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The Price of Happiness

What is the shape of the relationship between money and happiness, and what are its implications?

Matthew A. Killingsworth

Date: October 3, 2024

Abstract

People typically think about money in raw units such as dollars. Yet research on money and happiness typically examines the association between happiness and the logarithm of income, or Log(income). This logarithmic association between income and happiness is frequently either overlooked or misunderstood. To help address this, the present report examines this association and makes five key points. First, in a large U.S. sample, the shape of the association between happiness and Log(income) was extremely systematic: from $10,000/y to over $500,000/y, average happiness rose almost perfectly linearly with Log(income), with group-level correlations of 0.98-0.99 across a range of happiness measures, including both in-the-moment experience and overall life satisfaction. Second, a linear association between happiness and Log(income) implies that the marginal utility of additional dollars diminishes exponentially, though never mathematically plateaus. It also implies that a proportional difference in income, such as a 10% raise, would be associated with the same difference in happiness regardless of income level. Third, real-world incomes varied exponentially in size, effectively offsetting the declining marginal utility of dollars. Perhaps counterintuitively, while dollars exhibited sharply declining marginal utility for happiness, real-world incomes exhibited no decline at all. Fourth, by contrast, if trade-offs are made between people with unequal incomes - as could occur in philanthropy, compensation decisions, or tax policy - effects on collective happiness are predicted to be exponentially larger when lower-income people benefit. When it comes to money, this highlights a potential tension in the geometry of individual and collective happiness. Fifth, money’s diverging implications for happiness, linear in some contexts but exponential in others, may also help explain why income inequality persists as societies get richer, why the income distribution is shaped the way it is, and why happiness in the U.S. has not seen more improvement in recent decades. Reasoning linearly in a situation that calls for exponential thinking, or vice versa, is likely to lead to conclusions that are flawed. Knowing when to think linearly and when to think exponentially about money is crucial for understanding its relationship to happiness.

Bernoulli’s Revenge

Imagine you were given the option to play the following game: you flip a coin until it comes up heads, at which point the game ends. If you get heads on the first flip, you win $2, on the second flip, $4, on the third flip, $8. Each time you flip tails, the reward doubles, with extremely large payoffs if you manage to flip tails many times in a row before getting heads. Generically, if it takes N flips until you get heads, you’ll receive a reward of $2 raised to the Nth power, or $2N (and if you’re the flipper, you’d like N to be as large as possible). Assuming the game can be played out instantly, what’s the most money you would be willing to pay to play this game?

One way to decide is to calculate the expected monetary value of the game, which can be easily calculated. Just multiply the value of each potential outcome by the probability that it occurs, and then add them up. But once you do that, you’ll see that the expected value approaches infinity. Why? The expected value of each flip equals the odds of getting the first heads on that flip multiplied by its payoff. Therefore, the expected value of the first flip is (50%*$2), the expected value of the second flip is (25%*$4), and so on. In other words, each flip n has an expected value of 2n / 2n = $1. Since the number of potential flips approaches infinity, the expected monetary value of this game ($1 + $1 + $1 + …) also approaches infinity. Yet would you be willing to pay your entire life’s savings to play this game, as appears to be the “rational” choice? Most people would say, “No.” Why is that?

Daniel Bernoulli attempted to solve this paradox in 1738 with a simple solution that has influenced scholarly thought ever since (1). He proposed that people don’t attempt to maximize their expected wealth, but instead attempt to maximize their expected utility. And utility, he argued, scales with the logarithm of wealth, not with wealth itself. It therefore makes sense that people are unwilling to pay an exorbitant price for a tiny chance of a large payoff, because the marginal value of money declines when you have more of it. Bernoulli’s speculation was not based on any data, as far as I’m aware. But it proposed a way to...