We model a vertically differentiated duopoly with quantity-setting firms as an extended game in which firms noncooperatively choose the timing of moves at the quality stage, to show that at the subgame perfect equilibrium sequential play obtains, with the low-quality firm taking the leader's role.

A. Tampieri

L. Lambertini

English

Research Papers in Economics

 
 
 
 
   
 
 
 
   
Low-Quality Leadership in a Vertically 
Differentiated Duopoly with Cournot 
Competition 
 
   
Luca Lambertini 
Alessandro Tampieri 
 
 
Quaderni  - Working Paper DSE N° 750   
   
   
 Low-Quality Leadership in a
Vertically Di⁄erentiated Duopoly
with Cournot Competition
Luca Lambertinix;# and Alessandro Tampierix
§ Department of Economics, University of Bologna
Strada Maggiore 45, 40125 Bologna, Italy
luca.lambertini@unibo.it; alessandro.tampieri@unibo.it
# ENCORE, University of Amsterdam
Roetersstraat 11, WB1018 Amsterdam, The Netherlands
May 20, 2011
Abstract
We model a vertically di⁄erentiated duopoly with quantity-setting
￿rms as an extended game in which ￿rms noncooperatively choose
the timing of moves at the quality stage, to show that at the subgame
perfect equilibrium sequential play obtains, with the low-quality ￿rm
taking the leader￿ s role.
JEL codes: C73, L13
Keywords: endogenous timing, product quality
11 Introduction
A well established result produced by the theory of vertical di⁄erentiation
is that the ￿rst entrant ￿lls the highest quality niche, letting newcomers
locate further down along the quality spectrum. This result is commonly de-
rived under Bertrand competition (see Gabszewicz and Thisse, 1979; Shaked
and Sutton, 1982; Aoki and Prusa, 1997; and Lehmann-Grube, 1997, inter
alia). The opposite may apply if the role of time is accounted for, in such a
way that low-quality leadership emerges when the exploitation of ad interim
monopoly power matters more than skinning the cream (see van Dijk, 1996;
and Lambertini and Tedeschi, 2007a,b).
Here we take a di⁄erent angle to address the same issue, exploiting an
idea dating back to d￿ Aspremont and GØrard-Varet (1980) and then further
developed by Hamilton and Slutsky (1990), according to whom a game is
Stackelberg-solvable if there exists a Stackelberg equilibrium that Pareto-
dominates the Nash solution. We use this approach in a vertically di⁄er-
entiated duopoly in which ￿rms bear a convex cost of quality improvement
and then behave ￿ la Cournot-Nash. From the analysis of the quality stage,
there emerges that, while the high-quality ￿rm￿ s best reply is increasing, that
of the low-quality ￿rm is decreasing, and therefore the choice of roles con-
cerning the timing of moves in the quality space univocally selects sequential
play with low-quality leadership as part of the subgame perfect equilibrium.
This framework may indeed ￿t real-world episodes in which innovation from
below has been observed; one such instance is the introduction of solid state
(transistor) circuitry to replace vacuum tube designs in consumer electron-
ics at the turn of the Seventies, with a large production characterised by a
comparatively lower quality, higher quality versions being introduced later.
22 The model
We consider a duopoly market for vertically di⁄erentiated products supplied
by single-product ￿rms. The demand side is modelled ￿ la Mussa and Rosen
(1978). There is a continuum of consumers whose types are identi￿ed by
￿, uniformly distributed with density equal to one in the interval [0;￿] (so
that total demand is equal to ￿). Parameter ￿ represents the consumers￿
marginal willingness to pay for quality. Each consumer is assumed to buy at
most one unit of the vertically di⁄erentiated good in order to maximise the
following surplus function:
U = ￿qi ￿ pi; (1)
where qi 2 [0;Q] indicates the quality of the product and pi is the market
price at which that variety is supplied by ￿rm i = H;L; with qH ￿ qL:
Therefore, the consumer who is indi⁄erent between qH and qL is identi￿ed
by the level of marginal willingness to pay b ￿ that solves b ￿qH ￿pH = b ￿qL￿pL;
and therefore b ￿ = (pH ￿ pL)=(qH ￿ qL). Thus, market demand for the high-
quality good is xH = ￿ ￿ b ￿. We assume partial market coverage, so that
there exists a consumer indi⁄erent between buying qL or not buying at all,
identi￿ed by e ￿ solving e ￿qL ￿pL = 0; whereby e ￿ = pL=qL and the demand for
the inferior variety is xL = b ￿ ￿e ￿. This is what one needs to use in order to
model Bertrand behaviour, while inverse demands
pH = (￿ ￿ xH)qH ￿ qLxL
pL = (￿ ￿ xH ￿ xL)qL
(2)
are to be used under Cournot competition.
On the supply side, as in Motta (1993), inter alia, ￿rms incur in convex
￿xed costs of quality improvement Ci = cq2
i; i = H;L. Variable costs are
assumed away. Hence pro￿t functions are ￿H = pHxH ￿ cq2
H and ￿L =
pLxL￿cq2
L. Competition takes place in three stages. In the ￿rst, ￿rms choose
the timing to be followed in the second stage, where qualities are set, and
3then in the third stage simultaneous Cournot competition takes place. The
solution concept is the subgame perfect equilibrium by backward induction.
The ￿rst stage is a pre-play stage ￿ la Hamilton and Slutsky (1990), in
which, under complete, symmetric and imperfect information, ￿rms play a
discrete strategy game represented in Matrix 1.
L
F S
H F ￿N
H;￿N
L ￿SL
H ;￿SF
L
S ￿SF
H ;￿SL
L ￿N
H;￿N
L
Matrix 1
Actions F and S stand for ￿￿rst￿or ￿second￿ , and refers to the choice of
roles pertaining to the quality stage, while superscripts N, SL, and SF stand
for Nash, Stackelberg leader and Stackelberg follower, respectively. If ￿rms
select the same strategy - along the main diagonal - then the second-stage
quality game is simultaneous. Conversely, along the secondary diagonal, the
quality stage is going to be solved ￿ la Stackelberg. For future reference, it is
worth recalling that the ￿rms￿incentives as to the timing of moves is entirely
driven by the slope of their best replies (in this case, in the quality space), in
such a way that if a ￿rm has a decreasing (resp., increasing) reaction function,
it will prefer to move ￿rst (resp., second) (see Hamilton and Slutsky, 1990,
Theorem V, p. 38).
3 Results
To begin with, we characterise optimal outputs for any given quality pair:
x
N
H =
￿(2qH ￿ qL)
4qH ￿ qL
; x
N
L =
￿qH
4qH ￿ qL
(3)
4where superscript N stands for Nash equilibrium. The explicit derivation of
the Cournot equilibrium is omitted as it is known from Motta (1993).
We now turn to the second stage where the quality game takes place. The
relevant pro￿t functions are:
￿H =
qH
￿
￿2 (2qH ￿ qL)
2 ￿ cqH (4qH ￿ qL)
2￿
(4qH ￿ qL)
2
￿L =
qL
￿
￿2q2
H ￿ cqL (4qH ￿ qL)
2￿
(4qH ￿ qL)
2
(4)
The ￿rst order conditions for non cooperative pro￿t maximisation are:
@￿H
@qH
=
￿2 (16q3
H ￿ 12q2
HqL + 4qHq2
L ￿ q3
L) ￿ 2cqH (4qH ￿ qL)
3
(4qH ￿ qL)
3 = 0; (5)
@￿L
@qL
=
￿2q2
H (4qH + qL) ￿ 2c(4qH ￿ qL)
3
(4qH ￿ qL)
3 = 0: (6)
Given that the above FOCs do not allow for a fully analytical charac-
terisation of Nash and Stackelberg equilibria, we investigate the solution of
the quality stage by studying the map of the reaction functions, implicitly
revealed by (5-6). In particular, following Bulow et al. (1985), we know that
the nature of strategic interaction is entirely determined by the sign of the
partial derivatives of FOCs with respect to the competitor￿ s quality, which
ultimately indicate the slopes of reaction functions q￿
i (qj), i;j = H;L; i 6= j:
These derivatives are:
@q￿
H (qL)
@qL
_
@2￿H
@qH@qL
=
8￿2qHqL (qH ￿ qL)
(4qH ￿ qL)
4 > 0; (7)
@q￿
L (qH)
@qH
_
@2￿L
@qL@qH
= ￿
2￿2qHqL (8qH + qL)
(4qH ￿ qL)
4 < 0: (8)
The concavity/convexity of best replies is determined by the following deriv-
atives:
@3￿H
@qH@q2
L
= ￿
8￿2 (4q2
H ￿ 5qHqL ￿ 2q2
L)
(4qH ￿ qL)
5 ; (9)
5which is positive for all qH 2
 
0;
5 +
p
57
8
qL
!
and negative for all qH >
5 +
p
57
8
qL; and
@3￿L
@qL@q2
H
=
2￿2qL (64q2
H + 28qHqL + q2
L)
(4qH ￿ qL)
5 > 0: (10)
Hence q￿
H (qL) is always increasing and convex for all qH 2
 
0;
5 +
p
57
8
qL
!
and concave for all qH >
5 +
p
57
8
qL. On the other hand, q￿
L (qH) is every-
where decreasing and convex.
A Nash equilibium exists if the vertical intercept of q￿
H (qL) is lower than
the vertical intercept of q￿
L (qH) in the space fqL;qHg. We can see that:
@￿H
@qH
￿ ￿ ￿ ￿
qL=0
= ￿
￿2 ￿ 8cqH
4
= 0; (11)
so that q￿
H (qL)jqL=0 = ￿2=(8c), while there are no solutions w.r.t. qH to
@￿L=@qLjqL=0 = 0. Therefore, de￿ning b qH ￿ (q￿
L (qH))
￿1 ; we have limqL!0 b qH =
+1:
Figure 1 shows the map of the best replies and the respective isopro￿t
curves. The Nash equilibrium (point N) yields a lower pro￿t both to ￿rm L
and to ￿rm H as compared to the Stackelberg equilibrium where L acts as
the leader (point SL). As to the Stackelberg solution with ￿rm H leading,
the map of isopro￿t curves reveals that the high-quality ￿rm is better o⁄
w.r.t. the Nash equilibrium, while the opposite applies to the low-quality
￿rm. Overall, the following chains of inequalities hold:
￿SF
H > ￿SL
H > ￿N
H
￿SL
L > ￿N
L > ￿SF
L
(12)
6Figure 1. The map of best replies with the equilibrium points.
6
-
qH
qL 0
q￿
L(qH)
q￿
H(qL)
L isopro￿t
curves
H isopro￿t curves
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
SL
SH
N
Using these properties, one can go back to Matrix 1 and observe that the
￿rst (pre-play) stage can be solved by applying iterated dominance, since (i)
￿rm L drops strategy S as it is strictly dominated by F; then, (ii) ￿rm H
drops what remains of strategy F; and therefore (iii) the matrix reduces to
the cell (S;F): This discussion allows us to formulate
Proposition 1 The three-stage game has a unique subgame perfect equilib-
rium in pure strategies, where ￿rm L takes the lead in the quality stage.
7To this regard, it is worth remarking that the slopes of reaction functions
and the ranking of payo⁄s in (12) are the two sides of the same coin, the ￿rst
dating back to Hamilton and Slutsky (1990), the second to d￿ Aspremont and
GØrard-Varet (1980). Taken together, these two features of the second stage
of the game allow players to select the timing of moves so as to attain at
equilibrium the Pareto-e¢ cient outcome.
As a last remark, observe that, as switching from simultaneous to sequan-
tial play with ￿rm L leading involves a decrease in both quality levels. This
surely has negative consequences on consumer surplus
CS =
Z b ￿
e ￿
(kqL ￿ pL)dk+
Z ￿
b ￿
(zqH ￿ pH)dz; (13)
as can be ascertained from the following partial derivatives:
@CS
@qH
=
￿2 [4q2
H (4qH ￿ 3qL) + q2
L (2qH + qL)]
2(4qH ￿ qL)
3 ;
@CS
@qL
=
￿2q2
H (12qH ￿ 7qL)
2(4qH ￿ qL)
3
(14)
which are both positive. Hence, the balance between the increase in industry
pro￿ts and the decrease in consumer surplus is ambigous.
References
[1] Aoki, R., and Prusa, T. (1997). Sequential versus Simultaneous Choice
with Endogenous Quality. International Journal of Industrial Organiza-
tion 15: 103-121.
[2] Bulow, J., Geanakoplos, J. and Klemperer, P. (1985). Multimarket
Oligopoly: Strategic Substitutes and Complements. Journal of Politi-
cal Economy 93: 488-511.
[3] d￿ Aspremont, C. and GØrard-Varet, L.-A. (1980). Stackelberg-Solvable
Games and Pre-Play Communication. Journal of Economic Theory 23:
201-217.
8[4] Gabszewicz, J.J. and Thisse, J.-F. (1979). Price Competition, Quality,
and Income Disparity. Journal of Economic Theory 20: 340-359.
[5] Lambertini, L. and Tedeschi, P. (2007a). Would You Like to Enter First
with a Low-Quality Good? Bulletin of Economic Research 59: 269-282.
[6] Lambertini, L. and Tedeschi, P. (2007b). On the Social Desirability of
Patents for Sequential Innovations in a Vertically Di⁄erentiated Market.
Journal of Economics 90: 193-214.
[7] Lehmann-Grube, U. (1997). Strategic Choice of Quality when Quality Is
Costly: The Persistence of the High-Quality Advantage. RAND Journal
of Economics 28: 372-384.
[8] Motta, M. (1993). Endogenous Quality Choice: Price vs. Quantity Com-
petition. Journal of Industrial Economics 41: 113-131.
[9] Hamilton, J. and Slutsky, S. (1990). Endogenous Timing in Duopoly
Games: Stackelberg or Cournot Equilibria. Games and Economic Be-
havior 2: 29-46.
[10] Mussa, M. and Rosen, S. (1978). Monopoly and Product Quality. Jour-
nal of Economic Theory 18: 301-317.
[11] Shaked, A. and Sutton, J. (1982). Relaxing Price Competition through
Product Di⁄erentiation. Review of Economic Studies 49: 3-13.
[12] van Dijk, T. (1996). Patent Height and Competition in Product Im-
provements. Journal of Industrial Economics 44: 151-167.
9 

Low-Quality Leadership in a Vertically Differentiated Duopoly with Cournot Competition

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics