Economic Models - Convex Optimization

108 Chirstophe Deissenberg and Pavel Ševčík
takers, the optimal symmetric production decision is:
x B = x NB = x = p − t
2c x
. (7)
Note that the optimal production is the same for Bs and NBs, as it
exclusively depends upon the realized taxes t.
In the previous stage, R chooses t given v B . Maximizing v NB with
respect to t, with x is given by (7) gives the optimal reaction function:
t(v B ,v NB ) = p − c x[2(πv B + (1 − π)v NB ) + 1]
1 + c x
. (8)
When the firms determine their investment v, Eqs. (7) and (8), and t a
are known, but the actual tax rate t is not. Using Eq. (7) in Eq. (5), one
recognizes that the firms choose v in order to maximize
That is, they set:
(p − t e 2
)
+ t e v − 1 4c x 2 v2 . (9)
v = t e (10)
The Bs predict that t will be equal to its announced value, t e = t a . Thus,
v B = t a . (11)
The NBs know that the regulator will act according to Eq. (8). Thus, they
choose:
v NB = p − c x[2(πv B + (1 − π)v NB ) + 1]
1 + c x
. (12)
Solving this last equation for v NB gives for the equilibrium value:
v NB = v NB (t a ) p − c x(1 + 2πt a )
1 + c x (3 − 2π) . (13)
The first decision to be taken is the choice of t a by R. The regulator’s
instantaneous objective function φ, using the results from the previous
stages, i.e., Eq. (7) for x, Eq. (11) for v B and Eq. (13) for v NB , is concave