Economic Models - Convex Optimization

110 Chirstophe Deissenberg and Pavel Ševčík
Figure 1. The profits of the Bs (lower curve), of the NBs (middle curve), and the
regulator’s utility (upper curve) as a function of π for δ = 0.00025, c x = 0.6, p = 2.
As one would intuitively expect, the regulator’s instantaneous utility φ
too, increases with π. Most remarkable, however, is the fact that the profits of
both the Bs and NBs are also increasing in π (see Fig. 1 for an illustration).
An intuitive explanation is the following: For π = 0 the outcome is the
Nash solution of a game between the NBs and the regulator. This outcome
is not efficient leaving room for Pareto-improvement. As π increases, the
problem tends towards the solution a standard optimization situation with
the regulator as the sole decision-maker. This solution is efficient in the sense
that it does maximize the regulator’s objective function under the given
constraints. Since the objectives of the regulator are not extremely different
of the objectives of the firm, moving from the inefficient game-theoretic
solution to the efficient control-theoretic solution one improves the position
of all actors. Specifically, as π increases, the Bs enjoy the benefits of a lower
t, while the investment remains fixed at t a$ . Likewise, the NBs benefit from
the decrease in t. And R also benefits, because a raising number of Bs
are led by R’s announcement to invest more than they would, otherwise
and, subsequently, to produce more, to make higher profits, and to pollute
subsequently, to produce more, to make higher profits, and to pollute less.
In dynamic settings, however, the impact of the tax announcements
on the π dynamics may incite the regulator to reduce the spread between
t and t a , and in particular to choose t a