International Journal of Industrial Organization 5 (1987) 175-192. North-Holland

B E I N G A L E A D E R O R A F O L L O W E R

Ref lec t ions on the Distribution of Roles in Duopoly

M a r c e l B O Y E R *

Universitk de Montrkal. Montreal, Que., Canada H3C 3J7

M i c h e l M O R E A U X *

Universitd de Toulouse, 31042 Toulouse Cedex, France

Final version received November 1986

This paper deals with the analysis of conflicts in duopoly situations, in particular with the distribution of roles in the von Stackelberg framework: which firm will be the leader, which firm the follower? Under what circumstances will firms find a mutually advantageous distribution of roles? Will they fight for leadership or followship? In von Stackelberg's analysis of duopoly, either firms do not choose their respective roles or they fight for leadership. This situation is due to the particular strategy space, namely the quantity to produce of an homogeneous good. We extend here the strategy space to price-quantity pairs. We show that if costs are identical or similar, then both firms will prefer the role of follower; if there is a significant cost differential between the firms, then the non-cooperative equilibrium can only be of two types: either the less efficient firm will act as the leader, selling a limited quantity at a low price, and the more efficient firm as the follower, selling to the residual demand at a higher price, o1" the more efficient firm acting as leader will drive the less efficient firm out of the market by adopting a limit pricing strategy, but in so doing that firm makes less profits than if it acts as follower. We suggest that it is not unreasonable to expect that firms in a duopoly framework will tend to coordinate on their mutually advantageous role distribution even if they compete in a non-cooperative strategic way in terms of both prices and quantities.

1. Introduction

This p a p e r dea l s w i t h t he d i s t r i b u t i o n o f ro les in d u o p o l y s i t u a t i o n s . I t is a n e v i d e n c e t h a t in a n y t w o p e r s o n s g a m e , t h e r e will e i the r be a conf l i c t o v e r

t he d i s t r i b u t i o n o f t he ro le o f l e ade r a n d fo l l ower o r a c o n s e n s u s o n w h o s h o u l d be the l e a de r a n d w h o t h e f o l l o w e r ) In d u o p o l y t heo ry , i n c l u d i n g the bas ic ana lys i s o f v o n S t a c k e l b e r g (1934), 2 m o s t m o d e l s a s s u m e o r de r i ve t h a t d u o p o l i s t s will prefer , a n d t h e r e f o r e f ight , to be t h e leader . In p a r t i c u l a r , in

t he d u o p o l y t h e o r y w i t h an h o m o g e n e o u s g o o d , t h e r e is a c l ea r a d v a n t a g e to

*We want to thank C. Crampes, A. Hollander, R.E. Kihlstrom and J.-J. Laffont for their comments. Of course we remain solely responsible for the content of this article.

~Moulin (1982) gives many examples of these different possibilities. 2See also Fellner (1949).

0167-7187/87/$3.50 O 1987, Elsevier Science Publishers B.V. (North-Holland)

176 M. Boyer and M. Moreaux, Being a leader or a follower

be the leader when the strategy space consists of quantities produced and marketed by the duopolists. Price competition alone ~t la Bertrand (1883), on the other hand, leads to misspecified models. In this article, we start with the basic duopoly theory of von Stackelberg but with a different strategy space, namely price-quantity pairs. We will consider the duopolists making decisions about price and quantity to be produced. 3

Industrial Organization economists have generally considered that a firm or firms in a duopoly context can choose either a price or a quantity but not both since the two variables are related by the consumers' demand function. However, casual observation of firms' behavior suggests that they do in fact choose both a price and a production level when putting a product on the market. The economists' standard model may allow a reasonable approxi- mation in terms of analysis and predictions even if it is not 'descriptive' of firms' behavior. But one would like to see a formal analysis of equilibria in models where firms can choose both prices and quantities; in such an approach, the consumers' demand function appears as a maximum or a bound on firms' choices which need not be on the demand function; they cannot of course be beyond that function but they may be wi th in it. One may conjecture that equilibrium prices and quantities will be on demand functions but such a result should not be assumed at the start by restricting firms' decision variables to be either prices or quantities. In fact, we will show in this paper that allowing firms to choose price-quantity pairs will have a profound impact on the nature and characteristics of the equilibrium.

For simplicity, we will consider a linear model. We will show that when the cost functions are identical, then each duopolist will prefer to be follower rather than leader, a result totally different from that of the usual theory with quantities as strategy space. This result is robust to moderate pertur- bations of the cost functions, i.e., the role of follower remains more advan- tageous, in case of not too large cost differential, both for the least efficient firm and for the most efficient one. For large cost differentials, however, the picture is totally different. It would then be mutually advantageous for the two firms that the least efficient one takes the leadership role, and not the most efficient as one might have expected. 4 Moreover, the leader and the

3This kind of competition has previously been suggested by Shubik (1959) and also by Shubik and Levitan (1980), but to briefly study the Nash equilibria only.

'~This convergence of interests has already been noticed by Ono (1978), but in a model where the strategy spaces of the firms are different from each other, giving rise to an analysis quite different from ours. In Ono's model, the leader determines a price and at that price the follower can choose to produce any quantity he wants to sell, that is, the quantity for which his marginal cost equals the price set by the leader. Why would the leader let the follower get the market share which he, the follower, wants to have at the leader's price? This is a privilege which is granted by the leader to the follower and which is contrary to the leader's interest. No proper justification of such philantropic behaviour by the leader is given by Ono. The same comment can be formulated for the competitive fringe model since it is basically the same model. But in a duopoly or oligopoly context, the question is even more relevant, given the small number of agents.

M, Boyer and M. Moreaux, Being a leader or a follower 177

follower will charge different prices although the product is basically homogeneous. However, since the leader, who charges the lower price, will choose a production level strictly smaller than the quantity demanded at that price, he will need to ration his customers. Facing the residual demand of consumers not served by the leader, the follower will charge a higher price, making more profits by doing so than by 'undercutting' the leader's price and capturing the whole market. Hence an outcome very different from the usual ones.

The paper is organized as follows: the basic model itself is presented in section 2 and the different cases are discussed in sections 3 and 4. In the conclusion, we discuss some stylized facts which tend to give support to our results on role distribution.

2. A model of competition through both prices and quantities

Consider the market of an homogeneous good with the demand function and its inverse being

Q=Q(p) and p=p(Q). (1)

We will assume a linear form of p(Q).

p = m a x { a - b Q , O}, a,b>O, Q>O. (2)

Consider two firms i, j ~ { 1,2}, i¢j; let Pi be the price proposed by firm i and qO the quantity the firm supplies at price pi, which is not necessarily equal to the quantity the firm will sell. At price p~, the firm will sell qO or less but not more. Let us characterize the demand facing firm i that is

d 0 0 qi(Pi,Pj, qj), a function of its price pi, the price pj and the quantity qj chosen by firm j. We will assume that the demand d o q~ (P~, Pi, q J) is the contingent demand defined by Shubik (1959). If buyers are perfectly informed about the proposed prices of each firm, then they will buy from the firm with the lower price, namely

d 0 qi (Pi, P j, q j) = Q(Pi) if pi < p j, Vq °.

If proposed prices are identical, then buyers are indifferent between the two firms since products are identical. Hence the only reasonable assumption here is that they will go at random to one of the two firms; with a large number of buyers, each firm will catch half the market. Therefore, for Pi = P j, and q°>=Q(pj)/2, we have q~(pi, pj, q~)=O(pi)/2. However, if p~=pj and qO < Q(pj)/2, then a number of buyers who go to firm j for the good will not be served; they will then patronize firm i and therefore in this case

J . l ,O.-- D

178 M. Boyer and M. Moreaux, Being a leader or a follower

q~(Pi, P j, o o qj)=Q(p~)-qj. If pi>p~ then all the buyers go to firm j and if qO > Q(pj) then they all buy from j and qf(p~, p j, qO) __ O. However, if qO < Q(p.i) then again some buyers cannot be served by j at price pj. Firm i can under those circumstances satisfy this residual demand at a price p~ > pj.

The specification of this residual demand depends on the rationing scheme by which the excess demand at firm j is turned away. We will assume in this article that firm j whose price is lower and whose quantity produced is smaller than Q(pj) willserve the total demand expressed by a fraction q~/Q(Pi) of demanders representing an unbiased sample of all demanders. 5 In those circumstances, the demand facing a duopolist will be the contingent demand of Shubik.

d 0 qi (Pl, P j, q j) = Q(Pi) if pi < p j,

=Q(p,)/2 if pi=pj

=Q(p,)_qO if pi=pj

=[1-q°/Q(pj)]Q(p~) if p,>pj

= 0 if p~>pj

and in linear model case (2), we get

d 0 qi (Pi, P.i, qJ ) = (a -- pl)/b if Pl < P j,

=(a-pi) /2b if pi=pj

= [ ( a - p~)/b] - qO i f p~ = pj

=[I qO l ( a _ p i ) (a---pj)/b] ~ if Pl > Pl

=0 if pi>pj

and qO>Q(pj)/2,

and qO < Q(pj)/2,

and qy<Q(p~),

and qO>Q(pj), (3)

and qO > ( a _ p y 2 b ,

and q° <(a-pj) /2b,

and qO < ( a - pj)/b,

and q°>(a-pj ) /b .

(4)

The different cases are illustrated in fig. 1. But other rationing schemes are possible. [For more on the subject, see Dixon (1984).]

Consider a firm which offers a quantity qO at a price Pl. This quantity < d 0 supplied will be effectively sold only if qO =q~(Pi, P j, q j). Therefore, the profit

SOur results depend in part on the particular form of rationing applied by the low price firm. The form of rationing also has important implications for the consumers' welfare.

M, Boyer and M. Moreaux, Being a leader or a follower t 7 9

" - T ~ i I I

l i ! N .

Q(Pi) Q(pi) 2

(a) q~ < Q(pi)/2

Pi

Q(Pi) Q(Pi) q 2

(b) q~ = Q(pj)/2

q~ Q(Pi) q

(c) Q(pj)/2 < q~< Q(pj)

r Pi . . . . .

I I i : \

Fig . 1

q

(d) qj >Q(pj)

of duopolis t i will be

p, ' rain {q0, q~(p~, p~, qO)} _ Ci(qO),

where C~(q °) is the cost of producing the quanti ty put on the market, sold or not.

180 M, Bayer and M. Moreaux. Being a leader or a follower

We can define a Nash equilibrium for such a duopoly:

Definition 1. NBE (Nash-Bertrand-Edgeworth) equilibrium. ~,Fl,k,2,~/1/-* n* ~ 0 . , qO,) is an NBE equilibrium if:

(1) Vi: (p*, q~') is a solution to

max Pi 'min{q°, d * O* qi (Pi , P j , q j ) } - C i ( q ° ) , (pi, qOi )

(2) Vi: p* ' o, a , , ' mln {q, , qi(P,, P j , qO,)} _ Ci(qO,) >= O.

In general, such an equilibrium will not exist; in particular in a linear model where

Ci(q i ) = ciqi, i= 1, 2, (5)

and where the demand functions are of type (4), there exists no non-trivial equilibrium, i.e., for which qO, > 0, i = 1, 2.

Proposition 1. Under demand functions (4) and cost functions (5), there exists no non-trivial N ash-Bertr and-Edgeworth ( N B E) equilibrium.

Proof. Note first that if w1,v2,nlrn* ,* ,,o,, qO,) is such an equilibrium, then:

V p*: ,7* < ad(n* n * ,'I0"~ "ai .~- "li'.Yi ~ ~ j ~ "lj 1~ (6a)

{p ,>c l }~{qO, d , , o, =qi(Pi ,Pj ,qj )}. (6b)

The first inequality is due to the fact that if qO,> d , , 0* qi(Pi ,P j , qj ) then profits could be increased, for given (p*, qO,), by holding p~' constant and avoiding

0* d , , 0 , the cost of q~ -q~(p~, p j , qj ) which is offered by i but not sold. Considering (6a), if (6b) is not satisfied then qO, d , * <qi(Pl ,pj,qO,), but since p*>cl, firm i could increase its profits by maintaining p* and increasing its supply. Given (6a) and (6b) and letting ~=max{cl ,c2}, the prices p~' and p* can either be equal above ~, equal at ~ or different from each other. Let us consider each case separately.

(a) p~(=p,3=p,>~=max,=l .z{cl} .By(6b)qO, d , , o, = qj ), construc- • qi (Pi , P J , By tion of functions qd(.), we have qi(Pa,P~,a, qE°*)q-q2(P2,Pl,qla , , o , )=Q(p ), and therefore qO,+qO,=Q(p,). Under those conditions, if O<qi*~Q(p* ) then firm j can capture the whole market by reducing slightly its price, With a small enough reduction in price, it increases its profits. So if Pl and P2 are equal in equilibrium, they cannot be larger than ~.

M. Boyer and M. Moreaux, Being a leader or a follower 181

(b) * - * - * P l - P 2 - P =& Then, either c1=e2=~ or say d=c~>c 2, In the first case, if q* > 0, i = 1, 2, then a firm can realize a positive profit by increasing its price since q~(P~,Pl, q~*) remains positive for some pj>p*. In the second case, if q*>O,i= 1,2, then the low cost firm can reduce slightly its price, capture the whole market, and increase its profits. Therefore prices cannot be equal in equilibrium. We are left with only one possibility.

(c) Px* =/:P2.* If q°*>O,i=l,2, then the low price firm can increase its price, capture the whole market and increase its profits. Therefore no such equilibrium exists• [ ]

Let us now assign index l to the leader and 2 to the follower, and define the follower's best reply correspondence, QP2(pl, qO), by

, o , , O* OP2(Pt,q°)={(P2,q2 )](P2,P2 ) is a solution to

• o d 0 max P2" mm {q2, q2(P2, Pl, ql)} -- Cz(q~)}. (P2, qO)

(7)

We can then define a v o n Stackelberg equilibrium with competition in both prices and quantitites as follows: 6

Definition 2. Stackelberg equilibrium with competition in both prices and quantities. (p*,p~, qO,,qO,) is a v o n Stackelberg equilibrium with competition in both prices and quantities if it is a solution to

max Pa min {ql, q~(Pl, P2, q~)} -- Cl(q°),

subject to

(P2, qO) e QP2(P~, qO).

Although a Nash equilibrium does not exist for the linear model (4)-(5), we will show in the next section that the von Stackelberg equilibrium does exist.

3. The firms have the same cost function

Let c be the average cost of each firm. Without toss of generality, one may consider a simple linear inverse demand function and zero cost, namely

p = l - Q and c=0 . (8)

Let us consider the reaction of the follower. First note that under (8) the monopoly price and output are both 1/2 giving a profit of 1/4. If the leader

6See Moulin (1982).

182 M. Boyer and M. Moreaux, Being a leader or a follower

sets his price p~ ~ 1/2 then whatever be the quanti ty the leader offers, the follower will set a smaller price, get the whole market and make monopoly profits or almost. Therefore the leader will have a positive market share only if pa < 1/2. But what market share can he get at best in this case? Suppose Pl < 1/2 and let x be the fraction of the demand at price Pl which the leader decides not to satisfy, i.e.,

q O = [ l _ x ] Q ( p l ) = [ 1 - x ] [ 1 - p l ] . (9)

In such a case, the follower has two options: either let the leader sell quantity qO at price p~, set a price P2>P~ and sell to that fraction of the consumers not satisfied by the leader or set a price slightly lower than-p~ and capture the whole market. 7 For a given Pl, there is a minimal fraction ~(pa) for which the two options will appear indifferent for the follower and such that for x>~(p~) he will let the leader sell at Pl by setting a price p2>p~ and for x<:~(pl) he will undercut the leader's price and get the whole market.

Let us characterize the function Y(px). The demand function facing the follower for p 2 > p l is

qa2(P2, Pl, [1 - -x ] [1 - - P l ] ) = x [ 1 --P2], (lO)

and his profit is then

x[-1 - P2]P2. (11)

This profit is maximized for P2 = 1/2, This is the monopoly price since the follower will behave as a monopolist with respect to the customers not served by the leader. His maximal profit is therefore x/4. If on the other hand the follower sets a price slightly less than Pl, say P l - e , he can sell to the whole market and realize a profit equal to ( p ~ - e ) ( l - ( P l - e ) ) = p x ( 1 - P l ) - ( 2 p 1 - 1 ) e arbitrarily close to ( 1 - p 0 p t . The function x(Pl) is therefore defined by the following condition:

and so

x / 4 = ( 1 - p l ) p i , (12)

Y(p~) =4(1 - P I ) P l , 0 < p t < 1/2. (13)

Let us denote by c]°(pl) the maximal quanti ty which the leader can offer at

7He could also set a price p2=pj and sell Q(p~)/2 if q°>Q(pl)/2 or Q(pl)-q ° otherwise, but this possibility is dominated by P2 = Pl - ~ and q2 = Q(P2) since then 172 ~ Pl Q(Pl) > Px [Q(p0 - qO]. Therefore, the follower will never choose to sell at the leader's price. Recall that in this leader- follower framework, the leader sticks to his choice.

M. Boyer and M. Moreaux, Being a leader or a follower 183

price p: without risking to see the follower undercut his price and capture the whole market; and let ~(pl)=(1-pl)-gl°~(pl)=~(p~)[1-p:] be the minimal quantity which he then should not serve. We have

~(pl )=4[1-p:]2p: , 0 < p t < l / 2 . (]4)

This function goes to 0 at p: = 0 and its derivative is zero at Pl = 1/3 where it attains a maximum. At p: =0, we have d~/dp~ = 4 and at Pt =1/2, d~ /dp : = - l=dQ(p l ) /dpv In fig. 2, we illustrate the demand function and the function ~(Pl). For a price Pl, the maximal quantity q°(pl) which the leader can sell without inducing the follower to undercut p~ and capture the whole market is the horizontal distance at price p~ between ~ and the demand function. The profit H1 realized by the leader at price p~ is the hatched area in fig. 2. It is that surface which the leader will try to maximize. But this implies the conclusion that it is always preferable to be the follower. Suppose, for instance, that the maximum is reached at price p~ shown in fig. 2. The follower's profit is given by (12) and clearly

I12=x/4=pl(l - p : ) > p:(1 - p : - ~(p:)) = / / : .

We have therefore proved:

Proposition 2. In the linear duopoly model with competition in both prices and quantities, each duopolist prefers to be the follower when they have the same cost function.

p

1/2

- Q , q ~

~(Pl) Q(Pl)

• 1 ,, Im

Fig. 2

184 M. Boyer and M. Moreaux, Being a leader or a follower

4. The firms have different cost functions

Again, without loss of generality, we can assume, in our linear model, that

p = l - Q , (15)

0 < c = a v e r a g e cost of the higher cost firm (c < 1), (16)

0 - average cost of the lower cost firm. (17)

p can then be interpreted as the average profit of the lower cost firm and c as the cost differential between the two firms. To clarify the presentation, we will now use a system of two subscripts: a and b representing, respectively, the lower cost firm and the higher cost firm, 1 and 2 representing as before the leader and the follower, respectively. So P.1 will stand for the price set by the lower cost firm acting as leader.

Consider the lower cost firm a. One may a priori expect that this firm will act as leader. But as we will see shortly, it may not be profitable for the lower cost firm to act as leader. It is clear that by setting a price slightly lower than the cost of firm b, firm a can drive it out of the market. Its profit is then close to (1 -c )c . But duopolist a acting as leader may not find it profitable to drive duopolist b out of the market, He must consider the possibility to set a price P.1 >c , let firm b take a positive market share just large enough so that firm b sets a price higher than Pat and does not undercut PaX and capture the whole market. Let 2at(P.t) be the minimal fraction of the demanders which at Pal the leader must not satisfy, i.e., leave to the follower. If the follower sets a price Pb2 > P.~ and sells to that fraction x of the demanders, then his profits will be

x[1 --Pb2][Pb2--C], Pb2>P.t ,

which is maximized for Pb2 =((1 + c)/2), that is the monopoly price. His profit then is

/1 -c'¢

In order for this profit from setting Pbz > Pat to be larger than or equal to his profit from undercutting P.1 and capturing the whole market, it is necessary that

M. Boyer and M. Moreaux, Being a leader or a follower 185

whose solution is x.~(P.1),

)~al(Pat)---'~ ~ [ 1 - - p a ~ ] [ P a l - - C ] • (18)

Price Pal cannot be set higher than ((1 +c)/2), the monopoly price of firm b, unless X.l(pal)=l [one can check that in (18) Y . I = I if p. t=(( l+c) /2)] . Therefore, the minimal quantity demanded which the lower cost firm acting as the leader must leave unsatisfied, f°l(p.1), is

~al(Pal)= 1--p. t if p.1>=[1+c]/2,

= ~ - c [I -P"I]2[P"I--c] if [ l + c ] / 2 > p . t > c ,

= 0 if c>p~ 1. (19)

The function [2 /1 - c ]2 [1 -p.112[p.1 - c ] goes to 0 on the domain [c, (1 +c)/2] at p.~=c. On the other hand, _-0 + dr.l(p.1)/dp.l is equal to 4 at P.1 = c and to - 1 =dQ(p.t) /dp. t at P.t =(1 +c)/2. The maximum of the function is reached at Pat = 1/3 +(2/3)c.

If we consider the cost differential c as a parameter, then the two limit- cases to consider are given by c=1/2 and c = 0 because for c>1/2, firm a is setting the price p.a at its monopoly level and still drives firm b out of the market. For c = 1/2 we have

~(p~ l ) = 1611 --p.t]2[p,, a -- 1/2], 3/4>p.~ > 1/2,

which takes value 1/4 for p . t=3/4 , i.e., 1 -p .1 , goes to 0 at p .~=I /2 and reaches a maximum at p.~ =2/3, For the second limit case, c = 0 we have

~1 (P.a) =411 --pal]2p.l,

which is nothing else but (14), The lower cost firm acting as the leader will therefore set p.~ for which

//.x = [[1 --P.1] -- ~ t (P.t)JP.a

is a maximum. For c close to 0 the maximum is reached for Pat>C. We are then in a situation similar to the situation when both firms had identical costs and so firm a acting as leader will not drive the competitor out of the market, But if c is close to 1/2 then firm a acting as leader will set a price slightly lower than c and capture the whole market. For which value of c will

186 M. Boyer and M. Moreaux, Being a leader or a follower

P

3/4

2/3

1/2

1/3'

2/9

q

~ ( P a l ) as a function of c

Fig. 3

the lower cost firm acting as leader go from one strategy to the other? To find such a critical value, let us consider first an arbi trary value c. If firm a drives firm b out of the market, its profit will be [ 1 - c ] c , Consider now a price p.l>c. In order for firm a to make a profit at least as large as in the first case, it must be able to sell a quantity at least equal to [1-c]c/p , ,1 . The curve [1--C]C/Pal gives therefore the minimal quanti ty which the lower cost firm must sell at price Pal to make a profit at least equal to the profit it would realize by driving firm b out of the market. On the other hand, the maximal quantity which the lower cost firm can supply at price Pal without inducing firm b to undercut the price and capture the whole market is given by [1--Pal]--~al(Pal)" Therefore, the lower cost firm acting as leader will not drive firm b out of the market (by setting a price slightly lower than c) if and only if there exists a price level Pal in the interval (c, 1 - c ) such that

[1 - p.l-I - ~l(Pal) > I-1 -c]c/pax, (20)

Note that for P.1 =c, both sides of inequality (20) take the same value. Fig. 4 shows the two functions [1-- p.1] -- ?aOl (pal) and [1--c]c/p.1 for different values of c. For c = 1/2, it is impossible to sell more than [1 -c]c/p.1, since that function is then tangent to the demand function at Pal = 1/2. For c=c' close to 0, inequality (20) is verified for some interval p,,l~(c',d'), Finally, there exists a critical value ~ of c at which both functions in (20) are tangent. For c>? , inequality (20) cannot be satisfied, and for c<~ , it is satisfied for some interval of p.~. To determine ~, first note that we must have at P.1 = g the same slope for [l--pal]--~aal(Pal) and for (1--C)C/Pal. Since for paa=c, both

M. Boyer and M, Moreaux, Being a leader or a follower 187

1-5

3/4

(1 + 5/2) (1 + c')/2

1/2

d'

5 = 0 , 1 6 6

C'

C = C '

• ~ . . q (1-c) c/pal

N o t e : ( 1 - p a l ) - r ° a ; ( P a l ) is '1~ ~ . ~ - cu r ve ~ i f c = l / 2

~, ~ , ~ . ~ . - c u r v e /~ i f C = - c u r v e 3' if C = c '

_ _ ~ q

Fig. 4

functions reach the same value, it means that

and the condition that both slopes be equal implies

- 5 = ( 1 - e ) e = ( 1 - e ) e pa21 ~2 '

and therefore

e=]/6. For c = 1/6 the profit of the lower cost firm a acting as leader is maximized

on (~,(1 +~)/2) at p.~ = 1/6 since its profit is given by

r/°l(p.,) = f f l - p°~ ] - ~ (p .~) ]p . l

=[rl-p°/1- [T~eJ2 E1 _p°~]2[p+,-c-3]p°1,

188 M. Boyer and M, Moreaux, Being a leader or a follower

which, for PaX =C= 1/6, satisfies

d2/"/a 1 dH~----L = 0, <0. dp.1

Therefore, for the critical value c = 1/6 of the cost differential, there is only one optimal policy for the lower cost firm acting as leader: set a price Pat = 1/6, produce and sell q° 1 =5/6. The higher cost firm acting as follower cannot then realize a positive profit and leaves the market.

Proposition 3. In the linear duopoly model with competiton in both price and quantities, there exists a critical value (c= 1/6) of the cost differential between the two firms such that:

(a) if c < 1/6, the lower cost firm acting as leader will not drive out the higher cost firm acting as follower,

(b) /f c_> 1/6, the lower cost .firm acting as leader will drive out the higher cost firm acting as follower.

It is interesting to note that in case (b) above, the price set by the leader is p , t=c and not the monopoly price P,1 = 1/2. We have here a limit pricing situation.

The analysis above shows that when the cost differential between the two duopolists is small, we are, insofar as the distribution of roles is concerned, in a situation similar to that with identical cost functions: both duopolists prefer - and therefore fight for - being the follower. But the situation changes if the cost differential is high enough to induce the lower cost firm acting as leader to select a price-quantity pair which effectively drives the higher cost firm out of the market, or bars the entry into the market of the higher cost firm. I f the cost differential is that high, then the two firms would find a mutual advantage and therefore incentive to permute their roles: they can both make more profits! We have seen that if the lower cost firm acts as leader with c > 1/6, then the respective profits are

/ / , l = [1 - c ] c and /-/b2 = 0,

Let us permute the roles. We can then find the function ~l(Pb~) giving the minimal quantity demanded which firm b acting as leader should not satisfy in order to prevent the lower cost firm acting as follower from undercutting the price and capturing the whole market. Since the cost of firm a is assumed to be 0 here, the function ~bl(Pb~) is of the same form as (14).

r~l(pbl )= 1 -Pbl if Pbl> 1/2,

=4[Pb~--2p21 +pbal] if 1 /2>pbl>0.

M. Boyer and M. Moreaux, Being a leader or a follower 189

Firm b acting as leader maximizes

rib 1 -- [[1 --Pbl] -- ~1 (Pbl)] [Pbl --C].

The maximum is reached for a value/361 between c and I/2. Firm a acting as follower will choose a price P,2 = 1/2, its monopoly price, and sell to that fraction of demanders not served by firm b at price/361. By construction, firm a will realize a profit just equal to the profit it would realize by setting a price slightly smaller than /)bx and capturing the whole market, namely ./-/aZ=l'l--~bl']/)bl. Since O<c<Pbl<l /2 and since ( 1 - x ) x in an increasing function on the interval (0, 1/2), the profit of firm a as a follower will be higher than its profit as a leader [ 1 - c ] c . We have therefore shown that for c = ~ = 1/6,

/7°2>/7.~ = [ 1 - ? ] ~ > 0 , /761 >/762=0,

Consider a cost differential smaller than but close to 5: c = ~ - e , e > 0. For e sufficiently small, the relations between the gains of the duopolists in their different roles will be maintained. Therefore, the critical value ~ for which we go from a situation of mutual advantage on the distribution of roles to a situation of conflict, both duopolists prefering to be followers, is smaller than the critical value ~ for which the lower cost firm acting as leader finds it interesting to drive out of the market the higher cost firm. We therefore have shown:

Proposition 4. In the linear duopoly model with competition in both prices and quantities, there exists a critical value (6< i/6) of the cost differential between the firms such that:

(a) if c>~ then it would be more advantageous for the duopolists to reach the following distribution roles: the higher cost firm acts as the leader, the lower cost firm as the follower,

(b) if c <~ then both firms will prefer to be the follower.

5. Conclusion

In this article we have considered competition in both price and quantity, which appears from a casual observation of firms' behavior to be a kind of competition more representative or descriptive of that behavior than pure price competition or pure quantity competition. Firms are rarely ready to sell whatever quantity at some fixed price, or at whatever price some fixed quantity. But, at least in pure strategies, no Nash equilibrium exists in such a game. Hence one must resort either to mixed strategy Nash equilibria or to another solution concept. In this paper we explored the second avenue and

190 M, Boyer and M. Moreaux, Being a leader or a follower

we find that, contrary to the standard von Stackelberg model, firms prefer the more advantageous role of follower when their technologies are identical or at least similar. However, when technologies are sufficiently different, the firms would both be better off if the less efficient higher cost firm acts as the leader with the more efficient lower cost firm acting as the follower.

The practical relevance of theoretical models is not always self evident. But consider Scherer's discussion [Scherer (1980)] of leadership in different American industries. One learns that, for the cigarette industry, the big three (Reynolds, American Tobacco and Liggett and Myers) accommodated from 1921 to 1932 smaller companies selling at discount prices until those smaller companies increased their market share to 23~o. At that point, a price war was declared and by 1933, the smaller companies' market share had decreased to 6~o. What our model suggests is that the big three were acting as second movers, and the smaller companies as first movers, selling at discount prices and restricting their market share. But as soon as they overshot the equilibrium market share compatible with this distribution of roles, they induced a price undercut by the big three.

In the case of the steel industry, Scherer reports that U.S. Steel was considered to be the price leader up to 1958, although in our terminology it would appear as the second mover and the smaller discount firms as first movers. When overcapacity developed in the late fifities and early sixties and when cheaper imports among others had reduced U.S. Steel's market share to an all-time low of 21~, U.S. Steel could not tolerate or accommodate discount sellers anymore and substantial price cuts were introduced. The industry price structure was rather unstable afterwards. We would interpret this as follows. A stable leader-follower framework requires that the low price leader be disciplined through capacity limitation. Because of a drop in demand, what was a restricted capacity became a surplus capacity and U.S. Steel reacted to the new situation by undercutting the discount firms. We could have predicted that eventually some form of capacity limitation would be reintroduced. It was. In the sixties and seventies, price leadership began to be expressed on a product basis, a form of capacity restriction: a low price is announced for a given product only, preventing the follower from under- cutting the discount firm. This casual observation of the steel industry is not a clear cut example of our theoretical results. Others have claimed not without reasons that the steel industry is a classic example of barometric price leadership. But it may be a mix of the two models.

Although it may be in the interest of both firms that the higher cost firm takes the leadership role (that is moves first and takes into account the best- reply correspondence of the more efficient firm acting as follower), it does not necessarily follow that this particular distribution of roles will in fact emerge. One could mention, for instance, that although firms could maximize their joint profits by agreeing to form a cartel it does not follow that all

M. Boyer and M. Moreaux, Being a leader or a follower 191

industries will be cartellized even in an economy with no law preventing such agreements. The incentive to cheat on the cartel agreement will in general be high enough to make it unstable, especially if one requires sub-game perfectness in equilibrium choices. Moreover, unless side payments are introduced, it may not (and will not in general) be individually rat ional for all firms to participate in a joint profits maximization strategy. But in the model we developed in this paper, such a problem does not arise: we assumed non-cooperative behavior, and therefore imposed no restriction on 'cheating', and the agreement on the role distribution, when the cost differential is above the critical value, is indeed individually rational, both firms making more profits in equilibrium (with non-cooperative behavior).

We have shown that when the cost differential is large enough, only two equilibria may appear: a limit pricing equilibrium, with only the more efficient firm active, and a v o n Stackelberg equilibrium with the less efficient firm acting as leader. Profitwise, the latter equilibrium is Pareto superior to the former one; moreover, assuming the leadership role means survival for the less efficient firm, Although we did not offer a complete character izat ion of the decision process by which such an arrangement would appear, the fact that its emergence does not require an agreement, contrary to the cartel formation case, and the fact that its maintenance is individually rational, even with non-cooperat ive behavior, mean that two fundamental condit ions for a non-cooperative emergence of a specific role distr ibution are met. Hence, it does not appear unreasonable to expect that the latter market arrangement, with the less efficient firm acting as leader, will in fact emerge. If that is so, it implies that limit pricing is not likely to be observed, at least in the context of duopoly games of the type we analyzed in this paper.

The fact that the more efficient firm acting as leader will drive the less efficient firm out of the market when the cost differential is above the critical value, is a special case of the former firm having a 'differential movement advantage ' [e.g., Geroski and Jacquemin (1984)]. In a way, the more efficient firm is a dominant firm forcing the less efficient firm to act as leader. The source and persistence of such a cost differential could themselves result from strategic actions in a previous stage of the market game.

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192 M. Boyer and M. Moreaux, Being a leader or a foUower

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