There are four tax rates that are being discussed. Putting the definitions aside for a moment, we can describe what they are, ordered in increasing size of the denominator.
- Actual tax rate on $1 (or $0.01) of increased income, as calculated by the IRS. Because the IRS rounds figures to the nearest dollar, a $1 increase in income will usually (spikes excepted) result in either a $0 (0%) or $1 (100%) increase in the tax owed, like the example of going from $19 to $20 of earned income. If calculating with $0.01's, the penny will result in either $0 (0%) or $1 (10,000%) increase in tax owed. If the tax tables are being used, they are quantized in $50 increments, so a $1 increase in income will cause either a $0 (0%), $5 (500%), $6 (600%), $11 (1,100%), or $12 (1,200%) increase in tax due. (Those figures correspond to the 10%, 12%, 22%, and 24% brackets respectively). Multiply those rates by 100 if you want to use pennies instead of dollars. These numbers are "exact" but there are reasons why they are not useful for making decisions. We usually can't control our annual income down to the $1 to $0.01 level, so we usually can't optimize around these small little jumps. It would also wreak havoc on any kind of optimization algorithm to try to evaluate behavior over such an un-smooth curve and needing to constantly round off numbers. The only time this calculation should be done is when actually preparing taxes. Otherwise, the actual rates on $1 or $0.01 are basically useless.
- Tax rate on a "small" increase in income, which in this context means a change large enough to smooth out rounding to the nearest dollar, and if applicable, tax table jumps. $100 is usually a convenient "small" increase in income, because it's an even multiple of the tax table increments (2 x $50), and it will give you a precision down to 1%, which is usually good enough for most purposes. If you increase income by $100, and 24 of those dollars cause your tax to jump by $1, and the other 76 cause no change in tax, that's a tax rate of 24%. This is a "smoothed" version of rate #1 on a small scale. This is also analogous to the "slope" or "derivative" of the curve at a particular point. Unlike tax rate #1, this value is very useful for making decisions. When you are dealing with progressive areas of the tax curve, the well-known simple algorithm is contributing to pre-tax accounts until your predicted #2 withdrawal tax rate equals or exceeds your current #2 tax rate, then contribute any remaining money to Roth. Dealing with a smoothed rate also has advantages for software, as values can be represented easily by floating point numbers, and can be operated on without regard to small rounding.
- Tax rate on an entire incremental change in income. For example, if you get a $100,000 bonus at work, and owe $26,400 more in taxes, your #3 tax rate is 26.4%. This rate is also useful in several situations. When you're dealing with a chunk of income that can't be broken down into smaller pieces (like, getting a fixed-size bonus), then this is what you care about, more so than the #2 rate on the first bit or last bit of income, which may be very different. The #3 rate is usually a composite (weighted average) of the #2 rates over the various sections tax curve that this change spans. In this example, if $70,000 of the bonus was in the 24% bracket and $30,000 was in the 32% bracket, that would give $26,400 ($70k x 24% + $30k x 32%) change in tax and the 26.4% rate. If you were choosing to take the bonus on this year's or next year's tax return, the better choice is the one with the lower #3 rate. This #3 rate is also useful when making traditional/Roth decisions when the tax curve is not progressive and smooth. The more complex algorithm becomes: (a) calculate a #3 tax rate on all possible pre-tax contributions starting from $0, (b) find the point of maximum rate, (c) if this rate is greater than the predicted withdrawal #3 rate, then contribute pre-tax, go back to (a) and recalculate starting from this new point; if at any point the rate is less than the predicted withdrawal rate, contribute any remaining money to Roth. The wiki contains a worked example.
- Tax rate on the entirety of one's income (total tax / total income). Many people misleadingly use this rate to try to make T vs R decisions; it's not directly useful for that purpose. It does have a few uses, however, such as tax-weighting asset allocations between pre-tax and Roth accounts. A pre-tax account can be thought of as two separate accounts: an account owned entirely by the government which you manage the investments for, and a Roth account that belongs completely to you. The size of the "government account" equals the predicted #4 tax rate times the total account balance. Eg. if you have a $1M pre-tax IRA and you expect a #4 tax rate of 16%, you can think of it as a $840,000 Roth account and rebalance accordingly.
- This tax rate isn't commonly used and doesn't have a common name. Maybe "exact marginal tax rate" or "exact rate on the next dollar" would be appropriate, but should be accompanied by a description to avoid confusion.
- This tax rate should be called the "marginal" tax rate. This is the generally accepted term used in economics, and is also the term used by the "Personal Finance Toolbox" aka Case Study Spreadsheet.
- To my knowledge there is no generally accepted definition for this tax rate. "Incremental", "cumulative", "cumulative marginal", or "average marginal" may be appropriate, but should be accompanied by a clear explanation to about confusion. The PFT/CSS calls this "cumulative", which I thought was good enough that this was the term I used in the T vs. R tool I'm working on. I suggest avoiding "marginal tax rate" for this rate, to prevent confusion with #2.
- "Average" tax rate is used commonly enough that I think it's okay for this purpose.