Periodic Price Reduction as a Way to Boost Diminishing Demand

In this paper, I offer a new theory for why price reductions take place on a regular basis in some industries. I suggest that the demand for a firm’s product drops over time because of the erosion of consumers’ brand recall, and that price discounts are utilized to boost the diminishing demand. A dynamic model is then constructed to demonstrate the theory for both monopoly and duopoly competition. I show that it is optimal for a monopolist to alternate between a constant high (normal) price and a constant low (discount) price with fixed frequency, and that competing firms offer discounts at the same time in duopoly competition.


Introduction
Price discounts are pervasive in the business world. Simply at the retailing level, there are various forms of price reductions, for various reasons. This paper focuses on a particular form of price discounts, namely a periodic price reduction which consists of long periods of a constant (high) price and short periods of a discounted (low) price. This kind of intertemporal pricing is typical in the fast food and service industries.
For a time in Durham, North Carolina, all Burger King customers can enjoy burgers at $0.99 a piece on Wednesdays, while the normal price is $2.99. More interestingly, in the same time period, Wendy's also o¤ers its price discounts every Wednesday. Another example is the discount price of HK$40 that all movie theatres in Hong Kong simultaneously charge on Tuesdays, vis-a-vis a regular ticket price of HK$60 on any other days. Free admission is available each Wednesday in all public museums in Hong Kong. 1 In many European countries, public transportation is free on a particular day each year. For Figure 1: Daily price time path of color TV in Sears, San Diego (adopted from Gerstner, 1985). durable goods, similar pattern also exists. Figure 1 shows the daily price time path of color TV in Sears, San Diego (adopted from Gerstner, 1985).
Existing literature has several explanations on why price discounts are o¤ered. Some models focus on spatial price dispersion, while some others study price variations over time. Spatial price dispersion cannot persist unless there is market friction, otherwise a store charging a higher price than its competitors would not make any sale. Taking the form of consumer searching cost or switching cost, market friction typically leads to mixed strategy equilibria in which a store varies its price randomly over a continuous support, as shown in Varian(1980). Obviously, the time path of a particular store's price would not display any regularity.
When a grocery store o¤ers price discounts on some items, it might serve the purpose of generating store tra¢c. The familiar marketing tactic of "loss leader" or "bait and switch" strategy is meant to use the low price of one good to lure consumers into buying the complements (Hess and Gerstner, 1987; Lal and Rao, 1997) or substitutes (Gerstner and Hess, 1990) for that good. However, both fast food restaurants and movie cinemas are highly specialized stores; each store sells only one product.
More relevant to this paper are models with a clear time dimension. Lazear(1986) presents a model of demand uncertainty where price is …rst marked up and then eventually lowered down until the good is sold out. While Lazear's model explains well the "clearance sale" of fashion clothing, for which demand uncertainty plays a central role, it is hardly applicable to generic goods such as fast foods, for which demand is relatively predictable. Besides, the auction-type of downward pricing is quite di¤erent from a cyclic price movement. Intertemporal price discrimination could happen when consumers have di¤erential costs of holding inventories, 2 which makes it optimal for stores to o¤er periodic price discounts (Blattberg, Eppen, and Lieberman, 1981;Jeuland and Narasimhan, 1985). Obviously, people do not buy large amount of fast food to save for future consumptions. Fast food, like services, is not storable.
Finally, price discounts could be a store's response to a changing demand or cost. In Conlisk, Gerstner, and Sobel(1984), Sobel(1984), and Pesendorfer(2000), a new cohort of consumers enters the market each period so that cyclic pricing is optimal for the …rms. These models are more suitable for durable goods: once a consumer buys the good, she vanishes forever from the market; while if a consumer does not buy at some time, she stays in the market until the next sale. Clearly, fast foods are not durable. It is also hard to believe there are new customers to enter the fast food market everyday. This paper is interested in the regular intertemporal price movement of a non-durable, non-storable good exempli…ed by fast food and other industries. There is clearly a cycle: a store charges a constant regular price most of the time, and a constant discounted price occasionally with the same frequency.
The timing of discounts is …xed and predictable, and the pricing pattern exists in both monopoly and oligopoly. In monopolistic competition, all stores o¤er discounts at the same time. 3 Now the question is, how to explain this pricing behavior? In particular, what determines the magnitude, frequency, duration, and timing of price discounts, and what is the incentive behind simultaneous discounts? Obviously, the fundamental question is why a store wants to o¤er price discounts.
When people think of the periodic price reductions such as those by Burger King and Wendy's on Wednesdays, the …rst response might be "peak-load pricing," for instance, the "Happy Hours" in Varian(1989), the seasonal price variation in Gerstner(1986), or the well-known cases of electricity, car rental, telephone calls, etc. where …xed capacity plays an important role. Presumably the mid-week demand for dining (or movie) service is smaller than that during weekends, which justi…es di¤erential and cyclic prices: it is optimal for a monopolist to charge a low price in the mid-week when demand is low. But then why does the price reduction not appear on Tuesdays or Thursdays? It is hard to believe that the demand stays the same for six days of the week, and then sharply drops on Wednesdays, as implied by a discount price of $0.99 versus a normal price of $2.99. A more fundamental problem with this explanation, though, is the requirement of market power: when there is competition, peak-load pricing is simply not sustainable. In fact, Warner and Barsky(1995) document strong empirical evidence of lower prices when demand is higher, for example, during weekends and holidays. price variation over time to sort consumers with di¤erential valuations of the product. But as pointed out by Stokey(1979) and Varian(1989), this practice is in general not pro…table. 3 Doyle(1986) claims that "most stores seem to hold sales simultaneously." Another plausible explanation of periodic price reduction is intertemporal price discrimination-high valuation consumers purchase on the spot, while price sensitive customers wait until the price is low enough. However, a necessary condition for any intertemporal price discrimination is consumers' intertemporal choice, which in turn requires each consumer to have unit demand each week, not everyday.
While this might be true for movies and video tape rentals, it's hardly the case with fast food or mass transportation, where people make the purchase on a daily basis. In such a situation, whether or not a consumer is served today, she still needs to eat and commute tomorrow. The seller is facing the same demand everyday, therefore the optimal pricing should stay the same.
The second objection to price discrimination is that it cannot explain why the seller o¤er the good for free (i.e. museums and European mass transportation). Price discrimination serves the only purpose of making more pro…t. A zero price obviously means a negative pro…t in the discount period. The seller should be better-o¤ closing the business on that day rather than o¤ering it for free.
Summarizing the above argument, I do not think the existing literature is able to explain fully the price cycles in the fast food and other industries, and will attempt to give a new explanation in this paper. My argument consists of two major points. First, without any stimulation, the demand for a particular product diminishes over time. This is because consumers eventually forget about this product.
For example, a customer might not remember the good taste of a Big Mac if there is nothing to remind her. Consequently the person just brings lunch from home. In fact a declining demand is the ultimate reason behind the need of continuous and endless advertising. Why does Nike repeatedly promote itself?
It's because otherwise people will go away. A new product might need advertising to inform people of the existence and function of the product, or to signal its quality (Milgrom and Roberts, 1986), but for well established brand names such as McDonald's, Burger King, Nike, Kodak, or Intel, obviously the persistent advertising campaign serves only one purpose: keep people hooked. The implication is straightforward: without any reminder, the consumer base gradually erodes away.
The second point of my argument is, periodic price discount is a tactic used by the sellers to boost the diminishing demand. A price reduction will certainly lead to a higher quantity demanded in the current period, which in turn brings about a higher demand in the future. The familiar habit persistency literature has well established the now standard positive relationship between current consumption and future demand. Basically a consumer's utility is a function of past consumption as well as current consumptions.
Then the marginal utility of current consumption would be increasing in past consumption, and we know marginal utility corresponds to demand curve, which will be shifting up and down accordingly. Blattberg and Neslin(1990, p.118) express the idea clearly: "...any purchase of a brand has implications beyond the immediate purchase occasion: The consumer forms a habit toward purchasing the brand, sustains that habit, or learns about the performance of the brand." Slade(1998) expresses similar idea in her empirical formulation of price changes: the demand function contains a term of consumer goodwill, which erodes over time and reacts positively to a price cut.
If a declining demand needs to be stimulated, then why do …rms not use advertising? After all, one of the major functions of advertising is to stimulate demand by constantly drawing consumers' attention.
For example, "[a] simple newspaper advertisement may remind the consumer that he or she needs co¤ee, and since the brand name is attached to the ad, the consumer goes to the store thinking, 'I need brand X co¤ee.' " Blattberg and Neslin(1990, p.114). I suggest that a price reduction conveys real bene…t to consumers, thus a more e¢cient way to keep consumers hooked. It certainly sounds much more exciting when Burger King announces "Big sale at unbelievable price!" rather than simply saying "I am here.
Please come." Another distinction between price reduction and advertising is the mechanism through which the demand is boosted. Price reduction increases current consumer base, while advertising builds favorable brand image. 4 If we understand advertising as any method that stimulates demand, then price reduction is a special form of advertising.
Given the basic idea of a shrinking demand and the possibility for a …rm to use price discounts to boost the demand, this paper studies the optimal and equilibrium choice of price discounts in both monopoly and duopoly settings. In the model, …rms have a large set of choice variables at their disposal: the normal price, discount price, as well as the duration, frequency, and timing of discounts. The major …nding is: it works. That is, it is optimal for …rms to o¤er a constant normal (high) price most of the time, and a constant discounted (low) price occasionally with the same frequency; and in duopoly competition the two …rms should o¤er discounts at the same time.
The paper is organized as follows. Section 2 sets up the model. Sections 3, 4, and 5 study the optimal choice of regular price, discount price, and the frequency of discounts, respectively. Section 6 derives duopoly equilibrium, and …nally Section 7 concludes.

The Model: Monopoly
Time is discrete with in…nite horizon. A time unit is called a period. Consider a monopolist who sells a standard product to the market in each period. Consumers are distributed uniformly along a straight line, one end of which is occupied by the …rm. In each period every consumer demands at most one unit of the product. Consumers' willingness to pay is ¹ and the unit transportation cost in period t is (1=x t ) for every consumer. Thus the demand in period t is determined by the marginal consumer: where p t is the price in that period, and q t is the quantity demanded. Consequently The central idea of this paper is that consumers have a tendency of forgetting about this product at any given moment, in the sense of a steady rise in the transportation cost, or equivalently a continuous drop in x t . Facing this natural erosion of consumer base, the seller regularly engages in sharp price reductions in order to stimulate the demand. A price reduction leads to a higher quantity demanded in the current period, and as said before, the habit persistency literature has established a positive relationship between current consumption and future demand. To capture the above idea, this paper works directly on the demand coe¢cient x t , which fully characterizes the demand function. Basically we want x t to drop over time, and to jump up when there is a stimulus, where the stimulus takes the form of a price reduction.
More speci…cally, Assumption 1. The demand coe¢cient x t changes in the following way: given x t¡1 in the previous period and the depth of price discount d t in the current period, where¸> 0, 0 < ± < 1. The depth of discount d t is de…ned as where p t is the actual price in period t, and r t is the reference price for that period.
A change in x t is a rotation of a straight line around the price intercept (p = ¹; q = 0), which is a natural result when the transportation cost changes. 5 When there is no price reduction, the demand drops at a …xed rate ±. 6 Any price change would cause the normal movement along a particular demand curve, but a price discount will have an additional e¤ect of shifting out the entire demand curve. For the sake of simplicity, in equation (1) the e¤ect of a price discount on x t is assumed to be separable from x t¡1 and is linear in d t . This assumption can be relaxed without changing the basic features of the model. 7 The monopolist is said to o¤er a price discount in period t if the actual price p t is strictly lower than the reference price r t . Otherwise the price is "normal" (p t¸rt ). The depth of discount d t is de…ned to be the di¤erence between r t and p t . When the …rm charges a normal price, d t = 0. 8 Assumption 2. The reference price in period t is where p r is the long-term regular price, which is obtained by taking the average of normal prices over a long time span. p r does not have a time subscript. (i.e. it's the same for every period.) Consumers form their reference price in period t, r t , as the lower value between the price in the previous period, p t¡1 , and the long-term regular price, p r , because they will not be fooled by two possible tricks used by the …rm. On the one hand, there is a general regular price p r . The …rm is not able to cheat by raising p t¡1 , charging a p t that is lower than p t¡1 , and then claiming that it is having a "discount" in period t. On the other hand, consumers have fresh memory about the price in the previous period.
They do not think any price that is lower than the regular price is always a discount. The demand is not stimulated if today's price is higher than or equal to yesterday's price, even if both are lower than the regular price. In other words, a lower price is thought to constitute a discount only when it is lower than both the long-term regular price p r and the last period price p t¡1 .
Consumers take the average of normal prices (recall the de…nition of a "normal" price) in order to form the long-term regular price p r : It should be emphasized that the average is taken over non-discount prices, not the price in every period. For example, $2.99 is both the long-term regular price of a burger and the "normal" price that is charged each day except Wednesday. Consumers have a clear understanding that 6 The …xed percentage drop of demand should be more realistic than a …xed absolute drop: xt = x t¡1 ¡ ± +¸dt, where there is nothing to prevent the demand from dropping below zero. 7 A more general speci…cation is to let xt asymptotically approach a certain level, say x 0 . That is, xt = x 0 + [±(x t¡1 ¡ x 0 ) +¸dt] with xt¸x 0 for every t. The model presented in this paper is a special case where x 0 = 0. 8 That is to say, the e¤ect of price change on the position of the demand function is asymmetric: a price reduction shifts out the demand curve, while a price rise does not have the opposite e¤ect. This is because people have a clear perception about a normal price and a discount price: a two dollar price cut is welcomed by everybody, while moving back to $2.99 is only interpreted as "returning to the normal price." $0.99 is a discount price and should not be included when the long-term regular price is calculated. 9 Recall that a normal price p i is a price that is not lower than the reference price r i , which in turn depends on p r as r i = minfp i¡1 ; p r g. A change to the normal price in a single period would have a negligible impact on p r , while a change to the normal price in many consecutive periods would make consumers adjust p r accordingly. p r is self-ful…lling: given consumers' perception and expectation about p r , the monopolist chooses every normal price to maximize its objective function; given the …rm's choice of normal prices, p r indeed equals the moving average of normal prices.
Finally in order to …nish the setup, some standard elements are assumed for the …rm: a marginal production cost c 2 [0; ¹) that is constant over time, a …xed cost F in period t if d t > 0, and an intertemporal time discount factor¯2 (0; 1). The monopolist chooses the price in each period to maximize the total present value of pro…t.

Normal Prices and Long-Term Regular Price
The long-term regular price p r is the moving average of normal prices over a long time span. An equilibrium about p r is a situation in which neither the consumers want to change their perception about p r , nor does the monopolist want to change its choice of every normal price.
A normal price p n i (the superscript n means "normal") a¤ects the …rm's objective function in three ways: it directly enters the pro…t of period i; it could be the reference price of period (i+1); and …nally it could change the long-term regular price p r . Let's look at the three e¤ects in turn.
. Since d i = 0 (recall that the …rm charges a normal price in period i), the coe¢cient x i = ±x i¡1 +¸d i = ±x i¡1 is not a¤ected by the choice of p i .
Then the single period pro…t ¼ i is maximized at the monopoly price p m = (¹ + c)=2. The pro…t ¼ i is monotonically increasing in p i when p i < p m and is monotonically decreasing in p i when p i > p m . Notice that p m does not depend on the time index i.
E¤ect 2: p i could be the reference price for period (i+1). We distinguish between two cases: p nn i (the next period price is also normal), and p nd i (the next period price is discounted).
E¤ect 3: A change of the normal price in a single period would have a negligible impact on p r . A simultaneous change of many normal prices could change p r , but the e¤ect of every normal price on p r is the same.
Although a normal price enters the objective function in three ways, it can be proved that the …rm should always charge the same price when it is not o¤ering any discount.
Proposition 1 In any p r -equilibrium every normal price is the same, and they are all equal to Consequently the long-term regular price is p r = p m = (¹ + c)=2.
All proofs are collected in the Appendix.
Although the demand is changing all the time, the change is in such a way that the single period pro…t is maximized at the same price-the monopoly price p m . This is the driving force of the uniform normal price. As for the possible impact on the pro…t of the next period through the formation of reference price, the conclusion can be understood as follows. First, if the next period is also a normal period, the pro…t does not depend on the reference price, so the current price should be p m . Second, if the next period is a discount period, then there are two cases. If the long-term regular price p r is greater than the monopoly price p m , then we can …nd a normal price that is higher than both p r and p m . But then that price should be lowered because it does not a¤ect the reference price in the next period. In the second case, if p r is less than p m , then there is a normal price that is lower than both p r and p m . A rise in this normal price will raise both the current period pro…t and the next period reference price, which in turn raises the next period pro…t. This means the original price again is not optimal.
One might think that a uniform normal price that is slightly higher than p m might be bene…cial to the …rm, as a higher reference price means a smaller distortion in the discounted periods. This is not true, because the monopolist should always charge p m in a normal period that is followed by another normal period.
The …rm's choice variables are the prices in each period. The above proposition greatly simpli…es the …rm's choice: when it is not o¤ering any discount, it should always charge the same normal price.
Despite this simpli…cation, what is left for the …rm to determine is still very complicated: the duration of discounts (i.e. whether or not it wants two or three consecutive discount periods), the depth of every discount, and the frequency of discounts.

Depth of Discount
For a wide range of parameter values, it can be shown that the …rm should choose at most one discount period between any two adjacent normal prices. 10 Intuitively, a price must be lower than the preceding price in order to constitute a discount. If the preceding price is already discounted, o¤ering a price reduction in the next period would be more costly, as the price will have to be even farther away from the single-period-pro…t-maximizing price. Most of the time the …rm would rather wait for at least one period after a discount before the next is o¤ered.
In principle the …rm has the freedom of never o¤ering any discounts, but in most cases this is not optimal. 11 Basically the cost of o¤ering a discount includes the …xed cost F and a lower current pro…t, while the bene…t is a higher demand in the future. If the …rm never o¤ers any discount, the demand would eventually drop to zero (recall that ± < 1), in which case o¤ering discounts for at least once is bene…cial, given that the …xed cost F is not prohibitively high.
It now becomes clear that the …rm's optimal strategy is to o¤er price discounts once at a time: a price discount for one period, then a normal price for several periods, followed by another discount period, and several other normal periods, so on and so forth. I call the price movement from one discount period to the next a cycle. To …x the idea, consider the case where exactly n periods form a cycle. The …rm chooses n …rst, commits to it, and then chooses the depth of discount in each cycle. In order to …nd the optimal choice, I will …rst determine the optimal depth of discount for any …xed n, then discuss the stability of the system, and …nally choose the optimal n. Making n a parameter …rst and then a choice variable implies that the discounts must be o¤ered at equal pace. 12 A cycle starts with a discount period, followed by (n ¡ 1) periods of normal prices where p = p m , after which a new cycle starts. Suppose that the demand coe¢cient is x when a cycle starts, and that the monopolist o¤ers a discount d in the …rst period of that cycle. Then the total present value of the pro…ts of the cycle is: where z = (¹ ¡ c)=2 is the pro…t margin when p m is charged, and y = z 2 (1¡¯n± n ) 1¡¯± > 0.

Optimal Depth of Discount for Given Demand
Starting from an arbitrary x, the monopolist chooses a series of d in each cycle in order to maximize lifetime pro…t. This is a typical dynamic programming problem. In each cycle, there is a state variable x i and a choice variable d i . (The notation has been slightly abused as the subscript now indicates cycles rather than periods.) The transition equation is Following standard procedures (see, for example, Stokey and Lucas, 1989), de…ne the value function as V (x). The above optimization problem becomes (omitting the subscript): where the pro…t of the current cycle ¼ is given by equation (2). It turns out that there is a unique optimal choice d > 0 for every possible initial demand x.

Proposition 2
The optimal choice of discount d is given by: The interior solution d ¤ does not depend on the …xed cost F because for …xed n, F is a sunk cost.
Looking at the expression of ¼, one can see that d has two opposite e¤ects on the pro…t: it increases the demand coe¢cient, thus raises the pro…t; and at the same time for a …xed demand coe¢cient, it pushes the actual price in the discount period farther away from the optimal single period price p m , thus reduces the pro…t. At d = 0 or equivalently p = p m , the second e¤ect is zero so that the …rst e¤ect dominates.
Therefore unless¸= 0, the discount d should always be greater than zero. In other words, the …rm should always o¤er discounts.
It is easy to verify the following comparative static properties of the interior solution d ¤ : The e¤ect of a discount on the demand coe¢cient is described by equation (1): When x t¡1 is bigger , the relative e¤ect of d t is smaller, therefore the …rm should o¤er a smaller discount.
That is, if the current business is already very good, there is little need to boost the demand.
A single period pro…t when normal price is charged is proportional to the demand coe¢cient in that period, with z 2 being the proportion factor. A bigger z means a bigger pro…t in the future when demand is boosted, and therefore motivates a bigger discount. For the same reason, since the future pro…t is increasing in either¯or ±, the depth of discount should also increase in the two parameters. Basically what a discount does is to sacri…ce current pro…t in return for a higher demand (and thus a higher pro…t) in the future. When any of z,¯or ± is higher, it means the possible gain in future pro…t is higher, therefore a bigger sacri…ce at present is worthwhile.
Finally, parameter¸represents the e¤ectiveness of a discount on stimulating demand. When¸is higher, the discount should be deeper.

Stability of the Dynamic System
Proposition 2 determines the optimal depth of discounts. For any given x, there is a corresponding optimal choice d ¤ (x). Then given d ¤ , at the beginning of the next cycle, the …rm faces a new demand g = ± n (x +¸d ¤ (x)) and needs to make another choice of d. We are interested in the stability of the dynamic system, namely if the …rm adheres to the optimal choice of d ¤ speci…ed in Proposition 2, will the demand coe¢cient become larger and larger, or will it converge to a speci…c level (the steady state)?
This question is important because it makes no sense to talk about the depth of discounts if the system explodes. It turns out that the dynamic system is globally stable. Starting from any x, not only does there exist a unique steady state, but also the system approaches its steady state monotonically.
Proposition 3 The dynamic system (3) converges monotonically to a unique steady state in which In order to study the dynamic stability, in principle the state variable in one cycle needs to be expressed as a function of that in the preceding cycle. In this model, the transition equation is g = ± n (x +¸d).
With d ¤ being optimally chosen as a function of x, g is in turn a function of x. According to standard textbooks, the dynamic system is stable if and only if the eigenvalue of g = g(x) is less than 1 in absolute value. However, since d ¤ as a function of x is already very complicated, it is impossible to …nd the eigenvalue of g(x). Another approach has been taken in the proof.
First express both x and g as functions of the optimal discount d (the superscript ¤ has been suppressed for simplicity of notation). It can be shown that the two functions have a unique intersection, which, by de…nition, is the steady state. Both functions are decreasing in d at the intersection, but x always has a steeper slope than g does. x decreases in d over the entire range, while g can be either monotonically decreasing in d (case 1 in Figure 2, where d m is the biggest value that d can take) or U-shaped (case 2 in Figure 2). Thus the transition from x to g is determined by using d as a linkage. In either case, it is obvious from Figure 2 that starting from any x, the system converges monotonically toward the steady state.
To give an idea of how the system evolves, a numerical example is constructed as follows. Leţ from an x that is either above or below x, the system converges quickly and monotonically towards the steady state. (Table 1) x  The steady state x and d are given in Proposition 3. Plug them into equation (2), and we get the pro…t of one cycle in steady state. To simplify, assume¯= ±. Let s =¯n = ± n , then, 3 + 2s 13 The steady state one cycle pro…t is ¼ dn = 1773 ¡ F . Here we can have an idea it performs against two consecutive discounts. For the same parameter values, if the seller o¤er two consecutive discounts, the steady state is given by x = 14:94; d 1 = 0:50; d 2 = 4:64, and the one cycle pro…t is ¼ dd = 1785 ¡ 1:95F . Two-discount is better than one-discount only when F < 12:28. This is very unlikely. A typical F would be in the thousands. (For example, for the speci…ed parameters, if the frequency is a choice variable, n ¤ = 6 corresponds to F = 1250.) In fact there is an even more pro…table arrangement. Let the …rm o¤er a discount d 1 in period one, a normal price pm in period two, a second discount ±d 2 in period three, and normal prices in periods four through six. The pro…t of one cycle would be ¼ dnd = 1837 ¡ 1:9F , and the demand coe¢cient at the end of the cycle would be the same as that in the two consecutive discounts arrangement. ¼ dd > ¼ dnd only when F < ¡1099. In other words, for any F¸0, two consecutive discounts in steady state always give a lower pro…t than two discounts separated by one normal price. We can see that this new arrangement also falls in the category of one discount once at a time.
By direct observation, it is easy to verify the comparative statics in Table 2.
The single period pro…t is proportional to z. When z is higher, the bene…t of using discounts to boost demand is higher, therefore d should be deeper. Consequently the demand coe¢cient x and the pro…t of a cycle ¼ are both higher.
indicates the e¤ectiveness of a discount on the demand. Previous discussion has shown that for …xed x, d ¤ is increasing in¸. However, when d increases, the demand coe¢cient of the next cycle increases so that d of the next cycle should be lower. The net e¤ect happens to be that the steady state d does not depend on¸. 14 As for the other two variables, x and ¼, it is quite intuitive that they are both increasing in¸.
The two parameters¯and ± measure how important the future is to the …rm. Since the monopolist o¤ers discounts in order to enjoy a higher demand in the future, when¯and ± are higher, the steady state d should be deeper, and x and ¼ are both higher.

Frequency of Discount
So far the model has analyzed the optimal choice of discount in each cycle and the dynamics of the system for a given n. In principle the frequency of discounts represented by n should also be a choice variable.
In order to …nd the optimal n, we need to express the lifetime pro…t as a function of n, given that the …rm chooses d optimally in each cycle, and that the dynamic system is in its steady state.
Remember that the discount is o¤ered in the …rst period of each cycle. Now the setting needs to be changed slightly so that the discount is o¤ered in the last period of each cycle. Consequently the steady state pro…t in each cycle is the gross pro…t minus¯nF instead of F . Doing so will not change any of the previous results regarding the optimal choice of d ¤ or the dynamics, as for …xed n and F , both F and¯nF are sunk costs and will not a¤ect the time path of either x or d. The advantage of making the above change is that o¤ering no discount now becomes a special case where n ¤ = 1. In contrast, if the discount period happens at the beginning of each cycle, choosing n = 1 does not correspond to a case of never o¤ering any discount, as making one discount at the beginning is embedded into the setting.
The optimal frequency of discount is an n ¤ that maximizes the present value of total pro…t with in…nite time horizon: Remember that¯= ± and s =¯n = ± n . The total pro…t becomes a function of s: For …xed¯and ±, choosing n is equivalent to choosing s 2 (0; 1).
Intuitively, the …xed cost F should play a crucial role in determining the optimal frequency of discount.
As shown previously, the optimal choice of the depth of discount should always be positive when F is not taken into account. In other words, if there is no …xed cost, the seller would o¤er discount whenever she has a chance. 15 One can correctly anticipate that when F is extremely high, the …rm might never want any discount; when F is very low, discounts are o¤ered as frequently as possible; and when F takes a moderate value, there should be an interior solution of n.
(iv) The interior solution n ¤ has the following comparative static properties: The sign of comparative statics is quite intuitive. When the …xed cost is bigger, discounts should be o¤ered less frequently. When either¸is bigger (a discount is more e¤ective in boosting demand) or z is bigger (a higher demand is more pro…table), the …rm should o¤er discounts more often.

Modi…cation of the Model
The analysis so far deals with monopoly. Naturally the model should be extended to duopoly competition where again the central issue is the possibility for each …rm to use price discounts to boost its own demand.
In addition to the depth and frequency of discounts, the two …rms have concerns about the timing: do they want to o¤er discounts at the same time or at di¤erent times?
The model in previous sections is slightly modi…ed to accommodate duopoly competition. Two …rms, designated a and b, sell di¤erentiated products to a market. The demand of …rm i in period t is: The interaction between the two …rms a¤ects only the demand coe¢cient: where 0 < ± < 1;¸> µ > 0, and d i t is …rm i's depth of discount in period t. It is straightforward to see that µ = 0 is a special case where the two …rms become two independent monopolists. As in the model of monopoly, d i t > 0 if …rm i o¤ers a discount in period t, and d i t = 0 if a normal price is charged.
Obviously, if neither of the …rms is o¤ering any discount, the demand for each …rm is shrinking at a constant rate ±. That is, over time a …rm loses customers, but it does not lose them to its competitor.
When a …rm o¤ers a discount, the demands of both …rms are a¤ected: its own demand is raised by a factor¸, while its rival's demand is hurt by a factor µ. In other words, consumers evaluate the relative attractiveness of the two …rms' products only when at least one …rm is o¤ering a discount. 16 When both …rms o¤er discounts at the same time, the net e¤ect will depend on the magnitude of the two discounts.
Notice that¸> µ so that when the two discounts are of the same depth, demands of both …rms are boosted.
Since the interaction between the two …rms only a¤ects the demand coe¢cients, as in the case of monopoly, the normal prices should always be …xed at p m = (¹ + c)=2 for each …rm, and the pro…t of a normal period is In duopoly competition our attention is restricted to a …xed frequency of discounting. Each …rm o¤ers a discount once every n periods. At the beginning of the game each …rm announces the timing of its own discount, and commits to the timing for the entire game. Given the …xed timing, in each cycle the two …rms choose the depth of their own discounts simultaneously and independently. Then …rm a's pro…t of that cycle is:

1¡¯±
. Given d b , …rm a chooses d a to maximize the lifetime pro…t of its own. The dynamic programming problem for …rm a is: There are two state variables: the demand coe¢cients of each …rm at the beginning of the cycle, x a and x b ; and there is one choice variable for …rm a: the depth of its own discount in the current cycle, d a . The transition equations of the system are given by: To …nd the duopoly equilibrium in the dynamic setup, we need to derive the best response function for each …rm using dynamic programming. 17 Since the two …rms are symmetric, the best response function needs to be derived only once.
Lemma Starting from any pair of x a and x b , the equilibrium depths of simultaneous discounts of the two …rms in the dynamic competition are fully characterized by ½¸' where ' and Á are given in equation (4), and´= 1 ¡¯n± n ¡¯n± 2n : It is easy to check that when the two …rms' businesses are equally good (i.e. x a = x b = x), there is a unique equilibrium in which the two …rms o¤er discounts with the same magnitude: 18 Moreover, the common choice d has the following properties: @d @x < 0; @d @z > 0; @d @± > 0; @d @¯> 0; while @d @µ > 0 ()¯n± n +¯n± 2n < 1 These signs are quite intuitive and are also consistent with the corresponding signs in monopoly. There is a new parameter µ in duopoly. For a large range of n; ± and¯, the depth of discount d is increasing in µ. When µ is bigger, a …rm's demand is smaller because it is negatively a¤ected by the existence of the other …rm. Since d is decreasing in x, both …rms should have deeper discounts. 19 Numerical examples show that starting from x a = x b , the system converges monotonically to a unique steady state. However, when the initial demands of the two …rms are very unbalanced, although there still exists a unique steady state, the process of approaching the steady state is not necessarily monotonic. In particular, the …rm that starts with a lower demand overshoots with respect to both the choice variable d and the state variable x before …nally settling down at the steady state. Figure 3 illustrates this overshooting behavior. In this example, the parameters are¸=1; µ=0:5; z = 5; ±=¯=0:95; n=6, and the steady state is described by x a = x b = 9:11 and d a = d b = 6:56. Firm a starts from x 0 a = 100 while …rm b starts from x 0 b = 0: The overshooting can be understood as follows. Due to its original high demand, …rm a's discount is …rst very small and eventually rises. In principle …rm b should be just the opposite: dropping its discounts 18 This is not simply because the two functions are symmetric. Basically, if we take the di¤erence of the two equations, and if x a = x b = x but d a 6 = d b , then after some manipulation, we require¸y ¡¸'d a d b < 0, which is never true because ' < 1; y > d 2 a , y > d 2 b . 19 Notice that the presence of µ a¤ects the choice of d through the demand coe¢cient x, not through the parameter¸. Some people might have the following misunderstanding. One might look at the demand coe¢cient x i t = ±x i t¡1 +¸d i t ¡ µd j t and think that since d i t = d j t , the demand becomes x i t = ±x i t¡1 + (¸¡ µ)dt, therefore the presence of µ has the e¤ect of reducing the discount coe¢cient¸. By the comparative statics in monopoly, d should increase in¸. Consequently d should decrease in µ. This is not true. In duopoly competition a …rm makes its own optimal choice, regarding the other …rm's choice as …xed. When deriving the …rst order condition for …rm a, d b should be constant and should not be regarded as equal to da at that point. Consequently The presence of µ is felt through a reduced x i t , and we know that d is decreasing in x.
(ii) the system is globally stable (i.e. it always converges).
When the two …rms start from the same demand level, it is easy to understand that they will keep symmetric at every step on the path to the steady state. When they start from di¤erent demands, the …rm with a higher demand will have a smaller discount, which leads to a smaller di¤erence between the two …rms' demands at the beginning of the next cycle. Eventually they converge to the same level.

Proposition 6
In the steady state, d is increasing in µ, and x is decreasing in µ.
µ is the parameter to measure the interaction between the two …rms. When µ is bigger, the negative impact of one …rm's discount on the other …rm's demand is bigger. Consequently the equilibrium demand level for both …rms is lower. And in order to partly o¤set this e¤ect, each …rm needs to o¤er a deeper discount.
Since µ = 0 represents the case of monopoly, the above proposition leads to: Corollary The steady state demand in duopoly is lower than in monopoly, while the discount in duopoly is deeper.

Non-Simultaneous Timing
Now consider the case where the two …rms o¤er discounts at di¤erent times, but the timing is still …xed for each …rm once it is determined. Without loss of generality, let …rm a's discount be in the …rst period of each cycle, while …rm b's is in the kth period. 2 · k · n.
Following the same procedure as in the simultaneous timing case, it can be shown that the equilibrium depths of discounts, d a and d b , are characterized by the two best response functions: 20 where w = z 2 =(1 ¡¯±), and ', Á are found in equation (4), while´is de…ned in the Lemma.
Starting from any combination of x 0 a and x 0 b , there is a unique steady state for every timing arrangement. Figure 4 shows the two …rms' steady state demand in the discount period (right before the discount is o¤ered) x, discount d, and pro…t in one cycle ¼ for every possible timing. 21 Remember that k is the timing of …rm b's discount in each cycle. Since …rm a always o¤ers discount in the …rst period, k = 1 is the case of simultaneous timing, while k = 2; 3; 4; 5; 6 are all cases of non-simultaneous timing. The following conclusion is reached: With regard to the steady state in non-simultaneous timing, Figure 4: Steady state variables in di¤erent timing (ii) For non-simultaneous timing, ¼ a is increasing in k, and ¼ b is decreasing in k.
In order to understand conclusion (i), …rst consider the case k < n=2 + 1. That is, in each cycle …rm a o¤ers the discount …rst, and very soon …rm b follows this action. Firm a's demand is boosted when its discount is o¤ered, while at the same time …rm b's demand is hurt. When it is …rm b's turn, its demand is lower than …rm a's for two reasons: the natural drop of its own demand, and the loss of customers due to a's discount. Since the depth of discount d is decreasing in demand level x, …rm b should o¤er a deeper discount than a. Another way to understand the relationship is to realize that in steady state, …rm a's demand (at the point right before its discount is o¤ered) has to be higher than …rm b's, because pretty soon this hoisted demand is going to be eroded by …rm b's action. Likewise for the depth of discount, …rm a's incentive is also smaller because it can only enjoy the bene…t for a very short time.
The symmetry in conclusion (i) should come at no surprise. After all, a cycle is de…ned arbitrarily in the dynamic process with in…nite time horizon. When k > n=2 + 1, …rm b's discount is closer to …rm a's next discount than the current one. Consequently the starting and ending points of a cycle can be rede…ned so that …rm b's discount comes …rst in a cycle, while …rm a's discount follows after (n ¡ k) periods. The conclusion of (i) can be compressed as follows: de…ne the "leader" to be the …rm whose discount is closer to the other …rm's next discount than to the previous discount. Then the steady state discount of the leader is smaller than that of the follower, while the demand of the leader is bigger.
To understand conclusion (ii), notice that …rm a would like …rm b's discount to be as late as possible so that it can enjoy the boosted demand for a longer time. For precisely the same reason, …rm b wants its discount to appear as early as possible.  Table 3: Pro…ts with di¤erent timing of discounts (in the thousands of units)

Equilibrium Timing
The total present value of the pro…ts of the two …rms are calculated for the …rst 20 cycles, 22 starting from di¤erent combinations of x 0 a and x 0 b . Table 3 shows the results. Again notice that k = 1 represents the simultaneous timing.
Obviously in both the case of x 0 a = x 0 b = 5 and x 0 a = x 0 b = 100, …rm b's best choice is to o¤er discounts at the same time when …rm a does. In the case of x 0 a = 0; x 0 b = 100, …rm b does want to choose a di¤erent timing (to o¤er a discount in the second period of each cycle). The main reason for …rm b's deviation is that it starts with a very high demand (x 0 b = 100). In general a …rm wants to defer the discount process when demand is high so that a discount will be more e¤ective.
On the other hand, when x 0 a and x 0 b are both very high, a simultaneous, immediate discount is still an equilibrium. This is because the interaction between two …rms provides some extra incentive for the …rms to coordinate their timing of discounts. A …rm does not want to advance the timing of its discount because the e¤ect of its discount will be immediately weakened when the other …rm o¤ers its discount a short time later. Within a certain range, a …rm does not want to defer the timing of its discount either, both because the gain of customers from the other …rm is smaller due to the natural erosion of demand, and because it has a shorter time to enjoy the boosted demand as the other …rm's next discount will come sooner.
Summarizing the above results, we conclude that a simultaneous timing by the two …rms in the …rst period is an equilibrium in the …rst two cases, and is not an equilibrium in the third case.
When a …rm decides to deviate from a simultaneous timing arrangement (this only happens when the two …rms' demands are extremely unbalanced), the most likely time to o¤er a discount is immediately after its rival. This does not mean that a non-simultaneous timing can be supported as an equilibrium.

Conclusion
This paper provides a new explanation of why price discounts are o¤ered. It postulates that demand is shrinking over time and that a …rm periodically uses price discounts to boost the diminishing demand.
While it is well-known that demand can be stimulated by advertising, my contribution is to point out that price reduction can indeed be regarded as a special form of advertising.
For a widely used business practice such as price discounts, naturally there are many di¤erent explanations. For example, the periodic price reduction in durable good markets can be well explained by Conlisk, Gerstner, and Sobel(1984) or Sobel(1984), while consumers' stockpiling behavior for storable goods is modeled in Pesendorfer(2000). I used the fast food and service sectors to represent the nondurable, non-storable industries, because these are the places where the existing literature is not able to provide a satisfactory theory. The model in this paper works well to explain all of the stylized facts.
It also provides some further insight into the interaction between …rms in a dynamic setup when price discounts are the choice variables.
Notice that the model itself does not need the assumption of a non-durable, non-storable good. The basic idea is applicable to any industry. The model's prediction of optimal intertemporal pricing has actually been found in many other industries: periodic price reduction with a long period of constant high price and a short period of constant low price for durable goods (Gerstner, 1985) or non-durable but storable good (Pesendorfer, 2000), and a simultaneous o¤ering of price discounts by competing …rms (Doyle, 1986;Lal, 1990). On the other hand, it is fully understandable that several forces could all play a role in explaining the periodic price reduction in a particular industry. For example, while my model can explain why Burger King o¤ers deep discounts once a week, it does not exclude element of price discrimination, and the exact timing of discount might well re ‡ect the concern of peak-load pricing. 23 Of course in these cases, which theory is more appealing becomes an empirical question. Pashigian(1988), Pashigian and Bowen(1991) have done useful work in this direction.
Although I propose periodic price reduction as a special form of advertising in that it boosts diminishing demand, there are important di¤erence between these two promotion strategies. Price discount has the danger of delivering an image of inferior quality, therefore it will not be utilized in cases where quality is a major concern, such as a new product or fashion clothing or an industry with important vertical product di¤erentiation. Fast food is a good place to execute the discount strategy because the quality is standard and well-known. Consumers have long perceived a price like $2.99 as a fair price, so that a discount price of $0.99 will only stimulate demand without damaging the product image.
A practical implication of this paper is that managers should always consider periodic price reduction as one of the options to stimulate demand. For example, any service industry such as barber shops or restaurant in general can follow this strategy. In the face of more and more intense competition, companies that provide service on the Internet such as searching engines need to continuously remind consumers of their service, and one tactic they can use is periodic price reduction. Many internet services are free, then they should charge a "negative price"-people get to earn money by simply using the service.
The validity of this research against existing theories can be tested empirically by observing whether quantity sold by a …rm diminishes over time when a normal price is charged. One can correctly anticipate that a discount price will produce a greater quantity sold, but one does not know if this increase of sale comes from a movement along a …xed demand curve or from a shift of the entire demand. However, one implication of this paper is that the sale at normal prices should be shrinking over time between two adjacent discounts, and this is readily testable. My theory would tend to prevail in non-storable industries where capacity is not a major concern, otherwise peak-load pricing or intertemporal price discrimination would probably play a role and complicate the picture.
Another possible extension is to explore the relative e¤ectiveness of price discounts versus other promotion tactics, in particular the optimal combination of price reduction and advertising. Milgrom and Roberts(1986) analyze the combination for quality signaling. Many discounts in the real world are accompanied by heavy advertising. One the other hand, we do see various advertisements without a price discount. It will be interesting to study in what situation a …rm should use single advertising and in what situation the …rm should use a price discount with advertisement. A price discount without advertising is also imaginable-for e¤ective price discrimination.
A Proof of Proposition 1 Proof. A change to the normal price in a single period will have a negligible impact on the long term regular price. The only way a normal price p n i can a¤ect the …rm's pro…t in periods other than period i is through e¤ect 2, namely by changing the reference price of the next period. If pi = p nn i , i.e. the next period price is also normal, this e¤ect is zero. Then in equilibrium p nn i = p m = (¹ + c)=2 for every i. Now let's look at p nd i , namely period i is normal, while period (i + 1) is discounted. If p nd i = pm for every i, the proof is …nished. Let p nd i 6 = pm for at least one i in equilibrium. Then minifp nd i g < pr < maxifp nd i g, as pr is the weighted average of every p nd i and many p nn i . Let j 2 farg min i fp nd i gg and k 2 farg max i fp nd i gg. If pr¸pm, then p nd k > pr¸pm. Then the monopolist can earn a higher pro…t by slightly lowering p nd k , because r k+1 = minfpr; p nd k g = pr is not a¤ected by p nd k , and the pro…t in period k is decreasing in p k when p k > p m .
If pr < pm, then p nd j < pr < pm. Then rj+1 = minfpr; p nd j g = p nd j . The pro…t in period j is increasing in p nd j when p nd j < pm. The pro…t in the next period: ¼j+1 = (pj+1 ¡ c)qj+1 ¡ F = (±xj +¸dj+1)(¹ ¡ pj+1)(pj+1 ¡ c) ¡ F , is increasing in p j+1 for …xed d j+1 > 0, because p j+1 = p nd j ¡d j+1 < p nd j < p m . But p j+1 is increasing in p nd j , so in turn ¼j+1 is also increasing in p nd j . Therefore the …rm could have enjoyed a higher pro…t in both period j and period (j+1) by slightly increasing p nd j , while the pro…ts in all the following periods are either not a¤ected (when period (j+2) is a normal period) or positively a¤ected (when period (j+2) is a discount period), as the reference price in period (j+2), rj+2 = pj+1, is higher, while the demand coe¢cient, xj+2 = ±xj+1 +¸dj+2 = ± 2 xj + ±¸dj+1 +¸dj+2, is the same when dj+1 and dj+2 are …xed.
With the above argument, it can be concluded that in any equilibrium p nn i = pm for every i, and p nd i = pm for every i. Consequently p r = p m , as p r is the moving average of p n i .

B Proof of Proposition 2
Proof. We need to solve the dynamic programming problem (3). Let g = ± n (x +¸d). Then there is a …rst order condition associated with the choice variable d: There is an envelope equation associated with the state variable x: Use the above two equations to eliminate V 0 (g) to get: Now use variable g = ± n (x +¸d) to substitute x in equation (13) to get the derivative of the value function evaluated at g: Plug (14) into (11), and there emerges the implicit function that characterizes the optimal choice of d: Using ' and Á speci…ed in the proposition, (15) becomes Notice that¸; x; Á; y > 0, but ' could be either positive or negative. Equation (16) fully characterizes the interior solution d ¤ . Although in general the expression has two roots, only d ¤ given in the proposition is optimal due to the second order condition H 0 (d) < 0.
One can easily verify that d ¤ > 0. When x 2¸( '¸2y)=Á 2 , the term within the square root is non-negative so that d ¤ is well de…ned. Otherwise H(d) is positive for every d, and d should take the highest possible value, namely pm.
For the requirement that d ¤ · pm, observe that d ¤ depends on z, which is (¹ ¡ c)=2, while pm = (¹ + c)=2. For any p m , z can be made arbitrarily small so that d ¤ will be close to zero, and d ¤ · p m will always be satis…ed.

C Proof of Proposition 3
Proof. In any steady state x = g(x; d(x)) = ± n (x +¸d). Therefore By plugging (17) into the expression H(d) = 0 in equation (15), the steady state x and d can be found as shown in the proposition. Notice that d depends only on z = (¹ ¡ c)=2 and therefore is independent of the requirement d · pm = (¹ + c)=2. For simplicity, let's ignore the unlikely case where d > pm.
Recall from Proposition 2 that in order for d = d ¤ , x 2 must be no less than ('¸2y)=Á 2 . It turns out that this requirement is always satis…ed at x = x: Expression H(d) = 0 of equation (16) fully characterizes the optimal choice d ¤ . There is a one-for-one mapping between the state variable x and the choice variable d ¤ . From H(d ¤ ) = 0, x can be expressed as a function of d ¤ : Consequently the state variable of the next cycle, g, can also be expressed as a function of the choice d ¤ in the current cycle: It is clear from previous discussion of comparative statics that d ¤ is decreasing in x: x 0 (d ¤ ) < 0. So At (x; d), the slope of g(d ¤ ) is negative: n 1 ¡ ± n ¡2 + 2¯n± n ¡¯n± 2n 1 ¡¯n± n < 0 The two curves x(d ¤ ) and g(d ¤ ) cross each other at (x; d), at which point the slopes of both curves are negative. d ¤ is monotonically decreasing in x. Corresponding to x = 0, there is a maximum d: d = dm = p ¡(y='). In general g(d ¤ ) is U-shaped. g(d ¤ ) achieves maximum slope at dm: The slope g 0 (dm) could be either positive or negative. However, g(dm) is never greater than x: The dynamics of the system are represented by the transition from x(d ¤ ) to g(d ¤ ). The two variables are brought together by d ¤ as a linkage. Figure 2 shows the only two possible cases of the dynamic system: g 0 (dm) < 0 and g 0 (dm) > 0.
(1) g 0 (d m ) < 0. Both x(d) and g(d) are monotonically decreasing in d, and x(d) has a steeper slope than g(d) for any valid d. Starting from any x1, the …rm chooses d1; which is found on curve x(d). Then corresponding to the same d1, g1 is found from the g(d) curve. The transition is a mapping from x1 to g1. In the next round g1 becomes x 2 and the whole process repeats itself. It is clear from the picture that the system will monotonically converge to (x; d).
(2) g 0 (dm) > 0. Starting from any x, the system again converges monotonically to (x; d). Proof. The objective function is given in equation (6), and the choice variable is s. The …rst order condition is With G and u(s) de…ned as in the proposition, the above equation turns to be F = Gu(s ¤ ). u 0 (s) is always negative for any s 2 (0; 1), so the second order condition is always satis…ed. Figure 5 shows u as a function of s. It is clear from the picture that the optimal s (and thus the optimal n) is found at the intersection of u(s) and F G = F 2¸z 3 (1 ¡ ± 2 ) 3 2 . When F=G > 2 9 p 3 = 0:385, there is no intersection () ± n ¤ = s ¤ = 0 () n ¤ = 1. The fact that n¸2 (remember that discounts are never o¤ered in consecutive periods) leads to the other boundary result in (iii).
Since n ¤ is decreasing in s ¤ , it is straightforward to verify from the picture the comparative statics.

E Proof of the Lemma
Proof. Regarding equation (8), for …rm a, there is a …rst order condition with respect to the choice variable da: (y ¡ d 2 a ) ¡ 2(xa +¸da ¡ µdb)da +¯n · ± n¸@ V @x a (ga; gb) ¡ ± n µ @V @x b (ga; gb)¸ (18) There are two envelope equations, corresponding to the two state variables: @V @xa (xa; x b ) = (y ¡ d 2 a ) +¯n± n @V @xa From equations (19) and (20), express the two partial derivatives evaluated at (ga; g b ) in terms of those evaluated at (x a ; x b ), and plug the results into equation (18), Using ga to substitute xa in the right hand side of the above equation, we get the left hand side expression evaluated at (g a ; g b ):¸@ V @xa (ga; g b ) ¡ µ @V @x b (ga; g b ) = 2¸d 2 a + 2da± n (xa +¸da ¡ µd b ) ¡ 2µdad b (22) Plug (22) into equation (18), and the best response function for …rm a emerges as an implicit function of da and d b :¸¡ ¡3 + 2¯n± n + 2¯n± 2n ¢ d 2 a + 2µ which is the expression in the Lemma. From the setup of the model it is easy to see that monopoly is a special case of duopoly competition where µ = 0. As it turns out, plug µ = 0 into the above expression, the result is exactly the characteristic equation in monopoly, equation (15).
When x a is switched with x b in equation (23), and at the same time d a is switched with d b , …rm b's best response function will emerge as: F Proof of Proposition 5 Proof. We have already shown that the system always converges to a unique steady state. The steady state should be characterized by four equations: the two best response equations (23) and (24), and two steady state transition equations: From (25) we solve Take the di¤erence between equations (23) and (24), and plug in (26), we have: therefore d a = d b . Consequently x a = x b . Drop the subscript. Plug (26) into either (23) or (24), we get the steady state discount and demand in the proposition. Notice that the denominator in the expression ¾ =¸(3 ¡ ± n ¡ 2¯n± n ) ¡ 2µ(1 ¡¯n± n ) >¸(3 ¡ ± n ¡ 2¯n± n ) ¡ 2¸(1 ¡¯n± n ) =¸(1 ¡ ± n ) > 0 so that the expression is well de…ned.

G Proof of Proposition 6
Proof. We know by direct observation that d is increasing in µ.