Monday, December 03, 2007

Property Tax Thoughts

The paper today reported Daniel’s property tax plan. "Daniels also wants the state to pick up the costs of schools' general funds, school transportation and child welfare and supports limiting local spending growth to the six-year average growth of personal income." Most people would think this sounds good. However, a mathematician would say it is not. The reason is simple and it comes down to the difference between an average and compound rate of change. The average of six years will always be higher than the compounded annual rate over that same six year period. In simple terms if they limit it to this, spending and ultimately taxes will grow far faster than our income.

Some will say, it is small, but in fact anything that is even slightly greater than the income growth of those being taxed will eventually become large. Why not just state it mathematically correct to begin with. It is not a difficult calculation. Anyone can perform it on a calculator.

The formula is:

Exp[ln(Fv/Pv)/# of years]-1 = compounded rate

Below are three examples. The first shows a small change, but the last to show a large difference. The reason is that a high first year will create a much larger difference.
The first column is the year, second column is a randomly generated rate of growth and the third column is the value of any given year.
0 $1.00
1 1.6% $1.02
2 5.5% $1.07
3 0.8% $1.08
4 5.7% $1.14
5 5.4% $1.20
6 1.8% $1.22

3.45% 3.42%

0 $1.00
1 9.9% $1.10
2 5.4% $1.16
3 1.3% $1.17
4 4.0% $1.22
5 4.9% $1.28
6 1.2% $1.29

4.44% 3.76%

0 $1.00
1 9.6% $1.10
2 3.9% $1.14
3 1.1% $1.15
4 7.6% $1.24
5 1.9% $1.26
6 7.7% $1.36

5.30% 4.48%

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1 Comments:

At 11:33 AM, Blogger Phil Marx said...

Bill,

You make a good point. Usually the difference is pretty small, but not always. I think that many times when math is used incorrectly by politicians they probably don't even recognize the error. I'm even more certain that most people that read about their proposal dont understand the difference either.

Here's a couple math twisters that could be used to take advantage of unwitting people.

Take any dollar amount, cut it by 10%, then increase it by 10%. Most people would assume you'd be back where you started.

Let's try it. Start with $1000. Cut 10% ($100) and end up with $900. Now increase this $900 by 10% ($90) and end up with $990. You got screwed because you lost $10.

Example #2: What's the difference (as a percentage) between $90 & $100?

Well, if you want this to appear low you say 10/100 = 0.1 or 10%.
If you want it to look high you say that 10/90 = 0.11 or 11%.
If you wanted to be neautral and honest you would say 10/[(90 + 100)/2] = 10/95 = 0.105 or 10.5%.


Now when I say most politicians use math improperly by accident, I am simply attesting to their ignorance. I'm sure that if they understood it correctly, they would still use whichever form benefitted their agenda.

 

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