Marginal Utility of Wealth

Wow.  Every so often, you read something that completely re-frames your thoughts on something, or makes clear both a paradox and its explanation.  If you're an economist, you probably think in terms of marginal utility to wealth: as Bill Gates put it, after a certain point "it's the same hamburger."  This generally (always?) holds when we're talking about more of a good thing.

But when it's not about making your life better, but instead about making a bad thing stop, we see the reverse effect.  Imagine a pile of bills.  A huge pile of bills on your kitchen table.  Not just any kind, but the pink kind, that mean you'll have your utilities shut off next week if you don't pay, and the debt collectors are already calling you about your 5 credit cards, or would if you still had a phone.  Imagine how terrible that makes you feel.  If you're reading this blog, you've probably never faced this, or at least not lately, but even imagining it is not pleasant.  Now imagine paying one of your bills.  Feels better?  Nope, didn't think so.  Now the lights are on and the phone works, so the terrible situation is just that much clearer.

NOW imagine paying the last bill.  Wow, major relief.  You just experienced increasing marginal return to wealth, something that 5 years or so of economics background prepared me to more or less expect to never exist.  Now please excuse me while I re-examine most of my thoughts on povert.  Thank you Blogosphere!

Now, because I'm an econ nerd, here's what's happened to my implicit model of how people value stuff:

let u(x) be some log-type function such that du/dx is decreasing, approaching zero.

Before:

Utility= u(income)
Which looks like:

After:

let P be some constant of income that is needed to make the bulk of misery go away.  Perhaps it's the poverty level?

Let C be a value chosen between 0 and 1 by the individual to maximize their utility level at any given income level.

Utility = u(C*income)- max{0, u[P-(1-C)*income]})

Which has varying returns to wealth: some areas will have increasing returns to wealth, and some will have decreasing returns to wealth.  

Additionally, since the first term is "stuff you want" and the second term is "stuff you don't want" we can see c as a "indulgence" variable.  A higher C value means more indulgence, and a lower C means less indulgence.  This fully explains why "indulgent" choices make sense: if your demand function looks like this, you may well get more utility out of an "indulgent" choice of a $100 pair of sneakers than paying that nagging $100 utility bill that's 90 days past due.

I'm going to read up and see if anything in the "serious" econ literature explores this idea, and probably come up with a more detailed toy model of how wealth and utility might be related.