Channel coordination over time: incentive equilibria and credibility

Journal of Economic Dynamics & Control 27 (2003) 801–822www.elsevier.com/locate/econbase

Channel coordination over time: incentiveequilibria and credibility

Ste'en JHrgensena ; ∗, Georges Zaccourb

aDepartment of Organization and Management, University of Southern Denmark, Campusvej 55,DK-5230 Odense M, Denmark

bDepartment of Marketing, "Ecole des Hautes "Etudes Commerciales and GERAD, Montr"eal, Canada

Accepted 31 October 2001

Abstract

The paper studies a channel of distribution, consisting of a manufacturer and a retailer. In adynamic game of pricing and advertising, the carry-over e'ects of channel member’s advertisinge'orts are summarized in a stock of “advertising goodwill”. Each channel member has perfectinformation about the other channel member’s price and advertising e'ort. If each 7rm employsmarketing strategies that are linear functions of the other member’s actions, and these strategiesare jointly carried out, a Pareto-optimal joint pro7t maximization outcome can be realized. Thepaper also addresses the question of credibility of the incentive strategies. ? 2002 ElsevierScience B.V. All rights reserved.

JEL classi,cation: C73; M31; M37; D92

Keywords: Marketing; Channel coordination; Pricing and advertising; Incentive equilibrium; Credibility

1. Introduction

Consider a two-member channel of distribution, consisting of a manufacturer anda retailer. In the absence of cooperation, channel members determine their decisionvariables independently and noncooperatively. It is well known, in literature and inpractice, that such uncoordinated decision making creates “channel ine@ciency”, that is,channel members’ marketing strategies are not at their joint pro7t maximization levelsand their pro7ts are inferior to what could be achieved with coordinated behavior.This creates an incentive for cooperation. If channel members agree to cooperate, they

∗ Corresponding author. Tel.: +45-6550-3249; fax: +45-6593-1766.E-mail address: [email protected] (S. JHrgensen).

0165-1889/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.PII: S0165 -1889(01)00072 -0

802 S. J0rgensen, G. Zaccour / Journal of Economic Dynamics & Control 27 (2003) 801–822

negotiate to 7nd joint decisions that eliminate channel ine@ciency. We suppose thatalthough channel members cooperate, they remain independent.

The problem of marketing channel coordination has been addressed in a variety ofmodels, although most often in a static context. It is well known that channel coordi-nation can remedy pricing ine@ciency; mechanisms can be implemented such that theretailer always chooses a consumer price that maximizes total channel pro7ts. To illus-trate, Jeuland and Shugan (1988) used an approach which relies on a channel member’sconjecture of the other member’s decision making behavior, exploiting the concept ofconsistent conjectural variations (see Basar, 1986 for the theoretical background of theconcept). Another coordinating device is pro7t sharing which can be implemented indi'erent ways, for example, by a quantity discount scheme. The idea of pro7t sharingis to induce a relationship between total channel pro7ts and the individual pro7ts suchthat if anyone of these is maximized, they are all maximized (Jeuland and Shugan,1983; Moorthy, 1987).

Only a few studies have addressed channel coordination in a dynamic setting. Theseworks typically study dynamic advertising problems. Examples are Chintagunta andJain (1992), JHrgensen and Zaccour (1999) who compared coordinated strategies andpro7ts with uncoordinated ones. The purpose of the paper at hand is di'erent. We wishto address a problem that has been left open in previous studies: how can a coordinatedsolution be sustained over time? We propose a dynamic pricing and advertising problemin which channel members apply decision-dependent marketing strategies, that is, onechannel member makes his current decision contingent upon the current decision ofthe other channel member.

Marketing literature (e.g., Moorthy, 1987) often has assumed an asymmetric re-lationship where one of the channel members, typically the manufacturer, assumesthe responsibility for establishing a coordinated outcome. The manufacturer acts as aStackelberg leader. In this paper we suppose that channel members are symmetric, inthe sense that no channel member can unilaterally enforce her strategy upon the other(cf. Jeuland and Shugan, 1983).

The purpose of coordinated decision making here is the maximization of the jointpro7ts of the channel members. We assume that an appropriate notion of intertemporalindividual rationality is satis7ed such that the channel members, for this reason, willnot refuse to act in accordance with the e@cient outcome. 1 Now, although it is inthe common interest of channel members to cooperate, and individual rationality issatis7ed, each channel member may have an incentive to cheat on the agreement, i.e.,to choose another course of action than the one prescribed by the agreement. Thepossibility of cheating exists when the 7rms are independent and the agreement is notbinding, and an incentive to cheat exists if a 7rm can improve its payo' by deviatingfrom the agreement. To stabilize the agreement, ways must be found to deter cheating.Illustrative accounts of the cheating problem are given in Osborne (1976) and Philips(1988).

1 The literature contains various notions of intertemporal individual rationality; see Kaitala and Pohjola(1990), Petrosjan (1998), JHrgensen and Zaccour (2001).

S. J0rgensen, G. Zaccour / Journal of Economic Dynamics & Control 27 (2003) 801–822 803

To eliminate the incentives to cheat on the agreement, di'erent mechanisms areavailable. In very rare cases it happens that the cooperative solution is a Nash equi-librium outcome (or, put in another way, a Nash equilibrium is an e@cient outcome),but see Chiarella et al. (1984) and RincLon-Zapatero et al. (2000). Then coopera-tion is self-enforcing, but equilibrium strategies may be incredible o' the equilib-rium path. To resolve the credibility problem, a usual requirement is subgame per-fectness. Then, after no history of the game, no player has an incentive to deviatefrom her equilibrium strategy. One way to enforce a cooperative agreement as a sub-game perfect equilibrium is by using trigger strategies. Such strategies are based onplayers’ observations of the history of the game, and their reactions on deviationsfrom the agreed path. Trigger strategies embody a threat to punish e'ectively andcredibly any player who defects on the agreement. (E'ectively means that a cheaterwill be worse o' by cheating than by sticking to her part of the agreement. 2 ) Thereare other issues related to enforceability. For instance, would the players still usetheir trigger strategies if they negotiated and restored cooperation after a punishmentphase?

It has been noticed that trigger strategies may embody large discontinuities, that is,a slight deviation from an agreed path triggers retaliation in the form of harsh pun-ishments that will generate a path much di'erent from the agreed one. An alternativeapproach, and the one to be employed here, is to use incentive strategies that arecontinuous in the information. An incentive equilibrium has the property that whenboth players implement (or are believed to implement) their incentive strategies, thecooperative outcome is realized as an equilibrium. Hence, no player should be temptedto break the agreement during the course of the game, provided that incentive strate-gies are credible. An incentive strategy is credible if it is better for a player who hasbeen cheated to use her strategy, than to stand pat on the coordinated solution. Ehtamoand HOamOalOainen (1993, 1995) used linear incentive strategies in a dynamic resourcegame and demonstrated that such strategies are credible when deviations are not toolarge.

We apply the methodology developed in Ehtamo and HOamOalOainen (1993, 1995) toa channel coordination problem. The paper proceeds as follows. Section 2 presentsa dynamic game model of pricing and advertising which generalizes the symmetricgame of JHrgensen and Zaccour (1999). (However, in our discussion of the credibilityproblem we need to revert to a symmetric setting.) Obtaining the coordinated solutionis a straightforward optimal control problem which is solved in Section 3. Section4 contains the main contributions of the paper: it identi7es incentive equilibria withtwo di'erent types of linear incentive strategies and discusses the credibility problem.Section 5 concludes.

2 Game theoretic oligopoly theory, based upon repeated games, shows many instances where trigger strate-gies sustain a collusive solution as an equilibrium. There are not very many applications of trigger strategiesin di'erential games, partly due to the technicalities involved, and partly because the intuition of triggerstrategies in a continuous-time setup may be di@cult to comprehend. See, however, Kaitala and Pohjola(1990), Tolwinski et al. (1986), Dockner et al. (2000, Chapter 6) for some applications.


2. A dynamic game of pricing and advertising

Consider a two-member channel with manufacturer M and retailer R. 3 The decisionvariables of each channel member are advertising e'ort and price (transfer price andretail price, respectively). Denote by aM(t) and aR(t) the 7rms’ advertising e'ort ratesat time t ∈ [0; T ]. An important dynamic e'ect of advertising is carry-over which meansthat the e'ect of advertising is not exhausted in the current period. Carry-over e'ectshave often been modeled by summarizing them in a stock of “goodwill” G(t). Thestock can be interpreted to represent brand goodwill at the retailer’s store, and can beinRuenced through both channel members’ advertising e'orts. (To illustrate, one canthink of the manufacturer’s e'ort as “national advertising” and the retailer’s as “localadvertising”.) Manufacturer and retailer advertising toward the 7nal consumers aim atenhancing goodwill, which in turn stimulates consumer demand. (An example is themarketing of automobiles.) The goodwill stock evolves according to the dynamics

G(t) = �MaM(t) + �RaR(t) − �G(t); G(0) = G0¿ 0: (1)

In (1), �j ¿ 0, j∈{M;R} are “advertising e@ciency” parameters, measuring the impactof the e'orts on the rate of change of the goodwill stock. If �M = �R consumers reactin the same way on advertising made by the manufacturer and the retailer: advertisinge'orts are perfectly substitutable (Doraiswamy et al., 1979).

We should note that the goodwill stock plays the role as a public good, which can beinRuenced by the actions (advertising e'orts) of both 7rms. Both 7rms can bene7t fromthe stock because—as will be assumed below—the size of the stock a'ects consumerdemand. In the absence of a binding agreement to cooperate, a free-rider problem occurssince the provision of advertising e'ort is voluntary. Then the individual incentives ofthe channel members may lead to underprovision of advertising e'orts and the goodwillstock will be smaller. (For a formal derivation of the result that e'orts and stock arelower under noncooperation than under cooperation, see Chintagunta and Jain, 1992;JHrgensen and Zaccour, 1999.)

We suppose that the demand rate q(t) also depends on the retail price pR(t). Formodeling purposes, multiplicatively separable demand speci7cation have been quitepopular: q(pR ; G) = f(pR)h(G). Let function f(pR) be linearly decreasing and leth(G) be quadratic and concave. Then the demand rate is given by

q(t) = f(pR(t))h(G(t)) = [�− �pR(t)][g1G(t) − g2[G(t)]2=2] (2)

in which g1; g2, �, and � are positive constants. The speci7cation (2) implies thatthe price-function f(pR) is shifted by the level of goodwill but, due to concavity offunction h, the marginal e'ect of raising the goodwill stock is diminishing (i.e., thereis an advertising saturation e'ect).

Denote by pM(t) the transfer price charged per unit of sales from the manufacturerto the retailer and let c = const:∈ (0; �=�) be the manufacturer’s unit production cost.The advertising cost functions are quadratic and convex in e'ort: wja2

j =2, j∈{M;R}.

3 To simplify we assume that R only sells M’s brand within the product class. Although the assumptionmay be somewhat restrictive, a number of industries meet this assumption, see Doraiswamy et al. (1979).


The objective functionals are accumulated pro7ts over a 7xed and 7nite planningperiod [0; T ], plus a salvage value at the horizon date T :

JM =∫ T

0e−�t{[pM(t) − c]f(pR(t))h(G(t)) − wM[aM(t)]2=2} dt

+ e−�T SMG(T ); (3)

JR =∫ T

0e−�t{[pR(t) − pM(t)]f(pR(t))h(G(t)) − wR[aR(t)]2=2} dt

+ e−�T SRG(T ) (4)

in which the demand rate f(pR(t))h(G(t)) is given by (2), Sj¿ 0, j∈{M;R} is theconstant salvage value of a unit of the terminal stock of goodwill, as assessed bychannel member j, and � is a constant and nonnegative discount rate.

The natural state constraint G(t)¿ 0 is automatically satis7ed for all t ∈ [0; T ] whenadvertising rates are constrained to be nonnegative. For technical reasons it is conve-nient to let the state space be upper bounded:

Assumption 2.1. A feasible goodwill stock satis,es G(t)∈ [0; g1=g2] ∀t ∈ [0; T ].

Assumption 2.1 implies that function h(G) in (2) always is increasing. It seemsplausible to restrict the state space to such situations in which an increase in thegoodwill stock leads to an increase (and not a decrease) in consumer demand. (Theassumption also guarantees positivity of costate variables in optimal control problemsto be solved below.)

3. The cooperative solution

The cooperative solution maximizes the sum of the objective functions in (3) and(4) and is Pareto optimal. Henceforth we refer to the joint maximization outcome asthe desired one, represented by superscript “d”. The objective functional of the jointmaximization problem is

J d = JM + JR =∫ T

0e−�T

[pR(t) − c]f(pR(t)h(G(t))

−∑

j∈{M; R}wj[aj(t)]2=2

dt + e−�T [SM + SR]G(T ) (5)

which must be maximized with respect to retail price pR and advertising e'ort ratesaj, subject to (1) and the control constraints aj(t)¿ 0; c6pR(t)6 �=� ∀t ∈ [0; T ].


Since the retail price does not appear in the dynamics (1), and the integrand of (5)does not depend explicitly on time, an optimal retail price is constant over time. It isthe static monopoly price

pdR = [� + �c]=2�: (6)

To save on notation, introduce the positive constant

�∗ , (pdR − c)(�− �pd

R) = (�− c�)2=4�: (7)

The dynamic advertising problem is an optimal control problem. Using (5) and (7) theproblem is:

maxaM ; aR

∫ T

0e−�t

�∗h(G(t)) −

∑j∈{M; R}

wj[aj(t)]2=2

dt + e−�T [SM + SR]G(T )

subject to aj(t)¿ 0 and (1). The solution is straightforward and is presented in Propo-sition 3.1.

Proposition 3.1. De,ne K ≡ �2M=(wM) + �2

R=(wR) = const:¿ 0. In the joint maxi-mization problem; the triple (ad

M(t); adR(t); Gd(t)) is optimal since for all t ∈ [0; T ] the

maximized Hamiltonian is concave in G and there exists a continuous and continu-ously di=erentiable costate �d(t) such that

adj (t) = �j�d(t)=wj; (8)

�d(t) = (� + �)�d(t) − �∗[g1 − g2Gd(t)]; �d(T ) = SM + SR ; (9)

Gd(t) = −�Gd(t) + K�d(t); Gd(0) = G0: (10)

Proof. See the appendix.

The appendix shows that the optimal advertising rates in (8) are strictly positivefor all t ∈ [0; T ] if at least one salvage value Sj is positive. The proposition con7rmsthe intuition that the optimal advertising rate of channel member j increases if thee'ectiveness of that member’s advertising (�j) increases and=or the shadow price ofgoodwill (�d) increases. On the other hand, advertising is decreased if the advertisingcost (wj) increases. The allocation of advertising e'ort between the channel membersdepends on the cost-e'ectiveness (wj=�j) of their advertising e'orts.

The di'erential equations (9) and (10) are linear and admit the explicit solutionstated in the appendix. Using (8) shows that the optimal advertising rates evolve inthe same way as the shadow price �d. Thus, a (Gd ; �d) phase diagram will look thesame as a (G; ad

j ) phase diagram. We illustrate the (Gd ; �d) diagram in Fig. 1 whichdepicts the stable path to the steady state (Gd

∞; �d∞). However, this path can only

be a solution if the problem has an in7nite horizon. In a 7nite horizon problem, anoptimal trajectory can be any of the types marked I–IV. The transversality condition


Fig. 1. Phase diagram. Coordinated solution.

�d(T )=SM +SR, the level of the initial stock G0, the steady state point, and the lengthof the planning horizon T jointly determine which trajectory is optimal.

Consider a case where �d∞ ¿SM + SR. It means that the channel members’ assess-

ments of the 7nal goodwill stock are “low”. If G0 ¡Gd∞ (the initial goodwill stock

is “low”), trajectories of type III apply, and optimal advertising rates are always de-creasing. Goodwill decreases steadily (IIIA), or increases intially and then decreases(IIIB). On the other hand, if G0 ¿Gd

∞, trajectories of type IV apply. Here, goodwillis always decreasing, and the advertising rates always decrease or increase initiallyand then decrease. The decreasing advertising rates and goodwill stock follow fromthe low assessment of the terminal goodwill stock. The intuition is that the incentiveto advertise and build up goodwill gradually decreases toward the horizon date. Onthe other hand, if �d

∞ ¡SM + SR (assessments of the 7nal goodwill stock are “high”),trajectories of the types I or II apply. Here, advertising e'orts always increase.

The appendix also provides the derivation of the steady state values of �d and Gd.They are positive and are given by

�d∞ = ��∗g1=[�(� + �) + K�∗g2]; Gd

∞ = K�∗g1=[�(� + �) + K�∗g2]:

Suppose that channel members become increasingly myopic. Then the discount rate israised such that future pro7ts have less impact in the objective function. The steadystate values of state and costate tend to zero for � → +∞ which shows that increasingmyopism leads to less goodwill accumulation, accompanied by a decreasing shadowprice of the goodwill stock. The intuition is that a myopic decision maker pays littleor no attention to the carry-over e'ects of advertising and accumulates only little, ifany, goodwill.


4. Incentive equilibria

Channel members agree that the desired outcome is the solution of Section 3, butcannot make binding commitments to act accordingly. In that case, each channel mem-ber may reasonably wish to be assured that if she complies with the desired outcome,her partner will also comply. This section shows how it can be accomplished.

The desired outcome in Section 3 is not an equilibrium (cf. JHrgensen and Zaccour,1999). In order to realize the desired outcome over time, we apply the concept ofan incentive equilibrium. The concept was used in resource games by Ehtamo andHOamOalOainen (1993, 1995), but so far not in marketing problems. Its origin is in Stack-elberg games where the leader designs an incentive for the follower such that the latterbehaves in a way which is desirable for the leader. An incentive equilibrium may beseen as a two-sided “Stackelberg incentive” problem; when one player implements herincentive strategy, the other player can do no better than to act in accordance with theagreement.

However, even if the coordinated outcome can be supported as an incentive equi-librium, the problem remains whether a player really should believe that the otherone will implement her strategy: are incentive equilibrium strategies credible? Credi-bility can obtain if the following condition is satis7ed. Suppose that player j cheatson the agreement during some period of time during the play of the game. (Cheat-ing here means that one uses less e'ort than what is prescribed by the coordinatedsolution.) An incentive equilibrium strategy of player i = j is credible if it is betterfor player i to use the equilibrium strategy, rather than to stand pat on the agreement.The intuition is simple. Suppose that player j departs from the coordinated strategy.Then she would not believe that player i will implement her equilibrium strategy ifplayer j knows that player i would be better o' by continuing to play according to theagreeement.

To implement the incentive equilibrium we need to suppose that player i can ob-serve the marketing actions of player j at any time t, and player i makes her actionsdependent upon those of player j. Thus, each player observes the current actions of theother player and employs a strategy based upon that information, in order to inducethe other player to comply with the agreement. 4 To make sure that channel membershave the necessary information, one could assume that they honestly report their ac-tions to the other player. It is evident that the retailer knows the transfer price and notimplausible to suppose that the manufacturer can observe the retail price. Moreover,each 7rm normally has the possibility to observe the other 7rm’s current advertisinge'orts, for instance, in newspapers, TV, and radio. In real-life, these observations mayonly be available after a time lag, and our assumption of instantaneous observabilityshould be seen as a mathematical abstraction. (It is possible to introduce a time lagin the observations of actions. This is done in the continuous-time model of Ehtamo

4 Note the di'erence between an incentive equilibrium and a consistent conjectural variations equilibrium.In the former, a player conditions her action upon the actual action of the other player. In the latter, eachplayer forms a conjecture of the other player’s reaction to her decision. Ultimately, conjectures must coincidewith actual reactions, that is, conjectures must be consistent (rational).


and HOamOalOainen (1993) for the case of a constant time lag. Ehtamo and HOamOalOainen(1995) use a discrete-time model which perhaps is a more natural way to handle laggedobservations.)

The desired decisions are adM and ud

R ≡ (adR ; p

dR) where pd

R ; adj (t) are given by (6)

and (8), respectively. To construct incentive strategies let

UM ={aM | 06aM} and UR = {uR = (aR ; pR) | 06aR ; pM6aR ; pM6pR 6 ��}

be the 7rms’ sets of feasible decisions. Denote by

"M = {#M | #M :UR → UM}; "R = {#R | #R :UM → UR}

the sets of feasible strategies. The incentive strategy #M of the manufacturer, for in-stance, makes her decision a function of the retailer’s decision. Note that #M is afunction of (aR ; pR) whereas #R represents the pair (pR ; aR) = (#1

R(aM); #2R(aM)).

The problem of specifying the sets of admissible incentive strategies is by no meanstrivial. Indeed, it is an instance of a more general problem: how players choose strate-gies in dynamic games. This involves the choice of which observations the strategiesshould be based upon, and which particular functional forms should link the obser-vations to the choice of actions? The standard procedure here is to let these choicesbe exogenous, that is, stated as rules of the game. To depart from this paradigm onewould need to introduce an endogenous choice problem, according to which both theinformation structure and the functional forms of strategies can be selected. This, inturn, requires the de7nition of criteria to guide the players in their choices. A theoryfor this has so far not been developed, and the choices made in a speci7c instance aremainly ad hoc.

Below we shall employ linear incentive strategies, being continuous in the informa-tion, but the reader should see this as an illustration only. The reason for choosinglinear strategies is that they are simple, intuitive, easily computed, and are unique inour setting. Notice that due to their continuity they will not have the threat propertyexhibited by, for instance, trigger strategies. Also note that any pair of strategies thatsatis7es the requirements of De7nition 4.1 below induces an incentive equilibrium.

The players communicate and negotiate before playing the game, and during thisprocess they agree that the coordinated solution in Section 3 is the desired one. Theyalso agree on a particular choice of incentive strategies. An incentive strategy is to beused by each player in an attempt to safeguard her against the possibility that the otherplayer deviates from the agreed course of action (viz., that player’s part of the desiredsolution).

Many other choices of incentive strategies are possible. For example, one can havestrategies that are nonlinear or discontinuous in the information (here: the other player’scurrent decision). Moreover, increasing the information that is available to the playerswould allow for the construction of more elaborate strategies that depend on the currentstate as well as the current decisions, and even on the entire histories of state anddecisions. The particular choice of incentive strategies will, among other things, beconditioned upon the informational assumptions that are appropriate in a given context.


De nition 4.1. Strategy pair #M ∈"M; #R ∈"R is said to be an incentive equilibriumat (ad

M; udR) if

JM(adM; ud

R)¿ JM(aM; #R(aM)) ∀aM ∈UM;

JR(adM; ud

R)¿ JR(#M(uR); uR) ∀uR ∈UR ;(11a)

adM = #M(ud

R); udR = #R(ad

M): (11b)

When incentive strategies are jointly carried out, the desired outcome is realized; thestrategies form a Nash equilibrium that provides the desired outcome. De7nition 4.1states in (11b) that player j designs the incentive strategy #j in such a way that ifplayer i selects the desired action, then player j will respond with the desired action.The inequalities in (11a) state that when player i implements her strategy, the bestchoice of player j is to employ her part of the desired solution. As mentioned, itmay not be evident why player j should believe that player i actually will implementher incentive strategy. Thus, credibility of incentive strategies is not implied by thede7nition. We deal with credibility in Section 4.3.

De7nition 4.1 is a natural extension of the one-sided incentive problem where, typ-ically, the manufacturer is the leader and the retailer the follower. In such a problem,the de7nition only involves the strategy #M, the second inequality in (11a), the 7rstequality in (11b), and the pair (ad

M; udR) is an outcome that is favorable to the leader.

Typically it is the one which maximizes the leader’s individual payo', but it could alsobe the joint maximization outcome if the leader takes the responsibility for coordinatingthe channel. (The topic of channel leadership is dealt with in JHrgensen et al., 2001.)

4.1. The pricing incentive problem

The solution of this static problem is straightforward and has been obtained else-where, for example, by two-part transfer pricing or pro7t sharing, cf. Moorthy (1987).There is only one incentive strategy, used by the manufacturer to induce the retailer toselect the desired retail price pd

R. The strategy is #M(pR) = pdM + $M(pR −pd

R) where$M¿ 0; pd

M is the “desired” transfer price. (Actually, in the cooperative solution thereis no transfer price, but the choice of pd

M will turn out to be immaterial.) For any pdM,

the function

�R(#M(pR); pR) = [pR − #M(pR)](�− �pR)

= [pR − pdM − $M(pR − pd

R)](�− �pR)

has a unique maximizer p∗R = [�(1 − $M) + �(pd

M − $MpdR)]=2�(1 − $M). To give the

retailer an incentive to select p∗R =pd

R, the manufacturer 7xes the incentive coe@cientas $M = 2�(pd

M − c)=(�− �c)¿ 0. For later use we de7ne

12 (pd

M − c)(�− �c) , �dM; (� + �c − 2�pd

M)(�− �c)=4�, �dR ; (12)

where �dM + �d

R = �∗ and �∗ is de7ned in (7).


To see how the pricing incentive strategy can be implemented, rewrite the demandfunction in (2) as pR = (�− q)=�. We suppress the h(G) term since it does not a'ectthe static pricing incentive problem. Inserting pR = (�− q)=� and pd

R from (6) into #M

yields

#M(q) =[pd

M +$M(�− �c)

2�

]− $M

�q

which is a linear quantity discount rule. When the manufacturer uses the rule, theretailer will select pd

R.

4.2. The advertising incentive problem

In the dynamic advertising incentive problem, the outcome of the pricing incentiveproblem enters through the “revenue rate” �d

j in (12). The individual objectives are

XJ j =∫ T

0e−�t{�d

j h(G(t)) − wj[aj(t)]2=2} dt + e−�T SjG(T ); j∈{M;R}:

Admissible advertising incentive strategies are given by

#M(aR)(t) = max{0; adM(t) + %M(t)[aR(t) − ad

R(t)]};#R(aM)(t) = max{0; ad

R(t) + %R(t)[aM(t) − adM(t)]}

(13)

in which we require %M(t) = %R(t)−1 ¿ 0 ∀t ∈ [0; T ]. The incentive mechanism wouldbe peculiar if the incentive coe@cients were negative: then, for instance, if the retailerchooses less e'ort than the desired rate, the manufacturer would respond with an e'ortbeing larger than the desired one!

The intuition of the incentive strategies in (13) is simple. By the 7rst one, themanufacturer tells the retailer that if she chooses her advertising rate in the desiredway, that is, as ad

R, the manufacturer will act accordingly and select adM. Should the

retailer’s advertising e'ort be less [greater] than the desired level, the manufacturer’sadvertising e'ort will also be less [greater] than the desired rate. Later on we shallmodify the incentive strategies in such a way that player i only reacts if player j usesless e'ort than the desired level. Such a modi7cation is reasonable since we wouldexpect no player to use more e'ort than the desired rate.

To construct an incentive equilibrium, one considers a pair of optimal controlproblems:

P0: max XJR with respect to aR ¿ 0; subject to (1) and aM = #M(aR):

P1: max XJM with respect to aM¿ 0; subject to (1) and aR = #R(aM):

Solving these problems is straightforward; their solutions are stated in Lemma 4.1.

Lemma 4.1. The pair (a0R(t); G0(t)) is optimal in P0 since for all t ∈ [0; T ] the max-

imized Hamiltonian is concave in G and there exists a continuous and continuously


di=erentiable costate variable �0(t) such that

G0(t) = �M[ad

M(t) + %M(t)(a0R(t) − ad

R(t))] + �Ra0R(t) − �G0(t); G0(0) = G0;

(14)

�0(t) = (� + �)�0(t) − �d

R[g1 − g2G0(t)]; �0(T ) = SR ; (15)

a0R(t) = �0(t)[�M%M(t) + �R]=wR : (16)

The pair (a1M(t); G1(t)) is optimal in P1 since for all t ∈ [0; T ] the maximized Hamil-

tonian is concave in G and there exists a continuous and continuously di=erentiablecostate �1(t) such that

G1(t) = �Ma1

M(t) + �R[adR(t) + %R(t)(a1

M(t) − adM(t))] − �G1(t); G1(0) = G0;

(17)

�1(t) = (� + �)�1(t) − �d

M[g1 − g2G1(t)]; �1(T ) = SM; (18)

a1M(t) = �1(t)[�M + �R%R(t)]=wM: (19)

Proof. See the appendix.

In Lemma 4.1, the optimal advertising rate of player j depends on the incentiveparameter %i; i = j. This is the way in which player i creates an incentive for playerj. Now we are ready to establish a main result of the paper:

Proposition 4.1. The advertising incentive strategies in (13) provide an incentive equi-librium at (ad

M(t); adR(t)) if the incentive coeAcients %M; %R are determined by

%M(t) = �R�M(t)=�M�R(t); %R(t) = %M(t)−1: (20)

In (20); �j(t) is de,ned as being the solution of

�j(t) = (� + �)�j(t) − �dj [g1 − g2Gd(t)]; �j(T ) = Sj; j∈{M;R}; (21)

where Gd(t) is the solution of (10) and �dj is given by (12). It holds that

�M(t) + �R(t) = �d(t): (22)

Proof. Let (adM(t); ad

R(t); Gd(t); �d(t)) be given by (8)–(10) and let �j(t) be the uniquesolution of (21). Using (9); (21); and �∗ = �d

M + �dR establishes (22). To prove that

�j(t) is strictly positive for all t ∈ [0; T ]; one evaluates the time-derivative in the sameway as was done in Proposition 3.1. The coe@cients %M; %R are well-de7ned for all t;as long as Sj is positive. The incentive strategies in (13) satisfy (11b) by construction.To prove that (11a) holds; choose %M; %R as in (20). By (20) and (22) we have

�R(t)[�M%M(t) + �R] = �R�d(t); �M(t)[�M + �R%R(t)] = �M�d(t): (23)


Using (9); (21); and (23) shows that the triple (adR(t); Gd(t); �R(t)) solves (14)–

(16) and the inequality JR(adM; ad

R)¿ JR(#M(aR); aR) ∀aR ¿ 0 follows. The second in-equality in (11a) is satis7ed. Using (9); (21); and (23) once more shows that thetriple (ad

M(t); Gd(t); �M(t)) is a solution to (17)–(19) and the inequality JM(adM; ad

R)¿JM(aM; #R(aM)) ∀aM¿ 0 follows. The 7rst inequality in (11a) is satis7ed and the proofis complete.

From (20) we see that the incentive coe@cient %j(t) depends on the advertisinge@ciency parameters �j as well as the costates de7ned by (21). If one assumes �d

M =�d

R ; SM = SR, these costate variables are identical, �M(t) = �R(t), and the incentivecoe@cients depend only on the ratio of the e@ciency parameters. Then, if (for instance)�M ¡�R, the slope of the manufacturer’s linear incentive strategy in (13) is greaterthan one. Should the retailer increase [decrease] her advertising e'ort by one unit, themanufacturer increases [decreases] her e'ort by more than one unit. The explanationclearly lies in the higher e@ciency of the retailer’s local advertising in raising thegoodwill stock. Reactions are symmetric, one-to-one, if the e@ciency parameters areequal.

The idea of the advertising incentive is the same as that of the pricing incentive.There is a di'erence, however, in the optimization problems that must be solved. In thepricing incentive problem, we solve a one-sided, static optimization problem to obtainan optimal pricing decision of the retailer. The manufacturer’s incentive strategy makesthe desired retail price the unique solution to this problem. In the advertising problemwe have two optimal control problems P0 and P1. The retailer optimizes XJR, given aM=#M(aR), whereas the manufacturer optimizes XJM, given aR = #R(aM). The solutions ofthese problems are (a0

R(t); G0(t)) and (a1M(t); G1(t)), respectively. To induce the retailer

to select adR(t), and the manufacturer to select ad

M(t), the incentive coe@cients aredetermined as the time-functions in (20). The implication is that (ad

R(t); Gd(t); �R(t))solves P0 and (ad

M(t); Gd(t); �M(t)) solves P1.The formulation of the admissible strategies in (13) reRects that there is no a priori

guarantee that the natural constraints #j(ai)(t)¿ 0 (i; j∈{M;R}; i = j) will be satis-7ed. In a symmetric game we can establish such a guarantee. Inserting from (8) and(20) into, for instance, the inequality #M(aR)(t)¿ 0 to obtain

�d[�M

wM− �2

R�M

�M�RwR

]+ aR

�R�M

�M�R¿ 0 (24)

in which �d is given by (9), and �M and �R are given by (21). A su@cient conditionfor (24) to be satis7ed is that the bracketed term is nonnegative. Clearly, the bracketedterm is zero if �M = �R, wM =wR, and �M = �R. Now, (21) shows that �M and �R areidentical if SM = SR and �d

M = �dR. To have �d

M = �dR it su@ces to choose the desired

transfer price pdM accordingly.

Notice that the advertising incentive strategies in (13) are symmetric in the informa-tion; they call for a player’s reaction both if the other player uses less e'ort, and if sheuses more e'ort than desired. Since we would not expect a player to use more e'ortthan the desired rate, it may be more adequate to consider asymmetric strategies that


call for a reaction only if the other player uses less than the desirable e'ort (Ehtamoand HOamOalOainen, 1993, 1995). Hence, consider the strategies

#M(aR)(t) = max{0; adM(t) + H (ad

R(t) − aR(t))%M(t)[aR(t) − adR(t)]};

#R(aM)(t) = max{0; adR(T ) + H (ad

M(t) − aM(t))%R(t)[aM(t) − adM(t)]};

(25)

where H (s)=1 for s¿ 0, H (s)=0 for s6 0 is the unit step function. The speci7cation(25) implies that the manufacturer, for instance, reacts on the retailer’s action if theretailer’s e'ort is less than its desirable level. If it is above, the manufacturer does notreact. Clearly it holds that #j(ai)(t)6 ad

j (t) for i; j∈{M;R}; i = j.We shall show that the asymmetric strategy pair #M(aR); #R(aM) also provides an

incentive equilibrium at (adM; ad

R). First note that the (symmetric) strategies in (13) arerelated to those in (25) in the following way:

#M(aR)(t)6 #M(aR)(t); #R(aM)(t)6 #R(aM)(t): (26)

Thus, for any admissible advertising e'ort rate, a strategy in (25) provides an e'ortrate that does not exceed the rate generated by the strategy (13). The reason clearlyis that the strategies in (25) only call for a reaction if the other player uses less e'ortthan the desired level.

Second, it holds that

#M(a0R)(t) = #M(a0

R)(t); #R(a1M)(t) = #R(a1

M)(t) (27)

in which a0j (t) are given by (16) and (18). The result in (27) states that the symmetric

strategies in (13) coincide with the asymmetric ones in (25) when the retailer [themanufacturer] chooses her optimal e'ort rate in problem P0[P1], respectively. Theproof of (27) is simple: When the incentive coe@cients %j(t) are chosen as in (20),we know that (ad

R(t); Gd(t); �R(t)) solves P0 and (adM(t); Gd(t); �M(t)) solves P1; then

it holds that a0R(t) = ad

R(t) and a1M(t) = ad

M(t). Using (13) and (25) then producesthe result. Put in another way, (27) states that the desired e'ort rates are realized inequilibrium, irrespective of which kind (symmetric or asymmetric) of incentive strategyis used.

For the proof we 7nally need the following result.

Lemma 4.2. When (26) and (27) are satis,ed it holds that

XJM(a1M; #R(a1

M))¿ XJM(aM; #R(aM)) ∀aM¿ 0

XJR(#M(a0R); a0

R)¿ XJR(#M(aR); aR) ∀aR ¿ 0:(28)

Proof. The result of the lemma follows when one makes a slight modi7cation of theproof of Lemma 3:2 in Ehtamo and HOamOalOainen (1993).

Using the inequalities in (28) and recalling Proposition 4.1 then yields

XJM(adM; ad

R)¿ XJM(aM; #R(aM)) ∀aM¿ 0;

XJR(adM; ad

R)¿ XJR(#M(aR); aR) ∀aR ¿ 0(29)


and by De7nition 4.1 we see that strategies #M(aR); #R(aM) provide an incentive equi-librium at (ad

M; adR).

Employing asymmetric strategies seems more natural because player i should notreact if player j uses more e'ort than the desired rate. When player j uses more e'ort,goodwill increases and hence sales revenues will bene7t both players. Player j incursan extra advertising cost, but player i escapes an extra cost by standing pat on theagreement. On the other hand, if player j uses less e'ort than the desired rate, playeri should react since now player j is free-riding. The question remains if the cheatingplayer j has reason to believe that player i will react, i.e., are incentive equilibriumstrategies credible?

4.3. Credibility of incentive strategies

For the pricing incentive, it su@ces to look at the revenue term in the manufacturer’sobjective integral, cf. (3). Indeed, only the margin pM −c matters. Credibility amountsto verifying

#M = pdM + $M(pR − pd

R)¿pdM: (30)

Using the equilibrium strategy, the manufacturer gains more than if she stood pat onthe desired transfer price. The inequality (30) holds if pR ¿pd

R. The satisfaction ofthe inequality pR ¿pd

R is plausible since, in the absence of cooperation, the retailerwould not price below pd

R. It is well known that in a noncooperative game, the retailprice exceeds pd

R; see, e.g., JHrgensen and Zaccour (1999).For the advertising incentive problem, suppose that the retailer chooses her e'ort

according to

aR(t) = adR(t) − ((t) for t ∈ [0; )]; aR(t) = ad

R(t) for t ∈ (); T ] (31)

in which, ((t) is selected such that aR(t) is feasible. (All the subsequent argumentscan be repeated for the case where the manufacturer deviates.) Thus, the retailer cheatson the agreement by having a lower e'ort than the desired rate, during some initialperiod of time. For the manufacturer’s incentive strategy to be credible it must holdthat

XJM(#M(aR); aR)¿ XJM(adM; aR): (32)

The inequality (32) means that if the retailer cheats by choosing e'orts accord-ing to (31), it is better for the manufacturer to use her equilibrium strategy thanto stick to the desired rate. This should give credibility to the incentive strategy.It is important to note that it does not make the equilibrium strategy credible forany possible deviation; credibility is assured only against deviations of the type givenby (31).

Ehtamo and HOamOalOainen (1993, 1995) studied the credibility problem in a dynamicresource game. The 1993 paper deals with a continuous-time model where numericalsimulations are used to address the credibility problem. The model is symmetric withthe exception of the cost parameters. The 1995 paper proposes a discrete-time model


which admits an analytical approach to the credibility problem although the resultis only a local one. Ehtamo and HOamOalOainen (1993) rightly noted that a credibilityproblem as the one at hand is di@cult to study analytically. Here we try an analyticalapproach but no closed-form results can be derived in the general case. However, in asimpli7ed model some insights can be obtained.

Assume that players are symmetric, there are no salvage values, function h in (2)is linear, i.e., h(G) = g1G, and the deviation ( in (31) is constant. Thus we haveintroduced

Assumption 5.1. (i) �dM =�d

R , �d ; (ii) SM =SR =0; (iii) �M =�R , �; (iv) wM =wR ,w; (v) g2 = 0; (vi) ((t) = ( = const.

The 7rst assumption can be ful7lled by an appropriate choice of the (arti7cial) desiredtransfer price. The second means that the players put no value on the goodwill stockat the horizon date. This is the case if players care relatively little about the future, orbecause they are very far-sighted and have an in7nite planning horizon. (The assump-tion implies a minor technicality in the incentive coe@cients %j in (20) which are nolonger de7ned at the horizon date T . The problem can readily be overcome by applyingl’Hospital’s rule.) The third and fourth assumption imply that the players’ advertisinge'orts are equally e@cient in raising the goodwill stock, and the cost functions areidentical. Assumption (v) means that there are constant returns to goodwill, i.e., anadditional unit of goodwill generates the same increment of sales irrespective of thecurrent level of goodwill.

From (32) we obtain under Assumption 5.1 the following condition for credibilityof the manufacturer’s equilibrium strategy:

(e−�) − 1)[w\2�

+� dg1�

�(� + �)

]+ (e�) − 1)

� dg1��(� + �)

e−(�+�)T

+w∫ )

0e−�tad(t) dt¿ 0 (33)

which holds with equality for ( = 0 and=or ) = 0, as it should. From (33) we canderive the following observations:

(i) Using (8) one can replace ad by ��d(t)=w which shows that it is easier to satisfy(33) if the shadow price �d(t) is large for all t ∈ [0; )]. Thus, if the goodwill stockhas a large imputed value during the cheating period, it is likely that the retailer willbelieve that the manufacturer will employ her equilibrium strategy. Indeed, in the phasediagram of Fig. 1 we are in the situation where �d

∞ ¿SM + SR = 0; that is, trajectoriesof type III or IV apply. The 7gure shows that the advertising e'orts are largest initially.(The ultimate decrease toward zero is driven by the assumption of zero salvage values.)When the imputed value of the stock is high, it pays to add substantially to the stock,i.e., both 7rms should use large amounts of advertising e'ort. Should the retailer, inthis situation, lower her e'ort, it is better for the manufacturer to follow, rather than toallow the retailer to free-ride on the manufacturer’s high cooperative advertising e'ortrate.


(ii) De7ne the left-hand side of (33) as the function

f(); () = (e−�) − 1)w\2�

+ (e−�) − 1)� dg1�

�(� + �)

+ (e�) − 1)� dg1��(� + �)

e−(�+�)T + w∫ )

0e−�tad(t) dt:

It holds for any )¿ 0 that

f(); ()¿ 0 i'

(62�

w(1 − e−�))

[(e−�) − 1)

� dg1��(� + �)

+ (e�) − 1)� dg1��(� + �)

e−(�+�)T + w∫ )

0e−�tad(t) dt

], g()): (34)

Clearly, if g()) in (34) is negative, f cannot be positive for any feasible (. If g()) isnonnegative it holds that

f(); ()¿ 0 for (∈ [0; g())); f(); ()6 0 for (∈ [g()));∞):

Hence, credibility can be obtained for su@ciently small values of the deviation (.This result agrees with the 7ndings of Ehtamo and HOamOalOainen (1993, 1995); see alsoOsborne (1976). The implication is that a player should retaliate if confronted withdeviations that are not too large. It is not clear what would be the best strategy if thedeviation is large. Here it is not necessarily true that a player who has been cheatedwill lose more by standing pat on the agreement.

For g()) to be nonnegative it must hold that

(e�) − 1)� dg1��(� + �)

e−(�+�)T + w∫ )

0e−�tad(t) dt¿ (1 − e−�))

� dg1��(� + �)

: (35)

Both sides of (35) are increasing in ) but the right-hand side is upper bounded. Hencethere exists a 7nite ) for which (35) is satis7ed. The conclusion is that credibility canbe satis7ed for su@ciently large values of ), that is, the cheating period should besu@ciently long. The intuition here is that the retailer believes that the manufacturerwould use her equilibrium strategy if the retailer cheats for a longer period of time.Should she deviate on a shorter interval, it is not worthwhile for the manufacturer toreact; instead she can stand pat on the cooperative e'ort rate.

(iii) Cheating on a high cooperative advertising rate may give a sizeable cost saving,even if the deviation is small, due to the convexity of the cost function. For this reason,a player may wish to deviate from a high cooperative advertising level, but gives alsothe other player a reason to match the deviation. On the other hand, if advertisinge'orts and the deviation are small, it does not make much di'erence in costs if aplayer deviates (neither for the one who cheats nor for the other player).


5. Concluding comments

Quite many marketing science studies in pricing and advertising have been concernedwith the design of mechanisms that allow channel members to achieve a coordinatedoutcome in a static setting. Little e'ort has been devoted to design intertemporal in-centive mechanisms and addressing the problems of sustainability of cooperation. Thepaper has proposed incentive strategies which can support a coordinated outcome. Thecontributions of the paper are the following:

(i) Most marketing studies have used a single coordinating instrument, typicallytransfer pricing. We extended the literature by studying a setup in which price andadvertising are used simultaneously to create incentives for channel members to stickto the desired outcome.

(ii) We extended the static incentive approach to a dynamic setting in which adver-tising plays an intertemporal incentive-creating role. The use of dynamic incentives is,to the best of our knowledge, new in marketing science literature.

(iii) We departed from the often-used Stackelberg approach in which one of thechannel members is an exogeneously designated channel leader who designs an in-centive for the other channel member. Rather, we followed the “symmetric” approachsuggested by Jeuland and Shugan (1983), assuming that there is no natural channelleader. To achieve coordination in such a context, both channel members must applyappropriate incentive strategies. We constructed incentive equilibria for symmetric aswell as asymmetric strategies and discussed the problem of credibility.

A less desirable feature of our model is the independence between optimal advertisingand pricing decisions, which makes the pricing part a replication of results known fromstatic studies. Independence between pricing and advertising may be less satisfactory,but comes as a cost of having a tractable model. (Similar cases of disconnected pricingand advertising are the games considered in, for instance, Fertshtman et al., 1990;JHrgensen and Zaccour, 1999.) The problem here is that models which would providetruly dynamic pricing and advertising strategies are notoriously ill-behaved and admitonly few analytical insights. To study such models one has to resort to numericalsimulations.

We have used incentive strategies where, at each instant of time, the decision of oneplayer is dependent upon the other player’s current decision. The assumption is thateach player’s decisions are observable with no time lag at all, a plausible approximationin cases where decisions are observed with time lags that are insigni7cant compared tothe length of the planning period. As mentioned, it is possible to introduce incentivestrategies that are based upon lagged observations of the other player’s action.

The assumption of a 7nite planning horizon can be questioned. In real life, playersmay be unwilling or unable to 7x a planning horizon and it could be more naturalto have an in7nite time horizon. When playing an in7nite horizon game one can usestationary Markovian strategies (strategies that depend only on the current state G).This causes no di@culties in the determination of the desired solution. Here one canderive, using a Hamilton–Jacobi–Bellman equations approach, value functions that arelinear in G. In the incentive equilibrium, however, the application of state-dependentstrategies may be more complicated. Ehtamo and HOamOalOainen (1993, p. 674) wrote


that “the conditions for such strategies will be much more stringent than those for thedecision-dependent equilibrium strategies”. Nevertheless, the study of state-dependentincentive strategies may be a promising area for future research.

Acknowledgements

We wish to thank an anonymous reviewer for helpful comments. Any remainingerrors are the responsibility of the authors. Research supported by SSF, Denmark, andNSERC-Canada.

Appendix A.

A.1. Proof of Proposition 3.1

The current-value Hamiltonian is

H (G; aM; aR ; �d) = �∗[g1G − g2

2G2

]−

∑j∈{M;R}

wj

2a2j + �d

∑

j∈{M;R}�jaj − �G

:

Di'erentiation with respect to aj yields the optimal advertising rates adj (t) =�j�d(t)=wj

which always are strictly positive since �d(t)¿ 0 ∀t ∈ [0; T ]. The latter follows from

�d(t)|�d(t)=0 = −�∗[g1 − g2Gd(t)]¡ 0

and by using Assumption 2.1 and the transversality condition �(T )=SM +SR. Insertingthe optimal advertising rates into (1) and using the costate equation �d =��d − @H=@Gyields the linear, time-invariant and inhomogeneous di'erential equations stated in (9)and (10). The explicit solution of the system can be found by standard methods andis given by

�d(t) = �d∞ +

(s1 + �)K

Cd1 es1t +

(s2 + �)K

Cd2 es2t ;

Gd(t) = Gd∞ + Cd

1 es1t + Cd2 es2t (A.1)

in which

�d∞ =

��∗g1

�(� + �) + K�∗g2¿ 0; Gd

∞ =K�∗g1

�(� + �) + K�∗g2¿ 0

are the steady state values of costate and state. In (A.1), s1; s2 are characteristic rootsgiven by

s1 =�2

+12

[�2 + 4(K�∗g2 + �(� + �))]1=2 ¿ 0;

s2 =�2− 1

2[�2 + 4(K�∗g2 + �(� + �))]1=2 ¡ 0

and Cd1 ; C

d2 are constants of integration that can be found from the boundary conditions

in (9) and (10). Since the roots s1 and s2 are real and have opposite signs, the steady


state is a saddle point. The stable path to the steady state is obtained by setting Cd1 =0.

Noticing that the controls do not depend on the state, and using the fact that �d(t)¿ 0for all t ∈ [0; T ], shows that the maximized Hamiltonian is concave in G for all t ∈ [0; T ]and the proof is complete.

A.2. Proof of Lemma 4.1

In the 7rst part of the lemma, the current-value Hamiltonian is

H (G; aR ; �0) = �dR

[g1G − g2

2G2

]− wR

2a2

R

+ �0[�M(adM + %M(aR − ad

R)) + �RaR − �G]:

The costate equation in (18) follows by di'erentiation of H with respect to G. Thecostate is strictly positive for all t ∈ [0; T ] which can be seen by using an argumentsimilar to the one in the proof of Proposition 3.1. The optimal advertising rate a0

R(t) in(19) follows by di'erentiation of H and is strictly positive for all t ∈ [0; T ]. Since theoptimal advertising rate does not depend on G, and the costate always is nonnegative,the maximized Hamiltonian is concave in G for all t ∈ [0; T ]. The proof of the secondpart of the lemma follows by interchanging the subscripts R and M.

A.3. Credibility

The inequality in (32) is equivalent to∫ )

0e−�t{�d

Mh(Gls1 (t)) − wM[#M(aR)(t)]2=2} dt

+∫ T

)e−�t{�d

Mh(Gls2 (t)) − wM[(ad

M)(t)]2=2} dt + e−�T SMGls2 (T )

¿∫ )

0e−�t{�d

Mh(Grs1 (t)) − wM[ad

M(t)]2=2} dt

+∫ T

)e−�t{�d

Mh(Grs2 (t)) − wM[ad

M(t)]2=2} dt + e−�T SMGrs2 (T ) (A.2)

where function h is given in (2) and Gls1 (t); Grs

1 (t) are the unique solutions for t ∈ [0; )]of the di'erential equations

Gls1 (t) = −�Gls

1 (t) + �M#M(aR)(t) + �R aR(t); Gls1 (0) = G0;

Grs1 (t) = −�Grs

1 (t) + �MadM(t) + �R aR(t); Grs

1 (0) = G0

and Gls2 (t); Grs

2 (t) are the unique solutions for t ∈ (); T ] of

Gls2 (t) = −�Gls

2 (t) + �MadM(t) + �Rad

R(t); Gls2 ()) = Gls

1 ());

Grs2 (t) = −�Grs

2 (t) + �MadM(t) + �Rad

R(t); Grs2 ()) = Grs

1 ()):


The implications of Assumption 5.1 are the following: (i) and (ii) make the costates�j de7ned by (21) identical and using (iii) then shows that the incentive coe@cients %jin (20) equal one (as expected). Moreover, using (iii) and (iv) shows that the desiredadvertising rates ad

j in (8) are identical and we put adM = ad

R , ad. The e'ect of allthis is that (A.2) simpli7es to∫ )

0e−�t{� dg1[Gls

1 (t) − Grs1 (t)] − w(

2[(− 2ad(t)]} dt

+∫ T

)e−�t� dg1[Gls

2 (t) − Grs2 (t)] dt¿ 0 (A.3)

in which the state trajectories are given by

Gls1 (t) = G0e−�t + �

∫ t

0e−�(t−z)[2ad(z) − 2(] dz;

Grs1 (t) = G0e−�t + �

∫ t

0e−�(t−z)[2ad(z) − (] dz

for t ∈ [0; )], and by

Gls2 (t) = Gls

1 ())e−�(t−)) + 2�∫ T

)e−�(t−z)ad(z) dz;

Grs2 (t) = Grs

1 ())e−�(t−)) + 2�∫ T

)e−�(t−z)ad(z) dz

for t ∈ (); T ]. Inserting

Gls1 (t) − Grs

1 (t) = −�∫ t

0e−�(t−z)( dz;

Gls2 (t) − Grs

2 (t) = −�e−�(t−))∫ )

0e−�()−z)( dz

into (A.3) and integrating yields (33).

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Documents

Channel coordination over time: incentive equilibria and credibility