Economic Models - Convex Optimization

Cheap-talk Multiple Equilibria and Pareto 105
period. Rather than expenditures in emission-reducing capital, γ can therefore
be interpreted as the additional costs resulting of a temporary switch
to a cleaner resource — for e.g., of a switch from coal to natural gas.
The firms attempts to maximize their profit. As we shall see at a later
place, one can assume without loss of generality that they disregard the
future and maximize in every τ, their current instantaneous profit. However,
the actual tax rate t enters their instantaneous profit function and is unknown
at the time when they choose v. Thus, they need to base their optimal
choice of v on an expectation (or, prediction) t e of t. In reality, a firm
can invest larger or smaller amounts of money in order to make or less
accurate predictions of the tax, the government will actually implement.
In this model, we assume for simplicity that each firm has two (extreme)
options for predicting the implemented tax. It can either suppose that R’s
announcement is truthful, that is, that t will be equal to t a , or it can do costly
research and try to better predict the t that will actually be implemented.
Depending upon the expectation-formation mechanism the firm chooses,
we speak of believer B or of a non-believer NB:
A believer considers the regulator’s announcement to be truthful
and sets
t e = t a (3)
This does not cost anything.
A non-believer makes a “rational” prediction of t, considering the current
number of Bs and NBs as constant. Making the prediction in any given
period τ costs δ>0.
Details on the derivation of the NBs’prediction will be given later. Note
that a firm can change its choice of expectation-formation mechanism at
any moment in time. That is, a B can become a NB, and vice versa. We
denote with π = π(t) ∈ [0, 1] the current fraction of Bs in the population.
Thus, 1 − π is the current fraction of NBs.
2.2. The Regulator, R
The regulator’s goal is to maximize with respect to t a ≥ 0 and t ≥ 0, the
inter-temporal social welfare function.
(t a ,t)=
∫ ∞
0
e −ρτ ϕ(t a ,t)dτ