International Journal of Industrial Organization 18 (2000) 595–614 www.elsevier.com / locate / econbase
Competition in a duopoly with sticky price and advertising Claudio A.G. Piga* Department of Economics, University of York, Heslington, York, YO1 5 DD, UK Received 12 May 1997; accepted 27 February 1998
Abstract This paper develops a differential duopolistic game where price is sticky and firms can invest in market-enlarging promotional activities which have a public good nature. One finding indicates that advertising, and not output as in Fershtman and Kamien (Econometrica 55 (1987) 1151–1164) is responsible for the higher stationary price found in the open loop equilibrium relative to the linear feedback one. That is, free-riding is more intense when firms play linear Markov feedback strategies. However, the collusive outcome can be approximated, and opportunism eliminated, if firms can engage in preplay negotiations where they select a nonlinear Markov perfect strategy for output and advertising. Achieving the collusive outcome requires (as in the Folk Theorem for infinitely repeated games) the discount rate to be sufficiently small. 2000 Elsevier Science B.V. All rights reserved. Keywords: Differential games; Advertising; Sticky price; Markov strategy; Linear and nonlinear JEL classification: C72; C73; L13; M37
1. Introduction This paper develops a model of duopolistic competition by building on Fershtman and Kamien (1987) (henceforth denoted as FK). A distinguishing * Present address: Business School, University of Nottingham, University Park, Nottingham NG7 2RD, UK. E-mail address:
[email protected] (C.A.G. Piga) 0167-7187 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 98 )00030-7
596
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
modification is the inclusion of advertising, which has the effect of increasing the size of the market.1 Following Friedman (1983), such a type of advertising, which presents the characteristics of a public good, is defined as perfectly cooperative.2 We maintain the assumption that price is sticky; that is, it does not adjust simultaneously to the level implied by the demand curve, for given levels of industry output and advertising. The conventional framework for analysing players’ behaviour in a dynamic market is as a differential game.3 Two classes of equilibrium are commonly analyzed within a differential game: open-loop and feedback. The latter is further divided into linear and nonlinear Markov-Perfect strategies, i.e. strategies in which the players’ actions depend, linearly or nonlinearly, on a state variable that summarises the direct effect of the past actions on the current state of the system (Tsutsui and Mino, 1990; Dockner and Van Long, 1993; Wirl, 1994). The paper is divided in two parts. The first investigates how the provision of a ‘public good’ in the form of advertising affects the behaviour of both price and output rate in the open-loop and linear feedback equilibria. In FK, the stationary linear feedback price is always below the open-loop counterpart: the opposite then holds with regard to the output rates. Considering that in the feedback case firms can adjust the level of advertising and output in each period, one would tend to think that the combined effect of advertising and output feedback strategies would yield a price higher than that in the open-loop, thus inverting the FK result. However, it turns out that prices follow the same pattern as in FK, at least when linear feedback Markov strategies are considered. The rationale behind this result now hinges around the different levels of promotional effort which are exercised in both equilibria, whereas output levels do not play the crucial role they have in FK. More precisely, the paper shows that: 1. open-loop output rate is greater than that in the feedback equilibrium, if the impact of advertising on price is significant or the cost of advertising is not too high; 2. because of advertising’s ‘public good’ nature, promotional effort is higher in the open-loop equilibrium and, consequently, the open-loop price is higher than the linear feedback;
1
By ‘advertising’ we refer to a broader class of non-price variables which increase consumers’ willingness to pay. 2 On the other hand, advertising is referred to as purely predatory if it only affects the distribution of market shares. 3 See Kamien and Schwartz (1991), Fudenberg and Tirole (1991), Clemhout and Wan (1994) and Friedman (1994). Driskill and McCafferty (1989), (1996) and Reynolds (1987) are examples of linear-quadratic games of dynamic competition. Dawid and Feichtinger (1996) find explicitly the feedback Nash equilibrium of a game that is not linear-quadratic.
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
597
3. because the incentive to ‘free-ride’ is less intense when firms adopt openloop decision rules, profits are higher in the open-loop equilibrium. Point 1 emphasises that models where firms use ‘multiple competitive tools’ (Roberts and Samuelson, 1988; Slade, 1995) can produce results different from those obtained when the same tools operate in isolation. The statements 2 and 3, although deriving from a completely different framework, are consistent with earlier findings in Fershtman and Nitzan (1991). They show that the ‘free-riding’ problem is aggravated if agents can make their contributions to a ‘public good’ dependent upon the observation of its accumulated stock (i.e. if the players can adopt linear feedback strategies). The second part of the paper demonstrates that it is possible to construct nonlinear feedback Markov strategies supporting steady state prices which are greater than the open-loop one. In general, these stationary prices are lower than the price in the collusive equilibrium where firms maximise joint profits. However, if firms adopt nonlinear Markov feedback strategies and the discount rate is small enough, then we obtain the interesting result that the collusive price can be supported as a subgame perfect equilibrium. Tsutsui and Mino (1990), who also use FK’s framework, were the first to identify the analogy between the Folk Theorem in repeated games and the equilibrium properties of nonlinear Markov feedback strategies. Our analysis demonstrates that firms can achieve the individual Pareto efficient outcome, thus avoiding the ‘prisoners’ dilemma’ situation identified above, not via the use of a discontinuous trigger strategy, but by constructing an appropriate continuous nonlinear Markov feedback strategy. The next section illustrates the main features of the model. Section 3 analyses the linear feedback equilibrium, while open-loop strategies are derived in Section 4. The implications arising from the assumption of a perfectly cooperative advertising are highlighted in Section 5. The equilibrium in nonlinear Markov feedback strategies is derived in Section 6. Some concluding comments are reported in Section 7. Appendix A contains proofs of selected propositions.
2. The model Consider a duopoly where both firms have the following cost function: 1 C(qi ) 5 cqi 1 ] q 2i ; i 5 1,2 2
(1)
where qi represents firm i’s output. Firms can run an advertising campaign which is made up of promotional efficiency units. A campaign containing A i units has a cost, for firm i, equal to G(A i )51 / 2m A i 2 . The choice of a quadratic cost function is justified by several studies which show the existence of decreasing returns to scale for advertising
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
598
(Chintagunta and Vilcassim, 1992; Erickson, 1992; Feichtinger et al., 1994). The parameter m depends on the governmental policies related to advertising; it, for instance, encompasses the effects of a tax on advertising or of laws which forbid the access to certain media. It is assumed that the intercept of the demand function varies with the sum of the promotional units by the two firms: p 5 a 1 a (A 1 1 A 2 ) 2 (q1 1 q2 ),
(2)
where a measures the effect on price of one unit of promotional activity. Eq. (2) highlights how the investment by one firm generates a positive externality for the other. The dynamic nature of the model derives from the assumption that the price is ‘sticky’. This entails a discrepancy between the current price and the price implied by (2) for a given level of output and advertising. The adjustment process, which is similar to that in FK, is governed by the following differential equation: dp p~ (t) 5 ] 5 s[a 1 a (A 1 (t) 1 A 2 (t)) 2 (q1 (t) 1 q2 (t)) 2 p(t)]; p(0) 5 p0 , dt
(3)
where 0,s,` denotes the speed with which the price converges to its level on the demand function. FK discuss how the specification of price stickiness in (3) is consistent with consumers’ utility functions that depend both on current and past consumption of a good. They show that the smaller s is, the greater is the effect of past consumption on the marginal utility of current consumption and therefore on today’s price.4 The assumption of sticky price is therefore relevant for markets such as those of addictive products, for which past consumption has by definition clear consequences on current consumption decisions. Under the above assumptions, the problem for the two firms is to maximise their discounted stream of profits, subject to (3): max J i 5 qi, A i
E
`
0
F
G
1 1 e 2 r t pqi 2 cqi 2 ] q i2 2 ] m A i2 dt; i 5 1,2 2 2
(4)
s.t. (3), qi > 0, A i > 0, where r denotes the intertemporal discount rate. Note that in the static game, when the demand function is specified by (2) and the cost functions for output and advertising are, respectively, C(qi ) and G(A i ), the Cournot equilibrium is given by the following expressions:
4 A different, but not alternative, explanation is that for s finite, current price is higher than the level implied by the demand function because consumers pay today for the utility that they derive both today and in the future.
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
m (a 1 c) 2 ca 2 (a 2 c)a m (a 2 c) p d* 5 ]]]]] , A *i 5 ]]]] , i 5 1,2; 2 2 , q* i 5 ]]]] 2m 2 a 2(2m 2 a ) 2(2m 2 a 2 )
599
(5)
whereas the ‘competitive’ equilibrium, whereby both firms behave according to the formula ‘price5marginal cost’ can be expressed as follows: a 1 2c a2c p *c 5 ]], A ic 5 0, q ic 5 ]], i 5 1,2. 3 3
(6)
When the price has to be equal to the marginal cost, there is no incentive for a firm to carry out a promotional investment which increases total marginal cost, since the other firm, by not investing, can charge a lower price and appropriate the entire market. Hence, advertising in the competitive equilibrium is equal to zero.
3. The feedback Nash equilibrium in linear Markov strategies We begin the analysis of the game by deriving the feedback strategies for the two control variables, assuming that the firms adopt Markov strategies which are linear functions of the state variable p. Proposition 3.1. There exists a unique feedback Nash equilibrium in linear and stable Markov Perfect strategies: ] 1 m ( r 1 6s) 2ŒD qˆ i ( p) 5 1 2 ] ]]]]] p2c 6 s(a 2 1 m ) ] m ((6sm 2ŒD)(a 1 c) 1 rm (a 1 2c) 2 6csa 2 ) 2 ] ]]]]]]]]]]]] (7) ] 3 (a 2 1 m )( rm 1ŒD)
S
D
] a ( m ( r 1 6s) 2ŒD) Aˆ i ( p) 5 ] ]]]]] p 6m s(a 2 1 m ) ] a ((6sm 2ŒD)(a 1 c) 1 rm (a 1 2c) 2 6csa 2 ) 1 ] ]]]]]]]]]]]] ] 3 (a 2 1 m )( rm 1ŒD)
D 5 [ m 2 ( r 1 6s)2 2 12m s 2 (a 2 1 m )] . 0.
(8) (9)
First of all, note that for a 50, advertising is zero and the output strategy is that indicated in FK. The above relationships show that the players’ actions are conditional on the observation of the current value of the price; i.e. the state variable summarising the latest information on how the dynamic system has evolved. It is straightforward to verify that the coefficients multiplying the price in (7)
600
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
and (8) are both positive. Hence, the output strategy shows that an output expansion will induce a reduction in the price level which will also reduce the rival’s output rate in the subsequent period. Therefore, in each period, both firms face the incentive to raise their output, despite the detrimental effect that such behaviour has on price. However, output is not the only strategic decision that firms have to take. Advertising can, in principle, be used to limit, or even eliminate, the reduction in price caused by the strategic use of output. A higher level of advertising at time t increases price and therefore advertising at t11. Firms can therefore decide to produce less because a drastic fall in price reduces the contribution in advertising by the rival firm. The actual relationship between the two decision variables can be obtained by solving for p in (8) and substituting it in (7):
m qˆ i 5 p 2 c 2 ] Aˆ i . a
(10)
As expected, advertising determines a reduction in the output rate. After observing the realisation of p, firms can then adjust their advertising level so as to counterbalance the incentive to raise output. However, the results obtained so far do not clearly show whether the combined action of both variables may result in a price which converges towards the joint profit maximisation steady state level p *jp : 5
rm (a 1 2c) 1 sm (3a 1 2c) 2 4sa 2 c * 5 ]]]]]]]]]] p jp . 3rm 1 5m s 2 4sa 2
(11)
In FK, the steady state feedback price when firms use a linear Markov output strategy is lower than the collusive price. To prove that this result also arises when advertising is considered, we first need to derive the steady state feedback price and then consider the ‘limit game’ in which firms do not discount future. Corollary 3.1. The feedback equilibrium identified in Proposition 3.1 is globally asymptotically stable. The expressions for the steady state equilibrium price, output and advertising are given by:
5
D)(a 1 2c) 1 12sm (a 1 c) 2 12a 2 scgsm f(5rm 1Œ] pˆ ss 5 ]]]]]]]]]]]]]] , ] s3sm 2 rm 1Œ] Dds rm 1ŒDd
(12)
] (a 2 c)f rm ( r 1 3s) 1 2s 2 (2m 2 a 2 ) 1 ( r 1 s)ŒDgsm 2 qˆ ss 5 6]]]]]]]]]]]]]]] ] s3sm 2 rm 1Œ] Dds rm 1ŒDd 2
(13)
To derive (11) entails a procedure similar to that used in Corollary 3.1, and is therefore not repeated here. The proof is available from the author on request.
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
s 2 am (a 2 c) ss Aˆ 5 6]]]]]]]] ] . s3sm 2 rm 1Œ] Dds rm 1ŒDd
601
(14)
Proof. Eq. (41) shows that there exists a unique stable path towards pˆ ss . Substituting pˆ ss in the optimal response functions (7) and (8) yields (13) and (14). Q.E.D. An analysis of advertising steady state equilibrium shows that no advertising campaign is carried out if r →`, i.e. when the firms heavily discount the future. As previously discussed, the absence of advertising implies a perfectly competitive market in which, from (10), qˆ i 5p2c, i51,2, which represents the well known rule MC5MR with both marginal values derived from the instantaneous, not the long run, profit function. An analogous result is obtained when the adjustment speed tends to zero. Indeed, if the price is highly insensitive to output and advertising, and at the limit does not vary at all to reflect the market conditions, firms consider price as given and behave as they would in a perfectly competitive market. In such a situation, the only Nash equilibrium for advertising is ‘not to invest’. Indeed, if one firm did invest, then the other, by not investing, could appropriate the entire market by charging a lower price. To simplify the comparison between the prices in (11) and (12), we analyze the equilibrium in the limit game, that is the game where we let the adjustment speed tend to infinity or the discount rate to zero.6 Proposition 3.2. Let s →` or r→0. Then the price in the feedback equilibrium in linear strategies converges to ]]]]] smf12ss m (a 1 c) 2 ca 2d 1œ12s 2 m (2m 2 a 2 )(a 1 2c)g pˆ l 5 ]]]]]]]]]]]]]]] ]]]]] ]]]]] œ12s 2m (2m 2 a 2 )s3sm 1œ12s 2m (2m 2 a 2 )d ]]]]] 12s 2 m (2m 2 a 2 )p d* 1 3sm p c* œ 5 ]]]]]]]]] (15) ]]]]] , s3sm 1œ12s 2m (2m 2 a 2 )d which is a convex combination of p *c and p d* . Hence:
m (3a 1 2c) 2 4a 2 c p *c , pˆ l , p d* ,lim p *jp 5 ]]]]]] . r →0 5m 2 4a 2
(16)
Proof. To obtain the first part of (15), simply evaluate the limit of (11) and (12)
6 ‘‘When s goes to infinity the dynamic structure disappears and the price jumps instantaneously to its level on the demand function for each level of output. Thus, for s→`, the game can be viewed as a repeated Cournot game in its continuous time version’’ (Fershtman and Kamien, 1987, p. 1159).
602
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
for r →0. After algebraic manipulation, it is possible to derive the second part of ˆ (15). For s→`, apply the l’Hopital rule to (12). Q.E.D. Evidently, it is never possible to construct linear Markov Perfect strategies for advertising and output that yield the collusive price. The introduction of nonlinear Markov strategies is considered in Section 6. Prior to this, we investigate how the open-loop equilibrium price, especially in the ‘limit game’, compares to both the linear feedback and the collusive price.
4. The open-loop Nash equilibrium An open-loop decision rule is independent of the current state of the state variable and the actions taken by the other players (Mehlmann, 1988). It therefore consists of a time path for the control variables which is decided at the outset of the game and cannot be changed later. Such a precommitment implies that open-loop rules are not, in general, subgame perfect. Proposition 4.1. There exists a unique globally asymptotically stable open-loop Nash equilibrium. The steady state equilibrium levels for output, advertising and price are given by
m (a 2 c)( r 1 s) q˜ i 5 ]]]]]] 2s(2m 2 a 2 ) 1 3rm
(17)
(a 2 c)sa A˜ i 5 ]]]]]] i 5 1,2; 2s(2m 2 a 2 ) 1 3rm
(18)
2 2 2sm (a 1 c) 2 2sca 1 rm (a 1 2c) 2ss2m 2 a d p *d 1 3rm p *c ]]]]]]] p˜ 5 ]]]]]]]]]] 5 2s( m 2 a 2 ) 1 3rm 2s(2m 2 a 2 ) 1 3rm
(19)
First note that if a 50, advertising is equal to zero and price and quantity take the value in FK. Moreover, it is straightforward to show that: 1. for s→0 or for r →`, the steady state open-loop equilibrium price converges to the static competitive price; 2. for r .0 and s.0, the price in the open-loop equilibrium is below that in the static Cournot equilibrium but in the ‘limit game’ case, for r →0 or for s→`, the two prices coincide. Thus, the open-loop limit game yields an equilibrium price which is higher than its linear feedback counterpart but is still below the collusive level. Although this holds true also in FK, the explanation of such a result in our setting lies with the different effects that advertising exerts in the two equilibria. In particular, it is worth investigating why, despite the relationship (10) between output and
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
603
advertising still being valid in the open-loop case (see Eqs. (43) and (44) in Appendix A), the feedback price still falls short of the other. As the next section shows, under an open-loop rule the provision of the ‘public good’ in the form of advertising always exceeds that from the feedback rule, i.e. the incentive to free-ride is stronger when the players adopt state contingent strategies.
5. The consequences of cooperative advertising A novelty of the model is that it is necessary to consider the joint action of output and advertising, which, as already discussed, have conflicting effects on price. Firms can step up their output but, at the same time, limit a fall in price by simply investing in advertising. We should then expect a situation in which firms are able to adjust their output and advertising decision period by period so as to enjoy a higher price. However, the ‘limit games’ above yield the opposite result that competition is stiffer if firms can base their actions upon the observation of the realised price. To ascertain which factor drives the results in the ‘limit game’, we will resort to a graphical analysis. More precisely, we will conduct a comparison of the price, output and advertising steady state levels in the two equilibria as a function of the new parameters of the model, namely a and m.7 An increase in m determines an increase in the cost of an advertising campaign: it is easy to predict that this will cause a reduction in advertising which, in turn, will bring about a lower price and, via (7), a smaller output. On the other hand, an increase in a is expected to boost all the variables under analysis. Fig. 1 illustrates what has already been stated in Propositions 3.2 and 4.1. Open-loop price is always greater than the feedback one: both decrease with m and are positively affected by a. Fig. 2 highlights the first original result of the paper. Although the price behaviour follows the same pattern identified in FK, in the present paper feedback output is not always greater than the open-loop output. In particular, side (a) shows that for small values of m the opposite holds true. When the advertising cost is low, firms which cannot deviate from the strategies decided at the outset of the game will tend to raise their output levels more than they would do if, in each period, they could condition their decision on the observation of the realized price. Similarly, FK’s prediction holds true when a is small, but when the impact of advertising on the market size is conspicuous, the open-loop output exceeds the feedback output. Considering that open-loop price is in any case higher, we can expect that in this
7
Such values are those reported in Corollary 3.1 for the feedback equilibrium and in Proposition 4.1 for the open-loop equilibrium.
604
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
Fig. 1. Stationary price in the linear feedback and open-loop equilibrium as a function of m (a) and a (b). a5100, c55, s51, l 50.1, m 51, a 51, r 50.1.
equilibrium firms invest more in advertising. Fig. 3 confirms such an expectation. It shows that the difference between open-loop and feedback advertising decreases with m and increases with a. The previous discussion identifies an explanation for Proposition 3.2 that hinges on the advertising rather than on the output strategies. Recalling advertising’s ‘public good’ nature, the pattern in Fig. 3 is consistent with the findings in
Fig. 2. Stationary output in the linear feedback and open-loop equilibrium as a function of m (a) and a (b). a5100, c55, s51, l 50.1, m 51, a 51, r 50.1.
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
605
Fig. 3. Stationary advertising in the linear feedback and open-loop equilibrum as a function of m (a) and a (b). a5100, c55, s51, l 50.1, m 51, a 51, r 50.1.
Fershtman and Nitzan (1991), where it is shown that voluntary dynamic contributions to a public good are smaller in the feedback equilibrium as compared to the open-loop one. Therefore the free-riding problem is exacerbated when players can observe the accumulated stock of the public good, although this behaviour reduces agents’ utility. A similar outcome also occurs here. Indeed, Fig. 4 shows that the difference in the steady state open-loop and feedback profits is always positive. It is noteworthy that in the differential game of a duopoly with product differentiation analyzed in Piga (1998), where advertising exerts both a cooperative and a predatory effect, linear feedback advertising and profit show a pattern which is opposite to that identified here. However, the assumption that promotional activities carry only the effect of increasing the size of the market can be used to describe many real-life situations. Consider, for instance, the literature on franchising arrangements (Norton, 1995; Dnes, 1996). ‘Horizontal free-riding’ arises because franchisees must invest in costly promotional activities which generate a positive externality for all the other members of the chain. The resulting under-investment can damage the reputation of the chain permanently. In order to prevent this from occurring, franchisers have to design an optimal monitoring strategy. Gal-Or (1995) tackles this problem by assuming that the investment in improving the quality standards is, like in the present setting, perfectly cooperative. Given the assumption of a sticky price, the market for addictive products provides another relevant example. Roberts and Samuelson (1988) find that advertising in the US cigarette market is highly cooperative. Furthermore, their
606
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
Fig. 4. Difference between the open-loop steady state profit, p˜ , and the stationary feedback profit pˆ , as a function of m and a. a5100, c55, s51, l 50.1, r 50.1.
estimates reject the hypothesis that the companies operating in the US cigarette market follow ‘naive’ open-loop strategies but, instead, reveal that these firms follow ‘sophisticated’ feedback strategies. Dawid and Feichtinger (1996) consider a differential game where drug dealers exert an effort in the attempt to enlarge their market. The model can be used to evaluate the effects on consumption of policies which increase the cost of advertising in these types of markets. Such policies have in this model the unambiguous effect of reducing consumption and firms’ profits. The next section demonstrates that the Pareto outcome can be achieved, and ‘free-riding’ eliminated, when firms adopt nonlinear Markov feedback strategies.
6. The Nash equilibrium in nonlinear Markov feedback strategies In this section we analyze whether the use of nonlinear Markov strategies for advertising, and, given (10), for the output rate, can modify the results discussed thus far. In particular, we will consider strategies of the form: nl Aˆ i 5 g1 1 g2 p 1 g3 h( p), i 5 1,2
(20)
where h( p) is a nonlinear function of price and gj , j51,2,3 depend on the
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
607
parameters of the model.8 Following Tsutsui and Mino (1990) and using the Bellman’s equation approach, it can be demonstrated that h( p) satisfies a nonlinear differential equation having the general solution: C 5 [h 2 z a x] j 1 [h 2 z b x] j 2 ,
(21)
where z a , z b , j 1 and j 2 are functions of the parameters of the model, x is a linear transformation of p and C is a constant of integration. In particular, for C50, the linear equilibrium in Proposition 3.1 can be obtained. In general, the presence of the constant of integration determines a multiplicity of nonlinear strategies (20) that satisfy (21). Proposition 6.1. Any equilibrium price, p nl in the interval 2rm (a 1 2c) 1 sm (3a 1 2c) 2 4sa 2 c p *c , p nl , pH ; ]]]]]]]]]] 6rm 1 5m s 2 4sa 2
(22)
can be supported as an asymptotically stable steady state equilibrium in nonlinear Markov strategies. A first implication is therefore that allowing firms to use nonlinear advertising and output Markov strategies eliminates the previously observed negative effects of underinvestment in advertising. Indeed, since p˜ ,pH , any feedback equilibrium ˜ pH ] can be achieved by choosing an appropriate nonlinear price in the interval [p, strategy for the control variables. However, the upper bound pH is, in the general case, lower than the joint profit maximisation price p *jp . Therefore a feedback Markov equilibrium supporting the collusive stationary price cannot be generally constructed. However, the ‘limit game’ is characterised by a property reminiscent of the Folk Theorem in repeated games. Proposition 6.2. In the limit game when r → 0, the collusive price p *jp can be supported as a steady state of nonlinear differentiable Markov strategies:
m (3a 1 2c) 2 4a 2 c * 5 ]]]]]] lim pH 5lim p jp r →0 r →0 5m 2 4a 2
(23)
Intuitively, the payoff of a stationary nonlinear Markov feedback equilibrium is positively correlated with the level of its supporting stationary price. Therefore
8 The proofs for the results in this section are not provided because they are similar to those in Tsutsui and Mino (1990) and Dockner and Van Long (1993). They are available from the author on request.
608
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
Proposition 6.2 agrees with the Folk Theorem: if the discount factor is sufficiently close to one, any individually rational payoff can be supported as a subgame perfect equilibrium of an infinite horizon dynamic game. Tsutsui and Mino (1990) were the first to find that the most efficient outcome can be supported by continuous and differentiable strategies. This implies that players can avoid using discontinuous strategies that allow the use of threat and trigger strategies. In the present context, where the provision of a public good has to be taken into account, the collusive outcome may be approximated, and the free riding problem eliminated, through an appropriate choice of nonlinear Markov perfect strategies.9 Such a choice, as in Dockner and Van Long (1993) who consider a ‘Tragedy of the Commons’ situation, is possible if players engage in preplay negotiation, which would presumably lead to the selection of the most efficient strategy. After the agreement has been reached, the self-enforcing property of Markov-perfect strategies would protect both players from reciprocal opportunism.
7. Conclusions This paper has developed a differential game based on Fershtman and Kamien (1987) whereby firms can also invest in market-enlarging promotional activities. Adding a dimension to the strategies space implies that firms can use multiple competitive tools. The linkages between output and advertising optimal strategies are discussed in relation to three different equilibrium concepts, namely open-loop and linear and nonlinear feedback Markov. It turns out that the choices of advertising and output mutually influence each other. In particular, advertising can be used to limit the incentive to expand output. In the linear feedback equilibrium, both depend linearly on price. We find that a greater output at time t lowers price at time t11, whereas the opposite is true for advertising. The overall joint effect of advertising and output on the linear feedback price is to make it smaller than the related open-loop price. As often pointed out, promotional activities in this model have a ‘public good’ nature and, in this respect, it is expected that firms face an incentive to free-ride. Such a problem is more intense in the linear feedback equilibrium relative to the open-loop one. Introducing nonlinear Markov perfect feedback strategies reveals the existence of multiple feedback stationary prices whose level is above that of the open-loop price but below the joint profit steady state price. However, the collusive outcome can be supported as a subgame perfect equilibrium if the discount rate is sufficiently small and players engage in preplay negotiations in
9
The problem of selecting among an infinite number of nonlinear strategies arises from the existence of multiple equilibria in Proposition 6.1.
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
609
which an appropriate nonlinear strategy — i.e. the one yielding the collusive outcome — is selected. We conclude by setting up an agenda for future research. First of all, recall that the specification of the model does not allow for the inclusion of a predatory role for advertising. It is not clear a priori whether the under-investment in the linear feedback equilibrium relative to the open-loop one would persist if advertising were mainly predatory. One should expect that firms would overinvest in advertising in the linear feedback equilibrium relative to the open-loop. Investigating such an issue would require the setting up of a model with two state variables, with firms selling differentiated products. Whether a joint profit maximisation outcome can be achieved through the use of nonlinear strategies depending on two state variables also deserves careful examination, because the technique used in Section 6, developed in Tsutsui and Mino (1990), is applicable to a class of differential games which have one state variable and an infinite time horizon.
Acknowledgements This paper was presented at the Annual Post-Graduate Economics Conference, Leeds University Business School, November 1997. I would like to thank Gianni De Fraja, Steve Lawford, Marco Vivarelli and an anonymous referee for useful comments on a previous version of the paper. The usual disclaimer applies. Financial support from the Royal Economic Society Fellowship Award is gratefully acknowledged.
Appendix A Proof of Proposition 3.1 The strategies (qˆ i , Aˆ i ) in the feedback equilibrium must satisfy the following set of Bellman’s equations:
H
rVi ( p) 5max pqi 2 cqi 2 1 / 2q i2 2 1 / 2m A i2 qi, A i
S
D
S
D
J
d 1 ] Vi ( p) s(a (A 1 1 A 2 ) 1 a 2 q1 2 q2 2 p) , i 5 1,2. dp
(24)
Take f.o.c.: d qˆ i 5 p 2 c 2 ] Vi ( p) s; dp
(25)
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
610
S
D
d ] Vi ( p) sa dp Aˆ i 5 ]]]] m
(26)
Substitute (25) and (26) into (24), obtaining:
S
a2 rVi ( p) 5 1 / 2s 2 1 1 ] m
SS
1 s
2
DS
D
d ] Vi ( p) dp
2
1 1 / 2p 2 2 pc 1 1 / 2c 2
D
D
a2 d d ] ] V ( p) 1 2cs 2 3ps 1 sa ] Vi ( p); i, j 5 1,2; i ± j. 11 m dp j dp (27)
The expression (27) constitutes a system of differential equations whose unknowns are the value functions (V1 ( p), V2 ( p)). Once these are found, it is possible to derive the optimal equilibrium strategies by substituting the value functions into (25) and (26). Given the linear-quadratic structure of the game, a quadratic value function is proposed: Vi ( p) 5 b1 p 2 1 b2 p 1 b3 ,
(28)
which implies that d ] Vi ( p) 5 2b1 p 1 b2 . dp
(29)
Substituting the two previous relations into (27) and rearranging terms with respect to p yields:
S
D
s 2 a 2 b 12 2 2 2 1 / 2 2 rb1 2 6sb1 1 6s b 1 1 6 ]]] p m
S S
D
s 2 a 2 b1 b2 2 1 6 s b1 b2 2 c 2 rb2 2 3sb2 1 4scb1 1 2sab1 1 6 ]]] p m
D
s 2 a 2 b 22 2 2 2 1 2scb2 1 sab2 1 3 / 2s b 2 2 rb3 1 1 / 2c 1 3 / 2]]] 5 0 m
(30)
This expression holds if its terms in round brackets multiplying p and p 2 and the constant term are simultaneously equal to zero. These make up a system of three equations in three unknowns, h b1 , b2 , b3 j, which have the following solutions: ] 1 rm 1 6sm 1ŒD b 11 5 ] ]]]]] , (31) 12 s 2 ( m 1 a 2 )
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
611
] 1 m ((a 1 2c)( rm 1ŒD) 1 6sm (a 1 c) 2 6csa 2 ) b 5 ] ]]]]]]]]]]]]] , ] 3 s( m 1 a 2 )( rm 2ŒD)
(32)
] 1 rm 1 6sm 2ŒD and b 21 5 ] ]]]]] , 12 s 2 ( m 1 a 2 )
(33)
] 1 m ((a 1 2c)( rm 2ŒD) 1 6sm (a 1 c) 2 6csa 2 ) b 22 5 ] ]]]]]]]]]]]]] , ] 3 s( m 1 a 2 )( rm 1ŒD)
(34)
1 2
where b3 s are omitted because not relevant for the proof. It is now necessary to ascertain the stability of these solutions. Substitute (25), (26) and (29) into (3), so as to obtain the differential equation which describes the optimal time path for the state variable p(t):
S
s 2 a 2 b1 p~ (t) 5 s(23 1 4sb1 ) 1 4 ]] m
D
s 2 a 2 b2 p(t) 1 s(2c 1 a 1 2sb2 ) 1 2 ]]. m (35)
~ The steady state price is obtained by imposing p(t)50: 2s(a 2 1 m )b2 1 2m c 1 am ss pˆ 5 2 ]]]]]]]] . 4s(a 2 1 m )b1 2 3m
(36)
The general solution of (35) is given by: ss
ss
p(t) 5 pˆ 1 ( p0 2 pˆ ) e
4s b 1 (a 2 1 m )23 m ]]]] t m
.
(37)
Substitute now the first set of solutions ( b 11 , b 12 ) into (36), obtaining pˆ
ss 1
D)(a 1 2c) 1 12sm (a 1 c) 2 12a 2 scgsm f(5rm 2Œ] 5 ]]]]]]]]]]]]]] . ] srm 1Œ] D 2 3smdsŒD 2 rmd
(38)
The general solution to (37) takes the form: p(t) 5 pˆ ss 1 1s p0 2 pˆ ss 1de 2
] 3s m 2 rm 2ŒD ]]] t m
.
(39)
] The necessary condition for convergence is that 3sm 2 rm 2ŒD .0, but this would imply a negative denominator and hence a negative price. The first set of solutions h b 11 , b 12 j does not therefore generate a stable solution. On the contrary, if we substitute the second set of solutions b 21 , b 22 into (36) and (37), we obtain, respectively:
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
612
pˆ
ss 2
D)(a 1 2c) 1 12sm (a 1 c) 2 12a 2 scgsm f(5rm 1Œ] 5 ]]]]]]]]]]]]]] , ] s3sm 2 rm 1Œ] Dds rm 1ŒDd
(40)
] 3s m 2 rm 1ŒD
]]]t m p(t) 5 pˆ ss 2 1s p0 2 pˆ ss 2de 2 .
(41)
] Now (41) converges stably, as the condition (3sm 2 rm 1ŒD) . 0 does not contrast with the requirement of non-negativity for the price pˆ ss 2 . To obtain the control variables’ strategies, substitute (29), (33) and (34) into the f.o.c. (25) and (26), so as to yield the expressions (7) and (8) in Proposition 3.1. Proof of Proposition 4.1 Form the current value Hamiltonians:
*i 5 p(t)qi (t) 2 cqi (t) 2 1 / 2q i2 (t) 2 1 / 2m A 2i (t) 1 ci (t)ssa 2 q1 (t) 2 q2 (t) 2 p(t) 1 as A 1 (t) 1 A 2 (t)dd, i 5 1,2;
(42)
where ci (t) represents the costate variable for firm i. Apply the necessary conditions for an equilibrium, that given the concavity of the problem are also sufficient: ≠* ]]i 5 p(t) 2 c 2 qi (t) 2 ci (t)s 5 0, i 5 1,2 ≠qi
(43)
≠* ]]i 5 ci sa 2 m A i (t) 5 0, i 5 1,2 ≠A i
(44)
c~ i (t) 5 ci (t)( r 1 s) 2 qi (t), i 5 1,2.
(45)
Differentiate (43) and (44) with respect to time t, so as to obtain the values for c~ i (t) and ci (t). Substituting them into (45) yields:
~ 2 q~ i (t) 5 ( r 1 s)( p 2 c 2 qi ) 2 sqi , i 5 1,2 p(t)
(46)
sa A~ i (t) 5 ( r 1 s)A i 2 ] qi , i 5 1,2. m
(47)
~ i (t)5q~ i (t)50, (3), (46) and (47) form a ~ Recalling that in the steady state p(t)5A system of equations whose solution is given by the expressions (17), (18) and (19) reported in Proposition 4.1. To prove global stability, differentiate (3) with respect to time t: ¨ 5 ssa (A~ 1 (t) 1 A~ 2 (t)) 2 u~ 1 (t) 2 u~ 2 (t) 2 p(t) ~ d. p(t)
(48)
By substituting the expressions for q~ i and A~ i from (46) and (47), we obtain:
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
613
p¨ (t) 5 s( r 1 s)fs A 1 (t) 1 A 2 (t)d 2s q1 (t) 1 q2 (t)dg
S
D
a 2s ~ 2 s s 1 ]] s q1 (t) 1 q2 (t)d 1 2s( r 1 s)( p(t) 2 c) 2 3sp(t). m
(49)
Evaluate the value of the first expression in squared brackets from (3), to obtain: ¨ 5 s( r 1 s)fp(t) ~ 1 sp(t) 2 sag p(t)
S
D
a 2s ~ 2 s s 1 ]] s q1 (t) 1 q2 (t)d 1 2s( r 1 s)( p(t) 2 c) 2 3sp(t). m
(50)
To eliminate the terms in qi (t) and obtain a differential equation with p(t) as the only unknown, differentiate again (50) with respect to time. The expression for q~ i (t) is to be substituted from (46) and the values for (q1 (t)1q2 (t)) from (50). The resulting third-degree differential equation in p(t) has the following solutions: p(t) 5 p˜ 1 C1 e ( r 1s)t , p(t) 5 p˜ 1 C2 e p(t) 5 p˜ 1 C3 e
(51)
]]]]]] m ( r 2s)1œm 2 ( r 15s) 2 28 m s 2 (a 2 1 m ) ]]]]]]]] t 2m
]]]]]] m ( r 2s)2œm 2 ( r 15s) 2 28 m s 2 (a 2 1 m ) ]]]]]]]] t 2m
,
(52)
,
(53)
where C1 , C2 , C3 are constants of integration. Only the third solution is globally asymptotically stable. After imposing the condition p(0)5p0 , it is straightforward to obtain p(t)5p˜ 1( p0 2p˜ ) e l1t, with l1 5[ m ( r 2 s) 2 ]]]]]]]] 2 2 2 2 œm ( r 1 5s) 2 8m s (a 1 m )] / 2m.
References Chintagunta, P.K., Vilcassim, N.J., 1992. An empirical investigation of advertising strategies in a dynamic duopoly. Management Science 38, 1230–1244. Clemhout, S., Wan, H.Y., 1994. Differential games — Economic applications. In: Aumann, R.J., Hart, S. (Eds.), Handbook of Game Theory, vol. 2. Elsevier, Amsterdam, pp. 802–825. Dawid, H., Feichtinger, G., 1996. Optimal allocation of drug control efforts: a differential game analysis. Journal of Optimization Theory and Application 91, 279–297. Dnes, A.W., 1996. The economic analysis of franchise contracts. Journal of Institutional and Theoretical Economics 152, 297–324. Dockner, E.J., Van Long, N., 1993. International pollution control: cooperative versus noncooperative strategies. Journal of Environmental Economics and Management 24, 13–29. Driskill, R., McCafferty, S., 1996. Industrial policy and duopolistic trade with dynamic demand. The Review of Industrial Organization 11, 355–373. Driskill, R.A., McCafferty, S., 1989. Dynamic duopoly with adjustment costs: a differential game approach. Journal of Economic Theory 49, 324–338. Erickson, G.M., 1992. Empirical analysis of closed-loop advertising strategies. Management Science 38, 1732–1749.
614
C. A.G. Piga / Int. J. Ind. Organ. 18 (2000) 595 – 614
Feichtinger, G., Hartl, R.F., Sethi, P.S., 1994. Dynamic optimal control models in advertising: recent developments. Management Science 40, 195–226. Fershtman, C., Kamien, M.I., 1987. Dynamic duopolistic competition with sticky prices. Econometrica 55, 1151–1164. Fershtman, C., Nitzan, S., 1991. Dynamic voluntary provision of public goods. European Economic Review 35, 1057–1067. Friedman, A., 1994. Differential games. In: Aumann, R.J., Hart, S. (Eds.), Handbook of Game Theory, vol. 2. Elsevier, Amsterdam, pp. 782–799. Friedman, J.W., 1983. Advertising and oligopolistic equilibrium. The Bell Journal of Economics 14, 464–473. Fudenberg, D., Tirole, J., 1991. Game Theory. MIT Press, Cambridge, MA. Gal-Or, E., 1995. Maintaining quality standards in franchise chains. Management Science 41, 1774– 1792. Kamien, M.I., Schwartz, N.L., 1991. Dynamic Optimization. Elsevier, New York. Mehlmann, A., 1988. Applied Differential Games. Plenum, New York. Norton, S.W., 1995. Is franchising a capital structure issue?. Journal of Corporate Finance 2, 75–101. Piga, C.A., 1998. A dynamic model of advertising and product differentiation. The Review of Industrial Organization, in press. Reynolds, S.S., 1987. Capacity investment, preemption and commitment in an infinite horizon model. International Economics Review 28, 69–88. Roberts, J., Samuelson, L., 1988. An empirical analysis of dynamic, nonprice competition in an oligopolistic industry. RAND Journal of Economics 19, 200–220. Slade, M.E., 1995. Product rivalry with multiple strategic weapons: an analysis of price and advertising competition. Journal of Economics and Management Strategy 4, 445–476. Tsutsui, S., Mino, K., 1990. Nonlinear strategies in dynamic duopolistic competition with sticky prices. Journal of Economic Theory 52, 136–161. Wirl, F., 1994. Pigouvian taxation of energy for flow and stock externality and strategic, noncompetitive energy pricing. Journal of Environmental Economics and Management 26, 1–18.