Strategic Choice of Quality When Quality is Costly: The Persistence of the
High-Quality Advantage
Ulrich Lehmann-Grube
The RAND Journal of Economics, Vol. 28, No. 2. (Summer, 1997), pp. 372-384.
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RAND Journal of Economics
Vol. 28, No. 2, Summer 1997
pp. 372-384
Strategic choice of quality when quality is
costly: the persistence of the high-quality
advantage
Ulrich Lehmann-Grube*
In a two-firm, two-stage model of vertical product differentiation, I show that for every
convex fixed-cost function of quality, the firm that chooses the higher quality at the
first stage earns the higher profits. The result holds for the pure-strategy equilibrium
in the simultaneous-quality game, and it holds as well if firms choose their qualities
in sequential order.
1. Introduction
One of the well-established results of the industrial organization literature on vertical product differentiation is that the high-quality firm will earn higher profits than
low-quality firms. This is not a very surprising result when one realizes that in most
models, the costs of improving quality are zero (Shaked and Sutton, 1982; Tirole, 1988;
Choi and Shin, 1992; Donnenfeld and Weber, 1992 and 1995) or small and decreasing
(Shaked and Sutton, 1983). This is clearly unrealistic, and one should ask whether the
high-quality advantage persists when the costs of quality improvement are substantial
and increasing. Recently, some authors have analyzed strategic quality choice with
increasing costs (Ronnen, 1991; Motta, 1993; Boom, 1995), but in their models they
were not concerned with the question of a high-quality advantage.
In this article I shall give a straightforward generalization of the high-quality advantage. I shall show in a two-firm setting that the high-quality advantage in fact
persists for any cost function that is increasing and convex in the chosen quality if the
costs of quality are independent of output, i.e., sunk during price competition. The
fixed-cost case is the relevant one for a two-stage game because firms have to commit
themselves to a certain quality level to prevent the Bertrand outcome during price
competition. The most natural commitment is sunk costs of quality. Variable costs of
quality as they are analyzed by Motta (1993) are therefore ignored.
The result is important because, first, it allows a better understanding of firms'
struggles for the more costly high-quality position in most markets. Second, national
minimum quality standards get a new strategic interpretation. They may not only serve
the interests of domestic consumers, they may also help domestic firms capture the
* University of Hamburg; [email protected].
I wish to thank Christian Gabriel, Heidrun Hoppe, Wilhelm Pfahler, Ralf Winkler, and an anonymous
referee for useful suggestions. This article is related to a recent article by Aoki and Prusa (1997), which
became known to me only after the analysis in my article was complete.
372
Copyright 0 1997, RAND
LEHMANN-GRUBE 1 373
more profitable high-quality position in international competition. Nonstrategic welfare
effects are analyzed by, for instance, Ronnen (1991) and Boom (1995).
In Section 2 I present the model. The solution of the price-competition game for
given qualities is taken from Choi and Shin (1992). In Section 3 I characterize the
solution of the quality game when qualities are chosen by both firms simultaneously. I
show that in a pure-strategy equilibrium, the high-quality firm will earn the higher profits.
One should regard sequential instead of simultaneous choice of quality as the more
realistic case. Hence, in Section 4 I analyze the sequential-quality game. I show that this
game has a unique solution: the firm that chooses quality first (the leader) chooses a
high quality, the follower chooses a lower quality, and the leader earns higher profits.
Section 5 contains my conclusions and an examination of the generality of the results.
2. The model and price equilibrium
There are two firms in the market, firm 1 and firm 2. At the first stage of the
game, firms choose their respective quality, s,, s,, either in sequential order or simultaneously. The costs of quality are independent of output and convex in the chosen
quality: F(s), with F' r 0, F " > 0. To ensure that the simultaneous-quality as well as
the sequential-quality game always has an interior and bounded solution with both firms
entering the market, I further assume, as in Ronnen (1991), that F(0) = 0, F1(0) = 0,
and lim,,, F1(s) = co.' Both firms are engaged in simultaneous price competition at the
second stage of the game. Variable costs are equal among firms and, to keep notation
simple, are assumed to be zero.
For the demand side, I use a model inspired by Tirole (1988) and worked out by
Ronnen (1991) and Choi and Shin (1992). N consumers buy at most one unit from
either firm 1 or firm 2. Consumers differ in a taste parameter q, and they get a net
utility from buying a quality si at price p,:
A consumer of "taste" q will buy if U r 0 for at least one of the offered pricelquality
combinations, and he will buy from the firm that offers the best pricelquality combination
for him. Consumers are uniformly distributed over the range [0, qO].Without loss of generality, the total number of (potential) consumers, N, as well as qO,are normalized to unity.
For given qualities the model differs in no way from the model of Choi and Shin.
Hence I use their explicit solution of the price game. Equilibrium revenues are
where it is assumed without loss of generality that s, 5 s,. Throughout the article the
first argument in Reefers to the lower quality and the second refers to the higher quality.
3. Simultaneous choice of quality
If firms choose qualities at the first stage of the game simultaneously, the setting
is equivalent to the one of Ronnen (1991). Necessary conditions for a pure-strategy
Nash equilibrium in qualities (st 5 s f ) are
'
In fact it is sufficient to assume that F ' ( 0 ) < Y,,and lim,,, F f ( s )> '/,. Deviations from these conditions
allow the analysis of entry and entry-deterrence considerations.
374
1 THE RAND JOURNAL OF ECONOMICS
Proposition 1. For s, 5 s,, the simultaneous-quality game has exactly one Nash equilibrium in pure strategies (ST < s,*) if there exists no > s; or -s < ST such that
s
The simultaneous-quality game has no Nash equilibrium in pure strategies if there is
a > s,* or a s < ST such that inequality (6) or (7) holds.
s
ProoJ: First, it is obvious that ST has to be smaller than s,* to satisfy (5). That is, firms
choose distinct qualities (ST < s$):Second, it has to be shown that the pair (ST, s;)
that satisfies (4) and (5) is unique. This can be taken from Ronnen (1991) or more
explicitly from Lemma 1 in the Appendix. What is left to check for (ST, s;) is that
neither the low-quality firm has an incentive to become a high-quality firm, given s;
(inequality (6)), nor that the high-quality firm has an incentive to become a low-quality
firm, given ST (inequality (7)). On the other hand, if there is a > s,* or a s < s? such
that one of the inequalities is satisfied, then (ST, s;) is not an equilibrium, &d it follows
directly from the fact that it is the unique solution to (4) and ( 5 ) that there is no purestrategy equilibrium in that case. Q.E.D.
s
The case of high-quality leapfrogging (inequality (6)) can be relevant only if the
low-quality firm earns lower profits with (ST, SF) than the high-quality firm. On the
other hand, the case of "backward" leapfrogging (inequality (7)) can be relevant only
if the opposite is true. As shown below in Proposition 2, at (ST, s;) the high-quality
firm always earns the higher profits, so the case of "backward" leapfrogging can be
ruled out. For a quadratic cost function F(s) = '/,s2, Motta (1993) has shown that a
high-quality leapfrogging situation does not exist as well, i.e., for a quadratic cost
function there is always an equilibrium in pure strategies. But this result does not hold
for the general case. It is relatively simple to construct a cost function F(s) such that
there exists a quality > ST that satisfies inequality (6).
The nature of the pure-strategy equilibrium is clear: One firm chooses the highquality s,* at the first stage of the game, facing high costs at the first stage and high
revenues from the price game of the second stage. The other firm chooses the lower
quality ST, facing lower costs and lower revenues. The natural question now is whether
it is always more profitable to be the high-quality firm. In the remainder of this section
I shall show that this is in fact the case.
s
Proposition 2. If firms choose their qualities simultaneously, then in a pure-strategy
equilibrium (ST < s,*) the high-quality firm will earn higher profits.
ProoJ: Defining 0 < a < 1 and 0 < z such that az = s? < s,* = z, the necessary
conditions for a Nash equilibrium in pure strategies can be rewritten to
LEHMANN-GRUBE 1 375
Then it follows from Lemma 2 (see the Appendix) that
and hence the high-quality firm earns the higher profits in equilibrium.
Q.E.D.
The main idea behind the proof of the result can be seen in Figure 1, which gives
a picture of any Nash equilibrium situation. The curve RT(s,, ST) visualizes the functional relationship between the low-quality firm's revenue varying its quality while the
quality of the high-quality firm is held fixed at the equilibrium level ST. On the other
hand, in R$(sT, s,) s, is variable and s , is held fixed at the respective equilibrium level
ST of the low-quality firm. F ( s ) is an arbitrary convex cost function. Obviously the
slope of F ( - ) must be equal to the slope of RT(sI,s,*) at point s , = ST and to the slope
of R,*(sT, s,) at point s, = s;. Further, R$(s?, s,) must be zero at point s, = ST, whereas
R ~ ( S ,ST)
, must be zero at point s , = ST. The trick now is to choose a worst-case cost
function G ( s ) which is simply a tangent to F ( s ) at point s;. What is shown in Lemma
2 is that the distance between A and B is not greater than the distance between C and
D, i.e., RT(sT, s;) - G(sT) 5 RT(sT, ST) - G(s,*).From the convexity of F ( s ) it follows
immediately that F(s7) > G(s?), hence the low-quality firm earns the lower profits.
FIGURE 1
376
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THE RAND JOURNAL OF ECONOMICS
4. Sequential choice of quality
In Section 3 I showed that in the simultaneous-quality game either there is no
pure-strategy equilibrium or the high-quality firm earns higher profits. In both cases
firms have an incentive to choose the quality previous to the choice of the rival. Moreover, a sequential choice of qualities is clearly the more realistic scenario.
Let s, = h(s,) be the optimal low-quality choice of the follower if the leader
chooses s, first, and let s, = k(s,) be the optimal high-quality choice if the leader
chooses s, first. In Lemma 1 it is shown that h(.) and k(-) are well defined over the
whole range of s, and s, respectively, and that h' > 0 as well as k' > 0.
If the firm that chooses the quality first chooses the high quality, it has to choose
s2 such that the following equation holds:
follower effect < 0
If, on the other hand, the leader chooses the low quality, it has to choose s, such
that the following equation holds:
I
,
I
I
follower effect > 0
Hence the next proposition follows from Corollary 1 (see the Appendix).
Proposition 3. If there exists a pure-strategy equilibrium in the simultaneous-quality
game (sf, s f ) , then in the sequential-quality game the high-quality leader will choose
a lower s,L < sf than in the simultaneous-quality game, while the follower chooses
sf < s f . On the other hand, the low-quality leader will choose a higher quality than
in the simultaneous game (sf > sf), while the follower chooses s,F > s f .
Next I want to show that the leader will in fact choose the higher quality whether
there is a pure-strategy equilibrium in the simultaneous game or not. The mathematical
core of the result is stated as Lemma 3 (see the Appendix). Using this lemma, it is
possible to prove the main result of this section. During the proof several definitions
are given. An overview of these definitions and their respective properties is given in
the Appendix.
Proposition 4. In a sequential-quality game, the firm that has the first choice of quality
(the leader) will choose the higher quality and will earn higher profits than the follower.
Proof. The firm that chooses quality first has two options: the best low quality or the
best high quality.
Let us denote the best low quality by sf and denote the optimal reaction of the
follower to sf by sg = k(sf). Note that sf has to satisfy condition ( l l ) , the necessary
condition for a maximum of the low-quality leader. The highest profits the leader can
earn from low-quality choice is denoted by ITk, = Rf(sf, s,F) - F(sf). Note further
that from Proposition 3 it is known that sf > sf and s[ > sz, where the pair (sf, sz)
is the solution to (8) and (9).
LEHMANN-GRUBE 1 377
The best high-quality choice of the leader is more complicated. It may choose the
quality that is the best one under the assumption that the follower chooses its optimal
low quality in reaction. This quality is denoted by s,L, where the optimal low-quality
reaction of the follower is denoted by sp = h(s,F), and s,L satisfies condition (10). Or
the leader may be forced to choose a quality that prevents the follower from leapfrogging it (the leader) in terms of quality. The quality level that just prevents leapfrogging
is denoted by 9, where 9, = h(9) is the optimal low-quality reaction of the follower,
and g2 = k(9) is the optimal high-quality reaction of the follower to that 9. Then 9, f , ,
f2 are determined by the equation
Inspection of RT and R? easily reveals that 9, f I , f2 must be unique. The best highquality choice of the leader that ensures the leader will in fact be the high-quality firm
is denoted by sg, while s; = h(sg) denotes the respective optimal reaction of the follower.
Obviously sg must satisfy s; r max(9, s,L) to prevent leapfrogging. The respective profits
of the leader are denoted by ITkigh R?(s;, s;) - F(s;), and the profits of the (low= RT(s;, sd - F(s;).
quality) follower are IIf-,,
What has to be shown in this proof is that ITk, < ITkigh and IT:," < n t ; i g h , which implies
that the leader will in fact choose to be the high-quality firm,and that it earns higher profits
than the follower. This is done in three steps. In the first step it is shown-using Lemma
3-that the leader's profits are higher if it chooses s,F instead of sf. This ensures that the
leader will choose 4 5 instead of sf if 9 5 $. Next it is shown that the leader will earn
higher profits in choosing 9 instead of sf if 9 > s,F. Both steps imply that in fact
<
ITkigh. Finally, in the third step of the proof, it is shown that
<
Step I . It is shown that R?(&, s,3 - F ( s 3 > RT(sf, s,3 - F(sf), where F, = h(s9
denotes the optimal reaction of the follower if the leader chooses SF. And SF = k(sf) is
the follower's optimal reaction if the leader chooses sf. First, sf must satisfy condition
(1 1):
nkigh.
nk,
From Proposition 3 it is known that 5, < sf. Now, using the definitions
with 0 < 8 < a and
reformulated to
z > 0, the two first-order conditions of the follower can be
Then it follows from Lemma 3 that
378
1 THE RAND JOURNAL OF ECONOMICS
as stated.
Step 2. It is shown that R,*(s^,,9) - F(9) > >?(sf, SF) - F ( s f ) if s^ > s,F. First,
it follows from 3, < s^ and the fact that R,* is decreasing in the first argument that
R,*(s^,,f2)- F ( f 2 )> Rf(s^, s^), - F(s^,).It is known from Corollary 1 that
for all s, > SF.
Then it follows from sf < s: < 3, from the concavity of R,* with respect to the second
s^), - F(s^,),
argument, and from the convexity of F ( . ) that R,*(s^,,s^) - F(3) > RR,*(s^,,
and hence that Rf(s^,,3 ) - F ( f ) > R,*(s^,f,) - F(9,) = RF(b,, s^) - F(s^,)(by the definition
of s^,, s^, f,). What is left to show is that
This is readily satisfied, as RF is increasing in the second argument, s^ > s: by assumption, and s^, is the optimal reaction to s^ on the left-hand side while s; is not the
optimal reaction to s: on the right-hand side of the inequality. Hence
as stated.
Step 3. Steps
instead of s f . Let
E Rf(s;, s;)
leader will always
nkigh
1 and 2 established that the leader will choose s," 2 max(s,F, f)
s; be the optimal reaction of the follower. Now it is shown that
- F(s;) > R?(s;, s;) - F(s;) =
First, let SF < k Then the
choose the smallest possible s,, i.e., s," = f, because
nLw.
[dR,*(h(s,),s , ) ] l d ~<
, [aR,*(h(s,),s , ) l l ~ s 2< F1(s2) for all s2 > s f
(from Corollary 1). It has already been shown in step 2 that
R,*(s^,,s^) - F(s^)> Rf(s^,s)^,
- F(s^,) = RF(s^,,s^) - F(s^,)=
nCw
for a l l 3 > s,*
Now let s^ 5 SF. It is known already from Proposition 2 that for the pair (SF, s f ) , the highquality firm will earn higher profits than the low-quality firm. Hence if the leader chooses
s; = SF, it will earn the higher profits. The only reason for it to choose s$ < SF is that it
might earn even higher profits than with s$ = s f . On the other hand, the profits of the
follower are decreasing in that case. Hence if s^ 5 s f , as in the case where s^ > SF, the
leader earns higher profits than the follower. That finally proves the proposition. Q.E.D.
The main ideas behind the proof of the result can be seen in Figures 2 and 3.
Consider first Figure 2. Similar to Figure 1, R?(s,, s,F) are the low-quality firm's revenues if s , is varying and s, is fixed to s:. s; denotes the optimal low-quality reaction
given s, = s,F. In R,*(s;, s,) and R,*(s;, s,) the first arguments are fixed to S, and s f
respectively while s, is varying. It is clear that the slope of the cost function F ( s ) must
be equal to the slope of RF(sI,s:) at point s , = S, and equal to the slope of R f ( s f ,s,)
at point s, = s,F. What is proved in Lemma 3 is that the distance between A and B
LEHMANN-GRUBE I
379
FIGURE 2
cannot be greater than the distance between C and D. As it is clear from the convexity
of F(-) that the leader's profits from choosing the best low-quality sf are strictly smaller
than the distance between A and B, this proves that it is better for the leader to choose
sg instead. As long as the quality that just prevents leapfrogging 3 is not greater than
sg, the leader is free to choose s$ 5 s,F. This was step 1 in the proof.
For the case where 3 > s,F (step 2), two arguments are used. First observe in Figure
2 that the follower's profits from choosing 5, while the leader chooses s,F are greater
than the profits the low-quality leader gets when choosing sf. Further, the higher the
leader has to choose f, the better for the low-quality follower, i.e., for f > s,F it is true
that
Now consider Figure 3. By the definition of $, Rf(k 3,) - F(f2) = R?(3,, 3) - F(f,).
What is left to show now is that Rf(3,, 3) - F(f) > Rf($ 3,) - F(f2). Because Rf(3,, s,)
always lies above Rf(f, s,), it is sufficient to show that the distance between the two
curves Rf(f,, s,) and F(s) is always getting smaller to the right of 3. This in turn is
true for all s, > sf (see Corollary 1). This was step 2 in the proof.
5. Conclusions
The advantage of quality leadership is a stable result in the model presented above.
Future research should address the question of whether the high-quality advantage also
holds for a more general setting and could hence be seen as a structural property of
strategic quality competition. Several generalizations of the model are conceivable. I
sketch out some of them below.
The model applied to the analysis above is characterized by four crucial assumptions. First, variable costs of quality are ignored. The justification of this assumption
lies in the basic two-stage character of quality competition. To prevent simple Bertrand
380
/
THE RAND JOURNAL OF ECONOMICS
FIGURE 3
competition in the second stage, it has to be assured that firms do not change their
product quality during this price stage of the game. But firms will find it difficult to
commit themselves to a certain quality level if the costs of quality are variable. The
most natural commitment to a quality level are sunk costs of quality as they are modelled here. The straightforward generalization would be a setting where product quality
is a function of both fixed and variable cost, i.e., s := S(F, c), and firms choose F at
the first (quality) stage of the game while they choose the variable costs c at the second
(price) stage.
Second, the demand side of the model is of a rather special type. Basically it is
assumed that consumers' tastes are uniformly distributed, that their utility is additively
separable in the consumption of the good in question and all other goods (income),
and that they are not allowed to buy from more than one firm (unit demand). The last
assumption is crucial for all models of product differentiation because, otherwise, product differentiation by firms would not prevent the Bertrand outcome of price competition. The assumption of uniformly distributed taste ensures that market demand is
linear for each single quality. It should be interesting to check whether the advantage
of quality leadership also holds to all taste distributions, which ensures that market
demand is concave (p" 5 0). The assumption of additive separability, which was introduced by Tirole (1988), is reasonable as long as the price of the product has only a
small impact on the total budget of the consumers. For the (as it seems) less relevant
case of multiplicative separability, which has been introduced by Shaked and Sutton
(1982) and is still widely used in the literature (for instance in Boom (1995) and
Donnenfeld and Weber (1992, 1995)), the high-quality advantage can at least be confirmed in the context of the simultaneous game (Proposition 2).2
Third, the analysis is restricted to the duopoly case. Donnenfeld and Weber (1992)
have shown in a no-cost framework with more than two firms that the firm that is last
to enter the market will always choose a quality between the two other firms and earn
higher profits than the established low-quality firm but lower profits than the established
The proof of Proposition 2 for the Shaked and Sutton case is available to the interested reader on
request.
LEHMANN-GRUBE 1 381
high-quality firm. It would be interesting to investigate whether this result holds in the
presence of quality costs.
Fourth, the model presented here is static, whereas in fact one should assume that
the costs of quality depend on time: F(s, t ) , with F decreasing in time. The strategic
advantage of the firm that chooses quality first may then become a disadvantage in the
long run, as, contrary to the static setting, the leader may not be able to prevent
leapfrogging by a late entrant. This may lead to dynamic cycles in quality leadership.
Appendix
Statements and proofs follow for Lemmas 1-3 and Corollary 1 , which are used in the text. Following
them is an overview of definitions 1 - 1 1 , which are used in the text.
Lemma I . Let F: R + + R,, with F'(0) = 0 , F"(s) > 0 for all s E R,, and lim,,,
F'(s) = m. Let
D = { ( s , ,s,) # Ols, > s , > 0 ) . Let f , g: D + R, with f ( s , , s,) = s+[4s2- 7 s , ] / [ ( 4 s 2- s , ) ? ] - F 1 ( s , )and
g(s,, s,) = 4s,[as: - 3s2s, + 2s:]/[(4s2- s J 3 ] - F'(s,). Then there exists a unique pair ( s , , s,) that satisfies
f ( s , , s,) = 0 and g ( s , , s,) = 0 .
Proo$ F i t observe that F ( 0 ) = 0 ensures that there exists at least one s , for any given s, such that f = 0 . On
the other hand, lim,,, F'(s) = ensures that there exists at least one s, for any given s , such that g = 0 .
Next, to apply the implicit function theorem the partial derivatives of f and g have to be examined:
The signs of the partial derivatives are unambiguous. Hence the solution to f = 0 for each s, is unique as
is the solution to g = 0 for each s , . S o one may define the two functions s , = h(s,) and s, = k(s,), which
0 = g(s,, k(s,)).
are well defined over the whole range of s, and s , respectively, and satisfy f(h(s,), s,)
What is left to check is whether s , = h(k(s,))and s, = k(h(s,)) have a unique solution. It is known from the
fixed-point theorem of Banach that from h'k' < 1 it follows that s , = h(k(s,))and s, = k(h(s,)) have in fact
a unique solution. Now observe that
-
) B = 8 s , { ( 5 s 2+ s,)/(4s2- s , ) ~ )Then
.
from the implicit function
with A = 2s2((8s2+ 7s,)1(4s2- s , ) ~and
theorem we have
Q.E.D.
382 1 THE RAND JOURNAL OF ECONOMICS
Corollary 1 follows immediately from this lemma.
Corollary 1. Let Rf(s,, k(s,)) = [slk(sl)(k(sl)- s,)]l[(4k(sl)- sJ2] and Rf (h(s2),s2) = [4s2(s2- h(s2))]l
))~.
(4s2 - h ( ~ ~ Then
for all s, > s:
aR,*(h(s2),s2) = 4s2as:
as2
-
3s2h(s2) + 2h(s2I2
(4% - h(s2))'
> F1(s2) for all s2 > s f
for all s,
for all s2.
Lemma 2. If
4(4
-
3a
(4
-
+ 2a2) = F1(z),
a)'
4 - 7a
( 4 - a)'
2
0,
and 0 < a < 1 and z > 0 , then from F" > 0 it follows that
42-
1-a
1-a
- F(z) > az( 4 - a)2
(4 -
Prooj The following definitions are used: b = [4(4 - 3a
Now it is shown that
-
F(uz).
(A31
+ 2a2)]1[(4- a)'], A = bz - F(z); and G(s) = bs - A.
Using the definitions above,
baz
+A
4(4 - 3a + 2a2)
1 -a
(1 - a ) 2 O
(4 ( 4 - a)"
a ( 4 - a)-
This is readily satisfied because by assumption [4 - 7a]l[(4- a)" 1 0. Hence, in fact
4z- 1 - a
( 4 - a)'
-
F(z) = 4z- 1 - a
(4 -
By construction, G(.) is tangential to F(.) at point
for az < z. Hence it is true that
as stated by the lemma.
-
G(z) 2 az- 1 - a - G(az) is true.
(4 - a)2
z. From the convexity of F(.) it follows that F(az) > G(az)
Q.E.D. Lemma 3. Let 0 < d < a < 1 , 0 < z, and F"(s) > 0. Then from LEHMANN-GRUBE 1 383
4
4
3a + 2a2
= F'(z)
( 4 - a)'
-
4 - 78
2 0,
( 4 - 8)"
it follows that
-
Prooj The following definitions are used: b
G(s) = bs - A.
First it is shown that
1-8
(4-8)2
o 4-
-
4[4
-
3a
rnlnnrnum a1 a = -
=
Ff(z),A = bz
-
F(z), and
-
1-a
4 - 3a + 2a2
a2 4
(1
( 4 - ~ ) ~ ( 4 - a)"
u
w
decreasnng nn A
+ 2a2]/[(4- a)']
-
a),
decrearlng ~n a
(i) Let a 2 Y,. Then, elementary calculations yield that the terms in inequality ( A 9 ) are monotonic on the
desired range. Hence it is easily verified that
(note, 8
5
4/, from ( A 6 ) )and
and
1
8
-24
4
3a + 2a2
( 4 - a)'
-
Hence, as 7/,, - 1/48 = YE, inequality ( A 9 ) is verified for a 2 Y,.
(ii) Now let a < Y,. Then inequality ( A 9 ) is verified for sure if a
4 - a
(4 -
>4
=
8:
4-3a+2a2
4
-a<-.
( 4 - a)"
7
Hence it has been shown for any 0 < 8 < a < 1 that in fact ( A 8 ) is true:
1-8
4~~
(4 -
-
G ( z ) 2 az-
1-a
- G(az).
( 4 - a)2
By construction G(.) is tangential to F(.) at point z. Hence F(z) = G(z). From the convexity of F(.) it
follows that F(az) > G(az) for az < z. Hence
4
as stated by the lemma
1-8
2
7
- a)
-
(4
F ( Z )2 a
1-a
(4
-
1-a
z - G(az) > aza)2
(4 -
-
F(az),
Q.E.D. Definition 1. The pair (sr < ST) is the only solution to ( 4 ) and (5). See Proposition 1 : 384
/
THE RAND JOURNAL OF ECONOMICS
Definition 2. s,
=
h(s,) with h' > 0 is the unique solution to (4) for any given s, (Lemma 1 and Corollary 1)
Definition 3. s2 = k(sl)with k' > 0 is the unique solution to (5) for any given s, (Lemma 1 and Corollary 1).
Definition 4. The pair (sf, s,F = k(sf)) is the solution to the problem From Proposition 3, s: < sf and sf < SF. Definition 5. The pair (sf
=
From Proposition 3, sf <
ST
Definition 6.
h(s$),s f ) is the solution to the problem and s; < s f .
nk, = RT(sf, s,3
-
F(sf).
Definition 7 . The triple (9, = h(9) < 9 < 9,
Definition 8. The pair (sf
=
=
k(9))is the only solution to (12)
h(s;), s;) is the solution to the problem
max Rf(h(s2),s,)
-
F(s,), subject to s,
2
d
s1
Definition 9.
nkigh
Definition 10.
= Rf(sf, s;)
nc, = R?(s;, s;)
-
F(s;).
F(s;).
Definition 11. f, = h(s,3.
References
AOKI,R. AND PRUSA,T.J. "Sequential Versus Simultaneous Choice with Endogenous Quality." International
Journal of Industrial Organization, Vol. 15 (1997), pp. 103-121.
BOOM,A. "Asymmetric International Minimum Quality Standards and Vertical Differentiation." Journal of
Industrial Economics, Vol. 43 (1995),pp. 101-119.
CHOI,C.J. AND SHIN,H.S. ''A Comment on a Model of Vertical Product Differentiation." Journal of Industrial
Economics, Vol. 40 (1992), pp. 229-232.
DONNENFELD,
S. AND WEBER,S. "Vertical Product Differentiation with Entry." International Journal of
Industrial Organization, Vol. 10 (1992), pp. 449-472.
AND . "Limit Qualities and Entry Deterrence." RAND Journal of Economics, Vol. 26 (1995),
pp. 113-130.
MWITA, M. "Endogenous Quality Choice: Price vs. Quantity Competition." Journal of Industrial Economics,
Vol. 41 (1993), pp. 113-131.
RONNEN,U. "Minimum Quality standards, Fixed Costs, and Competition." RAND Journal of Economics,
Vol. 22 (1991), pp. 491-504.
SHAKED,
A. AND SUTTON,J. "Relaxin,g Price Competition Through Product Differentiation." Review of Economic Studies, Vol. 49 (1982), pp. 3-13.
AND . "Natural Oligopolies." Econometrica, Vol. 51 (1983), pp. 1469-1483.
TIROLE,J. The Theory of Industrial Organization, Cambridge, Mass.: MIT Press, 1988.
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You have printed the following article:
Strategic Choice of Quality When Quality is Costly: The Persistence of the High-Quality
Advantage
Ulrich Lehmann-Grube
The RAND Journal of Economics, Vol. 28, No. 2. (Summer, 1997), pp. 372-384.
Stable URL:
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References
Asymmetric International Minimum Quality Standards and Vertical Differentiation
Anette Boom
The Journal of Industrial Economics, Vol. 43, No. 1. (Mar., 1995), pp. 101-119.
Stable URL:
http://links.jstor.org/sici?sici=0022-1821%28199503%2943%3A1%3C101%3AAIMQSA%3E2.0.CO%3B2-Z
A Comment on a Model of Vertical Product Differentiation
Chong Ju Choi; Hyun Song Shin
The Journal of Industrial Economics, Vol. 40, No. 2. (Jun., 1992), pp. 229-231.
Stable URL:
http://links.jstor.org/sici?sici=0022-1821%28199206%2940%3A2%3C229%3AACOAMO%3E2.0.CO%3B2-D
Limit Qualities and Entry Deterrence
Shabtai Donnenfeld; Shlomo Weber
The RAND Journal of Economics, Vol. 26, No. 1. (Spring, 1995), pp. 113-130.
Stable URL:
http://links.jstor.org/sici?sici=0741-6261%28199521%2926%3A1%3C113%3ALQAED%3E2.0.CO%3B2-L
Endogenous Quality Choice: Price vs. Quantity Competition
Massimo Motta
The Journal of Industrial Economics, Vol. 41, No. 2. (Jun., 1993), pp. 113-131.
Stable URL:
http://links.jstor.org/sici?sici=0022-1821%28199306%2941%3A2%3C113%3AEQCPVQ%3E2.0.CO%3B2-%23
http://www.jstor.org
LINKED CITATIONS
- Page 2 of 2 -
Minimum Quality Standards, Fixed Costs, and Competition
Uri Ronnen
The RAND Journal of Economics, Vol. 22, No. 4. (Winter, 1991), pp. 490-504.
Stable URL:
http://links.jstor.org/sici?sici=0741-6261%28199124%2922%3A4%3C490%3AMQSFCA%3E2.0.CO%3B2-K
Relaxing Price Competition Through Product Differentiation
Avner Shaked; John Sutton
The Review of Economic Studies, Vol. 49, No. 1. (Jan., 1982), pp. 3-13.
Stable URL:
http://links.jstor.org/sici?sici=0034-6527%28198201%2949%3A1%3C3%3ARPCTPD%3E2.0.CO%3B2-D
Natural Oligopolies
Avner Shaked; John Sutton
Econometrica, Vol. 51, No. 5. (Sep., 1983), pp. 1469-1483.
Stable URL:
http://links.jstor.org/sici?sici=0012-9682%28198309%2951%3A5%3C1469%3ANO%3E2.0.CO%3B2-3