Economic Models - Convex Optimization

with respect to t a for all π ∈ [0, 1], if,
Cheap-talk Multiple Equilibria and Pareto 109
c x > 1/ √ 3. (14)
We shall assume in the following that this inequality holds. Then, the
regulator will choose
t a$ =
1 + p . (15)
1 + 3c x
Using this last equation in Eq. (8) with v NB given by Eq. (13), one obtains
for the equilibrium value of t
t $ =
1 + p 1 + c x
−
1 + 3c x c x (3 − 2π) + 1 . (16)
Both t a$ and t $ will be non-negative if
which we assume from now on.
p ≥
3.2. Some Properties of the Solution
3
3c x − 2 , (17)
For all c x > 0 and π ∈ [0, 1], the optimal announcement t a$ is a constant,
and always less than t a$ . It is optimal to announce a high tax in order to
elicit a high investment from the believers, and to implement a low tax to
motivate a high production level. Moreover, t $ decreases in π: When the
proportion of Bs augments, the announcement t a becomes a more powerful
instrument making a recourse to t less necessary.
The difference between the statically optimal (i.e., evaluated at t $ ) profits
of NBs and Bs
g NB$ − g B$ = (1 + c x) 2
− δ, (18)
2c x (2π − 3) 2
is likewise increasing in π. Modulo δ, the profit of the NBs is always higher
than the profit of the Bs whenever π>0, reflecting the fact that the latter
make a systematic error about the true value of t. The profit of the Bs,
however, can exceed the one of the NBs if the prediction costs δ are high
enough.
Since t a$ is constant and t $ decreasing in π and since t $ ≤ t a$ , the
difference t a$ − t $ , increases with π. Therefore, the difference Eq. (18)
between the profits of the NBs and Bs is also increasing in the difference
between the announced and the implemented tax, t a$ − t $ ,