Economic Models - Convex Optimization

104 Chirstophe Deissenberg and Pavel Ševčík
(and thus, on the labor income) and the profits. Thus, R has to choose the
tax level in order to achieve an optimal compromise among tax revenue,
private sector income, emission reduction, and employment levels. The
following sequence of event (S) occurs in every τ:
— R makes a non-binding announcement t a ≥ 0 about the tax level t ≥ 0
it will choose.
—Givent a , the firms form expectations t e about the true level of the
environmental tax. As will be described in more detail later, there are
two different ways for an individual firm to form its expectations.
— Each firm decides about its level of investment v based on its expectation
t e .
— R choose the actual level of tax t ≥ 0. The tax level t is the amount
each firm has to pay per unit of its emissions.
— Each firm produces a quantity x.
— Each firm may revise the way it forms its expectations depending on its
relative profits.
All firms are identical, except for the way they form their expectations t e .
We assume full information: the regulator and the firms perfectly know
both the sequence, we just presented and the economic structure will be
(production and prediction technology, objective functions ...) described
in the remainder of this section.
2.1. The Firms, Bs and NBs
Each firm produces the same homogenous goods using a linear production
technology: the production of x units of output, requires x units of labor and
generates x units of environmentally damaging emissions. The production
costs are given by:
c(x) = wx + c x x 2 , (1)
where x is the output, w>0 the fixed wage rate, and c x > 0 a parameter for
simplicity sake, the demand is assumed infinitely elastic at the given price
˜p >w. Let p := ˜p − w>0. At each point of time, each firm can spend
an additional amount of money γ in order to reduce its current emissions
x. The investment
γ(v) = 1 2 v2 (2)
is needed, in order to reduce the firm’s emissions from x to max [x − v, 0].
The investment in one period has no impact on the emissions in future