# Dual Tax-Free Taxes

Now, suppose you have two taxing bodies, how would you set tax for a system where income going into tax should be tax-free? $F$ is nominal federal tax, $S$ is the nominal state tax, and $p$ is the proportion of income paid to either taxes and $A$ is the income pretax. Let $p_f$ be proportion paid to federal government, and $p_s$ be amount paid to state.

If both governments refuse to tax money going towards government, then the effective tax rate as a function of the two taxes using the formula from the tax-free-tax system would be:

$p= \frac{F+S}{1+F+S}$

Let’s consider a bullying federal government that stipulates that California cannot tax its residents unless United States exists, therefore the federal tax takes priority. California still refuses to tax income paid to any government as tax. We apply the formula in the tax-free-tax system twice.

$p_f = \frac{F}{1+F}$

$p_s*A = S(1-p_f)(1-p_s)A$

$p_s = S \frac{1-p_f}{1 +S - S*p_f}$

Producing this expression:

$p= p_f + (1-p_f)p_s$

Finally, what happens when California retaliates by taxing the money paid into federal tax, but maintains that it does not tax income used to pay Californian income tax?

$p_s = \frac{S}{1+S}$

$p_f = \frac{F}{1+F}$

So the effective tax rate under inconsiderate governments will be:

$p = p_s + p_f = \frac{S + F + 2SF}{1 + S + F + SF}$

Understandably, the effective rate increases as the government ignore each other’s taxes.

There are, of course many other ways to skin this cat. We can consider one simplest one. Suppose California and USA disagree very strongly that California says it must have $s$ of the incomes of individuals of the state. (The state income $I$, If we were doing VAT, in which case we tax the gdp) and the federal government demands $f$ of that income. The argument proceeds until California decides that it is give me fair taxes or death and begins its secession from the union.

President Trump calculates that USA will save annually $Y$ from not patrolling California coast, etc. Governor Newsom stipulate that California contributes to non-California USA income (again, easier if talking about GDP) by some $Z$. Trump then retorts, yeah but you spend $\gamma$ of your state tax money on interstate commerce that you will not be spending. (This is arguable post facto assertion will stick, since all calculations are made not only to secess from union but also to dissolve California. In this case, there will be no $\gamma$ or $Y_{CA}$)

The loss USA stands to experience is $fI - Y + fZ$ and the loss California stands to experience is $s*I - \gamma + Y$. So, dividing the saved loss in half, both side may agree to share loss $\frac{(s(1-\gamma) + f)I + fZ }{2}$ of the tax money. This means

$p = f+s(1-\gamma)$

And federal government gives California $fZ/2$ of the taxes it collects from other states in addition to providing the existing $Y$ services. But in reality, this type of analysis is complicated by the fact that the state and federal budget may both exceed tax revenue! And of course we’d never agree on what $Y$ and $\gamma$ are.