One of the most useful bits of mathematics I ever learned was imparted, almost in passing, by a Cal Engineering professor I studied with. He was constantly checking his work on the back of loose envelopes on his desk by jotting down approximations of formulas. He called this method "Back of the Envelope Calculation," and I have been using it ever since to check whether or not certain things make sense.

For example, let's say you're trying to decide whether to lease or buy a new car. You know the car will cost you $30,000 new but the salesman enticingly offers a $500 a month lease. Looking at the terms of the lease you realize that it's for 48 months. Quick back of the envelope calculation shows you that the lease will cost approximately $24,000 (500 x 48), and that buying it will cost approximately $625 a month (30,000 / 48). Is the extra $125 a month worth it? Well that depends on a lot of other factors, but now you're comparing apples to apples.

The other day I received my yearly property tax assessment from the Los Angeles County Tax Assessor and I started doing some back of the envelope calculations of my own. Most Americans take it for granted that we pay far less in taxes than many of the "socialist" European countries. Most of our taxes, however, are somewhat hidden in the form of fees and other "built-in" costs. I started thinking: Just how much of our income do we pay in taxes?

"Back of the Envelope" calculations are all about keeping things simple so that you can calculate quickly and get a basic idea of what's going on. There will always be details and subtleties that you will not compute, but they are usually irrelevant to the big picture. In order to keep things simple, let's take a look at an above average couple living in Los Angeles and earning approximately $100,000 a year (median joint income in California is $61,084).

Our couple will expect to pay approximately $18,115 in Federal Income Taxes (~ 18.1%), and approximately $9,300 in California state income taxes (~9.3%). Add to that another approximately $7,650 in Social Security and Medicare Taxes (~7.65%).

If our couple owns their own home, they can expect the following: The median home price in Los Angeles in 2005 was $540,000. In West L.A. that number jumps to $837,000. Let's use an easy number in the middle: $700,000. Californians pay 1% of their home values in property taxes, however bond measures and special assessments have virtually doubled that to between 1.5% to 2% depending on your district. At 2%, our couple will expect to pay approximately $14,000 in assorted property taxes, or an additional 14% of their income.

For those of you keeping track, our erstwhile couple has now allocated about 49% of their income to the 4 largest taxes and we haven't yet begun to talk about the smaller taxes. In L.A., sales tax on any goods you purchase (after being taxed on your income and property) is an additional 8.25% on practically everything you buy. Add additional taxes for your car(s), cigarettes, alcohol, gasoline, road tolls, and any government permits you might need and you've got quite a bit more. Rounding to make things even, it's fairly safe to say that about 60% is spent on assorted taxation.

So our quick "back of the envelope calculations" tell us that our couple that earned $100,000 actually kept about $40,000 (or $3,333 a month) to pay for their mortgage, home owner's association fees (another tax?), food, entertainment, etc. If they've managed to save or invest any of that left over money, that is taxed again either as income or capital gains. Of course, anything left over at the very end of the day will be taxed again as "estate tax" before it can be willed to children or any other dependents.

Using our earlier figure of $700k for this couple's home, if we assume they put $200k down (a rather optimistic 28%) and mortgaged $500k, then they can expect to pay approximately $3,000 a month in mortgage (for a 30 year fixed at 6%), leaving our couple with less than $350 a month to live.

Sound unrealistic? Back of the envelope calculations are necessarily approximations. There are, of course, certain things that we left out like a fairly substantial home mortgage deduction from income tax. For our couple this actually amounts to a deduction of about $19k a year, putting an additional $5k back in our couple's pockets, but even that only helps with about $400 more a month. Clearly our "above average" couple cannot afford their "average" home.

So back to our previous question of do Americans pay far less in taxes than our European counterparts? The answer is a resounding "No." The average European tax rate in the original EU-15 counties was approximately 52.4%. This is supposedly as compared to an average American tax rate of 34.6%, however that only takes into account Federal taxes and not the state and local taxes. As we have seen from our previous calculations, those state and local taxes can easily be as much as our federal taxes, especially in urban areas of the country.

This did not used to be true. The current American system of taxation by payroll deduction and then a yearly "filing for refund," was not instituted until World War II and was the brain child of the Nobel-prize winning economist Milton Friedman. Prior to that time, Americans were quite literally sent a tax bill in the mail each year. Imagine our couple receiving a tax bill for $60,000 in April? In fact, prior to 1913, it was unconstitutional in the United States to tax individual incomes at all. That was changed with the 16th amendment to the constitution and the Federal government grew significantly in importance with it.

Are we being overtaxed in America? Perhaps we are, and perhaps we are not. The answer to that question depends on how well you think your tax money is being spent and how well you think the burden is distributed on the population (two interesting questions in their own rights). With mid-term elections coming up all over America in the next few weeks, you need to ask yourself these questions when asked to approve the litany of tax hikes, fee increases, and bond measures (hidden tax hikes) that pepper your ballot. Hopefully some quick "back of the envelope" calculations of your own can help you make a more informed choice.