Discrete Choices Under Social Influence: Generic Properties

Abstract

We consider a model of socially interacting individuals that make a binary choice in a context of positive additive endogenous externalities. It encompasses as particular cases several models from the sociology and economics literature. We extend previous results to the case of a general distribution of idiosyncratic preferences, called here Idiosyncratic Willingnesses to Pay (IWP).Positive additive externalities yield a family of inverse demand curves that include the classical 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 the adoption 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 adoption rates, but increasing within some intermediate range. Depending on the price there are thus either one or two equilibria.Beyond this first result, we exhibit the generic properties of the boundaries limiting the 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). The main results are summarized as phase diagrams in the space of the model parameters, on which the regions of multiple equilibria are precisely delimited.

Compartmental Modelling of Social Dynamics With Generalised Peer Incidence

“…Though frequently cited as important factors in the spread of many behavioural trends, these kinds of influences tend to feature in mathematical models only as part of initial uptake (cf. infection) 10,19,22,23,31 ; their impact on other aspects of behavioural transmission is rarely considered, and then limited to rates of behaviour relapse (see, for example, the heroin and bulimia models of White-Comiskey 34 and González et al 9 respectively). Indeed, to our knowledge, the effect of peer influence on the rate at which individuals abandon a given practice is yet to be considered, an omission we shall give special attention.…”

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Publisher's copyright statement:Electronic version of an article published as Mathematical models and methods in applied sciences, 24,4, 2014,10.1142/S0218202513500656 c 2014 World Scientic Publishing Co. http://www.worldscientic.com/worldscinet/m3asAdditional information:

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. A generalised compartmental method for investigating the spread of socially determined behaviour is introduced, and cast in the specific context of societal smoking dynamics with multiple peer influence.

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“…Electronic version of an article published as Mathematical models and methods in applied sciences, 24,4, 2014,10.1142/S0218202513500656 c 2014 World Scientic Publishing Co. http://www.worldscientic.com/worldscinet/m3as…”

“…is marginal. Secondly, when R 0 > 1, then it is possible for the set of square brackets in equation (4.8) to be zero, that is 10) so that the S.F.E. is unstable by λ + > 0.…”

Heterogeneity and increasing returns may drive socio-economic transitions

“…• People vary in their extent to which they want to pay for environmental benefits, i.e., heterogeneity of WTP (see Gordon et al 2005Gordon et al , 2007Nadal et al 2005). …”