bidding into electricity markets a transmission constrained residual demand derivative approach this paper proposes a novel approach to optimize a generator s profit in an offer based electricity market cleared by an optimal power flow opf program a generator s offer is optimized based on its transmission constrained residual demand derivative tcrdd which avoids representing the full network model in the optimization the tcrdd can be easily calculated based on a solved opf without changing existing opf algorithms and programs as demonstrated with an ieee 118 bus example the tcrdd approach is computationally efficient the tcrdd approach can help market participants to bid into electricity markets and help market monitors to diagnose bidding behaviors in the presence of transmission constraints

the welfare effects of third degree price discrimination with nonlinear demand functions the welfare effects of third degree price discrimination are analyzed when demand in one market is an additively shifted version of demand in the other market and both markets are served with uniform pricing social welfare is lower with discrimination if the slope of demand is log concave or the convexity of demand is nondecreasing in the price the demand functions commonly used in models of imperfect competition satisfy at least one of these sufficient conditions

european electricity market integration with mixed market designs part ii solution algorithm and case studies this paper presents the solution algorithm and test results of the market splitting approach proposed in the first part of this two paper series for the clearing of the forthcoming internal electricity market the market splitting approach is implemented in a europe wide level indicatively defining three power pools and 22 power exchanges comprising 42 bidding areas in total with each local national market respecting the standard market regulatory framework of power pools and power exchanges pxs the performance and computational requirements of this implementation is presented the model is tested in terms of problem solvability to its limits considering a day ahead market clearing time threshold equal to one hour the results demonstrate that a viable solution to the clearing of the internal electricity market can be attained with the current state of the art of computers processing power further discussion is made concerning the viability of the proposed approach and other harmonization issues that should be considered for successful market integration

european electricity market integration with mixed market designs part i formulation the integration of the day ahead electricity markets in europe shall lead to one multi area market that will substitute the local national ones in view of the target model that will be enforced in all european markets in conjunction with their forthcoming coupling integration a centralized market splitting algorithm is implemented in this paper respecting the standard market regulatory framework of power exchanges pxs and power pools including 1 block offers bids linked block offers bids flexible hourly offers bids and convertible block offers in power exchanges and 2 unit technical commitment constraints and system operating constraints in power pools the problem is formulated as a mixed integer linear programming milp model which can be solved using commercial milp solvers the performance and computational requirements of an implementation in pan european level is presented in the companion paper the model is tested in terms of problem solvability to its limits considering a day ahead market clearing time threshold equal to one hour

the variable rate of growth effect in the life cycle saving model the rate of growth in real gnp g affects the aggregate saving rate positively in life cycle saving models this rate of growth effect is invariably estimated as the coefficient of g in the saving function we show that other determinants of the saving rate influence the rate of growth effect in other words the rate of growth effect is not constant specifically the real rate of return on financial assets and the population dependency ratio determine the timing of saving over the life cycle and hence alter the rate of growth effect the dependency ratio also has an effect on the saving rate which is independent of g because it together with the foreign saving rate determines the level of saving over the life cycle

the efficiency of ireland s renewable energy feed in tariff refit for wind generation ireland s renewable energy feed in tariff refit for wind generation has some unusual features making it different from other refit schemes around the world by utilising an annual floor price element the scheme presents an option value to the contract holder which to date has gone unnoticed or unvalued in the market by employing an option pricing framework this paper has quantified for the first time in the public domain the expected costs and value of the irish refit support scheme for wind generation while the cost of the refit scheme to the electricity consumer appears to be lower than the cost of schemes in other countries significant inefficiencies exist as a result of the structure of the scheme the irish refit scheme is contrasted with a single fixed price support scheme and the analysis suggests that the fixed price scheme can provide a similar or greater incentive to the wind sector at half the cost to the end electricity consumer and may also prove more compatible with consumers desire to reduce inter year electricity portfolio cost volatility

electricity price forecasting using sale and purchase curves the x model abstract our paper aims to model and forecast the electricity price by taking a completely new perspective on the data it will be the first approach which is able to combine the insights of market structure models with extensive and modern econometric analysis instead of directly modeling the electricity price as it is usually done in time series or data mining approaches we model and utilize its true source the sale and purchase curves of the electricity exchange we will refer to this new model as x model as almost every deregulated electricity price is simply the result of the intersection of the electricity supply and demand curve at a certain auction therefore we show an approach to deal with a tremendous amount of auction data using a subtle data processing technique as well as dimension reduction and lasso based estimation methods we incorporate not only several known features such as seasonal behavior or the impact of other processes like renewable energy but also completely new elaborated stylized facts of the bidding structure our model is able to capture the non linear behavior of the electricity price which is especially useful for predicting huge price spikes using simulation methods we show how to derive prediction intervals for probabilistic forecasting we describe and show the proposed methods for the day ahead epex spot price of germany and austria

price maker bidding in day ahead electricity market for a retailer with flexible demands this study develops a short term planning model to determine the bidding curves on a day ahead market for a price maker retailer with flexible power demand it concerns the interactions between the spot price and the flexible demand both conditional value at risk and volume deviation risk are taken into account a numerical study using the data from the nordic electricity market is performed to investigate the influence of risk factors on the retailer s profit risk levels average spot price and total consumption three types of price elasticity are compared to show that the retailer can benefit from the flexibility in demand side in some cases the flexibility also leads to lower spot prices so that the customers in real time price based demand response can face a lower electricity price for per unit power consumption

aggregate spending and the terms of trade is there a laursen metzler effect this paper investigates the spending and current account effects of terms of trade shifts in a model where households maximize utility over an infinite planning period in the framework developed here an economy specialized in production must experience a fall in aggregate spending and a current surplus as a result of an unanticipated permanent worsening in its terms of trade the paper s model thus provides a setting in which the current account deficit predicted by laursen and metzier harberger and others fails to materialize

price discrimination and patent policy patent and antitrust policy are often presumed to be in conflict as an important example there is ongoing controversy about whether price discrimination by a patent holder is an illegal or socially undesirable exploitation of monopoly power in this article we show that no conflict exists in many price discrimination cases even ignoring the dynamic effects on incentives for innovation third degree price discrimination by patent holders can raise static social welfare in fact pareto improvements may well occur welfare gains occur because price discrimination allows patent holders to open new markets and to achieve economies of scale or learning further even in cases where discrimination incurs static welfare losses it may be efficient relative to other mechanisms such as length of patent life for rewarding innovators with profits

monopoly price discrimination and demand curvature this paper presents a general analysis of the eects of monopolistic third degree price discrimination on welfare and output when all markets are served su cient conditions involving straightforward comparisons of the curvatures of the direct and inverse demand functions in the dierent mar kets are presented for discrimination to have negative or positive eects on

the terms of trade and the current account the harberger laursen metzler effect the paper examines the effect of terms of trade changes on a small country s spending and current account assuming optimizing behavior in an intertemporal framework with perfect international capital mobility a temporary future terms of trade deterioration implies a deterioration improvement of the trade balance whereas a permanent terms of trade deterioration has an ambiguous effect depending on the rate of time preference nominal and real variables are considered via exact price indexes two periods and an infinite horizon are examined

bounding the welfare effects of third degree price discrimination under pigouvian third degree price discrimination a profit maximizing monopolist typically charges different groups of customers different prices resale between groups is assumed to be impossible these differences in price entail a loss of efficiency because marginal valuations of the output are not equal across buyers the net welfare effect of allowing price discrimination is ambiguous though because the total output under discrimination may exceed that under uniform pricing nevertheless under quite general conditions third degree price discrimination by a monopolist can increase static welfare only if total output is greater under discriminatory pricing than under uniform pricing richard schmalensee 1981 hal r varian 1985 marius schwartz 1990 although there has been much analysis of whether price discrimination raises or lowers total output or welfare the size of these effects has remained largely unexplored varian 1985 compares discriminatory pricing to uniform pricing for a monopolist and provides some bounds for the change in welfare in terms of market prices and outputs but these relationships give little sense of the relative size of welfare changes resulting from third degree price discrimination i address this issue by asking what can be said about the ratio wd wu where wd and wu denote welfare under discriminatory and uniform pricing respectively welfare is measured as the sum of producer surplus and marshallian consumer surplus section i provides two examples showing that this ratio can range from zero to infinity in these examples all markets are served under uniform pricing and demand in one of the markets is strictly convex considering a two market model in which a monopolist with constant marginal cost serves two independent markets under uniform pricing i show in section iii that if demands in both markets are concave then wd w is bounded below by 2 the assumptions of constant marginal cost and independent demands allow producer surplus and consumer surplus to be identified separately for each market separability across markets is central to the approach taken in this paper following joan robinson 1933 i call a market strong weak if the discriminatory price in that market is at least as great as no greater than the profit maximizing uniform price surprisingly the key to bounding wd wu is that the demand function in the weak market be concave if one market is strong the other is weak and demand in the weak market is concave then wd w can be bounded above by 2 5 or even 1 75 or 1 6 with additional assumptions on the demand functions these bounds arise for several reasons first the monopolist s profit under uniform pricing must be at least as large as the profit in the weak market under discrimination moreover if at all prices demand in the strong market is as great as in the weak market then profit under uniform pricing must be at least twice as great as profit in the weak market under discrimination otherwise the monopolist could do better by charging the monopoly price for the weak market second this relationship between profits in the two markets is useful in bounding welfare because welfare in each market can be related to profit in that market section ii shows that if demand is con department of economics and a b freeman school of business tulane university new orleans la 70118 i thank marius schwartz and two anonymous referees for their comments on earlier drafts of this paper

price discrimination and social welfare studies the effect of price discrimination on social welfare using methods from duality theory features of the reservation model presented quasi linear utility and consumers surplus upper and lower bounds on welfare change bounds on welfare change with optimal price discrimination from ebsco

the regulating power market on the nordic power exchange nord pool an econometric analysis abstract what differentiates the structure of nord pool from other power exchanges around the world is the way the balance from the spot market is maintained until the actual physical delivery takes place via the regulating power market in norway this paper reveals the pattern of the prices on the regulating power market by analysing the cost of being unable to fulfil the commitments made on the spot market some power producers with unpredictable fluctuations e g wind will need to buy regulation services the disclosed pattern implies that these producers must pay a limited premium of readiness in addition to the spot price this premium is independent of the amount of regulation the level of the premium of readiness for down regulation is shown to be strongly influenced by the level of the spot price on the other hand it is demonstrated that the premium for up regulation is less correlated to the spot price furthermore it is found that the amount of regulation affects the price of regulating power for up regulation more strongly than it does for down regulation the disclosed cost of using the regulating power market is a quadratic function of the amount of regulation this asymmetric cost may encourage bidders with fluctuating production to be more strategic in their way of bidding on the spot market by using such strategies the extra costs for example wind power needed to counter unpredictable fluctuations may be limited

strategic bidding in secondary reserve markets electricity markets are based on several sequential energy and reserve trading mechanisms to constantly maintain the balance between generation and demand during the last years reserve markets are getting much importance all around the world with the increasing social awareness of the renewable energy benefits additional reserve quantities and larger remunerations are being implemented by the regulators since this renewable energy is highly dependent on weather conditions uncertainties utilities are therefore demanding more and more powerful models to better optimize their reserve bidding curves to be sent to the system operators this paper describes a new methodology to obtain an optimized real bidding curve for the secondary spinning reserve market the approach is based on the maximization of a company profit assuming a set of residual demand curves scenarios for its competitors representation and considering the day ahead market opportunity cost to build the reserve cost curve the case study validates the main features of the proposed methodology by its application to the spanish secondary reserve market this methodology is being daily used by one of the most important electricity companies in spain

bounding the relative profitability of price discrimination we derive bounds on the ratio of a monopolist s profit from third degree price discrimination to that from uniform pricing if the monopolist serves n independent markets demand is continuous and the cost function is superadditive then the profit ratio is bounded by n a linear demand example is provided coming arbitrarily close to this bound we provide examples showing the profit ratio can be unboundedly large when marginal cost is decreasing demand is discontinuous or fixed cost is positive if the monopolist has access to certain demand rationing strategies under uniform pricing we can bound the profit ratio even for discontinuous demand functions and multiproduct cost functions

continuous time marginal pricing of electricity the current practice of discrete time electricity pricing starts to fall short in providing an accurate economic signal reflecting the continuous time variations of load and generation schedule in power systems this paper introduces the fundamental mathematical theory of continuous time marginal electricity pricing we first formulate the continuous time unit commitment problem as a constrained variational problem and subsequently define the continuous time economic dispatch ed problem where the binary commitment variables are fixed to their optimal values we then prove that the continuous time marginal electricity price equals to the lagrange multiplier of the variational power balance constraint in the continuous time ed problem the proposed continuous time marginal price is not only dependent to the incremental generation cost rate but also to the incremental ramping cost rate of the units thus embedding the ramping costs in calculation of the marginal electricity price the numerical results demonstrate that the continuous time marginal price manifests the behavior of the constantly varying load and generation schedule in power systems

debt dependence and the demographic transition latin america in to the next century abstract in the decades ahead world demographic trends indicate that our planet s share of the population of working age will be gradually diminishing the regional incidence of the trends however shows a marked bias thus the general equilibrium of the world saving investment nexus may be fundamentally realigned in the years to come via dependency effects for example favorable demographic prospects in latin america and elsewhere in the developing world offer poorer countries the chance to use the potential of their own demographic transitions to propel financial transitions to domestically financed accumulation even capital export

savings and nutrition at low incomes while many development theorists assume saving propensities vary among different groups there has been limited theoretical justification for this view this paper analyzes two complementary theories stressing the effect on saving of physiological consequences of poor nutrition at low incomes one influence is on the probability of survival another is on workers productivity either effect means that the average propensity to save can rise with income issues considered include annuities exogenous changes in survival and efficiency implications for the efficiency wage theory of unemployment and distinctions between wage and nonwage income implications of nutritional effects for econometric specification of saving functions are stressed

parallel imports demand dispersion and international price discrimination parallel imports goods imported by unauthorized resellers are advocated worldwide for undermining international price discrimination for a continuum of markets we find that uniform pricing by a monopolist yields lower global welfare than third degree discrimination if demand dispersion across markets is large though uniform pricing avoids output misallocation too many markets go unserved mixed systems permitting discrimination across but not within designated groups of markets yield significantly higher welfare than uniform pricing or unrestricted multimarket discrimination and can pareto dominate uniform pricing thus while parallel imports might benefit some countries our results weaken the multilateral case for allowing them

the merit order effect a detailed analysis of the price effect of renewable electricity generation on spot market prices in germany the german feed in support of electricity generation from renewable energy sources has led to high growth rates of the supported technologies critics state that the costs for consumers are too high an important aspect to be considered in the discussion is the price effect created by renewable electricity generation this paper seeks to analyse the impact of privileged renewable electricity generation on the electricity market in germany the central aspect to be analysed is the impact of renewable electricity generation on spot market prices the results generated by an agent based simulation platform indicate that the financial volume of the price reduction is considerable in the short run this gives rise to a distributional effect which creates savings for the demand side by reducing generator profits in the case of the year 2006 the volume of the merit order effect exceeds the volume of the net support payments for renewable electricity generation which have to be paid by consumers

transmission constrained inverse residual demand jacobian matrix in electricity markets a generation firm in an electricity market may own multiple generators located at multiple locations this paper generalizes the concept of transmission constrained residual demand from a single generator s perspective to that of a generation firm we calculate the derivative of a generation firm s inverse residual demand function i e the jacobian matrix based on a multi parameter sensitivity analysis of the optimal power flow solution and characterize some of its properties this jacobian matrix provides valuable information such as in characterizing a generation firm s profit maximizing strategy we apply the bundle newton method utilizing the jacobian matrix to find a generation firm s maximum profit the effectiveness and performance of the algorithm is demonstrated with the ieee 118 bus system example the jacobian matrix and the profit maximizing algorithm are helpful for market participants to bid into electricity markets and for market monitors to analyze firm based strategic behaviors

price discrimination and efficient distribution this paper focuses on third degree price discrimination by an upstream firm with some degree of monopoly power downstream firms fall into two categories efficient and inefficient according to their relative costs of transforming a unit of the upstream good into a unit of final product under ordinary static conditions price discrimination favors the inefficient firms which have more elastic demands we consider however the possibility that discrimination in the opposite direction can alter the downstream market structure toward greater efficiency discriminatory pricing then involves charging a higher price to the less efficient firms such pricing is shown to be both potentially profitable for the upstream firm and welfare improving as average consumer prices fall

expectations and attitudes toward gender based price discrimination this study explores consumer expectations and attitudes related to gender based price discrimination although much research has focused on pay inequalities and gender diversity considerably less attention has been focused on situations in which men and women are charged different prices based on gender in two studies expectations and attitudes toward gender based price discrimination are examined in study 1 two scenarios related to prices at hair salon and dry cleaning services were manipulated to measure expectations and attitudes toward gender based price discrimination we found that the nature of the service results in expectations of price differences between men and women we also found men expect gender based pricing more than women in study 2 qualitative research was conducted to reveal the cognitions that men and women experience when exposed to gender based price discrimination

welfare increasing third degree price discrimination when demand functions in different markets are derived from distributions of reservation prices that differ only in their means conditions exist such that third degree price discrimination leads to greater total output and greater total welfare welfare is higher with discrimination than with a uniform price when demand functions are derived from logistic distributions with different means welfare and consumer surplus are higher with discrimination for demands derived from a distribution related to the pareto in general whether discrimination increases total output depends on demand being more convex in markets in which prices fall with discrimination than in those in which prices rise

market opening under third degree price discrimination there are frequently regulatory and antitrust pressures for firms to cease price discrimination and practice uniform pricing such pressures however generally have negative welfare consequences when they lead to weaker markets not being served this paper derives conditions that determine when price discrimination will induce service to a market market opening that would not be served under a regime of price uniformity the factors that favor market opening under price discrimination are a large market share for the strong market profit margins in the two markets that are far apart and concave rather than convex demand curves copyright 1994 by blackwell publishing ltd

terms of trade and national saving rates in asia abstract in recent theoretical work obstfeld and svensson and razin re examine the harberger laursen metzler effect of terms of trade changes on the balance of payments current account their models of intertemporal utility maximization indicate that individuals saving responses to a change in the terms of trade are ambiguous in theory from this they infer that the corresponding movements in the current account are also ambiguous however they ignore the positive rate of growth effect incorporated in the life cycle theory on the aggregate saving rate this paper presents an empirical analysis of the terms of trade effect on aggregate national saving rates in 14 asian developing countries over the period 1961 1981 it turns out to be significantly positive as the life cycle model predicts

modeling the impact of a wind power producer as a price maker wind energy is present in many countries throughout the world the main types of wind sales in electricity markets are via regulated tariffs or pool based markets production companies choose cost effective options for selling wind energy and some markets like the irish electricity market use regulated tariffs to remunerate wind production this paper aims to provide some answers to explain what effect wind offers may have in an electricity market if wind power producers participated in the day ahead market without receiving any premium or aid a price maker optimization model is used to detect its effect on prices the model encompasses energy offers by other technologies using residual demand curves and detailed modeling of wind imbalances it is observed that wind acting as price maker reduces electricity prices and the imbalance penalties help the system operator to reduce imbalances a realistic case study using data from the irish electricity market illustrates the methodology used comparing the effect of imbalance penalties in the models in terms of profit and total imbalance of the system

price maker self scheduling in a pool based electricity market a mixed integer lp approach this paper addresses the self scheduling problem faced by a price maker to achieve maximum profit in a pool based electricity market an exact and computationally efficient mixed integer linear programming milp formulation of this problem is presented this formulation models precisely the price maker capability of altering market clearing prices to its own benefits through price quota curves no assumptions are made on the characteristics of the pool and its agents a realistic case study is presented and the results obtained are analyzed in detail

third degree price discrimination in linear demand markets effects on number of markets served and social welfare author affiliation victor kaftal department of mathematical sciences 839 c old chemistry building university of cincinnati cincinnati oh 45221 usa victor kaftal uc edudebashis pal department of economics 1204 crosley tower university of cincinnati cincinnati oh 45221 debashis pal uc edu corresponding author acknowledgment we are grateful to rick harbaugh laura razzolini david sappington and two anonymous reviewers debashis pal is also grateful to taft research center for financial support 1 introductiondoes monopolistic third degree price discrimination reduce social welfare the question has continued to intrigue economists and policy makers for more than half a century the seminal work of robinson 1933 shows that if a monopolist with a constant marginal cost sells in two distinct and independent markets having linear demands then welfare falls with third degree price discrimination almost half a century later schmalensee 1981 reexamines robinson s result and demonstrates that for any number of markets with linear demands third degree price discrimination lowers welfare schmalensee 1981 also establishes more generally that price discrimination increases welfare only if it increases aggregate output subsequently using ingenious duality approaches varian 1985 and schwartz 1990 generalize the results for correlated demands and nonlinear marginal costs 1it is therefore well known that for linear market demands and constant marginal costs monopolistic third degree price discrimination indeed lowers welfare it is important to note however that the welfare reducing effect of third degree price discrimination is derived under a crucial assumption it is assumed that if the monopolist is forced to charge the same price in all markets uniform pricing the monopolist would sell a positive quantity in each market in a recent article cowan 2007 considers various forms of nonlinear demands and presents the necessary and sufficient and or sufficient conditions for price discrimination to reduce welfare cowan 2007 however also assumes that all markets are served under uniform pricing yet when this assumption does not hold the welfare effects of monopolistic third degree price discrimination turn out to be significantly more intricate observe that when all markets are not served under uniform pricing price discrimination may increase welfare because price discrimination may serve markets that would not be served under uniform pricing thus relative to uniform pricing price discrimination has two countervailing effects on welfare aggregate welfare goes down in those markets that are originally served under uniform pricing whereas welfare goes up in those markets that are not served under uniform pricing therefore the additional welfare from the new markets under price discrimination may offset any loss of welfare in the markets that are originally served under uniform pricing the point is strongly emphasized in schmalensee s 1981 seminal work in the context of prescription drugs an identical sentiment is recently echoed in varian 2000 in the context of two markets the possibility of price discrimination opening up new markets is first analyzed graphically by battalio and ekelund 1972 layson 1994 also considers two markets but includes nonlinear demands and presents conditions for opening of new markets based on endogenous features such as elasticities in the context of patent policy opening of new markets by price discrimination is explicitly incorporated by hausman and mackie mason 1988 using two markets they show that price discrimination by a patent holder is socially desirable in many cases because it allows the patent holder to open new markets there is a related literature in spatial economics beginning with greenhut and ohta 1972 and holahan 1975 that deals with spatial price discrimination and market opening with linear demand

residual demand curves for modeling the effect of complex offering conditions on day ahead electricity markets residual demand curves rdcs can be used to represent the strategic interaction of participants in electricity markets rdcs relate the energy that an agent can buy or sell in one hour with the clearing market price that would be obtained in such hour assuming the market is organized as simple bid independent auctions despite the fact that they have been widely used in the literature the existence of time and or spatial constraints in the market clearing algorithm makes the rdcs not directly applicable this paper tries to overcome these difficulties by extending the concept of rdcs to zonal pricing markets where complex offering conditions and transmission constraints are taken into account therefore the rdcs are redefined in order to capture such effects which are usually neglected or oversimplified a new method for computing the redefined rdcs is established and its application to the iberian electricity market is presented the results show that modeling complex conditions and transmission constraints in rdcs can have a significant effect when compared to the standard approach found in the literature therefore the method presented in this paper modeling the effect of firm s decisions on market prices in a more accurate way

forecasting functional time series with a new hilbertian armax model application to electricity price forecasting a functional time series is the realization of a stochastic process where each observation is a continuous function defined on a finite interval these processes are commonly found in electricity markets and are gaining more importance as more market data become available and markets head toward continuous time marginal pricing approaches forecasting these time series requires models that operate with continuous functions this paper proposes a new functional forecasting method that attempts to generalize the standard seasonal armax time series model to the l 2 hilbert space the structure of the proposed model is a linear regression where functional parameters operate on functional variables the variables can be lagged values of the series autoregressive terms past observed innovations moving average terms or exogenous variables in this approach the functional parameters used are integral operators whose kernels are modeled as linear combinations of sigmoid functions the parameters of each sigmoid are optimized using a quasi newton algorithm that minimizes the sum of squared errors this novel approach allows us to estimate the moving average terms in functional time series models the new model is tested by forecasting the daily price profile of the spanish and german electricity markets and it is compared to other functional reference models

customer class pricing parallel trade and the optimal number of market segments abstract we consider the optimal market segmentation problem of a monopolist that faces a continuum of customers when it is costly to prevent resale or parallel trade among groups in our framework the monopolist chooses the number k 1 of market segments but also their design and the discriminatory prices all these quantities are chosen to maximize the total profit we solve the profit maximization problem when demands are linear and parallel as a function of the cost of separating markets we show that market segmentation and prices cannot be chosen independently and we also show that it is optimal to create only a few market segments we then turn to the welfare analysis and show that the socially optimal number of market segment is equal to three