The Price of Happiness

Date: October 3, 2024

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:

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

A Thought Experiment: The Coin Flip Game

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 $2^n / 2^n = $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 quantitatively translate an amount of money into its value to humans.

Bernoulli’s original aim with this theory was to explain why people are risk-averse. We now know that, as a theory of how humans actually make decisions, Bernoulli’s explanation is, at a minimum, incomplete. Expected Utility Theory grew out of Bernoulli’s thinking, and argued that people weight utility payoffs (rather than monetary payoffs) by their probabilities to calculate expected utility, and then make the choice that maximizes expected utility. Kahneman and Tversky showed that Expected Utility Theory cannot account for people’s actual decisions under uncertainty, and famously proposed a different solution, Prospect Theory, they argued could explain people’s actual decisions in conditions where Expected Utility Theory failed miserably (2). Much of the revolution in decision-making research and behavioral economics over the past 40 years has built on this foundation. That research has typically sought to explain people’s real-world behavior, often by pointing out systematic inconsistencies or other deviations from apparent rationality in how people decide what to do.

At the same time, we should note something critical: the calculations that describe people’s decisions (“decision utility”) and the calculations that describe their actual utility outcomes (“experienced utility”) are not necessarily the same (3). Bernoulli’s data-free hypothesis that utility scales with the logarithm of wealth was, at least under certain conditions, quite wrong as a theory of risk-aversion (decision utility). But, as I’ll show, it appears to be exceptionally accurate as a theory describing human happiness (experienced utility). This proposition and its implications are the subject of the rest of this paper.

Money and Happiness: Research Background

Few social science topics engender as much interest as the association between money and happiness. Are people who earn more money happier? Almost everyone has an opinion, even if it is based only on intuition. The research literature is large and complex, but virtually all research agrees on at least two facts: (a) the correlation between income and happiness is positive in sign; (b) the marginal value of money declines as incomes rise (4-12). In other words, richer people tend to be happier, but each additional dollar matters slightly less than the one before it. Causal evidence is much rarer, but there is compelling evidence that having more money does, on average, cause people to be happier (13-16).

In my own research, I have found that earning more money is associated with greater experienced happiness, as measured with large-scale experience sampling (17, 18). The more money people earn, the happier they tend to be in the moments of life. And this upward association appears to extend across a wide range of income levels, including well beyond a previously-accepted satiation threshold of $75,000/y. The fact that experienced happiness rises with income provides some of the most direct evidence yet that higher incomes are associated with genuinely better life experience (and genuinely better “experienced utility” in what is arguably the most literal sense we can currently measure it). A follow-up study comparing people with ordinary incomes to people with high net-worths provides extended evidence against satiation: the positive association between money and happiness appears to continue much farther, including well beyond incomes of $500,000/y, and possibly beyond the range of virtually all previous studies of money and happiness (19).

How to make sense of these findings, including my own findings as well as the broader literature, is a subject of some disagreement. Amongst researchers, educators, journalists, and laypeople, a particular issue seems especially prone to being misunderstood or overlooked: what to make of the fact that, for people’s happiness, each additional dollar appears slightly less valuable than the one before it. After all, in our lives, we are typically used to thinking in dollars. We navigate choices quantified in dollars (or whatever one’s local currency is; here, I will refer to dollars for simplicity). But if dollars have a variable value for happiness, it’s not necessarily obvious how to approach the decisions we make with happiness in mind. Knowing well-established facts (a) and (b) noted above provide little guidance, even if one assumes that having more money really does cause people to be happier.

This report makes five key points. Collectively, they show the association between income and happiness is highly systematic, but that its implications are linear in some contexts yet exponential in others. As is probably obvious, reasoning linearly in a situation that calls for exponential thinking, or vice versa, could lead to conclusions that are wrong. Not just a little wrong, but very wrong. For example, people sometimes assume that the diminishing value of dollars for happiness means that larger incomes have diminishing value for happiness. In my experience, this is not uncommonly asserted even by experts. While this might sound true given what I’ve said so far, it appears to be false in an important sense: as I’ll show, incomes vary exponentially in size, to a degree that appears to completely offset the declining marginal value of dollars for happiness.

Most of this paper is devoted to considering the implications of the association between income and happiness. In other words, what effects would we predict in the real-world if we extrapolate from the shape of the association? And how might our thinking need to change in different contexts?

To establish a foundation for this analysis, I first show that the average trend in happiness across income levels fits Bernoulli’s theory with almost perfect precision. This trend builds upon data on money and happiness from trackyourhappiness.org, as reported recently in other research (17-19). This includes ~1.7 million real-time reports of experienced happiness from 33,391 employed, working-age adults (ages 18 to 65) living in the United States. Participants received notifications on their smartphones at randomly-selected times during daily life, and were asked to report on their experiences at the moment just before the notification. Experienced happiness was measured with the question “How do you feel right now?” on a continuous response scale with endpoints labeled “Very bad” and “Very good.” Household income was measured with the question, “What is your total annual household income before taxes?” with answers collected in income bands. Happiness was also measured in evaluative terms via life satisfaction, including a continuous life satisfaction scale, a four-level life satisfaction scale, and the five-item seven-level Satisfaction With Life Scale (20).

The Five Key Points in Detail

Point 1: The association between Log(income) and happiness was almost perfectly linear

The most common way to account for the declining marginal utility of dollars is to log transform income before correlating it with happiness. We will consider the implications of this transformation later. But first let us simply ask: how systematic is the shape of the association between happiness and Log(income)? When I previously reported this association (17) I did not calculate how well (or poorly) a linear model could explain the overall shape of the association. To quantify it, let us calculate the group-level association between Log(income) and the average happiness of the people in each income category. This averages out the substantial variation in individual happiness that income cannot explain, and isolates the underlying shape of the association between income and happiness.

Results show that Log(income) and happiness were not merely positively related. From $10,000/y to over $500,000/y, they were correlated at the aggregate level to an extremely high degree, with group-level r’s = 0.98-0.99 across a range of happiness measures, including both experienced happiness and overall life satisfaction, see Fig. 1. These group-level correlations remain high when controlling for demographic variables (age, gender, education level, and marriage), r’s = 0.95-0.99.

Fig. 1. Mean levels of happiness, including experienced happiness (real-time feelings) and overall life satisfaction, for each income category. The group-level correlations between happiness and Log(income) is displayed. Income axis is log scaled, and error bars indicate 95% confidence intervals. Because the top 3 income categories have progressively fewer people and progressively wider confidence intervals, a composite point (displayed in gray) that combines them and is comparable in size to the other income categories is also plotted for reference (it is not included in the correlation or fit line calculations).

In other words, the average happiness of people in each income category was almost perfectly linearly associated with that category’s Log(income), explaining virtually 100% of group-level variation. This is just as Bernoulli’s nearly 300-year-old theory predicts, if we extend it from wealth to income and from “decision utility” to happiness. When explaining how happiness varies between income levels, a linear prediction from Log(income) was superbly accurate¹.

If we assume this remarkably systematic shape indicates the “price of happiness”, what are the implications for the rest o