Assume for a moment that the CPI accurately encompasses all prices. Also assume that I have $100 of outstanding debt. Let's now say that the CPI decreases by 25%. In real terms, this change in CPI effectively makes my debt now $133, correct? This is from a lecture I was watching. I understand there is a 33% jump between 75% of original CPI and 100%, but I would like to understand more clearly why it is not $125. Can someone please tell me the right way to think about this?
The CPI is dollars per goods/services. The value of a dollar is how much goods/services you can get per dollar. So that is the reciprocal; to get the value of a dollar, you divide by the CPI.
Suppose apples start out costing $1 a pound. Then your $100 debt corresponds to 100 pounds of apples. If the cost of apples decreases by 25%, the new price will be $0.75 a pound. This corresponds to 133 pounds of apples. So measured in past dollars, you debt is now $133.
When you subtract from the denominator, that's not the same as adding to the numerator. 1/(1-x) is not equal to 1+x. The more you subtract from the numerator, the greater further subtraction are relative to what's left; if you subtract one from 100, you're decreasing it by 1%, but once the numerator gets down to 50, subtracting one is subtracting 2%. because of this, subtracting from the denominator has more of an effect than adding to the numerator, and the effect is more pronounced the more you subtract. If the CPI decreased 99%, then your debt would be worth $10,000 in past dollars.