## 9.4 Income Taxation at the State Level

There are various systems of how income taxes are levied at the state level. In a first case, state income tax can be a percentage of the federal tax liability. In that case, revenue is sensitive to changes in the federal tax code. In a second case, the state tax base is equal to the federal taxable income but uses different tax rates. This system is sensitive to changes in the federal base but not the federal rates. In a third case, the state income tax base equals the federal AGI but uses different deductions, credits, and rates. This system has reduced revenue sensitivity as it relates to federal changes. And finally, state tax base and rates are unrelated to the federal taxes. This results in the state income tax system being completely disconnected from the federal system and associated changes. IN reality many states use a flat tax rate, e.g., 3.23% of Adjusted Gross Income in Indiana.

Income taxation at the federal and state level are characterized by some system of deductibility. Let \(s\) and \(f\) be the federal and state tax rate, respectively. And let \(x_s\) and \(x_f\) denote federal and state tax liability, respectively. For the following cases, assume that the taxable income is \(M=60,000\), the federal tax rate is \(f=0.25\), and the state tax rate is \(s=0.05\).

In the case of reciprocal deductibility, we have the following formulas:

State tax liability \[x_s=\frac{s-s \cdot f}{1-s \cdot f} \cdot M\]

Federal tax liability \[x_f=\frac{f-s \cdot f}{1-s \cdot f} \cdot M\]

Given the above example numbers, the state liability is calculated as follows: \[x_s=\frac{s-s \cdot f}{1-s \cdot f} \cdot M \Rightarrow \frac{0.05-0.05 \cdot 0.25}{1-0.05 \cdot 0.25} \cdot 60,000 = 2,279\]

And the federal liability is expressed as follows: \[x_f=\frac{f-s \cdot f}{1-s \cdot f} \cdot M \Rightarrow \frac{0.25-0.05 \cdot 0.25}{1-0.05 \cdot 0.25} \cdot 60,000 = 14,430\]

Thus, the total tax liability is $16,709 and the average tax rate is \(16,709/60,000=0.2785\). In the case of deductibility, we have the following for the state tax liability: \[s \cdot M \Rightarrow 0.05 \cdot 60,000 = 3,000\]

And for the federal tax liability: \[f \cdot M \Rightarrow 0.25 \cdot 60,000 = 15,000\]

Thus, the total tax liability is $18,000 and the average tax rate is \(18,000/60,000=0.3\). In the case of state deductability from the federal tax liability, we have the state liability: \[s \cdot M \Rightarrow 0.05 \cdot 60,000 = 3,000\]

And for the federal tax liability: \[f \cdot (1-s) \cdot M \Rightarrow 0.25 \cdot (1-0.05) \cdot 60,000 = 14,250\]

Thus, the total tax liability is $17,250 and the average tax rate is \(17,250/60,000=0.2875\).