George Rebane

This year I promised to write more stuff on technology and finances. In the winter I also start writing the year’s *TechTest2011* which will be taken in the spring by a select cadre of Nevada County’s high school seniors and juniors. Reviewing some of the problems from previous *TT*s, I came upon one that involves taxes and should be of interest to all of us.

Not many people have a clue about how income and sales taxes combine to present us with the total tax burden we have to bear. If we did, maybe more progressives would abandon their faith in ever higher taxes. So let’s consider a simple example of dollars and cents or sense.

*TI*= 20%, and you wanted to buy something that had an added sales tax

*TS*= 10%. How much would you have to earn,

*E*, in order to purchase something sold at a price

*P*? Knowing this would allow you to compute the actual tax burden you have to bear. Hint: It isn’t the simple sum of

*TI+TS*or 20% + 10% = 30%, it turns out to be significantly more. Here’s how to get the correct answer and understand why it’s hard to stretch your dollars.

When you earn an amount *E*, you pay *E*TI* income tax, and get to spend the amount *E**(1-*TI*). And when you want to buy something for a price *P*, you have to pay the amount *P**(1+*TS*) to take it home with you. So if we set the spendable amount from earnings *E* equal to the total amount we have to pay the merchant, we get the simple equation *E**(1-*TI*) = *P**(1+*TS*). Solving this for the amount *E* we have to earn gives us *E* = *P**(1+*TS*)/(1-*TI*).

Let’s put in the above tax rate numbers to get the message from this equation. We have to earn *E* = *P**(1+0.1)/(1-0.2) = *P**1.3750, an effective tax rate of 37.5%. The effective total tax burden is simply what is left over if we subtract price *P* from *E*, the amount we had to earn to make the purchase. This is the same as subtracting one from (1+*TS*)/(1-*TI*), the factor that multiplies the store’s price *P* for the item.

Doing this yields the true tax you must pay for everything that you purchase as the nifty little relation (*TI+TS*)/(1-*TI*) – simply add the two tax rates and divide by the complement (fancy word for ‘one minus’) of your income tax. Again putting in our example tax rates gives (0.2 + 0.1)/(1 – 0.2) = 0.375 or 37.5% as expected. As you can see, this adds another 7.5% of the price to what you must pay, and is significantly higher than the erroneous 20% + 10% = 30% tax burden that most people who even think about such things will compute in their head.

In fact, the actual tax burden is 7.5/30 = 0.25 or a whopping 25% higher than what you would commonly think. Multiple layers of taxes and fees have a pernicious effect that sneaks by most of us. They do not simply add, but combine in a more devious way to devastate your wallet. The government is betting that all this is beyond you.

BTW, this was part of Problem #4 on *TechTest2009* which you can download here and see what else the young people who take this annual merit scholarship test have to master. Not everyone can hack this stuff, but those who do will become the next wealth generators who will pay for our entitlements.

If you believe that supporting those in the next generation who have to acquire this kind of knowledge is important for our country, then you can make tax deductible donations to *SESF’s TechTest* scholarship fund. More on that here.

I have always wondered how it is the progressive tax, where you are taxed more for making more, and so beloved by the "progressives" ends up causing the top 5% of people paying 40% of the tax load and the "progressives" still complain. I would think their goals were met.

Posted by: Todd Juvinall | 25 January 2011 at 06:00 PM

With a marginal income tax rate of 40% and a 10% sales tax, the earner will pay an effective tax of (0.4+0.1)/(1-0.4) or over 83% tax for everything he buys. And we wonder why these people are trying to do everything they can to get out from under such a tax burden?!

Posted by: George Rebane | 25 January 2011 at 07:47 PM

That is why Walmart is so huge. Save a buck.

Posted by: Todd Juvinall | 25 January 2011 at 08:39 PM

Sales tax paid is a deductible item on your income tax return. By how much does that change your result if you include the tax savings due to the deduction?

Posted by: Wayne Hullett | 29 January 2011 at 02:51 AM

Good point Dr Hullett. Not all taxpayers can and/or do deduct sales tax from their income taxes - e.g. the 'rich' when buying big things like cars and boats. And with the current fiscal state of the states and the country overall, I think such remaining deductions will soon disappear. However, we can provide for sales tax deductions in the formula by appropriately adjusting (reducing) the aggregate income tax rate TI to compute the overall tax burden.

In the case where you do get the full amount of sales tax, P*TS, refunded, the formula for earnings then becomes E*(1-TI) + P*TS = P*(1+TS), and yields the amount earned to price ratio of E/P = 1/(1-TI). Subtracting one from both sides gives the total tax burden on the purchase as E/P – 1 = TI/(1-TI), or the same as if we would have entered TS = 0 into the original formula I gave in the post. This is what we would have expected. Again using our numerical example, the total tax burden in this case would then be 0.2/(1-0.2) = 0.25 = 25%, instead of the previous 37.5% with the non-deductible sales tax. We note that we still have to earn 25% more than we pay for the purchase, which is 5% more than the income tax rate TI = 20% that we pay on our earnings. Tricky business.

Posted by: George Rebane | 29 January 2011 at 09:45 AM