Mathematical Social Sciences 51 (2006) 153–161 www.elsevier.com/locate/econbase
Strategic advertising: The fat-cat effect and stability B Hamid Beladi a , Reza Oladi b,* a
b
Department of Economics, University of Texas at San Antonio, United States Department of Economics, Utah State University University,, 3530 Old Main Hill, Logan, Utah 84322-3530, United States
Received 1 September 2003; received in revised form 1 November 2004; accepted 1 August 2005 Available online 28 September 2005
Abstract
We use the theory of social situations (TOSS) to examine the stable advertising behavior and pricing. In contrast to the current literature, we show that the incumbent firm need not maintain a hungry-look by under-investing in advertising expenditure in order to deter entry. D 2005 Elsevier B.V. All rights reserved. Keywords: Advertising; Competition; Market structure; Theory of social situations JEL classification: C7; D4; M0
1. Introduction
The ef The effe fect ct of ad adve vert rtis ising ing on co comp mpet etit itio ion n ha hass be been en st stud udie ied d ex exte tensi nsive vely ly in mu much ch of th thee second half of the 20th century. The celebrated work of Bain Bain (1956) considered the effect of advert adv ertisi ising ng expe expendi nditur tures es on int intens ensity ity of com compet petiti ition on and con conclu cluded ded tha thatt suc such h exp expend enditur itures es have major anti-c anti-competit ompetitive ive conseq consequences. uences. Schmalense Schmalenseee (1974 (1974,, 1983) 1983) studied studied the str strateg ategic ic effe ef fect ct of ad adve vert rtis isin ing g on ba barr rrie iers rs to en entr try y. In a ga game me th theo eore reti ticc fr fram amew ewor ork, k, Schmalensee (1983) formulated a model of advertising game along with post-entry quantity competition. He co conc nclud luded ed th that at an in incu cumb mben ent’ t’ss ad adve vert rtis ising ing in inve vest stmen mentt en enco cour urage agess en entr try y ra rath ther er th than an deterring deterr ing it.
B
We would like to thank two anonymous referees and an associate editor for their helpful comments and suggestions. All possible errors are ours. *Corresponding author. Tel.: +1 435 797 8196; fax: +1 435 797 2701. E-mail address:
[email protected] (R. Oladi). 0165-4896/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2 doi:10.1016 /j.mathsocsci.2005.08.001 005.08.001
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H. Beladi, R. Oladi / Mathematical Social Sciences 51 (2006) 153–161
Fudenberg and Tirole (1984), by setting a simple model of advertising and price competition, confirmed the earlier results that heavy investment in advertising cannot deter entry. More specifically, their main result states that the incumbent should under-invest in advertising in order to deter entry, avoiding the fat-cat effect. That is, the incumbent must maintain a hungry-look. In this paper, we use the theory of social situations (Greenberg, 1990) to investigate the stability of advertising and price competition. Among other advantages of the theory of social situations (TOSS, hereafter), the solution concept offered by TOSS assumes forward-looking players. In comparison, the naive players of the Nash negotiation process assume the commitment of the opponents to the current strategies when contemplating a move. That is, Nash players are myopic. Instead, according to the stability approach offered by TOSS, when a player considers a move, he takes into account further moves by rivals. In our context of advertising and pricing, TOSS allows us to consider situations where management does not take the advertising investment and price of the rival firm as given, as it is assumed in Nash equilibrium. According to the TOSS notion of stability, a price pair is stable if any beneficial price change by any firm is necessarily unstable as it will result in another price change leading to another pair of stable prices. This is in contrast to the underlying assumption of Nash, where any beneficial price change is plausible regardless of the status of the resulting price pair. It is in this sense that the players in our setup are forward-looking and not myopic.1 It is more plausible that the managerial decisions made by firms on advertising expenditure are based on these stable outcomes, which are most likely to be the end results. We consider a game similar to Fudenberg and Tirole (1984) and show that the Pareto optimum prices, soft post-entry competition, as well as fierce competition are stable. More specifically, we show that in order to deter entry the incumbent need not maintain a hungrylook. That is, the fat cat who seeks stability may not be doomed by the fat-cat effect. This result is in sharp contrast to the current literature. Note that one may obtain similar results using an infinitely repeated game framework. However, we believe that the use of TOSS is more appealing as it does not involve the use of complex dynamic punishment strategies, commonly used in infinitely repeated games. The results of our model have important policy implications for managerial decision making. Specifically, we demonstrate that the competitive pricing strategy and advertising, to a large extent, depends on the nature of decisions made by firms, such as myopic and more insightful orientations. The battle for market share between incumbent and entrant depends significantly on the aggressive posture and persuasive advertising strategies of incumbent. Finally, our framework provides a theoretical foundation for empirical studies on advertising expenditure and its effect on competition. The layout of the paper is as follows. In Section 2 we set up the model. This section heavily depends on Fudenberg and Tirole (1984). We then analyze a price competition game with advertising in Section 3. Finally, Section 4 provides the concluding remarks. Proofs of Section 3 are presented in Appendix. b
b
Q
Q
b
Q
1
b
Q
Therefore, in some weak sense firms in our formulation have foresigh t. However, as one of the anonymous referees pointed out, this does not embody full farsightedness, considered by Chwe (1994) and Xue (1998). It is certainly interesting (but beyond the scope of this paper) to investigate the application of consistent sets, offered in Chwe (1994) and Xue (1998), to the current problem.
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2. The model
Following Fudenberg and Tirole (1984), we assume a unit population of consumers who buy a product if they are informed of it. There are two periods, 1 and 2. In period 1 an incumbent monopolist supplies the product to informed consumers. In the second period, however, an entrant may enter. Both firms advertise in newspapers and inform the consumers of both the existence of their products and their prices. We assume, as in Fudenberg and Tirole (1984), that once a consumer gets an ad in the first period, he will not bother to read the ads in the second period. Firm i , i =1, 2, supplies the good i .2 Let firm 1 be the incumbent. The demand functions for the two goods are given by, b
Q
D1 ðp1 p2 Þ
ð1Þ
D2 ðp1 p2 Þ
ð2Þ
;
;
where the consumers are aware of both products. When consumers are not aware of a good, e.g. good 2, we denote the demand function by D 1( p 1, ). This indicates that firm 1 is a monopolist in the (segment of the) market. However, we exclude the possibility of price discrimination. Accordingly, the revenue (net of costs) that an informed consumer brings to the firms are: l
R1 ðp1 p2 Þ
ð3Þ
R2 ðp1 p2 Þ
ð4Þ
;
;
2
i where R i is twice differentiable, quasi-concave, and BB pi R N0 i j ¼ 1 2 i p j . The last B p j assumption states that the marginal revenue of firm i increases with firm j ’s price. Note that, following our formulation of the demand functions, if the consumer does not know of good j , the revenue the consumer brings to the firm i will be R i ( p i , ). Let the cost of reaching a a [0,1] portion of population be u (a ), where u is assumed to be convex and u (1)= . In period 1, the incumbent advertises a by spending u (a ) and in the second period both firms spend a fixed amount u¯, allowing them to reach (1 a ) portion of the population jointly.The profit functions are as follows, ;
;
;
;
l
l
P1 ¼ ½a Rm uðaÞ þ d½a R1 ðp1
;
Þ þ ð1 aÞ R1 ðp1 P 2 Þ u¯
l
;
P2 ¼ d ½ð1 aÞ R2 ðp1 p2 Þ u¯ ;
ð5Þ ð6Þ
where R m is the monopolist revenue and d is the discount factor.3 The terms in the first brackets in Eq. (5) is firm 1’s profit in period 1, while the terms in the second brackets represent its profit in period 2. 3. The advertising and price competition game
Assume that the firms engage in post-entry advertised price competition. A strategy for firm i , i =1, 2, is a price p i a X i =[0, ]. Let X = X 1 X 2 be the set of all strategies. The advertising and l
2 3
We use subscript for firms (goods). This set up along with assumptions are borrowed from Fudenberg and Tirole (1984).
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price competition game is the triple G = ( N , X ,{P i }i N ), where N = {1,2} is the set of players and T i is the payoff function for player i , defined by Eqs. (5) and (6). Let w 1 and w 2 be the best response functions for firm 1 and 2, respectively, that is, w1 ðp2 Þ ¼ arg max p1 X 1 P 1 and w2 ðp1 Þ ¼ arg max p2 X 2 P 2 . We assume that the profit functions are symmetric around the best response functions, i.e., for all p V, p W, p a X such that p Vi = w i ( p j V), p j V = p j W = p j , p iW = p Vi e , p i = p Vi + e , e N 0, we have, P i ( p W) = P i ( p ), i , j =1, 2, i p j . In other words, profit functions are symmetric around the best response if a price deviation above or below the best response price by the same amount leads to the same loss in profit. This property is satisfied by linear demand functions. The following lemma addresses the well-known result that prices are strategic complements. That is, a firm responds with a higher price, the higher is its rival’s price. a
a
a
j
j
j
j
Lemma 1. w i is increasing in p j , i , j , =1, 2, i p j . Proof. See Fudenberg and Tirole (1984).
5
Definition 1. A pair of strategies (pˆ 1 , pˆ 2 ) is a Nash equilibrium if pˆ 1 = w 1 (pˆ 2 ) and pˆ 2 = w 2 (pˆ 1 ), i.e., pˆ 1 = w 1 ( w 2 (pˆ 1 )) .
We assume, as in Fudenberg and Tirole (1984), that such an equilibrium exists. The following lemma addresses the impact of advertising. Lemma 2. An increase in a increases the best response price of firm 1. Proof. See Fudenberg and Tirole (1984).
5
The above result states that the advertising expenditure increases firm 1’s best response price, for any given price of its rival. Proposition 1. An increase in advertising a increases the Nash equilibrium prices. Proof. See Fudenberg and Tirole (1984).
5
Lemma 2 and Proposition 1 are the essence of fat-cat effect. According to the Nash equilibrium of the game, an increase in incumbent’s goodwill (its market captivity, obtained by advertising expenditure) makes the incumbent unwilling to match the entrant’s price. In other words, market captivity makes the incumbent a pacifist. Therefore, based on the Nash equilibrium outcome, the incumbent should under-invest in advertising and maintain a hungry look in order to deter entry. We now formulate the advertising and price competition situation associated with the game G . Following Greenberg (1990), let p a X be any currently proposed price pair. Firm i can object to this proposal and induce another price pair by unilaterally selecting any other price from its set of prices X i . The set of prices that can unilaterally be induced by firm i a N is specified by an inducement correspondence, mapping X to X as follows, b
Q
b
n
V
Q
V
V
ci ð pÞ ¼ p j p j ¼ p j pi a X i ;
o
;
ð7Þ
i p j
:
The correspondence (7) states that firm i can induce the outcome p V from p = ( p 1, p 2) by setting a new price p Vi a X i , given that the firm j sticks to p j . The advertising and price competition situation is the quadruple ( N , X ,{P }i N ,{c i }i N ). A standard of behavior r is a mapping that assigns to every pair of prices p a X a subset of { p }, denoted by r ( p ). We call such a subset a solution. Note that the solution to the a
a
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outcome p is either { p } or the empty set F. In the former case, where the solution is non-empty, we say that the pair of strategies p is supported or accepted by the standard of behavior r. The above definition of the standard of behavior is arbitrary. Thus some stability criteria must be imposed on a standard of behavior if it is to be followed by a rational firm. b
Q
b
Q
Definition 2. A standard of behavior r is internally stable if for all p a X, p a r(p) implies that there do not exist a player i a N and a pair of strategies p V a c i (p) such that p Va r(p V ) and T i (p V ) N T i (p) .
The above criterion postulates that if the pair of strategies p is supported by a standard of behavior r , then no player should be able to induce another outcome p V, which is also supported by r and makes him better off. On the other hand, the following criterion states the condition for exclusion of an outcome from a solution. Accordingly, if an excluded pair of strategies were included in the solution, then internal stability would be contradicted. Definition 3. A standard of behavior r is externally stable if for all p a X \ r(p) there exist i a N and a pair strategies p Va c i (p) such that p Va r(p V ) and P i (p V ) N P i (p) . Definition 4. A standard of behavior r is stable if it is both internally and externally stable.
For any standard of behavior r, let r ( X ) be the set of strategy pairs supported by r, i.e., rð X Þ ¼ v p X rð pÞ. This is the set of all strategies permissible by the standard of behavior r. The following result states that a standard of behavior that supports the set of strategies identified by anyone of the best response functions is stable. a
Proposition 2. For any i = 1,2, let w i be a one-to-one function. Then the standard of behavior r , where r(X)={p a X|p i = w i (p j )}, i, j = 1,2 i p j, is stable.
Consider anyone of the best response functions, for example w 1. Because w 1 is a one-to-one function, none of the firms can induce a pair of prices, supported by r, from another pair of prices which is also supported by r. Thus, r is internally stable. On the other hand, from any pair of prices that is not supported by r, firm 1 can choose its best response price, inducing a pair of prices that is supported by r. By choosing such a price, firm 1’s profit also increases. Let p 2 = k ( p 1) be any strictly increasing function (as illustrated in Fig. 1) such that pˆ 2 = k ( pˆ 1), 1 w 1 ( p 1) b k ( p 1) b w 2( p 1) for p 1 b pˆ 1 and w 2( p 1) b k ( p 1) b w 1 1( p 1) for p 1 N pˆ 1.4 The assumption of existence of Nash equilibrium ensures the existence of such a function. Proposition 3. Consider the advertising and price competition game G and its associated situation. The standard of behavior r such that r(X)={p a X|p 2 = k(p 1 )} is stable.
The above proposition indicates that any strictly increasing function that lies between the two best response functions is stable. This proposition is at the heart of our contribution upon which the next two results are derived. Let b be the set of Pareto optimal prices, i.e. b = { p a X |Z p Va X such that P i ( p V) N P i ( p ) for every i a N }. Geometrically, these prices are the locus of tangent points of iso-profit curves of the two firms. We depict these points in Fig. 2.
4
We denote the inverse of w i ( p j ) by wi 1( p j ).
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ψ 1
2
λ ψ 2 N
ˆ2
pˆ 1
p1
Fig. 1.
Proposition 4. For every p a b there exists a stable standard of behavior r that supports p, i.e., p a r(p).
The above result is a general contribution to the price competition literature. Accordingly, assuming product differentiation, the Pareto optimal prices are stable. More specifically, there exist some stable prices p˜ a b , such that p˜ i N pˆ i , and P i ( p˜ ) N P i ( pˆ ),i =1, 2. That is, there exist some stable prices that Pareto dominate the Nash equilibrium prices. As stated earlier, Proposition 3 indicates that all outcomes between the two best response functions are supported by some stable standards of behavior. That is, the stable post-entry set of prices includes soft as well as fierce competition. Soft competition relates to the pairs of stable prices that are greater than the Nash equilibrium prices while fierce competition is associated with pairs of stable prices that are less than the Nash equilibrium prices. Thus, we have the following corollary which is in sharp contrast to the generally held view in the literature that heavy advertising encourages entry. More specifically, it b
Q
ψ 1
p 2
β
ψ 2 N
p1
Fig. 2.
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states that an incumbent can have a captive market and yet could select a low stable price in order to deter.5 Corollary 1. The incumbent need not maintain a lean and hungry look in order to deter entry. More specifically, the fat-cat who seeks stability can deter entry. b
b
Q
Q
4. Concluding remarks
We use the theory of social situations (Greenberg, 1990) to examine the fat-cat effect and stability in an advertising and price competition game. In this game, an incumbent invests in advertising in the first period and faces possible entry of a new firm and price competition in the second period. As in Fudenberg and Tirole (1984), we consider both exogenous and endogenous advertising. We derive the stable prices supported by stable standards of behavior. As the main result of this paper, we show among other things that the incumbent need not maintain a hungry-look (under-invest in advertising expenditure) in order to deter entry. This is in sharp contrast to the common conclusion in the literature on advertising that the incumbent cannot deter entry by heavy advertising due to the fat-cat effect. Our new approach in modeling advertising and its impacts on competition can be used in an alternative form of competition. One could use post entry quantity competition rather than price competition. Moreover, the new model presented in this paper can be extended in many other directions. Using our framework, one may examine strategic advertising and competition in a vertically differentiated market. Finally our analysis provide an alternative theoretical foundation for empirical studies of advertising expenditures and its effects on competition. Appendix A. Proofs Proof of Proposition 2. Internal stability : Since w i is a one-to-one function, for any p a X , p a r X ( p ), no player can induce another strategy pair, p V a X , p p p , such that r ( p V) p F. External stability: For any p a X \ r ( p ), p i p w i ( p j ), i , j =1,2 i p j , due to definition of r . Then, player i can induce p V = ( p Vi , p j ) such that p Vi = w i ( p j ), where P i ( p V) N P i ( p ). This inequality is due to the definition of best response function. 5 Proof of Proposition 3. We have to show that r as defined in the proposition is both internally and externally stable. Internal stability: Since k is strictly increasing, if p V a r ( p V), implying that p V2 = k ( p V1), then for any p W p p V such that pWi a X i and p j W = p j V, i , j =1,2, i p j , we have p 2W p k ( p 1W), i.e., all strategies that player i can induce have empty solutions. The internal stability follows immediately. External stability: Define p i = g i ( p j ) i , j =1,2, i p j such that for p V, p W a X where p j V = p j W, W), we have P i ( p V) = P i ( p W). The function g i maps all points symmetric p V2 = k ( p V1), pW i = g i ( p j (around w i ) to points mapped by k. Now assume that p a X \ r ( p ) and consider the following cases: Case 1 : p 2 b pˆ 2 Sub-case 1.1: p 1 b k 1( p 2). By definition of k , p 1 b k 1( p 2) b w 1( p 2), which implies that for p˜ where p˜ 2 = p 2 and p˜ 1 = k 1( p˜ 2) N p 1, P i ( p˜ ) N P i ( p ). The last inequality is due to the definition
5
All our results analysis will also be valid if one considers an endogenous choice of a .
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H. Beladi, R. Oladi / Mathematical Social Sciences 51 (2006) 153–161 BP
ð Þ
p of w 1, i.e. B p1 1 N 0 if p 1 b w 1( p 2). Note also that p˜ a c 1( p ) since this position is induced by a unilateral price increase by firm 1. Moreover, by construction we have p˜ a r ( p ). Sub-case 1.2: p 1 N g 1( p 2). By symmetric assumption of P 1 around the best response function w 1, we have P 1( p˜ ) N P 1( p ), where p˜ 2 = p 1 and p˜ 1 = k 1( p˜ 2) b p 1. Moreover, we have p˜ a r ( p˜ ) as in Sub-case 1.1, and p˜ a c 1( p ) since firm 1 decreases its price unilaterally. Sub-case 1.3 : k 1( p 2) b p 1 V g i ( p 2). Construct p˜ where p˜ 1 = p 1 and p˜ 2 = k ( p˜ 1). Consider the following possibilities: (1) p 1 b pˆ 1 : By definition of k , k ( p 1) b w 2( p 1) for all p 1 b pˆ 1. Also k 1( p 2) b p 1 implies that k ( p 1) N p 2 since k ( p 1) is strictly increasing. Therefore, we have ( p 2) b k ( p 1) b w 2( p 1). It follows by construction that ( p 2) b pˆ 2 b w 2( p 1). Next, recall from the BP ð pÞ definition of w 2 that for any given p 1 B p2 2 N0 if p 2 b w 2( p 1) This implies that P 2( p˜ ) N P 2( p ), as we already established that pˆ 2 = k ( pˆ 1) b w 2(pˆ1). That is, as firm 2 increases its price closer to its best response price, its profit increases. (2) p 1 N pˆ 1 :Then p 2 b g 2( p 1) since g 1 and g 2 intersect only at Nash equilibrium due to definitions of g i and k . It follows from construction of p˜ and definition of g 2 that P 2( p˜ ) N P 2( p ) because of symmetric assumption of P 2 around the best response function w 2. Finally, under both possibilities 1 and 2, p˜ a c 2( p ) since p˜ is induced through a unilateral increase in price by firm 2. Case 2 : p 2 N pˆ 2 Sub-case 2.1: p 1 N k 1( p 2). By definition of k , p 1 N k 1( p 2) N w 1( p 2). This implies that for p˜ such that p˜ 2 = p 2 and p˜ 1 = k 1( p˜ 2) b p 1, P 1( p˜ ) N P 1( p ). The reason for the last inequality is similar to that of Sub-case 1.1. Also, p˜ a c 1( p ), as firm 1 unilaterally reduces its price. Sub-case 2.2 : p 1 b g 1( p 2). By symmetry of P 1, we have P 1( p˜ ) N P 1( p ) where p˜ 2 = p 2 and p˜ 1 = k 1( p˜ 2) N p 1. In addition, p˜ a c 1( p ) since firm 1 increases its price unilaterally. Sub-case 2.3 : g 1( p 2) V p 1 b k 1( p 2). Construct p˜ where p˜ 1 = p 1 and p˜ 2 = k ( p˜ 1). Consider the following two possibilities: (1) p 1 N pˆ 1 : p 1 b k 1( p 2) implies that p 2 N k ( p 1) since k ( p 1) is strictly increasing. Also for all p 1 N pˆ 1, we have k ( p 1) N w 2( p 1) due to definition of k ( p 1) (see Section 3). Therefore, we conclude that p 2 N k ( p 1) N w 2( p 1). Then, by construction of p˜ , we have BP ð pÞ p 2 N p˜ 2 N w 2( p 1). This implies that P 2( p˜ ) N P 2( p ) because for any given p 1, B p2 2 b 0 if p 2 N w 2( p 1). That is, if firm 2’s price is greater than its best response price, the firm could increase its profit by lowering its price closer to its best response price. (2) p 1 b pˆ 1 :Since g 1 and g 2 intersect only at Nash equilibrium due to definitions of g i and k , we have p 2 N g 2( p 1). This, along with the symmetry of P 2 around the best response function w 2, implies that P 2( p˜ ) N P 2( p ). Finally, with both possibilities, p˜ a c 2( p ) since p˜ is induced by firm 2 reducing its price unilaterally. Cases 1 and 2 imply external stability of k . 5
Proof of Proposition 4. 8 p Va b choose a function k , as defined earlier, such that p V2k ( p V1). Then based on Proposition 3, the standard of behavior r with r ( X ) = { p a X | p 2 = k ( p 1)} is stable. 5 Proof of Corollary 1. Directly from Proposition 3.
5
References Bain, J.S., 1956. Barrier to New Competition: Their Character and Consequences in Manufacturing Industries. Harvard University Press, Cambridge. Chwe, M.S., 1994. Farsighted coalitional stability. Journal of Economic Theory 63, 299–325. Fudenberg, G., Tirole, J., 1984. The fat-cat effect, the puppy-dog ploy, and the lean and hungry look. American Economic Review, Papers and Proceedings 74, 361–366.
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Greenberg, J., 1990. The Theory of Social Situations: An Alternative Game Theoretic Approach. Cambridge University Press, Cambridge. Schmalensee, R., 1974. Brand loyalty and barriers to entry. Southern Economic Journal 40, 579–588. Schmalensee, R., 1983. Advertising and entry deterrence: an explanatory model. Journal of Political Economy 91, 636–653. Xue, L., 1998. Coalitional stability under perfect foresight. Economic Theory 11, 603–627.
