Goodwill and Market Share in Oligopoly

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Goodwill and Market Shares in OligopolyAuthor(s): Chaim FershtmanSource: Economica, New Series, Vol. 51, No. 203 (Aug., 1984), pp. 271-281Published by: Blackwell Publishing on behalf of The London School of Economics and Political Scienceand The Suntory and Toyota International Centres for Economics and Related Disciplines

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Economica, 51, 271-281

Goodwill and Market Shares in Oligopoly

By CHAIM FERSHTMAN

Northwestern University, Evanston, Illinois

INTRODUCTION

Economic literature treats price as the traditional competition variable.

However, Schmalensee (1976, p. 493) argued that prices change infrequently,

and that firms compete mainly through non-price variables. The most impor-tant non-price competition variables are advertising and product variation.Although there are many works dealing with non-price competition, they have

not devoted much attention to the multi-period effect of these variables andthe models have been mainly static, or a sequence of static models. (Furtherdiscussion of this argument can be found in Kydland (1977), in which dynamicdominant player models are discussed.)

In an earlier paper, Arrow and Nerlove (1962) pointed out that advertising

expenditure should be. treated in the same way as investment in a durablegood. They assumed that there is a stock of goodwill that determines thecurrent demand. This stock of goodwill summarizes the advertising in the past

and, like capital stock, depreciates over time.Arrow and Nerlove analysed the optimal advertising strategy for the

monopolistic firm. In this paper we extend the Arrow-Nerlove model tooligopolistic competition, in which the firms compete among themselves via

goodwill. The game we describe takes place over time and discounted profit

is used as a criterion. The current advertising of any player affects future

demand, and therefore the model has an intertemporal dependence structure.Extending an assumption by Schmalensee (1978), in Section I we allow

market shares to depend on the firm's goodwill and not merely on current

advertising. Using this definition of market share, we discuss in Section II the

optimal advertising policy for a single firm. In Section III we prove the existenceof a unique stationary equilibrium point for oligopolistic competition. Sincewe allow firms to have different cost functions, the equilibrium point is not a

symmetric one; i.e. different firms have different market shares and different

goodwill at the equilibrium point. In Section IV we discuss this asymmetricsolution by explaining the relation between production cost and market shares.

The impact of the number of firms on the advertising expenditure is discussedin Section V. We prove that the firms' advertising expenses tend to decline

as the number of firms increases. However, if one of the firms is a leadingfirm in the industry and has a market share above one-half, we show that this

firm might increase its advertising as other firms enter into the industry. As

an example, we discuss in Section VI the duopoly case and offer an explicitstatement of the market shares at the equilibrium point. In Section VII we

analyse an industry consisting of identical firms, and explore the influence ofthe interest rate and the depreciation rate on the firms'goodwill and advertising

policy.



272 ECONOMICA [AUGUST

I. THE MODEL

Consider a market in which there are N non-identical sellers indexed byi. Let G(t) = {G1(t), .. , GN(t)} be the firms' stock of goodwill at time t, andlet a (t) = {a, (t), ... ., aN(t)} be the advertising expenditures at time t. We

assume that the stocks of goodwill accumulate according to the Nerlove-Arrowcapital accumulation equation:

dG(t) = G(t) = a(t) - 8G(t)dt

where 8 is a constant proportional depreciation rate.Let Si(t) denote the ith firm's market share at time t. We assume that the

firms' market shares are functions of the firms' goodwill. For every G(t) ={G1(t), . . ., GN(t)} such that Ej Gj(t) > 0 it is assumed that the firms' marketshares are:

N

(1) Si(t) = Ga(t) Y Ga(t)j=1

where a is a non-negative constant. The larger a is, the more the demand isinfluenced by goodwill. If a = 0 the consumers ignore the goodwill and choosea brand in a random way. In this case market shares will be

Si= 1/N for every

i and sellers will not advertise.This assumption is similar to that in Schmalensee (1978), in which it is

assumed that

N

Si(t) = al a(t)/ E al a(t)

j=1

is the probability that any buyer who becomes dissatisfied with any brand inperiod t will purchase brand i in period t+ 1. Assuming no differences in the

products' quality and a large number of consumers, according to Schmalenseethe market shares are

Si = ai / a,

The difference between Schmalensee's assumption and the assumption that ismade in this work is that we assume that goodwill, and not the currentadvertising, determines the market shares.

For a > 1 equation (1) implies that there are economies of scale in goodwill.Firms with twice as much goodwill as their rivals will have market shares morethan double that of their rivals. Thus, like Schmalensee we assume that0 < a < 1, which means that consumers respond positively to goodwill but thatthere are diminishing returns to goodwill. (In other words, Si(t) is assumedto be a concave function of Gi(t).)

Given the market price p and the market share at time t, the rate at whichsales are made (qi(t)) is

(2) qi4(t)= Sit)h (p) i = 1, . .d, N

where +(p) is the market demand function.



1984] GOODWILL AND MARKET SHARES 273

Let C1{qi(t)} be the ith firm's production cost. We assume in this work

constant returns to scale, so Ci{qi(t)} = miqi(t) where mi is a positive constantand mi< p for every i. Under these assumptions the single-firm instantaneous

profit function can be written as(3) iT{Si(t), ai(t)} = (p- mi)4'(p)Si(t) - ai(t) i = 1, . . , N.

The advertising path that maximizes the expected present value of the

firm's profits is a function of the expected goodwill path of the rivals. Therefore,

in dealing with the multi-firm problem we will adopt the open-loop equilibrium

concept. Each firm maximizes its discounted profits, given the goodwill pathsof its competitors. These paths are also optimal and are correctly anticipated

by each firm.

Under these assumptions, the aim of each firm is to maximize its discountedprofit:

(4) max e-rt{(p- mi)(p)Si(t) - ai(t)} dt0

subject to

Gi(t) = ai(t) - 8Gi(t);

G(O) = Go (the initial level of goodwill is given);

Gj(t) are given for every j i;

ai(t) 2: 0

where r is the discount rate.The market can now be described as N non-identical firms, seeking to

maximize their discounted profits (4). In order to find the equilibrium point

we must investigate the single firm's optimal policy.

II. THE SINGLE FIRM'S OPTIMAL POLICY

Proposition 1. For every given 0 < Si < 1, there is a unique optimal stock of

goodwill Gi.

Proof. The maximization problem (4) can be solved directly using the

optimal control theory. The current-value Hamiltonian associated with this

problem is:

(5) H(ai, Gi, Si,A) = {(p- mi)+(p)Si(t) -ai(t)} +A(t){ai(t) -5G(t)}.

Since the Hamiltonian is a linear function of ai, it is well known that when

aH/aia >0 the optimal policy is ai = oo, and when aiH/aai <0 the optimal

policy is ai = 0. So it remains to investigate the optimal policy for alH/lai = 0.The other equations that must be satisfied by the optimal policy are:

(6)n(p mi) (P) -k6 +rA

and




The condition aiH/aai = 0 implies that A(t) = 1 and thus A t) =0. Equation

(6) can therefore be written after simple manipulations as:

(8) (P- mi)(P) as.= r+ 5.aGi

Differentiating (1) yields

a-1 a

aG( ,G G)

(9) aG, N

'GS(1S1

Therefore, using equations (8) and (9) the optimal stock of goodwill Gi can

be described as a function of the current market share Si(t):

(10) Gi(Si) = (p-me)(p) S(+

- SQ.E.D.

Since at time t the market share Si(t) is given, the optimal advertising

policy for the ith firm can be summarized as:

fai(t) = 0 if Gi(t) > Gi{Si(t)}(11) t ai(t) = 6if{Si(t)} if G(t) =Gi{Si(t)}

ai(t) = 00 if Gi(t) < GifSi(t)}.

The optimal advertising pQlicy derived from this model has the same

characteristics as the optimal policy described by Arrow and Nerlove (1962,

p. 135).

III. MARKET EQUILIBRIUM

Consider an industry in which there are N non-identical firms, each ofwhich is faced with the problem described in Section II. In this section we will

prove the existence and uniqueness of an open-loop stationary Nash equi-librium point for this differential game.

Denote (S*, G*) as the stationary Nash equilibrium point. There are then

two conditions that must be satisfied:(i) For every firm i equation (10) must be satisfied; i.e. every firm has its

optimal stock of goodwill according to its market share Si*:

(12) G*=(p-mi)4(p) aS*(1-S*) i=l,... ,N.

(ii) (S*, G*) must satisfy the definition of market share:

o*(13) S G= i=l,...,N.

j=1Every (S*, G*) that satisfies equations (12) and (13) is a stationary equilibrium

point. Each firmhas its optimal stock of goodwill according to its market share




and the stock of goodwill remains unchanged, as for each firm the optimal

advertising policy is a* = 8G* and therefore G* =0.

Proposition2. Under the assumptions of the model, there is a unique stationary

equilibrium point (S*, G*).

Proof. For every i c {1, . . . , N} define a constant ki> 0 such that

a(14) ki-(p - mi) (p) ++, 1 N.

For every c c R we define a point (S, G) c R2N such that

(S, G) = {GI ), . . , (SN, GN)}

and

(15) Gi=kicGN (1-cG a) i = 1, . . , Ni

(16) Si =cG a i =-1, ........N.

In order to prove that this is a good definition, we have to show that

equation (15) and (16) define, for every i, a unique pair (Si, GC) hat differs

from (0, 0).

For every Gi ? 0, equation (15) can be written as

(17) GI-a + kic2Ga = kic.

Since the left-hand expressions is increasing in Gi (accepting values below and

above kic), there is a unique solution in Gi. Moreover, it is clear that the

solution of (17) is such that cGa < 1 and therefore Si < 1.

Notice that for every i the pair (Si, Gi) satisfies equation (12); in order to

prove the existence of an equilibrium point it remains to find such c* that

defines (S, G) that satisfies equation (13).According to Appendix 1, Sii= 1, . . . , N is a continuous monotonically

increasing function of c.

Let m(c)- -j= Sj(c); then m(c) is a continuous monotonically increasing

function of c (as a final sum of such functions). For

i=

() ( N 1)

the appropriate Si(ci) equals 1/N. Choosing c =mmcl,n . . , CN} assures us

that for c < c we get Si(c) < 1/N for every i. Therefore for c < c, we get

N

m(c)= E Si(c)<1.j=1

On the other hand, choosing c = max {cl1, , CN} assures us that for c> c,

Si(c) ? 1/N for every i; therefore for c > c we get m(c) ? 1. From the con-

tinuity and monotonicity of m(c), it is clear (according to the intermediate

value theorem) that there is a unique c* such that m(c*)= 1.

Denote the point that c* defines as (S*, G*). Since (S*, G*) satisfiescondition (12), we need only to show that it also satisfies condition (13) to

prove that this is the equilibrium point.




Since (S*, G*) satisfies equation (16), S*= c*G*a for every i; therefore

N N

e = Gc .

j=1 jl1

However, since

N

j=1

we can conclude that

N

c Wj=1

and therefore

N

c=1 Yj=1

which implies that

V = c*G*a= , -1.,N.

E G*j=1

Therefore condition (13) is met and (S*, G*) is the unique equilibrium point.Q.E.D.

From equation (11) the advertising policy at the equilibrium point is

a*(t) = SGW for i-1. . ., N. Therefore

(18) a* =8kiS(l - S*) i = 1,... ,N.

Equation (18) gets its maximum value at Si =. Therefore for each firm

the advertising at the equilibrium point is bounded from above by ai = $(ki/4)

and the goodwill of each firm at the equilibrium point is bounded from above

by Gi = ki/4.

IV. THE PRODUCTION COST, ADVERTISING AND MARKET SHARE

The equilibrium point discussed in Section II is not a symmetric one.Different firms get different market shares at the equilibrium point. Moreover,the advertising expenditures at the equilibrium point are not identical for all

firms. The only difference between firms is the production cost. Therefore,although the market shares are assumed to be determined by the vector of

goodwill, the cause of the asymmetric equilibrium must be differences in thecost function.

Proposition3. There is a systematic connection between the production cost

and the market shares at the equilibrium point. Firms that have a lowerproduction cost will have a higher market share and will advertise more

extensively.






expenses; but, as described in Appendix 2, if one of the firms is a leading firm

in the industry and has a market share greater than one-half, this firm mayincrease its advertising expenditure as a response to the entry. However, as

discussed above, this increase in the advertising expenditure is not big enoughto keep the leading firm's market share.

VI. THE DUOPOLY CASE

Consider an industry in which there are two firms. Let mi (i = 1, 2) denote

their constant marginal cost. Assume that there is a constant market price p,and the firms use advertising in order to compete with their rivals. According

to proposition 1, for every given Si there is a unique optimal stock of goodwill

Gi such that

(21) Gi = kiSi(1-Si) i = 1, 2

where ki is defined in (14).

Proposition4. The market shares at the duopoly stationary equilibrium pointare

ka

Sk

i=1, 2.Ik, +k2'

Proof. According to (21), if S* = k/l (k1 + kg), then the firm's optimal stock

of goodwill must be equal to

(22) G = kiS(1-S-*) = ki (k 2) i = 1, 2.

In order to prove that (S*, G*) is the equilibrium point, it is sufficient toshow that

G *a + G*a i = 1, 2.

By using (22), we get

ak k+k2}

(23)(k k)G* 2kaka)k+k)2J

( k l +1 2 ){( k a + k "a)2}

ka-ki _-=S* i=1,2.

(S*, G*) satisfies the two equilibrium conditions; therefore the market shares

at the equilibrium point are

S*= k' i = 1, 2 Q.E.D.




VII. IDENTICAL FIRMS

Using the first-ordercondition (8), it can be easily proved that if an industryconsists of N identical firms the equilibrium point is a symmetric one; namely,the market shares at the equilibrium point are equal to 1/N and all the firmshave the same stock of goodwill. Therefore, according to (12), the stock ofgoodwill at the equilibrium point is

(24) G*=(p-m)k(p)r+-(l-) i=1,...,N.

By using this explicit expression of G* we can investigate the effect of changesin r, a, Sand m on the firms' advertising and goodwill at the equilibrium point:

__ a 1I1\(25) _c _ -1-<

am r+ N( N

(26) a =(p-m) 0 (p) 1 (1-!) >0aa N(r +8) \ N~

a__ a i/l\(27) a- =-(P-m)+(p) 5)2N(1

- <0ar (r+) N\N/

_ _ a(28) aG- =-(p-m)W(P) +2S N1 1<0

a85 (r+) N\N/

Using equations (25)-(28), we can conclude that, in the symmetric case:(a) an increase in the production cost will be followed by a reduction in

the firms' goodwill and advertising expenditures;(b) the stock of goodwill (and therefore the advertising) at the equilibrium

point tends to decrease as the interest rate or the depreciation rate increases.

VIII. SUMMARY

It is commonly accepted that the influence of some non-price competitionvariables on demand extend beyond the current period. As advertising is an

important non-price competition variable, we have analysed some dynamicaspects of advertising in oligopoly. By using the concept of goodwill, as

introduced by Arrow and Nerlove (i.e. accumulated advertising, subject to

depreciation), we described an industry in which the firms'goodwill determine

their market shares; therefore firms can change their market sharesby advertis-ing. The paper generalizes existing works in two ways: The Arrow-Nerlovemodel is extended by allowing an oligopoly; and Schmalensee's argument is

extended by allowing market shares to depend on a measure of cumulated

advertising expenditures with depreciation (i.e. goodwill), and not merely on

current advertising.We derived the optimal advertising policy for an individual firm in an

oligopolistic market and proved that for each market share there is an optimallevel of goodwill. Since the equilibrium point is not a symmetric one, we

analysed the connection between production cost and market shares at theequilibrium point. We proved that firms that have a lower production cost

will obtain a higher market share.




Analysing the equilibrium point, we found that, as the number of firmsincreases, the firms' goodwill and advertising expenditures at the equilibrium

point tend to decrease. However, if one of the firms in this industry has a

market share greater than one-half, this firmmay increase its advertising whenentry occurs.

By restricting the model to the case in which all the firms are identical, weproved that in the symmetric case the firms' goodwill and advertising expen-

ditures tend to decrease as the interest rate, the depreciation rate or theproduction cost increases.

Finally, this work provides a plausible framework for analysing dynamic

non-price competition in which the variables' impact extends beyond the

current period. This is because goodwill can be generalized by a state variable

that summarizes the current impact of the non-price variables in the previousperiods.

ACKNOWLEDGMENTS

This paper is part of a PhD dissertation presented to the Hebrew University ofJerusalem. I am highly indebted to my instructors, E. Sheshinski and E. Muller, formany discussions, ideas and much encouragement. I am also indebted to the Editorsand to a referee of this Journal for comprehensive and useful comments of an earlier

version.

APPENDIX 1

Si as an increasing function of c

From equations (15) and (16) we obtain that

(Al) kiSi 1 i-)-cl/ai/a -0

Therefore

-c(1/a)-1g1/a

dS a~c(A2) dSi= ae

dc 1 laglalkcki(l-2Si) - c -l/a(1/a

a

The numerator in (A2) is positive. Thus in order to prove that dSi/dc > 0, it is sufficientto prove that the denominator is negative.

Using (Al), we can conclude that

ki(1 - 2Si) - c- l/la(l/a)-l = ki( 1-2S2i)--ki (1-S1i)a a

= ki(l -Si)( 1--)- kigi < ?

which is negative since a < 1 and Si < 1.

APPENDIX 2

Goodwill and a reduction in c

For every c E R, Gi is defined by equation (17) as

(A3) d1-a + kiC2Gda= kic.




Differentiating (A3) yields

(A4) ___ 2k1cG7-kac a)(1a)-aa+aki C2a-l

The denominator in (A4) is positive; therefore the sign of aGi/ac is the opposite

sign of the numerator. Since Si = cGi, the numerator can be written as (2kiSi-

which will be positive for S > 2 and negative for Si < 2. Therefore

(A5 Gad1>0 forSi<2

ac < o for S2>_

If we compare Gi(c1) and Gi(c2) when c1> c2, then

(A6) Gi(cl) - Gi(c2) = ( dc.

Using Appendix 1, it is clear that if S -(c ) 1 then for every c2 ccl, S(c)<21

Thus aGi(c)/ac>0 for every c2 c cl, which implies (using (A6)) that Gi(cl)>Gi(c2). On the other hand, if Si(cl) > 2, a reduction of c can be big enough so Si(c) <4for some c2< c < c1. Therefore, since the sign of adi(c)/ac can be changed as Si(c)

becomes less than one-half, we cannot conclude which of the two is bigger, Gi(c1) or

Gi(C2).

Since at the equilibrium point EN1 SJ(c*)= 1, we know that there is at most one

firm whose market share is greater than one-half; therefore we can conclude that, for

at least (N - 1) firms,a reduction of c causes a reduction of the optimal stock of goodwill.

REFERENCES

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DORFMAN, R. and STEINER, P. 0. (1954). Optimal advertising and optimal quality. AmericanEconomic Review, 44, 826-836.

FRANK, C. R. and QUANDT, R. E. (1963). On the existence of Cournot equilibrium. InternationalEconomic Review, 4, 92-96.

FRIEDMAN, J. W. (1977). Oligopoly and the Theory of Games. Amsterdam: North Holland.

GOULD, J. P. (1970). Diffusion processes and optimal advertising policy. In E. S. Phelps (ed.),Microeconomic Foundation of Employment and Inflation Theory.New York: W. W. Norton,pp. 338-368.

KYDLAND, F. E. (1977). Equilibrium solutions in dynamic dominant player models. Journal ofEconomic Theory, 15, 307-324.

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Goodwill and Market Share in Oligopoly