Economic Models - Convex Optimization

Cheap-talk Multiple Equilibria and Pareto 111
a high proportion of Bs to a low one, since it has one more instrument
(namely, t a ) to influence the Bs than to regulate the NBs. A high spread
t a − t, however, implies that the profits of the Bs will be low compared
to those of the NBs. This, by Eq. (6), reduces the value of ˙π and leads
over time to a lower proportion of Bs, diminishing the instantaneous utility
of R. Therefore, in the dynamic problem, R will have to find an optimal
compromise between choosing a high value of t a , which allows a low value
of t and insures the regulator a high instantaneous utility, and choosing a
lower one, leading to a more favorable proportion of Bs in the future. We
are going to explore these mechanisms in more details in the next section.
4. The Dynamic Optimization Problem
4.1. Formulation
Being atomistic, the firms do not recognize that their individual decisions
to choose the B or NB prediction technology influence π over time. Assume
that they consider π as constant. The regulator’s optimization problem is
then:
max (t a ,t) 0 ≤ t a (τ), t(τ)
(19)
such that Eqs.(6), (7), (11), (13), and π 0 = π(0).
Using
g := p − c x(1 + 2πt a )
1 + c x (3 − 2π) ,
h := p − t
(20)
.
2c x
The Hamiltonian of the above problem is given by
H(t a ,t,π,λ)= h + π
[h −
(p − t)2
4c x
+ t(h − t a ) − ta2
2
(p − t)2
+ (1 − π)
[h − + t(h − g) − 1 4c x
2 g2 − δ
+ t[π(h − t a ) + (1 − π)(h − g)]
− [π(h − t a ) + (1 − π)(h − g)] 2
+ λπ(1 − π)β
[tt a − ta2
2 − th + 1 ]
2 h2 + δ ,
where λ is the co-state.
]
]