# Optimal Lifetime Consumption

## "Optimal" Lifetime Consumption

[The following analysis is all provable via some calculus. I ignore the proofs because (1) I'm don't want to write them out and (2) no one else would care.Assuming your utility function looks like

$$ u = - c^{-\epsilon} $$how much you consume each year should grow exponentially throughout your life (with some exceptions discussed later). How quickly should it grow? That growth rate is given by

$$ \frac{r}{1 + \epsilon} $$

where $r$ is the risk-adjusted rate of return. I estimate that $\epsilon \approx 0.35$ and, based on this and SP500 data stretching back to 1871, I estimate the real risk-adjusted rate of return is 5.1%. From this, we can compute that optimal consumption should grow roughly 3.8% per year.

[As an aside, it might strike you as weird to use data that's ~150 years old in this analysis, but average SP500 returns and variance have actually not changed during this time in a statistically significant way.]

However, this ignores risk of death. Accounting for that specifies that your consumption should increase by

$$ 1.038 \cdot a^{1/(1+\epsilon)} $$per year where $a$ is your chance of being alive by the end of the year. Using my numbers, this becomes

$$ 1.038 \cdot a^{0.74} $$Suppose you start spending mostly your own money when you're 20. This implies, for most people, that your consumption should grow each year until peaking at roughly 6x your 20-year-old spending when you're in your late 70s.

This ideal actually fairly closely describes peoples spending until before their 40s. From the 40s onwards, however, spending flattens and then starts shrinking from the 60s onwards Foster.

Part of the reason is that people tend to spend more when they have kids, which is rational and something we consider in the next section. Nevertheless, the fact is that people spend basically as much in their late 70s as they do in their early 20s - a far cry from the idealized ~6x.

What's more, I think its all but certain that the only reason the model fits at all during the 20s and 30s is that peoples' incomes tend to go up then, which allows them to spend more, which makes the ideal curve more a result of coincidence than planning.

## Adjustments

One thing utility optimization suggests is that you should spend more during years when you're supporting someone else. For instance, suppose utility is given by

$$ u = - n \cdot \left( c / (2 + n) \right)^{-\epsilon} $$This suggests a couple should consume ~2.6x more during the years they have kids compared to the years they don't. Their idealized lifetime consumption curve should look something like

Another thing that needs adjusting for are "special" expenses.

For instance, one-time expenses (e.g. college tuition) shouldn't be counted as regular consumption but since they don't move your point along the utility-consumption curve. Likewise, if you live in (e.g.) the Bay Area but expect to move to a cheaper place later, the excess rent would fit in this category. Likewise, required health care expenses belong in their own category since that kind of spending also doesn't move you along the utility-consumption curve.

Finally, most people with kids put some value on leaving wealth to their descendants. If you care as much about your kids as yourself, the exponential growth in consumption should simply continue unabated by your mortality odds, growing at ~3.8% per year forever. In short, if you care as much about your descendants as you do yourself, you spend little in exchange for starting an economic dynasty under this idealized model.

Really, though, this entire model is such a far cry from actual human spending patterns, that it really only suggests three things: (1) pretty much everyone should save more, (2) people don't value their futures nearly enough, and (3) the model is probably wrong.

A priori, the most obvious point of departure is the utility function: $\epsilon$ might change with income level, people are loss-averse, and people's sense of "acceptable" increases with income - all of which would violate the utility function we're using. I've also heard people claim that consumption when we're young is more utility-producing - though I've never heard a good justification for it.