How much value do you get if I hand you a dollar? The answer to this question is way more complicated than it might seem at first.

The value of dollars isn’t linear, since marginal dollars are less valuable when you have a lot of dollars already. If you only have $10 to your name, I’ve just boosted your worldly wealth by 10%, whereas if you have a million dollars, the extra millionth isn’t going to change your fortunes too much. For this reason, my intro econ book claims that the utility curve of money is logarithmic.

This model leads to all sorts of interesting consequences. One of my favorites is that losses and gains of the same dollar amount have different value (and not just psychologically!). If you have $1000 and I take $900, the log value changes by way more than if I give you $900. That is,

\[ \left| \log_{1000} - log_{1000 - 900} \right| > \left| \log_{1000} - \log_{1000 +900} \right| \]

\[ \left| 6.91 - 4.61 \right| > \left| 6.91 - 7.54 \right| \]

\[ 2.30 > 0.64 \]

The logarithmic model is probably closer to the truth than linear, but doesn’t capture everything we care about. Let’s say there’s something you really want to buy that costs $20. If you’ve saved up $18, and I boost you to $19, I haven’t done that much for you. However, if I boost you from $19 to $20, I’ve increased your purchasing power enough that you can now buy the thing you really wanted. That’s more valuable, even though \[\frac{20}{19}\] is a bit less of a marginal improvement than \[\frac{19}{18}\].

This also means I might be able to increase the felt value of gifts simply by giving them in smaller chunks and at the right times. If I give you $1000 all at once, the value of the thousandth dollar isn’t as high as that of the first. If I instead gave you $100 ten months in a row, right after you’ve paid rent, then although the nominal value is still $1000, the value to you might be slightly higher because you were relatively poorer each time I gave you a gift.

We have to be careful here. Let’s quickly test out an obviously bad mental model for money’s value. If you have $100 and I give you $50, I’ve increased your wealth by 50%. However, if you have $150 and I take away $50, I’ve decreased your wealth by only 33%. Therefore, if I keep giving and taking $50, I should be able to increase your wealth arbitrarily high, right? Clearly not. The value of money, like any marginal value, depends on how much money you already have.

The value of money is also weird because it changes over time, in a way that isn’t perfectly predictable. Thanks to inflation, it’s a pretty good bet that if I stick a hundred-dollar bill under my mattress today, it’ll be worth less in twenty years. However, thanks to similar forces, there are ways that I can grow my money over time without doing anything—simply by investing it in the right place.

Naively, it seems that people often don’t take long enough on the biggest financial decisions. Spending hundreds of thousands on a house might warrant ten thousand times more deliberation than spending $10 on what to have for lunch, but people rarely spend that long. Say it takes 5 minutes to decide what to have for lunch. Scaling that up 10,000x is 833 hours, or around 35 solid days of decision-making time.

There are so many interesting considerations that go into valuing money. Even though it seems like it should be the one thing we have a really good handle on the value of (since it backs the value of so many other things) it’s way more slippery in practice. Although dollars are fungible, the value of any two is unlikely to be the exact same.