4.2 Probability of the Optimal Tax Being Implemented:
A Numerical Analysis
Recall that in our scenario, the lake is in a eutrophic state in spite of an existing
tax on phosphorous loading. Either the current state of the lake reflects the
preferences of all of the communities around the lake or the tax is too low to
keep it in an oligotrophic state. Knowing that the Pareto-optimal state of the
lake is as shown in Section 3.1, the benevolent politician promises if he is elected
to implement the optimal tax rate.
What is the likelihood of the tax policy being implemented, that is, of this
politician being elected, given the relative preferences of the green communities
and farming communities and their resulting lobbying efforts?
The following constant values are used in the Nash equilibrium with tax
equation (30) and evaluated for values of x between 0 and 3.5.
b = 0.6 - recall from Section 2.2 that this is the phosphorous recycling value
that gave rise to a hysteresis in the lake dynamics.
c1 = 0.2 - thus denoting the farming communities’ low relative preference for
lake ecosystem services.
c2 = 2 - thus denoting the green communities’ high relative preference for
a clean lake.
L τ = 1 - is selected as the current taxation that results in high phosphorous
loading and thus a eutrophic state of the lake.
The dynamic socially optimal equilibrium level of phosphorous loading is
given by the intersection of the optimal a* equation (19) and the lake dynamics
equation (3), as depicted in Figure 1. The optimal tax is such that the dynamic
non-cooperative equilibrium intersects the lake dynamics equation for the same
optimal (x* , a*) coordinates, as shown in Figure 3.
Varying values of n1 and n2 results in different (x*, a*) coordinates and
affects the amount of lobbying applied by the different communities to obtain
their desired outcome with respect to the proposed tax increase versus keeping
the current low tax. This in turn affects the probability of the benevolent
politician being elected and thus of the optimal tax policy being implemented.
Substituting these values back into the equations derived earlier in the chapter,
namely, equations (40), (41), (31) and (32) gives us the lobbying efforts of the
green and agricultural communities as well as the probabilities of the optimal
tax policy being implemented.
The results are summarized in Table 1 below.
Table 1: Summary of Results.
|
n1=5, n2=1 |
n1=4, n^1 |
______n,=3. nl=1 |
______n1g2, ι¾*2 | |
|
Pi |
1 |
00021 |
0 0492 |
0 0511 |
|
⅜ |
0 |
0.9979 |
0.9508 |
0.9488 |
|
(x'.a∙) |
(0 3932,0 102) |
(0.3934,0.102) |
¢0.3795,0.1018) |
(0 3472,0 1007) |
|
37 74 |
33 82 |
29 44 |
297792 |
21