Question

Federal income tax allows a deduction for any state tax paid during the year. In addition, the state of Alabama allows a deduction from its state income tax for any federal tax paid during the year. The federal corporate income tax is equivalent to a flat rate of 34% and the Alabama state tax rate is a flat 5%.

Both the Alabama and the federal taxable income for a corporation is $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 (assume for this problem that the company has no other deductions to consider). One procedure often used by tax accountants to find the tax due in this and similar situations is called iteration and it’s described by the following steps:

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

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

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

Repeat the process in (2) and (3) until the federal tax changes from one repetition to another by less than $1.

QUESTION 1

Use Excel (or your favorite spreadsheet application) to model the above iteration process. Then complete the table, using as many rows as needed (until the change in federal tax is less than $1). Please include a print-out of your table rather than filling in the one below (this is just the sample)

	Federal Taxable Income (Taxable Income - State Tax Deduction)	Federal Tax (34%)	State Taxable Income (Taxable Income - Fed Tax Deduction)	State Tax (5%)
1st Iteration	1,000,000	340,000	660,000	33,000
2nd Iteration	1,000,000-33,000= 967,000
3rd Iteration

(Hint: this process will not take a lot of iterations – be careful how you apply the deductions at each step, and keep in mind that the taxable income is always $1,000,000.)

QUESTION 2

We can model the above situation via a system of linear equations.

What are the unknowns in this system? (Hint: What do you want to find out?)

x=

y=

Write down two linear equations that describe this taxation situation.

Find the augmented matrix that corresponds to this system:

A=

Find the solutions of this system of linear equations (they better match what you found in Question 1 – you are computing the same thing!)

x=

y=

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

Answer 1

SOLUTION:

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%