GRADIENT INDICES AND THE
STRUCTURE-PERFORMANCE RELATIONSHIP
Hugo Pedro Boff (UFRJ/Brazil)
hugo@ie.ufrj.br
ABSTRACT
T h i s p ape r add re ss es an old but int ere sting qu esti on: h ow to d efi ne a n ind ex to me asu re
indu st ry pe rfo rman c e in ma rk ets whe n p rodu cts a r e d i ff e re n t i at ed an d f i r ms h av e d i f f e r en t
l evel s of e ffi ci ency . In th e an aly si s o f b ene fi ts-co s ts of p olicy in t erventi ons in the
p rodu ct sp ac e, Dansby a nd Wil lig , 1979 (DW ) used st anda rd met ri c s t o t rack t he soci al
c ost s o f th es e i nte rventi ons and der iv ed perfo r ma n ce g radie nt in di ces ( PGI ) ac co rdi ng to
th e met ri c u sed . Ho wev e r, th ei r indi ce s fai l to ac count a dequ at ely fo r th e p erfo rman ce
e ff ect s of ch ang es in th e mark et st ru ctu re of diff erentiate d i ndust ries. Fu rth er, th ey a re
un able t o anti ci pat e tha t th e do wn siz e of a fi rm may be re co mme nd ed , in so me c ase s,
f ro m a so cie tal st andpoi nt . In this pap e r, we use a g enerali z ed Euc lide an metri c al ong
with effi cien cy weight s enabl ing th e regul at ory ag ency to id ent ify "p ri ma facie " fi rms
a nd di re cti on s th at should d riv e th e in t erv enti on . W e re lax a lit tle th e o rigin al DW
a s su mp tions on the so ci al welf are fun ctio n and assu me t h at i t i s pseu do conc av e a nd th at
it s g radi ent i s p ropo rtion al to t h e m ar ket v alu e of p rodu cti on . Th e i mp li ed PGI is a
f un ction of ma r k et p r imi tiv es . T he s t ru ctu r e -p erfo r man ce r e lation s hip i s t h en analy z ed
und e r sp ecifi ed deman d fun ction s in ho mog ene ous and di ff e renti at ed p rodu ct mark et s. In
th e ex e rci se , sev eral inte resting result s e me rge: (i ) Th e diff ere nc e b et ween th e
Hi r sch amnn-Her f ind ahl ind ex and th e PGI app ea rs as a me a su r e of t h e p e rfo rma n ce o f
v a ri et y; (ii ) Imp rov ed t echn i cal effi ci en cy of the high (lo w) pe rfo rmi ng fi rm reduc e s
(in c rea se s ) indu st ry pe rfo rman c e . (i ii ) Th e do wn sizing eithe r of t he l a rg est o r th e small est
fi rms i s account ed fo r wh en th e go od s a re too li ttle diffe rent iat ed .
K ey wo rds : Pe rfo rma nc e, economi c su rpl u s, mark et st ru ctu re, p rodu ct di ffe rent i ation .
J EL: D63 , L11 , L4 0 .
I - INTRODUCTION
The static performance of an industry is usually analyzed by considering
the firms' profitability and total welfare. For theoretical reasons, most of the
attention is directed to price-cost margins. However, the allocative and
productive efficiencies of firms evolve within a market structure which,
ideally, is composed by many sy mmetric firms. Thus, a good performance
measurement should tie together the size distribution of firms and their
relative contribution to the economic surplus.
2
The main attempts to build performance indices were by Dansby and
Willig(1979)
and Blackorby, Donaldson and Weimark(1982). The latter
paper presents an axiomatic approach to focusing on the theory of equivalent
numbers. This paper elaborates on the Dansby and Willig ( DW) approach .
DW compare the benefits of policy interventions in the production space,
with the social costs of these interventions. In their approach, performance is
measured, at the industry output vector q , by the instantaneous rate of
*
change in total surplus along an optimal path q (t ) of the firms' product
vector. The marginal benefits are derived from the Marshallian surplus
function V and assumed as proportional to the price-cost margins. A metric
is used to bound the size of the output adjustments that are feasible. The
emphasis is put on the Euclidean metric which they presume to track social
costs
more
adequately.
The
index
obtained
is:
Φ=
∑ λ
n
i =1
2
i
,where
λi = ( Pi − φi' ) / Pi is the price-marginal cost of firm i and n is the number of
firms. The higher the industry performance is, the lower the values of Φ are,
meaning that the performance improves when prices are closer to marginal
costs. For homogeneous product industries, DW show that, under different
behavior assumptions, the performance gradient index (PGI) is directly
linked
to
well-known
concentration
indices
like
CR
and
Hischmann-
Herfindhal (H) indices.
One of the main shortcomings of the PGI is that it cannot be used to rank
different welfare levels. More precisely, if Φ A < Φ B we cannot say that the
allocation
qA
is
better
than
the
allocation
qB
in
the
sense
V (q A ) > V (q B ) ,unless these allocations are taken along the optimal path
q * (t ) . Moreover, Lerner indices are not
good measures of the industry
performance when the goods are differentiated and the firms' costs are
asy mmetric, as we will see below.
1
Nevertheless, the DW approach offers to economists a rich framework to
make theoretical analyses of the structure-performance relationship. Indeed,
3
the inverse demand Pi and the cost functions
φi depend on preferences and
technology, so that the performance measured by Φ can also be viewed as a
function of market primitives. On the other hand, as DW pointed out (p.257),
Φcan be interpreted as a local indicator of the potential benefit-cost ratio of
government intervention. Therefore, by analyzing the behavior of the index
under different parameter values of preferences and/or costs, we can infer
which are the market conditions under which local interventions in the
planned production of firms are beneficial the most. This will be, precisely,
the main focus of the present paper.
DW close their 1979 paper suggesting that, besides the firm behavior, a
key factor commanding future research would be the social costs of policy
interventions.
Curiously,
thirty
years
later,
their
challenge
remains
unmatched and, at our knowledge, no further research on the issue has been
published. In the next section we outline the main limitations of the indices
derived from the metrics used in the DW paper. Then, after a brief discussion
on the intervention costs, we will introduce in Section III a generalized
metric along with efficiency weights to track these costs. The welfare
gradient is assumed to be proportional to the gross revenue of firms rather
than to net revenue. Next, in Section IV we analyze the structureperformance relationship under specified demand functions in homogeneous
and differentiated product markets. This exercise identifies conditions under
which several interesting results emerge. In particular, a measure of the
performance of variety in the market is obtained by taking the difference
between the PGI and the H-index. The performance of technical innovations
is addressed in Section V. We found out that improved technical efficiency of
the high (low) performing firm reduces (increases) industry performance. In
Section VI, a summary of the main results concludes the paper.
II- METRICS, BEST DIRECTIONS AND COSTS
DW point out that "...the most appropriate metric is the one most closely
related to the costs of the governmental actions required to effect the
4
movements" (p.250). Next, they state (p.257) that the metric assign distances
to various output changes that should be viewed as monotonically increasing
with the intervention costs required to effect the changes. But which costs are
these? The authors did not detailed them, but one might think that they are of
a composite nature and should include not only the social value of the output
changes but also the agency costs of coordination of the firms’ quantity
adjustments,
informational costs, and other expenditures needed to enforce
the proposed changes and support the new market configuration.
In their article, DW derive the PGI under three different metrics: (i) the
standard
Euclidean
metric
applied
to
the
present
value
of
the
size
adjustments, which generates Φ; (ii) the standard Euclidean metric applied to
the percent output changes;
(iii) the city-block metric with base-price
weights. The Euclidean and the city-block metrics treat symmetrically equal
valued adjustments in the firms of disparate sizes. The second metric applies
when it is socially more costly to effect a large than a small percentage
change in the output of a firm. This implies that, for a unit distance, it is
costly to adjust the quantity supplied of a small than of a large firm. 2
The standard metrics mentioned above treat symmetrically all firms. So,
in assessing the intervention costs, the differences in the firms’ efficiencies
are not tracked on and, as consequence, the social value of the proposed
output changes
are not considered. Further, the orthogonality of these
metrics allows the public agency to contemplate the allocative dimension of
costs only, not its competitive side. However, the agency costs to coordinate
the quantity adjustment of firms are important in many situations and deserve
more attention.
The consequences of ignoring the contribution of firms to the economic
surplus and the costs of coordination in the competitive setting are enhanced
by analyzing the behavior of Φ. The comments below also applies to the
indices derived from the other metrics presented in the original DW paper.
(i) Differentiation intensity. Consider the linear representative consumer
model, in which the suppliers of differentiated goods maximize profits. At
the sy mmetric equilibrium, whichever strategic variable they chose, prices
5
are lower when the goods are more standardized than when they are more
differentiated. Therefore, as the differentiation parameter increases, if
marginal costs are constant, the index Φ reduces, so that the net benefits of
an optimal policy intervention are maximized when the goods are perfectly
differentiated. However, the economic theory suggests instead that there is a
nonmonotone relationship between the aggregate performance and the
differentiation intensity. See Lancaster(1979); Scherer and Ross(1990,600607) and the subsequent literature. In the representative consumer model,
when the differentiation of goods increases beyond its optimal degree, the net
surplus of consumers is progressively eroded by increasing market power of
firms.
(ii) Best directions. The standard Euclidean metric treats sy mmetrically
equal
valued
adjustment
d ( Pi qi* (t )) / dt |t =0 = λi / Φ
sizes
so
that
the
best
directions
are all positive as long as the firms' prices exceed
their marginal costs. This means that, in all relevant cases, increases in the
supply of all firms are virtually recommended. Such prescription is obviously
restrictive, since it excludes cases in which, the horizontal divestiture or
even the downsizing of a firm are recommended from a social point of view.
(iii) Cost efficiencies. The equilibrium analysis of the linear Cournot
market with product homogeneity or the linear Bertrand market with
symmetric product differentiation, both show that if a cost economy is
obtained by a large and efficient firm, concentration and total surplus both
increase. Yet, if the cost reducing firm is small and inefficient, concentration
reduces and surplus increases and thereby, there would be less incentive for a
public intervention in the industry. However, if we calculate Φ at the
equilibrium points, we may find that the PGI increases and hence, the net
benefit of the intervention increases, in both cases.
In this paper we will show that a better tracking of the intervention costs
is obtained if the distance function accounts also for the relative position of
firms in the market structure. As we noted before,
the social costs of the
intervention are viewed as depending on the efficiency of firms and the
competitive setting.
6
Firstly, for an adjustment of a given size, the costs to intervene in the
planned production of a large and efficient firm may be higher than those
incurred in taking the same action on a small firm. This conforms with the
common sense that only inefficient firms deserve the public scrutiny.
Further, monetary compensations, subsidies, or informational costs to unveil
firm's marginal cost, for example,
may be high when the firm is large and
efficient.
Second, if it is true that cases where competition is insufficient should
be first to fall under scrutiny, then the intervention cost function of the
regulatory agency should put a higher weight on the quantity adjustments
pointing at the competitive directions then at the noncompetitive ones. So,
when a simultaneous adjustment in the production plan of two firms is
recommended from a social standpoint, the intervention would be costly if
the goods supplied by these firms are strongly substitute. This is so because
the fierce competition among the producers hampers the
agency efforts to
harmonize social and private interests. On the contrary, if the goods are only
weak substitute one each other, then competition is likely soft, so that
the
proposed change would be more easily implemented with low costs of
coordination to the agency.
III – A GENERALIZED INDEX.
We
introduce
now
d ω (q, x) = ( x − q )' Σ( x − q )
a
generalized
Euclidean
metric
, where Σ is a positive definite and symmetric
(p.d.s.) matrix, with nonnegative entries
σ ij
. Such metric sums up all
orthogonal and non orthogonal projections of individual distances. According
to the rationale of the intervention costs, we will define ahead the efficiency
weights generated by each individual firm, which are then used to weigh the
direct and cross terms of the distance. By this way, the allocative and the
competitive dimensions of the intervention costs are both represented in the
metric.
7
In order to obtain the PGI under the above metric, we start with base-price
D p = diag ( P1 ,..., Pn )
weighted output adjustments, as DW did. Let the
diagonal matrix of
be the
market prices and q = (q1 ,..., q n ) the present industry
output vector. For a budget of a given siz t > 0 , the optimal production path
q * (t )
maximizes
d ( D p x, D p q ) ≤ t
2
the
welfare
function
lim
2
t →0
. The PGI is defined by
V ( x)
under
the
restriction
dV (q * (t ))
dt
. In the DW paper, the
welfare gain of a marginal increase in qi equals the mean revenue of good i
∂V
≡ Pi (q) − φ i' (q i ) = λi Pi (q)
net of marginal costs, that is: ∂q i
. Then, by using
the envelope theorem, the above limit is:
3
Φ ω ≡ λ ′Σ −1λ
Where
λ ′ = (λ1 ,..., λn ) .
Mahalanobis
distance
(1)
Φω
When Σ is a matrix of dispersion,
between
0
and
λ,
with
scale
is the
Σ −1
matrix
(Mahalanobis,1936). Also, Φ ω gives the greater increase in welfare obtained
from
the
intervention
d (q, x) = ( x − q )' Σ −1 ( x − q )
at
q,
under
the
conjugate
metric
. Notice that the standard index is a particular
case of Φ ω when Σ = I . Moreover, all six propositions derived from the
standard index Φ in the DW paper also apply to the generalized index
Φω
with the metric d . 4
The
vector
of
best
directions
D ≡ [d ( D p q * (t )) / dt ]|t =0 = Σ −1λ / Φ ω
of
the
adjustments
is
here:
. Differently from the standard case, the
direction of a price weighted output adjustment of any good i depends on the
profitability of all firms.³ Indeed, the i
th
component of the direction vector is
Di = σ i λ / Φ ω where σ i denotes the i th row of Σ −1 . Notice that some
8
directions may be negative, because there is components of
σ i that are
negative.
3.1 Efficiency weights and Intervention costs
From now on, the matrix Σ , which will be written as Σ = nΩ , where Ω is
a normalized matrix whose entries
ωij
are calculated from the inverse demand
P
function for each good i , i minus the marginal cost function
weights are nonnegative and sum 1.
φi' . The
Let Wii be the Marshallian surplus
calculated from a straight line integral under the price - marginal cost
from 0 to qi . In the homogeneous and differentiated
function for good i ,
product cases, we have, respectively:
qi
Wii   Px  Qi  
qi
 i xdx
2a ; Wii   P i x;q i    i xdx
0
where
2b
0
Q is the total output, Qi = Q − qi and q−i = (q1 ,...qi −1 , qi +1 ,..., qn ).
For the cross weights (i ≠ j ) consider the line integrals for the homogeneous
and differentiated product cases, respectively:
qj
qj
Wij   Px  Qj  
 i q i dx
3a ; Wij   P i x;q j    i q i dx 3b
0
0
The elements of the matrix Ω of weights
.The weight
are defined as:
ωij = Wij / ∑ij Wij
ωii is an allocative measure of the social efficiency of firm i
when it supplies qi . The cross weight
ωij
is a measure of competition in
j
consumption between goods i and , in the sense it accounts for the amount
of change of the firm’s i share in total surplus as the consumption of good j
q
increases from 0 to the present amount j .
9
In order to have a closer view on the role of these weights in social costs,
consider the money value of the distance between the
d ω ( D p x, D p q ) =
∑ nω
ij
ij
Pi Pj ( xi − qi )( x j − q j )
For i = j , nω ii Pi ( xi − qi )
2
2
is the i
th
outputs
x and q :
.
square component of the social cost,
when the actual output vector q adjusted to x . Given a unit distance, a
higher weight is put on the direction of the more efficient firms that is, those
having higher share
ωii of the Marshallian surplus. The share ωii is expected
to reduce as the good i becomes more and more substitute to the other goods.
The idea behind this effect is that “wrong” interventions are penalized: the
more efficient a firm is, the less it needs intervention.
For the cross term
nωij Pi Pj ( xi − qi ) ( x j − q j )
, assume first that
( qi , q j )
lies
on the rayon vector pointing to the optimal direction. Then, the maximu m
welfare criteria will prescribe a quantity move in the same direction for both
goods i and
j . The costs to coordinate the change are higher when the
goods are more substitute because the producers are engaged in harsh
competition. As a consequence, the costs of the agency to enforce the change
are higher. On the contrary, if
( qi , q j )
lies outside the rayon vector through
the optimal point, the optimality will prescribe a quantity increase of one
x < qj
good, e.g., xi > qi and a reduction of the other, j
. Now, costs reduce
with increasing competition because a higher substitution between goods i
and
j makes the quantity adjustments easier. The agency costs of
coordination will be lower and a transfer pay ments scheme between winner
and losers may be set up in this case.
Notice that our definition of the matrix Σ assumes that the square
distance equals n times a weighted mean of individual square distances. Such
formulation implies that for a fixed mean adjustment per firm, the total cost
of the public intervention increases with the number of firms under scrutiny.
This complies with the perception that heterogeneous characteristics of firms
10
will likely prevent the public authority from obtaining economies of scale
and scope in the regulatory activity.
3.2 The modified performance index
In order to analyze de behavior of the index
Φω =
1
n
λ ′Ω −1λ , we need, of
course, to make assumptions on the inverse demand functions
calculating the weights
ωij
. Further, the price-cost margins
Pi for
λi , are not
directly observable and need to be estimated. A natural way to do this, as DW
suggested in their paper, is introducing additional assumptions on the
behavioral mode of competition in the industry. For example, if si = qi / Q
stands for the quantity market share of firm i and
ε is minus the price-
elasticity of the demand, Cournot equilibrium for homogeneous product
λi = si / ε . Equilibrium in differentiated product markets,
market
leads to
requires
λi is a decreasing function of direct and cross-price elasticities of
the demand. However, that will not be the way we go through in this paper.
First, we want a performance index that enables us to analyze the structureperformance relationship independently of any assumption on conduct. To
this aim, we need an index that relies uniquely on the parameters of market
primitives and on observables, that is, the market shares s = ( s1 ,..., s n ) , the
industry output
Q and the number of firms n . Second, as we noted above,
price-cost margins are not good indicators for the industry performance, so
that it seems to us more advisable to modify the assumption DW made on the
welfare gradient.
Given the diagonal matrix of prices
Dp
, and a social welfare function V
∂V / ∂q ≡ D p z
of quantities q , the gradient vector is:
where z is some vector
function of
q . DW takes the objective function V as the Marshallian surplus
function and its gradient is defined as surplus net costs, that is: z = λ .
11
In this paper, V is assumed to be any suitable social welfare function of
quantities q . It needs not be made explicit, since the social benefits are
measured at the margin. However, we will assume V is C , increasing in
2
qi ; i = 1,..., n and we relax the original concavity assumption by assuming V
is pseudoconcave over an open and convex neighborhood
U ⊂ R+n of q . 5
Further, while net surpluses are used here to weigh the value of output
changes, the gradient of social benefits is assumed to be proportional to the
gross revenue of firms, that is:
∂V / ∂qi ≡ Pi qi / Q . From this assumption, the
marginal benefits obtained from an increase in qi is larger (smaller) if the
market of good i generates larger (smaller) revenues. Further, other things
equal, the benefits of quantity increases diminishes for all goods, when the
market size Q increases.
The gradient assumption and the pseudoconcavity property assumed for
V at q ensures that, if
∑
n
j
s j Pj ( x j − q j ) ≤ 0
, then V ( x1 ,..., xn ) ≤ V ( q1 ,..., q n ).
So, a meaningful effect describing the underlying societal preferences is
obtained: higher social values are assigned to output vectors whose market
weighted mean value are larger. 6 Further, the pseudoconcavity property of V
and the convexity of the constraint, both ensure that the first order solution
q * (t ) is a global maximum path on the constraint set. 7
Thus, the performance index used from now on is defined by Φ ω
2
evaluated at z = s , that is :
Πn =
If
1
s ′Ω −1 s
n
(4)
ς stands for the parameters of preferences and/or costs, the index
Π n ( s, Q, ς ) in (4) gives the instantaneous rate of the welfare change expected
from an optimal change in the firms' supply, when the market conditions are
represented by the triple
(n, s, Ω(ς )) .
12
General properties of Π n
1. Sy mmetric conditions should generate the best industry performances.
Π n fulfils this requirement. Indeed, under sy mmetric preferences and costs,
and equal market shares si = 1 / n ; i = 1,..., n , we prove that Π n = 1 / n . 8 A
rather intuitive argument is used to set that deviations from the symmetric
case lead to values of Π n larger than 1 / n .
2. Since Ω is p.d.s., there is a linear orthogonal transformation of market
shares allowing one to give a canonic representation to Π n . 9
Moreover, the
index can also be written as a mean of squares of all weighted concentration
ratios
Ck
; k = 1,..., n . 1 0
Performance evaluations and best adjustments
Let M n ( s ) ⊂ {s} × R+ × R
k
be the set of structural conditions generating a
p.d.s. matrix of efficiency weights Ω under market shares s and industry
output Q . A triple ( s, Q, ς ) is said measurable if it belongs to M s ( s ) . For
measurable changes in the market primitives, from
ς to ς * , the industry
performance effects are evaluated by : Π n ( s, Q, ς *) − Π n ( s, Q, ς ) .
The
best
directions
Di = s′ω i / n Π n
implied
by
the
modified
index
in
(4)
are:
ω i is the i th column of Ω −1 . Under sy mmetric
, where
preferences, costs and market shares, one can check that
s ′ω i = 1 and
Π n = 1/ n . This implies Di = 1 / n , for all i . Thus, from a societal
standpoint, a firm i may be thought as high performer (low performer) if
Di ≤ 1 / n
( Di > 1 / n )
firms to the intervention.
. These are the least (the most) likely candidate
11
13
In the next section we will analyze the behavior of the modified index
obtained under standard market demand functions in the homogeneous and
differentiated product cases. The functions are chosen in order to obtain
explicit expressions for the efficiency weights.
In a first step, the efficiency weights are calculated from the
consumers' standpoint. This enabled us to obtain weights that are independent
of the output level Q . By this way, under quasi-linear preferences and
symmetric differentiation, the efficiency weights obtained show a meaningful
overall efect of the industry concentration on the metric and hence, the social
costs. As the Hirschmann-Herfindhal index increases, the costs of the public
intervention reduces and hence the net benefits derived from that intervention
increases.
Then we analyze the behavior of the generalized index and the best
directions
Di as a function of the differentiation intensity. A demand
linearity assumption enables us to use the index Π n to compare one each
other,
the
net
benefits
of
local
interventions
in
homogeneous
and
differentiated product markets.
IV - STRUCTURE-PERFORMANCE RELATIONSHIP
Throughout this section, the efficiency weights are calculated according to
a consumers' criteria. This means that marginal costs
(2a,b) and (3a,b) are replaced by price
φi′ in the formulas
P (Q) in the homogeneous product
case (2a and 3a) and by the product price Pi (q ) in the differentiated product
case (2band 3b).
The index obtained, noted hereafter Π , is a suitable market performance
c
index, since it only depends on the market shares and the preference
parameters, not on technology and costs.
14
3.1 Homogeneous goods
In this case, the goods are perfect substitute, so that the cross terms
Wij
can be neglected from the consumers' standpoint. 1 2 There is no agency costs
to coordinate the output changes. Only allocative costs are present. Then, if
Wi c is the consumer surplus obtained from (2a) by replacing the marginal cost
ωic = Wi c / ∑ j =1W jc
by the price level P (Q ) , the surplus shares will be:
;
n
i = 1,..., n . The matrix of surplus weights is diagonal in this case:
Ω c = Diag (ω1c ,..., ω nc ) . So, the formula (4) gives:
Π cn =
1 n si
∑ ( ) si
n i =1 ωic
(4)
At the market equilibrium, the firms' profit margins are a function of
market shares and demand elasticity. Usually, high elasticity entails a
fragmented market structure in which consumers have ease to switch their
demand in response to changes in relative prices. In the two exercises made
below, the market shares are held constant as the elasticity parameter
changes. In the first, the demand function models a market for a nonessential
good. The reservation price is finite and another good is presumably
available outside the market. While inside switching is not allowed, the
market performance worsens as the price-elasticity increases because the
benefits of the intervention increases when the consumers are more and more
sensitive to the price changes. The second exercise models a market demand
for an essential good. The reservation price is infinite and there is no outside
good. In this case, the market performance improves as the elasticity
parameter rises because the benefits of an optimal intervention falls with the
ease with which consumers could effect the switch from one seller to another.
When the demand becomes perfectly elastic both indices converge to the
same value
Π cn ( s ) given in (6). Of course, under the same market conditions
( s, ς ) , the nonessential good market outperforms its essential good
counterpart.
15
Example 1(nonessential good):
Consider the market demand function P (Q ) = α − β Q
demand is linear when
on
α
β
and
ρ
;
α , β , ρ > 0 . The
ρ = 1 . The surplus shares in this case do not depend
ρ:
but
only
on
the
elasticity
parameter
n
 ci  1  s i  1  s i  1  1/1     j1
1  s j  1  n
. Thus, by replacing
the weights in (4) leads to:
n
1  s j  1  n n
1     j1
s 2i
c
n s,  
5

i1
n
1  s i  1  s i  1  1
ρ , a perfectly fragmented market, si = 1 / n , or
Notice that, for every
monopoly ( n = 1)
both lead to Π m ( n , ρ ) = 1 / n for all
c
1
ρ . By keeping market
shares constant, one can show that Π n is a decreasing function of
c
ρ → 0,
the
surplus
ωic
share
tends
ρ . For
to:
n
 oi  s i  1  s i  ln1  s i /  j1
s j  1  s j  ln1  s j 
Substituting that value in (5) leads to an upper bound for
on s 
n
s j  1  s j ln1  s j 
 j1
n
n
 i1
Π cn ( s, ρ ) :
s 2i
6
s i  1  s i ln1  s i 
Therefore, if the demand becomes more elastic, i.e., if
ρ decreases to 0
while the size distribution is held constant, the market performance is
Π on ( s ) . At that point, the consumers are perfectly
elastic to price changes and, given s , the market performance is at its worst.
worsening to the level
For the monopoly case we have:
lim Π on (1 / n) = 1
n →1+
,and we can check that
16
Π on (1 / n) = 1 / n , for n ≥ 2 . Notice that Π on ( s ) decreases as the market
fragmentation increases.
Taking
ρ → ∞ , the surplus share ωic tends to the market share si .
Substituting that value in (5) yields a lower bound for
 cn s  1
n
Π cn ( s, ρ ) :
7
Here, the demand is perfectly inelastic, and the performance is optimal
irrespective of market shares. This shows that even highly concentrated
markets can achieve optimal performance levels if the consumers are not
sensitive at all to price changes.
Example 2(essential good)
Consider
the
constant
good: P (Q) = AQ
index, say
Indeed,
δ
elasticity
demand
function
for
an
essential
; A > 0 ; 0 ≤ δ < 1 .This demand generates a performance
−
Π cn ( s, δ ) , which overvalues the indices of the previous example.
the
use
of
(4)
leads
to
n
 ci  1  1  s i  1  s i  1 / n  1     j1
1  s j  1 
can check that the index increases with
upper bound, say
weights:
. Using (4), one
δ . For δ → 1 , the index have a
Π *n ( s ) which is obtained using (4) with welfare weights:
n
s i  ln1  s i /1   i1
ln1  s i 
converges to
the
.
For
δ → 0,
the
weight
ωic
ωio of the previous example, so that the lower bound equals
Π on ( s ) , given in (6).
17
Performance and Concentration :
The Hirschmann-Herfindhal index H ( s ) equals
i.e.
∑
n
i =1
si2
. For linear demand,
ρ = 1 in example 1 one obtains:
ωic = si2 / H ( s)
(8)
Therefore, Π n ( s,1) = H ( s ) . The identity (8) states that the stake of each
c
firm in the consumers' surplus equals its weight in concentration. The best
Di =
directions are:
H /n
si . Thus, under demand linearity, i is a high
1
n
performer firm (low performer) from the consumer’s standpoint if the square
of its market share is higher (lower) than the industry mean, that is, if
si2 ≥ H / n (≤) . See Section 3.2.
Proposition 1: For a homogeneous product market industry, the H-index is
a market performance index from the consumers’ standpoint if and only if the
market demand function is linear.
Proof: See Appendix
The figure (1) below depict the values of the index given in the examples
1 and 2 as a function of the inverse of the elasticity parameter for a market
composed of
+
n = 5 firms with market shares s = (.4, .3, .1, .1, .1) with
H ( s + ) − .28 .
FIGURE 1
As the Figure 1 shows, under the same market conditions, the nonessential
good market outperforms the essential good one, as long as the price
elasticity of the demand is finite. Their performance coincides if the demand
is infinitely elastic to price changes.
18
4.2 Differentiated goods
According to Bain (1968), besides entry conditions and concentration,
product differentiation is an important dimension of the market structure.
Location and discrete choice models gave a theoretical evidence of the socalled
differentiation
principle:
firms
want
to
differentiate
to
soften
competition. In the representative model, convex consumers can long for
diversity, but not too much; as the goods’ differentiation augments, their net
surplus tend to be nullified by increased market power of firms. So, besides
the number of varieties, a performance measure must be sensitive also to
their specifications.
Example (linear demand).
A representative consumer has quasi-linear preferences and maximizes a
quadratic utility function. Assume that the following demand system obtains :
Pi (qi ; q−i ) = α i − 2 β i qi − 2γ i (∑ j ( j ≠i ) γ j q j )
and
where
β i > γ i2 ; i = 1,..., n . Goods i
j are substitutes or independent according to γ i γ j > 0 or γ i γ j = 0 . By
replacing marginal costs
φi' by prices P i (q ) in (2b) and (3b) for this demand
system, the following direct and cross surplus obtains:
Wiic = γ i γ j q 2j
.
By noting
W c = ∑ij Wijc
,
Wiic = β i qi2 and
we obtain the efficiency shares by
dividing the numerator and the denominator of
1
2
(Wijc + W jic ) / W c
by
Q 2 . To
simplify things we assume the goods are symmetrically differentiated, that is:
β i = β and γ i = γ
 cii
 cij
; i = 1,..., n . In this case, the efficiency weights are:
s 2i

1  n  1Hs
1
2
s 2i  s 2j 

  cji
1  n  1Hs
9
10
19
where
ξ ≡ γ 2 / β ; 0 ≤ ξ < 1 . This parameter is a degree of the product
differentiation
(Singh
and
differentiation increases as
Vives,1984,
p.548).
The
intensity
of
ξ tends to 0.
The formulas (9) and (10) show that, other things equal, the more intense
is the differentiation of goods in the industry, the more the role of firms´
efficiencies are enhanced and the role of competition is mitigated. Hence, if
goods are strongly differentiated, a lower level of coordination activity of the
agency is needed to effect the proposed changes.
The Proposition 2 below states that, for each market shares s , there is an
optimal degree of
product differentiation ξ s , that maximizes the market
performance. The figure 2 ahead show that extreme values of the own and
cross price-elasticity parameters, that is, high or low values of ξ , w.r.t. ξ s
are associated with "excessive" or "insufficient"
differentiation intensity.
Such result for the intensity matches a Spence's prediction (1976a,b) for the
number of varieties in monopolistic markets: the competitive equilibrium
likely generate too many or too few varieties w.r.t. the social optimum,
according to whether these elasticities are high or low. 1 3
Notice that the
consideration of nonlinear systems is also possible. 1 4
Recall that if firms' sizes are also symmetric, the performance is
maximum:
Di = 1 / n
Π n (1 / n ; ξ ) = 1 / n
and all firms are high performer firms, because
for all admissible values of ξ . When the goods are perfectly
s 2 / H ( s)
differentiated that is, when ξ = 0 , the shares in (9) and (10) equal i
c
and 0 , respectively. Then, it is easy to check that Π n ( s,0) = H ( s) for all s .
Performance gains of variety
Proposition 2 . (i) If the demand functions of symmetric varieties are
c
linear, Π n ( s, ξ ) approaches H (s) from below, as ξ → 0 . (ii) Given s , there is
20
an upper bound
ξ o and an optimal degree ξ s > 0 maximizing the market
performance.
Proof: See Appendix
The Proposition 2 implies the existence of a differentiation threshold
ξˆ ∈ (ξ s , ξ o ) such that Π cn ( s, ξˆ) = H ( s ) . From Proposition 1, Π cn equals H under
c
a linear demand for homogeneous goods. Since Π n ( s, ξ ) ≤ H ( s ) when
ξ < ξˆ ,
the market of varieties outperforms the homogeneous market if the goods are
c
sufficiently differentiated. The variety gain is measured by H ( s ) − Π n ( s, ξ ) .
When
ξ = ξ s , Π cn is minimum, so that Π cn ( s, ξ s ) gives the best performance
allowed under market shares s . If goods are less differentiated than the
minimum level, i.e. if ξ ≥ ξ o , the market conditions are not measurable. The
set
Μ cn
shrinks as the size inequality increases. 1 5
The market performance of variety
In order to illustrate the above results we calculated the performance curve
Π cn ( s, ξ )
s = (.4, .3, .1, .1, .1) with n = 5
, given s . The market of section 4.1,
firms
and
+
another
equally
concentrated
( H = .28)
with
10
firms,
s + = (.36, .36, .13, .03, .03, .03, .03, .01, .01, .01) are considered. The curves shape like
ˆ
an elbow . The critical points obtained are: ξ o = .311; ξ = .302; ξ s = .232 with
Π cn (ξ s ) = .228 for n = 5 and ξ o = .022; ξˆ = .021; ξ s = .016 with Π cn (ξ s ) = .210 for
n = 10 . The Figure 2 below depicts the curve for n = 10 , which summarizes all
other cases. In this figure and the others shown ahead,
scale.
FIGURE 2
Π cn
is depicted in log
21
When
0 < ξ < ξˆ = .016 , we have
.28 > Π cn (ξ ) > .21 . The goods are more
differentiated than the optimal level underlying the present market sharing. If
ξ decreases to 0 , the increased differentiation increases the market power of
firms and reduces the consumers' gain from the variety, because substitution
becomes difficult. As a consequence, the market performance worsens. At the
limit ξ = 0 , all variety gain is eroded by the monopoly power of firms and the
ultimate performance level equals that of the homogeneous market where no
c
variety gain is possible: Π n ( s,0) = H ( s) = .28 . The inspection of curves for the
best directions of adjustments - not depicted here - show Di (ξ ) > 0 for all
goods on the range ξ << ξ s meaning that higher performance can be achieved
by increasing the supply of all firms. The derivatives are small for the largest
firms and high for the smaller ones on this range.
ˆ
.21 < Π cn (ξ ) < .28
When 0 < ξ < ξ = .021 , we have again
and the goods are less
differentiated than the optimal level underlying the present market sharing.
ˆ
When ξ moves on the right side towards ξ , the performance worsens because
the present differentiation intensity is excessive. The examination of the best
direction curves Di (ξ ) show D1 = D2 < 0 for ξ > .015 and D3 < 0 for ξ > .021 ,
q
suggesting that q1 , q 2 and 3 can be reduced within these differentiation
ranges. Positive adjustments are indicated for the other goods. 1 6
If .021 < ξ < ξ o = .022 , the homogeneous market outperforms the market of
varieties, because
Π cn (ξ ) > H = .28
.
ξˆ
ξs
The quantities
As = ∫0 Πcn (s;ξ )dξ
, and
where the differentiation intensity is
Bs = ∫ξ Πcn (s; ξ )dξ
s
give the area
"excessive " and "insufficient",
ˆ
respectively. A mean index is given by Π n ( s ) ≡ ( As + Bs ) / ξ . The mean
c
performance gain from variety is:
markets are shown in Table 1.
H ( s) − Π nc ( s ) . The relevant values for both
22
TABLE 1
The mean variety gain is (.280 − .242) = .038 points ( .047 points) for n = 5
firms ( n = 10 ). Thus, variety generates a market performance improvement of
15.6% (.038/.28) when there is
n = 5 firms and 16.8% (.047/.28) in the
market with n = 10 . Given the market shares s , the optimal performance
level is .228
(.210) . The total deviation of the mean performance value
from the best (symmetric) value is ( .242 − .200 ) = .042
(.133) , which is the
sum of the deviations due to preferences ( .242 − .228 ) = .014
market concentration ( .228 − .200 ) = .028
the
same
concentration
(.023) and to
(.110) . Tough both markets have
H ( = .28 ), concentration accounts for 100 x
(.028/.042) = 66% of the total performance deviation in the case n = 5 and
82.7% (.110/.133) of the market performance deviation in the case n = 10 .
Asymmetric varieties and Gross substitution
Under asymmetric differentiation we calculated the index in the case
n = 5 , by assuming different values for the vector β with a fixed substitution
parameter
γ 0 . Defining ξ i ≡ γ 02 / β i we look at the vector ξ β = (ξ1 ,..., ξ n ) to
be used for calculating Π n in connection with the efficiency shares
c
obtained
in
the
linear
ξ β = (.05, .09, .12, .12, .12)
+
demand
example.
The
differentiation
ωijc
vector
is assumed. It is presumed that the most preferred
goods are more differentiated. The values chosen implement a mean
preserving spread of the differentiation, with mean = .10 , for comparing the
value of the performance index with the value
Π 5c ( s + ,.10) = .243 which was
obtained before under symmetric differentiation. We calculate the index also
under the reverse profile,
ξ βr = (.12, .12, .12, .09, .05)
+
. Percents in parenthesis
indicate change w.r.t. the value under symmetric differentiation intensity.
23
The market performance values obtained are:
Π 5c ( s + , ξ βr ) = .222 (−8.6%)
+
Π 5c ( s + , ξ β ) = .361 (+48.5%)
+
and
. As expected, the mean preserving spread of
differentiation making the most (least) preferred goods be less differentiated,
improves (harms) the market performance. 1 7
V - THE PERFORMANCE OF INDUSTRIES
A comparative static analysis is now carried out with the modified index
obtained when distances, and hence, the intervention costs, are weighted
according to a full efficiency criteria. That is, total efficiency shares
used, instead of
ω are
ω c . Since in this case the technical efficiency of firms are
also taken into account,
Π n is a suitable index for the performance of the
industry. In most cases, Π n will depend on the market size Q. Indeed, to have
an index independent of total output, all functions
Wij
must be homogeneous
in q1 ,..., q n . So, the price-marginal cost functions ( Pi − φi ) must also be
'
homogeneous in all their arguments. Since
Pi is decreasing in qi , φi' must be
also decreasing in qi . Thus, to have a performance measure independent of
the aggregate product level, economies of scale in the production of all firms
would be required. 1 8
The index Π n is now used to obtain the performance effects of changes in
the productive efficiency of firms holding s constant. Recall that increasing
cost asymmetries tend to increase concentration, and hence, to reduce the
industry performance.
5.1 Technical efficiencies and performance
Consider a firm i facing the cost function
parameter and
φi (qi ; ci ) , where ci > 0 is a
∂φi / ∂ci > 0 . Assume there are nonsunken costs: φi (0; ci ) = 0
24
for all ci . When ci is a fixed cost and qi > 0 we have ∂φi / ∂ci = 1 and
∂ 2φi / ∂qi ∂ci = 0 , for all qi .
In the homogeneous product case assume first, for simplicity, that
ωij = 0 ; i ≠ j
. That is, the agency costs of coordination are not considered. By
taking the derivative of (4) w.r.t. ci and using the derivatives taken from
(2a), we obtain:

s
 n
 1 n ni  2   n  i
i
W
c i
c i
11
The expression into brackets in (11) can be written as
where
Di = si / nωi Π n
nΠ n ( Di2 − 1 / n) ,
is the best direction of the price weighted output.
Thus, a reduction in firm i ' s cost increases or not the aggregate performance
according to i is a low performer firm ( Di > 1 / n ) or a high performer firm
( Di ≤ 1 / n ) . One can check that a high performer firm i (low performer) is
such that its share in total surplus
ωi equals or outweighs (falls short) its
( si2 / ωi ) / ∑ j =1 ( s 2j / ω j )
n
share in the "weighted concentration"
.
Cost savings
got by a firm increase its profit as well as its stake in economic surplus, but
reduce the surplus share of the other firms. So, two opposite effects are at
work: a pro-competitive internal effect related to the increased market power
of the cost-reducing firm; and an anti-competitive external effect due to the
fall of the surplus shares of the other firms. Therefore, an increase in the
productive efficiency of a high performing firm harms the aggregate
performance because the external effect dominates. If the cost-reducing is a
low performing firm, the internal effect prevails, so that the industry
performance improves.
25
If the costs of coordination are to be considered, that is, the
ωij
cross
weights are used, the sign of ∂Π n / ∂ci cannot be easily predicted, unless ci is
a fixed cost. In this case, the technical change no longer affects the marginal
cost. If Fi stands for nonsunken fixed cost of firm i , taking the derivatives
from (2b) and (3b) and using (4) we obtain:
 i
 n
s
1
 n n  2   n 
W
F i
where
12
ω i is the i th column of Ω −1 . Again, the expression into brackets in
2
Di = s'ω / n Π n
is the best
(12) can be written as nΠ n ( Di − 1 / n) , where
i
direction for firm i . The derivative in (12) is null, negative or positive
( Di ≤ 1 / n )
or low performance
according to firm i ′ s high performance
( Di > 1 / n )
. Therefore, a reduction in fixed costs of a high performer firm
(low performer firm), other things
equal,
worsens (improves) the industry
performance.
5.2
Industry size and performance
The index
Π n depends on market size also when the firms bear fixed costs.
It is useful to evaluate now the performance effect of a marginal change in
the output level of the industry. Assume that all price-cost margins
( Pi − φi' )
are homogeneous functions of degree k and that all firms incur nonsunken
fixed costs:
φi (qi ) = Fi + ϕ i (qi ) for qi > 0 and ϕ i (0) = 0 . Then:
n n k  1F n
 n

 i1 D2i f i  s i i 
QW
Q
s 
26
F = ∑i =1 Fi
n
where
; f i = Fi / F . Thus, an increase in the industry output
that leaves unchanged the market shares is socially beneficial, if the sum of
weighted deviations of costs ratios f i from market share ratios si / s 'ω is
i
positive.
VI - SUMMARY AND FINAL COMMENTS
By using the gradient approach of Dansby and Willig we proposed here a
performance index
Π N ( s;ς ) which enabled us to overcome some limitations
of the indices presented in their 1979 paper. A generalized Euclidian metrics
is used to set the output changes that are feasible. Further, we define surplus
weights that are used to track not only the social costs of the output changes
but also the agency costs of coordination. Unlike the original indices,
present index allows us: (a) to
the
account adequately for the performance
effects of changes in the product differentiation intensity and the technical
efficiency of firms; (b) to anticipate that downsizing or horizontal divestiture
of firms may be, in some cases, optimal actions to be virtually taken from a
societal standpoint.
Providing a gradient index that revealed to be a useful analytical tool
for theoretical evaluations of the structure-performance relationship without
making any assumption on the firms ’
contribution
of
the
paper.
The
competition is, perhaps, the main
present
approach
complies
with
the
methodological claim that the analysis of the industry performance should
focus on the social value of the market outcomes rather than on explaining
firms' profitability or the relative product allocation. According to that view,
market shares, as well as preferences and technology, were taken as given
throughout the paper. The former are observed, the latter are hypothesized.
Conduct can be assessed but only indirectly. 1 9 Of course, the present method
do not intend to substitute current approaches of the industry competition
based on the econometric estimation of surplus and deadweight losses
generated by virtual or real changes in the market structure. However, the
27
indices obtained in the examples of Sections 4.1 and 4.2 depend on a few
number of parameters, only one in the event, which indicates that, sometimes,
the index can be calculated more easily than the welfare gains themselves.
We will summarize now our main results.
1 . The paper allowed us to qualify the role of market shares, demand
elasticity and variety in measuring the local effects of optimal output
adjustments when the underlying intervention costs depend on efficiency
weights that are calculated according to a consumers' criteria.
(i) In homogeneous product markets, the market performance depends on
market shares, demand elasticity and the number of firms. Excepting the case
of perfectly inelastic demand, an increase in concentration always worsens
performance. The way the elasticity parameter affects performance hinges on
the nature of the goods traded.
In case of nonessential goods, there are substitute goods outside the
market. Other things equal, performance worsens as the elasticity parameter
raises. If the demand is linear, the aggregate performance is measured by the
H-concentration index. When the demand becomes perfectly elastic to price,
performance is entirely determined by market concentration and the number
of
firms.
As
the
aggregate
demand
becomes
inelastic,
the
role
of
concentration weakens gradually. At the limit, the market performance is
entirely determined by the number of firms. This provides a valuable insight
to markets with price-inelastic demand: policies facilitating entry of new
sellers are more effective to improve the market performance than policies
aimed to reduce concentration.
In case of essential goods to which no substitute goods are available, other
things equal, performance improves if the elasticity parameter raises. The
indices always depend on market shares, even when the market demand is
nonlinear. So, any policy reducing concentration may be effective in
improving performance.
28
Excepting the case of perfectly elastic demand, under the same market
conditions, the nonessential goods market outperforms the essential one. This
suggests that policies providing outside options for consumers, such as trade
liberalization, are always effective in improving the performance of the
market of an essential good.
(ii) For differentiated product markets, the linear model in which a
representative consumer value variety enables us to obtain the market gains
of variety. In that case, the index depends on the number of firms, the market
shares and the parameters of product differentiation. Under symmetric
preferences
and
for
each
product
allocation s ,
there
is
an
optimal
differentiation degree which maximizes the market performance. Excepting
limit cases, the performance is higher than indicated by the H-concentration
index, which is a market performance measure for homogeneous product
industries when the demand is linear. Thus, the difference between these two
indices provides a simple measure for the performance of variety. Under
asymmetric
differentiation,
the
aggregate
performance
is
enhanced
(worsened) if the larger (smaller) firms supply goods less differentiated.
2 . The performance index is sensitive to the trade-offs between allocative
and productive efficiencies often occurring in heterogeneous industries.
In
Section V we have obtained the performance effects of cost economies under
general demand and cost functions. A comparative static analysis shows that
the industry performance improves when a cost economy is obtained by a low
performing firm. This opens the possibility for mergers among such firms to
increase
the
aggregate
performance,
if
they
are
motivated
by
scale
economies. In such cases, the pro competitive effects induced by the costs
saved in the consolidation do outweigh the anticompetitive effects induced by
the increased concentration. This result provides another theoretical support
to policies of industry restructuring based on merging or taking-over of
inefficient firms.
29
3 . At each output vector q = ( q1 ,..., qn ) , we can indentify the best directions
Di
in the firms' base price weighted outputs that are implied by the
constrained welfare maximization. Given market shares and measurable
conditions ( s, ς ) ∈ M ( s ) , the examples of Section 4.2 suggest that there is a
large subset of structural parameters in M ( s ) , within which the directions
are positive, meaning that an increasing supply of all firms should improve
the aggregate performance. Outside that subset, the best directions are
negative for some group of firms, either among the larger firms or among the
smaller ones. In the first case, there is indication that the size of the largest
firms might be "excessive" in that setting, which would call for policies
aimed to reduce concentration, such as horizontal divestiture. The second
case suggests that the smaller firms underperform in that setup, meaning that
policies of industry restructuring favoring mergers or takeover could be
considered.
An immediate extension to the present approach might be to use
traditional models of vertical differentiation (Mussa and Rosen,1978) and
horizontal models (Hotelling or circular city) to see how the index changes
with taste parameters, not just elasticity or differentiation intensity. Finally,
it might not useless to say, the way the metric was chosen and the
intervention costs modeled in the paper, as a fresh reply to an old challenge,
only illustrates a (fruitful) possibility. Other solutions may be at the stake,
and the way on how to improve upon the issue will remain an open question.
30
FOOTNOTES
1 / By argui ng th at a good p e rfo rman ce index should be a ble to ra nk wel fa re l ev els,
Bl acko rby e t al . adopt a n axio mati c app roa ch to built
a p e rfo rmanc e ind ex b ased on
nu mb ers -equiv alent . Th ei r i nd ex r e p res ents a n evalu ation o rd eri ng p r eviou sl y d ef in ed
ov e r alt e rn ativ e o utput s th at cap tu res the t r ad eof f
bet we en con cent ration and wel fa r e,
wh e re wel fare is mea sure d by tot al output . If t h e ev aluator's p refe ren ces a re homoth etic
a nd th e out put o rd erin g ve rifie s a repl ic at ion p rin ciple en abli ng on e to d e al with non int ege r nu mb e r of fi rms, th ey p rove that such ind ex must be a g eo met ri c mean betwe en a
nu mb er equiv alent
tot al ou tput
Q
E (q) ,
whi ch i s an in v ers e mea sure of the in du st ry con cent rati on , and
in the follo win g way:
Q γ [ E (q )]1−γ .
to th e choic es of t he ev al uat o r betwe en co n cent ration
(γ = 0)
∆qi
c onst ant r etu rn s to th e s cale , s in ce
3 / B y as su mptio n , we h ave:
∆qi
in th e sense tha t
in cre a s es , but t h e thi rd met ri cs assu me s
∂d / ∂ ∆qi = Pi
c onst ant .
∂V (q) / ∂x = D p λ
L( x; µ ) = V ( x) + µ[t 2 − ( x − q )' D p ΣD p ( x − q )]
ma xi mu m
.
(a ).
Th e
Th e
fi rst -o rd e r
condit ions
µ[t − (q * (t ) − q)' D p ΣD p (q * (t ) − q )] = 0
2
d ω ( D p q*, D p q ) ≤ t.
On
th e
∂V (q * (t )) / ∂t = ∂L / ∂t = 2 µt
ma y
write , fro m
L ag range an
∂V (q * (t )) / ∂x = 2 µD p ΣD p (q * (t ) − q )
gi v e:
(b)
othe r
h an d ,
is l ink ed
(γ = 1) .
a nd t ot al welfa re
2 / The fi rst t wo met ri cs exhi bit d ecre a s i n g ret u rn s t o t h e s c a l e o f
th e m et ri cs (a nd h enc e, c ost s ) in c rea s es a s
0 < γ <1
T he p arame t er
(c)
by
the
(d ). Sinc e th e mat rix
wi t h
en velop e
D p ΣD p
is :
fo r
a
(b )
a nd
µ ≥0
a nd
theo re m
we
h av e:
is po siti ve definit e we
[ D p ΣD p ]−1 / 2 ∂V (q * (t )) / ∂x = 2 µ[ D p ΣD p ]1 / 2 (q * (t ) − q)
.
Squ arring both side s of this equ ation and by using (c) i n th e rh s , we arriv e t o:
[∂V (q * (t )) / ∂x]'[ D p ΣD p ]−1[∂V (q * (t )) / ∂x] = (2 µt ) 2
( e ) . L e t squ a r e n o w b o t h
si d es of equ at ion (d ). Th en equating it s lhs with th e lhs of (e) and ta king the li mit , fo r
t →0
on
bot h
sid es
Φ ω2 ≡ lim[∂V (q * (t )) / ∂t ]2
t →0
=
of
th e
resultin g
e quat ion
[∂V (q) / ∂x]'[ D p ΣD p ]−1[∂V (q ) / ∂x]
we
, s inc e
obt ain :
V
is
C1
31
with in an op en n eigh bo rh ood of
q.
Us i n g no w t h e
as s u mpt ion
(a ) we a rr iv e t o:
Φ ω2 = λ ' Σ −1λ .
In
o rd er
to
obt ain
t he
b est
di rections ,
f ro m
(b )
and
Σ D ∂V (q * (t )) / ∂x = [dV (q * (t )) / dt ][ D p (q * (t ) − q ) / t ]
−1
−1
p
si d es fo r
t →0
Σ −1λ = Φ ω Di
a nd u s ing (a ) on lhs we obt ai n:
(d )
we
can
writ e
. Taki ng th e li mit on bo th
.
4 / I n p a rt i c u l a r , f o r t h e i r Proposi tion 4 , th e money v al ue of t h e welfa r e ch an g e,
V ( x) − V (q )
a bove by
e xpe ct ed fro m an a rbit ra ry ad just ment v ect or
Φ ω ( x − q )' D p ΣD p ( x − q )
( x − q) ,
i s n o w bound ed
.
5 / L e t V a r eal valu ed C 1 f unctio n de fined over U , a n op en c onvex subset of R n :
V : U ⊂ R+n → R .
The
[∇V (q )]' ( x − q ) ≤ 0
fun ction
i mpl i es
V
is
said
p se udocon ca ve
q ∈U
at
if
:
V ( x) ≤ V (q ) , ∀x ∈U .
6 / Th e val u e of fi rms’ p rodu cti ons i s weighte d by the ac tual mark et share of fi rms , so
th at
∑
n
j =1
Pj ( s j q j )
ma y be vi e wed a s t h e ma rk et v alue o f th e c o mp o site s upply of firms.
7 / Se e Si mo n and Blu m (1994 ), Th eo rem 21 .22 p .532 .
8 / The bound ing oc cu rs be cau s e th e po sit ive d efini t en ess of Ω mak es th e inde x be a
d ecre asing fun ct ion of th e economi c su rpl us, whi ch is maxi mu m wh en fi rms a nd
c onsu me rs a re both symme t ri c . No w, if th e wa y o f co mp eti tion unde r sy mmet ry dist o rt s
Π n > 1/ n .
th e si ze dist ri bution , we should h ave
ma rk et obt ains:
s = (1 / n)1 .
If do es not , a p e rf ectly fra g mented
Πn
Th en , we only hav e to sh o w th at the valu e of
is
1/ n
in
thi s c ase. To thi s en d , c onsi d er th at th e sy mmet ri c h ypoth ese s i mply that th e di ag on al and
off diagon al el ement s of th e wei ghtin g mat rix mu st b e co nst ant , say
r es p ectiv ely .
Th u s ,
d =a −b.
The
we
i nv ers e
Ω −1 = (c / d )[ I n − (nb / c)11' ] .
Π n = 1/ n
Ω = c −1[d .I n + b11' ] ,
h ave:
can
be
T hen ,
f o r all adm i ssi bl e v alu es o f
pe rfo r med
it
fol lo ws
where
a/c
c = n(d + nb)
st r aightf o r wardly .
t h at
a nd
1' Ω −11 = n 2 ,
and
It
b/c,
a nd
gi v es :
the r efo r e,
a, b , n .
9/ By a cl assi cal t h eo re m of Lin ear Alg eb ra, t h ere i s a lin ear o rthogon al t ransformati on
of th e marke t sh a re coordin ate s
s1 ,..., sn ,
say
m = R' s ,
all o wing to write ( 4 ) i n t he n ew
32
c oo rdinat es
R = [r1 ,..., rn ]
e igenv al ues
Π n = ∑i =1 (mi / nν i ) 2
n
m1 ,...mn
as
in
the
c an oni c
fo r m:
ri
i s t h e o rthogon al mat rix of t he ei genv ecto rs
νi > 0
,
wh ere
a s soc iat ed to the
o f th e matrix Ω .
10 / Fro m th e posit iv e defin ite ness o f Ω , the re i s a lo wer t ri angul ar ma t rix
T = [t ij ] ; t ij = 0, j > i
s uch
Π n = (1 / n)∑k =1 (∑ j =1 t kj s j ) 2
n
Th i s
al lo ws
to
writ e
(4 )
as
k
c oncent r ation r at io o f o rd er
11/ Th e numb e r
Ω −1 = T 'T .
th at
1/ n
wh e re
th e
t erm
i n side
p a renth esi s
is
a
wei ghted
k.
h as a s pecial mea ning in fairer vot ing t h eo ry . It i s p ropo rtion al to
th e p rob abil ity th at a poll wit h a l a rg e nu mb e r
n
of ind ep ende nt vote rs finish es in a d ead
h eat . Su ch p rob abilit y d efine s th e voting powe r of a parti cipant . It i s c alled Pe n rose' s
squa re-root l a w, aft e r Pen rose (1946).
12/ If th e co sts of coo rdin ation are to be con s id e red , th e equi v al ent o f (3 ) would lead to:
Wij = Wii
. Thu s , a n on po sitive defini te mat rix obt ains .
13 / Cha mb erlin (193 3 ) find s that monopol i sti c ma rket s t end to ov e rp rovi d e p rodu ct
div e rsit y . By using the p roduct cha ract e ri sti c a pp roa ch , La nc as t er(1977 ) find s the s ame,
but th e el as ti city o f sub sti tuti on b et we en i nsid e and out sid e g ood s mu st not b e too high .
Th e is sue is discussed al so wi th a re p resent ati ve consu me r a pp roach , as i n Sp en ce (1976 a ),
Dixi t and Stigl it z (1977 ) and Manki w and Whi nston (1985 ) and wi th a di scre te choi ce
a pp roach . See And erson et al .(1 995).
14/ Fo r in st anc e, a ssu me th at th e con su mer h as qua si -lin ea r p refe rence s rep resent ed by
th e
sep arabl e
util ity
U (q; qo ) = qo + ∑i =1 nβ i qiθ
function:
;
0 < θ < 1 , qi > 0 .
T he
nonl in ear de ma nd fun ctions gen e rat ed b y the uti lity maxi mi zation lead to th e foll o wi ng
Π cn = (1 / n)(∑i =1 β i siθ )(∑i =1 β i−1 si2−θ )
n
ind ex:
Π cn = (1 / n)(∑i =1 siθ )(∑i =1 si2−θ )
n
to:
n
h ave:
. Fo r
n
. Th en , fo r
βi = β
con stant , t h e ind ex speci aliz es
c
θ → 1, Π n → 1/ n
a nd fo r
θ →0
we
Π →H.
c
n
Ω
15 / To se e thi s in th e si mpl est case n = 2 , t h e posit iv e definit ene ss condi tion fo r c i s:
0 ≤ ξ < 2 s1 s2 /( s12 + s22 ) ≡ ξ o .
(99 / 100 ; 1 / 100)
yi elds
So , t h e m a r k e t sh a r e s
ξ o = .02 .
(1 / 2 , 1 / 2)
yi eld
ξo = 1
but
33
16 / Anot h er ex a mple sho wing th e opti m al ity of redu cing th e supply of th e small er fi rms
wh en the p rodu ct sub sti tuti on i s high:
ma r k et we ob t ai n:
ξ .0325 ; ξˆ = .43.
f o r t h e s ma l l e st f i r m:
D4 (ξ )
n = 4 ; s = (.4; .3; .2; .1)
with
H = .30 .
Th e di re ctions a re p ositi v e f o r all fi rms, e x cepti ng
b e co m es n ega tive i f
ξ > .36 .
17 / Fo r ex ampl e , the same qu alit at iv e effect is obt ain ed by mu ltipl ying
2.8 ,
w i t h me an
ξ = .28 :
Fo r this
the p e rfo rma n ce in c rea s es by
39.6%
ξβ
+
(and
ξ βr
( a n d d ec r e a s e s b y
+
) by
14% )
w .r . t . th e symm et ri c v al u e ( = .241 ) ;
18 / Unde r scale economi es, an Indu st ry index indep end ent of Q is giv en as foll o ws. Th e
i th
fi rm
of
a
ho mog en eous
p ro duct
φi (q) = (ci / 1 − δ )q1−δ ; 0 < ci < (1 − δ ) siδ
P = Q −δ
c ost
;
0 < δ < 1, i = 1,..., n .
p ar a me te rs
in du st ry
Fo r
Mn
si mpli city ,
th e
agen cy
c oo rdination co st s , so th at the c ro ss weigh ts a r e negl e ct ed (
l ead s to:
th e in dex
g et:
co st
i ncl udes t h e v ect o r of
doe s
ωij = 0
a nd
lim
Π ( s, c , δ ) = H ( s )
δ →1
Π n ( s;ς ) clo s e
to
1/ n
for all
in cu r
in
. W e t h en o bta in
b y u sing (4 ) . On e can che ck t h at und e r sy mm etr i c cost s
lim
Π ( s, c , δ ) = 1 / n
δ →0
19 / Valu es of
not
) . T h e u se o f (2 a)
n
1
 i  1  1  s i  1  c i s 1
 c i s 1
i /n   i1 1  s i 
i 
Π n ( s, δ , c )
fun cti on
a nd faces t he inve rse agg regate d e mand
In t hi s e cono my , a po int of
c = (c1 ,..., cn ) .
h as
ci = c
we
c.
me an th at th e ob se rved marke t sh ares fit the
st ructu re of p ref erence s and co sts, so that l a rg e d evi ati ons of th e index fro m that
mi ni mu m valu e sugg est t h at oth e r ec ono mi c fo rc es not rep re s ented by p a ramte rs
wo rk , anti co mp eti tive c onduct includ ed .
ς
are at
34
REFERENCES
ANDERSON,S.P., A. de PALMA and
Y. NES T E R O V ( 19 9 5 ) O l i g o p o l i s t i c c o m p e t i t i o n
a n d t h e o p t i m a l p ro v i s i o n o f p r o d u c t s ,
E c on o m e t r i c a, 63, 6, 1281- 1 301;
B A I N , J . ( 1 9 6 8 ) I n d u s t r i a l O r g a n i z a t i o n, J oh n W i l e y &
Sons , Ne w York;
B E R R Y , S . T . a n d J . W A L D F OG E L ( 20 0 1) D o me r g e r s
i n c r e a s e p r o d u c t va r i e t y ? E vi d e n c e f r o m R a d i o
Br oa dc a sting, Qua r te r ly Jour na l of Ec ono mic s ,
CXVI,3,1009-1025;
B L A C K O R B Y , C . , D . DO N A L D S O N & J . A . WE Y M A R K
( 19 8 2) A N or m a t i v e a p p r o a c h t o i n d us tr ia l pe rf or ma nc e e va l ua t i on a nd c on c e nt r a t i on
ind i ces, Eur o pe a n E c on o m i c R e v i e w , 19, 89-121;
C H A M B E R L I N , E . H . ( 1 93 3 ) T h e T h e o r y o f
Mo n o p o l i s t i c C o m p e t i t i o n, Ha rwa r d Univ. Pre ss;
DANSBY, R.E. and R.D.WI LLI G ( 19 79) I nd us t r y
p e r f o r ma nc e gra di e nt i n d i c e s , T h e A m e r i c a n
E co n o m i c R e v i e w, 69( 3),249-260;
D I X I T , A . K . a n d J . E . S T I GL I T Z ( 1 9 7 7 ) M o n o p o l i s t i c
competition and o p t i m u m p r o d u c t d i v e r s i t y , T h e
A m e r i c a n E c o n o m i c R e v i e w, 6 7 , 3, 2 9 7 - 3 0 8 ;
H A U S M A N , J . A . a nd G . K. LEONARD(2002) The
competitive effects of a new product
i n t r o d u c t i o n : a case study, The Journal of
I n dus t r i a l E c o n o m i c s, L, ( 3 ) , 2 3 7- 6 3 ;
KÜ HN,K-U. and X. VIVES(1999) Exce ss entry,
v e r t i c a l i n t e gr a t i on , a n d w e l f a r e , R a nd J ou r n a l
o f E c o n o m i c s, 30, 4, 575- 603.
L A N C A S T E R , K . ( 1 97 9 )
V a r i e ty , E qu i t y a n d
E f f i c i e n c y, C ol u mbi a Unive r s i t y Pr e s s .
M A H A L A N O B I S , P . C. ( 1 9 3 6 ) O n t h e g e n e r a l i z e d
d i s t a n c e i n s ta t i s t i c s , P r o c . N a t . I n s t . S c i. I n d i a , 1 2 ,
4 9 - 55;
MANKIW,N.G. and M.D.WHINSTON(1986) Free
e nt r y a nd s o c i a l i n e f f i c i e n c y , R a n d J ou r n a l o f
35
E c ono mi c s , 1 7 , 1 , 4 8 - 5 8 ;
MUSSA, M.,ROSEN,S.(1978) Monopoly and Product
Q u a l i t y , J o u r n a l o f E c o n o mi c T h e o r y, 18, 301-17;
P E N R O SE , L . S . ( 1 9 4 6 ) T h e e l e m e n t a r y s t a t i s t i c s o f
m a j o r i t y v o t i n g , J o u r n a l o f the Royal Statistical
S o c i e t y, C IX , 53- 5 7 ;
S C H E R E R , F . M . a n d D . R O S S ( 1 9 9 0 ) I nd u s t r i a l M a r k e t
S t r u c t u r e a n d E c o n o mi c P e r f o r m a n c e, Houghton
M i ff l i n a n d C o . 3 n d e d .
SIMON, C.P. and L.BLUM (1994 ) Mathematics for
E c ono mi s t s , W . W . N o r t o n E d i t i o n
SINGH, N. and X.VIVES(198 4) Pr ic e a nd qua nti ty
c om p e t i t i o n i n a
differentiated
d uopoly, Rand
J o u r n a l o f E c o n o m i c s , 15, 4, 546- 54;
SPENCE,M.(1976a)
Product seclection, Fixed Costs
a n d M o n o p o l i s t i c C o mpe t i t i o n , R e v i e w o f
E c ono mi c S tu d i e s, 43, 217-235;
( 1 9 7 6 b ) P r o d u c t d i f f e r en t i a t i o n a n d w e l f a r e ,
A me r i c a n E c ono mi c R e v i e w , 6 6 , 2 , 4 07-14;
Performance
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Inverse demand elasticity
Fig.1: PERFORMANCE and PRICE-ELASTICITY Essential good (green); Nonessential (red)
performance
0.29
0.28
0.27
0.26
0.25
0.24
0.23
0.22
0.21
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
Inverse differentiation intensity
Fig.2 Performance in differentiated product markets
1
Table 1 : Performance gains from variety and deviations
s+ = (:4; :3; :1; :1; :1); (:36; :36; 13; :03; :03; :03; :03; :01; :01; :01)
Mean values
H = 0:28
Performance
Gain from variety
Total deviation
Preferences
Market shares
n=5
0:242
0:038
0:042
0:014
0:028
n = 10
0:233
0:047
0:133
0:023
0:110
1
APPENDIX
In d u str y P e r fo rm a n c e
W e ig h te d I n d e x
V
V (q * (t))
Φ = tg θ
θ
x2
q2
q *2
q * (t)
q*
q
q *1
q1
x1
1. Proof of Proposition 1:
For the sufficiency, the use of equation (8) in connection with (4) makes
the linearity of P to imply Πcn = H. For the necessity, by using the mean
c
value theorem in equation (1a) with φ0 = P, one obtains, W
Pin = ki qi , where
c
ki = P (xi + QiP
) − P (Q), 0 < xi < qi . Therefore, si /ω i = ( l ki si )/ki or,
putting ks ≡ nl ki si = W c /Q this yields: si /ωci = ks /ki . Thus, we have:
n
n
P
P
c
( ωsii )si = ( ksii ) WQ = ( kqii )( kQi q2i ) = ( kqii ) ( kqii )s2i . If P is not linear(affine),
i=1
i=1
all ki ≡ P (xi + Qi ) − P (Q) cannot be written as constant proportions of the
quantities, say ki = τ qi ; τ > 0, i = 1, ..., n. Therefore, the r.h.s. of the last
equality cannot be equal to H for every output vector q = (q1 , ..., qn )τ , as it
would be required. ¤
2. Proof of Proposition 2:
+
the weighting matrix can be written as: Ωc (ξ) =
Given, s ∈ Sn−1
1
[Ds2 + ξB] where K(ξ) = [1 + (n − 1)ξ]; Ds2 = Diag(s21 , ..., s2n ); B =
HK(ξ)
1
[1s2τ + s2 1τ ] − Ds2 and s2 is the vector of squares of the market shares.
2
For ξ = 1 the matrix Ωc (1) has rank 2, and for ξ = 0, Ωc (0) is a diagonal matrix with full rank. The positive definitness of Ωc (ξ) occurs for
2
all ξ < ξ o where ξ o < 1 is a value anulling the smallest eigenvalue of
Ωc (ξ). So, Mn (s) = {s} × [0, ξ o ) and since Ωc (ξ) is a continuous function
of ξ over that interval, the same can be said about Cnc (s, ξ). Therefore,
any sequence (s, ξ j ) converging to (s, 0) makes the index converging to
Πcn (s; 0) = H(s). The limit is approximated from below. Indeed, the derivPn
1
∂Πcn (s, ξ)
τ i 2
τ i
=
ative of the index w.r.t. ξ is:
i=1 (s ω )si [(s ω ) −
∂ξ
HK(ξ)
Pn
1
1
τ j
τ i
2
j=1 (s ω )]. For ξ = 0, we have Ωc (0) = H Ds2 so that s ω = H/si . Thus,
n
c
∂Πn (s, 0)
= (nH/h)(h − 1/n), where h = Sn n(1/sj ) is the harmonic mean of
j=1
∂ξ
market shares. The inequality h 6 1/n holds for every s, with equality only
if s = en . Hence, the derivative is always nonpositive at the origin. Finally,
the continuity of Πcn (ξ) and Weiestrass theorem ensure that there is a point
ξ s ∈ (0, ξ o − ], providing a minimum value to the index for some > 0 with
Πcn (s, ξ s ) < H(s). Since Ω is singular at ξ o , we have: lim− Πcn (s, ξ) = ∞. ¤
ξ→ξ o