Price setting in a differentiated-product duopoly with asymmetric information. about demand * Miguel Ángel Ropero García

Transcription

1 Price setting in a differentiated-product duopoly with asymmetric information about demand * Miguel Ángel Ropero García Department of Applied Economics University of Malaga El Ejido 6, Malaga Spain This article explores price-setting in a two-period duopoly model in which only one firm is uncertain about the degree of product differentiation and the intercept of the demand curve. In this context, the informed firm must choose whether to anticipate to its rival obtaining more profits in the first period or to fool its rival into thinking that the demand is high to obtain more profits in the second period. We find that the best informed firm sets a higher price in this duopoly context than in a monopoly market when the demand uncertainty of the uninformed firm is sufficiently high. Key words: Demand uncertainty, degree of substitutability, asymmetric information, Perfect Bayesian Equilibrium. JEL Codes: D43, D83, L Introduction * I am very grateful to Miguel Ángel Meléndez for his helpful advice. I also wish to thank Gerad Llobet for his useful comments.

2 Standard economic theory has been used to study price and quantity decisions by firms in oligopolistic markets based on restrictive assumptions, such as homogeneous products, symmetric firms and perfect information. However, there are a number of markets where these assumptions are not fulfilled. In particular, many firms try to compete with their rivals by differentiating their products and some firms have more information about market conditions than others. In this context, we study a duopoly model where each firm offers a differentiated product and sets its price simultaneously in two periods. One of the competitors is aware of the market demand conditions whereas the other one is not aware of the intercept of the demand curve and the degree of substitutability between products. For this reason, the uninformed firm is uncertain about the size of the market and about the distribution of consumers valuations. Each firm can observe the prices of both products and its own demand quantity achieved in the first period before choosing its price in period 2. Therefore, the uninformed firm updates its knowledge on the demand parameters in the second period based on market data observed in period 1. The use of these assumptions can be justified in several contexts. For instance, when a new firm enters a market offering an alternative brand, it has less information about market demand than the incumbent firm. Furthermore, e-commerce has provided customer data that can be used in developing and implementing informed and sophisticated pricing strategies (Elmaghraby and Keskinocak, 2003). Despite this availability of data, some firms have better information systems than others and can take advantage of their greater experience in online markets. On the basis of the assumptions of the model, if the informed firm increases its price in the first period when the degree of substitutability between products is higher than the expectation of its rival, the demand quantity of the uninformed firm will increase more

3 than it expected. Hence, this uninformed firm would believe that the demand intercept is higher than expected in the first period and it would increase the price in the second period. However, if the informed firm increases its price in period 1 when the degree of substitutability between products is lower than that expected by its rival, the demand quantity of the uninformed firm will increase less than it expected and it will update a lower demand intercept and reduce its price in the second period. Thus, this model can explain the strategies of some firms in certain markets. For example, Pels and Rietveld (2004) studied the pricing policies of airlines on the London-Paris route and observed that some of them lower their fares when other competitors raise theirs, whereas others might follow the price movements of a competitor. Nonetheless, they can only turn to the random nature of demand in this context to explain their results. The model also shows that the higher the uncertainty about demand faced by the uninformed firm, the higher the price it set in period 1. Nevertheless, the effect of demand uncertainty on the price set by the informed firm depends on the degree of substitutability between products. We distinguish two cases. First, when the true substitutability between products is lower than that expected by the uninformed firm, the informed one has two juxtaposed incentives. On the one hand, it wants to decrease its price in period 1 to face a less competitive rival in the next period. On the other hand, it wants to increase its price in the first period to take advantage of its better information and surprise its rival. Second, when the true substitutability between products is greater than that expected by the uninformed firm, the informed one has the following incentives. On the one hand, it wants to increase its price in order to provoke an increase in the one set by the uninformed firm in the second period. On the other hand, the informed firm wants to decrease its price in the first period because this will surprise its rival. Thus, the effect of uncertainty as regards

4 market demand on the part of the uninformed firm on the price set by the informed one depends on which of these juxtaposed incentives is higher. Furthermore, if the uninformed firm s demand uncertainty is sufficiently high, the model predicts that the price set by the informed one in a duopoly market would be greater than the price it would set under a monopoly. Hence, this model can be added to the priceincreasing competition literature and allows us to explain some empirical puzzles. For example, Caves, Whinston and Hurwitz (1991) and Grabowski and Vernon (1992) provided evidence that the entry of generic products triggers higher prices for corresponding brand-name drugs in the U.S. pharmaceutical industry. Perloff, Suslow and Seguin (1995) concluded that the arrival of a new competitor raises prices in the anti-ulcer drug market. In addition, Regan (2008) found that the entry of generic drugs has a positive effect on branded-drug prices and a negative impact on prices of generics. If these different effects were related to the degree of substitutability between each type of drug and to the differences between firms in their knowledge about market conditions, the model presented would help to explain these results. The empirical evidence about priceincreasing competition is not only found in pharmaceutical markets. For instance, Ward et al. (2002) also found that the entry of new private labels raised prices of national brands in the food industry, and Thomadsen (2007) found that prices may be higher under duopoly competition than under monopoly in the fast-food industry. This article is organized as follows. The next Section reviews the related literature and discusses the contribution of this article to previous models. The third Section describes the model, whereas the fourth deals with the equilibrium in the second period, which allows us to derive some economic predictions. Section 5 deduces the Bayesian Perfect Equilibrium of the game and illustrates the main implications of the model. Section 6 determines the conditions under which the effect of competition on prices is positive or

5 negative, and finally, Section 7 summarizes the main conclusions. The Appendix includes the proofs of each Proposition obtained. 2. Literature Review The effects of demand uncertainty on expected profits, expected prices and capacity in competitive or monopolistic industries have been extensively analyzed (Oi, 1961; Dreze and Gabsewicz, 1967; Smith, 1969; Sandmo, 1971; Leland, 1972; Grossman et al., 1977; Dreze and Sheshinski, 1976). For example, Grossman et al. (1977) established that a monopolist would charge a higher price than is called for by static profit maximization because this results in a more informative market experiment. That is, by setting a higher price, the monopolist learns more about its demand function. Sandmo (1971) showed that the level of production of a competitive risk-averse firm is lower under demand uncertainty than under certainty about the market conditions. Nevertheless, the effect of a marginal increase of demand uncertainty on firms quantities or on market price is indeterminate in this model. Leland (1972) analyzed the behaviour of firms under uncertainty about demand and imperfect competition and also found that the risk-averse firm will produce less under demand uncertainty than it would under certainty, but it is a necessary condition that the principle of increasing uncertainty is fulfilled. However, if firms set prices instead of quantities, the effect of uncertainty on prices depends on the firms attitudes towards risk and on the shape of their cost curves. In particular, Leland (1972) showed that if marginal cost rises at a nondecreasing rate, the optimal price set by risk-neutral firms will be higher under uncertainty than under certainty. The opposite holds if marginal cost decreases at a nonincreasing rate. Finally, the introduction of uncertainty does not affect the price decision of the price-setting risk-neutral firm with constant marginal costs.

6 Later models have been used to study firms decisions under demand uncertainty in oligopolistic markets. For example, Gabszewicz and Poddar (1997) defined a two-stage game where firms offer homogeneous products and choose capacity in the first stage without knowing which level of demand is realized. In the second stage, firms choose output levels constrained by capacity decisions taken in the first stage. Under the assumption of a linear demand with a random intercept, they proved that there exists a symmetric equilibrium at which firms operate at excess capacity, when compared to the capacity they would choose in the certainty equivalent case. From a different perspective, Klemperer and Meyer (1986) studied firms preferences between quantity and price as strategic variables in the presence of demand uncertainty when firms offer differentiated products in a one-stage duopoly game. They suggested that three factors should be taken into account: the slope of marginal costs, the effect of the random disturbance on demand, and the curvature of demand. In particular, when marginal costs are steep enough, setting quantity is preferred because the ex post optimal quantity is relatively stable compared to the output that would result from setting a price and producing a sufficient quantity to meet market demand. However, setting prices is preferred in industries with flatter or decreasing marginal costs. Moreover, if demand uncertainty affects the reservation prices of potential consumers and marginal costs are constant, firms prefer setting quantity. Nevertheless, if there is uncertainty in the size of the market and marginal costs are constant, firms choose to set prices. Finally, Klemperer and Meyer show that the more concave the demand, the steeper the marginal revenue relative to demand, and the smaller the fluctuations in the ex post optimal quantity relative to the fluctuations in output under a set price. Hence, concavity of demand leads to a preference for stabilizing quantity and convexity of demand leads to a preference for setting prices.

7 More recently, Reisinger and Ressner (2009) analyzed a duopoly model with stochastic demand and differentiated products in which symmetric firms choose a strategy variable and compete afterwards. They found that in equilibrium the relative magnitude of demand uncertainty and the degree of substitutability determines firms choice of variable. In particular, they demonstrate that firms prefer setting prices if uncertainty is high relative to the degree of substitutability and commit to a quantity if the reverse holds true. Other theoretical research has shown that demand uncertainty can explain intra-firm and inter-firm price dispersion under imperfect competition (Prescott, 1975; Dana, 1999, 2001). Prescott s model assumes that there is a stochastic demand for homogeneous goods and all consumers are identical. He proved that the consequence of demand uncertainty is that each firm has a distribution of prices in equilibrium rather than a single price. Under similar assumptions, Dana (1999) demonstrated that price rigidities and demand uncertainty lead not only to inter-firm price dispersion, but also to intra-firm price dispersion in markets with perishable goods. Dana (2001) presented a model of competition in price and availability in which demand is uncertain and consumers choose where to shop given the firms observable prices and their expectations regarding the firms unobservable inventories. Under these assumptions, he showed that firms use higher prices to signal higher availability and that the model can explain inter-firm price dispersion in the video rental industry. Several articles have paid attention to dynamic competition under demand uncertainty. For example, Eden (1990, 2009) and Lucas and Woodford (1993) presented a model where symmetric firms gradually change prices over time in response to observed increases in cumulative aggregate sales. On the other hand, Riordan (1985) proposed a dynamic Cournot framework under the assumptions that demand shifts are positively serially correlated, risk-neutral firms never directly observe the position of the demand curve, and

8 firms never observe the previous outputs of rivals. Thus, firms do draw inferences about the position of the demand curve from past observations on prices. Consequently, the current output of a firm will influence the current market price, which will influence its rival s inferences about future demand curves. The future output decisions of rivals will respond accordingly. Essentially, a firm perceives that an increase in its output will lower the current market-clearing price, which will cause rival firms to think that the demand curve has shifted down, and hence induce them to lower their outputs in the future. Thus, this model also analyses the firms perceived abilities to fool rivals into thinking that the demand curve is higher or lower than it really is. Nevertheless, Riordan did not take into account product differentiation; also, both firms have the same information about the market conditions in his model. Similarly, Mirman, Samuelson and Schlee (1994) presented a model where two firms produce a homogeneous product in two periods and face a demand function with a parameter that is unknown to both firms. As in Riordan s (1985) model, each firm chooses a level of production in each period, but both firms observe the market price as well as the industry output in period 1 before choosing the level of production in period 2. This information is used to update prior beliefs about demand, though there is a random noise in the demand function which prevents firms from perfectly observing the state of demand. Once again, firms adjust their first-period outputs sacrificing current profits but affecting the informational content of the market outcome in order to increase future expected profits. In this model, firms affect the informativeness of the commonly observed price signal and the degree to which its rival is likely to update expectations, but not the direction. Moreover, if the density function of the unknown parameter satisfies the monotone likelihood ratio property, the higher the market price, the higher the posterior beliefs that the state of demand is good.

9 Research is scarce on oligopolistic markets where firms offer differentiated products and face demand uncertainty about the degree of substitutability between products. Harrington (1992; 1995) and Aghion, Espinosa and Jullien (1993) analyzed how learning behaviour can substantially modify the outcome of competition in an oligopolistic industry facing demand uncertainty. They considered the case of a symmetric duopoly game where firms offer differentiated products and compete in prices. These firms have imperfect information about market demand and learn through observing the volume of their sales. These models focus on one specific aspect of demand uncertainty, namely, the substitutability of the two products offered by the two firms. Through learning by experimentation, the degree of substitutability of the two products gradually become more apparent. Harrington (1992) suggested that firms set prices above their marginal costs because each firm prefers not to undercut its rival s price to maintain ignorance about the degree of substitutability between products. Likewise, Aghion, Espinosa and Jullien (1993) showed that information about the elasticity of substitution between competing goods is best acquired if firms set different prices. Similarly, Harrington (1995) developed a price-setting duopoly model in which firms are uncertain about the degree of product differentiation. Firms learn by observing the difference in the quantities that are demanded given the difference in their prices. In this case, Harrington showed that firms create more price dispersion than is predicted by static profit maximization to create a more informative market experiment when the products are highly substitutable. However, firms compress their prices to reduce how much information is generated about product substitutability in markets for highly differentiated products. Once again, all these models consider symmetric firms. More recently, Chen and Riordan (2008) presented a discrete choice model of product differentiation in which consumers values for two substitute products have an arbitrary symmetric joint distribution. Each firm produces a single product and these authors found that the symmetric duopoly price can be higher than, equal to, or lower than the monopoly

10 price. In their model, there are two opposite economic effects. First, a duopoly firm has an incentive to reduce prices below the monopoly level to increase its market share. On the other hand, under product differentiation, a duopoly firm s demand curve may be steeper than the monopolist s, because consumers have a choice of products in the duopoly case and therefore are less motivated to purchase the duopolist s product in response to a price cut. The steeper the duopolist s demand curve, relative to the monopolist s, the greater the incentive of the duopolist to raise prices above the monopoly level. Chen and Riordan showed that a decreasing hazard rate over a relevant range is sufficient for price-increasing competition. More intuitively, duopoly competition raises prices if consumers preferences for the two products are sufficiently diverse and negatively correlated. Greater negative correlation between consumers preferences means that the two products are more differentiated. Thus, price competition is lower with more product differentiation and profits are higher. However, Chen and Riordan did not take into account the effect of dynamic interaction between firms because they presented a one-period model. In contrast to previous theoretical literature about oligopolistic markets with firms offering differentiated products under demand uncertainty, our model introduces asymmetric information in a dynamic game where two firms simultaneously set their prices in two periods, but where only one firm is aware of the degree of substitutability between products and of the intercept of the demand curve. As a result of this informational advantage, the best informed firm will choose its price in the first period to influence its rival s expectations about the demand parameters in the next period. This potential effect has interesting implications on price competition between firms. Unlike previous models about learning by experimentation in oligopolistic markets with differentiated products (Aghion, Espinosa and Jullien, 1993; Harrington, 1995), the amount of information provided by market data cannot be influenced by the firms behaviour in

11 the model presented here. The theoretical literature has used two types of strategies to introduce learning by experimentation. For example, Aghion, Espinosa and Jullien (1993) considered that firms choose their prices in each period without observing a parameter of the demand functions, which can only take two possible values, and a common random shock, which has a known distribution of probability. Under these assumptions, each firm can infer the true value of the unknown parameter in the second period when the volume of its sales observed in the first period takes values within a certain interval. Harrington (1995) proposed a similar duopolistic structure, but the unknown parameter can take values from a continuous interval. This article shows that each firm can use the difference between its own sales and that of its rival in light of the price differential in the previous period as an unbiased predictor of the unknown parameter. In our model, the uninformed firm cannot observe its rival s volume of sales in the previous period and each of the two unknown parameters can take values from a continuous interval. Under these assumptions there will be always a geometric locus of combinations of both unknown parameters consistent with market data observed by the uninformed firm in the first period. Hence, the interaction between experimentation and competition is not considered in this model. 3. The model Market structure. Consider a duopoly lasting two periods. There are 2 risk-neutral firms in this market: firms i and j. Each firm has a constant unit cost of production equal to zero, up to an exogenous capacity of k, which is not binding. The outputs of firms i and j at date t are denoted by. Each firm simultaneously chooses its price in each period. So denotes the prices of firms i and j at date t. Moreover, each firm sells a differentiated product. Over relevant ranges of output, the following system of linear inverse demand curves is assumed:

12 (1) (2) where t = 1, 2; ; and. is the slope of the inverse demand curve and measures the substitutability between both products. If, both products are perfect substitutes. It is assumed that firm i can observe a and θ before choosing its price in each period, but firm j does not observe them before deciding its price in each period 1. In period 1, firm j considers that the values of a and θ are drawn from the distribution functions F(a) and G(θ), where and, where and. As a higher value for θ is associated with higher cross-price elasticity, the substitutability of firms products is increasing in θ. To avoid unnecessary complications, it is a requirement that support of θ is sufficiently small such that no equilibria emerge in which a firm sells a negative quantity. The distribution functions, F(.) and G(.), are assumed to be twice differentiable on and, respectively, with associated density functions f(a) and g(θ). Moreover, firm j assumes that the random variables, a and θ, are independent and the distribution function of θ is symmetric around 1, which is the average of θ. Additionally, firm j observes the prices chosen in period 1,, and its demand quantity, 1 The system of linear demand curves specified is similar to that considered by Klemperer and Meyer (1986). They analyzed the effect of two types of demand uncertainty on the strategic variable chosen by duopolists. First, they only included a common random additive shock in the intersect of the demand curves. Second, they specified a system of linear demand curves with fixed vertical intercept but with common multiplicative uncertainty about the slope and the degree of substitutability between products. The demand functions considered in our model simultaneously includes both types of uncertainty in the same way as Reisinger and Ressner (2009). As Klemperer and Meyer (1986) showed, a rotation about a fixed vertical intercept for any demand curve represents a change in the total size of a market in which the distribution of consumers reservation prices remains unchanged. On the other hand, they pointed out that a rotation about a fixed horizontal intercept represents a particular type of change in the distribution of reservation prices, with the total size of the market remaining unchanged. Finally, a vertically additive shift of the demand function is an intermediate case involving a change in both the size of the market and the distribution of the reservation prices. Reisinger and Ressner (2009) argued that introducing a shock to the intercept and a shock to the slope is more relevant in reality because uncertainty usually affects both market size and reservation price distribution.

13 , and updates its beliefs about the demand parameters, a and θ, before choosing its price in period 2. It is assumed that, (3) (4) (5) (6) (7) (8) (9) Where denotes the firm j s expectations about the demand parameters. Equation (5) is derived from the definition of the variance of θ, considers that the distribution of θ is symmetric,. As we have assumed that firm j and equation (6) is a direct consequence of this assumption. Equations (7)-(9) are derived directly by the independence assumption of a and θ. As firm i has a better information system than its rival, the former knows the true values of a and θ. However, firm j makes its price decision with uncertainty about the demand curve. In each period, price decisions are made simultaneously and independently. Firm j never observes the realizations of a and θ, nor does it ever directly observe the output of its rival, but it does directly observe its own quantity sold and the price of its rival in period 1 before choosing the price in period 2. Hence, firm j is imperfectly informed about the market environment when making its price decision. For example, when firm j sets a higher price,

14 it cannot distinguish whether a high demand quantity is caused by a higher demand intercept, a, or by a lower substitutability degree (demand elasticity), θ. Equilibrium: Definition and interpretation. On the one hand, a strategy for firm i is the specification of a price in period 1,, and the specification of a function determining the period-2 price from its rival s realized quantity and from prices in period 1,. Firm i s price in period 2 depends on its rival s demand and on both firms prices in period 1 because firm j forms an expectation about the future demand curve from its realized demand and prices in the last period. Thus, firm i takes into account this expectation process when it chooses its price in period 2. This is possible because firm i knows a and θ and thus can infer the demand quantity of firm j in period 1 by the demand curves, given a, θ, and, before choosing its price in period 2. On the other hand, the strategy for firm j is the specification of a price in period 1,, and the specification of a function determining the period-2 price from prices and from firm j s quantity in period 1,. Firm j never observes its rival s quantity sold in period 1 before setting the price in period 2. This specification of a strategy is appropriate for the informational assumptions of the model. In period 1, firm i knows the demand curves, but firm j must choose its price with no direct information about market conditions. The only other information that firm j possesses in period 2 is an observation of its own realized quantity and the prices set in period 1. An equilibrium is defined by a strategy that maximizes the discounted expected profits of each firm, given that its rival is following this strategy. Definition. is an equilibrium strategy for firm i if and only if it solves

15 where is the discount factor, which is common for both firms. Similarly, is an equilibrium strategy for firm j if and only if it solves The expectation operator in the definition of equilibrium, E j [.] is defined with respect to induced probability over by firm j. The concept of solution is a Perfect Bayesian Equilibrium. This is usefully interpreted as a statement that the uninformed firm maximizes its expected discounted profits given its specified beliefs concerning the behaviour of firm i and the demand conditions. Moreover, the informed firm chooses its prices to maximize its discounted profits and these decisions are consistent with firm i s beliefs about its rival s decisions. Finally, firm j s beliefs are consistent with the decisions of firm i in the equilibrium. It is worthwhile examining the particular nature of these beliefs in more detail. At the beginning of the second period, firm j has observed its own quantity in the previous period,, and the prices set,,. Thus, this firm updates its expectations about the demand curves given this new information. However, it cannot distinguish directly between how the price decisions or how the demand conditions have contributed to the formation of the realization of its demand. In the language of econometricians, firm j faces an identification problem. Given these beliefs, both firms choose prices in a sequentially rational fashion. At period 2 each one chooses a price that maximizes expected period-2 profits, conditional on its observation of the period-1 prices and its own demand quantity. At period 1 each firm

16 chooses a price that maximizes expected discounted profits, given its period-2 decision rule. This behaviour defines an equilibrium strategy. 4. Equilibrium in the second period game The first step of the analysis is to characterize the second-period decision problem of each firm. On the one hand, firm i maximizes its profits in the second period, (10) However, as firm j does not know the demand conditions, it expects that firm i is facing the following problem, (11) On the other hand, firm j maximizes its expected profits in the second period, (12) We assume that the demand equations (1) and (2) are fulfilled. Solving problems (10), (11) and (12), we can obtain the reaction functions of firms i and j, respectively. Specifically, the first-order condition of problem (11) allows us to determine the reaction function of firm i expected by firm j, which must be substituted in the reaction function of firm j to obtain its price for period 2 in equilibrium. Then, this price of firm j is included in the true reaction function of firm i and we obtain its price for period 2 in equilibrium. We can then analyze the following Proposition: Proposition 1. Let, be the equilibrium strategies in the second period for both firms. Then,

17 (i) (13) (ii) (14) The solution of the maximization of the expected profits in period 2 and the demonstration of this Proposition are included in the Appendix, but its intuition is very simple. The degree of substitutability between products offered by both firms increases with θ. If firm i increased its price in the first period when θ is higher than the expectation of its rival, the demand quantity of firm j would increase more than it expected. Hence, firm j would believe that the demand intercept is higher than expected in the first period and it would increase the price in the second period. However, if firm i increased its price in period 1 when θ is lower than that expected by its rival, the demand quantity of firm j would increase less than it expected and it would update a lower demand intercept and would reduce its price. Similarly, if firm j increased its price in the first period when θ is greater than 1, its demand quantity would reduce more than it expected. Then, firm j would infer that the demand intercept is lower than it thought and it would reduce its price in period 2. Firm i would anticipate its rival s behaviour and would reduce its price in the second period. Finally, if firm j increased its price in the first period when θ is less than 1, its demand quantity would reduce less than it expected. Thus, firm j would update its expectations about the demand intercept and would increase its price in period 2. Knowing this, firm i could increase its price in the second period. 5. Perfect Bayesian Equilibrium

18 The following step is to analyze the decisions of firms in the first period, given the expectation process of firm j and given the decision rules in the second period. The problem of firm i in the first period will be (15) where is the profit obtained by firm i in the second period when both firms choose their equilibrium prices and δ is the discount factor, which is common for both firms. To obtain the first-order condition for this problem, we can apply the envelope theorem, that is, (16) where is the demand quantity for firm i in period 2 when both firms fix the equilibrium prices. If we substitute (A.1) from the Appendix for t=1 in (15) and use (16), the following first-order condition for firm i will be obtained, (17) We can calculate and from (A.14) and (A.10) (shown in the Appendix) and can be substituted by the expression (A.2) from the Appendix for t=1. After these substitutions and some operations, the reaction function of firm i in the first period will be, (18) where,

19 However, firm j expects that firm i faces the following maximization problem, (19) Once again, the following version of the envelope theorem can be applied to obtain the first-order condition for this problem, (20) Then, substituting (4) and (A.1) from the Appendix for t=1 in (19) and using (20), the firstorder condition for firm i expected by firm j will be, (21) By calculating and from (A.14) and (A.10) (shown in the Appendix), substituting by the expression (A.2) from the Appendix for t=1 and using (3)-(9), the reaction function of firm i in the first period expected by firm j can be expressed as, (22) where,

20 Finally, the problem of the risk-neutral firm j in period 1 will be, (23) where is the profit obtained by firm j in the second period when both firms choose their equilibrium prices. Then, substituting (4) and (A.2) from the Appendix for t=1 in (23) and using the envelope theorem, the first-order condition for firm j will be, (24) By calculating and from (A.14) and (A.10), respectively, (shown in the Appendix), substituting by the expression (A.2) from the Appendix for t=1 and using (3)-(9), the reaction function of firm j in the first period can be expressed as, (25) where, Then, if the reaction function of firm i expected by firm j from equation (22) is substituted in (25), the equilibrium price chosen by firm j in the first period is the following,

21 (26) By including this price in the true reaction function of firm i from equation (18), its equilibrium price set in period 1 can be expressed as, (27) Using (27), we can obtain the effect of an increase in demand uncertainty faced by firm j on the price set by firm i through the following Proposition, Proposition 2. depends on θ in the following manner, i. if (28) ii. if (29) iii. if (30) iv. if (31) v. if (32) vi. if or (33) The proof of this Proposition is included in the Appendix, but Figure 1 shows its intuition. The effect of an increase in demand uncertainty of firm j on the price set by its informed rival in the first period depends on the degree of substitutability between products, that is, it depends on θ. We can distinguish several cases. First, when the true substitutability between products is lower than that expected by firm j, the informed competitor has two opposite incentives. On the one hand, firm i wants to

22 decrease its price in period 1 to provoke an increase in the price of its uninformed rival in the second period as Proposition 1 shows. On the other hand, firm i wants to increase its price in the first period taking advantage of its better information and surprising its rival. As Figure 1 shows, if the degree of substitutability between both products is very small (see (28) in Proposition 2) or θ is very close to the uninformed firm s expectation (see (30) in Proposition 2), the second incentive dominates the first one and the higher the demand uncertainty of firm j the higher the price set by firm i in the first period. Nevertheless, if the degree of substitutability between products is higher but sufficiently far from the uninformed firm s expectation, as in part (29) of Proposition 2, the first incentive dominates the second one, and the price set by firm i will decrease with the demand uncertainty of firm j. Second, when the true substitutability between products is greater than that expected by the uninformed competitor, then the informed one has the following incentives. On the one hand, it wants to increase its price to obtain a rise in the price set by firm j in the second period as Proposition 1 shows. On the other hand, firm i wants to decrease its price in the first period because this will surprise its rival and firm i can take advantage of it. As Figure 1 shows, if the degree of substitutability between both products is sufficiently high and far from the uninformed firm s expectation (see (32) in Proposition 2) or θ is too low (see (30) in Proposition 2), the first effect dominates the second one, and the higher the demand uncertainty of firm j the higher the price set by firm i. Nevertheless, if the degree of substitutability between products is small but sufficiently far from the uninformed firm s expectation, as in part (31) of Proposition 2, the second effect dominates the first one. Thus, the price set by the informed firm decreases with demand uncertainty of its competitor. Finally, the first incentive and the second one are equal in absolute values in the extreme cases considered in part (33) of this Proposition.

23 The next step is to analyse the effect of an increase in demand uncertainty faced by firm j on its price set in the first period. Using (26), we find that this effect is unambiguous, as the following Proposition shows, Proposition 3.. (34) The Appendix includes the proof of this Proposition. When the uninformed firm is a monopolist in the market, the uncertainty about θ does not affect the price set by this firm because changes in θ only generate isoelastic demand shifts. However, under the presence of an informed firm and the possibility that this firm can influence the expectation about the demand parameters for the next period, the higher the uncertainty about θ, the higher the price set by the uninformed firm because it expects to face, on average, a less competitive rival. Another question we can answer concerns which of the firms sets a lower price in each period when the uninformed firm has an unbiased expectation of the demand parameters, that is, when and. In this case, firm i sets a lower price than firm j in the first period because it takes advantage of its better information, but both firms set the same price in the second period because firm j solves its disadvantage inferring the demand parameters from the information obtained in period 1. The following Proposition summarizes this last statement. Proposition 4. If and, and. (35) where and are the prices set by both firms in period t when firm j has unbiased expectations about the demand parameters. The proof of this Proposition can be found in the Appendix. Despite this Proposition, the informed firm can set a higher price than the uninformed one in the first period and both firms can set different prices in period 2 if the

24 expectations about the demand parameters of firm j are biased. However, this Proposition shows the incentives of the informed firm to set a lower price than its rival to take advantage of its better information when the uninformed firm has unbiased expectations about demand parameters. 6. The effect of competition on prices Finally, it is interesting to compare the price set by the informed firm in period 1, when firm j has unbiased expectations about demand parameters, to the price set by this firm in a monopoly market. This comparison can be helpful to understand the consequences of the entry of a firm in a market under certain circumstances. For example, when a drug patent expires, another rival can enter the market to sell a similar product. In this case, the incumbent usually has better information about the market demand because it has more experience in the market. Another example can be found in the airline market because there are certain routes with only one operator due to legal or economic barriers in the market. However, another airline could enter this route when the market is deregulated or when the demand increases sufficiently, but the incumbent would have more information about the demand parameters. We can then obtain some useful predictions about prices under these market conditions, comparing the price set by firm i in this model when it is the only firm in the market to the price set by this firm when it has an uninformed rival, that is, firm j. Lemma 1. There exists a threshold,, for such that if and, when. (36)

25 where is the price set by an informed monopolist in the first period under the demand conditions given by (1) and (2). If we denote the total market demand in each period as, it is easy to prove that substituting for and and for in the linear demand curves (1) and (2) and solving the resulting maximization problem of the monopolist. As is greater than when, does not depend on and when from Proposition 2, the demonstration of Lemma 1 is obvious. The result obtained in Lemma 1 has a direct implication which can suggest an alternative explanation to some empirical puzzles. Even when a new entrant (the uninformed firm) has unbiased expectations about the demand parameters, the incumbent will set a higher price in the first period after this entry than before if the new entrant s uncertainty is sufficiently high. 7. Conclusions This article presents a two-stage game model where two firms offer differentiated products and one of them faces demand uncertainty, which affects both the intercept and the slope of the demand curve. In each period, each firm sets its price simultaneously and noncooperatively. In contrast to previous theoretical literature, only one firm faces demand uncertainty because its rival knows the market conditions before choosing its price in each period. In the first stage of the game, the uninformed firm bases its price decision on its prior beliefs about demand parameters. We assume that this firm forms its beliefs from a symmetric distribution of the slope shock and considers that the distribution functions of this shock and the demand intersect are statistically independent. The uninformed firm can observe the prices set for both products and its own demand quantity realized in the first

26 stage before choosing its price in the second period. This additional information allows it to update its beliefs about demand parameters before making decisions in the second period, and the informed firm knows this. Using the concept of Perfect Bayesian Equilibrium, it is shown that the informed firm has an incentive to increase (decrease) its price in the first period when the degree of substitutability between products is sufficiently high (low) to fool its rival into thinking that the demand intercept is high. If this strategy is successful, the informed firm will face a less competitive rival in the second period. However, this informed firm has an opposite incentive in the first period due to its rival s demand uncertainty. In particular, the informed firm wants to surprise its rival decreasing (increasing) its price in period 1 if the degree of substitutability between products and the elasticity of demand are sufficiently high (low). We then show that the effect of an increase in demand uncertainty of the uninformed firm on the price set by its informed rival in the first period depends on the degree of substitutability between products. In particular, if the true substitutability between products is very low or very high, or if it is very close to that expected by the uninformed rival, the higher the demand uncertainty of the uninformed firm the higher the price set by the informed one in the first period. Nevertheless, if the degree of substitutability between products is low or high but sufficiently close to the uninformed firm s expectation, the price set by the informed firm decreases when the demand uncertainty of its rival increases. On the other hand, the consequence of an increase in demand uncertainty on the price set by the uninformed firm in the first period is unambiguous in this model. In particular, when the demand uncertainty rises, its price set increases.

27 Finally, this model adds an alternative explanation to the recent price-increasing competition literature. In fact, we show that even when a new entrant has unbiased expectations about the demand parameters, the incumbent, which is better informed on market conditions, will set a higher price after this entry than before if the uncertainty about demand is sufficiently high. We end by pointing out some limitations of the model. Firstly, the results of this article may depend on the functional forms of the demand and cost curves. Similarly, the assumptions of independence between unknown demand parameters can be too restrictive in some contexts. Furthermore, some firms can have imperfect but better information about market conditions than others. In this case, the quality of private information about demand parameters would be higher for the best informed firms. Although the robustness of the predictions of this model to these alternative assumptions is an open question for future research, it could help to explain pricing policies in some contexts and adds more arguments to the increasing-price competition literature. In particular, the implications of the model can be fulfilled in markets where one firm has much more experience than the new entrant and the latter has imperfect information about the demand conditions. Appendix Proof of Proposition 1. We begin with the analysis of the equilibrium in the second period, following the backward induction method. As firm i knows the market conditions, it chooses its price to maximize its profit in the second period, From the system of linear inverse demand curves given in equations (1) and (2), we obtain

28 (A.1) (A.2) where t = 1, 2. When t = 1, we can rearrange the equation (A.2) to obtain a depending on and, (A.3) As firm j observes and before it chooses its price in the second period, it can update its beliefs about the demand conditions using (A.3). Substituting a, obtained from equation (A.3), and using (4) and equation (A.2) when t=2, the problem of the risk-neutral firm j in the second period will be the following: (A.4) The first-order condition is (A.5) Then, the reaction function of firm j in the second period is (A.6) Firm i knows that firm j behaves according to this reaction function. As firm j does not observe the demand parameters, its expectations about firm i s reaction function have to be considered. Then, using (4), (A.1) and (A.3), firm j expects that the problem of its rival in period 2 will be (A.7)

29 The first-order condition that firm j will expect for this firm is (A.8) Then, the reaction function of the firm i expected by firm j is (A.9) If we substitute from equation (A.9) into (A.6), we obtain the equilibrium price of firm j for the second period,, (A.10) Firm i can observe a and θ. If we use (A.1) when t=2, the true problem of this firm will be (A.11) The first-order condition of firm i will be (A.12) Thus, the reaction function of firm i will be (A.13) Firm i knows that its rival chooses its price following equation (A.10). Thus, if from this equation is included in (A.13), the equilibrium price of firm i is obtained (A.14)

30 As firm i knows a, θ and before choosing its price in the second period, it can infer using the system of linear inverse demand curves. From (A.10), we can analyze the reaction of the uninformed firm s price in the second period due to a change in the informed firm s price in the first period given, (A.15) Equation (A.2) when t =1 is used to calculate given and this is included in (A.15). Hence, we have the following result, (A.16) As β>γ, part (i) of Proposition 1 has been demonstrated. We can now calculate the reaction of the informed firm s price in the second period due to a change in the uninformed firm s price in period 1 from equation (A.14) given, (A.17) Using equation (A.2) when t=1 to obtain given, we obtain (A.18) As β>γ, part (ii) of Proposition 1 has been demonstrated. Proof of Proposition 2. First, it is necessary to obtain the derivative of the price set by firm i in period 1 with respect to the variance of θ from (27). After some simplifications, the following expression is obtained,

31 (A.19) where, As, then,,,,,, and. Thus, the sign of depends on the sign of and solving it for zero,. If we substitute by X and by X 2 in the last expression (A.20) It is clear that the solutions of this equation are the following: (A.21) (A.22)

32 If we substitute these values of X in, we obtain the values of θ, which satisfies equation (A.20), that is, (A.23) (A.24) (A.25) (A.26) Thus, part (33) of Proposition 2 has been proven. It is clear that the sign on the left-hand side of (A.20) is positive when or or and negative when or. Hence, Proposition 2 has been proven. Proof of Proposition 3. The effect of an increase in the variance of θ on the optimum price set by firm j in period 1 is directly derived from (26). After some simplifications, the following result is obtained, (A.27) The denominator of (A.27) is positive and the numerator is compounded by three terms. The last two are positive. The sign of the first term depends on the sign of, and this expression is greater than zero because. Thus, Proposition 3 has been proven.

33 Proof of Proposition 4. To begin, we obtain the prices set by both firms when and from (26) and (27). We call these prices and because they are the prices set by both firms when the uninformed one has an unbiased expectation about the demand parameters. Under these conditions, firm i will set the following price, (A.28) and the price set by firm j will be (A.29) The denominator of in (A.28) is equal to the denominator of in (A.29) multiplied by. Thus, the following difference between both prices can be obtained, (A.30) As, and,,,,, then this difference is negative. Thus, the first part of Proposition 4 has been proven. To finish the proof of this Proposition, when and, it is easy to show that (A.31) References Aghion, P., Espinosa, M. P. and Jullien, B. Dynamic duopoly with learning through market experimentation. Economic Theory, Vol. 3 (1993), pp