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.

Calculus

My real love in math is for calculus, because I think its one of the most beautiful insights to be found, and its core is simple and intuitive.  If you have taken calculus, and you had a really awesome teacher, you probably agree with me on this, but there's also a good chance you walked away with the impression that calculus is a technical mess involving Riemann sums, limits, differentials and lots of detailed rules.  There is a technical side to calculus, but since I'm not an engineer or mathematician, I'll leave those parts for them--the underlying insights are both more important, and more fun.

The central idea of calculus is this: "you can talk about how quickly something is changing at any given instant, and if you add up all of those little changes, you get the total amount of change."

This is something we're already comfortable with.  When we talk about speed in a car we can talk about average speed, saying "I traveled 70 miles in an hour," but we're more likely to say "at 2:08 pm, as I crossed the state line into Texas, I was in traffic, so I was only going 25 miles per hour."  The important thing to make explicit here is that in the second case, we're not talking about a specific number of miles in a specific amount of time, but instead we're talking about our speed at one instant.

Perhaps our overall travel looks like this, with total distance from home on the Y axis, and time since you started on the X axis.

Distance Traveled

It's worthwhile to talk about how far you went overall and how long it took, but it's also interesting to figure out how fast you were going exactly where that arrow is pointed.  To do that, we can zoom in:

Distance Traveled

and again:

Distance Traveled

Notice that now we have a part of the graph that looks straight--the slope, which here represents the speed of your car, is constant, so it's easy to measure.  You measure distance, divide that by time, and you're done.  Sort of.  This isn't quite the speed at any one moment, instead its the average speed over a very short period of time.  This might not matter practically, but mathematically this is important.  We really want the speed at that instant.  To do this, we imagine what the slope we would measure if we zoomed infinitely close in--this is the derivative, or "instantaneous slope."

Now lets make a new graph--again time is on the X axis, but this time the Y axis is your speed, or the "instantaneous slope" at each moment:

Speed

Notice how the speed starts out high, which matches the steep slope we see at the beginning of the "Distance Traveled" graph above, and then it almost goes to zero where you're stuck in traffic, and then comes back up as the traffic clears.  

That's it for derivatives--at some point I'll write about integrals and how these connect together, but this is more than enough for one sitting.

Linear Algebra

Linear Algebra is a little bit of a funny subject-- virtually every linear algebra problem will encounter at least one matrix, but its really no more about matrices than literature is about paper and ink.

The other day I was thinking about my linear algebra class from a few years ago, and realized that, while I could discuss most of the important points coherently, I could not remember the first thing about how it related to a matrix.  

I think it's worthwhile to condense a subject to its central insight, so my best attempt for linear algebra is this: "one vector defines a set of all vectors that are just like it, but bigger or smaller, and this set of vectors is a line passing through the origin.  If you add in a second vector, that is not on this line, you have defined a plane, because all points on this plane can be described as some combination of the first two vectors."

To be a bit more concrete about this, imagine a town neatly laid out in a grid pattern (not needed, but conceptually useful), and imagine town hall sitting at the center of town.  Now imagine I give you the instruction "go three blocks north."  This is a vector--I've given a distance and a direction.  I've also defined a set of instructions though--the instructions "go 6 blocks north" or "go 4 blocks south" are both closely related to the first direction, because you could re-write them as "do the first instruction twice" or "do the opposite of the first instruction one and a third times."  This are linear transformations of the first instruction.  Now, if I give you a completely different instruction, say "go one block west," I have described a new line, in the East-West direction.  The really cool thing though is that we can take those two instructions ("go 3 blocks north" and "go 1 block west") and put them together to describe any point on the map.  We needed two separate vectors to describe a two dimensional space, and we'd need three separate vectors to describe a three dimensional space, and so on, but once we have these vectors (a "basis set"), we can add them together in different ways to describe the entire space.