A Simple Model of Pricing, Markups and Market. Power Under Demand Fluctuations

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1 A Simle Model of Pricing, Markus and Market Power Under Demand Fluctuations Stanley S. Reynolds Deartment of Economics; University of Arizona; Tucson, AZ Bart J. Wilson Economic Science Laboratory; University of Arizona; Tucson, AZ March 2001 Preliminary Draft: Please do not cite without authors ermission. Abstract: This aer formulates and analyzes a relatively simle model of duooly ricing under conditions of fluctuating demand. Price-setting firms roduce a homogenous roduct subject to increasing marginal cost of roduction. We focus on static Nash equilibria in rice choices and examine the imact of varying the level of demand on equilibrium rice levels, markus of rice over marginal cost, and rice variability. Price variability can arise because equilibria may involve mixed strategies in rices. For a articular class of demand and cost conditions, we find the following imact of increasing the level of demand: 1) The average level of rices rises, 2) Average markus of rice over marginal cost may rise or fall, and 3) Price variability falls. The class of demand and cost conditions for which these results aly is a secial class. But we believe that it is not so secial that it could not cature the essential demand and technology conditions that drive ricing decisions in a variety of different industries. Our main oint is that models much simler than otimal collusion models are caable of exlaining interesting atterns of ricing and markus as demand fluctuates.

2 1. Introduction Questions regarding market ower and how firms exercise market ower as demand fluctuates have received a great deal of attention in the Industrial Organization literature. A articularly influential hyothesis is that firms with market ower may actually set higher markus of rice over marginal cost when demand is low - e.g., in a recession - than when demand is high - e.g., in a boom eriod. Rotemberg and Saloner (1986) and Bagwell and Staiger (1997) show that countercyclical markus may arise in models in which firms utilize an otimal collusion mechanism. These mechanisms require firms to adot fairly comlex strategies that would unish rival firms that defect from an agreed uon ricing attern. This aer formulates and analyzes a relatively simle model of duooly ricing under conditions of fluctuating demand. Price-setting firms roduce a homogenous roduct subject to increasing marginal cost of roduction. We focus on static Nash equilibria in rice choices and examine the imact of varying the level of demand on equilibrium rice levels, markus of rice over marginal cost, and rice variability. Price variability can arise because equilibria may involve mixed strategies in rices. For a articular class of demand and cost conditions, we find the following imact of increasing the level of demand: 1) The average level of rices rises, 2) Average markus of rice over marginal cost may rise or fall, 3) Price variability falls. The reader will see that the class of demand and cost conditions for which these results aly is a secial class. But we believe that it is not so secial that it could not cature the 1

3 essential demand and technology conditions that drive ricing decisions in a variety of different industries. Our main oint in roviding these results is that models much simler than otimal collusion models are caable of exlaining interesting atterns of ricing and markus as demand fluctuates. 2. Model We analyze a duooly model of ricing in a homogeneous roduct market. Our analysis is in the sirit of the literature on Bertrand-Edgeworth ricing models in which firms have roduction caacity constraints. We relax the assumtions of fixed caacity and constant marginal cost u to caacity. Instead, we allow for general uward sloing marginal cost curves. This aroach will allow us to consider a firm s marginal cost at all outut levels, and to evaluate markus of rice over marginal cost. For conventional Bertrand-Edgeworth models, the marku is undefined if a firm is roducing at full caacity. On the demand side we secify a simle ste demand function. This assumtion on the form of demand yields a tractable model for which Nash equilibrium rice distributions may be characterized and calculated. Product demand is described by a simle ste function with two arameters, v and d. v indicates the maximum willingness to ay for each of the first d units roduced. For any units beyond d, the willingness to ay is zero. We will examine variations in the level of demand by varying arameter d. Each firm i has a differentiable cost function c x ) that indicates the total cost of roducing outut x i. The cost function is assumed to be increasing and strictly convex, which imlies that marginal cost is strictly increasing. In addition, we assume that 0 c (0) < v, so that firms have an incentive to roduce at least some outut. Let s( ) ( i 2

4 arg max be defined by, s( ) { x c( x)}, where K is a arameter that exceeds the x [0, K] largest value of d used in our analysis. This would be the suly function for a firm if the firm were to oerate as a rice-taker in a cometitive market; s( ) corresonds to the inverse of the marginal cost function. Our modeling orientation is that c x ) is a short run total cost function for a firm that has already chosen some fixed inuts, such as caital. Firms engage in rice cometition in the short run, given current demand conditions as summarized by arameter d. We assume that firms have identical cost functions. This ermits us to focus on symmetric equilibria in our analysis of the Nash equilibrium ricing game. Demand and suly conditions that result from our assumtions are illustrated in Figure One. The demand curve is a ste function, with a maximum of d units demanded at a ositive rice. The suly curve for a single firm is s; the market suly curve for two firms is 2s. For the demand and suly conditions in Figure One, a cometitive rice is determined by the intersection of 2s with the vertical segment of the demand. Throughout this discussion we assume that the level of demand, d, satisfies: s( v) < d 2s( v). That is, the level of demand is between the amount that a single firm would be willing to suly at the maximum rice, and the amount that two firms would be willing to suly at the maximum rice. ( i 3

5 Figure One Demand and Suly Conditions $ v s 2s c c (0) d outut 3. Benchmark Cases In this section we summarize the imact of demand fluctuations for three different modes of market behavior: erfect cometition, Cournot quantity cometition, and monooly. These results rovide benchmarks against which results from a Nash equilibrium model of rice setting firms may be comared. If firms behave as erfect cometitors then the rice is given by the intersection of demand and suly. This is illustrated by the rice c in Figure One. In a cometitive equilibrium the rice is equal to marginal cost for each firm, so that there is no marku of rice over marginal cost. As demand arameter d increases, the cometitive equilibrium rice rises, since the suly function 2s() is strictly increasing in. 4

6 Suose instead that the two firms form a cartel that maximizes joint rofit. That is, the firms oerate as a single rofit maximizing monooly with the combined assets of the two firms. 1 The marginal cost curve for the cartel corresonds to the market suly curve for the two firms (2s in Figure One). The marginal revenue for the cartel is equal to v for outut u to d. The rofit maximizing outut is equal to d, since MR MC for cartel outut less than d; each firm roduces outut d/2 to minimize cartel cost. The cartel rice is v. As demand increases, the rice remains constant at v and the marku of rice over marginal cost falls, since marginal cost is increasing in outut for each firm. Now suose that the two firms oerate as Cournot quantity-setting duoolists. For any fixed outut choice x of a rival firm, a firm has marginal revenue of outut equal to v for outut u to d - x. As long as rival outut x is greater than or equal to d - s(v) a firm has an incentive to exand its outut until total outut is equal to d. A Cournot equilibrium involves a rice v and total outut of the two firms that sums to d. The symmetric Cournot equilibrium involves each firm choosing outut d/2; the equilibrium outcome is identical to the cartel outcome. 2 As d increases, the symmetric Cournot equilibrium rice remains constant at v and the marku of rice over marginal cost falls, just as in the cartel case. The monooly/cartel and Cournot duooly results suggest that countercyclical markus of rice over marginal cost can arise in very simle models in which firms have market ower. The main features that contribute to this result are increasing short run marginal costs and a demand function that gives firms little or no ability to raise rices as demand increases. 5

7 4. Price-Setting Firms We analyze a static game of simultaneous rice choice by the two firms. Each firm i chooses rice i from the interval, [ c (0), v]. The individual demand function for firm i when it chooses rice i and its rival chooses j is, (1) D(, i j d, ) = 1 d 2 d s( j ) if if if i i i < = > j j j We assume that firms roduce to order. A firm s sales are given by, (2) x, ) = min{ s( ), D(, )}. ( i j i i j A firm with rice i would not want to sell more than s( i ) even if customers wished to urchase more, since the extra sales would involve marginal cost that exceeds the rice. The rofit function for firm i is, (3) π, ) = x(, ) c( x(, )). ( i j i i j i j This rofit function is continuous in both rices excet when rices are equal. When rices are equal, a small decrease in rice i would yield a discrete increase in sales and rofits for firm i. There is no ure strategy Nash equilibrium for this ricing game. If a firm sets its rice equal to the cometitive rice then the best resonse of its rival is to set its rice equal to v. If a firm sets its rice above the cometitive rice then its rival would earn more by undercutting this rice by a small amount than by matching this rice. For any air of rices, ), at least one firm would wish to change their rice, given the rice of their rival. ( 1 2 6

8 PROPOSITION ONE: A mixed strategy equilibrium exists. The roof of this is fairly standard and can be constructed along the lines of Dixon (1984) and Maskin (1986). 3 These two aers, along with Allen and Hellwig (1986), rovide mixed strategy equilibrium existence results for ricing games in which firms have continuous, increasing cost functions. Thus, they generalize existence beyond the Bertrand-Edgeworth framework in which firms have fixed caacity constraints and constant marginal cost u to caacity. The three aers cited above focus on existence of equilibria rather than on characterization or calculation of equilbria. The next roosition characterizes a symmetric equilibrium for the ste demand formulation. Following this, we reort calculations of equilibrium outcomes for varying demand levels. PROPOSITION TWO: A symmetric mixed strategy equilibrium exists with suort, [, v] and c.d.f. F() defined by: (4) s( ) + [ d s( y) s( )] F ( y) dy F ( ) =. s( ) c( s( )) [( d s( )) c( d s( ))] There are no mass oints for this distribution. The lower bound of the suort exceeds the cometitive rice and satisfies the conditions, F ( ) = 0 and F ( ) d = 1, where F () is defined by (4). Proof: (not included) There is a symmetric mixed strategy equilibrium with suort that begins above the cometitive rice and extends u to the maximum rice, v. Exected rofit for a firm must be constant for all rices in the suort, given that its rival lays the equilibrium distribution. Taking the derivative of exected rofit with resect to rice yields an v 7

9 integro-differential equation in F. This equation may be transformed into a Volterra equation of the second kind; this is equation (4). 4 We conjecture that the mean of the equilibrium distribution is increasing in arameter d, and that the variance of the distribution is decreasing in d, for d ( s( v),2s( v)). We do not as of yet have a general roof of this result. In Table One we reort the results of numerical calculations that are consistent with our conjecture. There are standard rocedures for numerical aroximation of the solution of an equation of this tye. 5 For each value of d, the lower bound of the suort of the rice distribution must be chosen so that the cdf that is calculated attains a value of unity at a rice equal to v. We also reort calculations of average rice-cost markus in Table One. For the case of rice-setting firms, we use a Monte-Carlo simulation based on 10,000 airs of rice draws from the equilibrium distribution to calculate the markus. The ercentage marku for the NE ricing model falls from 28 % to 22 % to zero as d increases from 30 to 40 to 50. This attern of falling exected markus as d rises does not always emerge. For other numerical examles we have calculated, the exected marku first rises and then falls as d increases. 8

10 Table One Numerical Examle: Imact of Changes in Demand (d)* Perfect Cometition Mixed Strategy NE Cournot/Monooly d Min(rice) E(rice) Var(rice) E((rice-MC)/rice) ρ 1/ ρ * This examle has v = 40 and cost function, c ( x) = w( x k ), with arameters w = 10, k = 50, and ρ = -1; this cost function is derived from a CES roduction function with variable inut rice w, fixed inut quantity k, and elasticity of substitution 1/( ρ 1). ρ 9

11 Endnotes 1. The cartel solution described in the text does not take incentives to defect from the agreement into account. The otimal collusion mechanism aroach could be alied to our model to address the issue of otential defection from a cartel agreement. 2. There will also be asymmetric Cournot equilibria as long as d < 2s(v). 3. Our market demand function is discontinuous at a rice equal to v; this is a violation of their assumtions about demand. But since v is the monooly rice and firms would never choose a rice greater than v, we can restrict attention to rices less than or equal to v. The market demand is continuous (constant, in fact) for rices less than or equal to v. 4. Brunner (1988) shows how a class of integro-differential equations may be transformed into Volterra equations of the second kind. The equation for F generated by the equilibrium conditions for our roblem fits within this class. 5. See chater 18, section 2 in Press, et al (1992). 10

12 References Allen, B. and Hellwig, M., "Price-setting firms and the oligoolistic foundations of erfect cometition", American Economic Review, vol. 76, May (1986). Bagwell, K. and Staiger, R., "Collusion over the Business Cycle", RAND Journal of Economics, vol. 28, Sring (1997). Brunner, H., "The numerical solution of initial-value roblems for integro-differential equations", in Numerical Analysis 1987, edited by D.A. Griffiths and G.A. Watson, John Wiley & Sons, Inc.: New York, Dixon, H., "The existence of mixed-strategy equilibria in a rice-setting oligooly with convex costs", Economics Letters, vol. 16 (1984). Maskin, E., "The existence of equilibrium with rice-setting firms", American Economic Review, vol. 76, May (1986). Press, W.H., Vetterling, W.T., Teukolsky, S.A., and Flannery, B.A., Numerical Reciies in FORTRAN 77, 2 nd edition, Cambridge University Press: New York, Rotemberg, J. and Saloner, G., "A suergame-theoretic model of rice wars during booms", American Economic Review, vol. 76, May (1986). 11

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