We consider a model of socially interacting individuals that make a
binary choice in a context of positive endogenous externalities. The model
encompasses, as particular cases, several models presented in the sociology
and economics literature. We mainly (but not only) decline the model
within a market context: the binary choice is to buy or not a given good
at a price posted by a monopolist, and the latter determines the price in
order to maximize his profit. Putting the price to zero in the customers
system allows to model adoption of norms and other collective issues. This
paper extends previous result to the case of a generic distribution of the
Individual Preferences or Willingnesses to Pay (IWP).
We show that the model properties depend on the modality class (the
number of maxima) of the IWP probability distribution function (pdf),
and on its smoothness properties. The main results are summarized on
phase diagrams whose axes are the parameters of the model, namely, the
difference between the average IWP over the population and the posted
price, and the strength of the social component (externality), both measured
in units of the variance of the IWP distribution. In such phase
diagram we represent the boundaries limiting regions where the system
presents different types of equilibria (e.g. unique or multiple).
We derive generic properties of the demand (or adoption) curve, which
are shown to depend only on qualitative features of the population heterogeneity:
the modality (number of maxima), the smoothness, the variance
(second moment) and the type of support (compact or infinite) of the underlying
IWP distribution. These alone determine the qualitative features
of the collective outcome.
The case of a monomodal pdf (a single maximum) is investigated in
details. When the social influence is small with respect to the width of the
IWP distribution, the aggregate demand exhibits features similar to the
one of “classical” demand curves in economics, with a continuously decreasing
adoption rate with increasing prices. However, beyond a critical
value of the ratio between social influence/variance of the IWP distribution
the inverse demand function exhibits a non classical behaviour: it
is a decreasing function at low and high adoption rates but has a non
monotonic behaviour in some intermediate range reaching a maximum at
some (possibly large) value of the adoption rate – and this even if the
distribution of preferences is monomodal –. As a result, depending on the
difference between the average IWP and the price, there are either one
or two stable equilibria. Thus, taking into account a positive (additive)
externality in the market context yields a more general family of demand
curves that the classical ones.
Finally, we study the generic properties of a monopolistic market. For
a large enough social influence strength (compared to the variance of the
IWP distribution), the optimal strategy for the monopolist exhibits a
drastic jump from one with a high price and a low number of customers,
to one of low price and a large number of customers. This discontinuity
arises even in the parameter region where the customers system exhibits
a single equilibrium.