## Inflation-adjusted incomes go from 63K to 64k over a period where cumulative inflation is 8.7%, what is nominal income change

Submitted by FormerSanDiegan on March 14, 2017 - 8:03am
1.6%
-7.1%
6% (1 vote)
> 10%
Prices are inversely proportional to rates because of stealth inventory

Happy pi day

I survived the ides of March, now we'll see about St. Patricks Day!

((64,000 / 63,000) * (1 + 0.87)) - 1 = 10.425%

On the other hand, if nominal change was from 63K to 64K while simultaneously having inflation of 8.7%, inflation adjusted wage change would be

(64,000 / 63,000) / (1 + 0.87) - 1 = -6.54%

ucodegen wrote:
((64,000 / 63,000) * (1 + 0.87)) - 1 = 10.425%

On the other hand, if nominal change was from 63K to 64K while simultaneously having inflation of 8.7%, inflation adjusted wage change would be

(64,000 / 63,000) / (1 + 0.87) - 1 = -6.54%

My math is rusty but I think the first case should be 10.28% and not 10.425?

((64000 - 63000) / 63000) + .087 = 10.28%

ltsdd wrote:
ucodegen wrote:
((64,000 / 63,000) * (1 + 0.87)) - 1 = 10.425%

On the other hand, if nominal change was from 63K to 64K while simultaneously having inflation of 8.7%, inflation adjusted wage change would be

(64,000 / 63,000) / (1 + 0.87) - 1 = -6.54%

My math is rusty but I think the first case should be 10.28% and not 10.425?

((64000 - 63000) / 63000) + .087 = 10.28%

No, you have to adjust to current dollars, not use percentage change calcs and add percentage changes. You don't add multiplicatives. For example a 10% change on top of a 10% change is not 0.1 + 0.1. It is (1.1 * 1.1)-1 = 21%. For example if one year you had a 50% gain and the next year was an addition 50% gain on previous year, the total gain would be (1 + 0.5)(1 + 0.5) - 1 = 125%. Remember that the second years gain started with a larger amount upon which the 50% gain was applied. That means:
P1 = (1 + 0.5)P0
P2 = (1 + 0.5)P1
combining gives
P2 = (1 + 0.5)(1 + 0.5)P0
--
P0 = starting principal
P1 = principal after year 1
P2 = principal after year 2

In the case of inflation, your income values are adjusted to current values already, so you need to use growth solely due to income and then multiply by inflation to see nominal - non inflation adjusted growth.

Another way to look it it is that if we were dealing with nominal numbers instead of inflation adjusted number for income it would simply be:
64,000 / 63,000 - 1 = 1.587%
But with inflation adjusted numbers, either the 64,000 is adjusted down for inflation (to get constant buying power) or 63,000 is adjusted up for inflation that occurred since. To correct, you have to reverse that adjustment.

Here I'll assume the 64,000 had been adjusted down for inflation to have same buying power as previous period's earnings. Results are the same, but the math looks simpler. To adjust 64,000 back up to current dollars, multiply by inflation. ie:
64,000 * (1 + 0.087) = 69,568
So you end up with
69,568 / 63,000 - 1 = 10.425%
or
(64,000 * (1 +0.87)) / 63,000 - 1 = 10.425%
Rearranging gives
(64,000/63,000) * (1 + 0.087) - 1 = 10.425%

To look at it from the view of 63,000 having been adjusted to be current dollars by adjusting up for inflation, you have to reverse by dividing out the inflation ie:
63,000 / (1 + 0.087) = 57,957.68
or
64,000 / 57,957.68 - 1 = 10.425% (note: I rounded on cents above)
or
64,000 / (63,000 / (1 + 0.087)) - 1 = 10.425%
Rearranging again gives
(64,000/63,000) * (1 + 0.087) - 1 = 10.425%

and hopefully I didn't introduce any typos...