Economic Models - Convex Optimization

116 Chirstophe Deissenberg and Pavel Ševčík
equilibrium E M as well as the Skiba point π s to the left. In other words,
if the firms react quickly to profit differences, the regulator will follow a
policy that leads to the upper equilibrium E U even if the initial proportion
of Bs is small. Indeed, the high speed of reaction of the firms means that the
accumulated costs incurred by the regulator on the way to E U will be small
and easily compensated by the gains around and at E U . Moreover, R does
not have to make Bs much better off than NBs to insure a fast reaction. As a
result, for β large, the equilibrium E U is characterized by a large proportion
of Bs and thus insures high stationary gains to all actors.
Not surprisingly, an increase of the discount factor ρ has the opposite
effect. Impatient regulators want to build up confidence, only if the initial
proportion of Bs is high. The resulting equilibrium value π U will be relatively
low, since the time and efforts needed now for an additional increase
in the stock of Bs weighs heavily compared to the expected future benefits.
An increase of the prediction cost δ finally, shifts E M to the left and
E U to the right: It is easier for R to incite the firms to use the B strategy,
favoring the convergence to equilibrium with a large number of believers.
The scenario with two stable equilibria separated by a Skiba point can
change qualitatively, when the parameter values are sufficiently altered.
For small values of β and/or δ and/or for large values of ρ, the equilibrium
E U collides with E M and disappears. Only one (stable) equilibrium
remains, E L . Thus, in the long-run, the proportion of believers tends to zero.
On the other hand, for large values of δ trying to outguess the regulator is
never optimal. The only remaining equilibrium is the one where everybody
is a believer.
6. Conclusion
The starting point of this paper is a static situation, where standard rational
behavior by all agents leads to a Pareto-inferior outcome, although
there is no fundamental conflict of interest either among the different firms
or between the firms and the regulator, and although the firms are perfectly
informed and perfectly predict the regulator’s actions. In addition
to the environmental problem studied here, such a situation is common for
instance in monetary economics, in disaster relief, patent protection, capital
levies, or default on debt.
The present paper and previous work strongly suggest that, in such a
situation, the existence of believers who take the announcements of a macroeconomic
regulator at face value typically, Pareto-improve the economic