.
39
n=1$
n=2$
n=3$
n=4$
n=5$
Figure 9: Welfare loss for the case of ⇢-linear demand, constant average
costs, as a function of ⇢ when one firm shuts down and n firms remain in the
market.
40
n=1$
n=2$
n=3$
n=4$
n=5$
Figure 10: Consumer surplus loss of the Cournot equilibrium for the case of
⇢-linear demand, constant average costs, as a function of ⇢ with n firms in
the market.
41
n=1$
n=2$
n=3$
n=4$
n=5$
Figure 11: Welfare loss of the Cournot equilibrium for the case of ⇢-linear
demand, constant average costs, as a function of ⇢ with n firms in the market.
42
Figure 12: Average prices (AP), average costs (AvC) and size of the market
for PT and ZON-Multimedia in 2005 and 2009.
43
Figure 13: Average price (AP), average cost (AvC) and number of RGUs
served by PT and ZON-Multimedia in the Portuguese wireline market in the
period 2005-2009 and quarterly rate of decline in price.
44
β0.3%
ρd+$
≤0.3%
>$
≤0.3%
≥0.1%
ρ$
β0.2%
≤0.2%
β0.1%
≤0.1%
β0.1%
≤0.2%
≥0.1%
β0.2%
≥0.1%
≤0.3%
≥0.2%
≥0.2%
β0.3%
≥0.3%
0.74%
1.88%
3.38%
ρd#$
Figure 14: Welfare gain associated with the split of ZON-Multimedia from
PT as a function of ⇢d and ⇢+
d using the profits and slopes for the demand
curve as observed in the market before and after the split.
45
Figure 15:
w
P CSLn (⇢) is increasing in ⇢.
Figure 16:
w
P W Ln (⇢) is increasing in ⇢.
46
Guo, X. and H. Yang (2005). The Price of Anarchy of Cournot Oligopoly.
Berlin, Germany: Springer-Verlag.
Harberger, A. (1954). Monopoly and resource allocation. The American
Economic Review 74 (2), 77–87.
He↵ord, C. and D. Round (1978). The welfare cost of monopoly in australia. Southern Economic Journal 44 (4), 846–860.
Jenny, F. and A. Weber (1983). Aggregate welfare loss due to monopoly
power in the french economy: Some tentative estimates. Southern Economic Journal 32 (2), 113–130.
Johari, R. and J. Tsitsiklis (2005). A scalable network resource allocation
mechanism with bounded ef?ciency loss. IEEE Journal on Selected Areas in Communications 24 (5).
Novshek, W. (1985). On the existence of cournot equilibrium. The Review
of Economic Studies 52 (1), 85–98.
Posner, R. (1975). The social costs of monopoly and regulation. Journal
of Political Economy 83, 807–827.
Schwartzman, D. (1960). The burden of monopoly. The Journal of Political
Economy 68 (6), 627–630.
Siegfried, J. and J. Tiemann (1974). The welfare cost of monopoly: An
inter-industry analysis. Economic Inquiry 12, 190–202.
Stigler, G. (1956). The statistics of monopoly and merger. The Journal of
Political Economy 64 (1), 33–40.
Tullock, G. (1967). The welfare costs of tari↵s, monopolies and theft. Western Economic Journal 5, 224–232.
Worcester, D. (1973). New estimates of the welfare loss to monopoly,
united states: 1956-1969. Southern Economic Journal 40 (2), 234–245.
Worcester, D. (1975). On monopoly welfare losses: Comment. The American Economic Review 65 (5), 1015–1023.
47