redding.dev
	economics personal-finance The FIRE Model The goal of many of those in the FIRE community is to save enough money to be able to live on the interest indefinitely. This model is obviously wrong (or at least incomplete), but it has some neat consequences I want to explore. [For a more complete model see here.] Suppose you save $s$ per year while working, it grows at interest rate $r$, and you retire after $t$ years. Your retirement nest-egg will be $$ \frac{(1+r)^t - 1}{\ln(r)} s $$ The derivative of this relative to time is $$ (1+r)^t s $$ Roughly speaking, this means your nest egg will increase by $ (1+r)^t s $ in the last year. This has some interesting consequences for how we estimate the value of our time. The naive way to do this is to take your wage (or your salary divided by hours worked). However, this has two serious drawbacks. The first is that it ignores taxes. The second is that it ignores the fact that money is worth more if you have it sooner. We can see this by noting that Saving one additional dollar now will result in $(1+r)^t$ dollars upon retirement. The derivative of our savings upon retiring is $s (1+r)^t$ From this, it is clear that saving one dollar now will let you retire $1/s$ years sooner. In other words, the value of your time isn't what you earn per hour worked, it's what you save per hour worked.

economics personal-finance

The FIRE Model

The goal of many of those in the FIRE community is to save enough money to be able to live on the interest indefinitely. This model is obviously wrong (or at least incomplete), but it has some neat consequences I want to explore. [For a more complete model see here.]

Suppose you save $s$ per year while working, it grows at interest rate $r$, and you retire after $t$ years. Your retirement nest-egg will be

$$ \frac{(1+r)^t - 1}{\ln(r)} s $$

The derivative of this relative to time is

$$ (1+r)^t s $$

Roughly speaking, this means your nest egg will increase by $ (1+r)^t s $ in the last year.

This has some interesting consequences for how we estimate the value of our time. The naive way to do this is to take your wage (or your salary divided by hours worked). However, this has two serious drawbacks. The first is that it ignores taxes. The second is that it ignores the fact that money is worth more if you have it sooner.

We can see this by noting that

Saving one additional dollar now will result in $(1+r)^t$ dollars upon retirement.
The derivative of our savings upon retiring is $s (1+r)^t$

From this, it is clear that saving one dollar now will let you retire $1/s$ years sooner. In other words, the value of your time isn't what you earn per hour worked, it's what you save per hour worked.