# Steps to calculate reduction in spending vs. savings till retirement

I'm stuck and I I think it's an algebra issue. I have a typical what do I need to save (today) so I can retire question for the first part.

Let's say you need \$1,239,926.43 in savings at retirement (age 67) in order to fund your lifestyle (80%) till death (age 95). With a salary of \$50,000 in present dollars (age 30), zero savings to keep it simple and an inflation rate of 2% and a pre-retirement return of 6% compounding annually. Using the formula for PMT I can figure it out like this.

PMT = FV * r / (1 + r)^n - 1

First I get the geometric return of inflation and pre-retirement return since I have two rates to add.

let ri= (1+0.06)*(1+0.02)-1

ri=.0812

For PMT I get; PMT = 1,239,926.43 * 0.0812 / ( 1 + 0.0812 ) ^37 - 1

PMT = \$5,933.30

So here is where I am getting stuck. I would like to express this question differently. Instead of saying what do I need to save today (\$5,933.30) which assumes I will stop saving when I retire. I would rather say how much of a reduction in spending would I need to make to my lifestyle over my whole life (65 periods) to make it to death ending with zero dollars. This way instead of stopping my savings at age 67, I would continue to not spend a percentage of my income, inflation adjusted over my lifetime. That difference would remain in the bank and I would earn additional interest on it at a post-retirement return of 5%.

The problem I run into is if I save normally I have the \$1,239,926.43 FV at age 67. But if I reduced my spending I should have less than that since I would need less in retirement. But how much less is the question? How would you solve this problem without using a solver? 