Can Money Buy Happiness?
May 07, 2021
How much happiness can money buy?
Thought Experiment
Imagine that you are fresh off the boat with just the clothes on your back and a dollar in your pocket. In this new land, you have no family or friends to rely on.
How hard would you be willing to work for that first $100? What kind of jobs would you be willing to work?
How much happiness or freedom would that first $100 give you? Maybe you would still be miserable but at least you could be miserable on a full stomach and thinking about how to make the next $100.
Now imagine getting off that same boat with $1M to your name in a savings account.
Would the next $100 mean the same to you? Would you be willing to dig as deep for minimum wage now that you aren’t threatened with starvation and you can afford some shelter?
St. Petersburg lottery
Now imagine a new scenario where a casino offers a game of chance.
The initial stake begins at 2 dollars and is doubled every time heads appears.
The first time tails appears, the game ends and the player wins whatever is in the pot.
Thus the player wins 2 dollars if tails appears on the first toss, 4 dollars if heads appears on the first toss and tails on the second, 8 dollars if heads appears on the first two tosses and tails on the third, and so on. [1]
How much would you pay to play this game?
In the past, my mathematical mind might have suggested only entering a lottery if my expected value was positive.
In this case, the expected value is $2 * (1/2) + $4 * (1/4) + $8 * (1/8) + … = 1 + 1 + 1 + … = Infinity
Since the expected value is infinite, would you stake your entire net worth on this game?
Utility Function
In economics, utility represents the satisfaction or pleasure that consumers receive for consuming a good or service. Utility function measures consumers’ preferences for a set of goods and services. [2]
Hopefully, the thought experiments above illustrate that our utility (happiness or satisfaction) doesn’t increase linearly with money. In other words, the more money we have, the less marginal utility we get from acquiring more money.
Daniel Bernoulli (his cousin Nicolas invented the St Petersburg Paradox problem) suggested that a logarithmic function could describe the relationship between utility and money.
Analysis of international survey data in the 21st century have shown that insofar as utility represents happiness, as in utilitarianism, it is indeed proportional to log income. [3]
Here we want to show that, while less strong, there is also a correlation between income and happiness across time. [4]
Surveys seem to indicate that there is correlation between happiness and income but with more income buying exponentially less happiness.
So when we analyze risky investment opportunities, we must keep in mind the diminishing marginal utility of money.
References and Footnotes
[1] Wikipedia: St Petersburg Paradox
The St Petersburg Paradox also breaks down when you consider that the money supply is limited and that your winnings will be limited by the net worth of the casino.
Assuming Warren Buffett is offering you this game of chance, your expected value is now close to $2 * (1/2) + $4 * (1/4) + $8 * (1/8) + … + $(2^36) * (1/2^36) + $(96 billion) * (1/2^37) since Buffett has a $96B net worth. That is less than $37.
(Some might still pay more than $37 for the excitement of playing this game but that is a discussion for another day.)
[2] Investopedia: What Is the Utility Function and How Is It Calculated?
[4] Our World in Data: Happiness and Life Satisfaction
[5] Stack Overflow: Is the Utility of Money Actually Logarithmic
“Essentially, all models are wrong, but some are useful” George Box, Empirical Model-Building and Response Surfaces