Effect of Tax Rate on Tax Revenue: An optimization model
It is assumed in the following section that it is of interest to maximize the tax revenue generated through a Tax Rate imposed on a Population.
Tax Revenue (T) = Population (P) x Tax Rate (r_T)
Furthermore,
Population (P) = Maximum Population (P_M) x Loyalty (L)
and
Loyalty (L) + Public Grievance (PG) + Tax Rate (r_T) = 1 (L, PG, and r_T are expressed as fractional values -- i.e., 0 <= fractional value <= 1)
Accordingly, Tax Revenue can be rewritten as a function of Tax Rate as follows:
T = P * r_T = P_M * L * r_T = P_M * (1 - PG - r_T) * r_T
Accordingly, for a given Maximum Population and Public Grievance (we treat these as external or fixed parameters), we can locally optimize for Tax Revenue with respect to Tax Rate as follows:
dT / dr_T = P_M * ( (1 - PG) - 2 * r_T)
Which arrives at the following *unconstrained* or local maxima value for T when dT/dr_T = 0 occuring at r_T|max:
r_T|max,local = (1 - PG)/2
The global optimization model further considers the following constraint: If your population drops below your minimum required labor, you will arrive at productivity shortages and this outcome is undesirable. Accordingly, r_T|max,global can be determined as follows:
Let P_m = Minimum Population required to sustain city productivity, and r_T|upper-bound = the upper bound tax rate that must not be exceeded in order to mantain the population above P_m. Then,
P_m = P_M * (1 - PG - r_T|upper-bound)
Therefore,
r_T|upper-bound = 1 - PG - P_m/P_M
and the global optimization solution for the ideal Tax Rate that will maximize your Tax Revenue while maintaining a non-negative Idle Population (to prevent labor shortages) can be expressed as follows:
r_T|max,general = min { (1 - PG)/2 , 1 - PG - P_m/P_M }
The limiting criterion can be therefore established as follows:
If P_m/P_M >= (1 - PG)/2, then your maximum tax rate is limited by your minimum required labor. In the simple case that Public Grievance is non-existent, then if the productive non-idle population exceeds half your maximum capacity, then your tax rate is limited to the ratio of non-idle population to the population capacity; otherwise, the tax rate is optimal at 50%.
The corresponding Tax Revenue is simply given as:
T|max,general = P_M * (1 - PG - r_T|max,general) * r_T|max,general, where r_T|max,general is the optimal tax rate established above.