What's a Dollar Worth?

The elasticity of marginal utility of income (η) determines how quickly extra money loses its subjective value as income rises. Getting η right matters enormously for cross-income comparisons — the difference between η = 1.0 and η = 1.6 is roughly an order of magnitude when comparing rich and poor.

What η means, concretely

η is defined so that a 1% increase in income reduces the marginal utility of the next dollar by η%. (Economists report η as a positive number; the negative sign is implicit.)

η	$1 at $20k vs. $200k	Character
0.5	~3× more valuable	Mild diminishing returns
1.0	10× more valuable	Log utility — the lower bound of most estimates
1.3	~20× more valuable	Moderate central estimate
1.6	~40× more valuable	Meta-analytic consensus; this tool's default
2.0	100× more valuable	Strong diminishing returns

Three ways to estimate η

1. Ask people how happy they are (subjective well-being)

Correlate self-reported life satisfaction or emotional well-being with income. The relationship is strikingly log-linear across countries and income levels.

• Stevenson & Wolfers (2013) found no satiation point: the log-income/happiness relationship holds from $500/yr to $500,000/yr. Implies η ≈ 1.0.
• Layard, Mayraz & Nickell (2008) used six large surveys across 50+ countries, estimating η = 1.19–1.34.
• The Killingsworth–Kahneman–Mellers (2023) adversarial collaboration found happiness continues rising with log income for ~80% of people, with a plateau only for an "unhappy minority."

2. Look at how people behave (revealed preference)

Tax progressivity, risk aversion, and labor supply decisions reveal the curvature of utility indirectly.

• Evans (2018) used four revealed-preference methods on UK data: η ≈ 1.5.
• Tax-based estimates typically give η = 1.2–1.8.
• Risk-aversion estimates are highly variable (η = 0.5–3.0+), partly because they capture both utility curvature and pure risk preference, which are conceptually distinct.

3. Study consumption patterns (demand systems)

How spending shifts across categories as income rises constrains the shape of utility.

• Loebbing (2024) used PSID data and found η ≈ 0.58 — notably lower than other approaches.
• Demand-based estimates tend to be lower, possibly because they capture substitution effects that other methods don't isolate.

Why estimates disagree

Different methods measure slightly different things. Happiness surveys capture hedonic experience. Tax progressivity reflects social preferences embedded in policy. Risk aversion mixes utility curvature with risk attitudes. Demand systems capture consumption value. None is "wrong" — they're different lenses on a concept (utility) that isn't directly observable.

The meta-analysis

Acland & Greenberg (2023) synthesized US and UK estimates across all methodologies in the Journal of Benefit-Cost Analysis. Central estimate: η ≈ 1.6 (sensitivity range 1.2–2.0). Notably, they found methodology did not systematically shift estimates — the variation is mostly within methods, not between them.

Practical guidance

If you want…	Use	Rationale
A conservative lower bound	η = 1.0	Log utility; Stevenson & Wolfers; most intuitive
A moderate central estimate	η = 1.3	Between SWB and revealed-preference estimates
The meta-analytic consensus (default)	η = 1.6	Acland & Greenberg (2023)
A strong-redistribution case	η = 2.0	Upper end of empirical range; implies very steep diminishing returns

Values below 1.0 or above 2.0 are outside the range most economists would defend, though individual studies have produced them.

The problem

The utility model takes income as its input, but income alone understates the economic position of the wealthy. Someone earning $100k/year with $10 million in assets has a very different economic reality than someone earning $100k with nothing saved.

There is no standard model for combining income and wealth into a single utility input — this is a genuine gap in the literature. What this tool offers is a transparent approximation, not established theory.

The approach: annuitization

We convert net wealth into an effective annual income by computing what fixed annual payment the wealth could fund over a chosen time horizon, accounting for investment returns. This is added to actual income:

where W = net wealth, r = real annual return, T = annuitization period.

What the parameters mean

Annuitization period (T) — default: 40 years

How many years to spread the wealth over. 40 is a rough remaining adult lifespan for someone in their 20s–40s. A shorter period (e.g., 20) treats wealth as more immediately "income-like." As T → ∞, this approaches a perpetuity (annual flow = W × r).

Real rate of return (r) — default: 3%

The inflation-adjusted return on invested wealth. 3% is a standard long-run assumption for a diversified portfolio. Higher rates mean each dollar of wealth contributes more to effective income. Setting r = 0 reduces to simple division: W/T.

What this is not

• Not empirically grounded the way η is. The η estimates come from decades of research. The wealth-to-income conversion is a modeling convenience — a reasonable one, but an assumption nonetheless.
• Not a claim that wealth and income affect well-being identically. Wealth provides security, optionality, and social power in ways that annual income flow doesn't fully capture. For the ultra-wealthy (>$100M+), even this approach likely understates their economic advantage.
• Not accounting for illiquid wealth. Home equity, private business ownership, and restricted stock are not as fungible as the annuity formula assumes. Adjust accordingly.

When to use it

The wealth adjustment is most useful when comparing people whose income-to-wealth ratios are very different — e.g., a salaried worker vs. a retired person living off assets, or an average earner vs. someone with large inherited wealth. If both people have similar wealth-to-income ratios, it won't change the result much.

The utility function

This tool uses the standard isoelastic (CRRA) utility function:

where Y is annual income (or effective annual resources, if wealth is included) and η is the elasticity of marginal utility. When η = 1, this simplifies to U(Y) = ln(Y).

The equivalence calculation

We find the exact amount Z such that receiving Z at income Y₂ yields the same utility gain as receiving X at income Y₁:

Solving for Z:

For η = 1 (log utility): Z = Y₂ × X / Y₁.

Exact vs. marginal approximation

The common shorthand Z ≈ X × (Y₂/Y₁)^η is a marginal approximation that assumes X is small relative to Y₁. This tool uses the full integral above, which remains correct even when X is large relative to income (e.g., $1,000 to someone earning $700/year).

Key assumptions

• Everyone has the same utility function with the same η and the same scaling — a strong simplification.
• The amounts are evaluated as one-time changes to annual income: "what is it like to have Y+X instead of Y for a year?"
• No hedonic adaptation, aspiration effects, or reference dependence.
• When wealth is included, the "effective income" is used as Y in the formula above. This treats the annuitized wealth as equivalent to earned income, which is an additional modeling choice.

References

• Acland, D. & Greenberg, D. (2023). The Elasticity of Marginal Utility of Income. J. of Benefit-Cost Analysis, 14(2).
• Layard, R., Mayraz, G., & Nickell, S. (2008). The Marginal Utility of Income. J. of Public Economics, 92(8–9).
• Stevenson, B. & Wolfers, J. (2013). Subjective Well-Being and Income: Is There Any Evidence of Satiation? AER, 103(3).
• Evans, D.J. (2018). New Estimates of the Elasticity of Marginal Utility for the UK. Env. and Resource Economics.
• Loebbing, J. (2024). The Marginal Utility of Income and Homogeneous Demand Systems. J. of Econ. Behavior & Org.
• Killingsworth, M., Kahneman, D. & Mellers, B. (2023). Income and emotional well-being: A conflict resolved. PNAS, 120(10).