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Vertical product differentiation with entry
International
Journal
of Industrial
Organization
10 (1992) 449-472.
Vertical product differentiation
North-Holland
with entry
Shabtai Donnenfeld and Shlomo Weber* York University, North York, Ont., Canada Final version received
November
1991
This paper examines the non-cooperative behaviour of established firms that compete in product quality and prices while foreseeing the effects of their rivalry on the decisions made by the later entrants. We prove an existence and uniqueness of the equilibrium and show that the dominant firms engage in maximal product differentiation, i.e., select the highest and lowest technologically feasible qualities. Later entrant, however, always selects an intermediate quality. Moreover, we show that the equilibrium configuration is identical to that generated by perfect foresight equilibrium in a model of sequential entry. Furthermore, the profits earned by each firm are ranked in the same order as the qualities, implifying that the later entrant, despite being the last mover, ends up with higher profits and larger market share than one of the incumbents.
1. Introduction
Ever since the classic Hotelling ‘Principle of Minimum Differentiation’ has been introduced, a great deal of effort was devoted to study the effects of competition on product differentiation. Empirical evidence reveals that some industries are characterized by a slight degree of product differentiation whereas in other industries product differentiation is starker.’ From the theoretical point of view one can think of several reasons which explain the extent to which product differentiation will be observed: Diversity in consumers tastes, disparities in their incomes, timing of firms’ entry into the industry, product proliferation to deter further entry, etc. In this paper we wish to focus on the role that the sequence of entry has in determining the configuration of the firms’ product qualities. We embrace the paradigm of non-simultaneous entry; we consider the case where some firms are assumed to be the first to enter the industry and select qualities, thus enjoying the advantage of incumbency relative to firms which enter Correspondence to: Professor Weber, Department of Economics, University of York, 4700 Keele Street, North York, Ontario, Canada M3J lP3. *We are ihankful to two anonymous referees for their helpful suggestions. We are grateful to the Social Sciences and Humanities Research Council of Canada for its financial support. ‘See e.g. Hjorth-Andersen (1988) for analysis of the variance of product positions in quality space, based on the data for 175 German commodities, as well as Shaw (1982) and Swann (1985) for empirical evidence on product differentiation in the British fertilizer and microprocessor industry, respectively. 0167-7187/92/%05.00
0
1992-Elsevier
Science Publishers
B.V. All rights reserved
4.50
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
later. We capture this sort of competition in a model with several dominant players. Unlike the models with simultaneous entry Cd’Aspremont et al. (1979); Gabszewicz and Thisse (1986); Shaked and Sutton (1982)], where firms locate themselves in the product space at the same time, in our setup the estalished firms compete among themselves for positioning in the quality spectrum while foreseeing the effects of their competition on the decisions to be made by later entrants. Non-simultaneous entry encompasses several variants; firms may either enter sequentially, one after another, or initially several firms enter simultaneously and are viewed as incumbents by later entrants. The first case was originally examined by Prescott and Visscher (1977) and Hay (1976) in the context of spatial competition where the firms enter the industry and select location in a predetermined order. In both papers it is assumed that firms have perfect foresight, i.e., when a firm selects product location it is able to predict how its selection will affect the decisions of later entrants. Lane (1980) extended the original model by allowing for endogenous prices, which are determined as the outcome of a non-cooperative game. Different variants of sequential entry have been further studied by Bernheim (1984), Harris (1983, Eaton and Wooders (1985), Eaton and Ware (1987), Dewatripont (1987), Vives (1988), McLean and Riordan (1989), Kats (1989).2 Sequential entry models were explored and extended to encompass the effects of potential entry in industries with dominant firms which face the threat of potential entry. Palfrey (1984) studied spatial political competition of two established parties, which while competing for voters, foresee the effects of their competition on the location of a third party that enters later on. Palfrey demonstrates that, given the threat of entry, incumbents relax competition among themselves by choosing sharply distinctive platforms and positions in the political spectrum. Gilbert and Vives (1986) examined a similar model of structural hierarchy in the framework of an industry with multiple dominant firms, facing potential entry and competing in a homogeneous product market.3 In this paper we first consider an industry which consists of two established firms which compete in product quality while foreseeing the effects of their competition on the decisions made by a later entrant. We adopt the multiple dominant players (MDP) model, previously analyzed by Palfrey (1984) and Gilbert and Vives (1986). After proving existence and uniqueness of equilibrium of the associated non-cooperative game (Proposition l), we show that the equilibrium configuration of qualities in the MDP model, is not altered in perfect foresight equilibrium under sequential entry ‘Vives (1988) provides an excellent analysis of its welfare implications. 3The results of Palfrey and Gilbert and Waldman (1987), respectively.
review of the literature
on sequential
and Vives have been further
entry,
generalized
as well as an
by Weber (1992)
S. Donnenfeld and S. Weber, Vertical product dz@entiation
with entry
451
(SE) (Proposition 2). In both models the dominant firms, or the first two firms to enter the industry, select the extreme qualities that are technologically feasible, thus exhibiting maximal product differentiation among incumbents. This assertion could be supported by casual empiricism. Indeed, in some industries, like the personal computers there are early established firms, e.g., Compaq, that produce top of the line personal computers, which perform high-speed computations and provide high-resolution graphics images, whereas other early established firms, e.g., Compu-add, produce less advanced models which have a slower speed and lower resolution images. There are, in addition, later entrants in the industry, e.g., AST, which produce ‘intermediate’ quality PC’s with moderate speed of computation and reasonable resolution graphics. Our conclusions show that the threat of later entry further relaxes the degree of product competition. Indeed, in the model with simultaneous entry, Shaked and Sutton (1982) and Gabszewicz and Thisse (1986) showed that firms engage in product differentiation. However, the degree of product differentiation in their model is not maximal as one of the firms does not select the extreme quality. We also show that the entrant (in the MDP model), or the last firm to enter (in the SE model), always chooses an intermediate quality, that is, the entrant locates himself between the qualities chosen by the incumbents. The profits that the firms earn are ranked in the same order as their qualities. Interestingly, the entrant, despite being the last mover, ends up earning higher profits than one of the incumbents.4 This result differs signticantly from the observation of Palfrey that, in the context of horizontal product differentiation, the entrant ends up with a payoff (votes) which is lower than the payoffs of both incumbents. It highlights once more that competition in horizontally differentiated products differs in a significant way from competition in vertically differentiated markets. In the former mode of competition there is no particular advantage for a firm to be located to the right or to the left of the centre of the product spectrum. In the latter, being located farther up in the quality range is more desirable since profitability rises with quality. This stems from the consumers’ preferences since all of them evaluate a higher quality as more desirable and are willing to pay more for higher quality products. In the case of horizontally differentiated products, consumers vary with regard to their preferences for different varieties, according to the location of their most desirable variety in the product spectrum. The remainder of the paper is organized as follows: In section 2 we present the basic model and state our results on existence, uniqueness and characterization of equilibrium. In section 3 we examine the effects of technical change ‘In a different framework Gal-Or (1985), Eaton and Ware (1987), and McLean and Riordan (1989) have also shown that later entrants may enjoy higher profits than earlier entrants.
452
S. Donnenfeld and S. Weber, Vertical product dtfirentiation with entry
on equilibrium qualities, prices, market shares, market size and profitability of all firms. Section 4 contains the proofs of several results used for the proof of our main propositions that are presented in section 5. The technical derivations are relegated to appendix. We conclude the paper with a brief summary. 2. The model Consider a good that can be produced in a continuous array of quality levels, denoted by Q=[q,,,,qJ, where q,,, is the lowest and qM is the highest possible quality level. We assume that a consumer buys no more than one unit of the product from the quality array available in the market place. Consumers differ in their income; they will be identified by the income parameter t, which is assumed to be uniformly distributed over the interval [O,I]. A consumer’s utility is represented by U(x, q) where x denotes the quantity of the numeraire good with per unit price equal to one. Following Gabszewicz and Thisse (1979), we shall assume that the utility function takes the form u=zq,
where x = t-p is the consumer’s t disposable income devoted to the consumption of the numeraire good and p is the per unit price of the differentiated good. This specification of preferences implies that individuals with higher income enjoy quality improvements more than low income consumers. There are three tirms,5 each producing a single quality with zero costs. The firms compete for consumers by offering packages of price and quality (pi, qi), i = 1,2,3. Each consumer selects the best available package of price and quality that will maximize his utility without violating his budget constraint. That is, consumer’s t choice problem is to find the pair (pi, qi) that maximizes his utility, i.e., i(t) = arg max qi(t - pi).
If the resulting utility tqi(t)-pi(t) does not exceed zero, the consumer will stay out of the market. The sequence in which the firms make their decisions is the following: First, they decide (not all simultaneously) on the qualities to be offered. Once the triple of qualities, o=(ql,q2, q3), is chosen, the firms engage in price sAlthough we examine here a three firms industry, the analysis and most of the conclusions we draw hold for an industry with any finite number of firms.
S. Donnenfeld and S. Weber, Vertical product di$erentiation with entry
453
competition which generates a unique Nash equilibrium (p1,p2,p3), where pi =pi(@, i= 1,2, 3.6 In equilibrium the profits of the ith firm are
The firms differ in the order in which they enter the industry and select quality. The early entrant may have a strategic advantage (or disadvantage) of incumbency relatively to later entrants. In order to examine the various facets of this asymmetry we consider two models: (i) The multiple dominant players (MDP) model: There is a hierarchical structure in which two firms enter first whereas the third firm enters later. The first two incumbents compete with each other while foreseeing the repercussions of their rivalry for the entrant. The equilibrium of this game is called a MDP equilibrium [cf. Palfrey (1984) and Weber (1992)]. (ii) The model of sequential entry (SE): All three firms enter sequentially. When a firm decides to enter the industry, it takes into account the choices already made by the established firms and correctly anticipates the prolitmaximizing decisions of the firms yet to enter. The equilibrium notion in this model is called a perfect foresight equilibrium [cf. Prescott and Visscher (1977)]. 2.1. The MDP model There are two incumbent firms and one entrant. The incumbents play a Nash-Cournot game in qualities and behave as Stackelberg leaders with respect to the entrant by foreseeing his profit-maximizing response. Once all three qualities have been chosen, the firms compete in prices. The equilibrium prices give rise to the firms’ payoff functions, which are depicted in (1). By using backward induction, these payoff functions are used to determine the firms optimal choices of qualities. Since the ranking of the chosen qualities is not predetermined, we do not know ex ante which of the three firms will select the low, intermediate or high quality. We shall denote the incumbent firms by A and B, the entrant by E and their choices of qualities by qA, qB and qE, respectively. Since we do not impose any ranking on the firms’ quality choices, the assignment of the numbers 1,2,3 in eq. (1) to firms A,B,E cannot be ascertained before the game is played. Before proceeding with the formal definition of the game, we examine more closely the behaviour of the entrant. For a given pair of the incumbents’ quality choices, qA and qB, we shall denote by RE(q,,q,) the set of the entrant’s best responses %ee Shaked and Sutton (1982) for a similar approach with regard competition. They however assume that all firms move simultaneously.
to quality
and
price
454
S. Donnenfeld and S. Weber, Vertical product differentiation
Fig.
with entry
I
Suppose that qm< qB< qA< &,; firm E may consider entry in three distinctive regions: Below qB, between qB and qA and above gA (see fig. 1). It will be shown later in Lemmata 1-3, that the entrant’s profit function n(qA, &, .), associated with each of the three regions, is continuous and strictly quasiconcave. This guarantees the existence and uniqueness of the entrant’s best response in each of the regions. Moreover, the entrant’s highest attainable profits in region II are always higher than the local maximum in region I (Lemma 5). However, the ranking of the maximal profits in regions II and III cannot be ascertained. Thus, the set of the entrant’s best responses consists of the peak in region II or/and the peak of region III, where the latter, by Lemma 3, is equal to qhl. If E is indifferent between these two peaks, we assume, as is customarily done, that the entrant picks either of his best responses with equal probability.7 We are now in position to formally define a game I’, whose Nash equilibrium represents a MDP equilibrium in our model. ‘See Dewatripont (1987) for a discussion responses on the set of equilibria.
on the effect of multiplicity
of the entrant’s
best
S. Donnenfeld
and S. Weber, Vertical product differentiation
with entry
455
Definition 1. The game r is defined as follows: The players are the firms A,B and the strategy set of both players is Q; for each pair of strategies (qA,qB) the payoff of player I‘=A,B fi(qA,qB), is given by
where ITI stands for the cardinality of the set T. Definition 2. A pair (qA,qB) is called a MDP equilibrium (MDPE forth) if it constitutes a Nash equilibrium of the game r.
hence-
In the proposition below we state the main result of this paper. The proof of the proposition is presented in section 4. Proposition I. There is a unique (up to a permutation) MDPE (qA,qB). In this equilibrium the incumbents are engaged in maximal product dtfferentiation, i.e., qA=qM and qB=qm, while correctly anticipating the entrant’s location between them, qA > qE> q,,. Intuitively, the incumbents offer two distinctive products (different qualities) in order to relax the intensity of price competition. For any given pair Of qualities chosen by incumbents, (qA,&), as long as &, >qA> qB, the entrant may select either qE> qA or qA> qE> qB. (It Will be shown that qE< qB is always an inferior choice.) If the entrant finds it optimal to select qE> qA, then firm A anticipates the entrant’s choice and preempts him from capturing the high end of the quality range by setting qA=qM. If entrant prefers qA> qE> qB then firm A would be better off relaxing the price competition and choosing qA= qhl from the outset. Thus, in both cases qA=qhl. As the entrant will choose a quality qE to be located between the qB 2 qrn, selections of the incumbents. However, the low quality incumbent B, foreseeing the entrant’s location in the middle, finds it optimal to select the lowest feasible quality by setting qB=qm. By doing so firm B relaxes the intensity of price competition with the entrant. The hierarchical structure of the game generates incentives for stronger product differentiation among established firms than that generated in an environment where product selection decisions are made simultaneously by all firm~.~ Furthermore, it is noteworthy that this model suggests a different reason for the possibility that earlier movers may have a strategic advantage ‘Shaked and Sutton (1982) and Gabszewicz and Thisse (1986) examined the case where two firms move simultaneously and in equilibrium offer distinct qualities. We however show that under the non-simultaneous entry the firms engage in starker product differentiation.
J.I.O.
E
456
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
or disadvantage relative to later movers. Here, the high quality incumbent, who chooses product quality before the entrant does enjoy a strategic advantage which is reflected in higher profits. The low quality incumbent has a strategic disadvantage relative to the entrant; he has to differentiate his product from that provided by the other incumbent and thus opens a gap which will later be exploited by the enrant. Unlike the high quality incumbent, his low quality counterpart cannot preempt the entrant and thus ends up with lower profits. Preemption is not viable because profitability rises with product quality.
2.2. Sequential entry The MDP model considered in the previous subsection, envisioned an environment in which two incumbents enter early but simultaneously. We now examine a different sequence of entry where all firms enter the industry in a purely sequential manner. The equilibrium concept that we employ is that of a perfect foresight equilibrium (PFE) [Prescott and Visscher (1977)]. We shall demonstrate that replacing the entry sequence used in the MDP model with the assumption of sequential entry does not alter the equilibrium outcome; the firms choose the same strategies as in MDP equilibrium. There are again three firms, denoted A, B, E, that enter the industry sequentially (in this order) and select qualities from the interval Q = [qm, qM]. Each firm takes into account the choices of the firms that have already entered and anticipates the optimal selections of quality by firms yet to enter. Specifically, given the choices of the first two firms, A and B, the last entrant E chooses the quality which maximizes its profit. Firm B makes its profit-maximizing decision, taking as given the location of firm A and anticipating the choice of firm E. Firm A behaves as a Stackelberg leader with respect to the other two entrants, B and E, and anticipates their profit-maximizing responses. The question, which we address in this subsection, is whether A’s leadership position yields an additional gain relatively to its role of a Nash-Cournot player in the MDP model. Proposition 2 below, provides a negative answer to this question. It shows that equilibria in both models give rise to the same set of equilibrium selections of quality. Under the assumption of sequential entry, we use backward induction to determine the equilibrium strategies. Suppose that the first entrant, A, chooses quality qA. Then, assuming profit-maximizing behaviour of the entrant, we first define the set of the best responses of the second entrant firm B, RB(qA), to a given choice of the first entrant qA
S. Donnenfeld and S. Weber, Vertical product dtfirentiation with entry
457
We shall show later that this set is always nonempty and consists of a finite number of points. If the best response of B is not unique, we assume, as before, that firm B selects either of its best responses with equal probability. Thus the payoff of firm A is defined as the average over the set of the best responses of firm B. Formally, for a given qA, the payoff function of firm A, g*(& is defined by (3) We now define a concept of perfect foresight equilibrium (PFE). A triple (&, qg, qg), is called PFE if:
Definition 3.
(i) g(&)
tii) RB(&i)
&kA) =
vqA
E
Q;
&
(iii) R(qX, 4;) = qg. Condition (i) states that, correctly anticipating the actions of the other two firms, firm A maximizes its profit by choosing 41. According to conditions (ii) and (iii), qi is the unique best response of firm B to q%, whereas qg is the unique best response of firm E to the choices of the first two firms, qX and qg. Our next result, the proof of which is presented in section 4, implies the existence and the uniqueness of a PFE. Proposition 2.
There
exists a unique PFE
(qi,qg,qg).
Moreover,
qi=qMyI)
q$ = qm and 4: = z(qX, qg).
This proposition shows that, although in a model with sequential entry, firm A enjoys strategic advantage, being a Stackelberg leader rather than a Nash-Cournot player with respect to firm B, the equilibrium qualities of the first two firms are identical to those chosen in the MDPE. The reason is that in MDPE firm A ends up producing the highest quality, and this choice cannot be improved upon even if firm 1 is endowed with the Stackelberg leadership. Thus, in a model with sequential entry, firm A also chooses the highest quality qhl and firm B replies by selecting the lowest quality qm as its best response to the choice of firm A, which yields the same set of equilibrium strategies as in MDP equilibrium.
3. Comparative statics: Technical change We now examine the situation where, as a result of technical change, firms
458
S. Donnenfeld and S. Weber, Vertical product dzfirentiation with entry
are able to expand the quality spectrum Q to qualities higher than the original top quality qM. Indeed, in the previous section we have shown that the equilibrium quality that each firm chooses in MDP equilibrium, q;, i= A, B, E, is a function of the exogenously determined boundaries of the quality range Q = [qm, qM]. We assume now that the lower bound of Q, q,,,, is fixed, but the top quality, qhl, may vary and can take any value from the interval Q =(q,, CO). Thus, for any qm E @ and each i=A, B, E, we write the equilibrium qualities, prices, market shares and profits as q:(qJ, pT(qJ, $(q,) and ZT:(q& respectively. The properties of these equilibrium functions are examined in Propositions 3-7, the proofs of which are relegated to appendix. From Proposition 1 we know that, for given boundaries of the quality range the incumbents select the extreme positions, qA=qM and qB =qm. For the entrant we find the following. Proposition 3. The entrant’s equilibrium quality function qg( *), is strictly increasing and concave on the interval Q. Moreover,
4;1;(4d > &hf) > qB*(ch)forall qM E Q. In the next proposition we study the effect of a change in the upper bound of the quality spectrum on the equilibrium price functions. Proposition 4. For each i=A, B,E, the equilibrium price function pT( .) is strictly increasing, concave with lim 4M+4m pZ(qJ =O. Moreover, pB*(qr,J< pz(qJ
The effects on the firms’ equilibrium market shares are more intricate. Proposition 5. The market share of the low quality incumbent sg*(*) is strictly increasing and concave, whereas the market shares of the entrant and the highquality incumbent, .sg( .) and sX( .), are strictly decreasing and convex on the interval Q. Moreover, sg*(qU)
Technical change adversely affects the market size; the total number of consumers who purchase the product from one of the three firms decreases. Proposition 6. In equilibrium, the market size, M*(q,) strictly decreasing and convex on the interval 0.
= Z(Ci=A, B,e sT (qhl)), is
Finally, as expected, technical change enhances the profits of all firms. The
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
459
90 Fig. 2
9”
90
Fig. 3
rate of improvement of profitability is a decrasing function of the rate of technical change represented by the shift in the upper bound of the quality spectrum. Proposition 7. The equilibrium profit functions, Ilf( .), i= A, B, E are strictly increasing and concave on the interval 0. Moreover, II: < lI,*(qM) < IIX(qJ for all qME Q and there exist ZIlg
i=A,B,E. The properties of the equilibrium qualities, prices, market shares, market size and profit functions are portrayed in figs. 2-5. The values that the equilibrium functions approach in the limits, i.e., when qhl+qm and qhl+oo are presented in tables 1, 2. Intuitively, as a result of technical change, reflected by a shift in the upper limit of the quality spectrum, qm, the array of qualities that firms can produce is enlarged. From Proposition 1 we know that for any given lower
460
S. Donnenfeld and S. Weber, Vertical product d@erentiation with entry
GM
q0
Fig. 4
and upper bounds (qm,qM) the incumbent firms will find it optimal to select the extreme qualities and thus entails a more extreme degree of product differentiation relative to the situation that prevailed before the change occurred. This entices the entrant to move up in the quality spectrum, following the high quality incumbent, as portrayed in fig. 2. Taking into account that the consumers’ willingness to pay increases with income, the high quality incumbent and the entrant are able now to set higher prices. The higher prices associated with higher qualities enable the low quality incumbent to set a higher price as well. (The equilibrium price functions are portrayed in fig. 3.) As the low quality incumbent raises his price without changing the quality of its product, it induces consumers with lowest willingness to pay to exit the market. These consumers would have ended up with a negative surplus if they were to purchase a product of any quality. This explains the overall shrinkage of market size as portrayed in fig. 5. Furthermore, the low quality incumbent, although losing some of his original clientele from the low end of the income range, ends up serving buyers with intermediate levels of income. These consumers, previously served by the
461
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
Fig. 5
Table Firms
Prices (&p _ (&/z
Incumbent
Entrant
Incumbent
A
E
B
1
Market 4qp
share + 3qzz
4(2qYZ + 422)
4(2&Z + qitj2)
q1tj2_ q&/z 2( 2qg2 + qltjZ)
432 + q:2 2(2# + qrh/Z)
(qltj2- qZZ)(4qtiZ + 3qZZ) 4qtiZ(2q$2 + 422)
4(2#
qti2 + qZ2)
Profits IZ(qiZ -q;2)(4qa*
+ 3qZZ)2
16qM(2qiz + q:‘)’ WI, - %J 4(2qgz + qi2)2 {2qb/2(qgz -qz2) 16(2qa2+q:2)2
entrant, prefer now the low quality good qm as the entrant’s upgraded quality qE and higher price makes his product less attractive to them. In addition, the rise in the price of the high quality incumbent enables the entrant to lure away some of the consumers previously served by the high quality incumbent. The detailed analysis of these trends yields an increase in the market share of the low quality incumbent and a decline of the market share of the
462
S. Donnenfeld and S. Weber, Vertical product differentiation with entry Table 2 Prices
Firms
&+qm
Incumbent
Entrant
Incumbent
0
A
Market dK+a
dhl--%
share
Profits bhl-‘~
&%l
&.+~
I
5
E 1 8
B
entrant and the high quality incumbent, as portrayed in fig. 4. Finally, the profits of all firms rise as a result of technical change: The low quality incumbent enjoys a higher market share and a higher price even though his quality is unchanged. For the entrant and the high quality incumbent the beneficial effect of higher prices outweighs the negative effect of the reduced market share.
4. Preliminary
results
In this section we present lemmata, the proof of which is relegated to the appendix, We shall first examine the properties of the profit functions ni(41, q2, q3), i= 1,2,3, defined in (1). This will enable us later to establish the existence and uniqueness of MDPE and PFE. Assume, without loss of generality, the the qualities, chosen by the firms, are such that q1 < q2 < q3. In the first three lemmata we determine, given the rivals’ qualities, the peak of each profit function. Lemma
1.
II,( .,q2,q3)
For each pair (qz, q3) with q,,,< q2
the point y(q,, qd E CqmYqd. Lemma 2. For each pair (ql, q3) c Q with q1 < q3, the function I12(ql, ., q3) is strictly quasi-concave in the interval [ql,q3] with the peak at the point z(q1,q3)E(q1,q3) where the function z(., .) is strictly increasing in both arguments. Lemma 3. For each pair (ql, q2) with qmSql < qZ < qm, the function Z13(ql,q2, *) is strictly increasing on the interval [qz,qJ. Thus, its peak is qM.
S. Donnenfeld and S. Weber, Vertical product dzyerentiation with entry
463
Lemmata 3 and 4 are slight modifications of results obtained by Shaked and Sutton (1982) and presented here for the sake of completeness. Specifically, Lemma 4 shows that for any given triple of qualities, the ranking of profits is identical to that of qualities, so that a higher quality firm always earns higher profits than that of its lower quality rivals. Lemma 4. For inequalities hold
any triple (ql,q2, q3) t Q with q1
The next results shows that for given quality chosen by rivals, a firm could achieve higher profits by choosing a quality that lies between the locations of its competitors, rather than below this range. Lemma 5.
For each pair (ul, u2) c Q with u1 < u2, we have
max q,~u~U
n,(u, ui, u2) < max Z12(u’, v, u2) U’d”S&
In the following three lemmata we extend the analysis of the properties of the profit functions for cases that go beyond the standard Nash framework. We shall consider the situations where the firms, while deciding on product quality, take into account the impact of their selections on the rivals choices of quality. Lemma 6. Let q3 EQ be given. Then the function ZT,(q1,z(q1,q3), q3) is decreasing in q1 on the interval [q,,qJ. (Recall that z(q,, q3), defined in Lemma 2, is the optimal choice of a firm on the interval [ql, qs].) Lemma 7. Let q1 EQ be given. Then the function increasing in q3 on the interval [ql,qM].
I13(ql,z(qI,q3),
q3) is
Lemma 8. Let q3 E Q be given. Then the function decreasing in q1 on the interval [qm,q3].
Z13(ql, z(q,, q3), q3) is
Lemmata 68 deal with the situation where two incumbents, one producing the higher and the other the lower quality, correctly anticipate the entrant’s optimal response in the area between their product choices. Lemmata 6 and 7 provide the basis for the sharp product differentiation between two incumbents: The higher quality firm would be better off by raising its quality whereas the lower quality firm would gain by lowering its product quality. Lemma 8 shows that the principle of maximal differentiation
464
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
is incentive-compatible: The high quality firm is better off when its rival moves away towards the lower bound of the quality spectrum qm.
5. Proof of Propositions
1 and 2
We shall first state the lemma used in the proofs of the propositions. Lemma 9.
For each qA>qM
qM =
a% max fB(qA, qB). 4A54B5qM
Indeed, in game r let one of the incumbents, say A, choose a quality different from the top quality, i.e., qA
Let qA< qM be given. We shall show that no qB,qA< choice. Let f* = fB(qA, qM)= be the B’s optimal qB
(i) RE(q,, &) = Z(q,> qB)* Lemma 7 ITB(qA>
qB,
z(qA>
qB))
Then,
fB(qA,
Ch)
qM> z(qA,
=
nB(qAv
qM))
qB,
Z(4A,
qB))
and,
by
(4)
=f*.
Thus firm B would be better off by offering qM. (ii)
RE(qAyqB)
=
qM-
Then,
fBhA,
qB)
=
nB(qAy
qB? qM))
and,
by
z(qA,
qB))>
Lemmata
2
and 4 nB(qA,
qB> qM)
5
nB(qA,
hA>
qB)>
hl)
<
nB(qA?
qM,
(5)
and again, the choice of qM dominates qB. (iii) RE(qA,qB)= {gM,Z(q*, qB)}. In this CaSe fB(qA,
qB)
=i{nB(qA,
qBv z(qA,
qB))
+
nB(qA,
qB,
hl))*
(6)
However, (4) and (5) imply that each of the expressions on the right side of (6) is less than f *, yielding fB(qA, qB) < f*, which completes the proof of the lemma. 0 Proof of Proposition 1.
Let (qA, qB) be a MDP equilibrium. Then qA#qB and, without loss of generality, let qA>qB. Then, by Lemma 9, qA=qM and
S. Donnenfeld and S. Weber, Vertical product differentiation
465
with entry
Lemma 6 implies that qB= qM. Thus, the pair (qhl,q,J is the only candidate to constitute a MDPE. Applying Lemmata 9 and 6 again, we conclude that this 17 pair is, indeed, a unique MDP equilibrium. Proof
Let (qA,qB,qE) be a PFE and assume, in negation, of Proposition 2. that qA
dq,) = flA(% If
qB >
qA
then,
by
4%n,qhll).
4m
Lemma
9,
qB =
(7)
qM,
yielding
&A)
=nA(qA3
qM? z(qA?
qM)h
BY
Lemmata 2-6 nA(qA,
qM,
z(qA,
qM))
qM,
z(qm, qMkcnB(qA,
qM, z(qA,
qM)h
(8)
But the last expression in (8) is the payoff guaranteed to A whenever he chooses qM and, hence, is equal to g(%,$), yielding &A)
qhl),
If E selects z(q& qB) then imply &A)
%m
qB> qM)
g(q,)
z(qM,
<17A(qM~
=Z7,(qA,
%n))
=dqM).
qB, z(qM,
qB,z(qA,
qB)),
qB))
%%,d
(9)
and Lemmata 7 and 8
(10)
If the entrant is indifferent between entry at z(qA,qB) and qM, then (9) and (10) yield the choice of qM as the dominant strategy for A. Furthermore, by Lemmata 2 and 6, firm B chooses qm, whereas the entrant’s choice is z(%i, 4m). q 6. Concluding
remarks
In this paper we examined the implications of non-simultaneous entry into an industry where firms compete in markets with vertically differentiated products. Two variants of non-simultaneous entry were analyzed: The multiple dominant players model - where two firms enter first and are viewed as incumbents by a later entrant; and the model of sequential entry where firms enter the industry one after another. Both models generate maximal product differentiation between the equilibrium qualities of the first two firms to enter. Furthermore, both models shed some light on the first
466
S. Donnenfeld and S. Weber, Vertical product dl@rentiation with entry
(last) mover advantage issue. We show that under certain circumstances, being the last mover may entail a strategic advantage. The model presented here does not take into account the cost of production. However, if variable costs rise slowly with quality, our main result regarding the degree of product differentiation still holds; if variable costs rise rapidly with quality it will only relax product differentiation among the established firms. Fixed costs, on the other hand, are more intricate since they may endow the incumbents with the ability to prevent entry. In this paper we wished to focus on the effects of entry on vertical product differentiation and the examination of entry deterrence in this context is left for future research. Appendix A A.1. Proof of Lemmata l-8
In this part of the appendix we provide an explicit derivation of the firms profit functions given in (1) of section 2. For this purpose we need first to determine the assignment of consumers to firms’ products. Assume now that the price triple (p1,p2,p3) constitutes a Nash equilibrium for given product qualities choices (q1,q2,qJ)cQ, where q1
41@1 -PA
=o,
(A.11
and for i= 1,2, 64.2)
where consumer t, is indifferent between quality q1 at price p1 and staying out of the market, whereas ti + 1, i= 1,2 is indifferent between quality qi at price pi and quality 4i+l at price Pi+l. From (A.l) and (A.2), we have 64.3)
t1 =Pl,
t*=P2c12-Pl~l h--q1
’
t, = P3q3 -P242
q3--q2
.
(A-4)
(A.9
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
461
Thus the market segement of the first firm s1 = t,--tI, of the second firm s,=t,-tt,, and of the third firm sg =I- t, and for i= 1,2,3 the associated profits are
64.6)
ni=p,Si.
Substituting for ti, i= 1,2,3 from (A.3)-(A.5) into (A.6) and using the first order conditions for profit maximization of (A@, we obtain the following expressions for the equilibrium prices
pl_ztq2-qlm3-q2) 24
(A-7)
’
p2=zh2-ql)tq3-q2)
p3=zq3-q2 2q3
(A-8)
’
A
+p$
(A.9)
q3'
where A=4q2q3-q,q,-2q,q,-q:. Upon substitution of (A.7HA.9) in (A.6) yields the profit functions associated with the equilibrium prices ~l(ql,q2,q3)=12h2-41)(43-q2)2q2
44’
(A.lO)
’
n2(ql,q2,q3)=12(q2-ql)tq3-q2)tq3-o12
7
A2
(A.ll)
(A.12) Lemmata 1-8, stated in section 3, summarize the properties of the profit function presented in (A.lO)-(A.12). We now turn to the proof of these assertions. Proof of Lemma 1.
By (A.lO),
an
1 =sgn(-qlt2q2+q3)+5q~-22q2q3}. sgn i aql I Thus if qm>y=(5q:
-2q2q3)/(2q2
+q3)
(A.13)
then the profit function of the first
468
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
firm is declining on the interval [q,,,, qJ, and if q,S y then y is the peak of the profit function of the first firm. Overall, y(q,, q3) = max {y, qm} is its peak on the interval [q,,,,qJ. 0 Proof of Lemma 2.
By (A.ll),
an,_
-w~(q~-qlqd(4qlq~-q~q~-2q~q~-q~)~. sgn i aqz I
(A.14)
As 4qlq2-qlq3-2q2q3-q4:<4q,q2-q,q2-22q2q2-qqZ=3q,q2-3q~ 6, yielding firm on the interval [ql,q3].
is determined by the sign of the expression of the second firm rises for q2 <& and z(q,, q3) = 6 as the peak of the second 0
Proof of Lemma 3. Compare two triples of qualities (ql,q2,q3) and (ql, q2, ij3) where q1 < qz
open to the top quality firm, is to charge the same price as before, which gives rise to a higher profit. 0 One of the strategies feasible for the higher quality firm is to set the price equal to that of a lower quality competitor. Thus, the market share, and subsequently, the profits of the higher quality firm exceeds that of its rival in initial equilibrium. 0
Proof of Lemma 4.
Proof of Lemma 5. By Lemmata 1 and 2, it suffices to show that for any pair of qualities q, ij with qm
Substituting the expression for z(q,q) into (A.ll) we obtain
(A. 16)
By Lemma 1, there are two cases to consider: Y(%4) = 4m. (i)
By Lemma 2,
(i) y(q,ij) >q,;
and (ii)
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
711
=I2
4-4
469
(A.17)
36(q+2q)’
y > 0 implies that 5q > 24 and as 4 > q, we have
(ii)
1
1
“1 %i?
<
lo@
Substituting y=q,
=2
(A.18)
4(,,/% + l)2q < m’
<
in (A.lO) yields
@i-d2
,1<12(wd242<12
4a2
In order to demonstrate
(A.19)
4(4q- q)2’
that rrl
4-4
(A.20)
2(2&&)@ Simple algebra shows that (A.20) is equivalent to 3q+ 2q-4&>0. can easily verify that the latter inequality holds for all positive 4 > q. Proof
of Lemmata
substitution q2 =z(ql,
6-8. Let the pair of qualities q1
One 0
be given. Upon
(A.21)
For any given q3 the last expression, clearly, declines in ql, yielding the assertion of Lemma 6. Furthermore, subsituting q2 =z(ql, q3) in (A.12) we obtain
~3hz(q1,
q3),
q3)
=
z2(4,,&3,,h2 &-& (4&+2&G
4&
(A.22) ’
470
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
Let a > 0 be such that q3 = a2q,. Then (A.22) could be rewritten as 2(4cl+3)2 a-l
fl3(41,d41,43)> 43)=I ___
(4a+2)2
-*
(A.23)
4a
It is easy to verify that the last expression is increasing in a. Thus the profit of the top firm, H3(ql, z(q,, q5), q3), is increasing in q3 for any given qr and is decreasing in q1 for any given q3. 0 A.2. Proof of Propositions 3-7 Proof of Proposition 3. Proposition 1 implies that the entrant’s choice of quality, qE, lies between the incumbents’ selections, qA and qB. Furthermore, which is increasing and concave by Lemma 2, !?E=%/&ii=&iii in qM. 0 Proof of Proposition 4. To generate positive profits with a lower quality would charge a price that is higher quality rival, yielding &(qM) cc@(q,)
in equilibrium, no firm higher than that of its for all qME Q. Substitutwe get
p,*(q,)~I~
(A.24)
pE*(q,)=lS
(A.25)
4(2&L+&)’
2(2&+&A’ px(qM)_I(&-Jz)(4JG+3&) J&2&+JzA
(A.26) .
One can easily check that the price functions in (A.24)-(A.26) are increasing and concave, satisfying limnM,llmpi*(qM)= 0, i = A, B, E. Furthermore, and lim,,, mP!&) = I/g, lim,,, mp; (qM)= U4 lim,,, mpX(qM)= U2. q Proof of Propositions 5 and 6.
Again substituting q1 =qm, q2 =z(q,,,,qM) and q3=qM into (A.3)3A.5), we obtain the following expressions for market shares and market size
S&)=--L
4(2&+&Y
(A.27)
S. Donnenfeld and S. Weber, Vertical product differentiation
with entry
471
(A.28)
(A.29)
M*(&~S 4(2&+&)’
(A.30)
One can easily check that the market share of the lower quality incumbent is increasing and concave, whereas the market shares of the entrant and the high quality incumbent, as well as the total market size are decreasing and convex. Furthermore, lim4hI+oosg*(& = l/8, lim,,, msZ(qhl)= l/4, 0 lim4M+ms;l;(qM)= l/2 and lim4M+mM*(qn) = 7X/8. Proof of Proposition 7. Upon substitution of the price functions (A.24)(A.26) and the market shares (A.27)-(A.29) into (A.lO)-(A.12), we obtain the expressions for the profit functions of all three firms. Simple inspection reveals that the first derivative of each profit function with respect to the top quality qr,, is positive, whereas the second derivative with respect to the same argument is negative. Moreover, Propositions 4 and 5 imply that lim,, -t mG%,) = I’/643 lim,, + a, G(& hm,,, a, IlX(qJ = P/4. =Z2/16, 0
References Bernheim, B.D., 1984, Strategic deterrence of sequential entry into an industry, The Rand Journal of Economics 15, l-l 1. d’Aspremont, C., J.J. Gabszewicz and J.F. Thisse, 1979, On Hotelling’s stability in competition, Econometrica 47, 1145-1150. Dewatripont, M., 1987, The role of indifference in sequential models of spatial competition, Economics Letters 23, 323-328. Eaton, B.C. and R. Ware, 1987, A theory of market structure with sequential entry, The Rand Journal of Economics 18, 1-16. Gabszcwicz, J.J. and J.F. Thisse, 1979, Price competition, quality, and income disparities, Journal of Economic Theory 20,34Q-359. Gabszewicz, J.J. and J.F. Thisse, 1986, On the nature of competition with differentiated products, Economic Journal 96, 160-172. Gal-Or, E., 1985, First and second movers advantages, International Economic Review 26, 649-653. Gilbert, R. and X. Vives, 1986, Entry deterrence and the free rider problem, Review of Economic Studies 53, 71-83. Harris, C., 1985, Existence and characterization of perfect equilibrium in games of perfect information, Econometrica 53, 613-616. Hay, D.A., 1976, Sequential entry and entry-deterring strategies in spatial competition, Oxford Economic Papers 28, 2&257.
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S. Donnenfeld and S. Weber, Vertical product differentiation with entry
Hjorth-Andersen, C., 1988, Evidence on agglomeration in quality space, The Journal of Industrial Economics 37, 209-223. Hotelling, H., 1929, Stability in competition, Economic Journal 39, 41-57. Kats, A., 1989, Sequential entry in a model of spatial competition, Virginia Polytechnic Institute and State University Discussion Paper. Lane, W., 1980, Product differentiation in a market with sequential entry, Bell Journal of Economics 11,237-260. McLean, R.P. and M.H. Riordan, 1989, Industry structure with sequential technological choice, Journal of Economic Theory 47, 1-21. Palfrey, T.R., 1984, Spatial equilibrium with entry, Review of Economic Studies 51, 139-156. Prescott, E.C. and M. Visscher, 1977, Sequential location among firms with foresight, Bell Journal of Economics 8,378-393. Shaked, A. and J. Sutton, 1982, Relaxing price competition through product differentiation, Review of Economic Studies 49, 3-13. Shaked, A. and J. Sutton, 1983, Natural oligopolies, Econometrica 51, 1469-1484. Shaw, R.W., 1982, Product proliferation in characteristics space: The U.K. fertilizer industry, The Journal of Industrial Economics 31, 69-82. Swann, G.M.P., 1985, Product competition in microprocessors, The Journal of Industrial Economics 34,33-54. Vives, X., 1988, Sequential entry, industry structure and welfare, European Economic Review 32, 1671-1687. Waldman, M., 1987, Non-cooperative entry deterrence, uncertainty and the free rider problem, Review of Economic Studies 54, 301-310. Weber, S., 1992, On hierarchical spatial competition, Review of Economic Studies 59, 407425.
Journal
of Industrial
Organization
10 (1992) 449-472.
Vertical product differentiation
North-Holland
with entry
Shabtai Donnenfeld and Shlomo Weber* York University, North York, Ont., Canada Final version received
November
1991
This paper examines the non-cooperative behaviour of established firms that compete in product quality and prices while foreseeing the effects of their rivalry on the decisions made by the later entrants. We prove an existence and uniqueness of the equilibrium and show that the dominant firms engage in maximal product differentiation, i.e., select the highest and lowest technologically feasible qualities. Later entrant, however, always selects an intermediate quality. Moreover, we show that the equilibrium configuration is identical to that generated by perfect foresight equilibrium in a model of sequential entry. Furthermore, the profits earned by each firm are ranked in the same order as the qualities, implifying that the later entrant, despite being the last mover, ends up with higher profits and larger market share than one of the incumbents.
1. Introduction
Ever since the classic Hotelling ‘Principle of Minimum Differentiation’ has been introduced, a great deal of effort was devoted to study the effects of competition on product differentiation. Empirical evidence reveals that some industries are characterized by a slight degree of product differentiation whereas in other industries product differentiation is starker.’ From the theoretical point of view one can think of several reasons which explain the extent to which product differentiation will be observed: Diversity in consumers tastes, disparities in their incomes, timing of firms’ entry into the industry, product proliferation to deter further entry, etc. In this paper we wish to focus on the role that the sequence of entry has in determining the configuration of the firms’ product qualities. We embrace the paradigm of non-simultaneous entry; we consider the case where some firms are assumed to be the first to enter the industry and select qualities, thus enjoying the advantage of incumbency relative to firms which enter Correspondence to: Professor Weber, Department of Economics, University of York, 4700 Keele Street, North York, Ontario, Canada M3J lP3. *We are ihankful to two anonymous referees for their helpful suggestions. We are grateful to the Social Sciences and Humanities Research Council of Canada for its financial support. ‘See e.g. Hjorth-Andersen (1988) for analysis of the variance of product positions in quality space, based on the data for 175 German commodities, as well as Shaw (1982) and Swann (1985) for empirical evidence on product differentiation in the British fertilizer and microprocessor industry, respectively. 0167-7187/92/%05.00
0
1992-Elsevier
Science Publishers
B.V. All rights reserved
4.50
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
later. We capture this sort of competition in a model with several dominant players. Unlike the models with simultaneous entry Cd’Aspremont et al. (1979); Gabszewicz and Thisse (1986); Shaked and Sutton (1982)], where firms locate themselves in the product space at the same time, in our setup the estalished firms compete among themselves for positioning in the quality spectrum while foreseeing the effects of their competition on the decisions to be made by later entrants. Non-simultaneous entry encompasses several variants; firms may either enter sequentially, one after another, or initially several firms enter simultaneously and are viewed as incumbents by later entrants. The first case was originally examined by Prescott and Visscher (1977) and Hay (1976) in the context of spatial competition where the firms enter the industry and select location in a predetermined order. In both papers it is assumed that firms have perfect foresight, i.e., when a firm selects product location it is able to predict how its selection will affect the decisions of later entrants. Lane (1980) extended the original model by allowing for endogenous prices, which are determined as the outcome of a non-cooperative game. Different variants of sequential entry have been further studied by Bernheim (1984), Harris (1983, Eaton and Wooders (1985), Eaton and Ware (1987), Dewatripont (1987), Vives (1988), McLean and Riordan (1989), Kats (1989).2 Sequential entry models were explored and extended to encompass the effects of potential entry in industries with dominant firms which face the threat of potential entry. Palfrey (1984) studied spatial political competition of two established parties, which while competing for voters, foresee the effects of their competition on the location of a third party that enters later on. Palfrey demonstrates that, given the threat of entry, incumbents relax competition among themselves by choosing sharply distinctive platforms and positions in the political spectrum. Gilbert and Vives (1986) examined a similar model of structural hierarchy in the framework of an industry with multiple dominant firms, facing potential entry and competing in a homogeneous product market.3 In this paper we first consider an industry which consists of two established firms which compete in product quality while foreseeing the effects of their competition on the decisions made by a later entrant. We adopt the multiple dominant players (MDP) model, previously analyzed by Palfrey (1984) and Gilbert and Vives (1986). After proving existence and uniqueness of equilibrium of the associated non-cooperative game (Proposition l), we show that the equilibrium configuration of qualities in the MDP model, is not altered in perfect foresight equilibrium under sequential entry ‘Vives (1988) provides an excellent analysis of its welfare implications. 3The results of Palfrey and Gilbert and Waldman (1987), respectively.
review of the literature
on sequential
and Vives have been further
entry,
generalized
as well as an
by Weber (1992)
S. Donnenfeld and S. Weber, Vertical product dz@entiation
with entry
451
(SE) (Proposition 2). In both models the dominant firms, or the first two firms to enter the industry, select the extreme qualities that are technologically feasible, thus exhibiting maximal product differentiation among incumbents. This assertion could be supported by casual empiricism. Indeed, in some industries, like the personal computers there are early established firms, e.g., Compaq, that produce top of the line personal computers, which perform high-speed computations and provide high-resolution graphics images, whereas other early established firms, e.g., Compu-add, produce less advanced models which have a slower speed and lower resolution images. There are, in addition, later entrants in the industry, e.g., AST, which produce ‘intermediate’ quality PC’s with moderate speed of computation and reasonable resolution graphics. Our conclusions show that the threat of later entry further relaxes the degree of product competition. Indeed, in the model with simultaneous entry, Shaked and Sutton (1982) and Gabszewicz and Thisse (1986) showed that firms engage in product differentiation. However, the degree of product differentiation in their model is not maximal as one of the firms does not select the extreme quality. We also show that the entrant (in the MDP model), or the last firm to enter (in the SE model), always chooses an intermediate quality, that is, the entrant locates himself between the qualities chosen by the incumbents. The profits that the firms earn are ranked in the same order as their qualities. Interestingly, the entrant, despite being the last mover, ends up earning higher profits than one of the incumbents.4 This result differs signticantly from the observation of Palfrey that, in the context of horizontal product differentiation, the entrant ends up with a payoff (votes) which is lower than the payoffs of both incumbents. It highlights once more that competition in horizontally differentiated products differs in a significant way from competition in vertically differentiated markets. In the former mode of competition there is no particular advantage for a firm to be located to the right or to the left of the centre of the product spectrum. In the latter, being located farther up in the quality range is more desirable since profitability rises with quality. This stems from the consumers’ preferences since all of them evaluate a higher quality as more desirable and are willing to pay more for higher quality products. In the case of horizontally differentiated products, consumers vary with regard to their preferences for different varieties, according to the location of their most desirable variety in the product spectrum. The remainder of the paper is organized as follows: In section 2 we present the basic model and state our results on existence, uniqueness and characterization of equilibrium. In section 3 we examine the effects of technical change ‘In a different framework Gal-Or (1985), Eaton and Ware (1987), and McLean and Riordan (1989) have also shown that later entrants may enjoy higher profits than earlier entrants.
452
S. Donnenfeld and S. Weber, Vertical product dtfirentiation with entry
on equilibrium qualities, prices, market shares, market size and profitability of all firms. Section 4 contains the proofs of several results used for the proof of our main propositions that are presented in section 5. The technical derivations are relegated to appendix. We conclude the paper with a brief summary. 2. The model Consider a good that can be produced in a continuous array of quality levels, denoted by Q=[q,,,,qJ, where q,,, is the lowest and qM is the highest possible quality level. We assume that a consumer buys no more than one unit of the product from the quality array available in the market place. Consumers differ in their income; they will be identified by the income parameter t, which is assumed to be uniformly distributed over the interval [O,I]. A consumer’s utility is represented by U(x, q) where x denotes the quantity of the numeraire good with per unit price equal to one. Following Gabszewicz and Thisse (1979), we shall assume that the utility function takes the form u=zq,
where x = t-p is the consumer’s t disposable income devoted to the consumption of the numeraire good and p is the per unit price of the differentiated good. This specification of preferences implies that individuals with higher income enjoy quality improvements more than low income consumers. There are three tirms,5 each producing a single quality with zero costs. The firms compete for consumers by offering packages of price and quality (pi, qi), i = 1,2,3. Each consumer selects the best available package of price and quality that will maximize his utility without violating his budget constraint. That is, consumer’s t choice problem is to find the pair (pi, qi) that maximizes his utility, i.e., i(t) = arg max qi(t - pi).
If the resulting utility tqi(t)-pi(t) does not exceed zero, the consumer will stay out of the market. The sequence in which the firms make their decisions is the following: First, they decide (not all simultaneously) on the qualities to be offered. Once the triple of qualities, o=(ql,q2, q3), is chosen, the firms engage in price sAlthough we examine here a three firms industry, the analysis and most of the conclusions we draw hold for an industry with any finite number of firms.
S. Donnenfeld and S. Weber, Vertical product di$erentiation with entry
453
competition which generates a unique Nash equilibrium (p1,p2,p3), where pi =pi(@, i= 1,2, 3.6 In equilibrium the profits of the ith firm are
The firms differ in the order in which they enter the industry and select quality. The early entrant may have a strategic advantage (or disadvantage) of incumbency relatively to later entrants. In order to examine the various facets of this asymmetry we consider two models: (i) The multiple dominant players (MDP) model: There is a hierarchical structure in which two firms enter first whereas the third firm enters later. The first two incumbents compete with each other while foreseeing the repercussions of their rivalry for the entrant. The equilibrium of this game is called a MDP equilibrium [cf. Palfrey (1984) and Weber (1992)]. (ii) The model of sequential entry (SE): All three firms enter sequentially. When a firm decides to enter the industry, it takes into account the choices already made by the established firms and correctly anticipates the prolitmaximizing decisions of the firms yet to enter. The equilibrium notion in this model is called a perfect foresight equilibrium [cf. Prescott and Visscher (1977)]. 2.1. The MDP model There are two incumbent firms and one entrant. The incumbents play a Nash-Cournot game in qualities and behave as Stackelberg leaders with respect to the entrant by foreseeing his profit-maximizing response. Once all three qualities have been chosen, the firms compete in prices. The equilibrium prices give rise to the firms’ payoff functions, which are depicted in (1). By using backward induction, these payoff functions are used to determine the firms optimal choices of qualities. Since the ranking of the chosen qualities is not predetermined, we do not know ex ante which of the three firms will select the low, intermediate or high quality. We shall denote the incumbent firms by A and B, the entrant by E and their choices of qualities by qA, qB and qE, respectively. Since we do not impose any ranking on the firms’ quality choices, the assignment of the numbers 1,2,3 in eq. (1) to firms A,B,E cannot be ascertained before the game is played. Before proceeding with the formal definition of the game, we examine more closely the behaviour of the entrant. For a given pair of the incumbents’ quality choices, qA and qB, we shall denote by RE(q,,q,) the set of the entrant’s best responses %ee Shaked and Sutton (1982) for a similar approach with regard competition. They however assume that all firms move simultaneously.
to quality
and
price
454
S. Donnenfeld and S. Weber, Vertical product differentiation
Fig.
with entry
I
Suppose that qm< qB< qA< &,; firm E may consider entry in three distinctive regions: Below qB, between qB and qA and above gA (see fig. 1). It will be shown later in Lemmata 1-3, that the entrant’s profit function n(qA, &, .), associated with each of the three regions, is continuous and strictly quasiconcave. This guarantees the existence and uniqueness of the entrant’s best response in each of the regions. Moreover, the entrant’s highest attainable profits in region II are always higher than the local maximum in region I (Lemma 5). However, the ranking of the maximal profits in regions II and III cannot be ascertained. Thus, the set of the entrant’s best responses consists of the peak in region II or/and the peak of region III, where the latter, by Lemma 3, is equal to qhl. If E is indifferent between these two peaks, we assume, as is customarily done, that the entrant picks either of his best responses with equal probability.7 We are now in position to formally define a game I’, whose Nash equilibrium represents a MDP equilibrium in our model. ‘See Dewatripont (1987) for a discussion responses on the set of equilibria.
on the effect of multiplicity
of the entrant’s
best
S. Donnenfeld
and S. Weber, Vertical product differentiation
with entry
455
Definition 1. The game r is defined as follows: The players are the firms A,B and the strategy set of both players is Q; for each pair of strategies (qA,qB) the payoff of player I‘=A,B fi(qA,qB), is given by
where ITI stands for the cardinality of the set T. Definition 2. A pair (qA,qB) is called a MDP equilibrium (MDPE forth) if it constitutes a Nash equilibrium of the game r.
hence-
In the proposition below we state the main result of this paper. The proof of the proposition is presented in section 4. Proposition I. There is a unique (up to a permutation) MDPE (qA,qB). In this equilibrium the incumbents are engaged in maximal product dtfferentiation, i.e., qA=qM and qB=qm, while correctly anticipating the entrant’s location between them, qA > qE> q,,. Intuitively, the incumbents offer two distinctive products (different qualities) in order to relax the intensity of price competition. For any given pair Of qualities chosen by incumbents, (qA,&), as long as &, >qA> qB, the entrant may select either qE> qA or qA> qE> qB. (It Will be shown that qE< qB is always an inferior choice.) If the entrant finds it optimal to select qE> qA, then firm A anticipates the entrant’s choice and preempts him from capturing the high end of the quality range by setting qA=qM. If entrant prefers qA> qE> qB then firm A would be better off relaxing the price competition and choosing qA= qhl from the outset. Thus, in both cases qA=qhl. As the entrant will choose a quality qE to be located between the qB 2 qrn, selections of the incumbents. However, the low quality incumbent B, foreseeing the entrant’s location in the middle, finds it optimal to select the lowest feasible quality by setting qB=qm. By doing so firm B relaxes the intensity of price competition with the entrant. The hierarchical structure of the game generates incentives for stronger product differentiation among established firms than that generated in an environment where product selection decisions are made simultaneously by all firm~.~ Furthermore, it is noteworthy that this model suggests a different reason for the possibility that earlier movers may have a strategic advantage ‘Shaked and Sutton (1982) and Gabszewicz and Thisse (1986) examined the case where two firms move simultaneously and in equilibrium offer distinct qualities. We however show that under the non-simultaneous entry the firms engage in starker product differentiation.
J.I.O.
E
456
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
or disadvantage relative to later movers. Here, the high quality incumbent, who chooses product quality before the entrant does enjoy a strategic advantage which is reflected in higher profits. The low quality incumbent has a strategic disadvantage relative to the entrant; he has to differentiate his product from that provided by the other incumbent and thus opens a gap which will later be exploited by the enrant. Unlike the high quality incumbent, his low quality counterpart cannot preempt the entrant and thus ends up with lower profits. Preemption is not viable because profitability rises with product quality.
2.2. Sequential entry The MDP model considered in the previous subsection, envisioned an environment in which two incumbents enter early but simultaneously. We now examine a different sequence of entry where all firms enter the industry in a purely sequential manner. The equilibrium concept that we employ is that of a perfect foresight equilibrium (PFE) [Prescott and Visscher (1977)]. We shall demonstrate that replacing the entry sequence used in the MDP model with the assumption of sequential entry does not alter the equilibrium outcome; the firms choose the same strategies as in MDP equilibrium. There are again three firms, denoted A, B, E, that enter the industry sequentially (in this order) and select qualities from the interval Q = [qm, qM]. Each firm takes into account the choices of the firms that have already entered and anticipates the optimal selections of quality by firms yet to enter. Specifically, given the choices of the first two firms, A and B, the last entrant E chooses the quality which maximizes its profit. Firm B makes its profit-maximizing decision, taking as given the location of firm A and anticipating the choice of firm E. Firm A behaves as a Stackelberg leader with respect to the other two entrants, B and E, and anticipates their profit-maximizing responses. The question, which we address in this subsection, is whether A’s leadership position yields an additional gain relatively to its role of a Nash-Cournot player in the MDP model. Proposition 2 below, provides a negative answer to this question. It shows that equilibria in both models give rise to the same set of equilibrium selections of quality. Under the assumption of sequential entry, we use backward induction to determine the equilibrium strategies. Suppose that the first entrant, A, chooses quality qA. Then, assuming profit-maximizing behaviour of the entrant, we first define the set of the best responses of the second entrant firm B, RB(qA), to a given choice of the first entrant qA
S. Donnenfeld and S. Weber, Vertical product dtfirentiation with entry
457
We shall show later that this set is always nonempty and consists of a finite number of points. If the best response of B is not unique, we assume, as before, that firm B selects either of its best responses with equal probability. Thus the payoff of firm A is defined as the average over the set of the best responses of firm B. Formally, for a given qA, the payoff function of firm A, g*(& is defined by (3) We now define a concept of perfect foresight equilibrium (PFE). A triple (&, qg, qg), is called PFE if:
Definition 3.
(i) g(&)
tii) RB(&i)
&kA) =
vqA
E
Q;
&
(iii) R(qX, 4;) = qg. Condition (i) states that, correctly anticipating the actions of the other two firms, firm A maximizes its profit by choosing 41. According to conditions (ii) and (iii), qi is the unique best response of firm B to q%, whereas qg is the unique best response of firm E to the choices of the first two firms, qX and qg. Our next result, the proof of which is presented in section 4, implies the existence and the uniqueness of a PFE. Proposition 2.
There
exists a unique PFE
(qi,qg,qg).
Moreover,
qi=qMyI)
q$ = qm and 4: = z(qX, qg).
This proposition shows that, although in a model with sequential entry, firm A enjoys strategic advantage, being a Stackelberg leader rather than a Nash-Cournot player with respect to firm B, the equilibrium qualities of the first two firms are identical to those chosen in the MDPE. The reason is that in MDPE firm A ends up producing the highest quality, and this choice cannot be improved upon even if firm 1 is endowed with the Stackelberg leadership. Thus, in a model with sequential entry, firm A also chooses the highest quality qhl and firm B replies by selecting the lowest quality qm as its best response to the choice of firm A, which yields the same set of equilibrium strategies as in MDP equilibrium.
3. Comparative statics: Technical change We now examine the situation where, as a result of technical change, firms
458
S. Donnenfeld and S. Weber, Vertical product dzfirentiation with entry
are able to expand the quality spectrum Q to qualities higher than the original top quality qM. Indeed, in the previous section we have shown that the equilibrium quality that each firm chooses in MDP equilibrium, q;, i= A, B, E, is a function of the exogenously determined boundaries of the quality range Q = [qm, qM]. We assume now that the lower bound of Q, q,,,, is fixed, but the top quality, qhl, may vary and can take any value from the interval Q =(q,, CO). Thus, for any qm E @ and each i=A, B, E, we write the equilibrium qualities, prices, market shares and profits as q:(qJ, pT(qJ, $(q,) and ZT:(q& respectively. The properties of these equilibrium functions are examined in Propositions 3-7, the proofs of which are relegated to appendix. From Proposition 1 we know that, for given boundaries of the quality range the incumbents select the extreme positions, qA=qM and qB =qm. For the entrant we find the following. Proposition 3. The entrant’s equilibrium quality function qg( *), is strictly increasing and concave on the interval Q. Moreover,
4;1;(4d > &hf) > qB*(ch)forall qM E Q. In the next proposition we study the effect of a change in the upper bound of the quality spectrum on the equilibrium price functions. Proposition 4. For each i=A, B,E, the equilibrium price function pT( .) is strictly increasing, concave with lim 4M+4m pZ(qJ =O. Moreover, pB*(qr,J< pz(qJ
The effects on the firms’ equilibrium market shares are more intricate. Proposition 5. The market share of the low quality incumbent sg*(*) is strictly increasing and concave, whereas the market shares of the entrant and the highquality incumbent, .sg( .) and sX( .), are strictly decreasing and convex on the interval Q. Moreover, sg*(qU)
Technical change adversely affects the market size; the total number of consumers who purchase the product from one of the three firms decreases. Proposition 6. In equilibrium, the market size, M*(q,) strictly decreasing and convex on the interval 0.
= Z(Ci=A, B,e sT (qhl)), is
Finally, as expected, technical change enhances the profits of all firms. The
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
459
90 Fig. 2
9”
90
Fig. 3
rate of improvement of profitability is a decrasing function of the rate of technical change represented by the shift in the upper bound of the quality spectrum. Proposition 7. The equilibrium profit functions, Ilf( .), i= A, B, E are strictly increasing and concave on the interval 0. Moreover, II: < lI,*(qM) < IIX(qJ for all qME Q and there exist ZIlg
i=A,B,E. The properties of the equilibrium qualities, prices, market shares, market size and profit functions are portrayed in figs. 2-5. The values that the equilibrium functions approach in the limits, i.e., when qhl+qm and qhl+oo are presented in tables 1, 2. Intuitively, as a result of technical change, reflected by a shift in the upper limit of the quality spectrum, qm, the array of qualities that firms can produce is enlarged. From Proposition 1 we know that for any given lower
460
S. Donnenfeld and S. Weber, Vertical product d@erentiation with entry
GM
q0
Fig. 4
and upper bounds (qm,qM) the incumbent firms will find it optimal to select the extreme qualities and thus entails a more extreme degree of product differentiation relative to the situation that prevailed before the change occurred. This entices the entrant to move up in the quality spectrum, following the high quality incumbent, as portrayed in fig. 2. Taking into account that the consumers’ willingness to pay increases with income, the high quality incumbent and the entrant are able now to set higher prices. The higher prices associated with higher qualities enable the low quality incumbent to set a higher price as well. (The equilibrium price functions are portrayed in fig. 3.) As the low quality incumbent raises his price without changing the quality of its product, it induces consumers with lowest willingness to pay to exit the market. These consumers would have ended up with a negative surplus if they were to purchase a product of any quality. This explains the overall shrinkage of market size as portrayed in fig. 5. Furthermore, the low quality incumbent, although losing some of his original clientele from the low end of the income range, ends up serving buyers with intermediate levels of income. These consumers, previously served by the
461
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
Fig. 5
Table Firms
Prices (&p _ (&/z
Incumbent
Entrant
Incumbent
A
E
B
1
Market 4qp
share + 3qzz
4(2qYZ + 422)
4(2&Z + qitj2)
q1tj2_ q&/z 2( 2qg2 + qltjZ)
432 + q:2 2(2# + qrh/Z)
(qltj2- qZZ)(4qtiZ + 3qZZ) 4qtiZ(2q$2 + 422)
4(2#
qti2 + qZ2)
Profits IZ(qiZ -q;2)(4qa*
+ 3qZZ)2
16qM(2qiz + q:‘)’ WI, - %J 4(2qgz + qi2)2 {2qb/2(qgz -qz2) 16(2qa2+q:2)2
entrant, prefer now the low quality good qm as the entrant’s upgraded quality qE and higher price makes his product less attractive to them. In addition, the rise in the price of the high quality incumbent enables the entrant to lure away some of the consumers previously served by the high quality incumbent. The detailed analysis of these trends yields an increase in the market share of the low quality incumbent and a decline of the market share of the
462
S. Donnenfeld and S. Weber, Vertical product differentiation with entry Table 2 Prices
Firms
&+qm
Incumbent
Entrant
Incumbent
0
A
Market dK+a
dhl--%
share
Profits bhl-‘~
&%l
&.+~
I
5
E 1 8
B
entrant and the high quality incumbent, as portrayed in fig. 4. Finally, the profits of all firms rise as a result of technical change: The low quality incumbent enjoys a higher market share and a higher price even though his quality is unchanged. For the entrant and the high quality incumbent the beneficial effect of higher prices outweighs the negative effect of the reduced market share.
4. Preliminary
results
In this section we present lemmata, the proof of which is relegated to the appendix, We shall first examine the properties of the profit functions ni(41, q2, q3), i= 1,2,3, defined in (1). This will enable us later to establish the existence and uniqueness of MDPE and PFE. Assume, without loss of generality, the the qualities, chosen by the firms, are such that q1 < q2 < q3. In the first three lemmata we determine, given the rivals’ qualities, the peak of each profit function. Lemma
1.
II,( .,q2,q3)
For each pair (qz, q3) with q,,,< q2
the point y(q,, qd E CqmYqd. Lemma 2. For each pair (ql, q3) c Q with q1 < q3, the function I12(ql, ., q3) is strictly quasi-concave in the interval [ql,q3] with the peak at the point z(q1,q3)E(q1,q3) where the function z(., .) is strictly increasing in both arguments. Lemma 3. For each pair (ql, q2) with qmSql < qZ < qm, the function Z13(ql,q2, *) is strictly increasing on the interval [qz,qJ. Thus, its peak is qM.
S. Donnenfeld and S. Weber, Vertical product dzyerentiation with entry
463
Lemmata 3 and 4 are slight modifications of results obtained by Shaked and Sutton (1982) and presented here for the sake of completeness. Specifically, Lemma 4 shows that for any given triple of qualities, the ranking of profits is identical to that of qualities, so that a higher quality firm always earns higher profits than that of its lower quality rivals. Lemma 4. For inequalities hold
any triple (ql,q2, q3) t Q with q1
The next results shows that for given quality chosen by rivals, a firm could achieve higher profits by choosing a quality that lies between the locations of its competitors, rather than below this range. Lemma 5.
For each pair (ul, u2) c Q with u1 < u2, we have
max q,~u~U
n,(u, ui, u2) < max Z12(u’, v, u2) U’d”S&
In the following three lemmata we extend the analysis of the properties of the profit functions for cases that go beyond the standard Nash framework. We shall consider the situations where the firms, while deciding on product quality, take into account the impact of their selections on the rivals choices of quality. Lemma 6. Let q3 EQ be given. Then the function ZT,(q1,z(q1,q3), q3) is decreasing in q1 on the interval [q,,qJ. (Recall that z(q,, q3), defined in Lemma 2, is the optimal choice of a firm on the interval [ql, qs].) Lemma 7. Let q1 EQ be given. Then the function increasing in q3 on the interval [ql,qM].
I13(ql,z(qI,q3),
q3) is
Lemma 8. Let q3 E Q be given. Then the function decreasing in q1 on the interval [qm,q3].
Z13(ql, z(q,, q3), q3) is
Lemmata 68 deal with the situation where two incumbents, one producing the higher and the other the lower quality, correctly anticipate the entrant’s optimal response in the area between their product choices. Lemmata 6 and 7 provide the basis for the sharp product differentiation between two incumbents: The higher quality firm would be better off by raising its quality whereas the lower quality firm would gain by lowering its product quality. Lemma 8 shows that the principle of maximal differentiation
464
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
is incentive-compatible: The high quality firm is better off when its rival moves away towards the lower bound of the quality spectrum qm.
5. Proof of Propositions
1 and 2
We shall first state the lemma used in the proofs of the propositions. Lemma 9.
For each qA>qM
qM =
a% max fB(qA, qB). 4A54B5qM
Indeed, in game r let one of the incumbents, say A, choose a quality different from the top quality, i.e., qA
Let qA< qM be given. We shall show that no qB,qA< choice. Let f* = fB(qA, qM)= be the B’s optimal qB
(i) RE(q,, &) = Z(q,> qB)* Lemma 7 ITB(qA>
qB,
z(qA>
qB))
Then,
fB(qA,
Ch)
qM> z(qA,
=
nB(qAv
qM))
qB,
Z(4A,
qB))
and,
by
(4)
=f*.
Thus firm B would be better off by offering qM. (ii)
RE(qAyqB)
=
qM-
Then,
fBhA,
qB)
=
nB(qAy
qB? qM))
and,
by
z(qA,
qB))>
Lemmata
2
and 4 nB(qA,
qB> qM)
5
nB(qA,
hA>
qB)>
hl)
<
nB(qA?
qM,
(5)
and again, the choice of qM dominates qB. (iii) RE(qA,qB)= {gM,Z(q*, qB)}. In this CaSe fB(qA,
qB)
=i{nB(qA,
qBv z(qA,
qB))
+
nB(qA,
qB,
hl))*
(6)
However, (4) and (5) imply that each of the expressions on the right side of (6) is less than f *, yielding fB(qA, qB) < f*, which completes the proof of the lemma. 0 Proof of Proposition 1.
Let (qA, qB) be a MDP equilibrium. Then qA#qB and, without loss of generality, let qA>qB. Then, by Lemma 9, qA=qM and
S. Donnenfeld and S. Weber, Vertical product differentiation
465
with entry
Lemma 6 implies that qB= qM. Thus, the pair (qhl,q,J is the only candidate to constitute a MDPE. Applying Lemmata 9 and 6 again, we conclude that this 17 pair is, indeed, a unique MDP equilibrium. Proof
Let (qA,qB,qE) be a PFE and assume, in negation, of Proposition 2. that qA
dq,) = flA(% If
qB >
qA
then,
by
4%n,qhll).
4m
Lemma
9,
qB =
(7)
qM,
yielding
&A)
=nA(qA3
qM? z(qA?
qM)h
BY
Lemmata 2-6 nA(qA,
qM,
z(qA,
qM))
qM,
z(qm, qMkcnB(qA,
qM, z(qA,
qM)h
(8)
But the last expression in (8) is the payoff guaranteed to A whenever he chooses qM and, hence, is equal to g(%,$), yielding &A)
qhl),
If E selects z(q& qB) then imply &A)
%m
qB> qM)
g(q,)
z(qM,
<17A(qM~
=Z7,(qA,
%n))
=dqM).
qB, z(qM,
qB,z(qA,
qB)),
qB))
%%,d
(9)
and Lemmata 7 and 8
(10)
If the entrant is indifferent between entry at z(qA,qB) and qM, then (9) and (10) yield the choice of qM as the dominant strategy for A. Furthermore, by Lemmata 2 and 6, firm B chooses qm, whereas the entrant’s choice is z(%i, 4m). q 6. Concluding
remarks
In this paper we examined the implications of non-simultaneous entry into an industry where firms compete in markets with vertically differentiated products. Two variants of non-simultaneous entry were analyzed: The multiple dominant players model - where two firms enter first and are viewed as incumbents by a later entrant; and the model of sequential entry where firms enter the industry one after another. Both models generate maximal product differentiation between the equilibrium qualities of the first two firms to enter. Furthermore, both models shed some light on the first
466
S. Donnenfeld and S. Weber, Vertical product dl@rentiation with entry
(last) mover advantage issue. We show that under certain circumstances, being the last mover may entail a strategic advantage. The model presented here does not take into account the cost of production. However, if variable costs rise slowly with quality, our main result regarding the degree of product differentiation still holds; if variable costs rise rapidly with quality it will only relax product differentiation among the established firms. Fixed costs, on the other hand, are more intricate since they may endow the incumbents with the ability to prevent entry. In this paper we wished to focus on the effects of entry on vertical product differentiation and the examination of entry deterrence in this context is left for future research. Appendix A A.1. Proof of Lemmata l-8
In this part of the appendix we provide an explicit derivation of the firms profit functions given in (1) of section 2. For this purpose we need first to determine the assignment of consumers to firms’ products. Assume now that the price triple (p1,p2,p3) constitutes a Nash equilibrium for given product qualities choices (q1,q2,qJ)cQ, where q1
41@1 -PA
=o,
(A.11
and for i= 1,2, 64.2)
where consumer t, is indifferent between quality q1 at price p1 and staying out of the market, whereas ti + 1, i= 1,2 is indifferent between quality qi at price pi and quality 4i+l at price Pi+l. From (A.l) and (A.2), we have 64.3)
t1 =Pl,
t*=P2c12-Pl~l h--q1
’
t, = P3q3 -P242
q3--q2
.
(A-4)
(A.9
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
461
Thus the market segement of the first firm s1 = t,--tI, of the second firm s,=t,-tt,, and of the third firm sg =I- t, and for i= 1,2,3 the associated profits are
64.6)
ni=p,Si.
Substituting for ti, i= 1,2,3 from (A.3)-(A.5) into (A.6) and using the first order conditions for profit maximization of (A@, we obtain the following expressions for the equilibrium prices
pl_ztq2-qlm3-q2) 24
(A-7)
’
p2=zh2-ql)tq3-q2)
p3=zq3-q2 2q3
(A-8)
’
A
+p$
(A.9)
q3'
where A=4q2q3-q,q,-2q,q,-q:. Upon substitution of (A.7HA.9) in (A.6) yields the profit functions associated with the equilibrium prices ~l(ql,q2,q3)=12h2-41)(43-q2)2q2
44’
(A.lO)
’
n2(ql,q2,q3)=12(q2-ql)tq3-q2)tq3-o12
7
A2
(A.ll)
(A.12) Lemmata 1-8, stated in section 3, summarize the properties of the profit function presented in (A.lO)-(A.12). We now turn to the proof of these assertions. Proof of Lemma 1.
By (A.lO),
an
1 =sgn(-qlt2q2+q3)+5q~-22q2q3}. sgn i aql I Thus if qm>y=(5q:
-2q2q3)/(2q2
+q3)
(A.13)
then the profit function of the first
468
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
firm is declining on the interval [q,,,, qJ, and if q,S y then y is the peak of the profit function of the first firm. Overall, y(q,, q3) = max {y, qm} is its peak on the interval [q,,,,qJ. 0 Proof of Lemma 2.
By (A.ll),
an,_
-w~(q~-qlqd(4qlq~-q~q~-2q~q~-q~)~. sgn i aqz I
(A.14)
As 4qlq2-qlq3-2q2q3-q4:<4q,q2-q,q2-22q2q2-qqZ=3q,q2-3q~
is determined by the sign of the expression of the second firm rises for q2 <& and z(q,, q3) = 6 as the peak of the second 0
Proof of Lemma 3. Compare two triples of qualities (ql,q2,q3) and (ql, q2, ij3) where q1 < qz
open to the top quality firm, is to charge the same price as before, which gives rise to a higher profit. 0 One of the strategies feasible for the higher quality firm is to set the price equal to that of a lower quality competitor. Thus, the market share, and subsequently, the profits of the higher quality firm exceeds that of its rival in initial equilibrium. 0
Proof of Lemma 4.
Proof of Lemma 5. By Lemmata 1 and 2, it suffices to show that for any pair of qualities q, ij with qm
Substituting the expression for z(q,q) into (A.ll) we obtain
(A. 16)
By Lemma 1, there are two cases to consider: Y(%4) = 4m. (i)
By Lemma 2,
(i) y(q,ij) >q,;
and (ii)
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
711
=I2
4-4
469
(A.17)
36(q+2q)’
y > 0 implies that 5q > 24 and as 4 > q, we have
(ii)
1
1
“1 %i?
<
lo@
Substituting y=q,
=2
(A.18)
4(,,/% + l)2q < m’
<
in (A.lO) yields
@i-d2
,1<12(wd242<12
4a2
In order to demonstrate
(A.19)
4(4q- q)2’
that rrl
4-4
(A.20)
2(2&&)@ Simple algebra shows that (A.20) is equivalent to 3q+ 2q-4&>0. can easily verify that the latter inequality holds for all positive 4 > q. Proof
of Lemmata
substitution q2 =z(ql,
6-8. Let the pair of qualities q1
One 0
be given. Upon
(A.21)
For any given q3 the last expression, clearly, declines in ql, yielding the assertion of Lemma 6. Furthermore, subsituting q2 =z(ql, q3) in (A.12) we obtain
~3hz(q1,
q3),
q3)
=
z2(4,,&3,,h2 &-& (4&+2&G
4&
(A.22) ’
470
S. Donnenfeld and S. Weber, Vertical product differentiation with entry
Let a > 0 be such that q3 = a2q,. Then (A.22) could be rewritten as 2(4cl+3)2 a-l
fl3(41,d41,43)> 43)=I ___
(4a+2)2
-*
(A.23)
4a
It is easy to verify that the last expression is increasing in a. Thus the profit of the top firm, H3(ql, z(q,, q5), q3), is increasing in q3 for any given qr and is decreasing in q1 for any given q3. 0 A.2. Proof of Propositions 3-7 Proof of Proposition 3. Proposition 1 implies that the entrant’s choice of quality, qE, lies between the incumbents’ selections, qA and qB. Furthermore, which is increasing and concave by Lemma 2, !?E=%/&ii=&iii in qM. 0 Proof of Proposition 4. To generate positive profits with a lower quality would charge a price that is higher quality rival, yielding &(qM) cc@(q,)
in equilibrium, no firm higher than that of its for all qME Q. Substitutwe get
p,*(q,)~I~
(A.24)
pE*(q,)=lS
(A.25)
4(2&L+&)’
2(2&+&A’ px(qM)_I(&-Jz)(4JG+3&) J&2&+JzA
(A.26) .
One can easily check that the price functions in (A.24)-(A.26) are increasing and concave, satisfying limnM,llmpi*(qM)= 0, i = A, B, E. Furthermore, and lim,,, mP!&) = I/g, lim,,, mp; (qM)= U4 lim,,, mpX(qM)= U2. q Proof of Propositions 5 and 6.
Again substituting q1 =qm, q2 =z(q,,,,qM) and q3=qM into (A.3)3A.5), we obtain the following expressions for market shares and market size
S&)=--L
4(2&+&Y
(A.27)
S. Donnenfeld and S. Weber, Vertical product differentiation
with entry
471
(A.28)
(A.29)
M*(&~S 4(2&+&)’
(A.30)
One can easily check that the market share of the lower quality incumbent is increasing and concave, whereas the market shares of the entrant and the high quality incumbent, as well as the total market size are decreasing and convex. Furthermore, lim4hI+oosg*(& = l/8, lim,,, msZ(qhl)= l/4, 0 lim4M+ms;l;(qM)= l/2 and lim4M+mM*(qn) = 7X/8. Proof of Proposition 7. Upon substitution of the price functions (A.24)(A.26) and the market shares (A.27)-(A.29) into (A.lO)-(A.12), we obtain the expressions for the profit functions of all three firms. Simple inspection reveals that the first derivative of each profit function with respect to the top quality qr,, is positive, whereas the second derivative with respect to the same argument is negative. Moreover, Propositions 4 and 5 imply that lim,, -t mG%,) = I’/643 lim,, + a, G(& hm,,, a, IlX(qJ = P/4. =Z2/16, 0
References Bernheim, B.D., 1984, Strategic deterrence of sequential entry into an industry, The Rand Journal of Economics 15, l-l 1. d’Aspremont, C., J.J. Gabszewicz and J.F. Thisse, 1979, On Hotelling’s stability in competition, Econometrica 47, 1145-1150. Dewatripont, M., 1987, The role of indifference in sequential models of spatial competition, Economics Letters 23, 323-328. Eaton, B.C. and R. Ware, 1987, A theory of market structure with sequential entry, The Rand Journal of Economics 18, 1-16. Gabszcwicz, J.J. and J.F. Thisse, 1979, Price competition, quality, and income disparities, Journal of Economic Theory 20,34Q-359. Gabszewicz, J.J. and J.F. Thisse, 1986, On the nature of competition with differentiated products, Economic Journal 96, 160-172. Gal-Or, E., 1985, First and second movers advantages, International Economic Review 26, 649-653. Gilbert, R. and X. Vives, 1986, Entry deterrence and the free rider problem, Review of Economic Studies 53, 71-83. Harris, C., 1985, Existence and characterization of perfect equilibrium in games of perfect information, Econometrica 53, 613-616. Hay, D.A., 1976, Sequential entry and entry-deterring strategies in spatial competition, Oxford Economic Papers 28, 2&257.
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S. Donnenfeld and S. Weber, Vertical product differentiation with entry
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