Calculus of Frugality

I'm teaching calculus this semester -- actually, I teach calculus just about every semester.  I've been teaching calculus longer than my students have been alive.  Calculus colors the way I look at the world.

Here's an example of non-calculus thinking.  Most of us think about wealth with a picture that looks like this:  bar graphs.  We picture wealth as a snapshot of how we're doing right now.

A picture of wealth: poor Pete is down-and-out, wealthy Walter is rich and happy;
middle-class Marv is comfortable but looking to improve.

But calculus says that life is all about change.  And if you think about living a long time, you could see why a snapshot of where you are right now isn't all there is to the picture -- a good picture shows where you're at and where you're headed, as well. It's about looking at your wealth as you move through time.  Here's the same picture as the beginning of a graph.

Wealthy Walter enjoys spending his money on nice stuff.
Middle-class Marv is cutting coupons.
Poor Pete is darning his own clothes (darn; wish I could draw better).

From this snapshot, you can see not only where these guys are at, but you also get a hint at a trend.  It's not just how much we have, but also what we're doing with it, that matters.  If we catch up with these guys again 10 years later, the graph might stretch out like this.

Walter is still shopping boutiques, and Marv is still shopping the sales.
Pete has started hanging his own laundry,
and he's learning other ways to cut costs, too.

The trend lines show that a cost-conscious middle-class person like Marv can certainly set aside enough money to become comfortable.  Pete doesn't have as many options for saving money as Marv does (because he doesn't have as much to start with), but don't let the fact that his graph is still lower than the others' fool you.  For the long term, it's the trend, not the bar graph, that matters.   Because if this keeps up, the next ten years gives us a very different picture.

Whoops! Walter still has a lot of money, but he's in over his head anyway.
Marv is still comfortable clipping coupons.
And Pete can finally rest easy, thanks to his penny-pinching strategies. 

Pete's graph is exponential (as in, "prices are rising exponentially").  In the long run, exponential growth wins out just about any other kind . . . even though it's frustratingly slow early on.  It's why experts tell you to start investing in retirement as early as possible.  But it's also the reason why developing good habits and constantly learning new things can be so powerful . . . eventually.

It's these three graphs, seared deep in my psyche, that keep me from worrying much (if at all) about the occasional one-time expensive purchase -- that's a single snapshot in time.  But it's these same three graphs that keep me washing my plastic bags, packing my lunch for work, patching my son's backpacks.  Because habits and rituals are the parts of our lives that determine the shape -- and because of that, the height -- of that graph.