Question

Federal income tax allows a deduction for any state income tax paid during the year. In addition, the state of Alabama allows a deduction from its state income tax for any federal income tax paid during the year.

The federal corporate income tax is equivalent to a flat rate of 34% of the taxable federal income and the Alabama rate is 5% of the taxable state income. Both the federal and Alabama taxable income for a certain corporation are $1,000,000 before either tax is paid.

Because each tax is deductible on the other return, the taxable income will differ for the state and federal taxes. To find the tax due in this situation, you will use and then compare two different methods.

Iteration:

1. Make an estimate of the federal taxes due by assuming no state tax is due. Deduct this amount from the taxable income, and then calculate an estimate of the state taxes due on the basis of this assumption.

Federal due:

State due:

2. Deduct the estimated state tax due as computed in (1) above from the taxable income, and calculate a new estimate of the federal tax due under this assumption.

Federal due:

3. Deduct the new estimated federal tax due as computed in (2) above from the taxable income, and calculate a new estimate of the state tax due on the basis of these calculations.

State due:

4. Repeat the process in (2) and (3) to get a better estimate of the taxes due to both governments.

Federal due:

State due:

5. Continue this process until the federal tax changes from one repetition to another by less than $1.

What Federal Tax is due?

What State Tax is due?

Matrix Equation:

6. Create two linear equations that describe this taxation situation. Convert the system into a matrix equation, and solve the system to see exactly what tax is due to each government.

7. Does your answer from number 6 above match your answer from number 5? (Hint: it should)

8. What is the effective tax rate that this corporation pays to each government?

Answer 1

1. Assuming No State Tax is payable for the purpose of Federal Tax Calculations:

Taxable Income - $ 1,000,000

Federal Rate - 34%

Federal Taxes Payable - $ 340,000 (1,000,000 * 34%)

Therefore, State Taxes payable on the above will be:

Taxable Income - $ 1,000,000

Less: Federal Taxes - $ 340,000

Taxable Income for State - $ 660,000

State Tax Rate - 5%

State Taxes Payable - $ 33,000 ( $ 660,000 * 5%)

2. Assuming that the above calculated State taxes are payable before calculating the Federal Taxes.

Therefore, Federal Taxes will be:

Taxable Income - $ 1,000,000

Less: State Taxes - $ 33,000

Taxable Income for Federal - $ 967,000

Therefore, Federal Taxes Payable - $ 328,720 ($967,000 * 34%)

3. Assuming the above calculated amount is due by federal:

Taxable Income - $ 1,000,000

Federal Taxes - $ 328,720

Taxable Income for State - $ 671,280

State Taxes Payable - $ 33,564

4. Repeating the steps for 2 and 3 above, the last calculated Taxes will be:

Federal Taxes - $ 328,720

State Taxes - $ 33,564

5.

Taxable Income	Federal Taxes	Difference	Taxable Income	State Taxes	Difference
1,000,000.00	340,000.00	0	660,000.00	33,000.00	0
967,000.00	328,780.00	11,220.00	638,220.00	31,911.00	1,089.00
968,089.00	329,150.26	(370.26)	638,938.74	31,946.94	(35.94)
968,053.06	329,138.04	12.22	638,915.02	31,945.75	1.19
968,054.25	329,138.44	(0.40)	638,915.80	31,945.79	(0.04)

Federal Taxes Due - $ 329,138.44

State Taxes Due - $ 31,945.79

6. Linear Equation:

Let the Federal Taxes due be x; Let the State Taxes due be y.

Therefore, State Taxes due will be y = 0.05(1,000,000 - x)

and, Federal Taxes due will be x = 0.34(1,000,000 - y)

Therefore, Evaluating the above equation we will get

Y=0.05(1,000,000 – x)….. (a)

X=0.34(1,000,000-y)……(b)

Putting the value of y in equation b, we will get

X= 0.34[1,000,000 – 0.05(1,000,000 – x)]

X=0.34[1,000,000 - 50,000 + 0.05x]

X=0.34(950,000 + 0.05x)

X=323,000 + 0. 017x

0.983x=323,000

X=323,000 / 0.983

Therefore, x= $ 328,586

Therefore, Y = 0.05(1,000,000 – 328,586)

Y = 50,000 – 16,429

Y = $ 33,571

8. Effective Rate of Taxes will be:

Federal Taxes = 328,586 / 1,000,000 * 100 = 32.86%

States Taxes = 33,571 / 1,000,000 * 100 = 3.57%