Tax Math

A person on another forum asked this question:

Quote:

I can figure out the appropriate % to gross up based on a fixed marginal tax rate based on trial and error, but I'd like a formula to get me there no matter the marginal rate.

The problem.

Someone is paid $100,000 and pays taxes on it so they net a certain amount.
We want to pay them an additional amount (tax free), say $10,000.
Uncle Sam says 'no', they will have to pay taxes on the $10,000.
So we pay them an additional amount on top of the $10,000 to cover the taxes on the $10,000.
Then they pay taxes on that additonal amount, so we give them some more, which they owe taxes on, and so on.

If their marginal tax rate (x) is 27%, we need to pay them $10,000 + 10,000 * (y) or 36.9863%
If their marginal tax rate (x) is 28%, we need to pay them $10,000 + 10,000 * (y) or 38.8889%

What I need is the formual to plug x% in to get y%

Thanks

Any ideas?

For the benefit of others, let me rephrase the question a bit, for hopeful clarity.

The government decides you were overtaxed, and gives you $10,000. However, you have to pay tax on that $10,000. Let's say your marginal tax rate is 27%. So you they conveniently deduct $2700, and give you $6300. Here is where is doesn't make much sense, but it is decided you should pay tax on the tax, or 27% of the $2700, which is $729. Then you pay 27% on the tax on the tax, which is $196.83. This goes on infinitely. What would be the effective tax rate paid on the $10,000?

The answer for the general case of a tax rate of t is t/(1-t). So for 27% it is 0.27/0.73 = 0.3699.

Quote: Wizard

For the benefit of others, let me rephrase the question a bit, for hopeful clarity.

The government decides you were overtaxed, and gives you $10,000. However, you have to pay tax on that $10,000. Let's say your marginal tax rate is 27%. So you they conveniently deduct $2700, and give you $6300. Here is where is doesn't make much sense, but it is decided you should pay tax on the tax, or 27% of the $2700, which is $729. Then you pay 27% on the tax on the tax, which is $196.83. This goes on infinitely. What would be the effective tax rate paid on the $10,000?

The answer for the general case of a tax rate of t is t/(1-t). So for 27% it is 0.27/0.73 = 0.3699.

The tax laws (in the US) do not work in the manner posed in the question.

The solution for the original question is pretty straightforward. To obtain d after tax-dollars which are taxed at a pretax rate of t, the employer should pay out d/(1-t) dollars. For $10k takes at 27%, this yields 10000/.73 = 13698.63. As a check, 27% tax on 13698.63 is 3698.63, taking us back to an after-tax amount of $10k. I'm not sure how recursiveness fits into this here; you don't owe tax on that $3698.63, not in the United States real world at least.

This is quite a common calculation, as it turns out. Sometimes benefits that companies offer to employees are taxable. The benefits have a tangable value but aren't part of the base salary. So the company has to "gross up" the employee's salary to an amount that covers the taxes on the benefit.

Quote: 7outlineaway

I'm not sure how recursiveness fits into this here; you don't owe tax on that $3698.63, not in the United States real world at least.

Agreed.

There could also be other factors at work. Assume, for instance, the indiviudal files a joint return with their spouse, who makes 109,250 per year, for a total pre-tax income of 209,250, paying the marginal tax rate of 28%. However above the 209,250 threshold, every additional dollar in compensation is then taxed at the marginal rate of 33%. So the "bonus" amount should now be 10000/(1-.33)=$14,925.37.

So if the employer assumes a 27% rate and thinks they're giving the person 10,000 after-taxes, really the worker is only pocketing 8,773.

So really, the employer should pick a nice round number big enough to cover most of the taxes.

Or if they didn't mind, a cashier's check for $9,999.99 would work as well.

Quote: redsox

Agreed.

There could also be other factors at work. Assume, for instance, the indiviudal files a joint return with their spouse, who makes 109,250 per year, for a total pre-tax income of 209,250, paying the marginal tax rate of 28%. However above the 209,250 threshold, every additional dollar in compensation is then taxed at the marginal rate of 33%. So the "bonus" amount should now be 10000/(1-.33)=$14,925.37.

So if the employer assumes a 27% rate and thinks they're giving the person 10,000, really the worker is only pocketing 8,773.

7outlineaway got the correct formula, though, with "t" being the tax rate. My friend was just looking for the standard gross up formula, but he had worded it so that some read it as an infinite sequence.