A Model of Optimal Advertising Expenditures in a Dynamic Duopoly.

PETR MARIEL [*]

This paper develops a dynamic model of oligopolistic advertising competition. The model is general enough to include predatory advertising and informative advertising as particular cases. The analysis is conducted in a differential game framework and compares the open-loop

and feedback equilibria to the efficient outcome. It is found that for the informative advertising competition game, advertising levels are closer to the collusive outcomes in a feedback equilibrium. In the case of predatory advertising, expenditures are inefficiently high in a feedback equilibrium and the open-loop solution is more efficient. (JEL L13, M37)

Introduction

In this paper, a dynamic model of oligopolistic advertising competition is developed. Compared to previous literature, an important feature of this work is that the competitive and informative contents of advertising are explicitly considered, allowing advertising to have market size and business-stealing effects. When advertising effort is directed toward consumers who have not bought the product before, it has the effect of increasing total market size (providing information about the very existence of the product, its price, and general characteristics). However, when advertising is directed at a rival's clientele, it has a business-stealing effect. It increases the advertiser's market share only at the expense of competitors. In fact, the results obtained depend on the relative strength of these two factors. In their empirical analysis of the cigarette industry, Roberts and Samuelson [1988] find that advertising has a strong cooperative content. [1] In other industries, however, the business-stealing eff ect could be the relevant aspect. Slade [1995] finds that for saltine crackers, advertising is mildly predatory. For the soft drink industry, Gasmi et al. [1992] estimate a low advertising effect on market size and a strong predatory effect on market shares.

To study the problem of optimal advertising expenditure in a dynamic duopoly, a differential game framework is used and the open-loop and feedback equilibria are analyzed, the results being compared to the efficient outcome. Previous works have compared the open-loop and feedback outcomes. Fershtman and Kamien [1987] analyze a differential game of price competition assuming that price does not adjust instantaneously to its market clearing level. As the speed of price adjustment becomes instantaneous, with feedback strategies, the outcome is more competitive than in the openloop equilibrium, which approaches the Coumot equilibrium price. [2] In an advertising competition game, Sorger [1989] also finds that using feedback strategies instead of open-loop controls does not necessarily increase payoffs (see also Reynolds [1987]).

Even though the outcomes of feedback equilibria are usually less efficient than those of open-loop equilibria, the opposite result is obtained in this paper for an informative advertising competition game: advertising levels are closer to collusive outcomes in a feedback equilibrium. The intuition behind this result is that with feedback strategies, firms may take into account the stock of goodwill when deciding on their current level of advertising. Since current profits depend on past advertising by both firms, a strategy that positively depends on those stocks works as a promise to increase advertising expenses in the future, provided the opposing firm increases advertising in the current period. These strategies sustain higher advertising than the open-loop equilibrium, although the outcome does not always coincide with the joint profit maximization solution. In this present model, discontinuous strategies are ruled out so that trigger strategies, which could easily sustain collusion in a repeated game, are excluded.

For markets where advertising is mainly predatory, the results are reversed. In any Nash equilibrium, expenditures are inefficiently high and more so in a feedback strategy equilibrium. In this case, the flexibility provided by a feedback strategy is harmful. Firms would rather commit themselves than not to react to the state. When firms use a feedback strategy, there is a preemption effect that pushes them to overinvest in advertising for strategic purposes.

This paper differs from previous work in the Vidale and Wolfe [1957] tradition in that different potential effects of advertising are allowed for, that is, effect on the total market size (informative advertising) and effect on market shares (predatory advertising). Following the definition by Friedman [1983] and Slade [19951, advertising is predatory if an increase in advertising by one firm reduces its rival's discounted sales in the dynamic game. If advertising is informative (cooperative), the opposite is true.

This approach allows different advertising paths and steady state levels to be predicted, depending on the informative and competitive content of advertising. However, the model is simple enough to obtain closed-form solutions. Another advantage of this approach is that it provides a foundation for the evolution of sales as a function of the advertising levels, which enables the parameters of the model to be interpreted and empirical predictions to be obtained.

This paper is organized as follows. The second section presents a dynamic model for the determination of optimal advertising expenditure. The third section assumes that firms have the ability to commit to advertising paths in advance, then calculates the open-loop equilibrium and compares advertising levels and profits to the static solution and to the cooperative outcome. In the fourth section, there is no precommitment, and equilibria in feedback strategies are presented for three specific cases: informative advertising, predatory advertising, and an intermediate case. Results are then compared with the open-loop and efficient solutions. The fifth section offers concluding comments.

A Dynamic Model of Advertising Competition

Consider a duopoly market. The two firms are denoted by i and j and advertising levels, by [u.sub.i] and [u.sub.j], respectively. First, market competition at a given period is described, then the dynamics of the model are analyzed.

The Static Game

Any advertising message contains general information about the product as well as brand information. The impact of a message on a given consumer depends on whether the consumer is previously uninformed about the product and whether he is a customer of the advertiser's brand or a customer of the rival's brand. The content of the message cannot be specialized according to the recipient. At any period, the market can be divided into three segments: [3]

1) Segment 1: Consumers who prefer brand i, due either to their preferences or to their having consumed it in the past (there are switching costs of changing brands) are referred to as firm i's clientele.

2) Segment 2: Consumers who prefer brand] are referred to as firm j's clientele.

3) Segment 3: Consumers who are uninformed about brand differences or have no preference for either brand are neither firm i's nor firm j's clientele.

For the sake of simplicity, assume there is no price competition, that price, p, is the same for both firms. This is a simplifying assumption based on the symmetry of the model. Note that advertising could allow a firm to set a different price from the rival and not lose its clients. However, assuming both firms set the same price is consistent with a symmetric equilibrium. The price-cost margin, p - [c.sub.i] [greater than] 0, will be denoted as [q.sub.i].

The decision variable for firm i is the level of advertising, [u.sub.i]. This variable could be, for example, a measure of the number and duration of advertising messages m the media. For mathematical convenience, the cost of advertising is assumed to be a convex function of the advertising level, c([u.sub.i]) = [[u.sup.2].sub.i] .The assumption of a convex advertising cost function is used by, among others, Gould [1970], Sorger [1989], and Roberts and Samuelson [1988]. The justification for the convexity of the cost function is based on the increasing costs of converting current advertising into advertising goodwill. The argument is that the variation in advertising capital is a concave function of advertising messages. This is due to the fact that the marginal effectiveness of advertisements decreases as the quantity of messages increases, because either they reach consumers who already have information about the product or they are placed in less efficient media (see also Comanor and Wilson [1974]).

In the absence of any advertising, the total demand for the product is a decreasing function of the price f(p), f'(p) [less than equal to] 0. Denote as [s.sub.i] [epsilon] [0, 1] the proportion of the total demand, f(p),in Segment 1, and 1-[s.sub.i] - [s.sub.j], the proportion in Segment 3. In the absence of advertising, firm i's demand is the sum of sales in Segment 3 and sales in Segment 1: [X.sub.oi] = (1 - [s.sub.i] - [s.sub.i]) [[eta].sub.i]f(p) + [s.sub.i]f(p), where it is assumed that a proportion, [[eta].sub.i], [epsilon] [0, 1], of consumers in Segment 3 buys from firm i. Advertising will be modeled as having an additive effect on sales.

For firm i, the effect on its own sales of its advertising, [u.sub.i], and the rival's, [u.sub.j], on Segment 3 is (1 - [s.sub.i] - [s.sub.j]) [[eta].sub.i] f(p)([u.sub.i] + [u.sub.j]), that is, merely informative. It is assumed here that advertising messages by firm i bring useful information to consumers about the existence of the product, price, location, quality, and product characteristics so that total demand for the good and, in particular, demand for firm j increases.

The effect of advertising on the firm's own clientele is [s.sub.i]f(p)([u.sub.i] - [[theta].sub.i] [u.sub.j]), where E [[theta].sub.i] [epsilon] (0, 1] is a measure of the switching costs from brand i to brand j. The higher [[theta].sub.i] is, the lower the switching costs will be. Finally, the effect on firm i's sales of advertising to firm j's clientele is [s.sub.j]f(p)([[theta].sub.j][u.sub.i] - [u.sub.j]). Note that the switching costs here work against firm i.

Taking into account the effect of advertising on the three market segments, demand for firm i is (i, j = 1, 2; i [neq] j):

[X.sub.i] = [X.sub.oi] + (1 - [s.sub.i] - [s.sub.j])[[eta].sub.i]f(p)([u.sub.i] + [u.sub.j])

+ [s.sub.i]f(p)([u.sub.i] - [[theta].sub.i][u.sub.j]) + [s.sub.j]f(p)([[theta].sub.j][u.sub.i] - [u.sub.j]). (1)

The last three terms represent the additive effect of advertising on Segments 3, 1, and 2, respectively. Expression (1) takes a particularly simple form in the case of [[theta].sub.i] = [[theta].sub.j] = 1:

[X.sub.i] = [X.sub.oi] + (1 - [s.sub.i] - [s.sub.j])[[eta].sub.i]f(p)([u.sub.i] + [u.sub.j]) + ([s.sub.i] + [s.sub.j])f(p)([u.sub.i] - [u.sub.j])

i,j = 1,2; i [neq] j. (2)

Defining [z.sub.i] = (1 - [s.sub.i] - [s.sub.j])[[eta].sub.i]f(p) and [w.sub.i] = ([s.sub.i] + [s.sub.j])f(p), (2) can be expressed as:

[X.sub.i] = [X.sub.oi] + [z.sub.i]([u.sub.i] + [u.sub.j]) + [w.sub.i]([u.sub.i] - [u.sub.j])

i,j = 1,2; j [neq] j. (3)

In this case, advertising to the rival's clientele has a strong competitive effect: only the excess advertising ([u.sub.i] - [u.sub.j]) matters. However, advertising in the uninformed segment of the market has the effect of increasing market size. This increase is distributed between the firms in proportion to [[eta].sub.i], a parameter that could measure brand quality, tastes of uninformed consumers, and the like. This formulation is used for its simplicity and because it contains the market size and market share effects of advertising, which are the focus of this analysis. [4]

According to (3), when most consumers are already informed about the product because they have consumed it in the past (where [s.sub.i] + [s.sub.j] is large), advertising mainly has a business-stealing effect, increasing market share at the expense of the competitor (where [z.sub.i] is small and [w.sub.i] is large). This situation would correspond to the later stages of the product life cycle. For mature industries, advertising has a strong business-stealing effect. However, when consumers are not informed about the product (at the earlier stages of the product life cycle), [s.sub.i] + [s.sub.j] is low, and advertising has the effect of increasing the total demand for the product (where [z.sub.i] is large and [w.sub.i] is small). Thus, [s.sub.i] + [s.sub.j] is a measure of the degree of maturity of the market and also of the strength of the market share effect relative to the market size effect.

If firm i maximizes instantaneous profits, [q.sub.i] [X.sub.i] - [([u.sub.i]).sup.2], the first order conditions for a Nash equilibrium can be used to obtain the optimal advertising levels:

[[u.sup.*].sub.i] = [z.sub.i] + [w.sub.i]/2 [q.sub.i] i = 1,2. (4)

Refer to [[u.sup.*].sub.i] as the solution of the static game. Note that if firms could cooperate and maximize joint profits, the optimal static advertising levels would be:

[[u.sup.c].sub.i] = ([z.sub.i] + [w.sub.i])[q.sub.i] + ([z.sub.j]- [w.sub.j]) [q.sub.j]/2

i,j = 1, 2; i [neq] j. (5)

Comparing this solution to [[u.sup.*].sub.i], it can be seen that when the informative content of advertising is greater than the competitive content ([z.sub.j] [greater than] [w.sub.j]), firms would like to agree on higher advertising levels. Whereas, when the business-stealing effect is dominant ([z.sub.j] [less than] [w.sub.j]) firms would increase their profits if they could commit themselves to lower advertising efforts. In the static game, Nash equilibrium advertising is inefficiently low for informative advertising and inefficiently high for predatory advertising. In the former case, firms do not internalize all the positive external effects that advertising expenditures have on market sales (a problem of nonappropriability of total surplus). In the latter case, the external effects on rivals are negative. One of the reasons for analyzing dynamic advertising paths is to check whether firms are able to sustain higher profits per period than in the static solution, in other words, if firms get closer to the eff icient solution over time.

The Dynamics

A dynamic model captures strategic behavior that arises when firms engage in repeated market interaction. Furthermore, only in a dynamic framework can advertising expenses be modeled as investments that increase not only current sales, but also future sales. These potentially long-lived effects of advertising are compensated for in this model by a decay factor (-[a.sub.i], [X.sub.i]) to take into account that some of firm i's clients may leave the population or forget the information about the product they may have had in the past. Parameter [a.sub.i] may also contain a word-of-mouth effect: consumers buying the product provide information about the brand to other consumers. It will be assumed that depreciation and word-of-mouth effects are proportional to sales. Parameter [a.sub.i] might have a positive or negative sign, depending on which of the two effects dominates. However, it is usually assumed to be positive, that is, the depreciation effect dominates (it is probably only in the early stages of the li fe cycle of a product that [a.sub.i] has a negative sign).

It is assumed that the effect of advertising on sales lasts beyond the current period and only disappears through depreciation. For the sake of simplicity, it will also be assumed that all the other factors affecting the growth of the market are collected in a constant term, [K.sub.i]. With this specification (given (3)), the following differential equations give the evolution of sales as a function of advertising levels:

[X.sub.i](t) = [w.sub.i]([u.sub.i](t) - [u.sub.j](t)) + [z.sub.i]([u.sub.i](t) + [u.sub.j] (t)) + [K.sub.i] - [a.sub.i][X.sub.i](t)

[X.sub.i](0) = [X.sub.i0] [greater than] 0 i, j = 1, 2; i [neq] j, (6)

where (t) denotes time, [a.sub.i][X.sub.i](t) with [a.sub.i] [epsilon] (0, 1) is the decay term, and parameters [a.sub.i], [K.sub.i], [w.sub.i], and [z.sub.i] are assumed to be nonnegative and constant over time.

The system dynamics in (6) express two different effects of advertising. The first term on the right-hand side of (6) is the business-stealing effect. Advertising activities by firm j have a negative impact on firm i sales volume ([w.sub.i] [greater than] 0). This term represents purely competitive advertising. The second term on the right-hand side is the total demand effect. Advertising by firm i provides general information about the product (its main features are shared by all brands), and it may increase total market size ([z.sub.i] [greater than] 0), that is, it may also increase sales volume for firm j. [5] The condition [K.sub.i] [greater than] 0 assures the nonnegativity of sales in the limit. In the absence of advertising ([u.sub.i] = [u.sub.j] = 0), sales adjust exponentially ([X.sub.i](t) = ([X.sub.i0]- [K.sub.i]/[a.sub.i])[e.sup.-[a.sub.i]t] + [K.sub.i]/[a.sub.i]) to a nonnegative constant, [K.sub.i]/[a.sub.i], which can be considered the limit size of the market in that situation.

The objective of each firm i is to maximize the discounted sum of its instantaneous profit over an infinite time horizon:

[J.sup.i] = [[[integral].sup.[infinity]].sub.[t.sub.0]] [e.sup.-rt][q.sub.i][X.sub.i](t) - [[u.sup.2].sub.i](t)]dt, (7)

where [q.sub.i], the price-cost margin, is assumed to be constant over time and r is the discount rate, common to both firms.

Firm i faces the problem of maximizing (7) subject to (6), [u.sub.i](t) [greater than or equal to] 0 and [X.sub.i](t) [greater than or equal to] 0, with the initial state ([t.sub.0], X(0)) = [t.sub.0], ([X.sub.10], [X.sub.20])). The appropriate framework for analyzing this problem is a differential game. Two major strategy spaces are usually examined in literature and will be used in this paper: open-loop and feedback. [6] Denote as [[gamma].sub.OL] ([[gamma].sup.F]) the differential game, defined by the set of players 1 and 2, open-loop (feedback) strategy space, payoff functions given by (7), and the system dynamics in (6).

Open-Loop Nash Equilibria

The optimal open-loop strategy for firm i is the best response to the strategy chosen by the opposing firm. In this case, both firms commit themselves to maintaining the strategies ([[u.sup.OL].sub.i], [[u.sup.OL].sub.j]) over time. Such a solution may be reasonable if firm i cannot observe sales of firm j and vice versa. [7] If open-loop strategies are played and one of the firms changes strategy or there is an exogenous shock to the state variable X, the chosen path need not be the best response to this change and the open-loop strategy need not be subgame perfect. The following proposition characterizes the open-loop equilibrium for the game.

Proposition 1: There is a unique open-loop Nash equilibrium for the differential game [[gamma].sup.OL]. The equilibrium sales volumes are:

[[X.sup.OL].sub.i] = 1/[a.sub.i] [[([w.sub.i] + [z.sub.i]).sup.2]/2 [[gamma].sub.i] + ([w.sub.j] + [z.sub.j]) ([z.sub.i] - [w.sub.i]/2 [[gamma].sub.j] + [K.sub.i]]

i, j = 1, 2; i [neq] j, (8)

and advertising levels are:

[[u.sup.-OL].sub.i] = ([w.sub.i] + [z.sub.i])/2 [[gamma].sub.i] i = 1, 2, (9)

where [[gamma].sub.i] = [q.sub.i] / (r + [a.sub.i]).

Proof: Firm i maximizes (7), subject to (6) and given [u.sub.j](t). The Hamiltonian of this problem is:

[H.sub.i] = [e.sup.-rt]([q.sub.i][X.sub.i](t) - [u.sub.i][(t).sup.2]) + [[lambda].sub.i](t)[[w.sub.i]([u.sub.i](t) - [u.sub.j](t) + [z.sub.i]([u.sub.i](t) + [u.sub.j](t)) + [K.sub.i] - [a.sub.i][X.sub.i](t)], (10)

where [[lambda].sub.i] is the costate variable for firm i. The necessary conditions for an open-loop equilibrium are restrictions form (6) and:

[[lambda].sub.i](t) = -[e.sup.-rt][q.sub.i] + [[lambda].sub.i](t)[a.sub.i], (11)

and

0 = [[lambda].sub.i](t)[w.sub.i] + [[lambda].sub.i](t)[z.sub.i] - 2[u.sub.i](t)[e.sup.-rt]. (12)

The transversality conditions are [lim.sub.t[right arrow][infinity]] [[lambda].sub.i](t) [greater than or equal to] 0 and [lim.sup.t[right arrow][infinity]](t)[X.sub.i] (t) = 0. Differentiating (12) with respect to t and substituting [lambda](t) from (11) and [[lambda].sub.i](t) from (12) Yields:

[u.sub.i](t) = [u.sub.i](t) (r + [a.sub.i]) -[w.sub.i]+ [z.sub.i]/2 [q.sub.i]. (13)

The pair of differential equations, (13) and (6), determines the open-loop stationary equilibrium as the stationary point, [u.sub.i](t) = [X.sub.i](t) = 0.

The results in Proposition 1 are highly intuitive. The stationary level of advertising is decreasing in parameters r and [a.sub.i]. A low value of r indicates that firms put a high weight on future profits. Thus, they would be expected to make a big effort to invest and accumulate goodwill for the future. The higher the value of [a.sub.i], the lower the return on investment since advertising depreciates more quickly. So lower investment should be expected in equilibrium. Finally, the price cost margin [q.sub.i] and parameters [z.sub.i] and [w.sub.i] have a positive impact on the marginal revenue of advertising. When [z.sub.i] [greater than] [w.sub.i], that is, in industries where informative advertising dominates, the stationary sales volume [[X.sup.OL].sub.i] is increasing in [z.sub.i] and [z.sub.j] When [z.sub.i] [less than] [w.sub.i], that is, in industries where advertising is mainly competitive, then the second term of (8):

([w.sub.j] + [z.sub.j])([z.sub.i] - [w.sub.i])/2 [[gamma].sub.j],

is negative and the stationary sales volume of firm i is decreasing in ([w.sub.j] + [z.sub.j]) (the effect of firm j's advertising on its sales) and decreasing in the difference \[z.sub.i] - [w.sub.i]. The greater the difference between the informative content and the competitive content of advertising messages by firm i, the lower the stationary sales of firm i.

Corollary 1: The advertising level in the open-loop equilibrium, [[u.sup.OL].sub.i], is higher than the advertising level in the static game, [[u.sup.*].sub.i] (given by (4)) if and only if [a.sub.i] + r [less than] 1.

Usually, ([a.sub.i] + r) will be less than 1 unless advertising depreciates very quickly or the future is discounted heavily. The purpose now is to compare the open-loop solution to the joint profit maximization solution. The Hamiltonian for this problem is:

H = [[[sigma].sup.2].sub.k=1] {[e.sup.-rt] ([q.sub.k][X.sub.k](t) - [u.sub.k][(t).sup.2]} + [[lambda].sub.k](t) [[w.sub.k]([u.sub.k](t) - [u.sub.j] (t)) + [z.sub.k]([u.sub.k](t) + [u.sub.j](t)) + [K.sub.k] - [a.sub.k][X.sub.k](t)]} j = 1,2; j [neq] k. (14)

Solving the problem along the lines of the proof of Proposition 1, it is easy to get the value for:

[[u.sup.coop].sub.i] = [[gamma].sub.i]([z.sub.i] + [w.sub.i]) + [[gamma].sub.j]([z.sub.j] - [w.sub.j])/2.

Now the open-loop equilibrium values for advertising can be compared to the cooperative values.

Corollary 2: If [z.sub.j] [greater than] [w.sub.j], then [[u.sup.OL].sub.i] [less than] [[u.sup.coop].sub.i]; if [z.sub.j] [less than] [w.sub.j], then [[u.sup.OL].sub.i] [greater than] [[u.sup.coop].sub.i]; and if [z.sub.j] = [w.sub.j], then [[u.sup.OL].sub.i] = [[u.sup.coop].sub.i].

Corollary 2 implies that advertising in an open-loop equilibrium is efficient if and only if [z.sub.j] = [w.sub.j]. When advertising has a positive effect on market size greater than the market share effect ([z.sub.j] [greater than] [w.sub.j]), there is underinvestment in advertising relative to the joint profit maximization solution. The reason is the nonappropriability of the total surplus: firms do not internalize positive effects on industry sales. In the case of competitive advertising ([z.sub.j] [less than] [w.sub.j]), the net effect on the rival's sales is negative, and since firms do not internalize this negative effect, there is overinvestment in advertising at the open-loop equilibrium. The same result has been obtained for the static game. The only difference between the open-loop and the static solution is that the former takes into account the discounting of the future and the fact that advertising effects last more than one period. However, these considerations do not help firms to sustain more cooperative outcomes. Finally, when market size and market share effects are balanced, the open-loop controls sustain the joint profit maximization outcome. This is because under these conditions, the negative and positive external effects on the opposing firm cancel each other out and there is then no strategic effect. Thus, it is no surprise that when [z.sub.j] = [w.sub.j], the individual profit maximization coincides with the joint profit maximization solution. Now, turn to the issue of stability for a game that starts at [X.sub.i](0) [neq] [[X.sup.OL].sub.i].

Proposition 2: The open-loop Nash equilibrium sales trajectories are given by:

[X.sub.i][(t).sup.OL] = [[X.sup.OL].sub.i] + ([X.sub.i0] - [[X.sup.OL].sub.i]) [e.sup.-[a.sub.i]t] i = 1,2, (15)

where [[X.sup.OL].sub.i] is the stationary equilibrium sales volume.

Proof: The solution for (11) is:

[[lambda].sub.i](t) = ([[lambda].sub.i](0) - [[gamma].sub.i]) [e.sup.[a.sub.i]t] + [[gamma].sub.i][e.sup.-rt]. (16)

Define [A.sub.i] = ([lambda].sub.i](0) - [[lambda].sub.i]). Substitution of (16) in (12) yields:

[u.sub.i](t) = [w.sub.i] + [z.sub.i]/2 [A.sub.i][e.sup.t(r+[a.sub.i])] + [w.sub.i] + [z.sub.i]/2 [[gamma].sub.i]. (17)

Now, substitute (17) in the system dynamics of (6) and solve the resulting differential equation to get:

[X.sub.i](t) = ([X.sub.i0] - [[X.sup.OL].sub.i] - [[micro].sub.i][A.sub.i]/r + 2[a.sub.i] - [v.sub.i][A.sub.j]/r + [a.sub.i] + [a.sub.j]) [e.sup.-[a.sub.i]t]

+ [[X.sup.OL].sub.i] + [[micro].sub.i][A.sub.i]/r + 2[a.sub.i] [e.sup.t(r + [a.sub.1])], (18)

where [[micro].sub.i] = [([w.sub.i] + [z.sub.i]).sup.2]/2 and [v.sub.i] = ([w.sub.j] + [z.sub.j]) ([z.sub.i] - [w.sub.i]) /2. Applying the transversality conditions, the optimal trajectory, [X.sub.i][(t).sup.OL], is easily obtained.

The open-loop Nash equilibrium is stable as [a.sub.i] is assumed to be nonnegative. The optimal advertising spending of both firms is constant over time. Sales volumes converge exponentially to the stationary equilibrium [[X.sup.OL].sub.i], [[X.sup.OL].sub.j]), increasing or decreasing according to [X.sub.i0] and [X.sub.j0], respectively.

Note that from economic considerations, [X.sub.i][(t).sup.OL], should be nonnegative. Since [X.sub.i0] [less than or equal to] [X.sub.i][(t).sup.OL] [less than or equal to] [[X.sup.OL].sub.i] (or [[X.sup.OL].sub.i] [less than or equal to] [X.sub.i][(t).sup.OL] [less than or equal to] [X.sub.i0] and [X.sub.i0] [greater than or equal to] 0), this condition is fulfilled if the stationary equilibrium [[X.sup.OL].sub.i] [greater than or equal to] 0. There are two cases. First, if [z.sub.i] [greater than or equal to] [w.sub.i], then it is always the case that [[X.sup.OL].sub.i] [greater than] 0, as all terms of (8) are positive. Second, if [z.sub.i] [less than] [w.sub.i], then the following restriction on [K.sub.i] must be imposed:

[K.sub.i] [greater than or equal to] \([w.sub.j] + [z.sub.j])([z.sub.i] - [w.sub.i])/2 [[gamma].sub.j] \ - ([[w.sub.i] + [z.sub.i]).sup.2]/2 [[gamma].sub.i]. (19)

This assures the nonnegativity of [X.sub.i][(t).sup.OL]. It is easy to show that in the symmetric case ([w.sub.i] = [w.sub.j], [z.sub.i] = [z.sub.j], [q.sub.i] = [q.sub.j], and [a.sub.i] = [a.sub.j]), the right-hand side of (19) is negative so that given the initial assumption, [K.sub.i] [greater than] 0, (19) always holds. Only in an extreme asymmetric case ([w.sub.i] [much less than] [w.sub.j], [z.sub.i] [much less than] [z.sub.j]) could the right-hand side turn out to be positive. [8]

Feedback Equilibria

Here, Nash equilibria in feedback strategies are calculated, where the strategies are differentiable state-dependent rules. In this model, the durability of advertising creates intertemporal links between the firms. The payoff-relevant history (past choices affecting current profits) is the vector of sales in the previous period. To see this, note that current advertising affects the increase in sales (from the level in the previous period) rather than the absolute value of sales. Hence, sales in the previous period contain all the relevant information about past choices of advertising levels and the stock of goodwill. Thus, current profits depend on current advertising and the previous periods' sales.

In contrast to the open-loop Nash equilibrium, in this case, a firm cannot commit itself in advance to any given advertising spending path. The optimal advertising levels [[u.sup.FS].sub.i](X) change in response to changes in the state variables X(t) = ([X.sub.i](t), [X.sub.j](t)). The optimal feedback strategy indicates the best response for each value of the state variables, X, at each point in time. In an equilibrium in feedback strategies for the game, players are using optimal paths for the control variables, [u.sub.i], which are also optimal in every subgame.

The value function approach allows the optimal paths for advertising levels to be derived. The equilibrium feedback strategies ([[u.sup.FS].sub.i], [[u.sup.FS].sub.j]) must satisfy the following Hamilton-Jacobi-Bellman equation [Starr and Ho, 1969]:

r[V.sup.i](X) = [max.sub.[u.sub.i][epsilon][[S.sup.F].sub.i]] {[q.sub.i][X.sub.i] - [[u.sup.2].sub.i] + [[V.sup.i].sub.[X.sub.i]][[w.sub.i]([u.sub.i] - [[u.sup.FS].sub.j]) + [z.sub.i]([u.sub.i] + [[u.sup.FS].sub.j])

+ [K.sub.i] - [a.sub.i][X.sub.i]] + [[V.sup.i].sub.[X.sub.j]][[w.sub.j]([[u.sup.FS].sub.j] - [u.sub.i]) + [z.sub.j]([[u.sup.FS].sub.j] + [u.sub.i]) + [K.sub.j] - [a.sub.j][X.sub.j]]}, (20)

where [V.sup.i](X) is the value function for firm i (i = 1, 2), [X.sub.0] is initial sales, and subscripts on [V.sup.i] denote partial derivatives. As the right-hand side of (20) is concave in [u.sub.i], the value of [u.sub.i] that maximizes the expression is:

[[u.sup.FS].sub.i] = [w.sub.i] + [z.sub.i]/2 [[V.sup.i].sub.[X.sub.i]] + [z.sub.j] - [w.sub.j]/2 [[V.sup.i].sub.[X.sub.j]] i = 1,2, (21)

Substituting [[u.sup.FS].sub.i] and [[u.sup.FS].sub.j] in (20) yields a system of two partial differential equations (i = 1, 2):

r[V.sup.i](X) = [q.sub.i][X.sub.i] + [([[alpha].sub.i]).sup.2]/4 [([[V.sup.i].sub.[X.sub.i]).sup.2] + [([[beta].sub.j]).sup.2]/4 [([[V.sup.i].sub.[X.sub.j]).sup.2] + [([[alpha].sub.j]).sup.2]/2 [[V.sup.j].sub.[X.sub.j]] [[V.sup.i].sub.[X.sub.j]]

+ [([[beta].sub.i]).sup.2]/2 [[V.sup.j].sub.[X.sub.i]] [[V.sup.i].sub.[X.sub.i]] + [[alpha].sub.j][[beta].sub.i]/2 [[V.sup.i].sub.[X.sub.i]] [[V.sup.i].sub.[X.sub.j]] + [[alpha].sub.i][[beta].sub.j]/2 [[V.sup.i].sub.[X.sub.i]] [[V.sup.i].sub.[X.sub.j]]

+ [[alpha].sub.j][[beta].sub.i]/2 [[V.sup.j].sub.[X.sub.i]] [[V.sup.i].sub.[X.sub.j]] + [[V.sup.i].sub.[X.sub.i]][K.sub.i] + [[V.sup.i].sub.[X.sub.j]][K.sub.j] - [a.sub.i][X.sub.i][[V.sup.i].sub.[X.sub.i]] - [a.sub.j][x.sub.j][[V.sup.i].sub.[X.sub.j], (22)

where [a.sub.i] = [w.sub.i] + [z.sub.i] and [[beta].sub.i] = [z.sub.i] - [w.sub.i] (i = 1, 2).

The structure of the game and the form of system (22) leads to the following function being proposed as a solution for (22) (see, for example, Reynolds [1987] and Fershtman and Kamien [1987]): [9]

[V.sup.i](X) = [[c.sup.i].sub.1] + [[c.sup.i].sub.2][X.sub.i] + [[c.sup.i].sub.3][X.sub.j] + [[c.sup.i].sub.4] [[X.sup.2].sub.i]/2 + [[c.sup.i].sub.5][X.sub.i][X.sub.j] + [[c.sup.i].sub.6] [[X.sup.2].sub.j]/2

i = 1,2; i [neq] j. (23)

Equation (23) and the corresponding partial derivatives of these functions may be substituted in (22). The two sides of the resulting quadratic polynomial must be equal for all possible values of [X.sub.i] and [X.sub.j]. Equating the coefficients of [X.sub.i], [X.sub.j], [[X.sup.2].sub.i], [[X.sup.2].sub.j], and [X.sub.i] [X.sub.j], as well as the constant terms, yields a system of 12 nonlinear algebraic equations for the constants [[c.sup.i].sub.j], j = 1, 2, ..., 6, and i = 1, 2 (see Appendix 1).

In general, there are multiple equilibria as solutions for (20). In particular, the following proposition states that the open-loop outcome is always an equilibrium of the game [[gamma].sup.F]. However, this is not the only equilibrium of the game. Later, other feedback equilibria will be characterized.

Proposition 3: Let:

[[u.sup.FS].sub.i] = ([w.sub.i] + [z.sub.i]/2 [[gamma].sub.i] i = 1, 2, (24)

where ([[u.sup.FS].sub.1](X), [[u.sup.FS].sub.2](X)) is an equilibrium for the differential game [[gamma].sub.F] The corresponding feedback sales trajectories are given by:

[[X.sup.FS].sub.i](t) = ([X.sub.i](0) - [[X.sup.FS].sub.i])[e.sup.-[a.sub.i]t] + [[X.sup.FS].sub.i] i, j = 1, 2; i [neq] j, (25)

where [[X.sup.FS].sub.i] = [[X.sup.OL].sub.i] is the steady state level of sales.

Proof: See Appendix 1.

Now look for equilibria in which feedback strategies are nondegenerate, that is, where advertising levels react to past sales. The case presented below is the symmetric case: [w.sub.1] = [w.sub.2] = w, [z.sub.1] = [z.sub.2] = z, [q.sub.1] = [q.sub.2] = q, [a.sub.1] = [a.sub.2] = a, and [K.sub.1] = [K.sub.2], K and closed-form equilibrium feedback strategies are obtained for both firms. The proposed solution for (22) is:

[V.sup.i](X) = [c.sub.1] + [c.sub.2][X.sub.i] + [c.sub.3][X.sub.j] + [c.sub.4] [[X.sup.2].sub.i]/2 + [c.sub.5][X.sub.i][X.sub.j] + [c.sub.6] [[X.sup.2].sub.j]/2 i, j = 1, 2; i [neq] j. (26)

The symmetry of the solution imposed by (26) is reasonable, as the state variables X(t) = ([X.sub.i](t), [X.sub.j](t)) are the same for firms i and j and players have identical information and parameter values [Reynolds, 1987]. This leads to six nonlinear algebraic equations for the constants [c.sub.i] and i = 1, 2, ..., 6 (see Appendix 1).

The purpose is to characterize the optimal advertising path for different types of markets. This paper will examine markets where advertising is mainly informative and increases demand for all firms.

Informative Advertising

Informative advertising corresponds to markets where the product is not strongly differentiated (from the consumers' point of view) or where consumers are largely uninformed about the product (typically immature markets). Analyzing the extreme case, suppose w = 0.

First, note that when advertising has a market size effect, the static game has the structure of a game with positive externalities, that is, promotion activities by one of the firms increase market size, but each one takes into account only the increase in its own sales. Under these conditions, the outcome of the static game is an inefficiently low advertising effort by the firms. Calculate explicitly the feedback equilibrium strategies for the dynamic game and characterize the solution. For simplicity, only the results for the specific case, K = 0, are presented here. That is, in the absence of advertising, sales drop to zero. General results for K [greater than] 0 are found in Appendix 2.

Proposition 4: Assume w = 0. Let:

[[u.sup.[FS.sub.I]].sub.i] = -zq/2a + 2a + r/6z ([X.sub.i] + [X.sub.j]) i, j = 1, 2; i [neq] j, (27)

When initial conditions satisfy:

[X.sub.1](0) + [X.sub.2](0) = [[X.sup.[FS.sub.I]].sub.1] + [[X.sup.[FS.sub.I]].sub.2] = 6[qz.sup.2]/(a(a + 2r),

then ([[[u.sup.[FS.sub.I]].sub.1](X), [[u.sup.[FS.sub.I]].sub.2](X)) constitutes a symmetric feedback equilibrium for the differential game, [[gamma].sup.F]. The corresponding feedback sales trajectories are given by:

[[X.sup.[FS.sub.I]].sub.i](t) = ([X.sub.i](0) - [[X.sup.[FS.sub.I]].sub.i]) [e.sup.-at] + [[X.sup.[FS.sub.I]].sub.i] i, j = 1, 2; i [neq] j, (28)

where [[X.sup.[FS.sub.I]].sub.i] is the steady state level of sales.

Proof: See Appendix 2.

The two solutions for (22) define two different feedback strategies. For informative advertising, the open-loop outcome (as well as the degenerate feedback) is [u.sub.i] = zq/2(a + r) (see solution 1 in Appendix 2). In this section, the analysis is centered on the solution that defines strategy in (27).

An important feature of equilibrium strategies in (27) is that firm i reacts positively to firm j's past sales as well as to its own past sales. These positive coefficients work as a promise to cooperate implicitly with the rival. When advertising levels are made dependent on a rival's sales, advertising has not only a direct effect on sales (it increases present and future sales as indicated by the dynamic state in (6), but also an indirect effect: it increases future advertising spending by the rival who is using a feedback strategy, hence, future sales. This enticing effect (each firm increases advertising to entice the opposing firm to increase its advertising levels) is the mechanism through which firms are capable of sustaining an implicit cooperation that brings them closer to the joint profit maximization outcome.

Given the symmetry imposed on the parameters a, r, q, and z, the two firms have in the steady state the same levels of sales and advertising spending. The steady state sales associated with feedback Nash equilibrium strategies are given by:

[[X.sup.[FS.sub.I]].sub.i] = 3[qz.sup.2]/a(a + 2r) i = 1, 2, (29)

and the stationary feedback Nash equilibrium level of advertising effort is given by:

[[u.sup.-[FS.sub.I]].sub.i] = 3qz/2(a + 2r) i = 1, 2. (30)

Table 1 summarizes the steady state values in the three cases under study for symmetric informative advertising. The steady state advertising level, [[u.sup.-[FS.sub.I]].sub.i], in this case exceeds the open-loop equilibrium level, [[u.sup.OL].sub.i]. The same is true for steady state sales, that is, [[X.sup.[FS.sub.I]].sub.i] [greater than] [[X.sup.OL].sub.i]. These results imply that in the case of feedback strategies, the dynamic nature of competition between the firms allows them to sustain more cooperative outcomes than in the open-loop solution. Using feedback strategies, firms make their advertising effort dependent on total market sales, and in so doing, they increase the revenue from any investment in advertising goodwill. Advertising today implies a positive reaction by the rival in the form of a higher level of advertising in the future. This enticing effect of feedback strategies (absent in the case of an open-loop equilibrium) is responsible for the higher equilibrium advertising activity and ca n be interpreted as an implicit promise to cooperate in advertising expense. Now compare this feedback equilibrium to the open-loop and efficient outcomes.

Proposition 5: In the case of symmetric informative advertising, w = 0, profits in the equilibrium defined by (27), [[J.sup.i].sub.[FS.sub.I]], are higher than profits in the open-loop equilibrium, [[J.sup.i].sub.OL], and lower than profits in the joint profit maximization solution, [[J.sup.i].sub.coop], for a [neq] r. If a = r, the equilibrium implements the efficient outcome ([[J.sup.i].sub.[FS.sub.I]] = [[J.sup.i].sub.coop]).

Proof: See Appendix 3.

For the case of informative advertising, Propositions 3 and 4 characterize two symmetric equilibria of the differential game [[gamma].sup.F]. Proposition 5 shows that profits for both firms are higher in the feedback equilibrium that makes advertising dependent upon past sales. This result may be useful in selecting between the two perfect equilibria as firms are likely to coordinate on the equilibrium yielding higher profits. [10]

Predatory Advertising

Competitive advertising corresponds to mature markets where advertising activities emphasize brand features, and every unit spent by firm i affects market share. So this has a negative impact on the rival's sales. In the extreme case, z = 0. Formulation of this extreme case is equivalent to the models of excess advertising [Jorgensen, 1982]. In these models, only the amount of advertising that exceeds the rival's expenditure has a net effect on sales.

When advertising is perfectly predatory, both firms would be better off if they could jointly commit themselves not to advertise. However, this is not sustainable in a Nash equilibrium as each firm has incentives to increase advertising expenditures as the best response to zero advertising. The open-loop solution does not sustain the efficient outcome as previously mentioned, but in this case, the feedback strategies are even worse. When advertising is predatory, the ability to react to past sales makes interaction between firms even more competitive, reducing equilibrium profits.

Proposition 6: Assume z = 0. Let:

[[u.sup.[FS.sub.p]].sub.i] = 3/2 qw/2(a + 2r) + 2a + r/6W ([X.sub.i] - [X.sub.j]) i, j = 1, 2; i [neq] j. (31)

When the initial conditions satisfy:

[X.sub.1],(0) = [X.sub.2](0), ([[u.sup.[FS.sub.p]].sub.1](X), [[u.sup.[FS.sub.p]].sub.2](X)),

constitutes a symmetric feedback equilibrium for the dynamic game [[gamma].sup.F]. The corresponding feedback sales trajectories are given by:

[[X.sup.[FS.sub.p]].sub.i](t) = ([X.sub.i])(0) - [[X.sup.[FS.sub.p]].sub.i]) [e.sup.-at] + [[X.sup.[FS.sub.p]].sub.i] i, j = 1, 2; i [neq] j, (32)

where [[X.sup.[FS.sub.p]].sub.i] = K/a.

Proof: The proof can be obtained by following the same steps as in the proof for Proposition 4 (see Appendix 2).

Note that given the sales trajectories (the same for both firms) and the saddle point condition ([X.sub.1](0) = [X.sub.2](0)), the steady state sales are equal to guaranteed sales in the absence of advertising. This is due to the extreme nature of the case analyzed here, a perfectly predatory market. The only factor that may offset sales depreciation over time is advertising, and in a symmetric equilibrium, the levels of advertising cancel each other out. In a less extreme case of predatory advertising, however, this feature should not be found in equilibrium.

From (31), it can be obtained that firm i reacts positively to its own past sales but negatively to firm j's sales. This is in contrast with the behavior of firms in the informative advertising case where both responses were positive. When advertising is predatory, a preemption effect appears. Firms have incentives to increase their expenditure on advertising not only because this increases sales in the period, but also because it elicits a response from the rival, which will decrease its advertising in the future. Each firm's advertising preempts to some extent future rival advertising. This preemption effect is precisely why the feedback equilibrium is now worse in terms of profits than the open-loop outcome. [11]

Table 2 compares the open-loop solution to the feedback equilibrium and the cooperative outcome for the case of symmetric perfectly predatory advertising. [12] Cooperation in the predatory case implies [[u.sup.coop].sub.i] = 0. The steady state advertising level, [[u.sup.[FS.sub.p]].sub.i], exceeds the open-loop equilibrium level, [[u.sup.OL].sub.i], as in the case of informative advertising but for different reasons and with different consequences. In the open-loop solution, advertising is already too high and feedback strategies reinforce this inefficiency.

Profits from the open-loop and feedback equilibria can be compared straightforwardly. [[J.sup.i].sub.OL] [greater than] [[J.sup.t].sub.[FS.sub.p]] because [[u.sup.[FS.sub.p]].sub.i] [greater than] [[u.sup.OL].sub.t] and steady state sales are equal. The feedback equilibrium is then less efficient than the open-loop equilibrium, due to the preemption effect, which introduces an extra incentive for advertising. The result is excessive expenditure, which is harmful for profits in a market where advertising only has a business-stealing effect.

Strategies in (31) are only one of the two solutions for (22). As in the previous case of informative advertising, the other symmetric solution for (22) is a pair of degenerate feedback strategies which coincide with the open-loop equilibrium discussed earlier. Furthermore, in the case of predatory advertising, the degenerate feedback strategies yield higher profits than the nondegenerate feedback. If the same criterion is used to choose between the equilibria as previously, the preferred symmetric equilibrium would be the constant advertising levels given by the open-loop solution.

Informative and Predatory Advertising

In most markets, advertising will have a business-stealing effect and will provide general information about the characteristics of the product. In these intermediate cases, the structure of the static game is a mix of a game with positive externalities and a game with negative externalities. When one of the two effects clearly dominates, the conclusions are the same as those presented in the two previous sections. However, what happens when both effects are approximately the same size? Here, it is assumed that w = z. First note that when w = z, there is no interdependence between firms. The dynamic state in (6) reduces to:

[X.sub.i](t) = (w + z)[u.sub.i](t) + K = a[X.sub.i](t). (33)

System dynamics in (33) imply that the evolution of sales for firm i is not affected by its rival's advertising. By solving the symmetric game under these system dynamics, along the lines of Appendix 2, the optimal feedback strategy does not depend on the rival's sales:

[[u.[FS.sup.IP]].sub.i] = -q(w + z)/2a - K(2a + r)/a + 2a + r/(w + z) [X.sub.i] i, j = 1, 2; i [neq] j. (34)

Sales trajectories are given by:

[[X.[FS.sup.IP]].sub.i](t) = ([X.sub.i](0) - [[X.[FS.sup.IP]].sub.i]) [e.sup.(a+r)t] + [[X.[FS.sup.IP]].sub.i] i = 1, 2, (35)

where:

[[X.[FS.sup.IP]].sub.i] = q[(w + z).sup.2]/2a(a + r) + K/a, is the steady state level of sales and the stability condition is [X.sub.i](0) = [[X.[FS.sup.IP]].sub.i].

Table 3 summarizes the results for the symmetric case. The reason why both the open loop solution and the feedback equilibrium sustain the cooperative outcome is that strategic interdependence between firms has disappeared so that the solution to individual profit maximization coincides with that of joint profit maximization. [13]

Concluding Remarks

The main result of this paper is that both the nature of the game played by firms and the outcomes of the open-loop and feedback strategy equilibria differ according to the informative and predatory content of advertising. It is shown that when advertising has a relatively strong market size effect, feedback strategies can be used to sustain outcomes that are more collusive than the open-loop solution. This result differs from previous literature in that the outcomes of feedback equilibria are usually less efficient than those of an open-loop equilibrium (see, for example, Fershtman and Kamien [1987] and Reynolds [1987]). The reason is that under informative advertising, feedback strategies have an enticing effect. Firms increase advertising to elicit a similar response from their rival. However, when the market share effect is prevalent, feedback strategies are more competitive and the open-loop solution yields higher profits. In this case, feedback strategies make advertising a decreasing function of the o pposing firm's sales and each firm increases advertising to preempt future rival advertising. Finally, when the informative and predatory contents of advertising are exactly balanced, the joint profit maximization outcome can be implemented with both open-loop and feedback strategies.

This present model can be extended in several directions. One possibility is to consider asymmetries between players. This would make it possible to obtain results comparing different types of firms. [14] In this model, the consideration of asymmetries would, for example, allow analysis of which type of firms would be forced by rivals to exit the market when advertising is mainly predatory. A possible extension of this work would also be to examine nonlinear strategies. Tsutsui and Mino [1990] show that the solution obtained by the conventional method will not generally be unique and will provide an alternative method for obtaining multiple solutions in games with one state variable. Another interesting problem for further study is the introduction of a short-run strategic decision variable (prices or quantities (see, for example, Mariel and Sandonis [1999])) and the analysis of the effect of competition in long-run variables on short-run decisions.

(*.) Universidad del Pais Vasco--Spain. Financial aid was generously provided by Direcci6n General de Ensenanza Superior del Ministerio de Educacion y Ciencia under research grant PB97-0603, Gobierno Vasco under research grant PI-1998-86, and Universidad del Pais Vasco under research grant UPV-38.321-G55/98.

Footnotes

(1.) For low-tar cigarettes, this mainly affects the size of the market, while advertising high-tar brands (generally older brands) does not affect total consumption. The hypothesis of no market share effects cannot be rejected in either market.

(2.) Karp and Perloff [1993] obtain results that point in the same direction.

(3.) This classification is similar to the distinction drawn by Wrather and Yu [1979] between nonusers of the product, users of their own brand, and users of the other brand.

(4.) Another case in which (1) can be expressed as in (3) is when firms are symmetric in market share and switching costs, so that [s.sub.i] = [s.sub.j] and [[theta].sub.i] = [[theta].sub.j] = [theta]. Demand for firm i can then be written as:

[X.sub.i] = [X.sub.0i] + (1 - [s.sub.i] -[s.sub.j])[[eta].sub.i]f(p)([u.sub.i] + [u.sub.j]) + [s.sub.i] (1 + [theta])f(p)([u.sub.i] - [u.sub.j].

Letting [v.sub.i] = (1 - [s.sub.i] - [s.sub.j])[[eta].sub.i]f(p) and [[epsilon].sub.i] = [s.sub.i](1 + [theta])f(p) leads to an equation similar to (3).

(5.) See Wrather and Yu [1979] for a study of these two effects from another point of view. In their model, a company can either emphasize the product features to increase its product market or it can emphasize the brand features to enlarge its market share at the expense of the competitor.

(6.) See Fershtman and Kamien [1987] for detailed definitions.

(7.) Although, a firm could always make advertising a function of its own sales.

(8.) Condition (19) comes from the general solution for (6):

[X.sub.i](s) = (([X.sub.i0] - [K.sub.i]/[a.sub.i]) [e.sup.-as] + [K.sub.i]/[a.sub.i]) + [w.sub.i] [[[integral].sup.s].sub.0]([u.sub.i](t) - [u.sub.j](t))[e.sup.-[a.sub.i](s - t)]dt

+ [z.sub.i] [[[integral].sup.s].sub.0]([u.sub.i](t) + [u.sub.j](t))[e.sup.-[a.sub.i](s - t)]dt.

The trajectories of sales [X.sub.i](t) should be nonnegative. Given the equilibrium strategies, it is assumed that [K.sub.i] is high enough to make the expression positive.

(9.) Tsutsui and Mino [1990] solve a similar system of two differential equations (but only one state variable) without guessing the value functions, using an auxiliary equation that is derived from the Hamilton-Jacobi-Bellman equation.

(10.) Note, however, that the feedback equilibrium characterized in Proposition 4 requires a condition on initial sales. This condition means that total market potential is constant, a basic feature of differential games in advertising [Jorgensen, 1982].

(11.) Deal et al. [1979] studied a combined Vidale-Wolfe and excess advertising model which includes a similar term ([w.sub.i]([u.sub.i] - [u.sub.j])([x.sub.1] + [x.sub.2], where [x.sub.i] is market share) to describe the effect of competitive advertising. Their optimal advertising expenditure paths (depending on different [w.sub.i]) show a similar reaction: advertising expenditure is reduced as an answer to the rival's increasing sales.

(12.) Note that in this extreme case of predatory advertising, the second term of (6) disappears. If both firms are playing the same strategy, the first term is zero and no additional condition on K is needed to keep sales nonnegative. However, in cases where firms play different strategies, an additional condition on K must be imposed to rule out negative sales (see Footnote 7). For example, if the first firm chooses [u.sub.1] = 0 and the second firm plays the equilibrium open-loop strategy, the following restriction is needed:

K [greater than or equal to] [w.sup.2]q/2(a + r).

(13.) Remember that the strategies in (34) are only one of the two solutions for (22). The other symmetric solution for (22) is a pair of degenerate feedback strategies that coincide with the open-loop equilibrium.

(14.) For instance, Sorger [1989] gets the result that a small firm is better off in an open-loop game and only a big firm benefits from a feedback game.

References

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                   Symmetric Informative Advertising
Steady State       Open Loop             Feedback
 [u.sub.i]       1/2 zq/a + r          3/2 zq/a + 2r
 [X.sub.i]    [z.sup.2]q/a(a + r)  3[z.sup.2]q/a(a + 2r)
Steady State        Efficient
 [u.sub.i]          zq/a + r
 [X.sub.i]    2[z.sup.2]q/a(a + r)
               Symmetric Perfectly Predatory Advertising
Steady State   Open Loop      Feedback     Efficient
 [u.sub.i]    1/2 wq/a + r  3/2 wq/a + 2r      0
 [X.sub.i]        K/a            K/a          K/a
                           The Symmetric Case
Steady State            Open Loop
 [u.sub.i]          q(w + z)/2(a + r)
 [X.sub.i]    q[(w + z).sup.2]/2a(a + r) + K/a
Steady State             Feedback
 [u.sub.i]          q(w + z)/2(a + r)
 [X.sub.i]    q[(w + z).sup.2]/2a(a + r) + K/a
Steady State            Efficient
 [u.sub.i]          q(w + z)/2(a + r)
 [X.sub.i]    q[(w + z).sup.2]/2a(a + r) + K/a

APPENDIX 1

Equilibrium Feedback Strategies and Proof of Proposition 3: Substitution of the partial derivatives of (23) in (22) yields two cumbersome quadratic polynomials for [X.sub.1] and [X.sub.2] in the form r[V.sup.i](X) = [polynomial.sub.i] (i = 1, 2) and from (21):

[[u.sup.FS].sub.i] = [w.sub.i] + [z.sub.i]/2 ([[c.sup.i].sub.2] + [[c.sup.i].sub.4] [X.sub.i] + [[c.sup.1].sub.5] [X.sub.j]) + [z.sub.j] - [w.sub.j]/2 ([[c.sup.i].sub.3] + [[c.sup.i].sub.5] [X.sub.i] + [[c.sup.i].sub.6] [X.sub.j])

i, j = 1, 2; i [neq] j.

As for a degenerate feedback strategy, [[c.sup.i].sub.4] = [[c.sup.i].sub.5] = [[c.sup.i].sub.6] = 0, substituting these restrictions in the above-mentioned polynomial and equating coefficients of [X.sub.i], [X.sub.j] and the constant term yields [[c.sup.i].sub.2] = [q.sub.i] / ([a.sub.i] + r) and [[c.sup.i].sub.3] = 0. This solution yields the open-loop strategy in (24). The rest of Proposition 3 can be proved by substituting (24) in (6) and solving the resulting differential equation.

For a nondegenerate strategy, the two sides of the quadratic polynomials for [X.sub.1] and [X.sub.2] must be equal for all possible values of [X.sub.i] and [X.sub.j]. Equating the coefficients of [X.sub.i], [X.sub.j], [[X.sup.2].sub.i], [[X.sup.2].sub.j], and [X.sub.i] [X.sub.j], as well as the constant terms, yields a system of 12 nonlinear algebraic equations for the constants [[c.sup.i].sub.j] (j = 1, 2, ..., 6; i = 1, 2), which is very difficult to solve for the general case. In the symmetric case, [w.sub.1] = [w.sub.2] = w, [z.sub.1] = [z.sub.2] = z, [a.sub.1] = [a.sub.2] = a, [q.sub.1] = [q.sub.2] = q, and [K.sub.1] = [K.sub.2] = K. Also denote [alpha] = w + z and [beta] = z - w. In this case, since the proposed solution for (22) contains only six unknown constants [c.sub.i] (i = 1, 2, ..., 6) (see (26)), the previous nonlinear system of 12 equations reduces to:

0 = -r[c.sub.1] + ([[alpha].sup.2]/4 + [alpha][beta]/2) [[c.sup.2].sub.2] + ([[alpha].sup.2]/2 + [[beta].sup.2]/2 + [alpha][beta]/2) [c.sub.2][c.sub.3] + ([alpha][beta]/2 + [[beta].sup.2]/4) [[c.sup.2].sub.3] + ([c.sub.2] + [c.sub.3])K, (A1)

0 = -(r + a)[c.sub.2] + ([[alpha].sup.2]/2 + [alpha][beta]/2) [c.sub.2][c.sub.4] + ([[beta].sup.2]/2 + [alpha][beta]/2) [c.sub.3][c.sub.4] + ([[alpha].sup.2]/2 + [alpha][beta]) [c.sub.2][c.sub.5] + ([[alpha].sup.2]/2 + [alpha][beta]/2 + [[beta].sup.2]/2) [c.sub.3][c.sub.5] + [[beta].sup.2]/2[c.sub.2][c.sub.6] + [alpha][beta]/2[c.sub.3][c.sub.6] + ([c.sub.4] + [c.sub.5])K + q, (A2)

0 = -(r + a)[c.sub.3] + [alpha][beta]/2[c.sub.2][c.sub.4] + [[alpha].sup.2]/2[c.sub.3][c.sub.4] + ([[alpha].sup.2]/2 + [alpha][beta]/2 + [[beta].sup.2]/2)[c.sub.2][c.sub.5] + ([[beta].sup.2]/2 + [alpha][beta]) [c.sub.3][c.sub.5] + ([[alpha].sup.2]/2 + [alpha][beta]/2) [c.sub.2][c.sub.6] + ([alpha][beta]/2 + [[beta].sup.2]/2) [c.sub.3][c.sub.6] + ([c.sub.5] + [c.sub.6])K, (A3)

0 = -(r/2 + a)[c.sub.4] + [[alpha].sup.2]/4 [[c.sup.2].sub.4] + [alpha][beta][c.sub.4][c.sub.5] + ([[alpha].sup.2]/2 + [[beta].sup.2]/4) [[c.sup.2].sub.5] + [[beta].sup.2]/2 [c.sub.4][c.sub.6] + [alpha][beta]/2 + [c.sub.5][c.sub.6], (A4)

0 = -(r + 2a)[c.sub.5] + [alpha][beta]/2 [[c.sup.2].sub.4] + ([[alpha].sup.2] + [[beta].sup.2]/2) [c.sub.4][c.sub.5] + 3[alpha][beta]/2 [[c.sup.2].sub.5] + [alpha][beta]/2 [c.sub.4][c.sub.6] + ([[alpha].sup.2]/2 + [[beta].sup.2]) [c.sub.5][c.sub.6] + [alpha][beta]/2 [[c.sup.2].sub.6], (A5)

and

0 = -(r/2 + a) [c.sub.6] + [alpha][beta]/2 [c.sub.4][c.sub.5] + ([[alpha].sup.2]/4 + [[beta].sup.2]/2) [[c.sup.2].sub.5] + [alpha][beta][c.sub.5][c.sub.6] + [[alpha].sup.2]/2 [c.sub.4][c.sub.6] + [[beta].sup.2]/4 [[c.sup.2].sub.6]. (A6)

The general explicit solution for the system of (A1) through (A6) is not easy to find. Nevertheless, in an empirical application, the solutions to the nonlinear system of (A1) through (A6) could be found, as (A1) through (A6) contain only three constants: [c.sub.4], [c.sub.5] and [c.sub.6]. Subsequently, the solution for (A1) through (A6) may be substituted in (A1) through (A3) to get [c.sub.1], [c.sub.2], and [c.sub.3].

APPENDIX 2

Proof of Proposition 4: As in the symmetric case of informative advertising, [alpha] = [beta] = z, system (A1) through (A6) is much simpler. Table A1 shows that there are two solutions to this system under these restrictions. Note that only constants [c.sub.2], [c.sub.3], ..., [c.sub.6] are needed for the definition of feedback strategies.

According to (21), the feedback strategy for firm i is:

[[u.sup.[FS.sub.I]].sub.i] = z/2([c.sub.2] + [c.sub.3]) + z/2([c.sub.4] + [c.sub.5])[X.sub.i] + z/2([c.sub.5] + [c.sub.6])[X.sub.j]. (A7)

The first optimal strategy, corresponding to solution 1, is the open-loop (degenerate feedback) strategy. The second solution for (A1) through (A6) defines the following feedback strategy:

[[u.sup.[FS.sub.I].sub.i] = -qz/2a - zK(2a + r)/2a + 2a + r/6z ([X.sub.i] + [X.sub.j]). (A8)

Substituting these strategies in the system dynamics of (6) yields the following pair of differential equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (A9)

which can be written as [Z.sub.t] = [AZ.sub.t] + B.

The steady state ([Z.sub.t] = 0) of this system is:

Z = -[A.sup.-1] B = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The system can be centered by defining [Y.sub.t] = [Z.sub.t] - Z. System (A9) can now be rewritten as [Y.sub.t] = [AY.sub.t]. If [gamma] is a matrix whose columns are eigenvectors of matrix A, and [lambda] is a diagonal matrix with eigenvalues ([[lambda].sub.1], [[lambda].sub.2]), associated to the eigenvectors of matrix [gamma] on the diagonal, then [[gamma].sup.-1]A[gamma] = [lambda]. Premultiplying the centered system [Y.sub.t] by [[gamma].sup.-1] leads to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (A10)

which can be also written as:

[Y.sub.t] = [lambda][Y.sub.t] where [Y.sub.t] = [[gamma].sup.-1][Y.sub.t] and [Y.sub.t] = [[gamma].sup.-1][Y.sub.t].

In this case:

[lambda] = (-a 0 0 a + 2r/3) and [gamma] = (-1 1 1 1).

The solution for (A10) is:

([y.sub.1t] [y.sub.2t]) = ([k.sub.1][e.sup.-at] [k.sub.2][e.sup.a + 2r/3 t]),

where [k.sub.1] and [k.sub.2] are constants. Hence, the solution for the centered system [Y.sub.t] = [AY.sub.t] is [Y.sub.t] = [gamma] [Y.sub.t] That is:

([y.sub.1t] [y.sub.2t]) = (-[k.sub.1][e.sup.-at] + [k.sub.2][e.sup.a + 2r/3 t]) [k.sub.1][e.sup.-at] + [k.sub.2][e.sup.a + 2r/3 t]).

As a+2r / 3 [greater than] 0, the saddle point condition [k.sub.2] = 0 must be imposed to get a stable solution. Then the solution for Z(t) is:

([X.sub.1](t) [X.sub.2](t)) = (-[k.sub.1][e.sup.-at] + [[X.sup.[FS.sub.I]].sub.1] [k.sub.1][e.sup.-at] + [[X.sup.[FS.sub.I]].sub.2]).

The constant [k.sub.1] can be determined using the initial conditions:

[[X.sup.[FS.sub.I]].sub.i](t) = ([X.sub.i](0) - [[X.sup.[FS.sub.I]].sub.i]) [e.sup.-at] + [[X.sup.[FS.sub.I]].sub.i] i = 1, 2, (A11)

and the initial conditions must obey:

[X.sub.1](0) + [X.sub.2](0) = [[X.sup.[FS.sub.I]].sub.1] + [[X.sup.[FS.sub.I]].sub.2] 6([qz.sup.2] + K([z.sup.2](2a + r) - a))/a(a + 2r).

APPENDIX 3

Proof of Proposition 5: If both firms play open-loop strategies, equilibrium profits can be written as:

[[J.sup.i].sub.OL] = [[[integral].sup.[infinity]].sub.0] [e.sup.-rt][q[[X.sup.OL].sub.i](t) - [([[u.sup.OL].sub.i](t)).sup.2]]dt

= [[[integral].sup.[infinity]].sub.0] [e.sup.-rt][q([[X.sup.OL].sub.i] + ([X.sub.i0] - [[X.sup.OL].sub.i])[e.sup.-at]) - [(zq/2(a + r)).sup.2]]dt

= q/a + r[X.sub.i0] + 3[z.sup.2][q.sup.2]/4r[(a + r).sup.2],

as:

[[X.sup.OL].sub.i] = [z.sup.2]q/a(a + r).

In the case of joint profit maximization:

[[J.sup.i].sub.coop] = [[[integral].sup.[infinity]].sub.0] [e.sup.-rt][q([[X.sup.coop].sub.i](t) - [([[u.sup.coop].sub.i](t)).sup.2]]dt

= [[[integral].sup.[infinity]].sub.0] [e.sup.-rt][q([[X.sup.coop].sub.i] + ([X.sub.i0] - [[X.sup.coop].sub.i]) [e.sup.-at]) - [(zq/(a + r)).sup.2]]dt

= q/a + r[X.sub.i0] + [z.sup.2][q.sup.2]/r[(a + r).sup.2],

since:

[[X.sup.coop].sub.i] = 2[z.sup.2]q/a(a + r).

Comparing [[J.sup.i].sub.coop] and [[J.sup.i].sub.OL], the first part of Proposition 5 can be easily checked. Now, suppose both firms chose the feedback strategies. Then for K = 0:

[[J.sup.i].sub.[FS.sub.I]] = [[[integral].sup.[infinity]].sub.0] [e.sup.-rt] [q[[X.sup.[FS.sub.I]].sub.i](t) - [([[u.sup.[FS.sub.I]].sub.i](t)).sup.2]]dt

= [[[integral].sup.[infinity]].sub.0] [e.sup.-rt] [q([[X.sup.[FS.sub.I]].sub.i] + ([X.sub.i0] - [[X.sup.[FS.sub.I]].sub.i])[e.sup.-at]) - [(- qz/2a + 2a + r/6z ([[X.sup.[FS.sub.I]].sub.i](t) + [[X.sup.[FS.sub.I]].sub.j](t))).sup.2]]dt

= [[[integral].sup.[infinity]].sub.0] [e.sup.-rt] [q([[X.sup.[FS.sub.I]].sub.i] + ([X.sub.i0] - [[X.sup.[FS.sub.I]].sub.i]) [e.sup.-at]) - [(- qz/2a + 2a + r/6z (6[qz.sup.2]/[a.sup.2] + 2ar)).sup.2]]dt

= q/a + r [X.sub.i0] + 3[z.sup.2][q.sup.2](a + 5r)/4r(a + r)[(a + 2r).sup.2],

because the initial conditions are:

[X.sub.1](0) + [X.sub.2](0) = [[X.sup.[FS.sub.I]].sub.1] + [[X.sup.[FS.sub.I]].sub.2] = 6[qz.sup.2]/[a.sup.2] + 2ar.

If [[J.sup.i].sub.[FS.sub.I]] is higher than [[J.sup.i].sub.OL], the difference, [[J.sup.i].sub.[FS.sub.I]] - [[J.sup.i].sub.OL], must be positive:

[[J.sup.i].sub.[FS.sub.I]] - [[J.sup.i].sub.OL] = 3[z.sup.2][q.sup.2](2a + r)/4[(a + r).sup.2] [(a + 2r).sup.2] [greater than] 0.

The last part of Proposition 5 can be verified by computing [[J.sup.i].sub.coop] - [[J.sup.i].sub.[FS.sub.I]]:

[[J.sup.i].sub.coop] - [[J.sup.i].sub.[FS.sub.I]] = [z.sup.2][q.sup.2][(a - r).sup.2]/4r[(a + r).sup.2][(a + 2r).sup.2] [greater than or equal to] 0.

If a [neq] r, then [[J.sup.i].sub.coop] [greater than] [[J.sup.i].sub.[FS.sub.I]], but if a = r, then profits are equal.

                          The System Solutions
Constants  Solution 1                Solution 2
[c.sub.2]   q/a + r    -qr - K(2[a.sup.2] + 3ar + [r.sup.2])/
                                     2a(a + r)
[c.sub.3]      0              -(K(a + r) + q)(2a + r)/
                                     2a(a + r)
[c.sub.4]      0                      2a + r/
                                     6[z.sup.2]
[c.sub.5]      0                 2a + r/6[z.sup.2]
[c.sub.6]      0                 2a + r/6[z.sup.2]