WELFARE LOSSES UNDER OLIGOPOLY
LUIS C. CORCHÓN
Departamento de Economía
Universidad Carlos III
c/ Madrid 126, Getafe, Madrid 28903, Spain
[email protected]
First version December 2005. This version, February 17th 2006.
Abstract
We …nd that welfare losses due to oligoplistic output setting in the Cournot
model are small in a number of situations with as few as three competitors. However,
welfare losses due to the wrong number of …rms operating under oligopoly are found
to be quite substantial.
I am grateful to Carmen Beviá for comments to a preliminary version of this paper and to the
Spanish Ministery of Education for …nancial support under grant BEC2002-02194.
1. Introduction
In his classical contribution Cournot (1837, Chapter 8) established that when the number of …rms in a market tends to in…nity, oligopoly equilibrium tends to perfect competition. As a corollary, Welfare Losses (WL), measured as the di¤erence between social
welfare in the optimal and the equilibrium allocation, tend to zero.1 Our paper focus
attention on WL when the number of …rms is small, i.e. under oligopoly. To the best
of our knowledge there are only a handful of theoretical papers dealing with this problem from the classical work of Hotelling (1938), see McHardy (2000) and the references
there.2 It seems to us that the question of the size of WL under oligopoly is crucial
for two reasons: First, were WL small there would be little to be gained by considering
oligopolistic behavior: A simple equilibrium concept like perfect competition may be
preferable. Second, the motive for public policies would be weakened under small WL.
In Section 2 we will present the Cournot model of oligopoly with a given number
of identical …rms. Demand and costs are linear. The model allows to calculate the
percentage of WL under oligopoly, denoted by PWL, as follows P W L = 1=(1 + n)2
where n is the number of …rms. Thus a market composed by 7 …rms would produce a
PWL of 1.56%, not a big number. In order to appraise the size of WL under oligopoly
and regulation we compare PWL in oligopoly with PWL under price regulation with
unkown costs. The latter is chosen because it is a realistic alternative to "free market".
We …nd that PWL under price regulation are larger under oligopoly except, possibly, in
the duopoly case. Thus, the results in this section point out to the fact that oligopoly
WL are small, an unpleasant result as we argued before. Thus, in the rest of the paper
we consider the robustness of this result by considering the following scenarios:3
1) More general form of the demand function (Section 3).
2) Non constant returns to scale (Section 4).
1
A number of papers have studied the conditions under which this result occurs in general in large
economies, Dasgupta and Ushio (1981), Fraysse and Moreaux (1981) and Guesnerie and Hart (1985).
2
On the contrary, there is a sizeable literature measuring WL in actual economies following the work
of Harberger (1954) (see Tullock (2003) p. 2 for a list of papers in this literature).
3
Other attempts to …nd higher WL focus on di¤erent issues: ”X-Ine¢ ciency” (a forerunner of moral
hazard), see Leibenstein (1966) and Rent-Seeking (a forerunner of contest theory), see Tullock (1967).
2
3) Fixed Costs and free entry (Section 5).
4) Non identical …rms (Section 6).
The consideration of Point 1 does not bring a great improvement in the values of
PWL. However we …nd large WL associated with overentry under increasing returns or
free entry. In these two cases, except for the case of two, or possibly three …rms, WL
due to the wrong output are also small.
Finally, in Section 7 we o¤er some thoughts about these results. Our main conclusion
is that the search for WL under oligopoly should not focus on output misallocation, as
it has been done so far, but to the wrong number of …rms in the market.
It goes without saying that our main …ndings should be taken with some caution before a more complete picture of WL is available. For instance, it would be interesting to
compare oligopolistic WL with those arising under other kinds of regulation like optimal
regulation. Also important causes of WL are not considered here, i.e. product di¤erentiation. Intuition suggest that this is, potentially, an important cause of WL because,
in limiting cases, i.e. when products are almost independent, we are essentially back to
the case of monopoly where WL can be substantial. Also we leave for future research
the study of the impact of variables like investment, R&D, quality, location, etc., on
WL. The analysis of the impact of these variables on WL requires the consideration of
games that are far more complicated than those considered here.
2. The Linear Model
We will beguin by considering the simplest model of oligopoly (see Yarrow (1985) for
a similar approach). Suppose that there is a representative consumer with a utility
function U = Ax
bx2
2
px where x is aggregate output, p is the market price and A
and b are positive numbers. The maximization of utility generates an inverse demand
function p = A
bx: There are n identical …rms each producing a single output denoted
Pn
by xi . Thus x
i=1 xi . Marginal cost is constant and denoted by c. Pro…ts for
…rm i are
i
(p
c)xi . De…ning a
A
c we have that
i
(a
bx)xi . Assume
a > 0. If …rms compete à la Cournot, …rst order condition of pro…t maximization
yields a = bx + bxi . It is easy to check that the second order condition holds and that
3
equilibrium is symmetric. Thus
x =
an
:
b(n + 1)
(2.1)
Social welfare, denoted by W , is the sum of pro…ts and the utility of the consumer,
W
(p
c)x + Ax
bx2
2
px = ax
bx2
2 .
W =
Thus, in a Cournot equilibrium,
a2 n(n + 2)
:
2b(n + 1)2
(2.2)
In an optimal allocation, social welfare is maximized. First order condition yields
xo =
a
:
b
(2.3)
Again, second order condition holds and the allocation is symmetric. Thus,
Wo =
a2
:
2b
(2.4)
De…ne now the percentage of welfare loss (PWL) as follows
PWL
Wo W
=1
W0
n(n + 2)
1
=
:
(n + 1)2
(n + 1)2
(2.5)
For the case of monopoly, P W L is undoubtedly large, .25. But PWL decreases sharply
with the number of competitors; under duopoly is .11, with three …rms is .0625, for four
…rms, P W L is .04 and for …ve …rms P W L is .027.4 A natural question is, are these
numbers large to justify regulation? In order to get an idea about that we compare these
WL with those arising under regulation when the regulator is not completely informed
about the characteristics of …rms. If the latter are larger, oligopoly produces small WL
relative to those produced by regulation. For simplicity we will say that in this case,
oligopoly WL are small.
Suppose that the regulator controls the market price.5 She is uncertain of the value
of the marginal cost which can take two values: y with probability
4
(> 0) and z (< y)
This result shows that once linearity is assumed, WL never comes up to big numbers except if the
number of …rms is very small. This may help to explain the small WL found by the empirical studies.
5
For the optimal regulation under oligopoly, see López-Cuñat (1995) who generalizes the analysis of
Baron and Myerson (1982) in the monopoly case. We consider here a kind of regulation that is simpler
and more in tune with real life practices.
4
with probability 1
(> 0). Once p is set, …rms decide output. If p is smaller than
the actual value of the marginal cost …rms do to produce. In this case social welfare is
zero. If p is larger or equal than the actual value of the marginal cost, they produce
an identical quantity each that satis…es demand x =
bx2
2
Ax
cx =
A p A+p
b ( 2
A p
b .
In this case social welfare is
c) where c = y; z.
First we will show that a regulator interested in expected social welfare chooses
either p = y or p = z. Suppose that p > y. In this case expected social welfare, denoted
A p A+p
b ( 2
by EW is
y) + (1
@EW
1
= [ (y
@p
b
) A b p ( A+p
2
p) + (1
z). Thus
1
p)] = ( y + (1
b
)(z
)z
p) < 0;
hence such a price can not be optimal. Suppose now that y > p > z. In this case
) A b p ( A+p
2
expected social welfare is EW = (1
@E(p)
= (1
@p
z). Thus
p+z
< 0:
b
)
Again we …nd that a price decrease improves expected social welfare. Thus the price
chosen by the regulator depends on the di¤erence between EW when p = y (
(1
2z)
) (A y)(A+y
)
b
De…ne: aH
2 (1
and EW when p = z ((1
y)2
(A
p=y,
b
A
2
) (A b z) ),
z; aL
+ (1
A
)
(A
y and
y)(A + y
b
aH
aL .
)+1
2z)
(A y)2
b
+
i.e.
> (1
)
z)2
(A
(2.6)
b
From (2.6) above, p = y , (1
)
2
2 < 0: There is only one value of > 1 for which this inequality is in
q
fact a equality, namely 1 + 1 . It is easily seen that for larger values of the left
hand side of the previous inequality is always positive. Thus, we can characterize the
optimal policy in terms of
p=y,
and , namely
r
<1+
1
p=z,
>1+
r
1
(2.7)
Consider …rst that p = y. Then, PWL of price regulation, denoted by P W LP R is
P W LP R =
aL 2
aL 2 + (1
)
5
2 2
aL
=
+ (1
)
2:
(2.8)
If the regulator achieves larger welfare than oligopoly, P W LP R < P W L. From (2.8)
this is equivalent to
2
>
(1 P W L)
(1 )P W L .
From the latter inequality and (2.7), large WL
under oligopoly require in this case that
<1+
r
and
1
>
s
(1
(1
P W L)
)P W L
(2.9)
For the duopoly case, P W L = 1=9 so the second inequality in (2.9) reads
Both inequalities imply that
inequality in (2.9) reads
q
> 2:83
1
.
< : 22897. For n = 3, P W L = 1=16 so the second
q
> 3:87 1 . Both inequalities imply < : 10821 which
looks like a small number. For larger number of …rms P W L is smaller so the upper
bound of
will be even smaller and so the likelihood of ocurrence of large WL under
oligopoly.
Let us now turn to the case where p = z. Here PWL under price regulation is
P W LP R = (1
)
(aH aL )2
=
a2L + (1
)a2H
(1
) 2
+ (1
)
2:
(2.10)
Again, if the regulator can achieve larger welfare than under oligopoly, P W LP R <
P W L. From (2.10) this is equivalent to
2
<
(1
PWL
)(1 P W L) .
From the latter inequality
and (2.7), large WL under oligopoly require in this case that
s
r
PWL
and <
:
>1+
1
(1
)(1 P W L)
(2.11)
It is easy to see that for P W L < :5 there are no values of ( ; ) solving the above
inequalities. Thus, in this case WL of oligopoly are never large.
Summing up, except, possibly, for the case of duopoly, WL of oligopoly are small in
the linear model compared with those arising under price regulation.
3. More General Demand Function
In this section we assume that the utility function of the representative consumer is
U = Ax
bx
reads p = A
+1
+1
px with A
0, b > 0 and
bx . Notice that if
>
1. The inverse demand function
< 0 and A = 0 we have an isoelastic function
6
p=
if
bx with b < 0 and
< 0. The linear case considered in the Introduction occurs
= 1. First order condition of pro…t maximization yields a = bx + b x
1x .
i
Again
it is easy to check that the second order condition holds. Thus Cournot equilibrium
output is
x =(
bx +1
1+
Social welfare is now W = ax
1
an
) :
b(n + )
(3.1)
and the optimal aggregate output is
a 1
xo = ( ) :
b
(3.2)
Social welfare in equilibrium and in the optimal allocation, are, respectively
W =
a
+1
n
1
(n +
+ 1)
and W =
1
1
a
o
b (n + ) (n + )( + 1)
+1
:
1
(3.3)
b ( + 1)
From there it follows that the percentage of welfare loss is
1
n (n +
PWL = 1
+ 1)
(n + )
+1
P W L(n; ):
(3.4)
How large this magnitude can be? It is easily shown that P W L decreases with n. Also,
since P W L can be written as
1
1 + =n + 1=n
(1 + =n)
+1
:
It is readily seen that when n ! 1, P W L ! 0 and that when
regardless on n. In order to study the function for other values of
!
1; P W L ! 0,
let us …rst compute
the following:
1
@P W L
n (n + + 1)
=
1+ (ln n
@
2 (n + )
We see that at any point where
by multiplying by
2 (n+
)
1+
1
n (n+ +1)
0 = ln (n + )
ln (n + ))
n
1
(n + )
@P W L
@
1+
(1
1+
):
(n + )
= 0, the previous expression can be simpli…ed
. Then
ln n
(1 + n + 2 )
n2 + 2n + 2 + n +
7
= F ( ), say:
(3.5)
Notice that
= 0 is a solution of (3.5) but is not a solution to
2.
have multiplied the latter by
@P W L
@
@P W L
@
= 0 because we
However any other solution to (3.5) is a solution to
= 0. Now compute,
2+
@F ( )
=
@
(n2 + 2n +
n2 + 1
2
(3.6)
+ n + )2
Notice the following facts:
1) There are only two values of
p
1+ 4n 3
2
> 0.
2) For
'
1, from (3.6) above,
@F ( )
@
)
3) For large values of ,
4) Finally, compute
n + n2
3
2
@2F (
+6
@
for which
@F ( )
@
'
@F ( )
@
1 n2
(n2 n)2
= 0, namely
< 0.
which amounts to
2
2 n2
+2
3n
+3
From the expression above, we get that
Points 1-4 above imply that
2n
@F ( )
@
2
@ 2 F (0)
@ 2
+ 3 n2 + 2n3
4
3
+ n + )3
=
n+n2 n4 n3
(n2 +n)3
0.625
0.5
0.375
0.25
0.125
0
=
(6+5 +
)
0
n3
:
looks like in Figure 1 below (the …gure is done
y 0.75
2
n4
< 0:
for the case n = 2 but the shape is valid for any n):
(3
=
> 0.
(n2 + 2n +
@F ( )
@
= 0 and
1.25
2.5
3.75
5
x
2 )2
FIGURE 1
8
@F ( )
@
Now F (0) = 0, F (n)n!1 ! 1, plus the shape of
obtained before, imply that F (
) looks like in Figure 2 below (the …gure is done for the case of n = 2, but the shape is
valid for any n).
y
0.25
0.125
0
0
2.5
5
7.5
10
x
-0.125
F ( ) = ln (2 + )
ln 2
(3+2 )
6+5 + 2
FIGURE 2
Notice that the fact that
= 0 is not a solution to
) obtained before imply that there is only one value of
@P W L
@
= 0; and the shape of F (
for which
@P W L
@
= 0.6 Next,
using the Maple calculator of Scienti…c Work we will …nd maximum values of P W L:
For n = 2 the maximum is almost .118, see Figure 3 below, only 6.3% higher than
the .11 obtained in the linear case.7 Notice that F ( ) can not be increasing for values
of
larger than 30 because this would imply the existence of another point, say
@P W L
@
,
= 0 = F ( ) which we saw before it was impossible so indeed the maximum in
the picture is a global maximum.
6
Notice that P W L is not the integral of F ( ) even though
due to the fact that we multiplied
7
@P W L
@
by
2
n
1+
(n+ )
1
@P W L( 0 )
@
= 0 implies F ( 0 ) = 0. This is
in order to obtain F ( ).
(n+ +1)
Using similar methods it can be shown that in the case of monopoly; the maximum P W L is,
approximately, .264, only 5.% higher than the .25 obtained under linear demand,
9
y
0.1
0.075
0.05
0.025
0
P W L (2; ) = 1
2
1
0
5
10
15
20
25
3+
(2+ )
30
x
1+
FIGURE 3
When n = 3 the maximum is about .076 -see Figure 4 below- which is now 21% higher
than the .0625 obtained in the case of linear demand.
y 0.075
0.0625
0.05
0.0375
0.025
0.0125
0
P W L(3; ) = 1
3
1
0
5
4+
(3+ )
10
15
20
25
30
x
1+
FIGURE 4
When n = 4 is approximately .056 which is 37.5% higher than the .04 obtained in the
linear case, see Figure 5 below.
10
y
0.05
0.0375
0.025
0.0125
0
0
1
P W L(4; ) = 1
5
10
15
20
25
4 (5+ )
(4+ )
30
x
+1
FIGURE 5
For n = 5 P W L is about .044, which is 58.8% higher than the .0277 obtained in the
linear case, see Figure 6 below.
y
0.04
0.03
0.02
0.01
0
0
1
P W L(5; ) = 1
5
10
15
20
5 (6+ )
(5+ )
25
30
x
+1
FIGURE 6
Table 1 below shows the maximum P W L obtained in this section, denoted by P W L,
and the values obtained in the linear model, denoted by P W L, for some values of n: n
between 2 and 10 capture most oligopolistic markets. When n = 20; many economists
would say that the market is very close to perfect competition
11
n
2
3
4
P W L :118 :076
P W L :11
5
6
7
:056 :044 :0357 :032
:0625 :04
:027 :02
8
9
10
20
:027 :024 :022 :012
:0156 :012 :01
:008 :003
TABLE 1
Notice that the relative di¤erence between P W L and P W L increases with n and
achieves very large numbers (for n = 10, 250%). However this e¤ect occurs for small
values of P W L and is not strong enough to obtain large numbers. In the cases which
we identify in the previous section as likely to yield WL, the maximum WL here and
the WL there are substantially identical. Given this and that in the case considered
here, WL can be much smaller than P W L, we have to conclude that the consideration
of a more general class of demand functions does not bring signi…cant WL associated
with oligopoly.8
4. Non Constant Returns to Scale
In this section we assume that the cost function is cxi + x2i d=2. If d > 0 we have
decreasing returns and if d < 0 we have increasing returns. In the latter costs become
negative for su¢ ciently large output. We assume that 2c(d+b) >
da which guarantees
that costs are positive in the optimum and in equilibrium and implies b >
d. The latter
guarantees that second order conditions of pro…t and welfare maximization hold. Now
2
d Pn
2
(a bx dxi =2)xi and W ax bx2
i
i=1 xi . First order condition of pro…t
2
maximization is a
bx
(b + d)xi = 0. From there, it is clear if, say, k …rms are active,
each produces an identical output
a(b+d)
b(k+1)+d
8
a
d+b(k+1) .
Also, marginal pro…ts evaluated at zero are
> 0, so in equilibrium all …rms are active. Thus equilibrium is unique and it is
McHardy (2000) studies a model with quadratic demand and presents numerical calculations for
several values of the parameters. He …nds that in the case of duopoly WL can increase with respect to
the linear case as much as 33%. This result di¤er from our’s because in our case the maximum increase
is small (5.6%). In another case reported there, n = 5 and WL can increase as much as 30%, which is
in tune with our results.
12
given by
x =
an
:
d + b(n + 1)
(4.1)
Similar arguments establish that the optimum is unique and is given by
an
a
if d > 0 and xo =
if d < 0.
d + bn
d+b
xo =
(4.2)
From there we calculate that if d > 0
W
d + bn + 2b
a2 n
o
and
W
=
:
2(bn + d)
2 (d + bn + b)2
1
b2
=
:
2
[d + b(n + 1)]
(d=b + n + 1)2
= a2 n
PWL =
(4.3)
Notice that WL are smaller than in the linear case. However if d < 0
Wo =
a2
:
2 (d + b)
From this expression and W we …nd that
PWL =
d2
dbn + 2db + b2 nd2
(d + bn + b)2
dbn2
It is easy to see that for n > 1, P W L is decreasing on d. Thus maximal P W L, denoted
by P W L obtains when d attains its minimum value.9 Since 2c(d + b) >
bound for d obtains when
d=
2cb
a+2c :
PWL =
da, a lower
Thus, the maximal P W L is
2cna + a2 + 2cn2 a + 4c2 n2
:
(na + 2cn + a)2
We see that when n ! 1;
PWL !
4c2 + 2ca
2c
2c
=
=
.
2
2
a + 4c + 4ac
a + 2c
A+c
In this case we need no comparison with other resource allocation mechanisms to state
that P W L can be quite substantive: For instance for 2c = A, P W L ! 2=3.
9
Minimum P W L occurs for the maximum value of d, namely, d = 0. In this case P W L equals the
value obtained in (1.5).
13
A natural question is: Are these WL due to the wrong output being chosen by
oligopolists or to the fact that oligopoly entails too many …rms? In order to disentangle
these two e¤ects let us maximize social welfare with the constraint that all …rms produce
the same output. Call the solution Second Best. Second best output is identical to the
case d > 0 considered before and WL are given by (4.3) above. Again the maximum
value of P W L in the second best, denoted by P W LSB , is obtained when d =
2cb
a+2c ,
which amounts to
P W LSB = (
1
n+
a
a+2c
)2 :
Clearly when n ! 1 P W LSB ! 0 so when the number of …rms is large there will be
a huge discrepancy between total WL and WL due to the wrong output. Also we see
that
P W LSB 2 [
1
1
; 2 ]:
2
(n + 1) n
Notice that the upper (resp. lower) bound of M P W LSB is not far, especially for n > 3
(resp. identical) to the expression obtained under constant returns.
Summing up, increasing returns may bring quite substantial WL. However, WL
arising from output misallocation are not large.10 It is the fact that all …rms except one
are redundant from the social point of view what makes WL signi…cant.
5. Free Entry
Let us go back to the model with linear demand and costs considered in Section 2. In
this section we will assume that in order to produce, …rms must incur in a …xed cost,
denoted by K, and that there is an in…nite number of potential …rms. The number of
active …rms in equilibrium is denoted by n : Notice that for a given n , output under
free entry is identical to that calculated in the Introduction. Now we explain how n
is determined. We will assume that the decision of entry is prior to the decision on
output.11 Thus, equilibrium under free entry implies that if n …rms are in the market,
10
The fact that WL are small in large economies under increasing returns has been shown by Dasgupta
and Ushio (1981), Fraysse and Moreaux (1981) and Guesnerie and Hart (1985)
11
Lopez-Cuñat (1999) has shown that, under conditions that are met here, the equlibrium considered
in this paper is a subset of equilibrium when both decisions are simultaneous (for the latter, see Novshek
14
…rm n has non negative pro…ts but …rm (n + 1) has non positive pro…ts, formally
a2
b(n + 1)2
a2
:
b(n + 2)2
K
(5.1)
The expression on the left (resp. right) hand side provides an upper (resp. lower) bound
to the number of active …rms, denoted by n (resp. n). Denoting Q a2 =bK, we have
p
p
n = Q 1 and n = Q 2. Clearly, the distance between n and n is 1, thus the
number of active …rms in equilibrium is unique so they are all other variables.12 Welfare
in a Cournot equilibrium with free entry is
W =
a2 n (n + 2)
2b(n + 1)2
n K:
(5.2)
where n is the unique number ful…lling (5.1). In an optimal allocation, aggregate
output equals the one in (2.3) and, clearly, the number of active …rms is one.13 Thus,
social welfare is
Wo =
a2
2b
K:
(5.3)
The percentage of WL is
PWL = 1
n
a2 (n +2)
2b(n +1)2
a2
K
2b
K
=1
n
Q(n +2)
2(n +1)2
Q
1
2
1
(5.4)
It is easy to see that P W L is decreasing on Q. Thus the maximum value of P W L
occurs for the minimum value of Q; namely, (n + 1)2 . Then (5.4) can be written as
a function of n which is an observable variable. This gives us an upper bound to the
WL, denoted by P W L; which amounts to
n2
2n
1
= 2
:
2
n + 2n
1
n + 2n
1
Now we compute the minimum value of P W L, denoted by P W L, which occurs for the
PWL = 1
maximum value of Q, namely (n + 2)2 . Again, writing P W L in terms of n , we have
PWL =
2n 3 + 3n 2 + 2n + 2
:
(n + 1)2 (n 2 + 4n + 2)
[1980]).
12
If entry and output setting were simultaneous, Ushio (1983) has shown that there is a large number
of equilibria when K is small.
13
Overentry may occur even if the marginal cost is increasing, see von Weizsäcker (1980).
15
The next table gives us maximum and minimum P W L for selected values of n .
n
2
3
4
5
6
7
8
P W L :43 :36 :30 :26 :23 :21 :19
9
10
20
:17
:16 :089
P W L :27 :24 :22 :20 :18 :17 :155 :144 :13 :08
TABLE 2
Notice the large values of WL for small values of n. But even for ”large” values of
n WL are substantial: For instance for n = 10; the minimum P W L is 13% which is
larger than the WL in the case of no free entry with two …rms.
As in the previous section, let us disentangle the e¤ect from the misallocation of output from the e¤ect of misallocation due to overentry. Computing the optimal allocation
with the restriction that all equilibrium …rms are active,
P W LSB =
a2 n (n +2)
2b(n +1)2
a2
2b
a2
2b
n K
.
Clearly P W LSB is decreasing on K. Recall that the maximum and the minimum
values of K are
a2
b(n +1)2
and
a2
.
b(n +2)2
Thus, the maximum and the minimum values of
P W LSB , denoted by P W LSB and P W LSB are, respectively,
P W LSB =
1
1+n
2
and P W LSB =
(n + 2)2
(n + 1)2 (n 2 + 3n + 4)
Table 3 below summarizes the percentage of WL for selected values of n .
n
2
3
4
5
6
P W LSB :2
:1
:059 :038 :027
7
8
9
10
:02
:015
:012 :01
20
:0025
P W LSB :127 :071 :045 :003 :0023 :0017 :0013 :001 :0009 :0002
TABLE 3
16
The percentage of maximum (resp. minimum) WL that can be attributed to oligopolistic output setting is the ratio between P W LSB and P W L (resp. P W LSB and P W L).
Next table present these ratios for selected values of n :
n
2
P W LSB =P W L :46
3
4
:277 :20
5
6
7
8
9
:14 :117 :095 :079 :07
P W LSB =P W L :477 :296 :205 :15 :127 :1
10
20
:0225 :003
:083 :0694 :0692 :0025
TABLE 4
It is remarkable the similitude between the two columns. Only for the duopoly case
both ratios are substantial. From …ve active …rms on, the misallocation due to output
is really negligible.
The main conclusion of this section is that …xed costs and free entry imply a nonnegligible misallocation of resources. And except if the number of …rms is small, misallocation due to the wrong number of active …rms are far greater than misallocation due
to the wrong output being produced.
6. Non Identical Firms
Suppose that …rms have di¤erent productivities. Let ci be the marginal cost of …rm i.
Without loss of generality let c1 ci , for all i. Let ai A ci . We will assume that for
P
all i, nai > j6=i aj . We will see that his assumption guarantees that all …rms produce
a positive output in equilibrium (see equation (6.1) below). Pro…ts for …rm i can be
written as
bx)xi . It is easily calculated that in a Cournot equilibrium
Pn
Pn
ai
j=1 aj
j=1 aj
and x =
:
(6.1)
xi =
b
b(n + 1)
b(n + 1)
Pn
Pn
2
bx2
Social welfare is W = Ax bx2
j=1 cj xj =
j=1 aj xj
2 : In a Cournot equilibrium
W =
i
n
X
i=1
= (ai
ai
ai (
b
Pn
j=1 aj
b(n + 1)
)
" Pn
#2
n
X
a
j
b
a2i
j=1
=
2 b(n + 1)
b
i=1
17
P
(2n + 3)( nj=1 aj )2
2b(n + 1)2
;
(6.2)
which when all ai ’s are identical reduces to (2.2). In the optimal allocation only the
technology in the hands of Firm 1 is used and accordingly
xo =
a1
a2
and W o = 1 :
b
2b
(6.3)
Thus, the percentage of welfare loss can be written as
PWL =
a21
2
Pn
2
i=1 ai +
a21
P
2
(2n+3)( n
j=1 aj )
(n+1)2
:
(6.4)
In order to have a workable expression that depends on observable variables alone, let
us de…ne si as the market share of …rm i. Then,
P
Pn
(si + 1) nj=1 aj
ai (n + 1)
xi
j=1 aj
Pn
si
) ai =
=
x
n+1
j=1 aj
(6.5)
Plugging the last part of (6.5) into (6.4) we obtain that
PWL =
Dividing by (
P
(s1 +1) n
j=1 aj 2
(
)
n+1
Pn
PWL =
j=1 aj 2
n+1 )
2
P
(si +1) n
j=1 aj 2
(
)
i=1
n+1
Pn
(s1 +1) j=1 aj 2
(
)
n+1
Pn
+
Pn
aj
)2
(2n+3)( j=1
n+1
(n+1)2
:
(6.6)
P
+ 1)2 + 2n + 4
s21 + 2s1 2 ni=1 s2i
=
(s1 + 1)2
(s1 + 1)2
(6.7)
and rearranging, (6.6) simpli…es to
s21 + 2s1
2
Pn
i=1 (si
We see that in this case the percentage of welfare loss depends only on two things:
On the market share of the dominant …rm and on the Hirschman-Her…ndahl index of
P
concentration de…ned as H = ni=1 s2i . Notice that, for a given value of H, P W L is
increasing on s1 , which sounds reasonable: the larger the dominant …rm, the closer to
monopoly. However, for a given s1 , P W L is decreasing on H! The reason is that when
if H increases, production is being shifted to the less e¢ cient …rms which causes social
welfare to fall.14
14
In fact, social welfare is increasing in the marginal cost of small …rms see Lahiri and Ono (1988).
For a criticism of the idea that concentration is generally bad for social welfare see Daughety (1990)
and Farrell and Shapiro (1990).
18
We now study the extrema of (6.7). First we notice that since
can be written as
P
( i6=1 si )2
P
PWL =
4 + ( i6=1 si )2
1
Now compute
(4 + (
@P W L
@sj
P
2
i6=1 si )
4
2
4
for j = 2; :::; n, which equals to
P
P
Pi6=1
s2i
i6=1 si
Pn
2
i=1 si
:
(6.8)
P
+ 4sj ) + (2 i6=1 si + 4)(1
P
P
(4 + ( i6=1 si )2 4 i6=1 si )2
i6=1 si )(2
= 1, (6.7)
P
P
( i6=1 si )2
i6=1 si
2
P
2
i6=1 si )
From this equation we obtain the Kuhn Tucker inequalities corresponding to the extrema
P
of P W L with respect to (s2 ; s3 ; :::; sn ) with inequality constraints i6=1 si = 1 s1 , and
s1
sj
0, all j = 1; 2; 3; :::; n. These are for j = 2; 3; :::; n,
@P W L
@P W L
@P W L
< 0 ! sj = 0;
> 0 ! sj = 1, otherwise
=0
@sj
@sj
@sj
(6.9)
Comparing (6.9) corresponding to two di¤erent …rms, say, i and k, we see that the
solution to (6.9) must be such that si = sk = x, say for all i and k 6= 1. For easiness of
notation let m
n
1. Thus, FOC can be written as
8mx2 + 4m2 x2 16x 10mx + 4
(4 + m2 x2 4mx)2
The inequality constraints read in this case s1 = 1
x(m + 1)
to solving
and x =
1
m+1 .
4mx2 + 2m2 x2
1 or x
1
2m+4 :
to x =
1 m2 x2
4+m2 x2
x
0, which imply
Let us …rst …nd interior extrema of P W L. This is equivalent
8x
5mx + 2 = 0. This equation has two solutions: x =
The …rst solution is not acceptable since x must be not larger than
Thus extrema of P W L are, either x = 0, or x =
1
n or
2mx2
4mx
xm and s1
1
m+1
or x =
1
x = 2n+2
. Plugging these values in (6.8)
x2 n2 x2 +1
= 4x 4nx+x
2 2nx2 +n2 x2 +4 = P W L(x), say-
1
2m+4
m
2
1
m+1 .
which correspond
-which under symmetry reads
we obtain that
1
1
1
1
n+1
P W L(0) = , P W L( ) =
, P W L(
)=
:
2
4
n
(n + 1)
2n + 2
3n + 5
From these values of P W L we get the following:
1) The minimum P W L occurs when all …rms are identical.
2) The maximum P W L occurs when s1 =
(since 4(n + 1) > 3n + 5 for n > 1).
19
n+3
2n+2
and s2 = s3 =; :::; = sn =
1
2n+2
7. Conclusions
When one observes public policies on oligopolies one sees some concern about the number of active …rms. But the question of the output set by oligopolists is cause of little
or no concern at all. This paper provides some justi…cation to this attitude. In two
di¤erent settings, namely decreasing marginal costs with a given number of …rms and
constant marginal costs plus a …xed cost and free entry, we found that welfare losses
due to the wrong number of …rms can be quite substantive even with as much as ten
active …rms in the market. On the contrary welfare losses due to the divergence between
equilibrium and optimal output are small, even with as few as four …rms in the market.
This conclusion, though, is likely to be exaggerated by the fact that under our assumptions the optimal number of …rms is one. It is possible that if an U-shaped average
cost curve was assumed this conclusion would be weakened. Other factor likely to bring
down our estimates of welfare losses is the consideration of other solution concepts (e.g.
Bertrand). Thus our results are just a …rst cut to the problem.
Our results have a number of implications for the empirical literature that measures
welfare losses. Typically, this literature has attempted to measure welfare losses arising
from oligopolistic output setting. Our results suggest that this attempt is misguided
because welfare losses arising from this source are likely to be small. However welfare
losses due to the wrong number of …rms can be quite substantial. At the best of my
knowledge a measure of the latter has never been attempted.
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22