Economic Models - Convex Optimization

106 Chirstophe Deissenberg and Pavel Ševčík
with
ϕ(t a ,t)= (πx B + (1 − π)x NB ) + πg B + (1 − π)g NB
− (π(x B − v B ) + (1 − π)(x NB − v NB )) 2
+ t(π(x B − v B ) + (1 − π)(x NB − v NB )), (4)
where x B ,x NB ,v B ,v NB denote the production respectively, investment
chosen by the believers B and the non-believers NB, and where g B (.)
and g NB (.) are their instantaneous profits,
g B = px B − c x (x B ) 2 − t(x B − v B ) − 1 2 (vB ) 2
g NB = px NB − c x (x NB ) 2 − t(x NB − v NB ) − 1 2 (vNB ) 2 − δ.
The strictly positive parameter ρ is a social discount factor.
Thus, the instantaneous welfare φ is the sum of: (1) the economy’s
output, that is also, the level of employment (remember that output and
employment are one-to-one in this economy); (2) the total profits of Bs
and NBs; (3) the squared volume of emissions and (4) the tax revenue. This
function can be recognized as a reasonable approximation of the economy’s
social surplus. The difference between the results of Dawid et al. (2005) and
those of the present article stem directly from the different specification of
the regulator’s objective function. In the former article, this function does
not include the firm’s profit and is linear in the emissions.
2.3. Dynamics
The firms switch between the two possible expectation-formation mechanisms
(B or NB), according to a imitation-type dynamics, see Dawid (1999);
Hofbauer and Sigmund (1998). Specifically, at each τ the firms meet randomly
two-by-two, each pairing being equiprobable. At every encounter,
the firm with the lower current profit adopts the belief of the other firm with
a probability proportional to the current difference between the individual
profits. Thus, on the average, the firms tend to adopt the type of prediction,
N or NB, which may currently leads to the highest profits. Above
mechanism gives rise to the dynamics.
dπ
=˙π = βπ(1 − π)g B − g NB . (6)
dt
Notice that ˙π reaches its maximum for π =
2 1 (the value or π for
which the probability, for encounter between firms with different profits are
(5)