# Persuasive advertising and market competition.

1. Introduction

The basic decisions that an industrial organization has to face include price, output (quality as well as quantity), advertising, R & D expenditure, and so on. If the profit function is to be maximized in a global sense, these variables must be chosen congruently whether these decisions are instantaneous or sequential. When decisions are taken instantaneously in a one-period game, price and output decisions are likely to be influenced by the costs and the effectiveness of advertising, and perhaps also by R & D activities. Similarly, advertising and R & D expenditures will be affected by the costs of production. This decision process is further complicated by oligopolistic rivalry in terms of price, output, advertising and R & D.

Such rivalry reaches a new dimension when decisions on different variables are made sequentially. For example, advertising campaigns often precede the actual selling of new products. Classic examples of this include the distribution of free samples (or sample coupons) by Proctor and Gamble for its new product lines such as Rejoice shampoo,(1) the movie-distributor's pre-release ads for their forthcoming feature films,(2) and the product announcement efforts by various industries.(3) It is important to note that such sequential decisions making are not restricted to new product launch. For instance, the Cola War between Pepsi and Coke are characterized by spurts of advertising rivalry in one period, followed by relative serenity in the next. We interpret this as industrial firms tending to concentrate on competitive advertising in one period, and on another form of rivalry (e.g. price) in the subsequent period. We wish to offer a possible explanation to these phenomena in terms of the sunkeness of the advertising outlay.(4) Our major result is that by credibly committing to advertising in one period, each firm in this game can achieve greater profits by limiting the mutually damaging price/advertising rivalry in a subsequent period.

With the exception of recent models which focus on the role of pricing and advertising as information dissipating tools, the literature has traditionally paid little attention to the cross-effects between price and non-price variables. Thus many nonsignaling models of advertising such as Schmalensee (1976, 1977), as well as certain signaling ones such as Kihlstrom and Riordan (1984) and Cooper and Ross (1984), generally neglected this interdependence. More recently Schmalensee wrote,

"In models involving rivalry along two dimensions, cross-effects in the demand equation, like the effect of lower prices on advertising elasticities, assume central importance. ... Little evidence on such effects is available, however." (Schmalensee, 1986, p. 377, italics added)

While an increasing number of signaling papers have recently tackled the relationship between pricing and advertising (notable examples are Milgrom and Roberts (1986) and Bagwell and Ramey (1988)), they seem on a large part to have been concerned with establishing a formal link between prices and advertising in an information dissipating sense as first suggested by Phillip Nelson (1970, 1974), than to respond to the issue raised by Schmalensee above. In these models, price and/or advertising are usually purported to be dissipative signals concerning the quality of the product rather than being competitive tools for market shares in a more traditional sense of oligopoly rivalry. An overriding objective of these signaling models has been to study the existence of separating and pooling equilibria. They have found in general that price and advertising may signal product quality, but the ways in which they do depend on the shape of the various cost curves.

In this paper we will study pricing and advertising not as signaling tools but as competitive weapons for market shares in a linear spatial market in the sense of Hotelling (1929). It is well known that sunken commitment typically confers certain advantages to an incumbent monopoly (Eaton and Lipsey (1979), Gilbert and Newbery (1982), Reinganum (1983)). The literature has so far paid little attention to commitment games between co-existing oligopolists. In a series of comparative statics results we show that pricing and advertising are inter-dependent in a way that is affected by production and advertising costs. Moreover, we find that firms spend less on advertising, charge lower prices and earn greater profits when they can 'sink' advertising expenditure before commencing sales (or between different rounds of sales in the case of existing old products). In a super-game context players can choose between equilibria, the net profit incentive of the commitment equilibrium would tend to entice firms into advertising commitments prior to a period of competitive price rivalry.

It is well known that most advertisements are informative and persuasive at the same time. Our result suggests that a useful way to distinguish more sharply between signaling and persuasive advertising is their different patterns of timing.(5) It is obviously true that some industries engage in intermittent spurts of advertising wars (like the Cola wars), while others are more continuous (like some durable goods). Ads with intermittent spurts are much closer in their outlook to that being modeled here. Our results indicate that persuasive advertising, especially when the true quality differences (in our case costs) are not great, gives rise to incentives to commit to advertising in one period and focus on other forms of rivalry in the next. Thus persuasive advertising tends to be more intermittent in their timing, while informative or signaling ads, by contrast, are more uninterrupted and persistent.

Our basic model is presented in section 2. Sections 3 and 4 then examine the equilibrium without prior commitment. Equilibria with commitments are discussed in Sections 5 and 6. Section 7 contains a brief conclusion.

2. A Model of Persuasive Advertising

Let there be a commodity supplied by two duopolists competing for market share. Consumers, each of whom needs at most one unit of this commodity, are located uniformly over a linear market segment with unit length. Throughout this paper the duopolists, producing different brands of the same product, are assumed to be located at the two extremes of this linear product space.(6)

Define the utility of a consumer situated at distance x from firm i as

s - tx - |b.sub.i~|p.sub.i~ if he buys from firm i

U(|b.sub.i~,|b.sub.j~,|p.sub.i~,|p.sub.j~) = s - t(1 - x) - |b.sub.j~|p.sub.j~ if he buys from firm j

0 otherwise

(1)

for i |is not equal to~ j, i=1,2, j=1,2. The variables in (1) above may be interpreted in the following way. Firstly, s stands for some reservation price of the consumer, t the (constant) unit transportation cost, or a measure of how reluctant consumers are to try brands which are not their ideal choice. We assume that s,t |is greater than~ 0 holds. Firm i charges factory prices |p.sub.i~ for its products. The effects that firm i's advertising efforts have on consumer utility, and hence on their willingness to pay, is captured by the remaining variable |b.sub.i~. In general we assume that the more firm i spends on advertising, the smaller will be |b.sub.i~ and the more will consumers be willing to pay for its products. For tractability we specify(7)

|b.sub.i~ = 1/1 + |A.sub.i~, |A.sub.i~ |is greater than~ 0. (2)

Notice that |A.sub.i~ in (2) may be thought of as the frequency at which firm i's product appears on local media. Net consumer utility, therefore, is negatively related to transportation cost and positively related to advertising. It is perhaps more interesting to think of t as the 'unit-divergence' of a given brand from the consumer's 'ideal' brand. The terms tx and t(1 - x) in definition (1) then become the utility loss when one has to make do with such non-choicely brands. Advertising, on the other hand, persuades the consumers that the product is perhaps not so far from their 'ideal' cjpoce after all.(8) Now we can derive the duopolists' demand functions.

Define the marginal consumer as someone in the linear demand space who is indifferent between buying from firm 1 and firm 2. Assuming s to be large enough such that the utility of the marginal consumer is positive, we have from (1)

s - tx - |b.sub.i~|p.sub.i~ = s - t(1 - x) - |b.sub.j~|p.sub.j~. (3)

Firm i,s demand function is given by some x that satisfies equation (3) above. Noting however the option of non-production and the maximum unit length of x, we have, by letting |q.sub.i~ denote the demand for firm i,

|q.sub.i~ = Min{Max|0, t - |b.sub.i~|p.sub.i~ + |b.sub.j~|p.sub.j~/2t~, 1},

for i|is not equal to~ j, i = 1,2, j=1,2. (4)

Notice quite importantly that competitive advertising tends to cancel each other out. This is captured in (4) as, in a symmetric case for instance, simultaneous increases in |A.sub.i~ and |A.sub.j~ leave |q.sub.i~ unchanged. The firms would thus want to limit their advertising outlays collusively. Individually, of course, this is constantly challenged by a prisoners' dilemma type of incentive to cheat unless the collusion can somehow be made credible.

Let |c.sub.i~ denote the |i.sub.th~ firm's (constant) unit production cost. Also let |k.sub.i~ denote firm i's monetary cost per unit of advertising. Invoking assumption (2), firm i's profit function can be written as (ignoring fixed cost since it does not affect our analysis)

||Pi~.sub.i~ = (|p.sub.i~ - |c.sub.i~)|t - |b.sub.i~|p.sub.i~ + |b.sub.j~|p.sub.j~/2t~ - |k.sub.i~|A.sub.i~ (5)

and similarly for firm j. Notice that if there are any quality differences between the products supplied by the firms, the higher quality product would most probably have a higher unit cost c. This, however, is reflected in neither consumer utility nor demand. In other words, we deliberately rule out any possibility of ex ante signaling by the firms about their product quality. Our objective is to study persuasive advertising and its role as a collusive commitment.

3. Simultaneous, Non-Commitment Choice of Prices and Advertising

In a one-variable oligopoly game involving either prices or advertising, one could look for a Bertrand equilibrium in prices given advertising, or a Nash equilibrium in advertising given prices. When both variables are chosen simultaneously and no commitment is possible, we can proceed in the following way. Firstly we find a Bertrand equilibrium in prices given advertising choices. This Bertrand equilibrium then allows us to solve for a pair of advertising reaction functions from which we can solve for the Nash equilibrium in advertising. Naturally we are interested in the case where both firms are active in equilibrium. That is, we will examine the conditions under which the price-advertising pair |(|p.sub.i~, |A.sub.i~), (|p.sub.j~, |A.sub.j~)~ exists that characterizes the Nash advertising equilibrium in a Bertrand price equilibrium such that |q.sub.i~,|q.sub.j~ |is greater than~ 0 and ||Pi~.sub.i~,||Pi~.sub.j~ |is greater than~ 0.

Hence we begin by letting the firms choose prices to maximize ||Pi~.sub.i~ and ||Pi~.sub.j~, given A's. From the first order conditions for a maximum in (5) we have

|p.sub.i~ = t + |b.sub.j~|p.sub.j~ + |b.sub.i~|c.sub.i~/2|b.sub.i~ (6)

which is easily solved to give the Bertrand equilibrium in prices:

|p.sub.i~ = t(1 + |A.sub.i~) + |c.sub.j~(1 + |A.sub.i~)/3(1 + |A.sub.j~) + 2|c.sub.i~/3. (7)

Now we come to the Nash equilibrium in advertising. Among the ||p.sub.i~(|A.sub.i~,|A.sub.j~), |A.sub.i~~ pairs where |p.sub.i~(|A.sub.i~,|A.sub.j~) is optimal given |A.sub.i~ and |A.sub.j~, firm i chooses the |A.sub.i~ that maximizes profit. Thus we substitute (7) into (5) to give the profit functions in the Bertrand price equilibrium

|Mathematical Expression Omitted~

The first order conditions for a maximum in (8) when choosing advertising give an implicit advertising reaction function f(|A.sub.i~, |A.sub.j~) = 0 as

|Mathematical Expression Omitted~

which may be rewritten as

|Mathematical Expression Omitted~

3.1 Simultaneous Non-Commitment Equilibrium: The Symmetric Case

To fix ideas, let us first of all consider a symmetric equilibrium in which neither firm can make a committed advertising choice. Let |Mathematical Expression Omitted~ denote this symmetric non-commitment pair of price and advertising choice and let |Mathematical Expression Omitted~ be the corresponding profits. From equations (7) and (9) above we have immediately

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

Notice from (11) that if |Mathematical Expression Omitted~ holds, so will |Mathematical Expression Omitted~. To ensure |Mathematical Expression Omitted~, we require the following relationship to hold:

t/2 |is less than~ k |is less than~ t+2/2. (13)

As for symmetric duopoly profits, we require from (5) above that

|Mathematical Expression Omitted~

which, using (11) and (12), is easily reduced to k |is greater than~ c/2. Combining this and (13), the necessary and sufficient condition for a Bertrand-Nash equilibrium to exist in this symmetric case is

max |c/2, t/2~ |is less than~ k |is less than~ t + c/2. (15)

Our first set of comparative statics results emerges directly from (11) and (12) above:

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

Thus we find in this symmetric case the higher is production cost, the more both firms will spend on advertising (16) and, probably in a bid to cover advertising cost, more than passing this on to the consumer (17). Notice that if a higher c is indeed associated with higher product quality, then we find that the firms producing higher quality products will both spend more on advertising and charge a higher output price when advertising is persuasive rather than signaling. Secondly, an increase in the advertising cost k will reduce both advertising expenditure and price (18) and (19). Thirdly, a higher t, which carries the interpretation either of unit transportation cost or the perceived divergence from the consumer's ideal brand, will be associated with greater advertising efforts perhaps in a bid to overcome such consumer perceptions (20). Interestingly, this will lead to even higher output prices charged to the consumers in order to cover the higher advertising expenditure (21).

Finally, it is clear from the profit function (14) that higher production cost and/or higher advertising cost reduce profits. While a reverse association between production cost and profit seems natural, when costs are used as proxies for quality as we have done so far, a reverse association between quality and profit calls for a more careful explanation. Recall from the profit function (5) above that a higher c (quality) affects cost but not demand. As there is no way for the producer to credibly inform the customers of its higher output quality, it simply suffers the cost disadvantage and its profit is lowered accordingly. The findings of the signaling models are particularly relevant here. What should be pointed out, however, is that because of the higher profits earned, there is perhaps a tendency for an over-supply of low quality products in the absence of credible signaling.

3.2 Non-Commitment Equilibrium with Production cost Asymmetry

The duopolists in this model may differ from one another in either c, the production cost, or k, the unit advertising cost, or both. The case of c-asymmetry is more interesting than that of k-asymmetry for at least three reasons. Firstly, production cost differences traditionally lie at the heart of oligopoly rivalry and the corresponding equilibrium reflects the extent of resource allocation efficiency. Secondly, since it seems reasonable to assume a positive association between unit production cost and output quality, ceteris paribus, we may use cost differences as a proxy for quality differences. Thirdly, in actual oligopoly markets it is difficult to observe any systematic differences in advertising efficiency. A possible exceptions is the case where a large manufacturer enjoys unparalleled market power over advertising agencies or television channels. Such cases lie outside the main focus of our paper and we have opted to avoid them in what follows. That is, we will discuss exclusively the case of production cost (hence quality) differences under asymmetric equilibrium.

Thus we assume that the firms differ only in production cost and hence

|k.sub.1~ = |k.sub.2~ and |c.sub.2~ |is not equal to~ |c.sub.2~ (22)

hold. Let |Mathematical Expression Omitted~, i = 1, 2, denote the asymmetric equilibrium set of prices, advertising, output quantity, and profit, respectively. From (22) and the advertising reaction functions (9) it is easy to show that

|Mathematical Expression Omitted~

from which it is immediate that

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

This result echoes our first finding in the symmetric case (16) and (17). Using production costs |c.sub.i~ as a proxy for product quality, then the firm which produces the higher quality will spend more on advertising as well as charge a higher price than the low quality rival. Again this relationship between quality and price-advertising choice, so often found in the signaling literature, is reproduced here when price and advertising are used as a persuasive rather than an information dissipating tool. In our model, the firm with a higher c would want to maintain its market share by spending more on advertising since |k.sub.1~ = |k.sub.2~ holds. It then charges a higher price to cover the higher cost and the greater advertising expenditure. Further, using (23) and the demand equation (4) it is straight forward to show that in this equilibrium both firms end up with an identical market share (1/2).

In order to ensure that in this equilibrium prices, advertising and profits are all positive, we can derive from (23) and the price function (6) that

|Mathematical Expression Omitted~

Combining this with (23 and (9) above we have

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

The conditions for |Mathematical Expression Omitted~ are therefore

t/2 |is less than~ k |is less than~ t + min ||c.sub.1~, |c.sub.2~~/2. (29)

Since each of the duopolists occupies one half of the market, the profit function can simply be written as

|Mathematical Expression Omitted~

Together with (29), this yields the following existence conditions or a Nash equilibrium in the asymmetric case:

max|t, |c.sub.1~, |c.sub.2~~/2 |is less than~ k |is less than~ t + min||c.sub.1~, |c.sub.2~~/2. (31)

Moreover, notice that from (30) we have

|Mathematical Expression Omitted~

Invoking existence condition (31), this implies that

|Mathematical Expression Omitted~, or,

|Mathematical Expression Omitted~

by (24). Once again profit is lower because the high quality firm cannot credibly communicate this information to the customers.

To summarize our results under production cost asymmetry, if a firm has higher production cost (and higher output quality), it will (a) spend more on advertising; (b) charge a higher output price; (c) enjoy the same market share and (d) earn lower profit than its rival. Results (a) and (b) turn out to be rather robust as they are repeated again and again in the commitment cases that follow.

4. Commitment Equilibrium

Commitment typically arises from the need to elicit trust from certain parties on the credible enforcement of promises. In general, this can either communicate credible threats to ward off prospective intruders, or facilitate credible collusion between conflicting parties who are otherwise distrustful of one another. Sunken commitments such as fixed capital investment may confer an incumbency advantage to an existing firm. Theoretical models of oligopoly (with the exception of Fudenberg and Tirole (1986) have largely concentrated on an extreme form of such incumbency advantage in term of the pre-emptive behavior (credible threats) by a monpolist to keep out entrants.(9) The case of credible collusion, as much as we are aware, are much less frequently discussed in the literature.

Our model therefore distinguishes from the ones on the persistence of monopoly in two respects. Firstly, both firms in our model remain operational throughout the time periods being considered. This helps us to understand the commitment relationship in markets other than a pure monopoly. Secondly, and equally importantly, our model shows that sunken commitment in terms of past advertising outlay can facilitate credible collusion and increase profits all round. One suspects strongly that other commitments such as capacity investment or location on characteristics space may also be used to further the trust between rivals.

We will study a two period model in which the firms may, if they so wish, make certain commitments in the first period which cannot be altered subsequently. Notice that money spent on advertising in the first period is a credible commitment while promises to charge a certain price are not. In order to make the point about commitment equilibrium, it suffices to analyze only the case of an advertising commitment.(10)

Thus let the firms choose advertising levels in the first period, commit to them (i.e. simply spend them in the first period), and only choose prices in the second. Without loss of generality we assume that the effect of advertising to be non-depleting over the periods in question, and that the periods are short enough so that we can ignore the problem of discounting. Given advertising in the first period, prices form a Bertrand equilibrium in the second period. By backward induction, that is, given the price choices in the second period, we have from the optimal price equation (6)

|Mathematical Expression Omitted~

|b.sub.j~|p.sub.j~ - |b.sub.i~|p.sub.i~ = 1/3 ||c.sub.j~/1 + |A.sub.j~ - |c.sub.i~/1 + |A.sub.i~~ (34)

from which we can derive, using demand equation (4),

|q.sub.i~ = 1/6t |3t + |c.sub.j~/1 + |A.sub.j~ - |c.sub.i~/1 + |A.sub.i~~ for i |is not equal to~ j, i=1,2, j=1,2. (35)

From (34) we can also derive the per unit gross profits

|p.sub.i~ - |c.sub.i~ = 1 + |A.sub.i~/3 ||3t + |c.sub.j~/1 + |A.sub.j~ - |c.sub.i~/1 + |A.sub.i~~. (36)

The profit functions which the firms will maximize in period 1 by choosing advertising in period 1 are

|Mathematical Expression Omitted~

The first order conditions then give the implicit advertising reaction functions

||3t + |c.sub.j~/(1+|A.sub.j~)~.sup.2~ - ||c.sub.i~/(1 + |A.sub.i~)~.sup.2~ = 18|k.sub.i~t (38)

which in turn give the optimal first period advertising choices as

|Mathematical Expression Omitted~

4.1 Commitment Equilibrium: The Symmetric Case

Parallel to what we did in section 3 we now search for the symmetric and then the asymmetric equilibrium. Let |Mathematical Expression Omitted~ denote the symmetric commitment equilibrium values of price, committed advertising and profit. It is immediate from the advertising function (38) that the level of advertising commitment is given by

|Mathematical Expression Omitted~

The gross profit-margin equation (36) also gives the symmetric choice of prices in the second period, given |Mathematical Expression Omitted~ in the first, as

|Mathematical Expression Omitted~

which, on using committed advertising (40), becomes

|Mathematical Expression Omitted~

The conditions for |Mathematical Expression Omitted~ to exist are therefore

t/2 |is less than~ k |is less than~ 2c + 3t/6. (43)

Since each firm occupies one-half of the unit market space, the profit function can thus be written as

|Mathematical Expression Omitted~

Substituting and rearranging, the conditions for |Mathematical Expression Omitted~ |is greater than~ 0 to hold become

max |c/3, t/2~ |is less than~ k |is less than~ t/2 + c/3. (45)

Since this set of conditions differs from that of the non-commitment symmetric equilibrium (15) in section 3 above, we note that a Nash equilibrium may exist for the commitment equilibrium but not for the non-commitment case, and vice versa.

Assume however that both (15) and (45) are satisfied and thus a Nash equilibrium exists in both cases. Then by comparing the equilibrium choices in each case we have, firstly by using results (12) and (40),

|Mathematical Expression Omitted~

Results (11) and (44) also give

|Mathematical Expression Omitted~

Hence, when firms choose advertising levels in the first period and commit to it, they need only spend a lower level of advertising. This, of course, is very intuitive since persuasive advertising is mutually canceling and both firms would want to limit its expenditure collusively. Moreover, prices are lower because there is no need to charge high prices to cover advertising costs.

It remains to investigate the profit levels with and without commitment. Routine calculations show that

|Mathematical Expression Omitted~

Thus firms, by committing to advertising in the first period, realizes higher profits than without commitments. Profits are higher because excessive advertising outlays are avoided in a credible collusion.

4.2 Commitment Equilibrium with Production Cost Asymmetry

Once again we make assumption (22) which is repeated here for convenience:

|k.sub.1~ = |k.sub.2~ and |c.sub.1~ |is not equal to~ |c.sub.2~. (22)

From the conditions governing advertising commitment choice (38) we have, using (22),

|Mathematical Expression Omitted~

which implies that

|Mathematical Expression Omitted~

Using (49) in (34) gives the price choices as

|Mathematical Expression Omitted~

which implies

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

Results (50) and (52), namely that the higher cost (and perhaps higher output quality) producer advertises more and charges a higher price, repeat our results found previously (16), (17), (24), (25). This reinforces our belief that this association between high quality and higher advertising intensity is not unique to the signaling models.

The derivation of the conditions for the existence of a Nash equilibrium in this asymmetric commitment case, in particular that

|Mathematical Expression Omitted~

is a little tedious and is presented in Appendix 2. From Appendix 2 we also know that

|Mathematical Expression Omitted~

which implies that

|Mathematical Expression Omitted~

This is also similar to result (33) in the non-commitment case, namely that the higher cost (higher quality) firm suffers from its inability to credibly inform the customers of its quality. To summarize, we have found that whether or not commitment is possible, the firm that has a higher output cost (and hence higher quality) will (a) spend more on advertising, (b) charge a higher output price, (c) enjoy the same market share, and (d) earn lower profit than its lower cost rival.

Finally, using the equilibrium expressions for prices, advertising and profits in the asymmetric-cost commitment and non-commitment cases we derive the following set of results, again with usual notations:

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

and |Mathematical Expression Omitted~

That is, both the higher cost or the lower cost firm will (a) spend less on advertising, (b) charge a higher output price and (c) enjoy greater profits if it can make advertising commitments than otherwise. This again reinforces our results found in the cases under symmetric equilibrium. These results are clearly repeats of those found in the previous section. The relevant interpretations are again applicable here.

6. Concluding Remarks

We have tried to analyze the price-advertising interplay in a linear product space occupied by a simple duopoly. The linear market space allows us to sharpen our focus on the competitive rivalry for market shares. Market shares are affected by factory prices and advertising activities. We have emphasized that advertising in this context is largely persuasive rather than informative and this allows us to compare our results with a group of signaling models which purport advertising to be an information dissipating tool.

Moreover, persuasive advertising tend to cancel each other out. In this sense they are profit dissipating, and it makes sense for industrial rivals to collude so as to limit this mutually damaging act. We stress, however, any collusion must be made credible, perhaps by means of some sunken commitments.

Our results may be summarized as follows.

(a) In a symmetric duopoly when price and advertising are chosen simultaneously in the same period, the industry with higher production cost (which may be used as a proxy for higher output quality) will, ceteris paribus, advertise more and charge a higher price. They will charge a lower price, advertise less and charge a lower price when the advertising cost is high. They will advertise more and charge a higher price when the transportation cost is high. (Section 3.1)

(b) In a simultaneous equilibrium when firms differ in production costs, the firm with higher cost (quality) will advertise more, charge higher output price, have the same market share, but earn lower profit than its rival. (Section 3.2)

(c) In a symmetric equilibrium when firms choose advertising in the first period, commit to it, and only choose price in the second period, each firm will advertise less, charge a lower price, and earn larger profit than under case (a) above. (Section 4.1)

(d) In a commitment equilibrium of type (c) above but firms differ in production cost, the higher cost (quality) firm will advertise more, charge higher output price, have the same market share, but earn lower profit than its lower cost (quality) rival. (Section 4.2)

(e) In a commitment equilibrium when price rather than advertising is used as a commitment, equilibrium prices, advertising outlay, output, and profits are identical to the noncommitment case (a) above. There is little incentive to use price as an instrument for commitment. (Appendix 1)

In general, we have found theoretical evidence to support a positive association between production cost (and hence output quality), advertising activity, and output price (results (a) and (b)). This association exists even when prices and advertising are used as a persuasive rather than an information dissipative tool.

We have also found evidence where firms may attempt to credibly collude by sinking advertising expenditure in one period and compete in prices only in the next. Prices, however, cannot serve the same purpose in preserving a collusion.

There are a number of ways in which this model can be extended. One of these is to allow at least the threat of entry. Eaton and Lipsey (1979) offered an interesting starting point but they did not address the problem of advertising. Given that committed, persuasive advertising leads to credible collusion as we found in this paper, it opens the question whether it might be used as a credible entry deterrence strategy. It will also be interesting to allow advertising to increase total market demand by drawing customers from a substitute goods industry. Finally, we have in this paper used production cost as a proxy for output quality while the high-quality firms cannot credibly communicate this information to their customers. One way to overcome this limitation is to model product quality explicitly, rather than using variables such as cost to stand proxy for output quality.

Appendix 1

In this appendix we study the case where firms choose prices in the first period, and advertise only in the second period. We wish to substantiate our contention in the text that advertising is a more appropriate tool for credible collusion than output prices.

Using the induction argument in section 3, let there be an advertising Nash equilibrium (|A.sub.i~, |A.sub.j~), given the pair of prices chosen in period one. Given (|p.sub.j~, |A.sub.j~) and |p.sub.i~, firm i, i = 1,2, chooses |A.sub.i~ to maximize

|Mathematical Expression Omitted~

The first order condition readily gives

|Mathematical Expression Omitted~

Notice that each firm's choice of advertising strategy is dominant, i.e. it does not depend on the rival's price-advertising choice.

Now we can substitute (A2) into the profit function and let the firms choose prices in the first period. The first order conditions for price choices are

|Mathematical Expression Omitted~

for i = 1, 2, j = 1, 2.

In order to make our point about price commitments it suffices to examine the symmetric equilibrium. Let |Mathematical Expression Omitted~, |Mathematical Expression Omitted~ denote the symmetric equilibrium with price commitments in the first period. It is straight forward from (A2) and (A3) that

|Mathematical Expression Omitted~

and |Mathematical Expression Omitted~

which are identical to results (11) and (12) under symmetric simultaneous non-commitment equilibrium. It follows also that profits will be identical in the two cases under consideration. As we argued earlier, price is not a credible commitment and hence offers no profit incentive for firms to commitment to it. It turns out that the same result holds under production cost asymmetry.

Appendix 2

To establish the existence conditions for a Nash commitment equilibrium with production cost asymmetry, we notice firstly from the advertising equations (38), the price equation (36) and result (49) that

|Mathematical Expression Omitted~

and |Mathematical Expression Omitted~

Routine calculation also shows that the profit functions can be written as

|Mathematical Expression Omitted~

(B1), (B2) and (B3) together imply that the conditions for existence of a Nash equilibrium in this case are

max||c.sub.1~/3,|c.sub.2~/3,|t.sub.2~~ |is less than~ k |is less than~ t/2 + min||c.sub.1~/3,|c.sub.2~/3~ (B4)

which is the same as in the case of symmetric commitment equilibrium.

Notes

1. See "Newspaper and Shampoos", Editor and Publisher, Vol. 124, Aug. 24, 1991, p. 14.

2. According to a recent report in Advertising Age (Sept 27, 1989, pp. 47-54), 'New-product advertising and promotion rose 9.76% to US$236 million in 1988'. Some of these expenditures, of course, coincided with rather than strictly preceded the product launch. The actual lead times also vary from industry to industry, with that in the film distribution industry obviously longer than that for toiletries.

3. See J. Dagnoli, "Nabisco Entries get $100 Million Blitz (Product Announcement)", Advertising Age, Sept 11, 1989; and J. Liesse, "Big G's Big Intro's, New Cereals Arrive with $65 Million in Support", Advertising Age, Feb 4, 1991.

4. In fact all advertising outlay can be regarded as effectively sunk costs. To get a feel of the relative magnitudes of such outlays across countries, it was reported in the International Marketing Data and Statistics that the top three countries in terms of advertising expenditure as percentage of GDP were US (2.5%), Japan (1.7%) and South Korea (0.8%). Another well-known purpose for 'sinking' funds in advertising is of course: 'brand-name capital'. Thus Sunkist over the last 10 years licensed the use of its name on over 400 products over the world (see "Regional Profiles: Sunkist a Pioneer in New Products, Promotions", Advertising Age, Nov 9, 1988).

5. We are grateful to the referee for alerting us to this question.

6. Allowing relocation will bring up a new set of questions involving the stability of the equilibrium if one exists. We wish here to concentrate on the price-advertising interaction and its best to avoid location choice for the time being. A truly interesting problem concerns the price-advertising choice of the potential entrants and, relatedly, whether the incumbent would use price-advertising as a pre-emptive entry deterring tool in a manner similar to that described in Eaton and Lipsey (1979). However, this topic will have to be reserved for another paper.

7. In an earlier working paper we adopted a more general functional form: |b.sub.i~ = f(|A.sub.i~) where f|prime~ |is greater than~ 0 and f|prime~ |is less than~ 0 with similar but less sharp results.

8. It is in this sense that our model differs from the signaling ones in the literature. In practice most advertising tends to be both persuasive and informative at the same time. Notice also that we have so far refrained from using the term 'output quality' which will be reserved to refer to some commonly recognized, instead of individually perceived, characteristics of that commodity.

9. While pre-emptive commitments have been well documented in the entry deterrence literature, (Gilbert and Newbery (1982) and Reinganum (1983)), games with commitments between co-existing oligopolists, like the one described here, have rarely been discussed. For related issues see Fudenberg and Tirole (1986, p. 45).

10. For the sake of completeness we have in fact analyzed the case of price commitment. That is, we assume that firms choose prices in the first period and advertising in the second. This is presented in Appendix 1 below. The basic result there is that in this case the profits are identical to the case where prices and advertising are chosen in the same period. This confirms our belief that price is probably not the right tool to facilitate credible collusion.

References

Bagwell, K. and Ramey, G "Advertising and Limit Pricing." Rand Journal of Economics (1988), vol. 19, pp. 59-71.

Cooper, R. and Ross, T. "Prices, Product Qualities and Asymmetric Information: The Competitive Case." Review of Economic Studies (1984).

Eaton, B. and Lipsey, R. "The Theory of Market Pre-emption: The Persistence of Excess Capacity and Monopoly in Growing Spatial Markets." Economica (1979), vol. 46, pp. 149-58.

Fudenberg, D. and Tirole, J. Dynamic Models of Oligopoly. 1986) Harwood Academic Publishers.

Gilbert, R. and Newberry, D. "Pre-emptive Patenting and the Persistence of Monopoly." American Economic Review (1982), vol. 72, pp. 514-26.

Hotelling, H. "Stability in Competition." Economic Journal (1929), vol. 29, pp. 455-69.

Kihlstrom, R. and Riordan, M. "Advertising as a Signal." Journal of Political Economy (1984), vol. 92, pp. 427-50.

Milgrom, P. and Roberts, J. "Price and Advertising Signals of Product Quality." Journal of Political Economy (1986), vol. 94, pp. 796-821.

Nelson, P. "Information and Consumer Behavior." Journal of Political Economy (1970), pp. 311-29.

-----. "Advertising as Information." Journal of Political Economy (1974), vol. 82, pp. 729-54.

Reinganum, J. "Uncertain Innovation and the Persistence of Monopoly." American Economic Review (1983), vol. 73, pp. 741-48.

Schmalensee, R. "A Model of Promotional Competition in Oligopoly." Review of Economic Studies. (1976), vol. 43, pp. 493-507.

-----. "Comparative Static Properties of Regulated Airline Oligopolies." Bell Journal of Economics (1977), vol. 8, pp. 565-76.

-----. "Advertising and Market Structure." In Stiglitz, J. and Mathewson G. (eds.) New Developments in the Analysis of Market Structure (1986), MacMillan, pp. 373-96.

Steiner, R. "Does Advertising Lower Consumer Prices?" Journal of Marketing (1973), vol. 37, pp. 19-26.

The basic decisions that an industrial organization has to face include price, output (quality as well as quantity), advertising, R & D expenditure, and so on. If the profit function is to be maximized in a global sense, these variables must be chosen congruently whether these decisions are instantaneous or sequential. When decisions are taken instantaneously in a one-period game, price and output decisions are likely to be influenced by the costs and the effectiveness of advertising, and perhaps also by R & D activities. Similarly, advertising and R & D expenditures will be affected by the costs of production. This decision process is further complicated by oligopolistic rivalry in terms of price, output, advertising and R & D.

Such rivalry reaches a new dimension when decisions on different variables are made sequentially. For example, advertising campaigns often precede the actual selling of new products. Classic examples of this include the distribution of free samples (or sample coupons) by Proctor and Gamble for its new product lines such as Rejoice shampoo,(1) the movie-distributor's pre-release ads for their forthcoming feature films,(2) and the product announcement efforts by various industries.(3) It is important to note that such sequential decisions making are not restricted to new product launch. For instance, the Cola War between Pepsi and Coke are characterized by spurts of advertising rivalry in one period, followed by relative serenity in the next. We interpret this as industrial firms tending to concentrate on competitive advertising in one period, and on another form of rivalry (e.g. price) in the subsequent period. We wish to offer a possible explanation to these phenomena in terms of the sunkeness of the advertising outlay.(4) Our major result is that by credibly committing to advertising in one period, each firm in this game can achieve greater profits by limiting the mutually damaging price/advertising rivalry in a subsequent period.

With the exception of recent models which focus on the role of pricing and advertising as information dissipating tools, the literature has traditionally paid little attention to the cross-effects between price and non-price variables. Thus many nonsignaling models of advertising such as Schmalensee (1976, 1977), as well as certain signaling ones such as Kihlstrom and Riordan (1984) and Cooper and Ross (1984), generally neglected this interdependence. More recently Schmalensee wrote,

"In models involving rivalry along two dimensions, cross-effects in the demand equation, like the effect of lower prices on advertising elasticities, assume central importance. ... Little evidence on such effects is available, however." (Schmalensee, 1986, p. 377, italics added)

While an increasing number of signaling papers have recently tackled the relationship between pricing and advertising (notable examples are Milgrom and Roberts (1986) and Bagwell and Ramey (1988)), they seem on a large part to have been concerned with establishing a formal link between prices and advertising in an information dissipating sense as first suggested by Phillip Nelson (1970, 1974), than to respond to the issue raised by Schmalensee above. In these models, price and/or advertising are usually purported to be dissipative signals concerning the quality of the product rather than being competitive tools for market shares in a more traditional sense of oligopoly rivalry. An overriding objective of these signaling models has been to study the existence of separating and pooling equilibria. They have found in general that price and advertising may signal product quality, but the ways in which they do depend on the shape of the various cost curves.

In this paper we will study pricing and advertising not as signaling tools but as competitive weapons for market shares in a linear spatial market in the sense of Hotelling (1929). It is well known that sunken commitment typically confers certain advantages to an incumbent monopoly (Eaton and Lipsey (1979), Gilbert and Newbery (1982), Reinganum (1983)). The literature has so far paid little attention to commitment games between co-existing oligopolists. In a series of comparative statics results we show that pricing and advertising are inter-dependent in a way that is affected by production and advertising costs. Moreover, we find that firms spend less on advertising, charge lower prices and earn greater profits when they can 'sink' advertising expenditure before commencing sales (or between different rounds of sales in the case of existing old products). In a super-game context players can choose between equilibria, the net profit incentive of the commitment equilibrium would tend to entice firms into advertising commitments prior to a period of competitive price rivalry.

It is well known that most advertisements are informative and persuasive at the same time. Our result suggests that a useful way to distinguish more sharply between signaling and persuasive advertising is their different patterns of timing.(5) It is obviously true that some industries engage in intermittent spurts of advertising wars (like the Cola wars), while others are more continuous (like some durable goods). Ads with intermittent spurts are much closer in their outlook to that being modeled here. Our results indicate that persuasive advertising, especially when the true quality differences (in our case costs) are not great, gives rise to incentives to commit to advertising in one period and focus on other forms of rivalry in the next. Thus persuasive advertising tends to be more intermittent in their timing, while informative or signaling ads, by contrast, are more uninterrupted and persistent.

Our basic model is presented in section 2. Sections 3 and 4 then examine the equilibrium without prior commitment. Equilibria with commitments are discussed in Sections 5 and 6. Section 7 contains a brief conclusion.

2. A Model of Persuasive Advertising

Let there be a commodity supplied by two duopolists competing for market share. Consumers, each of whom needs at most one unit of this commodity, are located uniformly over a linear market segment with unit length. Throughout this paper the duopolists, producing different brands of the same product, are assumed to be located at the two extremes of this linear product space.(6)

Define the utility of a consumer situated at distance x from firm i as

s - tx - |b.sub.i~|p.sub.i~ if he buys from firm i

U(|b.sub.i~,|b.sub.j~,|p.sub.i~,|p.sub.j~) = s - t(1 - x) - |b.sub.j~|p.sub.j~ if he buys from firm j

0 otherwise

(1)

for i |is not equal to~ j, i=1,2, j=1,2. The variables in (1) above may be interpreted in the following way. Firstly, s stands for some reservation price of the consumer, t the (constant) unit transportation cost, or a measure of how reluctant consumers are to try brands which are not their ideal choice. We assume that s,t |is greater than~ 0 holds. Firm i charges factory prices |p.sub.i~ for its products. The effects that firm i's advertising efforts have on consumer utility, and hence on their willingness to pay, is captured by the remaining variable |b.sub.i~. In general we assume that the more firm i spends on advertising, the smaller will be |b.sub.i~ and the more will consumers be willing to pay for its products. For tractability we specify(7)

|b.sub.i~ = 1/1 + |A.sub.i~, |A.sub.i~ |is greater than~ 0. (2)

Notice that |A.sub.i~ in (2) may be thought of as the frequency at which firm i's product appears on local media. Net consumer utility, therefore, is negatively related to transportation cost and positively related to advertising. It is perhaps more interesting to think of t as the 'unit-divergence' of a given brand from the consumer's 'ideal' brand. The terms tx and t(1 - x) in definition (1) then become the utility loss when one has to make do with such non-choicely brands. Advertising, on the other hand, persuades the consumers that the product is perhaps not so far from their 'ideal' cjpoce after all.(8) Now we can derive the duopolists' demand functions.

Define the marginal consumer as someone in the linear demand space who is indifferent between buying from firm 1 and firm 2. Assuming s to be large enough such that the utility of the marginal consumer is positive, we have from (1)

s - tx - |b.sub.i~|p.sub.i~ = s - t(1 - x) - |b.sub.j~|p.sub.j~. (3)

Firm i,s demand function is given by some x that satisfies equation (3) above. Noting however the option of non-production and the maximum unit length of x, we have, by letting |q.sub.i~ denote the demand for firm i,

|q.sub.i~ = Min{Max|0, t - |b.sub.i~|p.sub.i~ + |b.sub.j~|p.sub.j~/2t~, 1},

for i|is not equal to~ j, i = 1,2, j=1,2. (4)

Notice quite importantly that competitive advertising tends to cancel each other out. This is captured in (4) as, in a symmetric case for instance, simultaneous increases in |A.sub.i~ and |A.sub.j~ leave |q.sub.i~ unchanged. The firms would thus want to limit their advertising outlays collusively. Individually, of course, this is constantly challenged by a prisoners' dilemma type of incentive to cheat unless the collusion can somehow be made credible.

Let |c.sub.i~ denote the |i.sub.th~ firm's (constant) unit production cost. Also let |k.sub.i~ denote firm i's monetary cost per unit of advertising. Invoking assumption (2), firm i's profit function can be written as (ignoring fixed cost since it does not affect our analysis)

||Pi~.sub.i~ = (|p.sub.i~ - |c.sub.i~)|t - |b.sub.i~|p.sub.i~ + |b.sub.j~|p.sub.j~/2t~ - |k.sub.i~|A.sub.i~ (5)

and similarly for firm j. Notice that if there are any quality differences between the products supplied by the firms, the higher quality product would most probably have a higher unit cost c. This, however, is reflected in neither consumer utility nor demand. In other words, we deliberately rule out any possibility of ex ante signaling by the firms about their product quality. Our objective is to study persuasive advertising and its role as a collusive commitment.

3. Simultaneous, Non-Commitment Choice of Prices and Advertising

In a one-variable oligopoly game involving either prices or advertising, one could look for a Bertrand equilibrium in prices given advertising, or a Nash equilibrium in advertising given prices. When both variables are chosen simultaneously and no commitment is possible, we can proceed in the following way. Firstly we find a Bertrand equilibrium in prices given advertising choices. This Bertrand equilibrium then allows us to solve for a pair of advertising reaction functions from which we can solve for the Nash equilibrium in advertising. Naturally we are interested in the case where both firms are active in equilibrium. That is, we will examine the conditions under which the price-advertising pair |(|p.sub.i~, |A.sub.i~), (|p.sub.j~, |A.sub.j~)~ exists that characterizes the Nash advertising equilibrium in a Bertrand price equilibrium such that |q.sub.i~,|q.sub.j~ |is greater than~ 0 and ||Pi~.sub.i~,||Pi~.sub.j~ |is greater than~ 0.

Hence we begin by letting the firms choose prices to maximize ||Pi~.sub.i~ and ||Pi~.sub.j~, given A's. From the first order conditions for a maximum in (5) we have

|p.sub.i~ = t + |b.sub.j~|p.sub.j~ + |b.sub.i~|c.sub.i~/2|b.sub.i~ (6)

which is easily solved to give the Bertrand equilibrium in prices:

|p.sub.i~ = t(1 + |A.sub.i~) + |c.sub.j~(1 + |A.sub.i~)/3(1 + |A.sub.j~) + 2|c.sub.i~/3. (7)

Now we come to the Nash equilibrium in advertising. Among the ||p.sub.i~(|A.sub.i~,|A.sub.j~), |A.sub.i~~ pairs where |p.sub.i~(|A.sub.i~,|A.sub.j~) is optimal given |A.sub.i~ and |A.sub.j~, firm i chooses the |A.sub.i~ that maximizes profit. Thus we substitute (7) into (5) to give the profit functions in the Bertrand price equilibrium

|Mathematical Expression Omitted~

The first order conditions for a maximum in (8) when choosing advertising give an implicit advertising reaction function f(|A.sub.i~, |A.sub.j~) = 0 as

|Mathematical Expression Omitted~

which may be rewritten as

|Mathematical Expression Omitted~

3.1 Simultaneous Non-Commitment Equilibrium: The Symmetric Case

To fix ideas, let us first of all consider a symmetric equilibrium in which neither firm can make a committed advertising choice. Let |Mathematical Expression Omitted~ denote this symmetric non-commitment pair of price and advertising choice and let |Mathematical Expression Omitted~ be the corresponding profits. From equations (7) and (9) above we have immediately

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

Notice from (11) that if |Mathematical Expression Omitted~ holds, so will |Mathematical Expression Omitted~. To ensure |Mathematical Expression Omitted~, we require the following relationship to hold:

t/2 |is less than~ k |is less than~ t+2/2. (13)

As for symmetric duopoly profits, we require from (5) above that

|Mathematical Expression Omitted~

which, using (11) and (12), is easily reduced to k |is greater than~ c/2. Combining this and (13), the necessary and sufficient condition for a Bertrand-Nash equilibrium to exist in this symmetric case is

max |c/2, t/2~ |is less than~ k |is less than~ t + c/2. (15)

Our first set of comparative statics results emerges directly from (11) and (12) above:

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

Thus we find in this symmetric case the higher is production cost, the more both firms will spend on advertising (16) and, probably in a bid to cover advertising cost, more than passing this on to the consumer (17). Notice that if a higher c is indeed associated with higher product quality, then we find that the firms producing higher quality products will both spend more on advertising and charge a higher output price when advertising is persuasive rather than signaling. Secondly, an increase in the advertising cost k will reduce both advertising expenditure and price (18) and (19). Thirdly, a higher t, which carries the interpretation either of unit transportation cost or the perceived divergence from the consumer's ideal brand, will be associated with greater advertising efforts perhaps in a bid to overcome such consumer perceptions (20). Interestingly, this will lead to even higher output prices charged to the consumers in order to cover the higher advertising expenditure (21).

Finally, it is clear from the profit function (14) that higher production cost and/or higher advertising cost reduce profits. While a reverse association between production cost and profit seems natural, when costs are used as proxies for quality as we have done so far, a reverse association between quality and profit calls for a more careful explanation. Recall from the profit function (5) above that a higher c (quality) affects cost but not demand. As there is no way for the producer to credibly inform the customers of its higher output quality, it simply suffers the cost disadvantage and its profit is lowered accordingly. The findings of the signaling models are particularly relevant here. What should be pointed out, however, is that because of the higher profits earned, there is perhaps a tendency for an over-supply of low quality products in the absence of credible signaling.

3.2 Non-Commitment Equilibrium with Production cost Asymmetry

The duopolists in this model may differ from one another in either c, the production cost, or k, the unit advertising cost, or both. The case of c-asymmetry is more interesting than that of k-asymmetry for at least three reasons. Firstly, production cost differences traditionally lie at the heart of oligopoly rivalry and the corresponding equilibrium reflects the extent of resource allocation efficiency. Secondly, since it seems reasonable to assume a positive association between unit production cost and output quality, ceteris paribus, we may use cost differences as a proxy for quality differences. Thirdly, in actual oligopoly markets it is difficult to observe any systematic differences in advertising efficiency. A possible exceptions is the case where a large manufacturer enjoys unparalleled market power over advertising agencies or television channels. Such cases lie outside the main focus of our paper and we have opted to avoid them in what follows. That is, we will discuss exclusively the case of production cost (hence quality) differences under asymmetric equilibrium.

Thus we assume that the firms differ only in production cost and hence

|k.sub.1~ = |k.sub.2~ and |c.sub.2~ |is not equal to~ |c.sub.2~ (22)

hold. Let |Mathematical Expression Omitted~, i = 1, 2, denote the asymmetric equilibrium set of prices, advertising, output quantity, and profit, respectively. From (22) and the advertising reaction functions (9) it is easy to show that

|Mathematical Expression Omitted~

from which it is immediate that

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

This result echoes our first finding in the symmetric case (16) and (17). Using production costs |c.sub.i~ as a proxy for product quality, then the firm which produces the higher quality will spend more on advertising as well as charge a higher price than the low quality rival. Again this relationship between quality and price-advertising choice, so often found in the signaling literature, is reproduced here when price and advertising are used as a persuasive rather than an information dissipating tool. In our model, the firm with a higher c would want to maintain its market share by spending more on advertising since |k.sub.1~ = |k.sub.2~ holds. It then charges a higher price to cover the higher cost and the greater advertising expenditure. Further, using (23) and the demand equation (4) it is straight forward to show that in this equilibrium both firms end up with an identical market share (1/2).

In order to ensure that in this equilibrium prices, advertising and profits are all positive, we can derive from (23) and the price function (6) that

|Mathematical Expression Omitted~

Combining this with (23 and (9) above we have

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

The conditions for |Mathematical Expression Omitted~ are therefore

t/2 |is less than~ k |is less than~ t + min ||c.sub.1~, |c.sub.2~~/2. (29)

Since each of the duopolists occupies one half of the market, the profit function can simply be written as

|Mathematical Expression Omitted~

Together with (29), this yields the following existence conditions or a Nash equilibrium in the asymmetric case:

max|t, |c.sub.1~, |c.sub.2~~/2 |is less than~ k |is less than~ t + min||c.sub.1~, |c.sub.2~~/2. (31)

Moreover, notice that from (30) we have

|Mathematical Expression Omitted~

Invoking existence condition (31), this implies that

|Mathematical Expression Omitted~, or,

|Mathematical Expression Omitted~

by (24). Once again profit is lower because the high quality firm cannot credibly communicate this information to the customers.

To summarize our results under production cost asymmetry, if a firm has higher production cost (and higher output quality), it will (a) spend more on advertising; (b) charge a higher output price; (c) enjoy the same market share and (d) earn lower profit than its rival. Results (a) and (b) turn out to be rather robust as they are repeated again and again in the commitment cases that follow.

4. Commitment Equilibrium

Commitment typically arises from the need to elicit trust from certain parties on the credible enforcement of promises. In general, this can either communicate credible threats to ward off prospective intruders, or facilitate credible collusion between conflicting parties who are otherwise distrustful of one another. Sunken commitments such as fixed capital investment may confer an incumbency advantage to an existing firm. Theoretical models of oligopoly (with the exception of Fudenberg and Tirole (1986) have largely concentrated on an extreme form of such incumbency advantage in term of the pre-emptive behavior (credible threats) by a monpolist to keep out entrants.(9) The case of credible collusion, as much as we are aware, are much less frequently discussed in the literature.

Our model therefore distinguishes from the ones on the persistence of monopoly in two respects. Firstly, both firms in our model remain operational throughout the time periods being considered. This helps us to understand the commitment relationship in markets other than a pure monopoly. Secondly, and equally importantly, our model shows that sunken commitment in terms of past advertising outlay can facilitate credible collusion and increase profits all round. One suspects strongly that other commitments such as capacity investment or location on characteristics space may also be used to further the trust between rivals.

We will study a two period model in which the firms may, if they so wish, make certain commitments in the first period which cannot be altered subsequently. Notice that money spent on advertising in the first period is a credible commitment while promises to charge a certain price are not. In order to make the point about commitment equilibrium, it suffices to analyze only the case of an advertising commitment.(10)

Thus let the firms choose advertising levels in the first period, commit to them (i.e. simply spend them in the first period), and only choose prices in the second. Without loss of generality we assume that the effect of advertising to be non-depleting over the periods in question, and that the periods are short enough so that we can ignore the problem of discounting. Given advertising in the first period, prices form a Bertrand equilibrium in the second period. By backward induction, that is, given the price choices in the second period, we have from the optimal price equation (6)

|Mathematical Expression Omitted~

|b.sub.j~|p.sub.j~ - |b.sub.i~|p.sub.i~ = 1/3 ||c.sub.j~/1 + |A.sub.j~ - |c.sub.i~/1 + |A.sub.i~~ (34)

from which we can derive, using demand equation (4),

|q.sub.i~ = 1/6t |3t + |c.sub.j~/1 + |A.sub.j~ - |c.sub.i~/1 + |A.sub.i~~ for i |is not equal to~ j, i=1,2, j=1,2. (35)

From (34) we can also derive the per unit gross profits

|p.sub.i~ - |c.sub.i~ = 1 + |A.sub.i~/3 ||3t + |c.sub.j~/1 + |A.sub.j~ - |c.sub.i~/1 + |A.sub.i~~. (36)

The profit functions which the firms will maximize in period 1 by choosing advertising in period 1 are

|Mathematical Expression Omitted~

The first order conditions then give the implicit advertising reaction functions

||3t + |c.sub.j~/(1+|A.sub.j~)~.sup.2~ - ||c.sub.i~/(1 + |A.sub.i~)~.sup.2~ = 18|k.sub.i~t (38)

which in turn give the optimal first period advertising choices as

|Mathematical Expression Omitted~

4.1 Commitment Equilibrium: The Symmetric Case

Parallel to what we did in section 3 we now search for the symmetric and then the asymmetric equilibrium. Let |Mathematical Expression Omitted~ denote the symmetric commitment equilibrium values of price, committed advertising and profit. It is immediate from the advertising function (38) that the level of advertising commitment is given by

|Mathematical Expression Omitted~

The gross profit-margin equation (36) also gives the symmetric choice of prices in the second period, given |Mathematical Expression Omitted~ in the first, as

|Mathematical Expression Omitted~

which, on using committed advertising (40), becomes

|Mathematical Expression Omitted~

The conditions for |Mathematical Expression Omitted~ to exist are therefore

t/2 |is less than~ k |is less than~ 2c + 3t/6. (43)

Since each firm occupies one-half of the unit market space, the profit function can thus be written as

|Mathematical Expression Omitted~

Substituting and rearranging, the conditions for |Mathematical Expression Omitted~ |is greater than~ 0 to hold become

max |c/3, t/2~ |is less than~ k |is less than~ t/2 + c/3. (45)

Since this set of conditions differs from that of the non-commitment symmetric equilibrium (15) in section 3 above, we note that a Nash equilibrium may exist for the commitment equilibrium but not for the non-commitment case, and vice versa.

Assume however that both (15) and (45) are satisfied and thus a Nash equilibrium exists in both cases. Then by comparing the equilibrium choices in each case we have, firstly by using results (12) and (40),

|Mathematical Expression Omitted~

Results (11) and (44) also give

|Mathematical Expression Omitted~

Hence, when firms choose advertising levels in the first period and commit to it, they need only spend a lower level of advertising. This, of course, is very intuitive since persuasive advertising is mutually canceling and both firms would want to limit its expenditure collusively. Moreover, prices are lower because there is no need to charge high prices to cover advertising costs.

It remains to investigate the profit levels with and without commitment. Routine calculations show that

|Mathematical Expression Omitted~

Thus firms, by committing to advertising in the first period, realizes higher profits than without commitments. Profits are higher because excessive advertising outlays are avoided in a credible collusion.

4.2 Commitment Equilibrium with Production Cost Asymmetry

Once again we make assumption (22) which is repeated here for convenience:

|k.sub.1~ = |k.sub.2~ and |c.sub.1~ |is not equal to~ |c.sub.2~. (22)

From the conditions governing advertising commitment choice (38) we have, using (22),

|Mathematical Expression Omitted~

which implies that

|Mathematical Expression Omitted~

Using (49) in (34) gives the price choices as

|Mathematical Expression Omitted~

which implies

|Mathematical Expression Omitted~

and

|Mathematical Expression Omitted~

Results (50) and (52), namely that the higher cost (and perhaps higher output quality) producer advertises more and charges a higher price, repeat our results found previously (16), (17), (24), (25). This reinforces our belief that this association between high quality and higher advertising intensity is not unique to the signaling models.

The derivation of the conditions for the existence of a Nash equilibrium in this asymmetric commitment case, in particular that

|Mathematical Expression Omitted~

is a little tedious and is presented in Appendix 2. From Appendix 2 we also know that

|Mathematical Expression Omitted~

which implies that

|Mathematical Expression Omitted~

This is also similar to result (33) in the non-commitment case, namely that the higher cost (higher quality) firm suffers from its inability to credibly inform the customers of its quality. To summarize, we have found that whether or not commitment is possible, the firm that has a higher output cost (and hence higher quality) will (a) spend more on advertising, (b) charge a higher output price, (c) enjoy the same market share, and (d) earn lower profit than its lower cost rival.

Finally, using the equilibrium expressions for prices, advertising and profits in the asymmetric-cost commitment and non-commitment cases we derive the following set of results, again with usual notations:

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

and |Mathematical Expression Omitted~

That is, both the higher cost or the lower cost firm will (a) spend less on advertising, (b) charge a higher output price and (c) enjoy greater profits if it can make advertising commitments than otherwise. This again reinforces our results found in the cases under symmetric equilibrium. These results are clearly repeats of those found in the previous section. The relevant interpretations are again applicable here.

6. Concluding Remarks

We have tried to analyze the price-advertising interplay in a linear product space occupied by a simple duopoly. The linear market space allows us to sharpen our focus on the competitive rivalry for market shares. Market shares are affected by factory prices and advertising activities. We have emphasized that advertising in this context is largely persuasive rather than informative and this allows us to compare our results with a group of signaling models which purport advertising to be an information dissipating tool.

Moreover, persuasive advertising tend to cancel each other out. In this sense they are profit dissipating, and it makes sense for industrial rivals to collude so as to limit this mutually damaging act. We stress, however, any collusion must be made credible, perhaps by means of some sunken commitments.

Our results may be summarized as follows.

(a) In a symmetric duopoly when price and advertising are chosen simultaneously in the same period, the industry with higher production cost (which may be used as a proxy for higher output quality) will, ceteris paribus, advertise more and charge a higher price. They will charge a lower price, advertise less and charge a lower price when the advertising cost is high. They will advertise more and charge a higher price when the transportation cost is high. (Section 3.1)

(b) In a simultaneous equilibrium when firms differ in production costs, the firm with higher cost (quality) will advertise more, charge higher output price, have the same market share, but earn lower profit than its rival. (Section 3.2)

(c) In a symmetric equilibrium when firms choose advertising in the first period, commit to it, and only choose price in the second period, each firm will advertise less, charge a lower price, and earn larger profit than under case (a) above. (Section 4.1)

(d) In a commitment equilibrium of type (c) above but firms differ in production cost, the higher cost (quality) firm will advertise more, charge higher output price, have the same market share, but earn lower profit than its lower cost (quality) rival. (Section 4.2)

(e) In a commitment equilibrium when price rather than advertising is used as a commitment, equilibrium prices, advertising outlay, output, and profits are identical to the noncommitment case (a) above. There is little incentive to use price as an instrument for commitment. (Appendix 1)

In general, we have found theoretical evidence to support a positive association between production cost (and hence output quality), advertising activity, and output price (results (a) and (b)). This association exists even when prices and advertising are used as a persuasive rather than an information dissipative tool.

We have also found evidence where firms may attempt to credibly collude by sinking advertising expenditure in one period and compete in prices only in the next. Prices, however, cannot serve the same purpose in preserving a collusion.

There are a number of ways in which this model can be extended. One of these is to allow at least the threat of entry. Eaton and Lipsey (1979) offered an interesting starting point but they did not address the problem of advertising. Given that committed, persuasive advertising leads to credible collusion as we found in this paper, it opens the question whether it might be used as a credible entry deterrence strategy. It will also be interesting to allow advertising to increase total market demand by drawing customers from a substitute goods industry. Finally, we have in this paper used production cost as a proxy for output quality while the high-quality firms cannot credibly communicate this information to their customers. One way to overcome this limitation is to model product quality explicitly, rather than using variables such as cost to stand proxy for output quality.

Appendix 1

In this appendix we study the case where firms choose prices in the first period, and advertise only in the second period. We wish to substantiate our contention in the text that advertising is a more appropriate tool for credible collusion than output prices.

Using the induction argument in section 3, let there be an advertising Nash equilibrium (|A.sub.i~, |A.sub.j~), given the pair of prices chosen in period one. Given (|p.sub.j~, |A.sub.j~) and |p.sub.i~, firm i, i = 1,2, chooses |A.sub.i~ to maximize

|Mathematical Expression Omitted~

The first order condition readily gives

|Mathematical Expression Omitted~

Notice that each firm's choice of advertising strategy is dominant, i.e. it does not depend on the rival's price-advertising choice.

Now we can substitute (A2) into the profit function and let the firms choose prices in the first period. The first order conditions for price choices are

|Mathematical Expression Omitted~

for i = 1, 2, j = 1, 2.

In order to make our point about price commitments it suffices to examine the symmetric equilibrium. Let |Mathematical Expression Omitted~, |Mathematical Expression Omitted~ denote the symmetric equilibrium with price commitments in the first period. It is straight forward from (A2) and (A3) that

|Mathematical Expression Omitted~

and |Mathematical Expression Omitted~

which are identical to results (11) and (12) under symmetric simultaneous non-commitment equilibrium. It follows also that profits will be identical in the two cases under consideration. As we argued earlier, price is not a credible commitment and hence offers no profit incentive for firms to commitment to it. It turns out that the same result holds under production cost asymmetry.

Appendix 2

To establish the existence conditions for a Nash commitment equilibrium with production cost asymmetry, we notice firstly from the advertising equations (38), the price equation (36) and result (49) that

|Mathematical Expression Omitted~

and |Mathematical Expression Omitted~

Routine calculation also shows that the profit functions can be written as

|Mathematical Expression Omitted~

(B1), (B2) and (B3) together imply that the conditions for existence of a Nash equilibrium in this case are

max||c.sub.1~/3,|c.sub.2~/3,|t.sub.2~~ |is less than~ k |is less than~ t/2 + min||c.sub.1~/3,|c.sub.2~/3~ (B4)

which is the same as in the case of symmetric commitment equilibrium.

Notes

1. See "Newspaper and Shampoos", Editor and Publisher, Vol. 124, Aug. 24, 1991, p. 14.

2. According to a recent report in Advertising Age (Sept 27, 1989, pp. 47-54), 'New-product advertising and promotion rose 9.76% to US$236 million in 1988'. Some of these expenditures, of course, coincided with rather than strictly preceded the product launch. The actual lead times also vary from industry to industry, with that in the film distribution industry obviously longer than that for toiletries.

3. See J. Dagnoli, "Nabisco Entries get $100 Million Blitz (Product Announcement)", Advertising Age, Sept 11, 1989; and J. Liesse, "Big G's Big Intro's, New Cereals Arrive with $65 Million in Support", Advertising Age, Feb 4, 1991.

4. In fact all advertising outlay can be regarded as effectively sunk costs. To get a feel of the relative magnitudes of such outlays across countries, it was reported in the International Marketing Data and Statistics that the top three countries in terms of advertising expenditure as percentage of GDP were US (2.5%), Japan (1.7%) and South Korea (0.8%). Another well-known purpose for 'sinking' funds in advertising is of course: 'brand-name capital'. Thus Sunkist over the last 10 years licensed the use of its name on over 400 products over the world (see "Regional Profiles: Sunkist a Pioneer in New Products, Promotions", Advertising Age, Nov 9, 1988).

5. We are grateful to the referee for alerting us to this question.

6. Allowing relocation will bring up a new set of questions involving the stability of the equilibrium if one exists. We wish here to concentrate on the price-advertising interaction and its best to avoid location choice for the time being. A truly interesting problem concerns the price-advertising choice of the potential entrants and, relatedly, whether the incumbent would use price-advertising as a pre-emptive entry deterring tool in a manner similar to that described in Eaton and Lipsey (1979). However, this topic will have to be reserved for another paper.

7. In an earlier working paper we adopted a more general functional form: |b.sub.i~ = f(|A.sub.i~) where f|prime~ |is greater than~ 0 and f|prime~ |is less than~ 0 with similar but less sharp results.

8. It is in this sense that our model differs from the signaling ones in the literature. In practice most advertising tends to be both persuasive and informative at the same time. Notice also that we have so far refrained from using the term 'output quality' which will be reserved to refer to some commonly recognized, instead of individually perceived, characteristics of that commodity.

9. While pre-emptive commitments have been well documented in the entry deterrence literature, (Gilbert and Newbery (1982) and Reinganum (1983)), games with commitments between co-existing oligopolists, like the one described here, have rarely been discussed. For related issues see Fudenberg and Tirole (1986, p. 45).

10. For the sake of completeness we have in fact analyzed the case of price commitment. That is, we assume that firms choose prices in the first period and advertising in the second. This is presented in Appendix 1 below. The basic result there is that in this case the profits are identical to the case where prices and advertising are chosen in the same period. This confirms our belief that price is probably not the right tool to facilitate credible collusion.

References

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Eaton, B. and Lipsey, R. "The Theory of Market Pre-emption: The Persistence of Excess Capacity and Monopoly in Growing Spatial Markets." Economica (1979), vol. 46, pp. 149-58.

Fudenberg, D. and Tirole, J. Dynamic Models of Oligopoly. 1986) Harwood Academic Publishers.

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-----. "Advertising as Information." Journal of Political Economy (1974), vol. 82, pp. 729-54.

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Schmalensee, R. "A Model of Promotional Competition in Oligopoly." Review of Economic Studies. (1976), vol. 43, pp. 493-507.

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Author: | Koh, Winston T.H.; Leung, H.M. |
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Publication: | American Economist |

Date: | Sep 22, 1992 |

Words: | 6387 |

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