- Email: [email protected]
Product differentiation, prices, and market structure
European Economic Review 22 (1983) 75-96. North-Holland
PRODUCT
DIFFERENTIATION, PRICES, AND MARKET STRUCTURE Yasuo KAWASHIMA*
Mriji
Gokuiu
University,
Shirokane,
Minato-ku,
Tokyo
108, Japan
We examine the effect of cost reductions and advertising on equilibrium prices and the equilibrium market shares. Our equilibrium has the following characteristics; the aggregate demand is the integral of individual demand over consumers, goods are dilferentiated in the sense or Novshek and Sonnenschein (1979), production functions are constant returns to scale. Cost reductions of a tirm, which are due to R&D activity of the firm, lead to declines of equilibrium prices. Advertising or a firm, which is supposed to influence consumers’ preferences in favor of the firm, cannot necessarily cause equilibrium prices to go up; one tirm can raise its prices, but the other cannot. Contrary to our intuition, cost reductions cannot always enable a tirm to capture a larger share.
1. Introduction In this paper, attempts are made to prove the existence of a stable Cournot-Nash equilibrium in prices in a differentiated market and to examine the effect of R&D and advertising on equilibrium prices in the market, through which we can discuss their effect on the equilibrium market shares. Since Bertrand (1883) found fault with the Cournot model, much effort has been made to demonstrate an equilibrium in prices in the model. The basic idea was already suggested by Sraffa (1926), who stated that in a differentiated market a firm can enjoy a position similar to that of an ordinary monopolist. This idea was further advanced by Hotelling (1929) in his spatial location model. Although many economists’ followed Hotelling and advanced his model, a product is still homogeneous in the model, but differentiated only by location; transportation costs make differences in the products of firms,’ and a consumer purchases the cheapest good. Furthermore, Roberts and Sonnenschein (1977) demonstrate, by examples which cannot be considered extreme, that a Cournot-Nash equilibrium fails to exist. *My greatest debt is to Daniel F. Spulber for discussions and valuable comments on the earlier draft. I wish to thank the Brown University ror its hospitality during my stay as a visiting associate professor. I have benelited from the comments of referees, K. Kawamata, K. Kunimoto, M. Ohyama, Y. Ono and R. Sato. Finally, I would like to thank C. d’Aspremont Ior his advice. I am solely responsible for any remaining errors. ‘For example, see Eaton (1976), Hay (1976), Prescott and Vischer (1977), and Salop (1979). 0014-2921/83/$3.00 0 1983, Elsevier Science Publishers B.V. (North-Holland)
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Recently, however, Gabszewicz and Thisse (1979) and Lane (1980) make great success on this subject; they can develop a model in which the equilibrium does exist. Especially, Gabszewicz and Thisse stimulate several recent works in the analysis of a differentiated market.2 However, their model seems a little restrictive; a consumer is assumed to purchase only a single unit of a brand, consumers’ preferences are represented by a special utility function, and a firm can produce output at no costs. When we consider a demand theory of differentiated brands, we should take the Novshek and Sonnenschein (1979) model into account, which generalizes consumers’ behavior in several ways; consumers can purchase an unlimited amount of a brand and their tastes are represented by a general utility function. Thus, their model includes the models mentioned above as special cases. In view of the result of Roberts and Sonnenschein, we have to specify a utility function when an attempt is made to demonstrate the existence of a Cournot-Nash equilibrium in prices. In our model, consumers’ preferences are represented by a quadratic utility function. Under these assumptions, if we assume further that production functions of a firm exhibit constant returns to scale, we can not only demonstrate the existence of an equilibrium in prices, but also examine the nature of differentiated markets. Differentiated markets are utterly different from competitive markets in that there exist some factors which play a key role in the determination of equilibrium prices and outputs. In the present model, our focus is on the effects of R&D activity and advertising on equilibrium prices, on the basis of which equilibrium market shares are taken up to clarify the importance of R&D activity in the markets. Our paper is organized as follows: Section 2 sets forth the basic model. Section 3 proves the existence of a Cournot-Nash equilibrium in prices in a differentiated markets, which is stable and unique. In section 4, we perform a comparative statics analysis to show the effects of R&D and advertising on equilibrium prices, through which we can develop the interesting relation between market structure and cost reductions, which come from R&D activity of a firm. Finally, our analysis is summarized in section 5.
2. The basic model Consider an economy with a continuum of consumers in which there exist a numeraire good and two differentiated goods labelled i=2,3. Consumers are indexed by a in the closed unit interval [O, 11. Each consumer is assumed to pick only one of the differentiated goods. A consumer is to maximize his 2See Gabszewicz and Thisse (1979), Gabszewicz, Shaked, Sutton and Thisse (1981), and Shaked and Sutton (1982).
I! Kawashima,
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77
utility subject to his budget constraint. Formally, we have
max4x1,x2, x3;4, subject to xl+p2x2+w3=m
and
x2x3 = 0,
where x1, x2 and xs denote consumption of the numeraire and differentiated goods, respectively, and all of them are continuous variables. Assume further that the partial derivative of U with respect to a is non-negative; a consumer with a high index feels more satisfication from consumption of brands. Let I/’ be the indirect utility function in the sense of Novshek and Sonnenschein (1979). Note, however, that an income variable is not explicitly included in the Novshek and Sonnenschein model, whereas in our model it is included explicitly. The marginal consumers are defined to be those consumers whose income levels and index satisfy V2(p2, m,a)= V3(p3, m, a) for given p2 and p3. These consumers are indifferent between purchasing brands 2 and 3. If a consumer is to pick brand 2, then his income and index must satisfy V2(p2,
m, a) > V3(p3, m, a)
for given p2 and
p3.
Let M, be the set of income levels and indices such that V2 is larger than V3, and M, the set of income levels and indices such that V3 2 V2. Then, market demand for each brand is defined by X2 = jf xr( .)p(m, a) dm da, M,
and X3 = jj xj( .)p(m, a) dm da, Mz
respectively, where p denotes the density of m and a, and xr(.) is individual demand for brand i. In what follows, we shall restrict our attention to the simple case in which income levels are identical for all consumers. In this case, M, will be of the form (a*, l] and M2 of the form [0, a*], where a* is the solution to V2(p2,m, a)= V3(p3,m,a) for a fixed m. Fig. 1 shows how a* is determined if we assume that Vz > V,“. The market demand functions will then be X2 = j xr( .)pl(a)da, a*
(1)
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structure
Vi
a 0
1
a*
Fig. 1. V’ =indirect utility function when a consumer is constrained to purchase diflerentiated brand i = 2,3; a = index for a consumer; a* = index for the marginal consumers.
and X3 =“jx:(
.)pl(a) dcr,
(2)
0
where p1 is the density of a. From the definition of a*, it follows that aa*/ap2 = - v;/( v: - v,“) >
0,
(3)
and aa*/ap3
Differentiation
=
vi/(
v,Z -
vi)
c
0.
of eq. (1) with respect to
axziap, =I (aWvdpd4
(4) pz
da-xf(p,,
yields a*h(a*)(aa*lah).
(5)
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19
The second term in (5) is negative by virtue of (3), while the sign of the first is ambiguous. Hence, we cannot say that demand function is negatively sloped with respect to prices. Routine calculations yield
aX2/ap3 = i (WIaP3Ma) da-xt(p2, a*h(a*)(aa*lap3). The second term in (6) is positive by virtue of (4). As brands 2 and 3 are substitute goods, the sign of (6) is also indetermlnate. Note that through the second terms in (5) and (6), the marginal consumers are shown to play a crucial role in the determination of the properties of the market dkmand functions. 3. Market demand and Cournot-Nash
equilibrium
In the analysis to follow, we assume that consumer’s preferences are represented by a quadratic utility function, and that the index a appears as a coefficient in this utility function. In particular, (7)
where u3, b, tj and a are positive constants.3 Thus, the marginal utility of brand 2 increases with the index a; in general, consumers with a high index will purchase brand 2 while those with a low index will pick brand 3. In the analysis to follow, we assume that b equals 1 and that consumers are uniformly distributed on the closed unit interval [0, 11. We are now in a position to derive the market demand functions and the index for the marginal consumers. We proceed to establish: Lemma 1. If consumers’ preferences are represented by (7), the market demand functions for the differentiated goods and the index for the marginal consumers are expressed us
(0
x3 = I@3
a*=(a3
- p3Ma3
- P3 + P2)la
- P3 + P2)1/4
(9)
(10)
if a-p,>u,-p,>O; 3A quadratic utility function is utilized in Dixit (1979). His utility function is a little more general than that used here.
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Y Kawashimo, Product diflerentiation, prices, and market structure
(ii)
x2=0,
(11)
X3=a3-ps,
(12)
a*=l,
(13)
ijaa,-p,>O, (iii)
ifa,-p,sO
Proof as4
tf3-p3>Cl-p2;
X’=(a-p2)‘/2a,
(14)
x3=0,
(15)
if a,-p,sO (iv)
tflld
ortrl a-p,>O;
x2=x3=0,
and a-p,SO.
In cases (i) and (ii) the indirect utility functions are given respectively V2=m+~(a2-p2)2
and
V3=m+-&(a3-p3)2,
where a2 =ua. In case (i), the index for the marginal consumers is determined by V2=V3
or
aa-p2=a3-p,,
which is reduced to (lo),
It follows from (1) and (2) that in case (i) the market demand functions [(S) and (9)] are X2=~x;(.)da=~(aor-p,)da = {(~-d2
-(%-h)2)/2%
% is assumed that m is large enough
for
demand for the numeraire to become positive.
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81
and
= {(a, - P3M3- P3+ P2W In case (ii) all consumers purchase brand 3 and no consumers pick brand 2, because a3 - p3 > a -p2. Thus, xf = 0 and xs = a3 -p3 > 0. By virtue of (I) and (2), market demand [( 11) and (12)] is
x2=0, and X3=jx$(.)dcc=a,-p,. 0
Note that the index a* is equal to 1 in that all consumers purchase brand 3. Thus, we have established case (ii). In case (iii) no consumers pick brand 3. Those consumers with an index higher than p2/a can pick brand 2, and those with an index lower than p,/a buy only the numeraire. Hence, we have (14),
where the index a* is determined by aa-p,=O.
As no one picks brand 3, we have (15), x3=0.
Finally, if u -p2 50 and a3 -p3 SO, nobody differentiated brands; i.e., x2 = x3 = 0.
can purchase either of the
Q.E.D.
In case (i) above, the Xi is positive and the index a* is positive but less than 1. These conditions will be used to determine restrictions on the set of strategies (prices) which yield an outcome as in case (i). First, the condition
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that 0 KU* < 1 is reduced to and
p3
p3>p2+a3-a.
Positivity of X2 requires a-p,>
+(a3-P3).
These inequalities can be rewritten as P3>P2+a3
-a
and
p3<-p2+a3+a.
As X3 is also positive in case (i), p3
In what follows, our focus is mainly on case (i). The other cases will be taken up later in this section. Consider the behaviour of duopolists, each of which is assumed to produce a differentiated brand at some cost. For simplicity, assume that the production function for each firm exhibits constant returns to scale, though tirms have different costs. Let ci be their respective average costs. Profit functions are given respectively by5 .i =(pi -Ci)X’,
i=2,3,
where pi zci, i=2,3. Firms are non-cooperative. Each firm is to maximize its own profits with respect to its price, taking the other firm’s price as given. We require some notations: 8, =(a, + 2c,)/3 and e2 = (a3 + c3)/2. Now we can establish: Lemma 2. Let the optimal strategy for firm i be the strategy (price pi) that maximizes firm i’s profits, given pi for j# i. If 3(a - c2) 2 2(a, -c,), then theit ‘We shall ignore the importance of fixed costs, to which Spence (1976), Dixit (1979), and Lane (1980) pay much attention. Spence discusses the relation between product and product characteristics, while Dixit and Lane take up the entry problem using lixed costs.
I! Kawashima,
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respective optimal strategies are defined for p3 in the open set (tl,, 0,) and are expressed as p2={cq+2a-J(a-c,)2+3(a3-p3)2}/3,
(17)
for firm 2, and for firm 3 as P2 = {(P,
Proof
- a,)@P,
- a3 - 2C,)}/@P,
- a3 -
C3).
(18)
The optimal strategies for firms are derived from i=2,3.
aK’/api=X’+(pi-Ci)(aX’/api)=O,
For firm 2, by virtue of Lemma 1, the equation above is reduced to
The maximum profits are obtained for (l7), p2 = {c2 +
2a - J(a - cz)’ + 3(a3 -~,)~}/3,
for p3 in (13,,0,). In fact, 1~’ is positive for p2 in (~~,a-(a,-~~)) a-((a3-p3) is not less than c2 at p3=e1 if3(a-c2)L2(a3-c3).
because
As for firm 3, the first-order condition yields &?/ap,
=x3
+(p3
-c3)(xf3/dp3)
=~(a3-p3)2+P2(a3-p3)~la-(p3-c3)~2(~3-P3)+P2~/~
(20)
={3pZ-2(2a3+c3+p~)p3+a~+a3p2+c3p2+2a3c3}/a=0.
Note that rr3 is positive for p3 in (c,, a3) and equal to zero at p3 =c3 and a3. Solving (20) for p2 yields (18), P2 = {(P,
for p3 in (e,, 0,).
-a3)(3P3
Q.E.D.
-a3
-2C3)}/(2P,
- a3 -C3),
Y. Kawashima,
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Eqs. (17) and (18) are reaction functions for the Cournot duopolists. They are continuous for p3 in (Or, 0,); they are depicted in fig. 2. Having deriven the reaction functions for firms we can now address the issue of existence of a Cournot-Nash equilibrium in prices.
2s+c2 3
a +2c;, 3
f2
I 0
/
p3 ,
“3
/
Fig. 2. 0, =(a3 + 2c,)/3; O2 =(a, + c,)/2; c2 =average costs of lirm 2; c) = average costs of firm 3; pi = initial price of good 3; RCI = reaction function of lirm 2; RC, = reaction function of h-m 3.
Proposition I. Let a Cournot-Nash equilibrium in prices be a combination of pf and p: such that pT maximizes xi given pi*, j# i. Then there exists a Cournot-Nash equilibrium in prices in a differentiated industry iJ 3(a-c,)12(a3-c,).
(21)
Moreover, the equilibrium point is unique and stable. Proof.
The equilibrium
set of prices (pf, ~3) is determined by (17) and (18).
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Both curves are continuous for p3 in the open interval (e,, 0,). The reaction curve for 3 starts from zero at p 3 =Or and is monotonically increasing to infinity as p3 approaches 13~.The reaction curve for 2 is also continuous and monotonically increasing for p3 in [f7,, S,]. Furthermore, the latter gives a higher pz at pa =8i than the former, and at p3=Oz the former provides a lower p2. Hence, both curves intersect at point E in fig. 2. Note that the intersection point is unique. Using (21), we can conclude that by virtue of (17), P2ZC2
at
p3=e1,
where p2 is firm 2’s reaction to p3. In fact, substituting
p3
=8, into (17) yields
which by virtue of (21) is reduced to
Since firm 2’s reaction function is increasing in
p3,
pt
is larger than c2,
Let 1=p,-p,-a,+a.
I equals zero at point (f7,, c2) if and only if 3(a - c2) = 2(a, - cJ. Bearing in mind that the slope of A=0 is larger than that of the reaction curve for 2 and 1 is non-negative at p3 =8, and p2 =c2, we can see that an equilibrium point (pf, pj) is in the set of strategies if (21) holds. Moreover we see in fig. 2 that any initial price pi in (e,, 0,) would trigger a reaction process which leads both lirms to the equilibrium point E; i.e., the point E is stable. Q.E.D. We now turn our attention to the cases which we have so far ignored. Depending upon the values of prices of brands, one of differentiated lirms can not sell its product to its customers. Hence, we may encounter the limit price. The limit price is defined to be a price which deters entry of a new firm. Then, we can establish: Proposition 2.
The limit price in a differentiated market is given by
pf* =(a, +c3)/2 pf* =(a $2c2)/3
in case (ii), in case (iii).
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Proof By virtue of Lemma 1, the market demand and profit functions in case (ii) are expressed respectively as (12), X3=a3-p3, and as 7t3 = (p3
- c3)(a3
The profit-maximizing
-p3).
price is set equal to
P:* =(a3+ Cd& where it should be noted that ps* is less than a3. To this price, if firm 2 sets reaction curve for 2 in case (i), an optimal price of 2 goes up to infinity as a price of brand 3 approaches to p j*. Then, for p3 which is close to pz*, a3
where tirrn.
-P~>~-P,(P~), is firm 2’s response to
p,(p3)
p3.
Thus pf* can deter entry of a new
In case (iii) the market demand for brand 2 is given by (14), X2 =(a-p,)‘/2a, from which we have 2 = (p2 - cz)(a -p2)2/2a. Then firm 2 can set its price equal to pf*
=(a
+ &)/3,
which is less than a. As long as the conditions in case (i) are satisfied, firm 3 sets prices according to its reaction curve; as a price of firm 2 approaches pf*, that of firm 3 goes to a3, so that for some p2 we have
where p3(.) is firm 3’s reaction to p2. Hence firm 3 cannot enter the market if p2 equal to pt*. Thus entry is deterred by pz* and/or pf*. Q.E.D.
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4. Comparative statics of markets
In this section, we examine the effect of parameter changes on equilibrium prices and market shares. Preliminary to an analysis of the effect on the equilibrium market shares of changes in demand and cost parameters is a comparative statics analysis of the equilibrium prices. Let pi*’ denote the partial derivative of pi* with respect to c2. From (19) and (20), we get (3p:-2a-@;‘+(a,--pj)pf’=pf-a,
or A,pf’+
A,p;‘=
B,pf’+
B,pf’=O,
A,,
(24 (23)
and B, = 2(3p$ - 2a, - cJ -pf),
B,=a,+c,-2p3,
where it should be noted that c,
(24)
Solving (24) for pz’ and pf’ yields
pf’ = A,B,ID,
(25)
pT’= - A,B,/D,
(26)
and
where D is the determinant of (24). The sign of (25) and (26) depends solely upon the sign of D. We can show that D is positive. Lemma
3.
The determinant D is positive.
Proof: By virtue of (19) and (20), the slopes of the reaction curves for firms 2 and 3 are given respectively by a~2/ap312
=
-(a3
-
pX3p,
-2a-c,),
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d@xvhation.
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and ap2/ap313 =
-
-
mP3
203
-
c3 -~~)/(a3
+ C3 -2P3).
As (19) can be rewritten as 3{p,-(c,+2a)/3)2-(a,-~,)2=(a-c,)~/3,
then the sople of the reaction curve for firm 2 is at most f. That for firm 3 is increasing with p3 in (Or, O,), and substituting p3=e1 and p2=0, we have
ap2iap313 > 6. Hence, at an equilibrium
point
ap2iap313 =- aP2iaP312r which is -&IA,
-E --MB,,
or A,B,-A,B,>O,
(27)
where it should be noted that A, is negative and B, positive.
Q.E.D.
By virtue of Lemma 3, we have Pf’>O,
(28)
p$‘>O.
(29)
and
For the effect of c3 on the pr, everything would go through Now we establish: Proposition 3. The effect of average cost changes on equilibrium positive. Formally, api*facj >O, Proof
i, j=2,3.
The effect of c3 on the pi” is expressed as
as we did.
prices is
(30)
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and B,=2($-a,)-pfO,
(30’)
and p:‘=A,B,/D>O.
Q.E.D.
(30”)
Thus, this proposition states that the cost change effect on the pi* is symmetric in the sense that cost reductions of a firm lead to reductions of the pi*. Taste parameters in the utility functions are supposed to lead the pr to change. We can summarize: Proposition 4. Changes of taste parameters a, a3 cause equilibrium prices pt to change according to the following:
ap;laa, <0, ap:iaa, i20,
(31)
and
Proof:
ap:laa>o,
ap: jaa > 0.
Differentiation
of (19) and (20) with respect to a3 yields
and A,=a,-pT>O, B,=4pS-2a,-2c,-pf
Solving for p”’ yields
(32)
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Y. Kawashima,
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and
Similarly, routine calculations would yield dpz/da > 0
and
apflda > 0.
Q.E.D.
Changes of taste parameter a has the positive effect on the pr; if firm 2 can succeed in raising its own absolute advantage by advertising, it then can increase not only its equilibrium prices but also those of firm 3. The difference between pf’ and ~3’ is pf’-pf’=A,B,/D+A,B,/D=(A,/D)(B,-B,),
(33)
and A,=2pf-a-c,
To the extent that Bz is negative and 8, positive, the difference is positive by virtue of Lemma 3. Hence, advertising by firm 2 leads its price increase to be higher than that of firm 3. On the other hand, a taste parameter a3 has the surprising effect on equilibrium prices. It is sure that a3 has the negative effect on his rival’s prices, but has the ambiguous effect on his prices. Using (32), we can derive the difference
(34) where the pr’ denotes the partial derivative of the pi* with respect to a3. Firm 3 can raise its prices higher than its rival’s ones. Let us turn our attention to market share. Let X2/X3 be y and the share for i be pi. Noting that X2=!
.+
(acr-pf)dcr
and
X3=y(a3-p:)da, 0
y depends upon a*, pf, a and a3. Formally,
y = Ha*, Pf, PS,a,a3).
(35)
1! Kawashimo,
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91
y with respect to ci, we get
dy/dCi = ay/auIa=.. aa*/ac, t ay/ap,)(apf/ac,) i=2,3.
+ &lap3)GWWci),
(36)
To determine the sign of (36), we will demonstrate the following lemma:
ay/a&=.. is negative.
Lemma 4. Proof:
Let Y’=ixz(.)da a
Differentiating
and
Y3=jx:(.)da. 0
( Y’/Y3) with respect to c( yields
4 y2/yw~ = Lb2
- aab - {(a/2 - p2) - ((a/2)a2 - p,~r)}-j
= b2 41 + ~‘NCa3 -p3b2. Substituting a=a* into equation above, the numerator is reduced to p2
-(a/N1
+u2)=P2
-(a/N(a3-~3
+p212 +a2)la2
= -{(a3-p3)2+2p2(a3-~3)+(a-~2)2}/2a<0. As (X2/X3) =( Y2/Y3) at an equilibrium,
we are done.
Q.E.D.
The index for the marginal consumers has the following properties. Then we have: Lemma 5. Cost changes lead the index for the marginal consumers to change according to the following: au*/&, >O for
i=2,
(37)
i=3.
(37’)
When a taste parameter changes, its effect on the index is
aa*/aa,> 0, EER-D
(38)
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rind
act*/aa=(i/a)(plr'-pS'-a*)~o. Proof:
Differentiating
(39)
(10) with respect to c2, we get
aa*/ac,=(l/~)(plS'-pS').
(40)
Together with (25) and (26), (40) is rewritten as (37) where B, +B,=4p3 -3a,-2c, Similarly, we can get 8a*/&,=
-(BS/D)(A1
-pf ~0
because
ps
is less than (a3 +cJ/2.
+A,)
(37’)
Note that A, +Pl,
=(l/aD){A,B,-A,B,+(a,-PS)(c,-a,
=(l/aD){(A,
+A,)(2p;-2a,-pf)}
-Pa-4&J
>O.
(38)
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The effect of a is written as act*/da = {a(~;’ --pg - (a3 - p: + p;)}/d = (l/a)(pt’ - ps’) - a*/a
where the pf’ a. Q.E.D.
denote the partial
derivatives
of the pf’ with respect to
We now establish the relationship between the market share of a firm and its cost reductions. First, we have the following: Proposition 5. An increase (or decrease) in average costs for firm 2 causes equilibrium output ratio y to decline (or go up) when at an equilibrium
-(l -a*)/a*)+y/6<0. Proof:
(40)
By virtue of (8) and (9),
a?
GP:‘+p,P;‘=
LY*-1
a7
xj
1
P:‘+YFa*P:’
u*
l-a*
=x3 PY -.+Y$
i
=p
u*
l-a* PT’ --i a*
I
B, yg. I
(41)
Note that at an equilibrium
ap2 ap3 3=-
2(2p;-2a,-c,-p;) a,+c,-2pS
B, = -g>6. I
From (41), we get
!* 3p2 *' X
(41’)
By virtue of Lemmas 4 and 5, the first term in (36) is negative for i=2. Hence, when (40) holds at an equilibrium, so that (41’) is negative, (36) is also negative for i = 2. Q.E.D.
94
I: Kawoshima,
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diferentiation,
prices,
and market
structure
According to this proposition, firm 2 can increase its market share by reducing average costs when (40) is true at an equilibrium. But this seems counter-intuitive in that firm 2’s cost reductions may lead declines of the market share of the firm. On the other hand, its rival can capture larger shares by R&D activity of the former. As for cost reductions of firm 3, we can establish: Proposition 6. Cost reductions ofJrm 3 has a positive e&t market shares when the following is met at the equilibrium:
on its equilibrium
-(l-a*)/a*+3y>O. Prooj:
(42)
For i= 3, the first term of (36) is, by virtue of Lemmas 4 and 5, ay/aal,=e &i*/ac, > 0.
(43)
The last two terms are
ci*
ayap:+ardp:=a*-1 FPt’+Y ap2 a+
x3P:’
apJ ac,
u*
l-CC*
=x3 Pf’
-7+Y$
i
a* ‘SPY
---1-c?* a* i
i A, T/ 1
(44)
where we make use of (30’) and (30”). Noting that - AI/A1 is the slope of the reaction curve for firm 2 at the equilibrium, which is at most f (44) is rewritten as a*
3
Pf’
-~-,~}>~p:~~-~+3y)>o, I
(45)
by virtue of (42). Together with (43) and (45), we finally conclude that
when the equilibrium with y, we are done.
satisfies (42). As the market share for firm 3 decreases Q.E.D.
I! Kawashima,
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and market
structure
95
This conclusion is also surprising in that even if a firm can succeed in developing more efficient ways of production by R&D activity, it may not necessarily capture more customers. Then it can increase its share only when the equilibrium satisfies the specific conditions, which are not considered extreme. In view of Propositions 5 and 6, price strategy of differentiated firms is not always essential to increase their respective market shares. 5. Conclusion This paper has examined when a Cournot-Nash equilibrium in prices in a differentiated market does exist. It has been assumed that consumers’ preferences are represented by a quadratic utility function, consumers are uniformly distributed, and that average costs are held fixed for both firms. The mode1 has also explored the cases in which the limit price pertains, depending upon the values of the net absolute advantages of both firms. R&D activity and advertising play key roles in a differentiated market, and their effects on the market have taken up on the basis of the model. Cost reductions by R&D of a firm lead not only to a decline of equilibrium prices, of the firm’s product, but also to that of rival’s product. On the other hand, advertising has the more complicated effect on equilibrium prices; taste changes, which come from advertising activity, may fail to raise equilibrium price of brand 3 even if firm 3 can make its absolute advantages of the brand higher, while firm 3’s advertising lowers prices of the rival’s brand. This seems counter-intuitive, but changes of prices of firm 3 are higher than those of firm 2. To the extent that a taste parameter is changed by firm 2’s advertising, both prices go up. Of course, an increase of prices of firm 2 is higher than that of firm 3. Hence, advertising effect is not symmetric. R&D and advertising are also expected to influence market shares of firms. The present mode1 has indicated that their effects on the shares are in general ambiguous; R&D can provide a larger share to a firm only when an equilibrium satisfies some specific conditions, which are considered extreme. Note, however, that these results have been derived in a specific model. References Bertrand, J., 1883, Theorie mathematique de la richesse sociale, Journal des Savant 48, 499-508. Dixit, Avinash, 1979, A model of duopoly suggesting a theory of entry barriers, Bell Journal of Economics 10,20-32. Eaton, B.C., 1976, Free entry in one-dimensional models: Pure product and multiple equilibria, Journal of Regional Science 16, 21-33. Gabszewicx, Jean Jaskold and Jacques-F. Thisse, 1979, Price competition, quality and income distribution, Journal of Economic Theory 20, 340-359. Gabszewin, Jean Jaskold and Jacques-F. Thisse, 1980, Entry (and exit) in a dil’lerentiated industry, Journal of Economic Theory 22, 327-338. Gabszewicx, Jean J., A. Shaked, John Sutton and Jacques-F. Thisse, 1981, International trade in a ditTerentiated industry, International Economic Review 22, 527-534.
Y. Knwashima,
96
Product
dilfprentiation,
prices,
and market
strwwe
Hay, D.C., 1976, Sequential entry and entry deterring strategies in spatial competition, Oxford Economic Papers 28,240-257. Hotelling, Harold, 1929, Stability in competition, Economic Journal 39, 41-57. Lane, W., 1980, Product digerentiation in a market with endogenous sequential entry, Bell Journal of Economics 11,237-260. Novshek, William and Hugo Sonnenschein, 1979, Marginal consumers and neoclassical demand theory, Journal of Political Economy 87, 1368-1376. Prescott, Edward E. and M. Vischer, 1977, Spatial location among tirms with foresight, Bell Journal of Economics 8, 378-399. Roberts, John and Hugo Sonnenschein, 1977, On the foundations of the theory or monopolistic competition, Econometrica 45, 101-l 13. Salop, Steve C., 1979, Monopolistic competition with outside goods, Bell Journal of Economics 10, 141-156. Shaked, Avner and John Sutton, 1982, Relaxing price competition through product differentiation, Review of Economic Studies 49, 3-13. Spence, A.M., 1976, Product selection, fixed costs and monopolistic competition, Review of Economic Studies 43, 217-235. SralTa, Piero, 1926, The law of returns under competitive conditions, Economic Journal 36, 535550.
PRODUCT
DIFFERENTIATION, PRICES, AND MARKET STRUCTURE Yasuo KAWASHIMA*
Mriji
Gokuiu
University,
Shirokane,
Minato-ku,
Tokyo
108, Japan
We examine the effect of cost reductions and advertising on equilibrium prices and the equilibrium market shares. Our equilibrium has the following characteristics; the aggregate demand is the integral of individual demand over consumers, goods are dilferentiated in the sense or Novshek and Sonnenschein (1979), production functions are constant returns to scale. Cost reductions of a tirm, which are due to R&D activity of the firm, lead to declines of equilibrium prices. Advertising or a firm, which is supposed to influence consumers’ preferences in favor of the firm, cannot necessarily cause equilibrium prices to go up; one tirm can raise its prices, but the other cannot. Contrary to our intuition, cost reductions cannot always enable a tirm to capture a larger share.
1. Introduction In this paper, attempts are made to prove the existence of a stable Cournot-Nash equilibrium in prices in a differentiated market and to examine the effect of R&D and advertising on equilibrium prices in the market, through which we can discuss their effect on the equilibrium market shares. Since Bertrand (1883) found fault with the Cournot model, much effort has been made to demonstrate an equilibrium in prices in the model. The basic idea was already suggested by Sraffa (1926), who stated that in a differentiated market a firm can enjoy a position similar to that of an ordinary monopolist. This idea was further advanced by Hotelling (1929) in his spatial location model. Although many economists’ followed Hotelling and advanced his model, a product is still homogeneous in the model, but differentiated only by location; transportation costs make differences in the products of firms,’ and a consumer purchases the cheapest good. Furthermore, Roberts and Sonnenschein (1977) demonstrate, by examples which cannot be considered extreme, that a Cournot-Nash equilibrium fails to exist. *My greatest debt is to Daniel F. Spulber for discussions and valuable comments on the earlier draft. I wish to thank the Brown University ror its hospitality during my stay as a visiting associate professor. I have benelited from the comments of referees, K. Kawamata, K. Kunimoto, M. Ohyama, Y. Ono and R. Sato. Finally, I would like to thank C. d’Aspremont Ior his advice. I am solely responsible for any remaining errors. ‘For example, see Eaton (1976), Hay (1976), Prescott and Vischer (1977), and Salop (1979). 0014-2921/83/$3.00 0 1983, Elsevier Science Publishers B.V. (North-Holland)
76
I! Kawashima,
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dtfirentiation,
prices,
and market
structure
Recently, however, Gabszewicz and Thisse (1979) and Lane (1980) make great success on this subject; they can develop a model in which the equilibrium does exist. Especially, Gabszewicz and Thisse stimulate several recent works in the analysis of a differentiated market.2 However, their model seems a little restrictive; a consumer is assumed to purchase only a single unit of a brand, consumers’ preferences are represented by a special utility function, and a firm can produce output at no costs. When we consider a demand theory of differentiated brands, we should take the Novshek and Sonnenschein (1979) model into account, which generalizes consumers’ behavior in several ways; consumers can purchase an unlimited amount of a brand and their tastes are represented by a general utility function. Thus, their model includes the models mentioned above as special cases. In view of the result of Roberts and Sonnenschein, we have to specify a utility function when an attempt is made to demonstrate the existence of a Cournot-Nash equilibrium in prices. In our model, consumers’ preferences are represented by a quadratic utility function. Under these assumptions, if we assume further that production functions of a firm exhibit constant returns to scale, we can not only demonstrate the existence of an equilibrium in prices, but also examine the nature of differentiated markets. Differentiated markets are utterly different from competitive markets in that there exist some factors which play a key role in the determination of equilibrium prices and outputs. In the present model, our focus is on the effects of R&D activity and advertising on equilibrium prices, on the basis of which equilibrium market shares are taken up to clarify the importance of R&D activity in the markets. Our paper is organized as follows: Section 2 sets forth the basic model. Section 3 proves the existence of a Cournot-Nash equilibrium in prices in a differentiated markets, which is stable and unique. In section 4, we perform a comparative statics analysis to show the effects of R&D and advertising on equilibrium prices, through which we can develop the interesting relation between market structure and cost reductions, which come from R&D activity of a firm. Finally, our analysis is summarized in section 5.
2. The basic model Consider an economy with a continuum of consumers in which there exist a numeraire good and two differentiated goods labelled i=2,3. Consumers are indexed by a in the closed unit interval [O, 11. Each consumer is assumed to pick only one of the differentiated goods. A consumer is to maximize his 2See Gabszewicz and Thisse (1979), Gabszewicz, Shaked, Sutton and Thisse (1981), and Shaked and Sutton (1982).
I! Kawashima,
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and market
structure
77
utility subject to his budget constraint. Formally, we have
max4x1,x2, x3;4, subject to xl+p2x2+w3=m
and
x2x3 = 0,
where x1, x2 and xs denote consumption of the numeraire and differentiated goods, respectively, and all of them are continuous variables. Assume further that the partial derivative of U with respect to a is non-negative; a consumer with a high index feels more satisfication from consumption of brands. Let I/’ be the indirect utility function in the sense of Novshek and Sonnenschein (1979). Note, however, that an income variable is not explicitly included in the Novshek and Sonnenschein model, whereas in our model it is included explicitly. The marginal consumers are defined to be those consumers whose income levels and index satisfy V2(p2, m,a)= V3(p3, m, a) for given p2 and p3. These consumers are indifferent between purchasing brands 2 and 3. If a consumer is to pick brand 2, then his income and index must satisfy V2(p2,
m, a) > V3(p3, m, a)
for given p2 and
p3.
Let M, be the set of income levels and indices such that V2 is larger than V3, and M, the set of income levels and indices such that V3 2 V2. Then, market demand for each brand is defined by X2 = jf xr( .)p(m, a) dm da, M,
and X3 = jj xj( .)p(m, a) dm da, Mz
respectively, where p denotes the density of m and a, and xr(.) is individual demand for brand i. In what follows, we shall restrict our attention to the simple case in which income levels are identical for all consumers. In this case, M, will be of the form (a*, l] and M2 of the form [0, a*], where a* is the solution to V2(p2,m, a)= V3(p3,m,a) for a fixed m. Fig. 1 shows how a* is determined if we assume that Vz > V,“. The market demand functions will then be X2 = j xr( .)pl(a)da, a*
(1)
78
I: Kawashima,
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d@rentiation,
prices,
and market
structure
Vi
a 0
1
a*
Fig. 1. V’ =indirect utility function when a consumer is constrained to purchase diflerentiated brand i = 2,3; a = index for a consumer; a* = index for the marginal consumers.
and X3 =“jx:(
.)pl(a) dcr,
(2)
0
where p1 is the density of a. From the definition of a*, it follows that aa*/ap2 = - v;/( v: - v,“) >
0,
(3)
and aa*/ap3
Differentiation
=
vi/(
v,Z -
vi)
c
0.
of eq. (1) with respect to
axziap, =I (aWvdpd4
(4) pz
da-xf(p,,
yields a*h(a*)(aa*lah).
(5)
I! Kawashima,
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and market
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19
The second term in (5) is negative by virtue of (3), while the sign of the first is ambiguous. Hence, we cannot say that demand function is negatively sloped with respect to prices. Routine calculations yield
aX2/ap3 = i (WIaP3Ma) da-xt(p2, a*h(a*)(aa*lap3). The second term in (6) is positive by virtue of (4). As brands 2 and 3 are substitute goods, the sign of (6) is also indetermlnate. Note that through the second terms in (5) and (6), the marginal consumers are shown to play a crucial role in the determination of the properties of the market dkmand functions. 3. Market demand and Cournot-Nash
equilibrium
In the analysis to follow, we assume that consumer’s preferences are represented by a quadratic utility function, and that the index a appears as a coefficient in this utility function. In particular, (7)
where u3, b, tj and a are positive constants.3 Thus, the marginal utility of brand 2 increases with the index a; in general, consumers with a high index will purchase brand 2 while those with a low index will pick brand 3. In the analysis to follow, we assume that b equals 1 and that consumers are uniformly distributed on the closed unit interval [0, 11. We are now in a position to derive the market demand functions and the index for the marginal consumers. We proceed to establish: Lemma 1. If consumers’ preferences are represented by (7), the market demand functions for the differentiated goods and the index for the marginal consumers are expressed us
(0
x3 = I@3
a*=(a3
- p3Ma3
- P3 + P2)la
- P3 + P2)1/4
(9)
(10)
if a-p,>u,-p,>O; 3A quadratic utility function is utilized in Dixit (1979). His utility function is a little more general than that used here.
80
Y Kawashimo, Product diflerentiation, prices, and market structure
(ii)
x2=0,
(11)
X3=a3-ps,
(12)
a*=l,
(13)
ijaa,-p,>O, (iii)
ifa,-p,sO
Proof as4
tf3-p3>Cl-p2;
X’=(a-p2)‘/2a,
(14)
x3=0,
(15)
if a,-p,sO (iv)
tflld
ortrl a-p,>O;
x2=x3=0,
and a-p,SO.
In cases (i) and (ii) the indirect utility functions are given respectively V2=m+~(a2-p2)2
and
V3=m+-&(a3-p3)2,
where a2 =ua. In case (i), the index for the marginal consumers is determined by V2=V3
or
aa-p2=a3-p,,
which is reduced to (lo),
It follows from (1) and (2) that in case (i) the market demand functions [(S) and (9)] are X2=~x;(.)da=~(aor-p,)da = {(~-d2
-(%-h)2)/2%
% is assumed that m is large enough
for
demand for the numeraire to become positive.
Y Kawashima,
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and market
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81
and
= {(a, - P3M3- P3+ P2W In case (ii) all consumers purchase brand 3 and no consumers pick brand 2, because a3 - p3 > a -p2. Thus, xf = 0 and xs = a3 -p3 > 0. By virtue of (I) and (2), market demand [( 11) and (12)] is
x2=0, and X3=jx$(.)dcc=a,-p,. 0
Note that the index a* is equal to 1 in that all consumers purchase brand 3. Thus, we have established case (ii). In case (iii) no consumers pick brand 3. Those consumers with an index higher than p2/a can pick brand 2, and those with an index lower than p,/a buy only the numeraire. Hence, we have (14),
where the index a* is determined by aa-p,=O.
As no one picks brand 3, we have (15), x3=0.
Finally, if u -p2 50 and a3 -p3 SO, nobody differentiated brands; i.e., x2 = x3 = 0.
can purchase either of the
Q.E.D.
In case (i) above, the Xi is positive and the index a* is positive but less than 1. These conditions will be used to determine restrictions on the set of strategies (prices) which yield an outcome as in case (i). First, the condition
82
I! Kawashima,
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and market
structure
that 0 KU* < 1 is reduced to and
p3
p3>p2+a3-a.
Positivity of X2 requires a-p,>
+(a3-P3).
These inequalities can be rewritten as P3>P2+a3
-a
and
p3<-p2+a3+a.
As X3 is also positive in case (i), p3
In what follows, our focus is mainly on case (i). The other cases will be taken up later in this section. Consider the behaviour of duopolists, each of which is assumed to produce a differentiated brand at some cost. For simplicity, assume that the production function for each firm exhibits constant returns to scale, though tirms have different costs. Let ci be their respective average costs. Profit functions are given respectively by5 .i =(pi -Ci)X’,
i=2,3,
where pi zci, i=2,3. Firms are non-cooperative. Each firm is to maximize its own profits with respect to its price, taking the other firm’s price as given. We require some notations: 8, =(a, + 2c,)/3 and e2 = (a3 + c3)/2. Now we can establish: Lemma 2. Let the optimal strategy for firm i be the strategy (price pi) that maximizes firm i’s profits, given pi for j# i. If 3(a - c2) 2 2(a, -c,), then theit ‘We shall ignore the importance of fixed costs, to which Spence (1976), Dixit (1979), and Lane (1980) pay much attention. Spence discusses the relation between product and product characteristics, while Dixit and Lane take up the entry problem using lixed costs.
I! Kawashima,
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83
respective optimal strategies are defined for p3 in the open set (tl,, 0,) and are expressed as p2={cq+2a-J(a-c,)2+3(a3-p3)2}/3,
(17)
for firm 2, and for firm 3 as P2 = {(P,
Proof
- a,)@P,
- a3 - 2C,)}/@P,
- a3 -
C3).
(18)
The optimal strategies for firms are derived from i=2,3.
aK’/api=X’+(pi-Ci)(aX’/api)=O,
For firm 2, by virtue of Lemma 1, the equation above is reduced to
The maximum profits are obtained for (l7), p2 = {c2 +
2a - J(a - cz)’ + 3(a3 -~,)~}/3,
for p3 in (13,,0,). In fact, 1~’ is positive for p2 in (~~,a-(a,-~~)) a-((a3-p3) is not less than c2 at p3=e1 if3(a-c2)L2(a3-c3).
because
As for firm 3, the first-order condition yields &?/ap,
=x3
+(p3
-c3)(xf3/dp3)
=~(a3-p3)2+P2(a3-p3)~la-(p3-c3)~2(~3-P3)+P2~/~
(20)
={3pZ-2(2a3+c3+p~)p3+a~+a3p2+c3p2+2a3c3}/a=0.
Note that rr3 is positive for p3 in (c,, a3) and equal to zero at p3 =c3 and a3. Solving (20) for p2 yields (18), P2 = {(P,
for p3 in (e,, 0,).
-a3)(3P3
Q.E.D.
-a3
-2C3)}/(2P,
- a3 -C3),
Y. Kawashima,
a4
Product
d@rentiation,
prices,
and market
structure
Eqs. (17) and (18) are reaction functions for the Cournot duopolists. They are continuous for p3 in (Or, 0,); they are depicted in fig. 2. Having deriven the reaction functions for firms we can now address the issue of existence of a Cournot-Nash equilibrium in prices.
2s+c2 3
a +2c;, 3
f2
I 0
/
p3 ,
“3
/
Fig. 2. 0, =(a3 + 2c,)/3; O2 =(a, + c,)/2; c2 =average costs of lirm 2; c) = average costs of firm 3; pi = initial price of good 3; RCI = reaction function of lirm 2; RC, = reaction function of h-m 3.
Proposition I. Let a Cournot-Nash equilibrium in prices be a combination of pf and p: such that pT maximizes xi given pi*, j# i. Then there exists a Cournot-Nash equilibrium in prices in a differentiated industry iJ 3(a-c,)12(a3-c,).
(21)
Moreover, the equilibrium point is unique and stable. Proof.
The equilibrium
set of prices (pf, ~3) is determined by (17) and (18).
Y Kawashima,
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and market
structure
85
Both curves are continuous for p3 in the open interval (e,, 0,). The reaction curve for 3 starts from zero at p 3 =Or and is monotonically increasing to infinity as p3 approaches 13~.The reaction curve for 2 is also continuous and monotonically increasing for p3 in [f7,, S,]. Furthermore, the latter gives a higher pz at pa =8i than the former, and at p3=Oz the former provides a lower p2. Hence, both curves intersect at point E in fig. 2. Note that the intersection point is unique. Using (21), we can conclude that by virtue of (17), P2ZC2
at
p3=e1,
where p2 is firm 2’s reaction to p3. In fact, substituting
p3
=8, into (17) yields
which by virtue of (21) is reduced to
Since firm 2’s reaction function is increasing in
p3,
pt
is larger than c2,
Let 1=p,-p,-a,+a.
I equals zero at point (f7,, c2) if and only if 3(a - c2) = 2(a, - cJ. Bearing in mind that the slope of A=0 is larger than that of the reaction curve for 2 and 1 is non-negative at p3 =8, and p2 =c2, we can see that an equilibrium point (pf, pj) is in the set of strategies if (21) holds. Moreover we see in fig. 2 that any initial price pi in (e,, 0,) would trigger a reaction process which leads both lirms to the equilibrium point E; i.e., the point E is stable. Q.E.D. We now turn our attention to the cases which we have so far ignored. Depending upon the values of prices of brands, one of differentiated lirms can not sell its product to its customers. Hence, we may encounter the limit price. The limit price is defined to be a price which deters entry of a new firm. Then, we can establish: Proposition 2.
The limit price in a differentiated market is given by
pf* =(a, +c3)/2 pf* =(a $2c2)/3
in case (ii), in case (iii).
86
I! Kawashima,
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dlfirentiation,
prices,
and market
structure
Proof By virtue of Lemma 1, the market demand and profit functions in case (ii) are expressed respectively as (12), X3=a3-p3, and as 7t3 = (p3
- c3)(a3
The profit-maximizing
-p3).
price is set equal to
P:* =(a3+ Cd& where it should be noted that ps* is less than a3. To this price, if firm 2 sets reaction curve for 2 in case (i), an optimal price of 2 goes up to infinity as a price of brand 3 approaches to p j*. Then, for p3 which is close to pz*, a3
where tirrn.
-P~>~-P,(P~), is firm 2’s response to
p,(p3)
p3.
Thus pf* can deter entry of a new
In case (iii) the market demand for brand 2 is given by (14), X2 =(a-p,)‘/2a, from which we have 2 = (p2 - cz)(a -p2)2/2a. Then firm 2 can set its price equal to pf*
=(a
+ &)/3,
which is less than a. As long as the conditions in case (i) are satisfied, firm 3 sets prices according to its reaction curve; as a price of firm 2 approaches pf*, that of firm 3 goes to a3, so that for some p2 we have
where p3(.) is firm 3’s reaction to p2. Hence firm 3 cannot enter the market if p2 equal to pt*. Thus entry is deterred by pz* and/or pf*. Q.E.D.
Y. Kawashima,
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and market
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87
4. Comparative statics of markets
In this section, we examine the effect of parameter changes on equilibrium prices and market shares. Preliminary to an analysis of the effect on the equilibrium market shares of changes in demand and cost parameters is a comparative statics analysis of the equilibrium prices. Let pi*’ denote the partial derivative of pi* with respect to c2. From (19) and (20), we get (3p:-2a-@;‘+(a,--pj)pf’=pf-a,
or A,pf’+
A,p;‘=
B,pf’+
B,pf’=O,
A,,
(24 (23)
and B, = 2(3p$ - 2a, - cJ -pf),
B,=a,+c,-2p3,
where it should be noted that c,
(24)
Solving (24) for pz’ and pf’ yields
pf’ = A,B,ID,
(25)
pT’= - A,B,/D,
(26)
and
where D is the determinant of (24). The sign of (25) and (26) depends solely upon the sign of D. We can show that D is positive. Lemma
3.
The determinant D is positive.
Proof: By virtue of (19) and (20), the slopes of the reaction curves for firms 2 and 3 are given respectively by a~2/ap312
=
-(a3
-
pX3p,
-2a-c,),
Y Kawashima,
88
d@xvhation.
Product
prices,
and market
structure
and ap2/ap313 =
-
-
mP3
203
-
c3 -~~)/(a3
+ C3 -2P3).
As (19) can be rewritten as 3{p,-(c,+2a)/3)2-(a,-~,)2=(a-c,)~/3,
then the sople of the reaction curve for firm 2 is at most f. That for firm 3 is increasing with p3 in (Or, O,), and substituting p3=e1 and p2=0, we have
ap2iap313 > 6. Hence, at an equilibrium
point
ap2iap313 =- aP2iaP312r which is -&IA,
-E --MB,,
or A,B,-A,B,>O,
(27)
where it should be noted that A, is negative and B, positive.
Q.E.D.
By virtue of Lemma 3, we have Pf’>O,
(28)
p$‘>O.
(29)
and
For the effect of c3 on the pr, everything would go through Now we establish: Proposition 3. The effect of average cost changes on equilibrium positive. Formally, api*facj >O, Proof
i, j=2,3.
The effect of c3 on the pi” is expressed as
as we did.
prices is
(30)
I! Kawashima,
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89
and B,=2($-a,)-pf
(30’)
and p:‘=A,B,/D>O.
Q.E.D.
(30”)
Thus, this proposition states that the cost change effect on the pi* is symmetric in the sense that cost reductions of a firm lead to reductions of the pi*. Taste parameters in the utility functions are supposed to lead the pr to change. We can summarize: Proposition 4. Changes of taste parameters a, a3 cause equilibrium prices pt to change according to the following:
ap;laa, <0, ap:iaa, i20,
(31)
and
Proof:
ap:laa>o,
ap: jaa > 0.
Differentiation
of (19) and (20) with respect to a3 yields
and A,=a,-pT>O, B,=4pS-2a,-2c,-pf
Solving for p”’ yields
(32)
90
Y. Kawashima,
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differentiation,
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and market
structure
and
Similarly, routine calculations would yield dpz/da > 0
and
apflda > 0.
Q.E.D.
Changes of taste parameter a has the positive effect on the pr; if firm 2 can succeed in raising its own absolute advantage by advertising, it then can increase not only its equilibrium prices but also those of firm 3. The difference between pf’ and ~3’ is pf’-pf’=A,B,/D+A,B,/D=(A,/D)(B,-B,),
(33)
and A,=2pf-a-c,
To the extent that Bz is negative and 8, positive, the difference is positive by virtue of Lemma 3. Hence, advertising by firm 2 leads its price increase to be higher than that of firm 3. On the other hand, a taste parameter a3 has the surprising effect on equilibrium prices. It is sure that a3 has the negative effect on his rival’s prices, but has the ambiguous effect on his prices. Using (32), we can derive the difference
(34) where the pr’ denotes the partial derivative of the pi* with respect to a3. Firm 3 can raise its prices higher than its rival’s ones. Let us turn our attention to market share. Let X2/X3 be y and the share for i be pi. Noting that X2=!
.+
(acr-pf)dcr
and
X3=y(a3-p:)da, 0
y depends upon a*, pf, a and a3. Formally,
y = Ha*, Pf, PS,a,a3).
(35)
1! Kawashimo,
Differentiating
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91
y with respect to ci, we get
dy/dCi = ay/auIa=.. aa*/ac, t ay/ap,)(apf/ac,) i=2,3.
+ &lap3)GWWci),
(36)
To determine the sign of (36), we will demonstrate the following lemma:
ay/a&=.. is negative.
Lemma 4. Proof:
Let Y’=ixz(.)da a
Differentiating
and
Y3=jx:(.)da. 0
( Y’/Y3) with respect to c( yields
4 y2/yw~ = Lb2
- aab - {(a/2 - p2) - ((a/2)a2 - p,~r)}-j
= b2 41 + ~‘NCa3 -p3b2. Substituting a=a* into equation above, the numerator is reduced to p2
-(a/N1
+u2)=P2
-(a/N(a3-~3
+p212 +a2)la2
= -{(a3-p3)2+2p2(a3-~3)+(a-~2)2}/2a<0. As (X2/X3) =( Y2/Y3) at an equilibrium,
we are done.
Q.E.D.
The index for the marginal consumers has the following properties. Then we have: Lemma 5. Cost changes lead the index for the marginal consumers to change according to the following: au*/&, >O for
i=2,
(37)
i=3.
(37’)
When a taste parameter changes, its effect on the index is
aa*/aa,> 0, EER-D
(38)
L Kawashima,
92
Product
differentiation,
prices,
and nwket
structure
rind
act*/aa=(i/a)(plr'-pS'-a*)~o. Proof:
Differentiating
(39)
(10) with respect to c2, we get
aa*/ac,=(l/~)(plS'-pS').
(40)
Together with (25) and (26), (40) is rewritten as (37) where B, +B,=4p3 -3a,-2c, Similarly, we can get 8a*/&,=
-(BS/D)(A1
-pf ~0
because
ps
is less than (a3 +cJ/2.
+A,)
(37’)
Note that A, +Pl,
=(l/aD){A,B,-A,B,+(a,-PS)(c,-a,
=(l/aD){(A,
+A,)(2p;-2a,-pf)}
-Pa-4&J
>O.
(38)
I! Kawashima,
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93
The effect of a is written as act*/da = {a(~;’ --pg - (a3 - p: + p;)}/d = (l/a)(pt’ - ps’) - a*/a
where the pf’ a. Q.E.D.
denote the partial
derivatives
of the pf’ with respect to
We now establish the relationship between the market share of a firm and its cost reductions. First, we have the following: Proposition 5. An increase (or decrease) in average costs for firm 2 causes equilibrium output ratio y to decline (or go up) when at an equilibrium
-(l -a*)/a*)+y/6<0. Proof:
(40)
By virtue of (8) and (9),
a?
GP:‘+p,P;‘=
LY*-1
a7
xj
1
P:‘+YFa*P:’
u*
l-a*
=x3 PY -.+Y$
i
=p
u*
l-a* PT’ --i a*
I
B, yg. I
(41)
Note that at an equilibrium
ap2 ap3 3=-
2(2p;-2a,-c,-p;) a,+c,-2pS
B, = -g>6. I
From (41), we get
!* 3p2 *' X
(41’)
By virtue of Lemmas 4 and 5, the first term in (36) is negative for i=2. Hence, when (40) holds at an equilibrium, so that (41’) is negative, (36) is also negative for i = 2. Q.E.D.
94
I: Kawoshima,
Product
diferentiation,
prices,
and market
structure
According to this proposition, firm 2 can increase its market share by reducing average costs when (40) is true at an equilibrium. But this seems counter-intuitive in that firm 2’s cost reductions may lead declines of the market share of the firm. On the other hand, its rival can capture larger shares by R&D activity of the former. As for cost reductions of firm 3, we can establish: Proposition 6. Cost reductions ofJrm 3 has a positive e&t market shares when the following is met at the equilibrium:
on its equilibrium
-(l-a*)/a*+3y>O. Prooj:
(42)
For i= 3, the first term of (36) is, by virtue of Lemmas 4 and 5, ay/aal,=e &i*/ac, > 0.
(43)
The last two terms are
ci*
ayap:+ardp:=a*-1 FPt’+Y ap2 a+
x3P:’
apJ ac,
u*
l-CC*
=x3 Pf’
-7+Y$
i
a* ‘SPY
---1-c?* a* i
i A, T/ 1
(44)
where we make use of (30’) and (30”). Noting that - AI/A1 is the slope of the reaction curve for firm 2 at the equilibrium, which is at most f (44) is rewritten as a*
3
Pf’
-~-,~}>~p:~~-~+3y)>o, I
(45)
by virtue of (42). Together with (43) and (45), we finally conclude that
when the equilibrium with y, we are done.
satisfies (42). As the market share for firm 3 decreases Q.E.D.
I! Kawashima,
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and market
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95
This conclusion is also surprising in that even if a firm can succeed in developing more efficient ways of production by R&D activity, it may not necessarily capture more customers. Then it can increase its share only when the equilibrium satisfies the specific conditions, which are not considered extreme. In view of Propositions 5 and 6, price strategy of differentiated firms is not always essential to increase their respective market shares. 5. Conclusion This paper has examined when a Cournot-Nash equilibrium in prices in a differentiated market does exist. It has been assumed that consumers’ preferences are represented by a quadratic utility function, consumers are uniformly distributed, and that average costs are held fixed for both firms. The mode1 has also explored the cases in which the limit price pertains, depending upon the values of the net absolute advantages of both firms. R&D activity and advertising play key roles in a differentiated market, and their effects on the market have taken up on the basis of the model. Cost reductions by R&D of a firm lead not only to a decline of equilibrium prices, of the firm’s product, but also to that of rival’s product. On the other hand, advertising has the more complicated effect on equilibrium prices; taste changes, which come from advertising activity, may fail to raise equilibrium price of brand 3 even if firm 3 can make its absolute advantages of the brand higher, while firm 3’s advertising lowers prices of the rival’s brand. This seems counter-intuitive, but changes of prices of firm 3 are higher than those of firm 2. To the extent that a taste parameter is changed by firm 2’s advertising, both prices go up. Of course, an increase of prices of firm 2 is higher than that of firm 3. Hence, advertising effect is not symmetric. R&D and advertising are also expected to influence market shares of firms. The present mode1 has indicated that their effects on the shares are in general ambiguous; R&D can provide a larger share to a firm only when an equilibrium satisfies some specific conditions, which are considered extreme. Note, however, that these results have been derived in a specific model. References Bertrand, J., 1883, Theorie mathematique de la richesse sociale, Journal des Savant 48, 499-508. Dixit, Avinash, 1979, A model of duopoly suggesting a theory of entry barriers, Bell Journal of Economics 10,20-32. Eaton, B.C., 1976, Free entry in one-dimensional models: Pure product and multiple equilibria, Journal of Regional Science 16, 21-33. Gabszewicx, Jean Jaskold and Jacques-F. Thisse, 1979, Price competition, quality and income distribution, Journal of Economic Theory 20, 340-359. Gabszewin, Jean Jaskold and Jacques-F. Thisse, 1980, Entry (and exit) in a dil’lerentiated industry, Journal of Economic Theory 22, 327-338. Gabszewicx, Jean J., A. Shaked, John Sutton and Jacques-F. Thisse, 1981, International trade in a ditTerentiated industry, International Economic Review 22, 527-534.
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