Quality and Quantity Competition in Multiproduct Duopoly Yi-Ling Chengy

Shin-Kun Pengz

Running head: Quality Choices in Multiproduct Duopoly JEL Classi cation: D21, D43, L11, L13

We are most grateful to three anonymous referees who generously o ered a number of insightful comments to improve this paper. We have also bene ted much from fruitful discussions with Takatoshi Tabuchi, Hong Hwang and Toshihiro Matsumura, which led to signi cant improvements in this paper. We are especially indebted to Takatoshi Tabuchi for his contributive comments and suggestions. Finally, we would like to thank conference participants at 2010 APET in Istanbul, SAET in Singapore, as well as seminar participants at IEAS and NTU. Financial support from Academia Sinica and the National Science Council, Taiwan, is gratefully acknowledged. The usual disclaimer applies. y Tunghai University, No. 181, Section 2, Taichung Port Road, Taichung, 40704, Taiwan. E-mail: [email protected] z Academia Sinica and National Taiwan University, 128 Academia Road, Section 2, Nankang, Taipei, 115, Taiwan. E-mail: [email protected] (corresponding author)

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Abstract: This paper proposes a Cournot model of two-stage competition to examine the patterns of vertical product di erentiation in a multiproduct duopoly. Firms simultaneously choose the number of products and their qualities at the rst stage, and compete in quantities at the second stage. We show that, when the xed setup cost of a product is high enough to result in a monopoly outcome, the monopolist always sells a single product. Moreover, in any equilibrium of a multiproduct duopoly, quality di erentiation between them will develop into a non-segmented pattern because each rm desires to avoid a strong e ect of cannibalization. The set of equilibria reveals the properties of quality di erentiation between multiproduct rms. In a multiproduct duopoly, the pro t from a high-quality product can be lower than that from a low-quality product. This nding sharply contrasts with the literature on single-product rms which nds the high-quality advantage. Keywords: multiproduct lines, Cournot competition.

rms, vertical product di erentiation, quality, product

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1. Introduction Most of the literature dealing with vertical product di erentiation, such as Bonanno (1986) and Motta (1993), assumes that each rm can produce only one product. However, multiproduct

rms abound in the real world. Firms in various industries produce multiple

quality-di erentiated goods. For example, Mercedes-Benz provides three di erent classes of sedans, namely, its C class, E class and S class, and BMW has its 3 series, 5 series and 7 series. In addition, both Lenovo and Sony supply around twenty kinds of laptops ranked in accordance with their central processing units and out ts.1 A casual survey of the empirical evidence also suggests that any two multiproduct oligopolies usually di erentiate the qualities of their products, and the patterns of their quality di erentiation can be classi ed into the segmented type and the non-segmented type. The former type implies that the products of one rm are always of higher quality and more highly-priced than those of the other rm, such as the sedans of BMW and Honda. All the sedans sold by BMW are more expensive and of higher quality than those of Honda. The latter type denotes completely di erent patterns where the products produced by the two rms have interlacing patterns of their prices/qualities. For example, the prices of the Mercedes-Benz C, E and S classes are, respectively, $32900, $52700 and $87800, while the prices of BMW's 3, 5 and 7 series are given as $33600, $45800 and $80300, respectively.2 The prices of their sedans are interlaced, and it is inferred that their qualities are interlaced and non-segmented. Other examples include the whiskey o ered by Johnnie Walker and Edrington, and the plasma televisions provided by Samsung and Sony. The non-segmented patterns of di erentiation between the multiproduct rms with

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similar technological capabilities are very common in reality. However, the related literature has not shed enough light on studying them. While the segmented type of quality di erentiation has been characterized by Champsaur and Rochet (1989, 1990) with a quality-then-price competition of multiproduct duopolists, the non-segmented one, so far, has not been captured and explained in the literature.3 Therefore, this paper intends to examine the equilibrium qualities as well as the number of products chosen by the multiproduct rms with identical/similar technological capabilities, such as Mercedes-Benz and BMW. Our main purpose is to characterize the patterns of vertical product di erentiation, and to reveal the properties of quality di erentiation between them. The contributive papers (Gal-Or 1983; De Fraja 1996; Johnson and Myatt 2003, 2006) examine a variety/quality-and-quantity competition between multiproduct rms.4 Most of them conclude that rms with equal technological capabilities will compete head-to-head by producing identical products/qualities, such as Gal-Or (1983), De Fraja (1996) and Johnson and Myatt (2003, Proposition 6).5 By contrast, Johnson and Myatt (2006) study a twostage game with a model of two-exogenous qualities, and show that the symmetric rms may choose di erent product lines for strategic reasons. However, because there are only two choices of product qualities and they are exogenously given, there is no discussion over the quality levels made by multiproduct duopolists and quality di erentiation between them. Moreover, although they reveal that there is a possibility of asymmetric equilibrium, where rms produce di erent products, they do not provide a worked-out example for a more comprehensive analysis. In the case of multiplicative preferences and uniformly distributed consumer valuations, they derive the complete equilibrium, but it is shown that rms will

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compete head-to-head by producing both qualities, i.e., all qualities in the domain (Johnson and Myatt 2006, Proposition 11). That is, there is symmetric equilibrium and no product di erentiation between the multiproduct rms. In previous studies, there were no restrictions on the number of products and no costs associated with building up a product line for a distinct quality. Accordingly, because increasing the number of products is costless, each rm provides a full product line (Johnson and Myatt 2006, Proposition 11). However, in reality, a rm usually restricts itself to produce less varieties rather than a full product line due to the limitation of its resources, and boosting the number of product lines often involves additional setup costs due to the requirements of the distinct or advanced facilities.6 Thus, this paper involves a xed setup cost for each kind of product; that is, a xed cost will be incurred each time a product is introduced.7 Moreover, since it is not the purpose of this paper to discuss product proliferation, we involve a restriction on the maximum number of products. For the simplicity of the analysis, we assume that each rm will not provide more than two products as long as there is a positive xed setup cost.8 In contrast to the symmetric outcome in Johnson and Myatt (2006, Proposition 11), we show that multiproduct rms will di erentiate the qualities of their products rather than providing identical qualities when there is a restriction on product proliferation. The set of equilibrium outcomes in this study is much richer compared to those in the literature, and it formalizes some patterns of non-segmented types of vertical di erentiation. Moreover, it reveals the properties of quality di erentiation in a multiproduct duopoly. The results sharply contrast with those of the literature on vertical product di erentiation when each rm can produce only one quality.

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In order to compare our result with that in the literature on vertical di erentiation, we employ a model a la Motta (1993) and generalize a standard model to multiproduct o erings while maintaining the remaining structure of the original model.9 We assume that the costs of quality improvement fall upon the variable costs, and specify a quadratic form for tractability as in Motta (1993), Cremer and Thisse (1994), Lambertini and Orsini (2000, 2001) and Brouhle and Khanna (2007), among others.10 Firms simultaneously choose the number of products and the qualities of their products at the

rst stage, and then

compete in quantities at the second stage. The choice of a game where rms make decisions simultaneously rather than sequentially is crucial in the derivation of the results. Some of the ndings are summarized as follows. First, compared with the multiproduct monopoly in the contributive literature (Mussa and Rosen 1978; Lambertini and Orsini 2000; Johnson and Myatt 2003), we consider a potential entry and show that, as the xed cost is high enough and this results in a natural monopoly, the monopolist always sells a single product.11 Second, in any equilibrium of a multiproduct duopoly, quality di erentiation between them will develop into a non-segmented pattern because each rm desires to avoid a strong e ect of cannibalization, which refers to the competition among the products provided by the same rm. Moreover, equilibrium outcomes depends on the xed cost of a product, and it is revealed that rms never produce more products as the xed cost is higher. Third, we nd that the pro t from a high-quality product can be lower than that from a low-quality product in a multiproduct duopoly. This nding is in sharp contrast to that of a high-quality advantage in the literature on single-product rms, such as Motta (1993) and Lehmann-Grube (1997), where each rm can produce only one product and they show that producing a higher quality product always brings a higher pro t.12 Fourth, while

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looking into the product di erentiation of neighboring products, we nd that the quality di erentiation of the two middle products in the equilibrium where both rms provide two products is quite large. In addition, product di erentiation in the high-quality market will be smaller than that in the low-quality one. Fifth, it is revealed that the impact of selling multiple products reduces the average quality. The remainder of this paper is organized as follows. Section 2 outlines the environment of the model. Section 3 characterizes the subgame perfect Nash equilibrium (SPNE). Section 4 provides a simple example to validate our main results. Section 5 discusses asymmetric patterns of quality di erentiation in reality. Section 6 concludes.

2. The Model In the economy, there are two rms, R = A; B, with equal technological capabilities, and each of them can produce at most two products of quality-di erentiated goods. The number of products provided by rm R is denoted by nR (

2); the quality, price and quantity of the

product indexed by s (= 1; 2) o ered by rm R are denoted by qrs , prs and xrs respectively.13 There is a continuum of consumers with di erent tastes for quality. Their willingness to pay for quality is distributed uniformly over the interval [0; ] with the density normalized to 1.14 Each consumer purchases one unit of the product either from rm A or B, or does not purchase at all. The utility of consumer

U =

8 > > < qrs > > : 0

prs

2 [0; ] is given by

if he buys the product s (= 1; 2) from rm R otherwise

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where qrs is the quality of the product s of rm R, and prs is the price of this product. This utility function implies that all consumers prefer higher quality at a given price, and the is willing to pay more for a unit quality.15

consumer indexed by a higher

Consider the demand for each product. Suppose that the total numbers of distinct products provided by the two rms are N (

4). According to the ranking of their quality

levels, we re-label these qualities qe1 ; qe2 ; :::; qeN where qe1 > qe2 > ::: > qeN , and their prices are also re-labled by pe1 ; pe2 ; :::; peN , respectively. Hence, marginal consumers indexed by

such that

ei iq

pei =

ei+1 iq

pei+1 are indi erent between purchasing quality qei at price pei

or quality qei+1 at price pei+1 . Any consumer with an index greater than qei+1 for all i = 1; :::; N eN Nq

i

i

will prefer qei to

1. In addition, the marginal consumer with an index

N

such that

peN = 0 is indi erent between buying quality qeN at peN and not buying at all, and any

consumer with an index greater than

N

will prefer to buy quality qeN than not to buy at all.

Accordingly, the demand x ei for the i-th highest quality qei can be expressed as follows.16 x ei =

8 > > > > > > < > > > > > > :

i

i 1

N 1

pe1 pe2 ; qe1 qe2

= i

= N

pei qei

=

1

1

peN qeN

pei qei 1

1

if i = 1.

peN qeN

pei pei+1 ; qei qei+1 peN ; qeN

if i = 2; :::; N

1.

(1)

if i = N .

In order to derive the equilibrium of a game where rms compete in quantities rather than in prices at the last stage, we invert the system of demand functions in (1) and thus obtain the following.

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Lemma 1 The inverse demand of (1) can be written as 8 > > > qei > > > > > < pei = qei > > > > > > > > : qeN

N P

j=1

qei (

qej x ej

i P

x ej )

j=1 N P

qeN (

for i = 1

j=1

N P

j=i+1

x ej )

qej x ej

for i = 2; :::; N

1

(2)

for i = N

Proof. See Appendix 1. We assume that the costs of quality improvement fall upon variable costs instead of xed costs, which means that xed costs are independent of quality. In addition, the production activities are fully additive so that the unit cost of products is independent of its quantities.17 The unit cost of quality improvement is a quadratic form and denoted by 2 c(qrs ) = qrs where

> 0 so that the marginal cost of quality improvement is increasing.

Furthermore, each product line involves a xed cost f > 0. The pro t of rm R denoted by R

is written as

R

=

nR X

[prs

2 qrs ]xrs

nR f

(3)

s=1

=

where

R

R

nR f

is the gross pro ts without subtracting the incurred xed costs. In addition, xrs

is the quantities of the product of quality qrs , and prs is the associated price which can be derived from (2) as all products are ordered by their qualities. For example, if rm A o ers two products of the highest and the second-highest qualities and rm B o ers a single product of the lowest quality, namely, (e q1 ; qe2 ; qe3 ) = (qa1 ; qa2 ; qb1 ) and (e x1 ; x e2 ; x e3 ) = (xa1 ; xa2 ; xb1 ), then

10 according to (2) the associated prices, (e p1 ; pe2 ; pe3 ) = (pa1 ; pa2 ; pb1 ), can be respectively written as

pa1 = qa1

(qa1 xa1 + qa2 xa2 + qb1 xb1 );

pa2 = qa2

qa2 (xa1 + xa2 )

qb1 xb1 ;

pb1 = qb1

qb1 (xa1 + xa2 + xb1 ):

(4)

Furthermore, if rm A o ers two products of the highest and the third-highest qualities and

rm B o ers the product of the second-highest quality, namely, (e q1 ; qe2 ; qe3 ) =

(qa1 ; qb1 ; qa2 ) and (e x1 ; x e2 ; x e3 ) = (xa1 ; xb1 ; xa2 ), then according to (2) the associated prices,

(e p1 ; pe2 ; pe3 ) = (pa1 ; pb1 ; pa2 ) can be respectively written as pa1 = qa1

(qa1 xa1 + qb1 xb1 + qa2 xa2 );

pa2 = qa2

qa2 (xa1 + xb1 + xa2 );

pb1 = qb1

qb1 (xa1 + xb1 )

(5)

qa2 xa2 :

Game Structure

Two rms, A and B, play a two-stage game. At the rst stage, they simultaneously choose the numbers of products nR and the qualities of their products, qrs . Moreover, a rm enters the market only if its pro t is strictly positive. At the second stage, they simultaneously decide the quantities of their outputs, having observed the numbers of products and qualities. In equilibrium, eighteen con gurations possibly arise. Excluding the con gurations which can be obtained by relabeling rms, we thus consider the following nine alternatives

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of vertically spatial con gurations:

f(a); (aa); (ab); (aab); (abb); (aba); (aabb); (abba); (abab)g;

(6)

where (a) is a single-product monopoly, (aa) is a two-product monopoly, (ab) is a singleproduct duopoly, and the three cases (aab), (abb), (aabb) are all referred to as segmentation because multiproduct rms produce neighboring qualities and thus the markets are vertically segmented. Moreover, (aba), (abba), (abab) are respectively referred to as sandwich, enclosure and interlacing. For example, if rm A produces two products of quality (qa1 ; qa2 ) and rm B produces a single product of quality qb1 such that qa1 > qa2

qb1 , this con guration is denoted by

(aab). Moreover, by substituting (4) into (3), the pro ts of rms in (aab) can be respectively expressed as

A (aab)

= [qa1 +[(qa2

B (aab)

= [qb1

(qa1 xa1 + qa2 xa2 + qb1 xb1 ) qa2 (xa1 + xa2 )

qb1 xb1 )

qb1 (xa1 + xa2 + xb1 )

2 qa1 ]xa1 2 qa2 ]xa2

2 ]xb1 qb1

2f;

f:

If rm A produces two products of quality (qa1 ; qa2 ) and rm B produces a single product of quality qb1 such that qa1

qb1

qa2 (but qa1 6= qa2 ), this con guration is denoted

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by (aba). By substituting (5) into (3), the pro ts in (aba) can be respectively expressed as

A (aba)

= [qa1 +[qa2

B (aba)

= [qb1

(qa1 xa1 + qb1 xb1 + qa2 xa2 ) qa2 (xa1 + xb1 + xa2 ) qb1 (xa1 + xb1 )

qa2 xa2

2 ]xa1 qa1 2 qa2 ]xa2 2 ]xb1 qb1

2f; f:

Similarly, the pro t functions of the other con gurations in (6) can be expressed in the same way.

3. Equilibrium Characterization As mentioned in the previous section, rms rst choose the number of products and associated qualities, and then decide on the associated quantities of outputs. We rst solve for the equilibrium qualities and quantities of products within each of the nine con gurations in (6). Then, in order to ensure that each of the derived equilibrium candidates can be selected as a SPNE of this two-stage game, we need to further check that both rms have no pro table deviation regardless of whether they alter the ordering of qualities, or alter the number of products and qualities. While looking into the two monopolistic cases, (a) and (aa), we can simply derive the equilibrium outcome of either case from the rst-order conditions of a monopolist that maximizes its pro t with respect to the quality and quantity of its products. The details of the derivations can be learned from Lambertini (1997).18 We thus omit the details, and summarize the outcomes of the two cases in Table 1. Note that they are presented in decimal

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gures to facilitate a comparison with the outcomes of the duopolistic ones. By examining the outcomes of the duopolistic cases, we follow the procedure of backward induction and rst solve the second-stage equilibrium of the rm's decision on quantities. By taking the numbers of products and product qualities as given, each duopolist maximizes its pro t

R(

) with respect to the quantity of its products, xrs . From the rst-

order conditions, we may derive the quantity of each product, xrs , as a function of the qualities qrs . In addition, by substituting the equilibrium quantities into the inverse demand in (2), the prices as well as the pro ts of rms,

R(

), are expressed as a function of the

qualities qrs . Then, by going back to the rst stage to solve the equilibrium qualities of the products provided, the best response correspondences of rm R, qr1 (q

r1 ; q r2 )

and qr2 (q

r1 ; q r2 )

are

characterized by the two rst-order conditions

@pr1 @xr1 @pr2 @ R = ( 2 q r1 )xr1 +(pr1 q 2r1 ) +( )x +(pr2 @qr1 @qr1 @qr1 @qr1 r2 @ R @pr1 @xr1 @pr2 = ( )xr1 +(pr1 q 2r1 ) +( 2 q r2 )xr2 +(pr2 @qr2 @qr2 @qr2 @qr2

@xr2 =0 @qr1 @xr2 q 2r2 ) =0 @qr2

q 2r2 )

(7) (8)

where xr1 , xr2 , pr1 and pr2 are the quantities and prices, respectively, derived from the optimal choices of the second stage and the inverse demand in (2), so each of them is a function of qualities. Note that, when rm R produces only a product, the values of qr2 , xr2 and pr2 in the above two equations are taken as zero. In addition, the qualities provided by the rms satisfy the speci cation of the con guration. For example, in (ab), all the values of qa2 , xa2 , pa2 , qb2 , xb2 and pb2 in equations (7) and (8) are zero, and the two qualities satisfy qa1 > qb1 . The equilibrium quantities in the

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second stage are derived as

2 2 ( 2qa1 + qb1 ) + (2qa1 4qa1 qb1

(xa1 ; xb1 ) =

qb1 )

;

qa1 [(qa1 2qb1 ) + ] 4qa1 qb1

(9)

By substituting them into the associated prices in (2), the prices can be written as

(pa1 ; pb1 ) =

2 qa1 [(2qa1

2 qa1 qb1 + qb1 ) + (2qa1 4qa1 qb1

2 2 + 2qa1 qb1 qb1 ) + qa1 ] qb1 ) ] qb1 [(qa1 ; 4qa1 qb1

(10) Substituting the above results into equations (7) and (8), we have

3 (24qa1 2 (4qa1

2 2 3 10qa1 qb1 + 4qa1 qb1 + qb1 ) 2 23qa1 qb1 + 2qb1 ) + (4qa1

2 (8qa1

2 2qa1 qb1 + qb1 ) =0

qb1 ) = 0

Solving the above two equations, we obtain the equilibrium qualities19

(qa1 ; qb1 ) = (0:369 ; 0:293 )

By substituting the equilibrium qualities into the equations in (9) and (10), the equilibrium quantities and prices are (xa1 ; xb1 ) = (0:219 ; 0:244 ) 2

2

(pa1 ; pb1 ) = (0:217 ; 0:157 ) Following the above steps to solve the rst-order conditions (7) and (8) of each case, we obtain the equilibrium outcomes of all cases.20 They are summarized in Table 1.21 It has been veri ed that each outcome satis es its second-order conditions, and is consistent with

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the assumption that rms do not cover the market. These equilibrium con gurations are the candidates for the SPNEs.

"Insert Table 1 approximately here"

According to Table 1, the equilibrium qualities are proportional to . This indicates that they decrease with the coe cient for the costs of quality improvement ( ) and increases with the coe cient , which is a constant proportion of the average willingness to pay for quality ( 2 ).22 In addition, the equilibrium prices are proportional to

2

; that is, the prices

are lower when the coe cient for costs ( ) increases. This is because the equilibrium quality is lower when the costs increase. Furthermore, the equilibrium quantities are in proportion to , which implies that they increase with the average willingness to pay for quality. Now, we are going to investigate which equilibrium con gurations in Table 1 are indeed the SPNEs. We must further check that in each equilibrium con guration both rms have no pro table deviation no matter by altering the ordering of qualities, or by altering the number of products and qualities. In the following analysis, we assume that any quality deviation is in proportion to

.

First, we nd that the equilibrium con guration (aa) in Table 1 can never be selected as the SPNE. This is because, if the xed cost f is small enough for rm A to produce two distinct qualities, it must be pro table for rm B to enter the market. Accordingly, the following proposition is established: Proposition 1 A multiproduct monopoly does not emerge in equilibrium. Proof. See Appendix 2.

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In comparison with the result of a multiproduct monopoly in Mussa and Rosen (1978), Lambertini and Orsini (2000) and Johnson and Myatt (2003, Proposition 1), Proposition 1 shows that a monopolist will not sell multiple products while a potential entry is considered.23 This is because, if the xed cost per product line is small enough such that a monopolist is willing to o er a second product, the potential competitor must enter the market since the pro t from his rst product is greater than the pro t from the second one of the monopolist.24 Note that, in the equilibrium con gurations (aab) and (aabb), there are corner solutions to qualities, so that we have qa2 = qb1 as shown in Table 1. However, in both cases, we nd that it is pro table for rm A to lower its quality qa2 such that qa2 < qb1 . Moreover, in regard to the other segmented candidate (abb), rm B has an incentive to deviate to a non-segmented pattern (bab). Thus, we establish the following proposition:

Proposition 2 The segmented type of di erentiation where all qualities produced by one rm are higher than those o ered by the other does not emerge when rms simultaneously compete in the number of products, qualities and then quantities.

Proof. See Appendix 3. Proposition 2 shows that no rm has any incentive to introduce a new product with neighboring quality. This is because each rm has an incentive to avoid a strong negative e ect of cannibalization. Cannibalization refers to a reduction in the market share of one product as a result of the introduction of a new product by the same producer. In other words, it implies the competition among the products provided by the same rm. In a segmented pattern, the competition between the products provided by the same rm are more serious because these products are next to each other. Thus, a multiproduct rm has

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an incentive to deviate to a non-segmented pattern to avoid the keen competition among its own products. Lastly, to ensure that the other candidates in Table 1, namely, (a), (ab), (aba), (abba) and (abab), can be selected as SPNEs of a two-stage game, we have to verify that no rm has a pro table deviation at the rst stage. Because the number of products and qualities are both chosen at the rst stage, we should check that there is no pro table deviation regardless of whether the rm alters the qualities of its products or alters both the number and qualities of its products. Moreover, a rm will only enter the market when its pro t is strictly positive. Formally, the three conditions for (ab) to be selected as a SPNE are

Single-product duopoly (ab). A (ab)

maxf

A (ab);

A (ba)g

A (ab)

maxf

A (aab);

A (ab)

> 0 and

B (ab)

and

A (aba);

B (ab)

A (baa)g

maxf and

B (ab);

B (ab)

B (ba)g:

maxf

B (abb);

B (bab);

B (bba)g:

> 0:

The rst one implies that both rms have no pro table deviation by altering their qualities. More speci cally, the former inequality means that, given qb1 = 0:293 , it is not pro table for

rm A to alter its qualities; the latter one is that, given qa1 = 0:369 ,

no incentive to alter its qualities, either. By the de nition of A (ab)

maxf

A (ab)g

A (ab)

maxf

A (ba)g

and and

B (ab) B (ab)

maxf maxf

B (ab)g, B (ba)g.

R (ab),

rm B has

it must hold that

so that we only need to check that That is, given qb1 = 0:293 , it is not

pro table for rm A to provide a product with qa1 < 0:293 ; given qa1 = 0:369 , it is not pro table for rm B to provide a product with qb1 > 0:369 .

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The second condition implies that neither rm has a pro table deviation by altering the number of its products as well as its qualities. That is, given qb1 = 0:293 , it is not pro table for rm A to provide two qualities regardless of whether qa1 > qa2 > 0:293

or

qa1 > 0:293 > qa2 or 0:293 > qa1 > qa2 . Similarly, the condition that, given qa1 = 0:369 , it is not pro table for rm B to provide two qualities regardless of whether 0:369 > qb1 > qb2 or qb1 > 0:369

> qb2 or qb1 > qb2 > 0:369 . In addition, the third/last condition ensures

that both rms can pro tably enter the market. The conditions for the other con gurations (a), (aba) (abba) and (abab) to be selected as a SPNE are placed in Appendix 4. Based on all of these no deviation conditions, which are in turn based on the Nash concept and entering conditions, we obtain the following proposition:25 Proposition 3 In a two-stage game where rms compete in the number of products, qualities and then quantities, we derive a SPNE depending on the range of xed costs as follows: 3

0:0016 , the top

(i) When 0 < f

rm, the

rm selling the highest quality, o ers

two products of quality (0:414 ; 0:173 ), and the other o ers two products of quality (0:386 ; 0:225 ) between those o ered by the top rm. /Enclosure (abba). 3

(ii) When 0 < f

0:00189 , the top rm o ers two products of quality (0:430 ; 0:239 )

and the other o ers two products of quality (0:379 ; 0:165 ) that are interlaced with those o ered by the top rm. / Interlacing (abab). (iii) When 0:0017

3

f

3

0:0031 , the top

rm sells two distinct products of quality

(0:424 ; 0:194 ) straddling the rival's unique product of quality 0:342 . / Sandwich (aba).

19

(iv) When 0:0028

3

3

f < 0:0175 . Both rms produce a single product. The associated

qualities are 0:0369 and 0:0293 . / Single-product duopoly (ab). (v) When 0:0168

3

3

f < 0:0370 , a single rm monopolizes the market by o ering a

product of quality 0:333 . / Single-product monopoly (a). (vi) When f

3

0:0370 , no rm enters the market.

Notice that Proposition 3 does not claim the uniqueness of the SPNE but presents a SPNE for this two-stage game, because the candidates (abba) and (abab) in Table 1 may not be unique solution in their con gurations.26 The equilibrium qualities and pro ts of the two rms for each equilibrium outcome in Proposition 3 can be found in Table 1, and they are respectively illustrated in Figures 1 and 2. In Figure 1, we nd that the types of equilibrium con gurations depend on the xed cost of a product and rms never produce more products as the xed cost per product is higher. Figure 2 indicates that the pro ts of rms are not monotonic with the xed cost. As the xed cost increases, the pro ts of rms fall but there are some jumps. It is revealed that the equilibrium derived by Motta (1993) is one of our equilibrium outcomes for a speci c range of xed cost, which is speci ed by (iv). There are other asymmetric equilibria of multiproduct duopolists which have not been found in the literature.

"Insert Figure 1 and Figure 2 approximately here" Three points are worth mentioning. First, in the multiproduct equilibrium, only the non-segmented patterns sandwich (aba), enclosure (abba) and interlacing (abab) may emerge.27 As mentioned in Proposition 2, none of the segmented patterns, (aab); (abb) and

20

(aabb), can be selected as a SPNE, because a multiproduct rm has an incentive to deviate to a non-segmented pattern to avoid the keen competition among its own products (i.e., the strong negative e ect of cannibalization) in the segmented pattern. Second, there are multiple equilibria in some ranges of the xed cost. For example, both the equilibrium (ab) and the sandwich equilibrium (aba) may emerge when 0:0028 f

3

3

0:0031 . However, according to Figure 2, both rms prefer the equilibrium (ab) to

(aba) because both of their pro ts are higher in the equilibrium of a single-product duopoly (ab). Thus, the equilibrium (ab) dominates the sandwich one (aba) in this area of multiple equilibria. Moreover, the interlacing equilibrium (abab) dominates the enclosure one (abba) in the entire range of the xed costs where the enclosure equilibrium may emerge. Therefore, all enclosure outcomes are dominated. Third, according to the sandwich (aba) and interlacing (abab) equilibrium in Table 1, we nd that the equilibrium pro t from the product associated with the highest quality is lower than that from the product associated with the second highest quality. Hence, we establish the following proposition:

Proposition 4 In a multiproduct duopoly, producing a higher quality does not imply a higher pro t.

This result is in sharp contrast to that of the high-quality advantage in the literature on single-product rms, such as Motta (1993) and Lehmann-Grube (1997), where each rm can produce only one product and it is concluded that producing a higher quality product always brings a higher pro t. However, Proposition 4 shows that the pro t from a high-quality product can be lower than that from a low-quality product in a multiproduct duopoly.28 This

21

is because the cannibalization e ect on the strategies of the rms may bene t a low-quality product. As shown in the two patterns, (aba) and (abab), because rm A would like to di erentiates the qualities of its two products to avoid the strong e ect of cannibalization, the product in the middle of them (that is, the product with the second highest quality) is able to acquire a large demand and enjoy a higher pro t.29 Moreover, according to Table 1, the pro t of rm A is lower than that of rm B in either of the enclosure (abba) or interlacing (abab) equilibria. This shows that the multiproduct rm that produces the highest quality product generates a lower pro t. Based on the interlacing equilibrium (abab), it can also be inferred that the rm that produces two qualities which are respectively higher than the two qualities of the other rm generates a lower pro t.

"Insert Table 2 approximately here" Table 2 summarizes the quality di erentiation between neighboring products and the average qualities in the market for each equilibrium con guration in Proposition 3. As shown in Table 2, the central segment of quality di erentiation in the interlacing (abab) equilibrium is 0:161 . It is quite large compared to the other two levels of di erentiation between neighboring products, which are 0:051 and 0:074 . This is because each rm would like to di erentiate the qualities of its own products from each other to avoid a strong cannibalization e ect. Accordingly, the two middle qualities become more di erentiated. Similarly, in the enclosure (abba) equilibrium, the di erentiation of the two middle qualities is much greater than the di erentiation of the other neighboring qualities. Moreover, in this con guration, it is implied that product di erentiation within a rm is greater than

22

that between rms. In the interlacing and enclosure equilibrium, we also nd that product di erentiation in the high-quality market (:051 =:028 ) is smaller than that in the low-quality market (:074 =:052 ).30 Intuitively, the product-di erentiation behavior is more costly in the highquality market than in the low-quality one because the marginal cost of quality improvement is increasing. According to the average qualities in Table 2, the average qualities in each of the equilibria (aba), (abba) and (abab) are lower than those in the single-product equilibrium (ab). That is, the impact of selling multiple products reduces the average quality provided in the market. This is because, when rms can produce multiple qualities, they provide more low-quality goods to capture the consumers with a low willingness to pay, and less highquality goods to discriminate more e ectively among the consumers with a higher willingness to pay. Accordingly, more lower quality and less higher quality products are produced.

4. Example In this section, we provide a simple example with three exogenously given qualities and with one of the two rms who is limited to sell only one quality at most. This example validates Propositions 1-4 and further clari es our main results. Let

= 1=2 and

= 10. Consider three given qualities fqH ; qM ; qL g = f7; 6; 2g, and

that rm A is an incumbent who can sell at most two of these qualities while the entrant, rm B, can sell only at most one of them. The two rms simultaneously choose the number of products and their qualities from the exogenously given qualities fqH ; qM ; qL g at the rst

23 stage, and simultaneously decide the quantities of their products at the second stage.31 All possible cases are classi ed and summarized as follows. First, there are three cases of a single-product monopoly where (na ; nb ) = (1; 0) (resp., (na ; nb ) = (0; 1)): fH; g; fM; g and fL; g (resp., f ; Hg; f ; M g and f ; Lg), which respectively denote that rm A (resp., B) sells a single product of quality qH , qM and qL while rm B (resp., A) is out of market. Because the quality is exogenously given, the monopolist's pro t in each case can be easily derived by maximizing its pro t in (3) with respect to the quantity of its product. Before subtracting xed costs, we get the gross pro ts in these three cases as below and the maximal one of them is

M axf

A (fH;

g);

A (fM;

g);

A f(L;

g)g = M axf

1183 1183 147 81 ; ; g= 16 2 2 16

73:94:

Accordingly, it is clearly optimal for a single-product monopoly to sell the quality qH , since these three cases have the same incurred xed costs (f ). Second, there are also three cases of a multiproduct monopoly where (na ; nb ) = (2; 0): fHM; g; fHL; g and fM L; g, which respectively denote that rm A sells two products of quality (qH ; qM ), (qH ; qL ) and (qM ; qL ) while rm B is out of the market. Similarly, solving the rst-order conditions of the monopolist maximizing its pro t in (3) with respect to the quantities of its two products, we derive the gross pro ts in these three cases as shown below, and the maximal one of them is

M axf

A (fHM;

g);

A (fHL;

g);

A (fM L;

g)g =M axf

1225 1253 153 1253 ; ; g= 78:31: 16 16 2 16

24

Accordingly, fHL; g is the optimal outcome of a multiproduct monopoly, since these three cases have the same incurred xed costs (2f ). That is, a multiproduct monopolist will sell the highest and the lowest qualities (qH ; qL ). Note that, to ensure that the multiproduct monopolist does not deviate to sell a single product, we must have 78:31 i.e., f

4:37. However, later we will show that, when f

2f

73:94

f,

4:37, rm B must have incentives

to enter the market, so that a multiproduct monopoly does not emerge in equilibrium as remarked in Proposition 1. Third, there are nine cases of a single-product duopoly where (na ; nb ) = (1; 1). Because the number and the qualities of their products are given, the pro ts of the two rms in each case can be easily derived by solving the two rst-order conditions of both rms who maximize their pro ts

R

in (3) with respect to the quantities of their products. The gross

pro ts of the two rms in these nine cases are summarized in Table 3.

"Insert Table 3 here" By looking into the pro ts in Table 3, the two outcomes fH; M g and fM; Hg are the Nash equilibrium under single-product duopoly.32 That is, one of them sells the highest quality qH and the other sells the middle quality qM in a single-product duopoly. Fourth, there are also nine cases of a multiproduct duopoly where (na ; nb ) = (2; 1). By solving the three rst-order conditions of both rms maximizing their pro ts in (3) with respect to the quantities of their products, the gross pro ts of two rms in these nine cases are as shown in Table 4.

"Insert Table 4 here"

25

In Table 4, the pro ts in box fHL; M g reveal that, in the (optimal) multiproduct monopoly fHL; g, if the xed cost is small enough for the multiproduct monopolist not to deviate to sell a single product, i.e., f

4:37, the entrant B must have incentives to

enter the market to provide qM and earn a positive pro t "32:87

f ". This validates that

a multiproduct monopoly can never be selected as a SPNE (Proposition 1). In addition, by looking into the pro ts in Table 4, the unique pure Nash equilibrium of a multiproduct duopoly is fHL; M g, which indicates that rm A sells the highest and the lowest qualities (qH ; qL ) and rm B provides the middle quality qM . This result validates that the segmented types of production di erentiation, like fHM; Lg and fM L; Hg, do not emerge in a multiproduct duopoly equilibrium (Proposition 2). To ensure that the four candidates fH; g (f ; Hg), fH; M g, fM; Hg and fHL; M g are indeed SPNEs, we further con rm no deviation conditions based on the Nash concept and positive pro ts for entry.33 Accordingly, a SPNE of this two-stage game is given by (i) The multiproduct duopoly fHL; M g is the SPNE when 0 < f

3:89, based on

the no deviation conditions below. A (fHL; M g) B (fHL; M g)

M axf =0

A (fHM; M g);

M axf

A (fHL; M g);

B (fHL; Hg);

A (fM L; M g)g

B (fHL; M g);

B (fHL; Lg)g

(ii) The single-product duopoly fM; Hg is the SPNE when 3:07

f < 34:17, based

on the no deviation conditions below. A (fM; Hg) B (fM; Hg)

M axf

A (fHM; Hg);

A (fHL; Hg);

A (fM L; Hg)g

>0

(iii) The single-product duopoly fH; M g is the SPNE when 3:89 on the no deviation conditions below.34

f < 34:17, based

26

A (fH; M g) B (fH; M g)

M axf

A (fHM; M g);

A (fHL; M g);

A (fM L; M g)g

>0

(iv) The single-product monopoly fH; g (or f ; Hg) is the SPNE when 34:17

f

0

M axf

B (fH; Hg);

B (fH; M g);

B (fH; Lg)g

Apparently, in this example, when f > 73:94, no rm will enter the market. The above SPNE shows that the types of equilibrium con gurations depend on the xed cost, f . Only fH; g (f ; Hg), fH; M g, fM; Hg and fHL; M g may emerge, which is in fact corresponding to the cases (a), (ab), (ba) and (aba) in Proposition 3. Moreover, as implied by Proposition 3,

rms never produce more products as the

xed cost is higher, and a

multiproduct duopoly emerges only when the xed cost is small enough. Furthermore, in the equilibrium of multiproduct duopoly fHL; M g, the gross pro t from the highest quality qH and the lowest one qL are respectively

eH =

eL =

1008337 35344 89205 8836

28:52 10:10

while the pro t of the middle quality, eM , is 32:87 which is clearly higher than the pro t of

the highest quality, eH . This result validates that a higher quality does not necessarily yield

a higher pro t (Proposition 4).

27

5. Discussion In contrast to the literature on multiproduct Cournot rms (Gal-Or 1983; De Fraja 1996; Johnson and Myatt 2003, 2006), this paper shows that, when the number of products produced by a rm is restricted for a certain reason, the multiproduct duopolists with identical technological capabilities will di erentiate their qualities. Moreover, quality di erentiation between them will develop into a non-segmented pattern. A large amount of empirical evidence supports the result of the non-segmented di erentiation between the rms with similar capabilities. For example, both Nikon and Canon have several classes of digital single lens re ex (SLR) cameras. Nikon's D7000, D700 and D3X are sold for $1199, $2699 and $7999, respectively, and Canon's 50D, 7D, 5D and 1D are provided at the prices $1099, $1699, $2499 and $6999, respectively. In addition, Sony Ericsson sells high-end cellphones which have higher quality than the high-end ones of Nokia, but it also sells low-end cellphones with quality lower than the high-end products of Nokia. Furthermore, the topmost sedan of BMW, the 7 series, has a higher price than Audi's foremost model, the A8, while Audi's A6 has a higher price than BMW's 3 series.36 However, the qualities are sometimes segmented as mentioned in Section 1. There are some cases of segmented di erentiation where one rm always provides products of higher quality than the other does, but they are not explained in this model. Thus, there is a possibility that they are portrayed by a case where rms have di erent technological capabilities as shown in Johnson and Myatt (2003, Proposition 7).37 Moreover, another possible reason is the presence of the positive externality on the costs of producing similar/neighboring qualities. As shown in the two previous sections, no segmented pattern emerges in equilib-

28

rium because each multiproduct rm has no incentive to produce products with neighboring qualities to avoid the keen competition among its own products (the cannibalization e ect). Therefore, if there is a positive externality on costs which reduces the cost of providing similar qualities and it is large enough to dominate the cannibalization e ect, it is intuitive that multiproduct rms may produce neighboring qualities and that their qualities will result in a segmented pattern.

6. Conclusion In this paper, we have analyzed the variety/quality-then-quantity competition when each rm can produce more than one product, in which the numbers of products, as well as the qualities and quantities of products are endogenously determined. The main contribution is that we present some equilibrium con gurations which have not been captured in the literature, and which formalize some non-segmented types of vertical di erentiation between the multiproduct rms in the real world. Compared to the ndings in the literature on single-product rms, the set of equilibrium outcomes in our study is much richer. The single-product equilibrium in Motta (1993) is only a special case in this study. In the equilibrium of a multiproduct duopoly, there are asymmetric outcomes with non-segmented patterns of di erentiation, such as sandwich, enclosure, and interlacing con gurations. The segmented patterns of di erentiation do not emerge because rms desire to avoid a strong cannibalization e ect. In addition, we nd that the producing a higher quality product does not imply a higher pro t, which in sharp contrast with the high-quality advantage of the literature on single product rms. Moreover,

29

the outcomes reveal the properties of the quality di erentiation between multiproduct rms. It is shown that the quality di erentiation of the two middle products in the equilibrium where both rms produce two products is much greater than that of the other neighboring products. This also implies that product di erentiation within a rm is greater than that between rms. Our ndings were obtained for a speci c model with several assumptions regarding the preferences of consumers and technology which are common in the related literature. This paper captures some equilibrium properties of competition between multiproduct rms and they sharply contrast with those in the literature on the vertically spatial competition. One drawback of this model is that the equilibrium outcomes do not comprehend the segmented con gurations. However, product qualities are sometimes segmented in reality. There are also some related extensions. Consider a positive externality on the production cost of producing similar qualities, or a non-uniform distribution of consumers' willingness to pay for quality. Besides, rms may compete in terms of prices rather than quantities as in the case of Champsaur and Rochet (1989, 1990).

References Brander, J. A., and J. Eaton. 1984. Product Line Rivalry. American Economic Review 74:323{334. Brouhle, K., and M. Khanna. 2007. Information and the Provision of Quality Di erentiated Products. Economic Inquiry 45:377{394.

30

Bonanno, G.. 1986. Vertical Di erentiation with Cournot Competition. Economic Notes 15:68{91. Champsaur, P., and J.-C. Rochet. 1989. Multiproduct Duopolists. Econometrica 57:533{557. Choi, C. J., and H. S. Shin. 1992. A Comment on a Model of Vertical Product Di erentiation. Journal of Industrial Economics 40:229{23. Cremer, H., and J.-F. Thisse. 1994. Commodity Taxation in a Di erentiated Oligopoly. International Economic Review 35:613{33. De Fraja, G.. 1996. Product Line Competition in Vertically Di erentiated Markets. International Journal of Industrial Organization 14:389{414. Doraszelski, U., and M. Draganska. 2006. Market Segmentation Strategies of Multiproduct Firms. Journal of Industrial Economics 54:125{149. Gabszewicz, J., and J.F. Thisse. 1979. Price Competition, Quality and Income Disparities. Journal of Economic Theory 20:340{359. Gabszewicz, J., and J. F. Thisse. 1980. Entry and Exit in a Di erentiated Industry. Journal of Economic Theory 22:327{338. Gal-Or, E. 1983. Quality and Quantity Competition. Bell Journal of Economics. 14:590{600. Gilbert, R. J., and C. Matutes. 1993. Product Line Rivalry with Brand Di erentiation. Journal of Industrial Economics 41:223{240. Johnson, J. P., and D. P. Myatt. 2003. Multiproduct Quality Competition: Fighting Brands and Product Line Pruning. American Economic Review 93:748{774.

31

Johnson, J. P., and D. P. Myatt. 2006. Multiproduct Cournot Oligopoly. Rand Journal of Economics 37: 583{601. Katz, M. 1984. Firm Speci c Di erentiation and Competition among Multiproduct Firms. Journal of Business 57:S149-166. Lambertini, L. 1997. The Multiproduct Monopolist under Vertical Di erentiation: An Inductive Approach. Recherches Economiques de Louvain 63:109{122. Lambertini, L., and R. Orsini. 2000. Process and Product Innovation in a Vertically Di erentiated Monopoly. Economics Letters 68:333{337. Lambertini, L., and R. Orsini. 2001. Network Externalities and the Overprovision of Quality by a Monopolist. Southern Economic Journal 67:969{982. 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. Motta, M. 1993. Endogenous Quality Choice: Price vs. Quantity Competition. Journal of Industrial Economics 41:113{131. Mussa, M., and S. Rosen. 1978. Monopoly and Product Quality. Journal of Economic Theory 18:301{317. Peng, S.-K. and T. Tabuchi. 2007. Spatial Competition in Variety and Number of Stores. Journal of Economics & Management Strategy 16:227{250. Schmidt, R. 2006. On the Robustness of the High-Quality Advantage under Vertical Di erentiation. Journal of Industry, Competition and Trade. 6:183{193.

32

Shaked, A., and J. Sutton. 1982. Relaxing Price Competition through Product Di erentiation. Review of Economic Studies 49:3{13. Shaked, A., and J. Sutton. 1983. Natural Oligopolies. Econometrica 51:1469-1483. Tirole, J. 1988. The Theory of Industrial Organization. Cambridge, MA: The MIT Press.

Notes 1

IBM sold its laptop division to Lenovo in 2004.

2

These prices are from their o cial websites; they denote the suggested retail price of each class with

basic equipment. 3

Champsaur and Rochet (1989) assume that the duopolists each provide a range of connected qualities

and compete with each other in prices. They show that, there is a gap between the two intervals of qualities to avoid the keen competition. 4

Katz (1984), Champsaur and Rochet (1989) and Gilbert and Matutes (1993) note that multiproduct

rms engage in price competition. 5

Johnson and Myatt (2003) show that multiproduct rms may not sell identical products when their

technological capabilities are not equal. 6

While we incorporate a xed setup cost into the case in Johnson and Maytt (2006, Proposition 11), it is

shown that, when the xed cost is high enough, rms no longer provide a full product line. Either rm has an incentive to restrict itself to produce only the high-quality product. 7

In the horizontally di erentiated literature on multiproduct rms, Brander and Eaton (1984), Doraszelski

and Draganska (2006), and Peng and Tabuchi (2007) have also speci ed a xed cost for a variety in production technology. 8

This assumption will rule out certain con gurations with more products. However, involving more

products will not derive much additional intuitions, but this will make the analysis much more complicated

33

because the number of the possible outcomes will increase greatly. For instance, we will need to work on 34 possibilities if each rm may produce one more product, i.e., three products at most. 9

In Motta (1993), there are multiplicative preferences and uniformly distributed consumer valuations.

Both rms have identical technologies of production. However, each of them is assumed to provide only one product. 10

Mussa and Rosen (1978), Lambertini and Orsini (2000, 2001), Gal-Or (1983) and De Fraja (1996) and

Johnson and Myatt (2003, 2006) assume that the quality improvement is associated with the variable costs, while Shaked and Sutton (1982, 1983) specify that the quality improvement is associated with the xed costs. 11

Without considering potential entry, Lambertini and Orsini (2000) present the conditions for a single-

variety monopolist to carry out process and product innovation to produce two varieties. Johnson and Myatt (2003) show that a monopoly will provide a single product if the average cost of quality is decreasing. If not, it will o er multiple products. 12

Note that Schmidt (2006) assumes single-product rms and a fully covered market, and he shows that

the high-quality advantage may not be persistent under some speci cations for the utility or the unit costs of quality, such as a convex unit cost-quality relation. 13

This analytical technique is similar to the approach used in Peng and Tabuchi (2007) where they propose

a location-then-variety competition for a multiproduct and multistore duopoly. 14

This paper focuses on uncovered market outcomes where not all consumers are served by rms. Thus,

we assume that the lowest willingness to pay for quality is zero so that we do not need to impose other speci c conditions for an uncovered outcome. 15

Some studies (Gabszewicz and Thisse 1979, 1980; Shaked and Sutton 1982, 1983) on vertically di eren-

tiated oligopoly assume that consumers are identical in taste but di er in income levels. 16

17

Note that, if N = 1, the demand is reduced to that where x e1 (p; q) =

1

=

p e1 qe1 .

This is because it is not our purpose to discuss the relationships between qualities from the cost inter-

actions. This assumption is quite common in the related literature such as Tirole (1988), Choi and Shin (1992) and Motta (1993).

34

18

Please refer to the section on partial market coverage in their paper.

19

The equilibrium qualities satisfy the second-order conditions as veri ed by

and 20

@2 B 2 @qb1

=

0:406722

@2 A 2 @qa1

=

0:416496

Shin-Kun Pengz

Running head: Quality Choices in Multiproduct Duopoly JEL Classi cation: D21, D43, L11, L13

We are most grateful to three anonymous referees who generously o ered a number of insightful comments to improve this paper. We have also bene ted much from fruitful discussions with Takatoshi Tabuchi, Hong Hwang and Toshihiro Matsumura, which led to signi cant improvements in this paper. We are especially indebted to Takatoshi Tabuchi for his contributive comments and suggestions. Finally, we would like to thank conference participants at 2010 APET in Istanbul, SAET in Singapore, as well as seminar participants at IEAS and NTU. Financial support from Academia Sinica and the National Science Council, Taiwan, is gratefully acknowledged. The usual disclaimer applies. y Tunghai University, No. 181, Section 2, Taichung Port Road, Taichung, 40704, Taiwan. E-mail: [email protected] z Academia Sinica and National Taiwan University, 128 Academia Road, Section 2, Nankang, Taipei, 115, Taiwan. E-mail: [email protected] (corresponding author)

1

2

Abstract: This paper proposes a Cournot model of two-stage competition to examine the patterns of vertical product di erentiation in a multiproduct duopoly. Firms simultaneously choose the number of products and their qualities at the rst stage, and compete in quantities at the second stage. We show that, when the xed setup cost of a product is high enough to result in a monopoly outcome, the monopolist always sells a single product. Moreover, in any equilibrium of a multiproduct duopoly, quality di erentiation between them will develop into a non-segmented pattern because each rm desires to avoid a strong e ect of cannibalization. The set of equilibria reveals the properties of quality di erentiation between multiproduct rms. In a multiproduct duopoly, the pro t from a high-quality product can be lower than that from a low-quality product. This nding sharply contrasts with the literature on single-product rms which nds the high-quality advantage. Keywords: multiproduct lines, Cournot competition.

rms, vertical product di erentiation, quality, product

3

1. Introduction Most of the literature dealing with vertical product di erentiation, such as Bonanno (1986) and Motta (1993), assumes that each rm can produce only one product. However, multiproduct

rms abound in the real world. Firms in various industries produce multiple

quality-di erentiated goods. For example, Mercedes-Benz provides three di erent classes of sedans, namely, its C class, E class and S class, and BMW has its 3 series, 5 series and 7 series. In addition, both Lenovo and Sony supply around twenty kinds of laptops ranked in accordance with their central processing units and out ts.1 A casual survey of the empirical evidence also suggests that any two multiproduct oligopolies usually di erentiate the qualities of their products, and the patterns of their quality di erentiation can be classi ed into the segmented type and the non-segmented type. The former type implies that the products of one rm are always of higher quality and more highly-priced than those of the other rm, such as the sedans of BMW and Honda. All the sedans sold by BMW are more expensive and of higher quality than those of Honda. The latter type denotes completely di erent patterns where the products produced by the two rms have interlacing patterns of their prices/qualities. For example, the prices of the Mercedes-Benz C, E and S classes are, respectively, $32900, $52700 and $87800, while the prices of BMW's 3, 5 and 7 series are given as $33600, $45800 and $80300, respectively.2 The prices of their sedans are interlaced, and it is inferred that their qualities are interlaced and non-segmented. Other examples include the whiskey o ered by Johnnie Walker and Edrington, and the plasma televisions provided by Samsung and Sony. The non-segmented patterns of di erentiation between the multiproduct rms with

4

similar technological capabilities are very common in reality. However, the related literature has not shed enough light on studying them. While the segmented type of quality di erentiation has been characterized by Champsaur and Rochet (1989, 1990) with a quality-then-price competition of multiproduct duopolists, the non-segmented one, so far, has not been captured and explained in the literature.3 Therefore, this paper intends to examine the equilibrium qualities as well as the number of products chosen by the multiproduct rms with identical/similar technological capabilities, such as Mercedes-Benz and BMW. Our main purpose is to characterize the patterns of vertical product di erentiation, and to reveal the properties of quality di erentiation between them. The contributive papers (Gal-Or 1983; De Fraja 1996; Johnson and Myatt 2003, 2006) examine a variety/quality-and-quantity competition between multiproduct rms.4 Most of them conclude that rms with equal technological capabilities will compete head-to-head by producing identical products/qualities, such as Gal-Or (1983), De Fraja (1996) and Johnson and Myatt (2003, Proposition 6).5 By contrast, Johnson and Myatt (2006) study a twostage game with a model of two-exogenous qualities, and show that the symmetric rms may choose di erent product lines for strategic reasons. However, because there are only two choices of product qualities and they are exogenously given, there is no discussion over the quality levels made by multiproduct duopolists and quality di erentiation between them. Moreover, although they reveal that there is a possibility of asymmetric equilibrium, where rms produce di erent products, they do not provide a worked-out example for a more comprehensive analysis. In the case of multiplicative preferences and uniformly distributed consumer valuations, they derive the complete equilibrium, but it is shown that rms will

5

compete head-to-head by producing both qualities, i.e., all qualities in the domain (Johnson and Myatt 2006, Proposition 11). That is, there is symmetric equilibrium and no product di erentiation between the multiproduct rms. In previous studies, there were no restrictions on the number of products and no costs associated with building up a product line for a distinct quality. Accordingly, because increasing the number of products is costless, each rm provides a full product line (Johnson and Myatt 2006, Proposition 11). However, in reality, a rm usually restricts itself to produce less varieties rather than a full product line due to the limitation of its resources, and boosting the number of product lines often involves additional setup costs due to the requirements of the distinct or advanced facilities.6 Thus, this paper involves a xed setup cost for each kind of product; that is, a xed cost will be incurred each time a product is introduced.7 Moreover, since it is not the purpose of this paper to discuss product proliferation, we involve a restriction on the maximum number of products. For the simplicity of the analysis, we assume that each rm will not provide more than two products as long as there is a positive xed setup cost.8 In contrast to the symmetric outcome in Johnson and Myatt (2006, Proposition 11), we show that multiproduct rms will di erentiate the qualities of their products rather than providing identical qualities when there is a restriction on product proliferation. The set of equilibrium outcomes in this study is much richer compared to those in the literature, and it formalizes some patterns of non-segmented types of vertical di erentiation. Moreover, it reveals the properties of quality di erentiation in a multiproduct duopoly. The results sharply contrast with those of the literature on vertical product di erentiation when each rm can produce only one quality.

6

In order to compare our result with that in the literature on vertical di erentiation, we employ a model a la Motta (1993) and generalize a standard model to multiproduct o erings while maintaining the remaining structure of the original model.9 We assume that the costs of quality improvement fall upon the variable costs, and specify a quadratic form for tractability as in Motta (1993), Cremer and Thisse (1994), Lambertini and Orsini (2000, 2001) and Brouhle and Khanna (2007), among others.10 Firms simultaneously choose the number of products and the qualities of their products at the

rst stage, and then

compete in quantities at the second stage. The choice of a game where rms make decisions simultaneously rather than sequentially is crucial in the derivation of the results. Some of the ndings are summarized as follows. First, compared with the multiproduct monopoly in the contributive literature (Mussa and Rosen 1978; Lambertini and Orsini 2000; Johnson and Myatt 2003), we consider a potential entry and show that, as the xed cost is high enough and this results in a natural monopoly, the monopolist always sells a single product.11 Second, in any equilibrium of a multiproduct duopoly, quality di erentiation between them will develop into a non-segmented pattern because each rm desires to avoid a strong e ect of cannibalization, which refers to the competition among the products provided by the same rm. Moreover, equilibrium outcomes depends on the xed cost of a product, and it is revealed that rms never produce more products as the xed cost is higher. Third, we nd that the pro t from a high-quality product can be lower than that from a low-quality product in a multiproduct duopoly. This nding is in sharp contrast to that of a high-quality advantage in the literature on single-product rms, such as Motta (1993) and Lehmann-Grube (1997), where each rm can produce only one product and they show that producing a higher quality product always brings a higher pro t.12 Fourth, while

7

looking into the product di erentiation of neighboring products, we nd that the quality di erentiation of the two middle products in the equilibrium where both rms provide two products is quite large. In addition, product di erentiation in the high-quality market will be smaller than that in the low-quality one. Fifth, it is revealed that the impact of selling multiple products reduces the average quality. The remainder of this paper is organized as follows. Section 2 outlines the environment of the model. Section 3 characterizes the subgame perfect Nash equilibrium (SPNE). Section 4 provides a simple example to validate our main results. Section 5 discusses asymmetric patterns of quality di erentiation in reality. Section 6 concludes.

2. The Model In the economy, there are two rms, R = A; B, with equal technological capabilities, and each of them can produce at most two products of quality-di erentiated goods. The number of products provided by rm R is denoted by nR (

2); the quality, price and quantity of the

product indexed by s (= 1; 2) o ered by rm R are denoted by qrs , prs and xrs respectively.13 There is a continuum of consumers with di erent tastes for quality. Their willingness to pay for quality is distributed uniformly over the interval [0; ] with the density normalized to 1.14 Each consumer purchases one unit of the product either from rm A or B, or does not purchase at all. The utility of consumer

U =

8 > > < qrs > > : 0

prs

2 [0; ] is given by

if he buys the product s (= 1; 2) from rm R otherwise

8

where qrs is the quality of the product s of rm R, and prs is the price of this product. This utility function implies that all consumers prefer higher quality at a given price, and the is willing to pay more for a unit quality.15

consumer indexed by a higher

Consider the demand for each product. Suppose that the total numbers of distinct products provided by the two rms are N (

4). According to the ranking of their quality

levels, we re-label these qualities qe1 ; qe2 ; :::; qeN where qe1 > qe2 > ::: > qeN , and their prices are also re-labled by pe1 ; pe2 ; :::; peN , respectively. Hence, marginal consumers indexed by

such that

ei iq

pei =

ei+1 iq

pei+1 are indi erent between purchasing quality qei at price pei

or quality qei+1 at price pei+1 . Any consumer with an index greater than qei+1 for all i = 1; :::; N eN Nq

i

i

will prefer qei to

1. In addition, the marginal consumer with an index

N

such that

peN = 0 is indi erent between buying quality qeN at peN and not buying at all, and any

consumer with an index greater than

N

will prefer to buy quality qeN than not to buy at all.

Accordingly, the demand x ei for the i-th highest quality qei can be expressed as follows.16 x ei =

8 > > > > > > < > > > > > > :

i

i 1

N 1

pe1 pe2 ; qe1 qe2

= i

= N

pei qei

=

1

1

peN qeN

pei qei 1

1

if i = 1.

peN qeN

pei pei+1 ; qei qei+1 peN ; qeN

if i = 2; :::; N

1.

(1)

if i = N .

In order to derive the equilibrium of a game where rms compete in quantities rather than in prices at the last stage, we invert the system of demand functions in (1) and thus obtain the following.

9

Lemma 1 The inverse demand of (1) can be written as 8 > > > qei > > > > > < pei = qei > > > > > > > > : qeN

N P

j=1

qei (

qej x ej

i P

x ej )

j=1 N P

qeN (

for i = 1

j=1

N P

j=i+1

x ej )

qej x ej

for i = 2; :::; N

1

(2)

for i = N

Proof. See Appendix 1. We assume that the costs of quality improvement fall upon variable costs instead of xed costs, which means that xed costs are independent of quality. In addition, the production activities are fully additive so that the unit cost of products is independent of its quantities.17 The unit cost of quality improvement is a quadratic form and denoted by 2 c(qrs ) = qrs where

> 0 so that the marginal cost of quality improvement is increasing.

Furthermore, each product line involves a xed cost f > 0. The pro t of rm R denoted by R

is written as

R

=

nR X

[prs

2 qrs ]xrs

nR f

(3)

s=1

=

where

R

R

nR f

is the gross pro ts without subtracting the incurred xed costs. In addition, xrs

is the quantities of the product of quality qrs , and prs is the associated price which can be derived from (2) as all products are ordered by their qualities. For example, if rm A o ers two products of the highest and the second-highest qualities and rm B o ers a single product of the lowest quality, namely, (e q1 ; qe2 ; qe3 ) = (qa1 ; qa2 ; qb1 ) and (e x1 ; x e2 ; x e3 ) = (xa1 ; xa2 ; xb1 ), then

10 according to (2) the associated prices, (e p1 ; pe2 ; pe3 ) = (pa1 ; pa2 ; pb1 ), can be respectively written as

pa1 = qa1

(qa1 xa1 + qa2 xa2 + qb1 xb1 );

pa2 = qa2

qa2 (xa1 + xa2 )

qb1 xb1 ;

pb1 = qb1

qb1 (xa1 + xa2 + xb1 ):

(4)

Furthermore, if rm A o ers two products of the highest and the third-highest qualities and

rm B o ers the product of the second-highest quality, namely, (e q1 ; qe2 ; qe3 ) =

(qa1 ; qb1 ; qa2 ) and (e x1 ; x e2 ; x e3 ) = (xa1 ; xb1 ; xa2 ), then according to (2) the associated prices,

(e p1 ; pe2 ; pe3 ) = (pa1 ; pb1 ; pa2 ) can be respectively written as pa1 = qa1

(qa1 xa1 + qb1 xb1 + qa2 xa2 );

pa2 = qa2

qa2 (xa1 + xb1 + xa2 );

pb1 = qb1

qb1 (xa1 + xb1 )

(5)

qa2 xa2 :

Game Structure

Two rms, A and B, play a two-stage game. At the rst stage, they simultaneously choose the numbers of products nR and the qualities of their products, qrs . Moreover, a rm enters the market only if its pro t is strictly positive. At the second stage, they simultaneously decide the quantities of their outputs, having observed the numbers of products and qualities. In equilibrium, eighteen con gurations possibly arise. Excluding the con gurations which can be obtained by relabeling rms, we thus consider the following nine alternatives

11

of vertically spatial con gurations:

f(a); (aa); (ab); (aab); (abb); (aba); (aabb); (abba); (abab)g;

(6)

where (a) is a single-product monopoly, (aa) is a two-product monopoly, (ab) is a singleproduct duopoly, and the three cases (aab), (abb), (aabb) are all referred to as segmentation because multiproduct rms produce neighboring qualities and thus the markets are vertically segmented. Moreover, (aba), (abba), (abab) are respectively referred to as sandwich, enclosure and interlacing. For example, if rm A produces two products of quality (qa1 ; qa2 ) and rm B produces a single product of quality qb1 such that qa1 > qa2

qb1 , this con guration is denoted by

(aab). Moreover, by substituting (4) into (3), the pro ts of rms in (aab) can be respectively expressed as

A (aab)

= [qa1 +[(qa2

B (aab)

= [qb1

(qa1 xa1 + qa2 xa2 + qb1 xb1 ) qa2 (xa1 + xa2 )

qb1 xb1 )

qb1 (xa1 + xa2 + xb1 )

2 qa1 ]xa1 2 qa2 ]xa2

2 ]xb1 qb1

2f;

f:

If rm A produces two products of quality (qa1 ; qa2 ) and rm B produces a single product of quality qb1 such that qa1

qb1

qa2 (but qa1 6= qa2 ), this con guration is denoted

12

by (aba). By substituting (5) into (3), the pro ts in (aba) can be respectively expressed as

A (aba)

= [qa1 +[qa2

B (aba)

= [qb1

(qa1 xa1 + qb1 xb1 + qa2 xa2 ) qa2 (xa1 + xb1 + xa2 ) qb1 (xa1 + xb1 )

qa2 xa2

2 ]xa1 qa1 2 qa2 ]xa2 2 ]xb1 qb1

2f; f:

Similarly, the pro t functions of the other con gurations in (6) can be expressed in the same way.

3. Equilibrium Characterization As mentioned in the previous section, rms rst choose the number of products and associated qualities, and then decide on the associated quantities of outputs. We rst solve for the equilibrium qualities and quantities of products within each of the nine con gurations in (6). Then, in order to ensure that each of the derived equilibrium candidates can be selected as a SPNE of this two-stage game, we need to further check that both rms have no pro table deviation regardless of whether they alter the ordering of qualities, or alter the number of products and qualities. While looking into the two monopolistic cases, (a) and (aa), we can simply derive the equilibrium outcome of either case from the rst-order conditions of a monopolist that maximizes its pro t with respect to the quality and quantity of its products. The details of the derivations can be learned from Lambertini (1997).18 We thus omit the details, and summarize the outcomes of the two cases in Table 1. Note that they are presented in decimal

13

gures to facilitate a comparison with the outcomes of the duopolistic ones. By examining the outcomes of the duopolistic cases, we follow the procedure of backward induction and rst solve the second-stage equilibrium of the rm's decision on quantities. By taking the numbers of products and product qualities as given, each duopolist maximizes its pro t

R(

) with respect to the quantity of its products, xrs . From the rst-

order conditions, we may derive the quantity of each product, xrs , as a function of the qualities qrs . In addition, by substituting the equilibrium quantities into the inverse demand in (2), the prices as well as the pro ts of rms,

R(

), are expressed as a function of the

qualities qrs . Then, by going back to the rst stage to solve the equilibrium qualities of the products provided, the best response correspondences of rm R, qr1 (q

r1 ; q r2 )

and qr2 (q

r1 ; q r2 )

are

characterized by the two rst-order conditions

@pr1 @xr1 @pr2 @ R = ( 2 q r1 )xr1 +(pr1 q 2r1 ) +( )x +(pr2 @qr1 @qr1 @qr1 @qr1 r2 @ R @pr1 @xr1 @pr2 = ( )xr1 +(pr1 q 2r1 ) +( 2 q r2 )xr2 +(pr2 @qr2 @qr2 @qr2 @qr2

@xr2 =0 @qr1 @xr2 q 2r2 ) =0 @qr2

q 2r2 )

(7) (8)

where xr1 , xr2 , pr1 and pr2 are the quantities and prices, respectively, derived from the optimal choices of the second stage and the inverse demand in (2), so each of them is a function of qualities. Note that, when rm R produces only a product, the values of qr2 , xr2 and pr2 in the above two equations are taken as zero. In addition, the qualities provided by the rms satisfy the speci cation of the con guration. For example, in (ab), all the values of qa2 , xa2 , pa2 , qb2 , xb2 and pb2 in equations (7) and (8) are zero, and the two qualities satisfy qa1 > qb1 . The equilibrium quantities in the

14

second stage are derived as

2 2 ( 2qa1 + qb1 ) + (2qa1 4qa1 qb1

(xa1 ; xb1 ) =

qb1 )

;

qa1 [(qa1 2qb1 ) + ] 4qa1 qb1

(9)

By substituting them into the associated prices in (2), the prices can be written as

(pa1 ; pb1 ) =

2 qa1 [(2qa1

2 qa1 qb1 + qb1 ) + (2qa1 4qa1 qb1

2 2 + 2qa1 qb1 qb1 ) + qa1 ] qb1 ) ] qb1 [(qa1 ; 4qa1 qb1

(10) Substituting the above results into equations (7) and (8), we have

3 (24qa1 2 (4qa1

2 2 3 10qa1 qb1 + 4qa1 qb1 + qb1 ) 2 23qa1 qb1 + 2qb1 ) + (4qa1

2 (8qa1

2 2qa1 qb1 + qb1 ) =0

qb1 ) = 0

Solving the above two equations, we obtain the equilibrium qualities19

(qa1 ; qb1 ) = (0:369 ; 0:293 )

By substituting the equilibrium qualities into the equations in (9) and (10), the equilibrium quantities and prices are (xa1 ; xb1 ) = (0:219 ; 0:244 ) 2

2

(pa1 ; pb1 ) = (0:217 ; 0:157 ) Following the above steps to solve the rst-order conditions (7) and (8) of each case, we obtain the equilibrium outcomes of all cases.20 They are summarized in Table 1.21 It has been veri ed that each outcome satis es its second-order conditions, and is consistent with

15

the assumption that rms do not cover the market. These equilibrium con gurations are the candidates for the SPNEs.

"Insert Table 1 approximately here"

According to Table 1, the equilibrium qualities are proportional to . This indicates that they decrease with the coe cient for the costs of quality improvement ( ) and increases with the coe cient , which is a constant proportion of the average willingness to pay for quality ( 2 ).22 In addition, the equilibrium prices are proportional to

2

; that is, the prices

are lower when the coe cient for costs ( ) increases. This is because the equilibrium quality is lower when the costs increase. Furthermore, the equilibrium quantities are in proportion to , which implies that they increase with the average willingness to pay for quality. Now, we are going to investigate which equilibrium con gurations in Table 1 are indeed the SPNEs. We must further check that in each equilibrium con guration both rms have no pro table deviation no matter by altering the ordering of qualities, or by altering the number of products and qualities. In the following analysis, we assume that any quality deviation is in proportion to

.

First, we nd that the equilibrium con guration (aa) in Table 1 can never be selected as the SPNE. This is because, if the xed cost f is small enough for rm A to produce two distinct qualities, it must be pro table for rm B to enter the market. Accordingly, the following proposition is established: Proposition 1 A multiproduct monopoly does not emerge in equilibrium. Proof. See Appendix 2.

16

In comparison with the result of a multiproduct monopoly in Mussa and Rosen (1978), Lambertini and Orsini (2000) and Johnson and Myatt (2003, Proposition 1), Proposition 1 shows that a monopolist will not sell multiple products while a potential entry is considered.23 This is because, if the xed cost per product line is small enough such that a monopolist is willing to o er a second product, the potential competitor must enter the market since the pro t from his rst product is greater than the pro t from the second one of the monopolist.24 Note that, in the equilibrium con gurations (aab) and (aabb), there are corner solutions to qualities, so that we have qa2 = qb1 as shown in Table 1. However, in both cases, we nd that it is pro table for rm A to lower its quality qa2 such that qa2 < qb1 . Moreover, in regard to the other segmented candidate (abb), rm B has an incentive to deviate to a non-segmented pattern (bab). Thus, we establish the following proposition:

Proposition 2 The segmented type of di erentiation where all qualities produced by one rm are higher than those o ered by the other does not emerge when rms simultaneously compete in the number of products, qualities and then quantities.

Proof. See Appendix 3. Proposition 2 shows that no rm has any incentive to introduce a new product with neighboring quality. This is because each rm has an incentive to avoid a strong negative e ect of cannibalization. Cannibalization refers to a reduction in the market share of one product as a result of the introduction of a new product by the same producer. In other words, it implies the competition among the products provided by the same rm. In a segmented pattern, the competition between the products provided by the same rm are more serious because these products are next to each other. Thus, a multiproduct rm has

17

an incentive to deviate to a non-segmented pattern to avoid the keen competition among its own products. Lastly, to ensure that the other candidates in Table 1, namely, (a), (ab), (aba), (abba) and (abab), can be selected as SPNEs of a two-stage game, we have to verify that no rm has a pro table deviation at the rst stage. Because the number of products and qualities are both chosen at the rst stage, we should check that there is no pro table deviation regardless of whether the rm alters the qualities of its products or alters both the number and qualities of its products. Moreover, a rm will only enter the market when its pro t is strictly positive. Formally, the three conditions for (ab) to be selected as a SPNE are

Single-product duopoly (ab). A (ab)

maxf

A (ab);

A (ba)g

A (ab)

maxf

A (aab);

A (ab)

> 0 and

B (ab)

and

A (aba);

B (ab)

A (baa)g

maxf and

B (ab);

B (ab)

B (ba)g:

maxf

B (abb);

B (bab);

B (bba)g:

> 0:

The rst one implies that both rms have no pro table deviation by altering their qualities. More speci cally, the former inequality means that, given qb1 = 0:293 , it is not pro table for

rm A to alter its qualities; the latter one is that, given qa1 = 0:369 ,

no incentive to alter its qualities, either. By the de nition of A (ab)

maxf

A (ab)g

A (ab)

maxf

A (ba)g

and and

B (ab) B (ab)

maxf maxf

B (ab)g, B (ba)g.

R (ab),

rm B has

it must hold that

so that we only need to check that That is, given qb1 = 0:293 , it is not

pro table for rm A to provide a product with qa1 < 0:293 ; given qa1 = 0:369 , it is not pro table for rm B to provide a product with qb1 > 0:369 .

18

The second condition implies that neither rm has a pro table deviation by altering the number of its products as well as its qualities. That is, given qb1 = 0:293 , it is not pro table for rm A to provide two qualities regardless of whether qa1 > qa2 > 0:293

or

qa1 > 0:293 > qa2 or 0:293 > qa1 > qa2 . Similarly, the condition that, given qa1 = 0:369 , it is not pro table for rm B to provide two qualities regardless of whether 0:369 > qb1 > qb2 or qb1 > 0:369

> qb2 or qb1 > qb2 > 0:369 . In addition, the third/last condition ensures

that both rms can pro tably enter the market. The conditions for the other con gurations (a), (aba) (abba) and (abab) to be selected as a SPNE are placed in Appendix 4. Based on all of these no deviation conditions, which are in turn based on the Nash concept and entering conditions, we obtain the following proposition:25 Proposition 3 In a two-stage game where rms compete in the number of products, qualities and then quantities, we derive a SPNE depending on the range of xed costs as follows: 3

0:0016 , the top

(i) When 0 < f

rm, the

rm selling the highest quality, o ers

two products of quality (0:414 ; 0:173 ), and the other o ers two products of quality (0:386 ; 0:225 ) between those o ered by the top rm. /Enclosure (abba). 3

(ii) When 0 < f

0:00189 , the top rm o ers two products of quality (0:430 ; 0:239 )

and the other o ers two products of quality (0:379 ; 0:165 ) that are interlaced with those o ered by the top rm. / Interlacing (abab). (iii) When 0:0017

3

f

3

0:0031 , the top

rm sells two distinct products of quality

(0:424 ; 0:194 ) straddling the rival's unique product of quality 0:342 . / Sandwich (aba).

19

(iv) When 0:0028

3

3

f < 0:0175 . Both rms produce a single product. The associated

qualities are 0:0369 and 0:0293 . / Single-product duopoly (ab). (v) When 0:0168

3

3

f < 0:0370 , a single rm monopolizes the market by o ering a

product of quality 0:333 . / Single-product monopoly (a). (vi) When f

3

0:0370 , no rm enters the market.

Notice that Proposition 3 does not claim the uniqueness of the SPNE but presents a SPNE for this two-stage game, because the candidates (abba) and (abab) in Table 1 may not be unique solution in their con gurations.26 The equilibrium qualities and pro ts of the two rms for each equilibrium outcome in Proposition 3 can be found in Table 1, and they are respectively illustrated in Figures 1 and 2. In Figure 1, we nd that the types of equilibrium con gurations depend on the xed cost of a product and rms never produce more products as the xed cost per product is higher. Figure 2 indicates that the pro ts of rms are not monotonic with the xed cost. As the xed cost increases, the pro ts of rms fall but there are some jumps. It is revealed that the equilibrium derived by Motta (1993) is one of our equilibrium outcomes for a speci c range of xed cost, which is speci ed by (iv). There are other asymmetric equilibria of multiproduct duopolists which have not been found in the literature.

"Insert Figure 1 and Figure 2 approximately here" Three points are worth mentioning. First, in the multiproduct equilibrium, only the non-segmented patterns sandwich (aba), enclosure (abba) and interlacing (abab) may emerge.27 As mentioned in Proposition 2, none of the segmented patterns, (aab); (abb) and

20

(aabb), can be selected as a SPNE, because a multiproduct rm has an incentive to deviate to a non-segmented pattern to avoid the keen competition among its own products (i.e., the strong negative e ect of cannibalization) in the segmented pattern. Second, there are multiple equilibria in some ranges of the xed cost. For example, both the equilibrium (ab) and the sandwich equilibrium (aba) may emerge when 0:0028 f

3

3

0:0031 . However, according to Figure 2, both rms prefer the equilibrium (ab) to

(aba) because both of their pro ts are higher in the equilibrium of a single-product duopoly (ab). Thus, the equilibrium (ab) dominates the sandwich one (aba) in this area of multiple equilibria. Moreover, the interlacing equilibrium (abab) dominates the enclosure one (abba) in the entire range of the xed costs where the enclosure equilibrium may emerge. Therefore, all enclosure outcomes are dominated. Third, according to the sandwich (aba) and interlacing (abab) equilibrium in Table 1, we nd that the equilibrium pro t from the product associated with the highest quality is lower than that from the product associated with the second highest quality. Hence, we establish the following proposition:

Proposition 4 In a multiproduct duopoly, producing a higher quality does not imply a higher pro t.

This result is in sharp contrast to that of the high-quality advantage in the literature on single-product rms, such as Motta (1993) and Lehmann-Grube (1997), where each rm can produce only one product and it is concluded that producing a higher quality product always brings a higher pro t. However, Proposition 4 shows that the pro t from a high-quality product can be lower than that from a low-quality product in a multiproduct duopoly.28 This

21

is because the cannibalization e ect on the strategies of the rms may bene t a low-quality product. As shown in the two patterns, (aba) and (abab), because rm A would like to di erentiates the qualities of its two products to avoid the strong e ect of cannibalization, the product in the middle of them (that is, the product with the second highest quality) is able to acquire a large demand and enjoy a higher pro t.29 Moreover, according to Table 1, the pro t of rm A is lower than that of rm B in either of the enclosure (abba) or interlacing (abab) equilibria. This shows that the multiproduct rm that produces the highest quality product generates a lower pro t. Based on the interlacing equilibrium (abab), it can also be inferred that the rm that produces two qualities which are respectively higher than the two qualities of the other rm generates a lower pro t.

"Insert Table 2 approximately here" Table 2 summarizes the quality di erentiation between neighboring products and the average qualities in the market for each equilibrium con guration in Proposition 3. As shown in Table 2, the central segment of quality di erentiation in the interlacing (abab) equilibrium is 0:161 . It is quite large compared to the other two levels of di erentiation between neighboring products, which are 0:051 and 0:074 . This is because each rm would like to di erentiate the qualities of its own products from each other to avoid a strong cannibalization e ect. Accordingly, the two middle qualities become more di erentiated. Similarly, in the enclosure (abba) equilibrium, the di erentiation of the two middle qualities is much greater than the di erentiation of the other neighboring qualities. Moreover, in this con guration, it is implied that product di erentiation within a rm is greater than

22

that between rms. In the interlacing and enclosure equilibrium, we also nd that product di erentiation in the high-quality market (:051 =:028 ) is smaller than that in the low-quality market (:074 =:052 ).30 Intuitively, the product-di erentiation behavior is more costly in the highquality market than in the low-quality one because the marginal cost of quality improvement is increasing. According to the average qualities in Table 2, the average qualities in each of the equilibria (aba), (abba) and (abab) are lower than those in the single-product equilibrium (ab). That is, the impact of selling multiple products reduces the average quality provided in the market. This is because, when rms can produce multiple qualities, they provide more low-quality goods to capture the consumers with a low willingness to pay, and less highquality goods to discriminate more e ectively among the consumers with a higher willingness to pay. Accordingly, more lower quality and less higher quality products are produced.

4. Example In this section, we provide a simple example with three exogenously given qualities and with one of the two rms who is limited to sell only one quality at most. This example validates Propositions 1-4 and further clari es our main results. Let

= 1=2 and

= 10. Consider three given qualities fqH ; qM ; qL g = f7; 6; 2g, and

that rm A is an incumbent who can sell at most two of these qualities while the entrant, rm B, can sell only at most one of them. The two rms simultaneously choose the number of products and their qualities from the exogenously given qualities fqH ; qM ; qL g at the rst

23 stage, and simultaneously decide the quantities of their products at the second stage.31 All possible cases are classi ed and summarized as follows. First, there are three cases of a single-product monopoly where (na ; nb ) = (1; 0) (resp., (na ; nb ) = (0; 1)): fH; g; fM; g and fL; g (resp., f ; Hg; f ; M g and f ; Lg), which respectively denote that rm A (resp., B) sells a single product of quality qH , qM and qL while rm B (resp., A) is out of market. Because the quality is exogenously given, the monopolist's pro t in each case can be easily derived by maximizing its pro t in (3) with respect to the quantity of its product. Before subtracting xed costs, we get the gross pro ts in these three cases as below and the maximal one of them is

M axf

A (fH;

g);

A (fM;

g);

A f(L;

g)g = M axf

1183 1183 147 81 ; ; g= 16 2 2 16

73:94:

Accordingly, it is clearly optimal for a single-product monopoly to sell the quality qH , since these three cases have the same incurred xed costs (f ). Second, there are also three cases of a multiproduct monopoly where (na ; nb ) = (2; 0): fHM; g; fHL; g and fM L; g, which respectively denote that rm A sells two products of quality (qH ; qM ), (qH ; qL ) and (qM ; qL ) while rm B is out of the market. Similarly, solving the rst-order conditions of the monopolist maximizing its pro t in (3) with respect to the quantities of its two products, we derive the gross pro ts in these three cases as shown below, and the maximal one of them is

M axf

A (fHM;

g);

A (fHL;

g);

A (fM L;

g)g =M axf

1225 1253 153 1253 ; ; g= 78:31: 16 16 2 16

24

Accordingly, fHL; g is the optimal outcome of a multiproduct monopoly, since these three cases have the same incurred xed costs (2f ). That is, a multiproduct monopolist will sell the highest and the lowest qualities (qH ; qL ). Note that, to ensure that the multiproduct monopolist does not deviate to sell a single product, we must have 78:31 i.e., f

4:37. However, later we will show that, when f

2f

73:94

f,

4:37, rm B must have incentives

to enter the market, so that a multiproduct monopoly does not emerge in equilibrium as remarked in Proposition 1. Third, there are nine cases of a single-product duopoly where (na ; nb ) = (1; 1). Because the number and the qualities of their products are given, the pro ts of the two rms in each case can be easily derived by solving the two rst-order conditions of both rms who maximize their pro ts

R

in (3) with respect to the quantities of their products. The gross

pro ts of the two rms in these nine cases are summarized in Table 3.

"Insert Table 3 here" By looking into the pro ts in Table 3, the two outcomes fH; M g and fM; Hg are the Nash equilibrium under single-product duopoly.32 That is, one of them sells the highest quality qH and the other sells the middle quality qM in a single-product duopoly. Fourth, there are also nine cases of a multiproduct duopoly where (na ; nb ) = (2; 1). By solving the three rst-order conditions of both rms maximizing their pro ts in (3) with respect to the quantities of their products, the gross pro ts of two rms in these nine cases are as shown in Table 4.

"Insert Table 4 here"

25

In Table 4, the pro ts in box fHL; M g reveal that, in the (optimal) multiproduct monopoly fHL; g, if the xed cost is small enough for the multiproduct monopolist not to deviate to sell a single product, i.e., f

4:37, the entrant B must have incentives to

enter the market to provide qM and earn a positive pro t "32:87

f ". This validates that

a multiproduct monopoly can never be selected as a SPNE (Proposition 1). In addition, by looking into the pro ts in Table 4, the unique pure Nash equilibrium of a multiproduct duopoly is fHL; M g, which indicates that rm A sells the highest and the lowest qualities (qH ; qL ) and rm B provides the middle quality qM . This result validates that the segmented types of production di erentiation, like fHM; Lg and fM L; Hg, do not emerge in a multiproduct duopoly equilibrium (Proposition 2). To ensure that the four candidates fH; g (f ; Hg), fH; M g, fM; Hg and fHL; M g are indeed SPNEs, we further con rm no deviation conditions based on the Nash concept and positive pro ts for entry.33 Accordingly, a SPNE of this two-stage game is given by (i) The multiproduct duopoly fHL; M g is the SPNE when 0 < f

3:89, based on

the no deviation conditions below. A (fHL; M g) B (fHL; M g)

M axf =0

A (fHM; M g);

M axf

A (fHL; M g);

B (fHL; Hg);

A (fM L; M g)g

B (fHL; M g);

B (fHL; Lg)g

(ii) The single-product duopoly fM; Hg is the SPNE when 3:07

f < 34:17, based

on the no deviation conditions below. A (fM; Hg) B (fM; Hg)

M axf

A (fHM; Hg);

A (fHL; Hg);

A (fM L; Hg)g

>0

(iii) The single-product duopoly fH; M g is the SPNE when 3:89 on the no deviation conditions below.34

f < 34:17, based

26

A (fH; M g) B (fH; M g)

M axf

A (fHM; M g);

A (fHL; M g);

A (fM L; M g)g

>0

(iv) The single-product monopoly fH; g (or f ; Hg) is the SPNE when 34:17

f

0

M axf

B (fH; Hg);

B (fH; M g);

B (fH; Lg)g

Apparently, in this example, when f > 73:94, no rm will enter the market. The above SPNE shows that the types of equilibrium con gurations depend on the xed cost, f . Only fH; g (f ; Hg), fH; M g, fM; Hg and fHL; M g may emerge, which is in fact corresponding to the cases (a), (ab), (ba) and (aba) in Proposition 3. Moreover, as implied by Proposition 3,

rms never produce more products as the

xed cost is higher, and a

multiproduct duopoly emerges only when the xed cost is small enough. Furthermore, in the equilibrium of multiproduct duopoly fHL; M g, the gross pro t from the highest quality qH and the lowest one qL are respectively

eH =

eL =

1008337 35344 89205 8836

28:52 10:10

while the pro t of the middle quality, eM , is 32:87 which is clearly higher than the pro t of

the highest quality, eH . This result validates that a higher quality does not necessarily yield

a higher pro t (Proposition 4).

27

5. Discussion In contrast to the literature on multiproduct Cournot rms (Gal-Or 1983; De Fraja 1996; Johnson and Myatt 2003, 2006), this paper shows that, when the number of products produced by a rm is restricted for a certain reason, the multiproduct duopolists with identical technological capabilities will di erentiate their qualities. Moreover, quality di erentiation between them will develop into a non-segmented pattern. A large amount of empirical evidence supports the result of the non-segmented di erentiation between the rms with similar capabilities. For example, both Nikon and Canon have several classes of digital single lens re ex (SLR) cameras. Nikon's D7000, D700 and D3X are sold for $1199, $2699 and $7999, respectively, and Canon's 50D, 7D, 5D and 1D are provided at the prices $1099, $1699, $2499 and $6999, respectively. In addition, Sony Ericsson sells high-end cellphones which have higher quality than the high-end ones of Nokia, but it also sells low-end cellphones with quality lower than the high-end products of Nokia. Furthermore, the topmost sedan of BMW, the 7 series, has a higher price than Audi's foremost model, the A8, while Audi's A6 has a higher price than BMW's 3 series.36 However, the qualities are sometimes segmented as mentioned in Section 1. There are some cases of segmented di erentiation where one rm always provides products of higher quality than the other does, but they are not explained in this model. Thus, there is a possibility that they are portrayed by a case where rms have di erent technological capabilities as shown in Johnson and Myatt (2003, Proposition 7).37 Moreover, another possible reason is the presence of the positive externality on the costs of producing similar/neighboring qualities. As shown in the two previous sections, no segmented pattern emerges in equilib-

28

rium because each multiproduct rm has no incentive to produce products with neighboring qualities to avoid the keen competition among its own products (the cannibalization e ect). Therefore, if there is a positive externality on costs which reduces the cost of providing similar qualities and it is large enough to dominate the cannibalization e ect, it is intuitive that multiproduct rms may produce neighboring qualities and that their qualities will result in a segmented pattern.

6. Conclusion In this paper, we have analyzed the variety/quality-then-quantity competition when each rm can produce more than one product, in which the numbers of products, as well as the qualities and quantities of products are endogenously determined. The main contribution is that we present some equilibrium con gurations which have not been captured in the literature, and which formalize some non-segmented types of vertical di erentiation between the multiproduct rms in the real world. Compared to the ndings in the literature on single-product rms, the set of equilibrium outcomes in our study is much richer. The single-product equilibrium in Motta (1993) is only a special case in this study. In the equilibrium of a multiproduct duopoly, there are asymmetric outcomes with non-segmented patterns of di erentiation, such as sandwich, enclosure, and interlacing con gurations. The segmented patterns of di erentiation do not emerge because rms desire to avoid a strong cannibalization e ect. In addition, we nd that the producing a higher quality product does not imply a higher pro t, which in sharp contrast with the high-quality advantage of the literature on single product rms. Moreover,

29

the outcomes reveal the properties of the quality di erentiation between multiproduct rms. It is shown that the quality di erentiation of the two middle products in the equilibrium where both rms produce two products is much greater than that of the other neighboring products. This also implies that product di erentiation within a rm is greater than that between rms. Our ndings were obtained for a speci c model with several assumptions regarding the preferences of consumers and technology which are common in the related literature. This paper captures some equilibrium properties of competition between multiproduct rms and they sharply contrast with those in the literature on the vertically spatial competition. One drawback of this model is that the equilibrium outcomes do not comprehend the segmented con gurations. However, product qualities are sometimes segmented in reality. There are also some related extensions. Consider a positive externality on the production cost of producing similar qualities, or a non-uniform distribution of consumers' willingness to pay for quality. Besides, rms may compete in terms of prices rather than quantities as in the case of Champsaur and Rochet (1989, 1990).

References Brander, J. A., and J. Eaton. 1984. Product Line Rivalry. American Economic Review 74:323{334. Brouhle, K., and M. Khanna. 2007. Information and the Provision of Quality Di erentiated Products. Economic Inquiry 45:377{394.

30

Bonanno, G.. 1986. Vertical Di erentiation with Cournot Competition. Economic Notes 15:68{91. Champsaur, P., and J.-C. Rochet. 1989. Multiproduct Duopolists. Econometrica 57:533{557. Choi, C. J., and H. S. Shin. 1992. A Comment on a Model of Vertical Product Di erentiation. Journal of Industrial Economics 40:229{23. Cremer, H., and J.-F. Thisse. 1994. Commodity Taxation in a Di erentiated Oligopoly. International Economic Review 35:613{33. De Fraja, G.. 1996. Product Line Competition in Vertically Di erentiated Markets. International Journal of Industrial Organization 14:389{414. Doraszelski, U., and M. Draganska. 2006. Market Segmentation Strategies of Multiproduct Firms. Journal of Industrial Economics 54:125{149. Gabszewicz, J., and J.F. Thisse. 1979. Price Competition, Quality and Income Disparities. Journal of Economic Theory 20:340{359. Gabszewicz, J., and J. F. Thisse. 1980. Entry and Exit in a Di erentiated Industry. Journal of Economic Theory 22:327{338. Gal-Or, E. 1983. Quality and Quantity Competition. Bell Journal of Economics. 14:590{600. Gilbert, R. J., and C. Matutes. 1993. Product Line Rivalry with Brand Di erentiation. Journal of Industrial Economics 41:223{240. Johnson, J. P., and D. P. Myatt. 2003. Multiproduct Quality Competition: Fighting Brands and Product Line Pruning. American Economic Review 93:748{774.

31

Johnson, J. P., and D. P. Myatt. 2006. Multiproduct Cournot Oligopoly. Rand Journal of Economics 37: 583{601. Katz, M. 1984. Firm Speci c Di erentiation and Competition among Multiproduct Firms. Journal of Business 57:S149-166. Lambertini, L. 1997. The Multiproduct Monopolist under Vertical Di erentiation: An Inductive Approach. Recherches Economiques de Louvain 63:109{122. Lambertini, L., and R. Orsini. 2000. Process and Product Innovation in a Vertically Di erentiated Monopoly. Economics Letters 68:333{337. Lambertini, L., and R. Orsini. 2001. Network Externalities and the Overprovision of Quality by a Monopolist. Southern Economic Journal 67:969{982. 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. Motta, M. 1993. Endogenous Quality Choice: Price vs. Quantity Competition. Journal of Industrial Economics 41:113{131. Mussa, M., and S. Rosen. 1978. Monopoly and Product Quality. Journal of Economic Theory 18:301{317. Peng, S.-K. and T. Tabuchi. 2007. Spatial Competition in Variety and Number of Stores. Journal of Economics & Management Strategy 16:227{250. Schmidt, R. 2006. On the Robustness of the High-Quality Advantage under Vertical Di erentiation. Journal of Industry, Competition and Trade. 6:183{193.

32

Shaked, A., and J. Sutton. 1982. Relaxing Price Competition through Product Di erentiation. Review of Economic Studies 49:3{13. Shaked, A., and J. Sutton. 1983. Natural Oligopolies. Econometrica 51:1469-1483. Tirole, J. 1988. The Theory of Industrial Organization. Cambridge, MA: The MIT Press.

Notes 1

IBM sold its laptop division to Lenovo in 2004.

2

These prices are from their o cial websites; they denote the suggested retail price of each class with

basic equipment. 3

Champsaur and Rochet (1989) assume that the duopolists each provide a range of connected qualities

and compete with each other in prices. They show that, there is a gap between the two intervals of qualities to avoid the keen competition. 4

Katz (1984), Champsaur and Rochet (1989) and Gilbert and Matutes (1993) note that multiproduct

rms engage in price competition. 5

Johnson and Myatt (2003) show that multiproduct rms may not sell identical products when their

technological capabilities are not equal. 6

While we incorporate a xed setup cost into the case in Johnson and Maytt (2006, Proposition 11), it is

shown that, when the xed cost is high enough, rms no longer provide a full product line. Either rm has an incentive to restrict itself to produce only the high-quality product. 7

In the horizontally di erentiated literature on multiproduct rms, Brander and Eaton (1984), Doraszelski

and Draganska (2006), and Peng and Tabuchi (2007) have also speci ed a xed cost for a variety in production technology. 8

This assumption will rule out certain con gurations with more products. However, involving more

products will not derive much additional intuitions, but this will make the analysis much more complicated

33

because the number of the possible outcomes will increase greatly. For instance, we will need to work on 34 possibilities if each rm may produce one more product, i.e., three products at most. 9

In Motta (1993), there are multiplicative preferences and uniformly distributed consumer valuations.

Both rms have identical technologies of production. However, each of them is assumed to provide only one product. 10

Mussa and Rosen (1978), Lambertini and Orsini (2000, 2001), Gal-Or (1983) and De Fraja (1996) and

Johnson and Myatt (2003, 2006) assume that the quality improvement is associated with the variable costs, while Shaked and Sutton (1982, 1983) specify that the quality improvement is associated with the xed costs. 11

Without considering potential entry, Lambertini and Orsini (2000) present the conditions for a single-

variety monopolist to carry out process and product innovation to produce two varieties. Johnson and Myatt (2003) show that a monopoly will provide a single product if the average cost of quality is decreasing. If not, it will o er multiple products. 12

Note that Schmidt (2006) assumes single-product rms and a fully covered market, and he shows that

the high-quality advantage may not be persistent under some speci cations for the utility or the unit costs of quality, such as a convex unit cost-quality relation. 13

This analytical technique is similar to the approach used in Peng and Tabuchi (2007) where they propose

a location-then-variety competition for a multiproduct and multistore duopoly. 14

This paper focuses on uncovered market outcomes where not all consumers are served by rms. Thus,

we assume that the lowest willingness to pay for quality is zero so that we do not need to impose other speci c conditions for an uncovered outcome. 15

Some studies (Gabszewicz and Thisse 1979, 1980; Shaked and Sutton 1982, 1983) on vertically di eren-

tiated oligopoly assume that consumers are identical in taste but di er in income levels. 16

17

Note that, if N = 1, the demand is reduced to that where x e1 (p; q) =

1

=

p e1 qe1 .

This is because it is not our purpose to discuss the relationships between qualities from the cost inter-

actions. This assumption is quite common in the related literature such as Tirole (1988), Choi and Shin (1992) and Motta (1993).

34

18

Please refer to the section on partial market coverage in their paper.

19

The equilibrium qualities satisfy the second-order conditions as veri ed by

and 20

@2 B 2 @qb1

=

0:406722

@2 A 2 @qa1

=

0:416496