When Firm-Level Demand with Price Competition and Inverse Demand
with Quantity Competition Coincide
William Kolberg
Department of Economics
Ithaca College
August, 2008
Introduction
This paper explores the conditions under which firm-level demand and inverse
demand coincide. In the case of duopoly, inverse demand is less price elastic than
demand regardless of the degree of product differentiation. (See Singh and Vives 1984).
It is well-known that the distinction between price competition and quantity competition
is removed when the product is sold by a single monopoly firm, or by the product group
as a whole choosing to set prices or quantities in consort, regardless of product type. This
is true in the single firm monopoly case because cross-price effects are absent. It is true
in the case of firms choosing prices or quantities in consort because firms will have the
incentive to act as a monopolist, effectively negating the impact of cross elasticities as
well. It is also well known that the distinction between price competition and quantity
competition shrinks and evaporates as the number of firms is expanded to a sufficiently
large number when the product is homogeneous. In each of these cases, demand and
inverse demand are identical.
Vives (1999) and (1987) suggests why the distinction between price and quantity
competition when the product is differentiated among many firms has not been addressed
extensively yet. If the residual price elasticity of demand faced by a firm is unbounded
with increased competition, it leads to the competitive result with no distinction. If the
residual price elasticity of demand remains bounded as the number of firms is increased,
it leads to monopolistic competition, but cross-price elasticities will vanish as will any
difference. The distinction would remain in kind as n is increased, but only diminish in
degree at most. Also, in the less technically limiting case which includes cross effects
terms, inverting the demand relationship may itself be daunting with increased numbers,
impeding any comparison at all.
.
Nevertheless, Hackner (2000) explores the distinction between price and
quantity competition with the possibility of many sellers when product quality varies
across sellers. With the aid of some restrictive assumptions, he shows that when there are
more than two firms, quality varies greatly, duopoly results may be reversed, suggesting
further exploration in this arena may be justified.
Kolberg (2008), shows that very small cross-price effects in the firm's demand
under price competition will exert an unexpectedly strong impact on the character of the
firm's corresponding inverse demand under quantity competition. This results in an
unexpected dramatic convergence, and then divergence, between the firm's demand under
price competition and the firm's inverse demand under quantity competition when the
product is differentiated, as the number of firms is increased beyond 2. As a result,
inverse demand, compared with demand may be similar, more price elastic or less price
1
elastic as the number of firms in the product group is expanded beyond the duopoly case.
This implies that the conditions under which demand and inverse demand coincide are
different than those proposed by Vives (1999). The objective of this paper is to identify
those conditions.
This study is conducted with the demand system used in Kolberg (2008).
Demand is specified with a log-linear function, in which demand and inverse demand
elasticities are explicitly transparent. The first part of the paper reviews an analytical
method for inverting demand in price competition to find the corresponding inverse
demand under quantity competition. The method yields a simple function relating
demand elasticities with inverse demand elasticities. The second part of the paper uses
this relationship to determine the conditions under which demand and inverse demand are
the same. It is found that demand and inverse demand coincide when the firm's price
elasticity of demand is proportional to the number of competitors the firm faces in the
product group. The constant of proportionality is found to equal the product group
elasticity of demand.
The Demand System
Demands faced by individual firms are assumed symmetrical. Log-linear demand
for a representative firm under price competition is as follows.
(1) ln qi = b ln pi + g n ln p j + ... + g n ln pn + ln C
Where:
pi is firm i's price;
qi is firm i's quantity demanded;
b is the firm's price elasticity of demand;
gn is pair-wise cross-price elasticity and
C is a constant, and
i≠ j.
We assume b < 0, gn > 0 for substitutes, and
(2)
b + gn(n-1) = m
Here m is the market demand elasticity for the good produced by the product group1 .
Equation (2) states the net response of buyers to a 1% change in firm i's price will equal
the market demand elasticity for the product group. This implies b ≥ g n (n − 1) . These
assumptions ensure that (1) may be inverted 2 .
1
Removing logs from both sides of (1), and (3) yields the linear demand system. When this is done, b, g,
gn, and m on the one hand, and βn, γn, and μ , on the other are coefficients. All of the logic and method
discussed in the log-linear example of the text transfers directly and generalizes to the linear case with
elasticiites replaced with coefficients. This is not done in the text for space considerations.
2
See Cheng (1985).
2
Inverting (1) yields:
(3) ln pi = β n ln qi + γ n ln q j + ... + γ n ln qn + ln c
Where:
qi is the quantity offered for sale by firm i;
pi is the willingness to pay for qi;
βn is firm i's quantity elasticity of demand;
γn is pair-wise cross-quantity elasticity; and
c is a constant.
With n = 2,
(4) β 2 =
(5) γ 2 =
b
<0,
(b − g 2 )
2
−g
< 0 , and
(b − g 2 )
2
With the number of firms, n, greater than 2, the analytical inversion process can
degenerate into unmanageable complexity.
Analytical Inversion of the Demand System with n > 2
For internal consistency, (2) implies that these elasticities must be unaffected by a
change in the number of firms in the product group. Otherwise, omissions or double
counting must be occurring. It may be that these elasticities are in fact affected by a
change in n, but the impact must be distributed on both sides of (2) to preserve the
equality 3 . This property is general and should apply no matter what functional form is
used to model the firm's demand. This means that the character of (2) is completely
determined by the nature of the product and is independent of n. We can then define
(6) g = g n (n − 1)
We will refer to g as the product group cross-price elasticity. It is the percentage change
in the demand of the firm's competitors, as a group, in response to a one percent change
in the firm's price holding other prices constant. This measures total substitution
possibilities inside the product group. It should move along with |b| in response to
changes in product differentiation among firms inside the product group. It is similar to
|b| in this regard, but unlike |b|, it does not include added substitution opportunities
outside the product group.
Applying the same reasoning used in the case of firm i's demand function,
3
Any impact on (1) or (3) tied to a change in n should therefore be restricted to an impact on the constant
in each case, since impacts on elasticities would cancel. The specific form of the impact is an interesting
issue, not explored here.
3
(7) β n + γ n ( n − 1) = μ
…holds under any conditions, where μ is the inverse market demand elasticity for the
product group. It follows that:
(8) μ = 1/m, also under any conditions.
It will be useful to define a criterion for measuring the degree of product
differentiation under quantity competition 4 . Set n = 2, divide both sides of (7) by μ:
⎛ β2 ⎞ ⎛ γ 2 ⎞
(9) ⎜ ⎟ + ⎜ ⎟ = 1
⎝ μ ⎠ ⎝μ⎠
This implies:
⎛
⎞ ⎛ γ2
⎞
⎟⎟ + ⎜⎜
⎟⎟ = 1
⎝ ( β2 + γ 2 ) ⎠ ⎝ ( β2 + γ 2 ) ⎠
(10) ⎜⎜
β2
The first term of both (9) and (10) is the own quantity portion of μ and the second term is
the cross quantity portion of μ.
With a perfectly homogeneous product, an increase in production from any firm
will impact buyer willingness to pay equally, and all of the quantity elasticities will be
equal. The own and cross quantity portion of μ will be 0.5 in (9) and (10).
With product differentiation, the firm will have some ability to isolate itself from
the impact of an increase in production by another firm in the product group. The own
quantity portion of μ will therefore be greater than 0.5 and the cross quantity portion of μ
will be less than 0.5 in (9) and (10) for the case of a differentiated product. The nature of
the product, therefore will be reflected in the relative size of shares in (9) and (10) for the
case of n = 2.
For a fixed and given product group quantity elasticity, μ, as n is increased, each
of the left side terms in (7) must be reduced proportionally, asymptotically approaching 0
as n approaches infinity. In the case of homogeneous products, this would correspond to
perfectly elastic firm demand under perfect quantity competition. With differentiated
products, the process would always preserve relatively larger absolute values for own
quantity elasticity as compared to any single pair-wise quantity cross elasticity as n is
increased.
With symmetrical demand across firms, the relationship, (10), between βn, and γn,
in this process of increasing n, is preserved in that:
4
For consistent notation, βn will be used to represent own quantity elasticity when n > 2. β2 will represent
own quantity elasticity when n = 2. Similarly, γn will represent pair-wise cross elasticity when n > 2. γ2
will represent pair-wise cross elasticity when n = 2.
4
(11)
(β
βn
n
+γn )
+
(β
γn
n
+γn )
=1
…for any pairing of firm i with one of the other n-1 firms. This is because with
symmetrical demands, the impact of increasing n is distributed proportionally across
firms. It follows that the own quantity portion of any two firms' share of μ when n > 2
will always be equal to the own quantity share of total μ when n = 2.
(12)
⎛
⎞ ⎛ β2 ⎞
βn
⎜⎜
⎟⎟ = ⎜ ⎟
⎝ ( βn + γ n ) ⎠ ⎝ μ ⎠
This will be true despite the fact that:
βn < β2
and β n + γ n < μ
This character of the firm's inverse demand will be completely determined by the nature
of the product and the degree of product differentiation. Therefore, it will be independent
of the number of firms. In addition, from (7) we know that all of the quantity elasticities,
own and cross, must sum to equal the product group quantity elasticity, μ.
(12) and (7) provide two conditions which must hold for each firm's demand
function under quantity competition with n firms in the product group. To solve (12) and
(7) for βn and γn, as a functions of the duopoly solution, first solve (12) for γn:
⎛ μ
⎞
− 1⎟
⎝ β2
⎠
(13) γ n = β n ⎜
Substitute (13) into (12), and factor out βn:
μ
(14) β n = ⎛
⎛⎛ μ ⎞ ⎞⎞
⎜ 1 + ( n − 1) ⎜ ⎜ ⎟ − 1⎟ ⎟
⎜ β
⎟⎟
⎜
⎝⎝ 2 ⎠ ⎠⎠
⎝
With βn from (14) γn may be found using (13). Equations (14) and (13) are the solution
to (12) and (7). These relationships are perfectly general and should hold for any inverse
demand system regardless of functional form.
The conversion of b to βn expressed entirely in terms of the n = 2 case under price
competition may be derived from (14) by substituting in (4) and (5) from the two firm
inversion. This is because in this particular functional form, (4) and (5) express inverse
demand elasticities in terms of the corresponding demand elasticities. Some algebra
yields:
(15)
βn =
1
g⎞
⎛
⎜ 1 − ( n − 1) b ⎟ m
⎝
⎠
5
In addition, given βn, from (15), γn may be most easily determined by using (13) and
substituting in (4) and (5) from the 2-firm inversion:
⎛ −g ⎞
⎟
⎝ b ⎠
(16) γ n = β n ⎜
So any firm-level log-linear demand function with price competition among n
firms, (1), may be converted into its corresponding inverse firm demand function with
quantity competition among n firms, (1), preserving the appropriate relationship between
firms and the nature of product as given by elasticities of the price competition demand
function 5 .
The Conditions for Coincidence of Firm-Level Demand and Inverse Demand
In order to provide some structure upon which to base a search for conditions
under which demand and inverse demand at the firm level coincide it is useful to briefly
set out the landscape over which such a search will be made. This is done in Figure 1.
Figure 1 lays out the space determined by two critical factors that determine the
underlying nature of competition among firms, the nature of the product, and the number
of firms that make up the product group under study. Figure 1 graphs the firm's elasticity
of demand, b on the horizontal axis and the number of the firm's competitors, (n-1) on the
vertical. The space plotted in Figure 1 is part of the space in which we find substitutes
sold by members of the product group. Moving left to right in Figure 1, the product
becomes more differentiated. At the left side b is -10. Further moves to the left would
eventually lead to b = −∞ indicating a homogeneous product The number of firms is
graphed on the vertical axis with n-1 = 10 at the top, continuing further, the extreme case
is (n − 1) = ∞ . Figure 1, then focuses on a subset of this space near the left side origin
where |b| is reduced to |m|, its lowest possible value, and n-1 = 1, its lowest possible
value.
We know that inverse demand and demand coincide under conditions of
competition among a large number of firms when the nature of the product produced by
the product group is homogeneous. This would occur beyond the extreme northwest
corner of Figure 1. We also know that under conditions of monopoly or collusion among
firms in the product group demand and inverse demand will coincide as well. This
occurs all along the vertical axis on the right side of Figure 1. This includes the limiting
case of monopolistic competition, where there are many firms and the firms are perfectly
isolated as discussed by Vives (1999), and of course this also holds regardless of the
number of firms along the vertical axis. In what follows we explore the possibility of
other conditions under which demand and inverse demand are likely to coincide.
A critical distinction between demand and inverse demand is the impact that a
change in the number of firms in the product group has on the firm's demand or inverse
demand elasticity. Holding m constant, an increase in the number of firms, will have no
impact on the firm's price elasticity of demand under price competition. This would
5
This would require additional information on the particular impact a change in n would have on C, and c,
so that the constant may be inverted as well. These issues have no influence on this study and are not
explored here.
6
entail a pure vertical move in figure 1. A change in the number of firms will, however,
impact on price elasticity of demand under quantity competition.
When the product group quantity elasticity is held constant, (7) implies βn and γn
will each decrease in proportion to (n-1) as the number of firms competing with the
representative firm increases. This means 1/βn will always increase in proportion to the
number of firms competing with the representative firm in the product group. This is one
of the critical differences between demand and inverse demand. To see this, take the
derivative of (15) with respect to (n-1):
(17)
⎛ 1 ⎞
∂⎜ ⎟
⎝ βn ⎠ = − ⎛ g ⎞ m
⎜b⎟
∂ (n − 1)
⎝ ⎠
It follows, then, that demand and inverse demand could coincide if, b could be made to
increases in proportion to (n-1) as the number of firms in the product group is increased
in the same way that 1/βn increases in proportion to (1-n). In price competition, as the
product type tends to homogeneous, both g and b will tend to infinity with m held
constant. Equation (15) implies the well-known result that with a homogeneous product,
as the number of firms is increased toward N, a very large number, 1/βn will also tend to
infinity, and demand and inverse demand will coincide. This is the expected result with a
homogeneous product and a large number of sellers. In this case, equation (17) implies a
decrease in firms by one would decrease 1/βn by m. In this range, therefore, a
corresponding reduction in b by m would preserve b(N-1) = 1/β(N-1) . At the other extreme,
when b is reduced to m, its lowest possible value with g = 0. b2 = 1/β2 = m because all
cross effects are absent. Since bn = 1/βn if bn = m(n-1) when (n-1), bn , and 1/βn are at
their maximum and minimum possible values, it seems that this relation is a good general
candidate for the condition under which demand and inverse demand coincide when the
product is differentiated and the number of firms in the product group is variable.
To check this conjecture, substitute b = m(n-1) in (15):
βn =
1
⎛
⎞
g
⎜1 − ( n − 1)
⎟m
m(n − 1) ⎠
⎝
And,
1/ β n = ( m − g ) = b
So 1/βn will always equal bn when bn is increased in proportion to (n-1) with the constant
of proportionality being m.
7
In the case of price competition, if we begin with b2 = 1/β2, at the right side origin
of Figure 1 where there are no cross effects, increasing the firm's price elasticity of
demand in proportion to the number of firms competing with the representative firm, with
product group demand held constant will ensure that bn = 1/βn as well. Since at the right
side origin, b2 = m, we must continue to hold product group demand elasticity constant
for this process to work properly.
Calculations are shown in Table 1 based on (15). With b2 = m = -1.5. In this
case, gn and g may be calculated with (2), and (6) respectfully. Then based on those
results, μ, βn, and γn are calculated using equations (8), (15), and (16) respectively.
So using (19), as long as b2 = m is held constant, and (n-1) is increased, bn will
also increase in proportion to the number of firms in the product group, (n-1). Since the
number of firms has no effect on the firm's own price elasticity in this demand
framework, this process of augmenting bn in proportion to the number of firms must
entail a diagonal move from the southeast origin in Figure 1 to the Northwest.
So a necessary condition for demand and inverse demand to coincide away from
the vertical axis in Figure 1 is for both bn and 1/βn to increase in proportion to the number
of the representative firm's competitors, (n-1). However this is not sufficient for such a
coincidence. This is because we know that 1/βn will also increase in proportion to (n-1)
when the nature of the product is held constant with a vertical move in Figure 1, where
(16) remains invariant regardless of the number of firms competing with the
representative firm in the product group. Such a move could never preserve the
coincidence of inverse demand and demand. The diagonal move from point A through
Point F involves a decrease in product differentiation as well as an increase in the number
of firms. We know this is happening with demand because bn is being increased in the
process. The associated adjustment with inverse demand is more subtle. Using (16), we
know that this diagonal move will require some special adjustments in inverse demand
besides the increase in the number of firms. A diagonal move however must require
concurrent adjustments in γn that are more than proportional to the change in (n-1) so that
the βn share in (16) decreases reflecting a reduction in product differentiation along with
the change in the number of firms. Table 1 also shows calculations for the pair-wise
share tied to βn for points A-F. We see the share decreases as expected from 1 to
0.545455. This indicates that for inverse demand like demand, the indicated moves A-F
in Figure 1 is diagonal, involving a move from extreme product differentiation toward a
homogeneous product where the pair-wise share tied to βn is reduced to 0.5. Since the
move is diagonal, both βn and γn are adjusted such that the pair-wise share tied to βn
declines as (n-1) is increased. This implies that for each point A-F, with a fixed μ, there
is a unique β2 along the horizontal axis below each discrete point A-F tied to the nature of
the product at that point. We can conclude that a necessary condition for the coincidence
of demand and inverse demand is that both bn and 1/βn must increase in proportion to the
number of firms competing with the representative firm, (n-1) as the number of firms is
increased, moving away from point A in Figure 1. A sufficient condition is that γn also
decreases more than proportionally to 1/(n-1), in order to guarantee the nature of the
product under inverse demand adjusts in the same way it does under demand.
To further place the coincidence locus A-F of Figure 1 in context, it is useful to
"decompose" the sources of the firm's demand under price competition. This can be done
by solving (2) for b.
8
(20)
b = m -gn(n-1)
Here b is broken into two components, the "internal" group cross-price elasticity
component, gn(n-1), and an external component, since both terms on the right side of (22)
are negative. The external component is due to the entry-exit of buyers to or from the
rest of the economy as the product group as a whole increases or decreases its price in
consort. The internal component is the movement to or from the firm to or from other
firms by buyers already inside the product group. Together, these two components make
up the price elasticity of demand, b, for the firm.
Contrast this with the case of quantity competition. In this case, solving (8) for
β n,
β n = μ − γ n ( n − 1)
(21)
Notice that γn in (21) unlike gn in (20), is negative with quantity competition. This means
the right side of (21) is not a component breakdown as in (20). Instead, it is just a net
amount. It is what is left of μ, after γn(n-1) has been removed. Here, βn can't be broken
down into any components, because a price, unlike a quantity can't be disaggregated in
any way. We can't speak of part of a price change coming from here and another part of
the price change coming from there. This is another critical difference between demand
under price competition as compared to inverse demand under quantity competition.
Notice that when m = -gn, (20) becomes
(22) b = -gn(n)
In this special case, it is as if there are no component parts tied to price competition,
because all the elements, each -gn and also m, in (22) have the same magnitude. This
may be another condition tied to the coincidence of demand and inverse demand.
Exploring demand and inverse demand in the neighborhood of where -gn = m should
also prove to be useful. Substituting 25 into (15):
βn =
(23)
1
, and
⎛
− g n ( n − 1) ⎞
⎜⎜1 + (n − 1)
⎟⎟ m
− gn n ⎠
⎝
( − gn =m)
βn =
1
m( n − 1) +
.
m
n ( − g n =m )
9
(23) states that as long as -gn = m, βn = (1/m) as expected, when n = 1. On the other
hand, as n becomes large, βn approaches 0 as expected. If (23) is a valid representation of
the relationship between βn and b when demand and inverse demand coincide, then in
that case,
m
(24) b( m, n) = m( n − 1) + n
( − g n =m )
The lower portion of Table 1 calculates b using (24), again holding m = -1.5 as
before. We find that these calculations are close to but not the same as those tied to (29).
Figure 1 shows b calculated using (24) with the solid line, with points a-f. (lower case).
Equation (24), traces a locus along which -gn = m. Along this -gn = m locus, the firm's
ability to protect itself from the actions of a representative competitor in the product
group is about the same relative to the rest of the economy. This is because the loss in
the firm's demand when an individual competitor drops its price with all else equal, is the
same as compared with the net impact on the firm when all n firms drop their prices
together.
To the NE of the -gn = m locus, gn < -m. In this region, the firm has the ability to
isolate itself from an individual representative competitor from inside the product group
to a greater degree than from the rest of the economy. This is because the loss in the
Figure 1
10
firm's demand when an individual competitor drops its price is less as compared with the
net effect on the firm's demand with all n firms in the product group dropping price
together. Just to the NE of the gn = -m locus we will refer to firms, therefore as "weakly
isolated". With further rotational moves to the NE, firms will become more isolated
among themselves relative to the rest of the economy. All along the vertical axis, firms
will be completely isolated monopolies. But note that in the limit just before reaching the
vertical axis where |b| is reduced to |m|, vertical moves will still entail greater degrees of
isolation relative to the rest of the economy.
To the SW of the -gn = m locus, gn > -m , and the firm has a more difficult time
protecting itself from the actions of an individual representative competitor relative to the
rest of the economy. This is because the loss in the firm's demand when an individual
competitor drops its price is more as compared with the net effect on the firm when all n
firms in the product group drop price together. To the SW of the -gn = m locus we will
refer to firms, therefore, as "weakly interdependent".
It turns out that according to table 1, along the -gn = m locus, b ≠ (1/ β n ) . This
locus is close but not exactly the same as the set of points along which demand and
inverse demand coincide.
11
Notice that Table 1 shows the error between b and 1/βn is equal to (m/n). It turns
out that this error must be there because when it is removed while -gn = m, (26) implies b
would equal -g = m(n-1), and b can never equal -g unless the product group demand is
perfectly inelastic, in which case m = 0. This means the -gn = m locus will always be
close but not the same as the coincidence locus we are searching for where b = 1/βn. We
will expect that this difference becomes smaller, as the product group demand becomes
less price elastic.
Notice that with points, A-F, b2(n-1) ≠ g. So (24) along the A-F locus does not
imply b = g. Finally, note that the calculations in Table 1 for points a-f show the
calculated value of b from (24). These values of b are then used for calculating gn, g, and
m along the -gn = m locus. Since m = -1.5 with A-F, m ≠ -1.5 along a-f. First gn is
calculated using b(24). Then m and g are calculated using (20). Note that in each case,
a-f, m = -gn as required for the border between weakly isolated and weakly
interdependent competition among product group firms. Finally, note that in Figure 1
demand and inverse demand coincide, and |1/βn| =|b| along the A-F dotted line. To the
NE of this locus, |1/βn| > |b|. and to the SW of this locus, |1/βn| < |b|.
We can conclude that coincidence of demand and inverse demand occurs on a
locus that lies near the locus of points for which -gn = m, with b = gn(n), but not exactly
there. Instead, it occurs where b =b2(n-1). Since b2(n-1) is less than gn(n), the
coincidence locus appears to be located in the region we have labeled "weakly isolated"
in Figure 1. However, changes in the nature of competition among firms are "tightly
packed in the neighborhood of the origin in Figure 1. At point A, even though close to
point a, firms are completely isolated monopolies, since point A is on the vertical axis.
Moves away from A to B-F entail significant reductions in isolation among firms in the
product group. Since the horizontal difference between these loci is (m/n) 6 , as the
number of firms increases, the coincidence locus approaches the -gn = m locus. We can
conclude that along the coincidence locus where b = 1/βn, firms in the product group
range from isolated to a situation where all remnants of isolation have been removed in
the extreme case of a perfectly homogeneous product, and sufficiently large number of
firms.
We can conclude that the coincidence of demand and inverse demand can be
found under conditions of perfectly isolated duopoly all the way continuously through to
the case of perfectly homogeneous competition among a very large number of firms
under perfect competition.
The determinant of the location of these loci in (b, (n-1)) space is the price elasticity of
the product group demand, m, along the coincidence locus. Tables 2 and 3 of the
Appendix show calculations when |m|, along the coincidence locus, is first reduced from
1.5 to 0.3, and then increased to 3. These changes are graphed in Figures 2 and 3. In
each case the minimum point on the b axis is set equal to m, as with Figure 1, since this is
always the minimum possible value for b. As the product group demand elasticity, m,
used in tracing points A-F becomes more inelastic, the -gn = m locus will rotate to the
right and the point where it intersects the horizontal axis, where (n -1) = 1, moves to the
right. The coincidence locus will also rotate to the right, but always intersect the
6
These are the differences a-A, b-B, c-C, d-D, e-E, and f-F.
12
horizontal axis at the origin where b = m. Also we see the error between the -gn = m
locus and the coincidence locus is decreased, as product group demand becomes more
inelastic. As product group demand along A-F becomes more elastic, both the A-F
coincidence locus and the a-f, -gn = m locus rotate down and to the left. The A-F locus
will continue to intersect at the origin where b = m, but the horizontal intersection for the
-gn = m locus will shift to the left. Also we see the error between the -gn = m locus and
the coincidence locus is increased, as product group demand becomes more elastic. We
can conclude, then, that other things equal, as the product group demand becomes more
(less) inelastic, firms will become more (less) interdependent or less (more) isolated.
Table 1: Calculations for Points A-F and a-f of Figure 1
n
(n -1)
b (27)
m
gn
g
n
(n -1)
A
2
1
-1.5
-1.5
0
0
B
3
2
-3
-1.5
0.75
1.5
C
4
3
-4.5
-1.5
1
3
D
5
4
-6
-1.5
1.125
4.5
E
6
5
-7.5
-1.5
1.2
6
F
7
6
-9
-1.5
1.25
7.5
βn
1/ β n
γ n (17)
γ n (n-1)
2
3
4
5
6
7
1
2
3
4
5
6
-0.666667 -0.333333 -0.222222 -0.166667 -0.133333 -0.111111
-1.5
-3
-4.5
-6
-7.5
-9
0
-0.166667 -0.148148 -0.125 -0.106667 -0.092593
0
-0.333333 -0.444444
-0.5
-0.533333 -0.555556
μ (9)
-0.666667 -0.666667 -0.666667 -0.666667 -0.666667 -0.666667
β n share (16)
n
(n-1 )
b (26)
off by
gn
g
m
m/n
1 0.666667
a
2
1
-2.25
-0.75
1.125
1.125
-1.125
-0.75
0.6 0.571429 0.555556 0.5454545
b
c
3
4
2
3
-3.5
-4.875
-0.5
-0.375
1.166667 1.21875
2.333333 3.65625
-1.166667 -1.21875
-0.5
-0.375
d
5
4
-6.3
-0.3
1.26
5.04
-1.26
-0.3
e
6
5
-7.75
-0.25
1.291667
6.458333
-1.291667
-0.25
f
7
6
-9.214286
-0.214286
1.3163265
7.8979592
-1.316327
-0.214286
13
Figure 2
Figure 3
14
Conclusions
The first part of the paper reviews an analytical method for inverting demand in
price competition to find the corresponding inverse demand under quantity competition.
The method yields a simple function relating demand elasticities with inverse demand
elasticities. The second part of the paper uses this relationship to determine the
conditions under which demand and inverse demand are the same.
It is found that demand and inverse demand coincide when the firm's price
elasticity of demand is proportional to the number of competitors the firm faces in the
product group. The constant of proportionality is found to equal the product group
elasticity of demand.
The coincidence of demand and inverse demand is found under conditions of
perfectly isolated duopoly all the way continuously through to the case of perfectly
homogeneous competition among a very large number of firms in which any degree of
isolation that may have existed elsewhere along the way has been removed.
References
Cheng, Leonard 1985. "Inverting Systems of Demand Functions" Journal of Economic
Theory, Vol. 37, pp.202-210, (1985)
Hackner, Jonas "A Note on Price and Quantity Competition in Differentiated
Oligopolies" Journal of Economic Theory 93, pp. 233-239 (2000).
Kolberg, W. "A Peak into a Pandora's Box: Comparing Firm-Level Demand with Price
Competition vs. Quantity Competition." Social Science Research Network
Working Paper Series #1198243. (http://ssrn.com/abstact=1198243)
Mas-Colell, A. 1975. "A Model of Equilibrium with Differentiated Commodities".
Journal of Mathematical Economics 2: 263-95.
Singh, V. and Xavier Vives "Price and Quantity Competition in a Differentiated
Duopoly." Rand Journal of Economics, Vol. 15, No. 4. (Winter, 1984),
pp. 546-555
Vives, X., 1987. "Small Income Effects: A Marshallian Therory of Consumer Surplus
and Downward Sloping Demand". Review of Economic Studies 54:87-103.
Vives, Xavier 1999. Oligopoly Pricing, The MIT Press.
15
Appendix
Table 2: Calculations for Points A-F and a-f of Figure 2
n
(n -1)
b (27)
m
gn
g
n
(n -1)
A
2
1
-3
-3
0
0
B
3
2
-6
-3
1.5
3
C
4
3
-9
-3
2
6
D
5
4
-12
-3
2.25
9
E
6
5
-15
-3
2.4
12
F
7
6
-18
-3
2.5
15
βn
1/ β n
γ n (17)
γ n (n-1)
2
3
4
5
6
7
1
2
3
4
5
6
-0.333333 -0.166667 -0.111111 -0.083333 -0.066667 -0.055556
-3
-6
-9
-12
-15
-18
0
-0.083333 -0.074074 -0.0625 -0.053333 -0.046296
0
-0.166667 -0.222222
-0.25
-0.266667 -0.277778
μ (9)
-0.333333 -0.333333 -0.333333 -0.333333 -0.333333 -0.333333
β n share (16)
n
(n-1 )
b (26)
off by
gn
g
m
m/n
1 0.666667
a
2
1
-4.5
-1.5
2.25
2.25
-2.25
-1.5
b
3
2
-7
-1
2.333333
4.666667
-2.333333
-1
0.6 0.571429 0.555556 0.5454545
c
4
3
-9.75
-0.75
2.4375
7.3125
-2.4375
-0.75
d
5
4
-12.6
-0.6
2.52
10.08
-2.52
-0.6
e
6
5
-15.5
-0.5
2.583333
12.91667
-2.583333
-0.5
f
7
6
-18.42857
-0.428571
2.6326531
15.795918
-2.632653
-0.428571
16
Table 3: Calculations for Points A-F and a-f of Figure 3
n
(n -1)
b (27)
m
gn
g
n
(n -1)
A
2
1
-0.3
-0.3
0
0
B
3
2
-0.6
-0.3
0.15
0.3
C
4
3
-0.9
-0.3
0.2
0.6
D
5
4
-1.2
-0.3
0.225
0.9
E
6
5
-1.5
-0.3
0.24
1.2
F
7
6
-1.8
-0.3
0.25
1.5
βn
1/ β n
γ n (17)
γ n (n-1)
2
3
4
5
6
7
1
2
3
4
5
6
-3.333333 -1.666667 -1.111111 -0.833333 -0.666667 -0.555556
-0.3
-0.6
-0.9
-1.2
-1.5
-1.8
0
-0.833333 -0.740741 -0.625 -0.533333 -0.462963
0
-1.666667 -2.222222
-2.5
-2.666667 -2.777778
μ (9)
-3.333333 -3.333333 -3.333333 -3.333333 -3.333333 -3.333333
β n share (16)
n
(n-1 )
b (26)
off by
gn
g
m
m/n
1 0.666667
a
2
1
-0.45
-0.15
0.225
0.225
-0.225
-0.15
0.6 0.571429 0.555556 0.5454545
b
c
3
4
2
3
-0.7
-0.975
-0.1
-0.075
0.233333 0.24375
0.466667 0.73125
-0.233333 -0.24375
-0.1
-0.075
d
5
4
-1.26
-0.06
0.252
1.008
-0.252
-0.06
e
6
5
-1.55
-0.05
0.258333
1.291667
-0.258333
-0.05
f
7
6
-1.842857
-0.042857
0.2632653
1.5795918
-0.263265
-0.042857
17