Discrete Choices under Social Influence: Generic Properties
Mirta B. Gordon, Jean-Pierre Nadal, Denis Phan, Viktoriya Semeshenko
DDiscrete Choices under Social Influence:Generic Properties
Mirta B. Gordon (1) , Jean-Pierre Nadal (2,3) ,Denis Phan (4,5) , Viktoriya Semeshenko (1)(1) Laboratoire Techniques de l’Ing´enierie M´edicale et de la Complexit´e(TIMC-IMAG, UMR 5525 CNRS-UJF), Universit´e Joseph Fourier, Grenoble(2) Centre d’Analyse et Math´ematique Sociales (CAMS, UMR 8557 CNRS-EHESS),Ecole des Hautes Etudes en Sciences Sociales, Paris(3) Laboratoire de Physique Statistique (LPS, UMR 8550 CNRS-ENS-Paris 6-Paris 7),Ecole Normale Sup´erieure, Paris(4) Centre de Recherche en Economie et Management(CREM, UMR 6211 CNRS-Universit´e de Caen-Universit´e de Rennes 1), Universit´e de Rennes 1(5) Groupe d’Etude des M´ethodes de l’Analyse Sociologique(GEMAS, UMR 8598 CNRS-Universit´e Paris Sorbonne - Paris IV), Paris
March 8, 2007
Abstract
We consider a model of socially interacting individuals that make a binary choice in acontext of positive additive endogenous externalities. It encompasses as particular casesseveral models from the sociology and economics literature. We extend previous resultsto the case of a general distribution of idiosyncratic preferences, called here IdiosyncraticWillingnesses to Pay (IWP).Positive additive externalities yield a family of inverse demand curves that include theclassical downward sloping ones but also new ones with non constant convexity. When j ,the ratio of the social influence strength to the standard deviation of the IWP distribution,is small enough, the inverse demand is a classical monotonic (decreasing) function of theadoption rate. Even if the IWP distribution is mono-modal, there is a critical value of j above which the inverse demand is non monotonic, decreasing for small and high adoptionrates, but increasing within some intermediate range. Depending on the price there arethus either one or two equilibria.Beyond this first result, we exhibit the generic properties of the boundaries limitingthe regions where the system presents different types of equilibria (unique or multiple).These properties are shown to depend only on qualitative features of the IWP distribution:modality (number of maxima), smoothness and type of support (compact or infinite). Themain results are summarized as phase diagrams in the space of the model parameters, onwhich the regions of multiple equilibria are precisely delimited. a r X i v : . [ phy s i c s . s o c - ph ] A p r ontents A.1 Pdfs with compact support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.2 Pdfs with fat tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33A.3 Aggregate demand for multimodal pdfs . . . . . . . . . . . . . . . . . . . . . . . 35A.3.1 Smooth pdfs: generic properties . . . . . . . . . . . . . . . . . . . . . . . 35A.3.2 A degenerate case: 2 Dirac . . . . . . . . . . . . . . . . . . . . . . . . . . 412
Introduction
There are many circumstances in social and economic contexts where, faced with differentalternatives, the best choice for an individual depends on the choices of other individuals in thepopulation. The decision of leaving a neighborhood [78], to attend a seminar [78] or a crowdedbar [2, 3], to participate to collective actions such as strikes and riots [46], are particularexamples taken from social sciences. It has been suggested that social interactions may explainthe school dropout [24], the persistence in the educational level within some neighborhoods [30]and the related consequences in the stratification of investment in human capital and economicsegregation [11], the large dispersion in urban crime through cities with similar characteristics[42], the emergence of social norms [71], the labor market behavior and related unemploymentpatterns [88, 86], the housing demand [96], the existence of poverty traps [34], the smokingbehavior [56, 55, 83], etc.Similarly, there is a growing economic literature that recognizes the influence on consumersof the social world they live in. In market situations like the subscription to a telephonenetwork [5, 75, 90, 25] or the choice of a computer operating system [52], the willingness topay generally depends not only on the individual preferences but also on the choice made byothers [81, 76]. If the externality is positive the utility of the most popular choice increases evenfor individuals who otherwise would never make this choice. In other words, the conformityeffect may dominate the heterogeneity of preferences , as pointed out by Bernheim [12]. Generalaspects of these issues have been discussed in the literature [10, 60]. Particular insightful papersare Becker’s note [9] about restaurants pricing, and the qualitative analysis by Granoveter andSoong [47] of the consequences of interpersonal influences (“bandwagon effects”[57, 76]) on theconsumers demand and on the supply prices.In the present paper we consider the general properties of a model of socially interactingheterogeneous individuals that make a binary choice in a context of positive endogenous ex-ternalities. The model encompasses, as particular cases, most of the above mentioned modelspresented in the sociology and economics literature. In a forthcoming paper [44] we explore theconsequences of the externalities on the economy, taking as an example the simplest market,i.e. that of a monopolist pricing a single good.In social sciences, the question of discrete (typically binary) choices with heterogeneousagents and positive externalities has been first addressed in the 70’s by Schelling [77, 78], whoborrowed from Physics the concept of critical mass : in a repeated-decisions setting, dependingon whether this critical mass is or not reached, the system may end up at very different equi-libria. Granovetter further develops Schelling’s model, applying it to particular problems suchas joining or not a riot [46], voting, etc [47]. The same topic is reconsidered within a statis-tical physics point of view in the early 80’s by Galam et al [39]. The notion of critical massis then related to the Physics concept of phase transition at a critical point, in the neighbor-hood of which the system may be extremely susceptible: by tipping effects, small microscopicchanges can lead to drastic changes at the macroscopic level. Similar tools have been appliedin 1980 by Kindermann and Snell [53] to the study of social networks. These authors intro-duced into the sociology and economics literature the equivalence between statistical physicsapproaches —that use the Boltzmann-Gibbs distribution— and Markov Random Fields. An-other physically-inspired approach for modeling social phenomena such as opinion diffusionhas been developed by Weidlich and Haag [93, 92] in 1983, through a master equation andthe Fokker-Planck approximation. Later, these physically inspired models of opinion contagionhave been exploited in economics by Orl´ean [69, 70] for the analysis of mimetic behaviors inthe context of financial markets. There is now a large and growing literature on opinion andinnovation diffusion (see e.g. [89, 49, 26, 91, 84]) closely related to the general discrete choice3odel considered in this paper. Since the beginning of the 90’s the general framework of socialinteractions in non-market contexts is reconsidered in a Beckerian way [8, 10], in particular byGlaeser et al [41, 42, 40].The first application of statistical mechanics approaches in economics may be traced backto the pioneering work of F¨olmer [36]. Introducing an economic interpretation of the Isingmodel of ferromagnetism at finite temperature, he shows that strong externalities may hinderthe stabilization of an economy. These models introduce Markov random fields (equivalentlyBoltzmann-Gibbs distributions) to model uncertainty in the decision making process, allowingfor the definition of a general equilibrium concept. According to F¨olmer, Hildebrand’s [48] justi-fication of the representative agent approach breaks down when agents’ decisions are correlateddue to their social interactions (for a discussion, see also [54]).A renewal of interest for models of binary decisions with externalities arose in economicsin the 90’s. On one side, Durlauf and collaborators [27, 28, 29, 30] and Kirman and Weis-buch [94] among others, consider agents that choose an action according to a Boltzmann-Gibbsdistribution, that is a logistic choice function, reflecting some random aspects in the agent’sutility. In this context Brock [19] and Blume [13, 14] explicit the links between Game Theoryand Statistical Mechanics, while Kirman and coworkers [67] show that the logistic choice func-tion may be seen as resulting from an exploration-exploitation compromise. These and otherrecent papers [4, 31, 32, 20, 21, 40, 95, 35, 45, 65, 16] analyze with statistical physics tools theconsequences of positive social (market and non-market) interactions in the aggregate behaviorof large populations (for a short introduction to statistical physics approaches see [43] and fortheir application to economics see [72]; see also [7] for a survey). Most of the above mentionedauthors restrict the analysis of the model to the case where all the individuals have the sameidiosyncratic preference. Heterogeneity in the population is introduced through the probabilis-tic decision-making process ( random utility model [58, 59]), like in F¨olmer’s work [36]. Then,the actual equilibrium reached by the system depends on the fixed points of the decision dy-namics, generally a myopic best reply. An interesting characteristic of these models is that theypresent multiple equilibria for some range of the parameters. Becker [9] pointed out importantconsequences of these multiple equilibria, induced by externalities, on the economy: he suggeststhat they could be the reason of seemingly suboptimal pricing in situations of persistent excessdemand.In this paper we consider intrinsically heterogeneous agents with fixed utilities, like in Mc-Fadden’s approach to Quantal Choice models [62, 63]. Each individual has an
IdiosyncraticWillingness to Pay (IWP) that remains fixed on time. We mainly (but not exclusively) declinethe model within a market context, in which the binary choice corresponds to buying or not agiven good at a posted price. This general setting allows us to generalize Becker’s qualitativeanalysis [9] of the optimal pricing problem. Putting the price to zero allows us to recover thesocial sciences models. We assume that these IWP are random variables that are distributedamong the population according to a given probability density function (pdf).We determine the possible equilibria of the system without assuming any precise decision-making dynamics. We show that the model’s properties depend on the strength of the ex-ternality and on qualitative properties of the IWP pdf, like its modality class (the numberof maxima), its smoothness properties and the kind of support. We display the main resultson a plane whose axes are the parameters of the model, namely, the average IWP and thestrength of the social component, both measured in units of the standard deviation of the IWPdistribution. Particular cases of our model have been published elsewhere [66, 45]. This paperextends those result to the case of a general IWP distribution. The particularly important caseof a uni-modal pdf (with a single maximum) is thoroughly studied, but we also discuss theconsequences of multi-modality. Our results are summarized on phase diagrams , that is, weplot in the parameters space the boundaries of the regions where different types of solutionsexist. 4efore entering into the details of our approach (section 2) we discuss the analogies anddifferences between our model and other models of social interactions (section 1.2) and wesummarize our main results (section 1.3).
In this section we briefly discuss the relationship between the model to be considered hereand other models studied in the literature. Let us first consider models of discrete choicesin the absence of externalities. According to the typology proposed by Anderson et al. [1],within the general framework of
Random Utility Models (RUM)[58, 59] with additive stochasticutilities, there are two distinct approaches to individual choices: a “psychological” one and an“economic” one. In the psychological perspective (Thurstone [87], Luce [58]) the randomness is atime-dependent i.i.d. random variable: the random components of the idiosyncratic preferencesare assumed to be independently drawn afresh by each individual from a given pdf, each timethe choice has to be made. They are interpreted as individual temporary changes, or mistakesin the estimated utilities. In the simplest case —actually, the only one treated in the socialand economic literature— the agents IWPs have identical deterministic parts and only differby this random time-varying term which is systematically assumed to be drawn from a logisticpdf. In practice many approaches like in [20] consider the choice rule as deriving from a randomutility model [58]. As shown by McFadden [62], in this context the logistic form is obtained ifthe random terms in the underlying Thurstone’s discriminant process are i.i.d. Weibull randomvariables, i.e. have a double exponential (extreme value, type I) distribution (see also [1]).In the presence of strategic complementarities ([22, 23]), the resulting model is well knownin statistical physics: it corresponds to the standard Ising model , i.e. with ferromagnetic in-teractions and annealed disorder, that is, at finite temperature T . The latter is the inverse ofthe standard logistic parameter β ≡ /T and is thus proportional to the standard deviation ofthe IWP distribution. The ferromagnetic interaction constant J corresponds to the strengthof the social externality. Introduced by F¨olmer [36] in the economics context, this standardIsing model has recently been reconsidered in the social and economic literature mainly byDurlauf and coworkers [31, 32, 20, 21] and by Kirman and Weisbuch [94, 67]. The correspond-ing equilibria are reminiscent of the Quantal Response Equilibria [64] used in the context ofexperimental economics and behavioral game theory. These are equilibria “on the average”,in the statistical sense (as in Physics): they do not correspond to the strict maximization ofthe utilities (that are random variables) but to that of their estimated or expected values. Inthe generally considered infinite population limit (where the variance of the expected valuesvanishes) the expected utilities are systematically smaller that the maximal ones.The standard Ising model is quite well understood [85]. Ising himself [51] gave an analyticdescription of its properties in the simplest case of a linear chain where each agent interacts onlywith his right and left nearest neighbors. There is also an analytical description of the stationarystates of the model, due to Onsager [68], in the case where the agents are situated on the verticesof a 2-dimensional square lattice, each having four neighbors. However, no analytic results existfor arbitrary neighborhoods except for the specific case of a global neighborhood, known as the mean field approximation in Physics. Accordingly, considering global neighborhoods, Brock andDurlauf [20] analyze the properties of the expected demand in the case of rational expectationsunder the assumption of a logistic distribution of such expectations (assuming thus doubleexponential random utilities). They find, in agreement with standard results in statisticalmechanics [85], that there exist either one, two or three solutions for the demand function,depending on the relative magnitudes of the idiosyncratic uniform social term, the variance ofthe stochastic term and the strength of the social effects.In the following we adopt instead McFadden’s [62] economic approach (see also [59, 1]):we assume that each agent has a willingness to pay invariable in time, that is different from5ne agent to the other. In statistical physics this heterogeneity is called quenched disorder.The particular model we study is analogous to the ferromagnetic Random Field Ising Model (RFIM) at zero temperature (corresponding to the fact that the agents make deterministicchoices). Thus, our modeling approach assumes the so called “risky” situation: an externalobserver (e.g. a seller) does not have access to the individual preferences, but may know theirprobability distribution. According to McFadden, “Thurstone’s construction is appealing to aneconomist because the assumption that a single subject will draw independent utility functionsin repeated choice settings and then proceed to maximize them is formally equivalent to a modelin which the experimenter draws individuals randomly from a population with differing, butfixed, utility functions, and offers each a single choice; the latter model is consistent with theclassical postulates of economic rationality” ([63], p 365). However, in the presence of socialinteractions this statement is in general incorrect. In a repeated choice setting, individualutilities evolve in time according to the others’ decisions. In fact, the equilibria of systemsreached through a dynamics that corresponds to an iterated game where agents make myopicchoices at each time-step is one of the main concerns of statistical physics. It is known thatthe equilibria of systems with annealed or quenched disorder are of very different nature. Thetime average on a single agent and the population average at a given time do not necessarilycoincide.In contrast to the standard Ising model at finite temperature (annealed disorder), the prop-erties of the RFIM with externalities, both at zero and at finite temperatures, are far from beingfully understood. The properties of quenched disordered systems have been and are still thesubject of numerous studies. Since the first studies of the RFIM, which date back to Aharonyand Galam [38, 37], a number of important results have been published in the physics literatureon this model (see e. g. [80]). Several variants of the RFIM have already been used in thecontext of socio-economic modeling, both by physicists and economists [39, 70, 18, 95].The quenched-utilities model (RFIM at zero temperature) and the annealed-disorder model(standard mean field Ising model at finite temperature) have the same aggregate behavior(i.e. demand function for the market case) and equilibria under the following conditions - butessentially only under such conditions:1. the choice function with annealed utilities is identical to the cumulative distribution ofthe quenched IWPs;2. in the annealed case, equilibrium is reached through repeated best reply choices, wherethe expected demand is myopically estimated;3. the population size is infinite, guaranteeing that the variance of the demand vanishes inboth models.However, the economic interpretation of these equilibria are very different: in the case ofquenched utilities these are standard Nash equilibria, while in the case of annealed utilitiesthese are similar (although not identical) to Quantal Response equilibria.
In the present paper we determine the equilibrium properties in the case of a global neighborhood with time invariant (quenched) random utilities, in the limit of an infinite number of agents.Since our paper focuses on equilibrium (static) properties, the social influence depends on the actual choices of the neighbors, in contrast with [20], where the social influence in the surplusfunction depends on the agent’s expected demand.Previous studies using annealed or quenched utilities consider specific probability distribu-tions, mostly a logistic or a Gaussian [87]. Some papers have determined conditions on the6hoice function for having multi-equilibria [67, 40]. From the Physics literature we expect thatspecific properties near a critical point (a bifurcation point, see Section 3.2.1 below) are inde-pendent of the details of the model: this is used in [80] for describing the hysteresis effects ina family of (physical) systems at such a critical point, and exploited in [65] for the analysis ofempirical socio-economic data in cases where the actual pdfs are not known. However, the fulldescription of the phase diagram for an arbitrary pdf has not been done yet. Here we presentthis detailed analysis for a typical probability distribution of the IWP. We show how uniquenessor multiplicity of equilibria, related to convexity properties of the inverse demand functions,result from modality and smoothness of the pdf, as well as from the strength of the externality.We show that for small enough strengths of the social influence (the case of moderate socialinfluence in [40]), the demand has a classical shape, that is, with a continuous decreasingadoption rate for increasing prices. However, if the ratio between the social influence strengthand the standard deviation of the IWP distribution exceeds a critical value, the inverse demandfunction exhibits a non-classical, non-monotonic, behavior. As a result, depending on the price,there are either one or two stable equilibria for the demand: the positive (additive) externalitiesin a market context may give raise to a family of non-monotonic demand curves generalizingthus the classical ones.Beyond this first main result, we exhibit the generic properties of the boundaries limitingthe regions where the system presents different types of equilibria (unique or multiple). Wecall these properties generic since we show that they depend only on qualitative features ofthe IWP distribution: modality (number of maxima), smoothness (continuity and derivabilityproperties) and type of support (compact or infinite). The main results are summarized as phase diagrams in the space of the relevant parameters of the model, namely (i) the socialinfluence strength and (ii) the difference between the population average of the IWP and theposted price, both parameters conveniently normalized by the standard deviation of the IWPdistribution (which measures the importance of the heterogeneity).
The paper is organized as follows. In Section 2 we present the model: we first (section 2.1)specify the agents (customers) model, then in 2.2 we introduce a normalized form of the basicequations which is convenient for analyzing the demand, and in 2.3 we show on two simpleextreme cases what to expect from these equations. In 2.4 we detail the families of probabilitydistributions of the IWP covered by this paper.In Section 3 we analyze the aggregate demand (its collective behavior) for a generic smoothpdf. In 3.1 we introduce and study the direct and inverse demand functions; the demandphase diagram is derived in Section 3.2: in 3.2.1 we obtain the domain of multiple solutionswhich allows to plot the phase diagram (section 3.2.3). We analyze in details the vicinity ofthe bifurcation point (section 3.2.4) and the question of Pareto optimality (section 3.2.5). Asummary of the demand properties is given in 3.3.We leave to the Appendix A the analysis of other kinds of pdfs, where we study the demandphase diagrams of IWP distributions with compact support (section A.1) and with fat tails(section A.2). The demand phase diagram for a pdf with an arbitrary number of maxima isstudied in A.3 —in details for a smooth multimodal pdf in Section A.3.1, and on a simpleexample of singular bimodal distribution in section A.3.2—.Finally we summarize the main results and give several perspectives for further research inSection 4. 7
Model of discrete choices with heterogeneous agentsand positive externalities
We consider a population of N agents ( i = 1 , , . . . , N ). Each individual i has to make a binarychoice. Depending on the context, this binary decision may represent the fact of buying ornot a good, adopting or not a given standard, adopting or not some social behavior such asjoining a riot [46], or a journal club [77], [78], etc. Formally each agent i must choose a strategy ω i in the strategic set Ω = { , } ( ω i = 1 denotes to buy/adopt/join, ω i = 0 otherwise) .Hereafter, without loss of generality, we will refer to the simplest market situation where theagents are customers who must choose whether to buy or not a single good at a price P . Ourmain concern is with the agents’ behaviors, and P is considered as an exogenous parameter—e.g. it is posted by a monopolist selling the good—. Non-market models like those recentlyconsidered by, i.e., Glaeser et al. [40] are obtained by setting P = 0 or by considering P as anexogenous social cost, common to all the agents. We are interested in the collective outcome ofthe agents decisions. In a following paper [44] focusing on the market context we will analyzethe consequences of the customers collective behavior on a monopolist’s program for fixing theoptimal price.The population is heterogeneous. Each individual i has an idiosyncratic preference or will-ingness to pay/adopt (hereafter IWP) H i , meaning that in the absence of social influences, anagent i adopts the state ω i = 1 if H i is larger than the price P . Following Mc-Fadden [62] andManski [59], we work within the framework of Random Utility Models (RUM): we assume thatthe H i are time independent random variables independently and identically distributed (i.i.d.)in the population. Denoting by H the mean and by σ the variance of the IWP distribution,hereafter we assume that the random variable ( H i − H ) /σ is distributed according to: P ( x < H i − Hσ < x + dx ) = f ( x ) dx, (1)so that f is a pdf with zero mean and unitary variance. In the RUM view point, the agentshave utilities H i and H i for not adopting and adopting respectively, with H and H thecorresponding population averages. For k = 0 ,
1, one writes H ki = H k + (cid:15) ki , where (cid:15) ki areindependent random variables with zero mean and standard deviation σ k , with not necessarilyidentical pdfs for k = 0 and k = 1. Then in our model we have H i = H i − H i , H = H − H and f is the pdf of the normalized difference x i = ( (cid:15) i − (cid:15) i ) /σ , with σ = ( σ ) + ( σ ) becauseof the additivity of the variances of independent variables. As particular examples, if (cid:15) ki areGaussian variables, then x i is also Gaussian; if (cid:15) ki are uniformly distributed on, say the intervals[ − a k , a k ], then x i has a trapezoidal pdf, which becomes a triangular distribution if e.g. a = 0(no uncertainty in the utility of not adopting). In the following we do not assume any specificform of the pdf of x . In Section 2.4 below, we present in details the class of pdfs considered inthis paper.If all the individuals had the same IWP, the outcome in absence of social interactions wouldbe very simple: either the price is below this common value, and everybody buys, or it is above itand nobody buys. All the individuals would behave in the same way, obtaining the same payoffs,and in the market aggregate analysis, they may be replaced by a fictitious representative agent [54]. In the case of a heterogeneous population considered here, only the agents with H i ≥ P would buy at price P , but getting different payoffs.The situation is more complex when the decision of each agent depends also on the decisionsof others ([59] and references therein). We assume that each agent is the more willing to pay Some authors use the notation s i = 1 and s i = −
1; both encodings are equivalent: it suffices to replace ω i = ( s i + 1) / if agent i buys at the posted price P , his surplus is S i = H i + Jη − P, (2)where η is the fraction of buyers in the population. Taking into account the definition of ω i : η ≡ N N (cid:88) i =1 ω i . (3)We assume also that the externality Jη corresponds to strategic complementarities, i.e. thatthe strength of the social influence is positive: J > .The actual surplus of agent i is: W i = S i ω i . (4)In order to maximize his surplus, agent i should buy/adopt ( ω i = 1) if S i >
0, but not ( ω i = 0)when S i <
0. Since the IWP are i.i.d., when N is very large (more precisely, in the limit N → ∞ ), by the law of large numbers, the fraction of buyers (3) —which is the average of ω i —converges to the expected value of ω i over the IWP distribution. Thus, η is given by the fixedpoint equation: η = P ( H i − P + Jη > . (5)The marginal customer m , indifferent between adopting or not, is defined by the condition ofzero surplus, S m = 0: H m − P + Jη = 0 (6)so that (5) may be written as η = P ( H i > H m ) . (7)For what follows it is more useful to write (5) as η = P ( H i − H > − S ) (8)where S = S ( J, H, P ; η ) ≡ H − P + Jη (9)is the population average of the ( ex ante ) surplus S i . It depends on the parameters J and H ,which are properties of the customers population, and P , the exogenous price. Notation.
Generally, upon manipulating functions, we put in parenthesis the parametersin front, separated with a semicolon (;) from what we consider the variable. Our notation S ( J, H, P ; η ) in (9) indicates that J , H and P are considered as parameters, whereas η is thevariable. Sometimes, when the context is clear, we drop down the parameters and keep onlythe variable, writing thus S ( η ). Whenever we consider functions of two variables, they areseparated by a simple colon, like in equations (37) and (38), and in Appendix A.3. More generally, the social term may be proportional to the fraction of buyers in an individual-dependingsubset of the population, called “neighbors” of agent i . In this paper, we consider a global neighborhood, whereevery agent has social connections with every other agent, mainly because this case can be studied analytically. .2 Aggregate behavior: normalized equations Clearly, the fraction of buyers η depends on the strength of the social influence J , the price P and the average willingness to pay in the population H , and on the distribution of thedeviation of the IWP H i from its population average H . The agents choices depend only onthe surplus sign, and they are invariant under changes of the surplus scale. Since the surplus islinear, we can formally multiply every term of the surplus by a same strictly positive numberwithout changing the agents’ choices. An adequate scale is given by the typical scale of theIWP distribution: it is convenient to measure each quantity ( J, H, P ) in units of the width σ ofthe IWP pdf. Hence instead of four parameters, we are left with three independent parameters.Hereafter we will thus work with the following normalized variables j ≡ Jσ , h ≡ Hσ , p ≡ Pσ (10)In addition, as it is obvious from equations (8) and (9), η depends on the price P and theaverage willingness to pay H only through their difference H − P . We introduce the normalizeddifference: δ ≡ H − Pσ = h − p, (11)which is the average ex-ante surplus in the absence of externality. For short hereafter we call δ the bare surplus . In non-market models ( p = 0) it is the average willingness to adopt. Remark.
In (almost) all the following we will work with the above reduced variables (10),(11), referring to them as the (normalized) strength of social influence, average willingness topay, price, and bare surplus. However one should keep in mind, especially when interpreting theresults, that they represent the ratios of the non normalized parameters to the width of the IWPdistribution. Clearly other normalizations are possible. An alternative of particular interest isthe normalization obtained by measuring every quantity in units of the social strength J : therelevant parameters are then˜ σ ≡ σJ , ˜ h ≡ HJ , ˜ p ≡ PJ , ˜ δ ≡ H − PJ = ˜ h − ˜ p (12)(equivalently one can do as if J = 1). Note that this choice of normalization is no more than anequivalent representation of the parameters space; indeed one has ˜ σ = 1 /j, ˜ h = h/j, ˜ p = p/j .It is also interesting to analyze the results in term of the set of parameters (12), which allow tostress the model properties as a function of the degree of heterogeneity (relative to the strengthof the social influence) —a homogeneous population corresponding to the limiting case ˜ σ = 0,a highly heterogeneous one to a large ˜ σ —.With the normalized variables (10), (11), equation (8) becomes η = (cid:90) ∞− s f ( x ) dx = 1 − F ( − s ) , (13)where F is the cumulative probability distribution and s = S/σ , with S defined by (9), dependson h and p through the bare surplus δ , that is s = s ( j, δ ; η ) ≡ δ + jη. (14)If the pdf has infinite support, F ( − s ) ≡ P ( x ≤ − s ) = (cid:90) − s −∞ f ( x ) dx. (15)10n the case of a compact support [ x m , x M ], one can write: η = 1 − F ( − s ) = (cid:90) max { x M , − s } max { x m , − s } f ( x ) dx. (16)Obviously, when − s < x m , we have η = 1, and when − s > x M we have η = 0. In the absence of social influence the problem is simple because after introduction of j = 0in (14) we obtain s = δ which does not depend on η . Then, due to the monotonicity ofcumulative distributions, the fraction of buyers (13) is a monotonically increasing function of δ (equivalently, at fixed h , a decreasing function of the price p ): η = 1 − F ( − δ ) . (17)Another extreme case is that of a homogeneous population: H i = H for every i —a sit-uation obtained in the singular limit ˜ σ = 1 /j → η = 0 or η = 1. For each agent the surplus in case of adoption would be H − P if no other agent adopt ( η = 0), whereas if η = 1 the surplus is H − P + J . In fact, η = 0is a solution for H < P , while η = 1 is a solution for H > P − J . Hence there is a domain, P − J < H < P , where the two solutions coexist. The whole population behaves as a singleagent who either does not adopt, η = 0, or adopts, η = 1, with a different surplus depending onwhether he is “in” or “out of” the market: this is analogous to the problem of multi-equilibriawith hysteresis in trade analyzed by Baldwin and Krugman [6], except that here the problemarises only at the collective level .We have thus on one side, for J = 0 and σ finite, a unique well behaved equilibrium, and for J > σ = 0 a situation of multiequilibria. The question addressed in the following aimsat understanding what happens “in between”. We will show that when the social interactionstrength is large enough compared to the heterogeneity width, the demand faces a complexproblem. More precisely, there is a critical value j B of j = J/σ . Below it, the fraction of buyers(equation (13)) follows monotonically the price variations. Beyond j B , equation (13) presentsmultiple solutions. Among them, the (possibly multiple) Nash equilibria are those solutionsthat have an economic meaning, i.e. for which the demand decreases when prices increase.The Section 3 of the paper is devoted to a detailed study of the nature of the solutionsof equations (13) and (14) with j ≥ Since we are interested in the generic properties of the model, we explicit the general charac-teristics of the idiosyncratic willingness-to-pay (IWP) distributions covered by our analysis.Since a pdf must be integrable, f ( x ) (equation (1) ) must vanish in the limits x → ±∞ .For sufficiently regular pdfs, this can happen in two different ways: either the pdf decreasescontinuously to 0 as x → ±∞ , or it is strictly zero outside some compact support [ x m , x M ].Most of the analysis in this paper is restricted to the class of pdfs obeying to the followinghypotheses:H1. Modality : f is unimodal , that is it has a unique maximum f B ≡ sup x f ( x ) . (18)112. Smoothness : f is non zero, continuous, and at least piecewise twice continuously differ-entiable inside its support, ] x m , x M [ , where x m and x M may be finite or equal to ±∞ .In the latter case f is stricly monotonically decreasing towards zero as x → ±∞ .H3. Boundedness : the maximum of f , f B (that may be reached at x m or x M if these numbersare finite), is finite : f B < ∞ (19)Within the class of pdfs satisfying H1, H2 and H3, we will consider more specifically theimportant following prototypical cases:1. Unbounded supports : The support of the distribution is the real axis; the pdf is continuousand twice continuously derivable on ] − ∞ , ∞ [, with a unique maximum. A typical ex-ample, relevant to economics (see e.g. [1]), is given by the logistic distribution, althoughmore generally we do not assume that the pdf is symmetric. We make the followingsupplementary hypothesis, that amount to impose that the pdf decreases fast enough for x → ±∞ :H4. Mean value : the pdf has a finite mean value. Then, the smoothness condition H2imposes that f decreases when x → ±∞ faster than | x | − .H5. Variance : the pdf has a finite variance. Then, the smoothness condition H2 imposesthat f decreases when x → ±∞ faster than | x | − .2. Compact supports : the support of the distribution is some interval [ x m , x M ], with x m and x M finite; the pdf is continuous on [ x m , x M ] and continuously derivable on ] x m , x M [, witha unique maximum on [ x m , x M ]. Note that, since f has zero mean, x m < < x M .Hypothesis H2 and H3 exclude cases where the pdf is not a function but a distribution —containing, e.g., a Dirac delta —. Clearly, if the pdf’s support is the real line, ] − ∞ , + ∞ [,the boundedness hypothesis H3 is a consequence of the smoothness hypothesis H2. In thecase of compact supports, H3 excludes pdfs diverging at a boundary of the support. Althoughhypothesis H3 is actually true under H2 if f is continuous on the closed interval [ x m , x M ], weexplicit it because some of our results are valid under H3 even for pdfs less regular than thosesatisfying H2.Although hypothesis H5 is not necessary for the study of the aggregate demand, it corre-sponds to a wide family of realistic distributions for which one can conveniently use the standarddeviation as the unit for measuring the relevant parameters (i.e., using normalization (10)).Generic results for unbounded support pdfs satisfying H1 to H5 are presented in the mainbody of the paper. They extend previous results obtained for a logistic distribution [66]. Theanalysis of other types of pdfs is left to Appendix A:- In Appendix A.1 we present general results for bounded support pdfs. The case of auniform distribution on a finite interval [ x m , x M ], which corresponds to an interestingdegenerate case ( f is maximal at every point within the interval), has been presentedelsewhere [45]. The particular case of general a triangular pdf is explicitly worked out forillustration.- In Appendix A.2 we extend the analysis to fat-tail distributions, which correspond toan important limiting case of pdfs with infinite variance (for such distributions, the nor-malization constant σ in (1) and in equations (10) and (11) is no longer the standarddeviation, but an arbitrary positive constant setting the units of H , J , P and C ).12 Finally, in Appendix A.3 we extend the discussion to the case of multimodal pdfs (distri-butions with an arbitrary number of maxima): we derive the demand phase diagram fora generic smooth multimodal pdf and we discuss the case of a singular pdf using as anillustrative example a pdf with two Dirac peaks. In this section we discuss the demand function, that is the relationship between price p andfraction of buyers (or adopters, in non market contexts) η , expressed by equation (13). Aswe have already seen, this means studying the relationship between η and the bare surplus δ = h − p , and how it depends on the externality parameter j . We show that, for a large rangeof values of the parameters j and δ , the demand presents two equilibria which can be qualifiedas Nash equilibria from a game-theoretic point of view. This result is valid for any pdf satisfyingthe general hypothesis described in the preceding section. The expected demand η d at a given value of δ is obtained as the implicit solution of (13) and(14). As we will see, the application η → − F ( − s ( η )) may be a multiply valued functionof η ; it is thus preferable to express δ , or p = h − δ , as a function of η , and determine the inverse demand function p d ( η ), that is, the price at which exactly N η units of the good wouldbe bought .Under the hypothesis H2 the cumulative distribution F is a continuous and strictly mono-tonic function on ] x m , x M [ and has a unique inflexion point. Hence it is invertible. Denoting Γthe inverse of 1 − F ( − s ), we have the following equivalence: η = 1 − F ( − s ) ⇐⇒ s = Γ( η ) , (20)with s defined by (14). For unbounded supports, Γ( η ) increases monotonically from −∞ to + ∞ when η goes from 0 to 1 (see figure 1 for an example, and Appendix A for other cases). In thecase of a compact support [ x m , x M ], Γ( η ) takes the finite values, Γ(0) = − x M , and Γ(1) = − x m for η respectively 0 and 1. Note that neither we assume f to be symmetric nor to have itsmaximum at x = 0.Replacing s in the r.h.s. of (20) by its expression (14) yields δ = D ( j ; η ) , (21)with D ( j ; η ) ≡ Γ( η ) − jη. (22)Interestingly, D ( j ; η ) depends on the parameter j but not on h . Actually, in Section 3.2 and inAppendix A.3, we will have to consider D ( j ; η ) as a function of the two variables, j and η . Inthe present subsection however, we consider D ( j ; η ) as a function of the single variable η , with j as a (fixed) parameter - hence according to our convention on notations introduced in Section2.1, we keep in mind the dependency of D on j by writing D ( j ; η ), and derivatives of D ( j ; η )with respect to η are denoted D (cid:48) ( j ; η ).Plots of D ( j ; η ) against η for different values of j are presented on figure 2 for the logisticdistribution. Solutions to (21) correspond to the intersections of these functions with horizontallines at y = δ . In non market models, where generally p = 0, results in this section give the aggregate choice Nη d ( j, h ) asa function of j and h . This is the relationship between the fraction of adopters, the average willingness to adoptof the population and the strength of the social interactions. . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0- 505 G ( h ) G ’ ( h ) G ’ ’ ( h ) h Figure 1: Γ( η ) and derivatives as a function of η for the logistic pdf of unitary variance. Remark:all these functions diverge at η = 0 and at η = 1 . ( h B , d B )( h U , d U ) h j = 1 j = j B j = 5 D ( h ) ( h L , d L ) Figure 2: D ( j ; η ) as a function of η in the case of a logistic distribution of the IWP, for threevalues of j : j = 1 < j B , j = j B = 2 . and j = 5 > j B . D (cid:48) ( j ; η ) ≥ . (23)Thus, the intersections of y = δ with D when D (cid:48) < j and a given value of δ , the solution η = η d ( j ; δ ) of equation(21) satisfying (23) gives the expected demand N η d ( j ; δ ) at a price p = h − δ .From the definition of δ and (22), the inverse demand function is thus p d ( η ) = h − D ( j ; η ) . (24)This function depends on both parameters h and j , and when necessary we will write p d ( η ) = p d ( h, j ; η ).As we will see, the demand η d ( j ; δ ) can be a multivalued function of δ for some range ofparameters. On the contrary, since the function Γ( η ) is a uniquely defined function of η , so is D ( j ; η ). This is the reason why, instead of considering (13), we prefer to obtain the propertiesof the demand η d ( j ; δ ) from the analysis of equations (21) and (22).Under assumption H2, Γ( η ) is at least piecewise three times continuously derivable on ]0 , (cid:48) is continuous and strictly positive. In particular, at any point η in ]0 , D (cid:48) ( j ; η ) = Γ (cid:48) ( η ) − j. (25)In the case of a compact support, the above equation also holds for the right and left derivativesat, respectively, η = 0 and η = 1.In terms of the pdf f , Γ (cid:48) ( η ) = 1 f ( − s ) , with s = Γ( η ) . (26)Under H1, Γ (cid:48) has a unique absolute minimum (qualitatively there is a unique point where thecurvature of Γ changes from convex to concave; if Γ is smooth, it has a unique inflexion point).Thus min η Γ (cid:48) ( η ) = 1 f B > . (27)This minimum is reached at some value η = η B : η B ≡ arg min η Γ (cid:48) ( η ) . (28)If f is smooth enough at its maximum, then Γ (cid:48)(cid:48) ( η B ) = 0: η B is the inflexion point of Γ. Forsymmetric pdfs, η B = 1 /
2, but we do not restrict to this case.As a consequence of the properties of Γ( η ), we see from equation (25) that D (cid:48) ( j ; η ) is strictlypositive for j < j B , with j B ≡ Γ (cid:48) ( η B ) = 1 f B . (29)The value j B separates two regions where the model presents qualitatively different behav-iors. When j < j B , the function D ( j ; η ) is strictly increasing from −∞ to + ∞ as η goes from0 to 1. As a result it is invertible: for any δ in ] − ∞ , + ∞ [, equation (21) has a unique solution η d ( δ ).If j > j B , there is a range of values of δ for which (21) has several solutions.The existence of multiple solutions in the demand is thus a generic property of discrete choicemodels with heterogeneous agents and social interactions (externalities). This is true whateverthe number of maxima of f , as shown in Section A.3 of Appendix A. Actually, the domainwhere there is a unique solution, that is 0 ≤ j ≤ j B = 1 /f B , is very narrow if f B is large: a15 . 00 . 20 . 40 . 60 . 81 . 0 - 8 - 6 - 4 - 2 0 2 4 d B d U d L h L h B h U dh d ( j ; d ) j = 1 j = j B j = 5 m u l t i p l ee q u i l i b r i a Figure 3:
Demand η d ( j ; δ ) as function of δ ≡ h − p in the case of a logistic distribution of theIWP, for three values of j : j = 1 < j B , j = j B = 2 . and j = 5 > j B . Notice that theorigin of the horizontal axis ( δ = 0 ) corresponds to h = p . Remark that prices increase from right to left . Unstable solutions: the negative slope curve joining ( δ L ( j ) , η L ( j )) to ( δ U ( j ) , η U ( j )) for j = 5 is not shown. leptokurtic distribution will have in general a narrower domain of unicity of the demand thana platykurtic distribution of same variance.In our case of unimodal pdfs, equation (21) may have three solutions for j > j B (seefigure 2). The intermediate solution, laying on a branch with D (cid:48) ( j ; η ) < η increasesas δ = h − p decreases— is sometimes called a critical mass point in the literature [77]: itcorresponds to a demand that would increase for increasing prices. Hence, in a tatonnement dynamics, this corresponds to an unstable solution separating the basins of attraction of thetwo stable equilibria. The marginal case, j = j B , is a bifurcation point (hence the subscript B )where multiple solutions to (21) appear on increasing j . The stable equilibria of the demandthat satisfy (23) are represented against δ on figure 3. Let us consider more in details the behavior of the application δ → η d ( δ ) in the case of asmooth unimodal pdf on ] − ∞ , + ∞ [. Considerations specific to compact support pdfs are leftto Section A.1 of Appendix A.The functions Γ( η ) corresponding to pdfs satisfying H1 to H5 are at least three times con-tinuously derivable on ]0 , −∞ and + ∞ as η goes to 0 and 1 respectively.We have already seen that for j < j B there is a unique solution, and η d goes from 0 to 1 as δ = h − p goes smoothly from −∞ to + ∞ .For j > j B , (21) has 3 solutions whenever δ U < δ < δ L (see figure 2), where δ L and δ U arethe values of δ that satisfy the equality (marginal stability condition) in equation (23). Thatis, the boundaries of the region with multiple solutions are the values for which D ( j ; η ) has a16orizontal slope (see figure 2): D (cid:48) ( j ; η ) = 0 (30)which is equivalent to dp d ( η ) dη = 0 . (31)Considering the definition (22) of D , this means that on these boundaries D ( j, η ( j )), as afunction of j , is the Legendre transform of Γ( η ). Under our hypothesis H1, Γ (cid:48) has a uniqueminimum, and necessarily tends towards + ∞ as η goes to either 0 or 1; Γ is strictly convex on] η B , , η B [, hence the Legendre transform is well defined and uniqueon each one of these intervals: equation (30) for j > j B has indeed two solutions η L ( j ) and η U ( j ), given by j = Γ (cid:48) ( η Λ ) , Λ =
U, L. (32)with η L ( j ) < η B < η U ( j ) . (33)From the knowledge of η U ( j ) and η L ( j ), using (21) one gets the marginal stability curves δ U ( j ) and δ L ( j ), that is, the extreme values of δ bounding the region where multiple solutionsexist: δ Λ ( j ) = D ( j ; η Λ ( j )) = Γ( η Λ ( j )) − jη Λ ( j ) , Λ =
U, L. (34)As already stated, for δ U ( j ) < δ < δ L ( j ), equation (21) has three solutions. The curve η d ( j ; δ ) has two stable branches (see figure 3): an upper one η dU ( j ; δ ) with η dU ( j ; δ ) > η U ( j ) > η B ,and a lower one η dL ( j ; δ ) with η dL ( j ; δ ) < η L ( j ) < η B ; they are joined by a branch of unstablesolutions —the above mentioned set of unstable equilibria (see figure 3)—. The upper branchexists for δ ≥ δ U ( j ), the lower one for δ ≤ δ L ( j ). At the end points dη d dδ | L,U = ∞ . In otherwords, solutions with large fractions of buyers, i.e. high- η solutions, only exist for δ ≥ δ U ( j )whereas low- η solutions exist only if δ ≤ δ L ( j ). Since δ U ( j ) ≤ δ L ( j ), the system has multiplesolutions for the demand η d whenever δ U ( j ) ≤ δ ≤ δ L ( j ).For j = j B , these marginal stability curves merge at a single (degenerate) point δ L ( j B ) = δ U ( j B ) = δ B with δ B ≡ − Γ (cid:48) ( η B ) η B + Γ( η B ) (35)This defines the bifurcation point B in the ( j, δ ) plane, B ≡ { j B , δ B } . (36)One should note that η U,L ( j ) and δ U,L ( j ), which depend on j (and on the function Γ( . )), areindependent of h and p . In fact, the preceding analysis can be made more general because the main results may beobtained only based on the continuity and the convexity properties of Γ, without assuming anysmoothness properties of the derivatives of f . Let us consider this alternative.First, whatever the smoothness properties of f , the demand η d must be a decreasing functionof the price: the economically acceptable values of the equilibrium demand, η d ∈ [0 , δ increases ( p decreases). Thus, among the solutions of (21), the equilibrialie on the branches where D (defined by equation (22)), is an increasing function of η (fordifferentiable pdfs, this condition is given by equation (23) ).Next, let us analyze D ( j ; η ) as a function of η (see figure 2). By continuity of the functionΓ( η ), D ( j ; η ) is a continuous function of η ∈ [0 , η → D → −∞ , and as η → D → + ∞ . Since Γ is concave on ]0 , η B ], on increasing η from 0 within [0 , η B ], D ( j, η ) has17 d U ( j ) d L ( j ) d = h - p j B d B j B Figure 4:
Demand phase diagram on the plane ( j = J/σ, δ = ( H − P ) /σ ) , for a smoothIWP distribution (here the logistic). In the shaded region the demand presents multiple Nashequilibria. Outside this region, the demand is a single valued function of j and δ . a maximum, δ L ( j ), on this interval. δ L ( j ) is by definition the Legendre transform of Γ( η )restricted to ]0 , η B ]. For η ≥ η B , Γ is convex, and thus D ( j, η ) has a minimum δ U ( j ) on [ η B , η ) restricted to [ η B , D ( j ; η )increases with η .Now, for j < j B , the maximum on [0 , η B ] and the minimum on [ η B ,
1] of D ( j ; η ) are bothreached at η B , hence D ( j ; η ) increases monotonically as a function of η ∈ [0 , η d to equation (21) are unique monotonically increasing functions of δ for each j .As a result, the inverse demand (24) is a uniquely defined continuously decreasing function of η ∈ [0 , j > j B , the maximum δ L ( j ) is reached at η = η L ( j ) ∈ ]0 , η B [. Beyond this maximum, D ( j ; η ) decreases as η increases. The minimum δ U ( j ) is reached at η = η U ( j ) ∈ ] η B , η L ( j ) , η U ( j )[ containing η B where D ( j ; η ) decreases with η , from δ L to δ U . No value of η within this interval can be a stable economic equilibrium. Hence,for δ ranging between these extrema of D ( j ; η ) the demand η d ( j ; δ ) as a function of δ has twobranches, a lower one for δ ≤ δ L , with η dL ( j ; δ ) ≤ η L ( j ) < η B and and an upper one for δ ≥ δ U ,with η dU ( j ; δ ) ≥ η U ( j ) > η B .In the case of a continuously differentiable function, the preceding results are recovered,since the Legendre transforms —the above mentioned minimum and maximum of D ( j ; η ) for j > j B — are reached at the values of η solutions of (30). All this discussion based on convexityarguments can be extended to multimodal pdfs, that is to cases where the distributions of theIWPs have more than one maximum. This is done in Appendix A.3. The results of the preceding section may be summarized on a customers phase diagram in theplane ( j, δ ), where we represent the boundary of the multiple solutions region, as in figure4. These boundaries are the functions δ Λ ( j ), (Λ = L, U ) defined by equations (34), which18 B σ / J(H−P) / J Figure 5:
Demand phase diagram in the plane (˜ σ = σ/J, ˜ δ = ( H − P ) /J ) , for a smooth IWPdistribution (here the logistic). Inside the dashed region the demand presents multiple Nashequilibria. Outside this region, the demand is a single valued function of ˜ σ and ˜ δ . η ), one for η < η B and the other for η > η B . Note that in term of prices, the extreme values δ L ( j ) and δ U ( j ) correspond to prices p L ( j, h ) < p U ( j, h ) given by p Λ ( j, h ) = h − δ Λ ( j ) , Λ =
U, L. (37)By construction of the Legendre transforms, the branch δ = δ U ( j ) is concave, and the branch δ = δ L ( j ) is convex. In addition, under the smoothness hypothesis, along each branch of themultiple solutions region in the phase diagram: dδ Λ ( j ) dj = d D ( j, η Λ ( j )) dj = − η Λ ( j ) , ; Λ ∈ { L, U } . (38)This property may be easily checked by deriving (34) with respect to j and making use of(32). This means that the tangents to the boundaries have a slope given by the value of η that is marginally stable on the corresponding boundary (i.e. by the η value of the solutionwhich appears/disappears as one crosses the boundary). Consequently, in the phase diagram,the width along the δ -axis of the multiple solutions region increases with j as a result of theconvexity properties of the functions δ Λ ( j ), (Λ = L, U ). This may also be seen from (33), sincethe slope of the L boundary (the one corresponding to η L ) is larger than that of the U boundary(defined through η U ). At the bifurcation point B , these two boundaries merge, and, accordingto (38), have a common slope − η B .Referring back to figure 2, upon increasing δ from −∞ , we have the following picture: if j < j B , the fraction η increases smoothly from 0 and reaches its upper value 1 for δ → ∞ .That is, to each value of the bare surplus δ , —or each value of the average willingness to adopt,in non-market situations— corresponds a unique fraction of buyers/adopters. In the phasediagram, figure 4, these solutions lie on the white region. On the other hand, if j > j B , when δ reaches the value δ U ( j ), a second, high- η solution appears besides the low- η one. These solutionsco-exist for δ U ( j ) ≤ δ ≤ δ L ( j ). The low- η solution disappears when δ increases beyond δ L ( j ),leaving only the high- η solution. The parameter values for which there are multiple equilibriais the grey region of the phase diagram, figure 4. Notice that in this region, there exists a thirdsolution that we neglected because it corresponds to the unstable situation where the demandwould increase with the price (or decrease with the bare surplus).As mentioned in Section 2.2, it is also useful to consider the same results in terms of theparameters ˜ σ = σ/J and ˜ δ = ( H − P ) /J (see (12). The phase diagram in the plane (˜ σ, ˜ δ )is shown on figure 5. For large heterogeneity ( σ/J larger than ˜ σ B ≡ /j B ), there is a singlesmooth solution. For weak enough heterogeneity (˜ σ B < /j B ), there is a domain with multiplesolutions. In the limit ˜ σ →
0, one recovers the simple results for a homogeneous population, asbriefly discussed Section 2.3.
The vicinity of the bifurcation point B in the phase diagram is of particular interest. Under thesmoothness assumption H2, we can study analytically its properties. Let us consider j = j B + (cid:15) with 0 < (cid:15) <<
1. Expanding (32) about j B , remembering that Γ (cid:48)(cid:48) ( η B ) = 0, one gets, to thelowest order in (cid:15) , the singular behavior η dL,U = η B ± (cid:115) (cid:48)(cid:48)(cid:48) ( η B ) (cid:15) / , (39a) δ L,U ( j ) = Γ( η B ) − η B j B − η B (cid:15) ∓ (cid:115) (cid:48)(cid:48)(cid:48) ( η B ) (cid:15) / . (39b)20he above singular behaviors are typical examples of scaling properties which are universal :the same scaling is obtained for any smooth distribution. From studies in statistical physics oneexpects the exponents (e.g., here, 1 / η ) to depend mainly on the structure ofthe network of interactions: the exponents would be different at the analogous critical point forthe model with agents situated on the vertices of a d -dimensional square lattice and interactingonly with their nearest neighbors. Typically the exponents depend on d up to some criticaldimension d c , above which they become identical to the “mean-field” exponents, which arethose obtained here with the global neighborhood. For the present model, other universalscaling properties have been obtained, in relation with the hysteresis effects [80], and thesehave been used in order to analyze empirical socio-economics data [65]. In a related work (witha generalization of the model to more than two choices), Borghesi and Bouchaud [16] analyzeempirical data for which the social strength can be estimated, and is found to be close to thecritical value (the analogous of j B ).In [44], where we consider the supply side, it will be seen that the bifurcation point B = { j B , δ B } in the ( j, δ ) plane gives a singular point { j B , h B ≡ δ B } in the ( j, h ) plane which playsan important role in the phase diagram associated to the optimal strategy for the monopolist. Each one of the equilibria η d ( j ; δ ) discussed in the preceding section is a Nash equilibrium forthe customers, at a posted price p . In this section we show that, whenever multiple solutionsexist, that is for j > j B , the solution with the largest η is Pareto optimal. This is the solution η d ( j ; δ ) that satisfies η d ( j ; δ ) ≥ η U ( j ).Let us recall that if a customer i decides to buy, it is because his (normalized) surplus s i = δ + jη + x i , (40)is positive. When s i < w i = s i ω i (see Section 2).Consider now the two equilibria η d ( j ; δ ) in the region δ U ( j ) < δ < δ L ( j ) (see the curve D ( j ; η ) for j = 5 on figure 2). Let’s denote by η dL ( j ; δ ) the low- η equilibrium ( η dL ( j ; δ ) ≤ η L ( j )),and by η dU ( j ; δ ) the high- η equilibrium ( η U ( j ) ≤ η dU ( j ; δ )). In either equilibrium, the agents whobuy are those with x i > − δ − jη d ( j ; δ ). Since η dL ( j ; δ ) < η dU ( j ; δ ), agents with x i < − δ − jη dU ( j )are not buyers in any of the equilibria whereas agents with x i > − δ − jη dL ( j ) are buyers inboth equilibria. Those with − δ − jη dU ( j ) < x i < − δ − jη dL ( j ) are buyers only in the high- η equilibrium, and their utility is thus larger (strictly positive instead of zero) in that case.Moreover, even those agents that would buy in both cases have a larger surplus if the realizedequilibrium is the high- η one. Hence, in the high- η equilibrium all these agents have a largersurplus than in the low- η one. This situation with two possible Nash equilibria, where thestrictly dominant one may be risk dominated, is reminiscent of coordination problems in gametheory. For a detailed analysis of this analogy see Phan and Semeshenko [74]. The presentanalysis shows that coordination problems may arise in systems with heterogeneous agentswhenever the externalities are strong enough.Whether a Nash equilibrium —and which one in the case of multiple equilibria— is actuallyrealized depends on the rationality of the agents and the information they have access to.In the context of bounded rationality and of repeated choices, a natural hypothesis is thatagents estimate what will be the fraction of adopters, and may base their estimate on previousobservations. In this paper we will not discuss these issues, that we are currently analyzing.Some partial results (dynamics with myopic agents and with various reinforcement learningparadigms) are discussed elsewhere [45, 79]. 21igure 6: Inverse demand p − h ( = − δ ) as a function of η for different externality strengthvalues j , illustrated on the case of a logistic IWP distribution. To summarize this section, if the social influence is small enough to satisfy the condition j < j B ,at each value of the bare surplus δ = h − p , which measures the gap between the populationaverage willingness to pay and the price, there is a unique solution η d ( j ; δ ) to equation (21).This demand is a monotonic increasing function of δ . However, if the social influence is largeenough ( j > j B ), there is a range of values δ U ( j ) ≤ δ ≤ δ L ( j ) for which two different (stable)solutions exist, a high demand one ( η d ( j ; δ ) ≥ η U ( j ) > η B ) and a low demand one ( η d ( j ; δ ) ≤ η L ( j ) < η B ). In this region, the customers are faced with a coordination problem. If δ ismodified dynamically within this range, the demand may jump abruptly between these twosolutions, a situation analogous to so called first order phase transitions in physics. Outsidethe range [ δ U ( j ) , δ L ( j )], there is a single solution, like in the small j case.We showed that the threshold j B , which corresponds to the onset of a bifurcation in thecustomers phase diagram, is determined by the maximum f B of the IWP pdf: j B = 1 /f B .Although most of the analysis has been done for smooth pdfs, we have shown that the genericbehavior stems only from convexity properties of the function Γ( η ), that is from the fact thatthe pdf f ( x ) is strictly increasing for x < x B and strictly decreasing for x > x B , where x B isthe mode of f .Some specific properties which arise for distributions with compact support are discussed inappendix A.1, and the case of distributions with infinite variance is studied in appendix A.2.In appendix A.3 we extend the results of this section to multimodal distributions.In market contexts (and in particular for the market analysis of [44]), it is useful to considerthe inverse demand p d instead of δ , as in standard approaches. In figure 6 we plot the valuesof δ = p − h as a function of the demand η and the strength j of the social externality for thecase of a logistic distribution. 22 Conclusion and perspectives
The model of collective behavior considered in this paper, under the general hypothesis detailedin Section 2, may be declined in both non-economic and economic contexts. In the first case,one is interested in the fraction of adopters, which corresponds to studying the demand functionfor an exogenous price in the second case.Like in many other models in the recent literature, we consider optimizing agents makingbinary choices, with willingnesses to adopt that depend additively on an idiosyncratic part(IWP) and on the choices of other agents. The population is intrinsically heterogeneous: theIWPs are drawn from some distribution of mean H and variance σ . In contrast with othermost studied models, here the idiosyncratic willingness-to-adopt heterogeneity is frozen: itdoes not result from (time varying) random shocks. In other words, each agent’s choice isdeterministic with a well known (to him) IWP, and we concentrate on the aggregate behaviors.We analyze the equilibrium properties (Nash equilibria) characterized by the emergence of acollective behavior resulting from the combined effect of externalities and heterogeneity.Our results, for global uniform interactions —a global social influence of uniform strength—in the limit of an infinite population —through the application of the central limit theorem—are summarized on phase diagrams . The axes of a phase diagram are the relevant modelparameters. In the case of the demand, the parameters are: δ , the bare surplus (that is, inthe economic context, the gap between the average IWP and the posted price), and j , thesocial influence strength, both parameters being measured in units of the standard deviation σ of the idiosyncratic term distribution (see 2.2). In this space of parameters, one drawsthe boundaries between regions (“phases”) of qualitatively different collective behaviors. Theboundaries, where “phase transitions” occur, are lines of non-analyticity (e.g. the demand isdiscontinuous on the boundary). In our problem, the main feature characterizing a given regionis the number of equilibria in this region.From a constructivist point of view, our model encompasses the classical downward slopingdemand curve as a particular case. Indeed, one of the main results for the demand is that, forvery general IWP distributions, there is a region in the phase diagram with multiple equilibria.More precisely, if the IWP distribution is mono-modal, there are two Nash equilibria for any j larger than a distribution-dependent value j B . For smaller values of j , the (Marshallian) demandcurves are, ceteris paribus , downward sloping (i.e. monotonically decreasing with increasingprices). For large externality strengths ( j > j B ), when the population average willingness topay is small enough, the demand becomes not-monotonic (as in Becker’s example [9]). This isa very general property of the model with additive externalities, and does not depend on theparticular statistical distribution of the idiosyncratic preferences. We also discuss (althoughmore briefly) the results for multimodal distributions —for which there exist several regionswith multiple equilibria, with possibly more than two equlibria for some of them—, and presentdetailed analysis of many illustrative examples.An important contribution of this paper is to exhibit the detailed properties of the bound-aries of the regions (in the parameters space) where multiple solutions exist. These propertiesare generic, in that they depend only on qualitative features of the IWP distribution.Future work may extend the results presented in this paper in several directions. First, theindividual preferences may include a stochastic (noise) term like in [36, 33, 94, 19, 13, 14, 67, 4],on top of the idiosyncratic one. Second, the present paper concentrates on the equilibriumproperties – that may be considered as the “static” analysis of long term equilibria in theMarshallian tradition. Further studies should focus on the process that makes the systemreach one or the other of the possible equilibria. A first study, implying revision of beliefsin an repeated choice setting, has already been published [79]: in the region with multipleequilibria, interesting complex dynamics occur with a large family of different equilibria beingreached, depending on the particular learning rule used by the agents. Third, literature on23arketing and studies of social psychology shows that choices very often depend on imitationeffects or social influence. For example, the existence of externalities in the Communicationand Information Technologies (CIT) sector is well established, and may result in a multiplicityof equilibria [76]. This may arise in other sectors also. Yet, empirical and econometric studiesallowing identification of the corresponding preferences distributions and the strength of thesocial influence are lacking [15]. Fourth, the influence of social networks topologies deservesfurther attention. Ioannides [50] reported results for the “Thurnstone model, i.e. homogeneousIWP and stochastic (logistic) choices, mainly for tree-like and one-dimensional networks withnearest-neighbor interactions. It would be interesting to explore how the phase diagrams forthe model considered here —heterogeneous IWPs and deterministic choices— are affected byshort range interactions. Analytical and simulation studies on the RFIM [80, 73] show thatthe heterogeneity introduces hysteretic effects in the dynamics, with interesting path-dependantproperties (return-point memory effect [80]). A statistical method to calculate the return pointsexactly, starting from an arbitrary initial state, has been recently proposed [82] for the simplecase of a one dimensional periodic network with nearest-neighbor interactions (called cyclictopology in [50]). The impact of such properties on both individual and collective economicbehavior remains to be investigated.In a companion paper [44], we will discuss within the market context the profit optimizationby a monopolist fixing the price. Other extensions of the results of this second part involve thestudy of how, with repeated choices, the long term equilibria depend on the entangled dynamicswhere customers and monopolist learn from each other. Finally, at least two directions areworth to be explored: the case of an oligopolistic competition and the consequences of Coaseconjecture in the case of choices with externalities involving a durable good – an issue alreadyaddressed in the literature [61], but not yet in the regime where multiple equilibria exist. Acknowledgements
This work is part of the project “ELICCIR” supported by the joint program “Complex Systemsin Human and Social Sciences” of the French Ministry of Research and of the CNRS. M.B. G.,J.-P. N. and D. P. are CNRS members. This work has been partly done while M.B. G. and V.S. were with the laboratory LEIBNIZ, CNRS-IMAG, Grenoble.24 eferences [1] S. P. Anderson, A. de Palma, and J.-F. Thisse.
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Triangular pdf of unitary variance and a maximum at x B = − . Left: Γ( η ) and itsfirst derivative. Right: D ( j ; η ) for different values of j . The values of η L , δ L , η U and δ U arerepresented for the particular value j = 5 . A Appendix: Demand for other distributions
In this Appendix we extend our analysis to more general distributions.We first (A.1) explicit the particularities introduced on the above generic results when thepdf has a bounded support. In Section A.2 we relax hypothesis H5, and consider pdfs withunbounded support called fat tail distributions in the literature. Finally, we extend our resultsto multimodal distributions in Section A.3.
A.1 Pdfs with compact support
We consider here pdfs f ( x ) with compact supports: x ∈ [ x m , x M ] presenting a unique maxi-mum (which may be located at one boundary). Clearly, such pdfs have finite variances. Thediscussion follows the same lines as that of the generic smooth distributions, except that inaddition one has to pay attention to the values of Γ and its derivatives at the boundaries η = 0, η = 1. A simple uniform distribution, analyzed in [45], is a particular case where the maximumof the pdf is degenerate.In this section, derivatives like Γ (cid:48) (1) or Γ (cid:48) (0), stand for the left and the right derivative ofΓ at η = 1 and η = 0, respectively. Due to the fact that the pdf strictly vanishes beyond itssupport, if the price is very high with respect to h (small δ ) there may be no buyers at all,and η = 0. On the contrary, if the price is very low (large δ ) the market may or not saturate,i.e. η = 1, depending on the behaviour of the pdf in the vicinity of x m . We represent thelines delimiting the regions where these solutions exist on the phase diagram. It should bestressed that these lines only indicate saturation and non-existence of a market. Their natureis different from that of the boundaries δ L and δ U . In the following we consider a triangulardistribution with a maximum f B at x B to illustrate our general results. The figures in thissection correspond to x B = −
1, i.e. a case where the maximum lies inside the support. Thecase x B = x m has been considered in the study of the learning dynamics [79].In the case of compact supports [ x m , x M ], Γ( η ) increases from Γ(0) = − x M <
0, to Γ(1) = − x m >
0. Like in the generic case of unbounded supports, Γ (cid:48) reaches a minimum at η B , andthere is a critical value j B = 1 /f B beyond which multiple solutions appear. Notice that ifthe maximum of the pdf lies at x m or at x M , η B lies at one of the boundaries of the [0 , η B = 1 / x B we have: f ( x ) = (cid:40) x − x m )( x M − x m )( x B − x m ) if x m ≤ x ≤ x B , x M − x )( x M − x m )( x M − x B ) if x B ≤ x ≤ x M , (A-1)with f ( x ) = 0 outside the support. At its maximum, f B = 2 / ( x M − x m ). The constraint ofzero mean and unit variance imposes a relationship between the support boundaries and thevalue of x B : x m = ( − x B − (cid:112) − x B )) / x M = ( − x B + (cid:112) − x B )) / x B = x M the distribution increases monotonically from 0 at x m , reaching its maximumat x M , and is defined only by the first equation in (A-1). Conversely, if x B = x m then f ( x )decreases inside its support, and is defined by the second equation in (A-1). In both cases, f ( x )presents a discontinuity at one boundary of its support.The function Γ and its first derivative for the triangular distribution are respectivelyΓ( η ) = (cid:26) − x M + (cid:112) ( x M − x m )( x M − x B ) η if 0 ≤ η ≤ η B − x m − (cid:112) ( x M − x m )( x B − x m )(1 − η ) if η B ≤ η ≤ (cid:48) ( η ) = √ ( x M − x m )( x M − x B )2 √ η if 0 ≤ η ≤ η B √ ( x M − x m )( x B − x m )2 √ (1 − η ) if η B ≤ η ≤ η B = ( x M − x B ) / ( x M − x m ) is the inflexion point of Γ. They are represented on figure 7(left) for the particular value x B = −
1. Notice that Γ (cid:48)(cid:48) ( η ) is discontinuous at η B , because themaximum of f ( x ) is a cusp.The corresponding inverse demand function D ( j ; η ) (equation (22 ) ) is represented on figure7 (right). Due to the finite range of the compact support, there are two new particular valuesof δ : δ ≡ D ( j ; 0) = Γ(0) = − x M , independent of j , and δ ( j ) ≡ D ( j ; 1) = Γ(1) − j = − x m − j :for δ < δ , η = 0 (no-market) is a solution, while for δ > δ ( j ) there is a solution η = 1 (marketsaturation). These extreme values of η may be reached upon finite values of δ (i.e. finite pricesand finite average IWP) only in the case of compact supports.For j < j B , D ( j ; η ) is strictly increasing and invertible on ]0 , δ in ] − x M , − x m − j [,equation (21) has a unique solution η d ( δ ) (cid:54) = { , } . One can easily check that j < j B implies j ≤ x M − x m , so that − x M < − x m − j and consequently D ( j ; 0) < D ( j ; 1). In the particularcase where f is the uniform distribution, one has precisely j B = x M − x m . For the triangularpdf (A-1), (29) gives j B = ( x M − x m ) /
2, the support’s half-width. Thus, for δ < δ < δ ( j ) thefraction of buyers/adopters is a monotonic increasing function of δ . For δ < δ , η = 0, and for δ > δ ( j ) the market saturates ( η = 1).For j > j B there are two stable solutions whenever δ U ( j ) ≤ δ ≤ δ L ( j ). Due to the existenceof the extreme solutions η = 0 and η = 1, the analysis is more cumbersome than for infinitesupports. If the maximum of the pdf lies inside the support, the solutions η L ( j ) and η U ( j )of equation (30) lie in ]0 ,
1[ and δ U ( j ) and δ L ( j ) both satisfy D (cid:48) = 0. On increasing δ from −∞ , there is no demand until δ = min { δ , δ U ( j ) } . If δ < δ U ( j ), when δ increases beyond δ the demand becomes finite and remains unique provided that δ < δ < δ U ( j ). For δ > δ U ( j )we enter the region of multiple solutions. On the other hand, if δ > δ U ( j ), the system stepsdirectly from the no-demand solution into a region where a finite demand equilibrium with η > η B coexists with the no-demand one. Notice that the high- η solution may correspondto either a fraction of buyers strictly smaller than 1 (if δ L ( j ) < δ ( j )) or to saturation (if δ L ( j ) > δ ( j )). In the case of the triangular distribution it is straightforward to check that themultiple solutions region sets in at j B = ( x M − x m ) / δ B = x B / d = - x M d U ( j ) - x M h < 1 h = 0 d = h - p j h = 1B j B d B d =-x m -j d L ( j ) Figure 8:
Triangular pdf of unitary variance and a maximum at x B = − : customers phasediagram. If the pdf has its maximum at one of the boundaries of its support, either η U or η L coincidewith η B . More precisely, if x B = x m then η U = η B = 1, if x B = x M , η L = η B = 0.Summarizing, the customers phase diagram for pdfs with compact supports have two sup-plementary lines with respect to that with unbounded supports. They indicate the boundary ofthe viability region (no market exists below this line) and the saturation boundary (above whichall the customers are buyers). Figure 8 presents the customers’ phase diagram for our examplecorresponding to the triangular pdf of unitary variance (A-1), with a maximum at x B = − A.2 Pdfs with fat tails
Fat-tail distributions are characterized by the fact that the pdf f has a slow decrease at largevalues of x , so that the variance is infinite - or even the mean is infinite. Equivalently thisoccurs if Γ( η ) diverges ’too fast’ to −∞ when η →
0. In the case of the logistic, Γ ∼ log η ; forthe Gaussian, Γ ∼ −√− η ; for a power law, Γ ∼ − η b . This suggest to consider the generalfollowing smooth behavior: as η →
0: Γ( η ) ∼ − K ( − log η ) a η b , (A-4)with the constant K > a ≥ b ≥ ab (cid:54) = 0). The fat-tail case corresponds then to a = 0and b ≥ x behaves like: f ( x ) ∼ x − (1+ µ ) (A-5)with µ ≥
0. Then, for small η , Γ ∼ − η b with b = 1 /µ , so that b ≥ µ ≤
1. For µ < x is infinite.33 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0- 4 0- 2 002 04 06 08 01 0 0 h G G ’ G ’ ’ Figure 9: Γ function and derivatives corresponding to the function (A-6).
For fat-tails distributions one has to look at finite size effects: it is no more possible totake directly the large N limit and make use of the central limit theorem: quantities like(1 /N ) (cid:80) i G ( x i ) for any function G will be dominated by rare events, that is by the largestvalues encountered in the population of (large but finite) size N . There is, however, no difficultyin doing this analysis: the results are obtained by doing as if the pdf had a finite support, theupper bound x M being given as an increasing function of N (for an introduction to statisticswith fat tails, see e.g. [17]).Since we considered compact supports in the preceding section, we concentrate here on themarginal case µ = b = 1, which can be analyzed as a limiting case of distributions with infinitesupport.For µ = 1, f ( x ) does not have a finite variance. Then, the value of σ that defines thenormalized variables (10) may be any (finite) measure of the width of the distribution, as forexample, the value of x at which f ( x ) is equal to half its maximum. Let us discuss this marginalcase on a simple example (see figure 9):Γ ≡ − η + 11 − η , (A-6)corresponding to the cumulative function: F ( z ) = 12 − z + sgn( z ) (cid:114) z + 14 . (A-7)The corresponding pdf is, f ( x ) = 1 x [1 − √ x ] , (A-8)as represented on figure 10. Since this is a symmetric distribution, η B = 1 /
2, and one finds(see equations (28), (29) and (35) ) that the point B in the customers phase diagram is j B =34 f ( x ) x Figure 10:
Pdf corresponding to equation (A-8). Γ (cid:48) ( η B ) = 8, δ B = −
4. Notice that, like for any monomodal distribution (satisfying thushypothesis H1), Γ( η ) is convex for η > η B and concave for η < η B , with as before η B being theinflexion point. The supply function has thus the generic behavior described in Section 3 evenfor fat-tail distributions. A.3 Aggregate demand for multimodal pdfs
A.3.1 Smooth pdfs: generic properties
In this section we consider the behavior of the application δ → η d ( δ ) in the case of a smoothmultimodal pdf with support on ] − ∞ , + ∞ [. The discussion Section 3.2, based on convexityarguments, can be extended to describe the phase diagram for the aggregate demand in themultimodal case.The minimal hypotheses we consider are the following. • HA0. The pdf f ( x ) is continuous with a finite number K ≥ x -values, −∞ < x KB
1] is the subset of the ensemble of solutions of (21) for which δ increases ( p decreases)as η increases. As for the monomodal case we study the function of η defined by (21) forany given j , δ ( η ) = D ( j ; η ). By continuity of the function Γ, δ ( η ) is a continuous function of η ∈ [0 , η → δ → −∞ , and as η → δ → + ∞ . Increasing η from 0, δ ( η ) increases.Similarly, decreasing η from η = 1, δ ( η ) decreases. Since f ( x ) is continuous, Γ( η ) is continuouslydifferentiable, with Γ (cid:48) ( η ) ≡ d Γ( η ) /dη = 1 /f ( x ) at x = − Γ( η ). Hence Γ (cid:48) ( η ) has (local) minimaat values of η given by Γ (cid:48) ( η kB ) = 1 f ( x kB ) k = 1 , ..., K (A-12)and (local) maxima at values of η given byΓ (cid:48) ( η kC ) = 1 f ( x kC ) k = 1 , ..., K − η kB and η kC s are inflexion points for Γ. Note that for any k =1 , ..., K , η k − C < η kB < η kC .The most important remark is that Γ( η ) is strictly concave on every interval ] η k − C , η kB [ , k =1 , ..., K , and strictly convex on every interval ] η kB , η kC [ , k = 1 , ..., K . Then as η varies on[ η k − C , η kB ], the function D ( j, η ) = Γ( η ) − jη has, at some value η kL ( j ), a maximum δ kL ( j ) whichis by definition the Legendre transform of Γ( η ) restricted to [ η k − C , η kB ]. Similarly, on [ η kB , η kC ], D ( j, η ) has, at some value η kU ( j ), a minimum δ kU ( j ), the Legendre transform of Γ( η ) restrictedto [ η kB , η kC ].Depending on the value of j compared to the values j kB , j kC , these min and max may bereached either at a boundary of an interval, or in the interior. More precisely: j < j kB , η kL = η kU = η kB (A-14) j kB < j < j kC , η kB < η kU < η kC (A-15) j kB < j < j k − C , η k − C < η kL < η kB (A-16) j kC < j, η k +1 L = η kU = η kC (A-17)(and η kU increases from η kB to η kC as j increases from j kB to j kC , whereas η kL decreases from η kB to η k − C as j increases from j kB to j k − C ). In the case of a continuously differentiable pdf, everyLegendre transform η k Λ ( j ) , Λ =
L, U satisfies the marginal stability equation, ∂ D ( j, η ) ∂η | η = η k Λ ( j ) = 0 . (A-18)One should note that η kU,L and δ kU,L depend on j (and on the function Γ( . )), but not on h or p .36ow for j < j B ≡ min k j kB , every min and max are reached at the corresponding value η kB :this means that there is no intermediate regime with a decreasing behavior of δ ( η ) as η increases,hence δ d ( η ) = D ( j, η ), uniquely defined, is a continuously increasing function of η ∈ [0 , j > j B , there is at least one k where the maximum δ kL ( j ) is reached for η = η kL ( j ) < η kB , andthe minimum δ kU ( j ) is reached for η = η kU ( j ) > η kB , so that there is at least one finite interval of η on which the function D ( j, η ) decreases with η , and thus does not correspond to an economicequilibrium. Hence the demand η d ( δ ) has at least two branches.In the plane ( j, δ ), the boundaries of the multiple solutions regions are thus given by thefonctions δ k Λ ( j ) = D ( j, η k Λ ( j )) , Λ =
L, U , which are the graphs of all the branches of theLegendre transform of Γ. By construction of the Legendre transform, every branch δ = δ kU ( j )is a concave curve, and every branch δ = δ kL ( j ) is a convex curve, and, under the smoothnesshypothesis HA1, along each branch Λ = L, U , dδ k Λ ( j ) dj = d D ( j, η k Λ ( j )) dj = − η k Λ ( j ) . (A-19)Recall that η k Λ is the value of η for the solution which is marginally stable on this boundary.These boundaries can be easily drawn for any distribution making use of a parameterizationby s (or equivalently x ≡ − s ): from the basic equations η = 1 − F ( − s ) where F is the cumulativeof the pdf f , s = Γ( η ), and Γ (cid:48) ( η ) = 1 /f ( − s ); with the marginal stability condition (A-18)which gives j = Γ (cid:48) ( η ), the locus of marginal stability is then given in the plane ( j, δ ) by theparameterized curve for x ∈ support( f ) ,j = 1 /f ( x ) (A-20) δ = − x − − F ( x ) f ( x ) (A-21)This is this representation that we have used to draw the phase diagram, figure 12, for theparticular example of the bimodal distribution shown on figure 11.The domain of multiple solutions can then be described as follows. The phase diagram is akind of superposition of diagrams associated to mono-modal phase diagrams, every maximum(every ’bump’ in the pdf) k being responsible of the appearance of a domain of multistability:when j becomes larger than j kB , a continuous solution split into two solutions, with a lowersolution η d ( j, δ ) ≤ η kL ( j ) < η kB and δ ≤ δ kL , and an upper one with η d ( j, δ ) ≥ η kU ( j ) > η kB and δ ≥ δ kU (see figure 12). When j becomes larger than j kC , this bump is no more ’seen’. Sincea minimum of the pdf, if not at a boundary, is in between two maxima, such an intermediatesolution may exist either because of one bump or the other - or both.The branch δ = δ kU ( j ) has thus as left end point, B k ≡ ( j kB , δ kB = D ( j kB , η kB )), and as rightendpoint (if j kC is finite), C k ≡ ( j kC , δ kC = D ( j kC , η kC )). B k is the merging point of δ kU and δ kL ,and C k the merging point of δ kU and δ k +1 L . Since δ kL and δ k +1 L must be both above δ kU , thesetwo branches must intersect one another for some value of j = j kBC between j kB and j kC : thereis thus coexistence of three solutions in the triangular-like domain bounded below by δ kU (ormax( δ kU , δ k +1 U ) if B k is below the branch δ k +1 U ), and above by δ kL for j ≤ j kBC , and by δ k +1 L for j ≥ j kBC .In the smooth case (HA1), at every bifurcation point B k , resp. C k where two boundariesmerge, according to A-19 there is a common slope − η kB , resp. − η kC .One may say that the pdf is probed at different scales for different values of j . Considerthe graph y = f ( x ). Every maximum below the line y = 1 /j is not seen (it does not changethe structure of the solution), whereas a set of maxima higher than 1 /j , but joined by minimawhere f is still higher than 1 /j , is seen as a single global bump. This gives in particular that for37 x f(x) B C B Figure 11:
An example of bimodal pdf. δ L1 δ U2 δ δ U1 δ L2 j B B C Figure 12:
Phase diagram (aggregate demand) for the case of the smooth bimodal pdf shown onfigure 11. Figure 13:
Examples of multimodal pdfs. At a given value of j = J/σ , the qualitative propertiesare obtained by looking at the intersection of the horizontal line y = 1 /j with the graph of thepdf, y = f ( x ) : for the particular value of j corresponding to the horizontal line on this figure,the two pdfs lead to the same qualitative properties of the Demand. h = 1 / 2 h = 0 h - p j h = 1 Figure 14:
Phase diagram (aggregate demand) for the case of a bimodal pdf composed of twoDirac peaks. j > j B , the number of solutions is equal to one plus the number of times the line y = 1 /j cutthe graph y = f ( x ) at points where f is increasing. Note that this does not give the number ofsolutions for a given value of δ . On figure 13, two pdfs are shown; the intersection of the graph y = f ( x ) with the line y = 1 /j gives the structure of the demand at this particular value of j (in the case illustrated on the figure, the demand has 3 solutions for the two pdfs). A.3.2 A degenerate case: 2 Dirac
Let us consider the particular case of an IWP distribution given by two Delta pics: x i = ± x with equal probability ( x = 1 / √ f is normalized to 1). For j = 0, onehas clearly η = 0 , / δ < − x , − x < δ < + x or δ > x . For j > η can still take only these three values. One gets easily the domain of existence andstability of these solutions, η = 0 , / ,
1, by direct inspection of the equation (13). The resultingphase diagram is shown on figure 14.This phase diagram for a singular distribution can also be understood by comparison withthe predicted phase diagram for a continuous distribution. In the present case, the two maximahave equal height, + ∞ , which gives j B = j B = 0, in agreement with the fact that boundarylines meet at j = 0. The minimum between the two maxima is at f = 0, hence j C = ∞ : thedomain of stability of the intermediate solution η = 1 / f ( x C ) = 0. The marginal stability lines are straight lines - hence,marginally concave and convex curves -, with slopes 0 , / B = (0 , − , B = (0 ,,