One + One < Two: Part II of a Series on Utility Theory

Jan 31, 2007 by Adam C. Powell, Ph.D.

You may recall the old jingle, “What would you do for a Klondike bar?” Imagine that I offered to give you a Klondike bar (an ice cream sandwich). Would you do twice as much to receive two Klondike bars as one Klondike bar? Probably not. While eating the first Klondike bar might make you happy, eating two Klondike bars in a row is unlikely to cause twice the increase in happiness of eating one Klondike bar. Sure, it might make you a bit happier than eating just one, but probably not twice as happy.

The same thing applies to money. When you are a little kid with not much money, and a relative gives you $10, it is a big deal. Say, you had $40 in the piggy bank, and then received another $10. Your wealth has just increased 25%. Now, imagine that you have an income of you own, and a savings of $1,000. Does receiving $10, 1% of your wealth, make you as happy as it did as when you were a kid. Probably not. As our level of wealth increases, the increase in utility that we get from receiving a fixed amount of money decreases. This is true whether wealth is measured in Klondike bars or dollars; it is a lot better to go from 0 Klondike bars to 1 Klondike bar than it is to go from 1 Klondike bar to two Klondike bars.

Now, imagine you had a box of a dozen Klondike bars in your freezer. If I stole one, you would be unhappy. If I stole two, would you be twice as unhappy? Probably not. There is a certain amount of unhappiness one faces from losing anything. Losing two Klondike bars is worse than losing one Klondike bar, but psychologically, it is not twice as worse as losing one Klondike bar.

What does this mean to an investor? Here are two hypothetical investments:

A. 50% chance of receiving a 5% return, 50% chance of receiving a 0% return

B. 25% chance of receiving a 10% return, 75% chance of receiving a 0% return

On average, both bets yield a 2.5% return. Most people would prefer option A, as they don’t value a 10% return twice as much as they value a 5% return. If a 5% return were worth the joy of eating one Klondike bar to you, a 10% return might be worth the joy of eating two Klondike bars. However:

(Joy of One Klondike Bar) * 2   <   (Joy of Two Klondike Bars)

For convenience, let’s call the Joy of One Klondike Bar “1 Util.” Thus, we can also call the Joy of Two Klondike Bars “1.8 Utils.” (The choice of 1.8 is arbitrary; I could have chosen any number between 1 and 2.)

So, we can reframe the two choices as:

A. 50% chance of receiving a 1 Util, 50% chance of receiving 0 Utils

B. 25% chance of receiving a 1.8 Utils, 75% chance of receiving 0 Utils

Now that we have reframed this problem in terms of Utils, we can see that the two options are not equivalent. Option A has an expected return of 0.5 Utils, while Option B has an expected return of 0.45 Utils.

As investors, we often are given an array of choices that give different levels of return at different levels of risk. We often pick investments with lower levels of return and lower levels of risk when offered other investments with the same expected value, but a higher level of return and a higher level of risk. We do this because we don’t usually receive twice the utility (joy) from earning twice as much money.