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Barry1974 · Oct 19, 2011, 06:27 AM

I"m a teacher an our step salary schedule is based on a 4.5% increase which INCLUDES both the step AND the yearly raise. It has 15 steps. When comparing step 1 of this year to step one of next year, the annual increase is 3.92%. When comparing step 15 of this year to step 15 of next year the annual increase is 3.57%. I tried to explain to our union that a salary scale calculated in this manner DOES NOT WORK, and over time the percentage difference between step 1 and step 15 is getting smaller. This percent is supposed to stay constant -- from what I've seen on EVERY other salary scale I have found online. Any help on explaining this to people who just "don't get it" would be greatly appreciated. Thank you, Barry

ArcSine · Oct 20, 2011, 02:11 PM

Could you clarify a bit on the terms of the arrangement? Are there 15 "pay grades", each one a specified percentage over the immediately preceding one? Or is each step a specified dollar amount over the preceding level? And are the annual raises the same percentage for each level?

Barry1974 · Nov 30, 2011, 07:34 PM

Hi Arc, I attached a link to the salary scale. We just got a new contract and it's the same problem. They call it a 4.5% increase but it's a 4.5% for everyone on the scale and this percentage includes both the annual and the step. I figured out the annual percentage alone -- check it out.

Barry1974 · Nov 30, 2011, 07:34 PM

http://imageshack.us/f/62/oldsalaryscale.jpg/

ArcSine · Dec 1, 2011, 05:47 AM

I see that the absolute amount of the raise is in the $1,830 - $1,850 range, roughly, for each of the 15 levels. But on a percentage basis, an 1,830 raise for someone currently at 45,000 (4.1%) is quite different from that same 1,830 for someone presently at 50,000 (3.7%).

I'm not sure if that's the point you're going for, but that much is clear.

(BTW, I think you've got a typo on the 4th level. The numbers as shown imply just a 0.65% increase.)

Barry1974 · Dec 1, 2011, 08:05 AM

Arc, that number should be $49,386. Sorry.

Barry1974 · Dec 1, 2011, 08:08 AM

He's the main problem I see: The percentage difference of step 1 and step 15 gets smaller each year with this method of calculation. Wouldn't a SET annual raise keep the percentage difference between step 1 and step 15 consistent as time goes on? According to my calculations, it would... I just want some expert advice! :) Thank you!

ArcSine · Dec 1, 2011, 10:54 AM

The percentage diff between any two levels would be kept constant if they received the same percentage raise, but not if they received the same dollar raise.

Generalizing, take any two arbitrary, unequal positive amounts a and b. If they are both increased by some percentage r (expressed as a decimal), then their post-increase ratio is

$\frac{a(1+r)}{b(1+r)} \ = \ \frac{a}{b}$ ; i.e. their post-increase ratio is identical to their pre-raise ratio. It's readily apparent that this would hold true for any number of periods as long as both levels received the same percentage raise each period.

Wrapping this around some illustrative numbers, take 40,000 and 44,000. That is, the higher level is 10% greater than the lower. Suppose each level gets a 7% raise, putting them at 42,800 and 47,080 respectively. The ol' calculator shows that the higher level remains precisely 110% of the lower.

It's also simple to prove that for any two positive numbers a and b with a > b > 0, if you add the same positive quantity x to both, then

$\frac{a+x}{b+x} \ < \ \frac{a}{b}$ ; that is, their post-raise ratio is less than their pre-raise ratio.

Again with the same starting levels (40K and 44K), give 'em both a 3K raise. After the raise the higher level is only 9.3% greater than the lower, whereas it was 10% higher before the raise.

Hence, the percentage diff between any two levels would shrink if they received the same dollar raise, but would remain unchanged if they received the same percentage raise.