Economic Models - Convex Optimization

Cheap-talk Multiple Equilibria and Pareto 107
maximized), and tends towards 0, for π → 0 and π → 1 (for extreme values
of π, all but very few firms have the same profits). The parameter β ≥ 0,
that captures the probability with which a firm changes its strategy during
one of these meetings, calibrates the speed of the information flow between
Bs and NBs. It can be interpreted as a measure of the firms’ willingness to
change strategies, that is, of the flexibility of the firms.
Equation (6) implies that by choosing the value of (t a ,t)at time τ the
regulator not only influences the instantaneous social welfare, but also the
future proportion of Bs in the economy. This, in turn, has an impact on
the future social welfare. Hence, there are no explicit dynamics for the
economy. This again has an impact on the future social welfare. Hence,
although there are no explicit dynamics for the economic variables v and
x, R faces a non-trivial inter-temporal optimization problem.
On the other hand, since the firms are atomistic, each single producer
is too small to influence the dynamics Eq. (6) of π, the only source of
dynamics in the model. Thus, the single firm does not take into account
any inter-temporal effect and, independently of its true planing horizon,
maximizes its current profit in every τ. This naturally leads us to examine
the (Nash) solution of the game between the regulator and the non-believers
that follows from the sequence (S) when, for an arbitrary, fixed value of π,
(a) R maximizes the integrand φ in Eq. (4) with respect to t a and t and (b)
the NBs maximize their profits with respect to ν NB and x NB . As previously
mentioned, we assume full information. In particular, the R and the NBs
are fully aware of the (non-strategic) behavior of the NBs.
The analysis of the static case is also necessary since, as previously
mentioned, it provides the prediction t e of the NBs.
3. The Sequential Nash Game
The game is solved by backwards induction, taking into account the fact
that the firms, being atomistic, rightly assume that their individual actions
do not affect the aggregate values, that is consider these values as given. We
consider only symmetric solutions where all Bs respectively, NBs choose
the same values for v and x.
3.1. Derivation of the Solution
The last decision in the sequence (S) is the choice of the production level x
by the firms. This choice is made after v and t are known. The firms being