Utility Growth and the VSL

Introduction

This post is about a way to estimate utility growth using nominal interest rates and the value of statistical life (VSL).¹

The idea is simple. Let u(c,t) be the function from nominal consumption per year (c) to flow utility over the course of a year (u) in year t, where u = 0 denotes the utility assigned to death during year t, for some person. (The time-dependency of the utility function can account for inflation, or any other changes in the relationship between utility and nominal consumption.) Then the value of a statistical life-year for this person at t (VSLY_t), given that she enjoys nominal consumption c_t at t, is given by

\(\text{VSLY}_t = \frac{u(c_t,t)}{\frac{d}{dc_t}u(c_t,t)}.\)

Letting u_t ≡ u(c_t,t) and u′_t ≡ d/dc_t [u(c_t,t)], and rearranging,

\(u_t = \text{VSLY}_tu'_t \quad \implies \quad g_{u,t} = g_{\text{VSLY},t} + g_{u',t}.\)

For this person to be on her Euler equation, we must have g_u',t = 𝛿 - r_t, where 𝛿 is the discount rate and r_t is the nominal interest rate at t. So

\(g_{u,t} = g_{\text{VSLY},t} + \delta - r_t;\)

or, in cumulative terms,

\(\frac{u_t}{u_0} = \frac{\text{VSLY}_t}{\text{VSLY}_0} \frac{e^{\delta t}}{R_t},\quad\quad(*)\)

where R_t is the gross return from 0 to t.

Note that (⁎) imposes no assumptions at all on u(c,t) except that, at each t, it is locally differentiable in c around c_t. It is compatible with any model of changing products, tastes, leisure, health, habits, price indices, and so on. It requires no definition of inflation or real consumption.

Finally, let EU_t denote what we might call the “utility of life”—our individual’s expected (discounted) lifetime utility from t onward—and recall that

\(EU_t = \text{VSL}_t u'_t.\)

Applying precisely the same rearrangement as above to this identity, observed VSLs and interest rates can let us track the growth in the utility of life. An attempt below to apply this calculation suggests that utility has grown much more quickly than is typically believed.

Contrast with the standard approach to estimating utility growth

When trying to put a number to how much better off we are now than at some earlier time, it is most standard to

1. assume something about the shape of the function from “consumption” to “utility”, u(c); and then

2. measure growth in “real consumption” over time.

If we believe that welfare has changed over time for reasons other than changes in “real consumption”, we adjust the number we plug into u(.) accordingly.

Even before making any such adjustments, this approach has at least two conceptual difficulties.

First: this approach does not tell us anything about how quickly utility is growing in the “ratio scale” sense where 0 is the utility of death.

Second, and much more importantly: however we define “real consumption”—except under the unrealistic condition that we all have unchanging, homothetic preferences (and they’re all the same)—the utility a person achieves at a given level of consumption may vary arbitrarily over time, especially over long stretches of time. So our utility is really given not by u(c) but by some u(c,t). For example, I don’t think it should be controversial to say that, for a person with typical preferences, having infinitely large quantities of the goods available in, say, 1500 London would put one on a lower indifference curve than having sufficiently large quantities of the goods available in 2025 London. If so, there is some c such that u(c,1500) < u(c,2025).

The unifying point is that data on (i) consumption growth and (ii) individual behavior that does not affect risks to life cannot in principle tell us anything about utility growth. Data on financial risk tolerance tells us only about the curvature of the relationship between c and u(c,t) at a given t, and data on saving tells us only about the relationship among u′_t across various t. To see that neither of these is sufficient to infer anything about utility growth, observe that for any definition of real consumption, the cases

\(u(c_t,t) = \ln(c_t) \quad \text{and} \quad u(c_t,t) = \ln(c_t) + t \)

have very different implications for utility growth but produce indistinguishable financial behavior.

Uncertainty about movements in marginal utility

At each t, we have mechanically

\(u_t = u_{t-1} \frac{\text{VSLY}_t}{\text{VSLY}_{t-1}}\frac{u'_t}{u'_{t-1}}.\)

The one-year nominal risk-free interest rate at t does not tell us about u′_t/u′_t-1 directly. We can only say that it maintains the Euler equation

\(\mathbb{E}_{t-1}\Big[\frac{u'_t}{u'_{t-1}}\Big] = e^\delta \frac{R_{t-1}}{R_t},\)

where E_t-1 is the expectation operator for investors at t-1. But if we trust that, at each t-1, people have not been biased in their expectations of the next-period movement in marginal utility, and if we have no information beyond the observed interest rate series about how marginal utility moved, then, letting E denote our own expectation operator,

\(\mathbb{E}\Big[\frac{u'_t}{u'_{t-1}}\Big] = \mathbb{E}_{t-1}\Big[\frac{u'_t}{u'_{t-1}}\Big].\)

In fact we do sometimes know something about whether marginal utility fell more or less quickly than expected in a given historical year. Unfortunately, we cannot quantify this knowledge without making further assumptions. For example, we know that growth failed to meet expectations if t marked the beginning of an unexpected recession. It seems reasonable to infer from this that marginal utility at t fell less quickly than expected (even rose), but we cannot infer this, let alone quantify it, unless we assume something further about the utility function.

Under “robustness check #5” below, I discuss the possibility that we happen to know in hindsight that marginal utility fell much more quickly than people were expecting. But for now just observe that, thankfully, discarding this information does not introduce a bias to our own mean estimate of utility growth. This is because, assuming we observe the VSLY over time,

\(\mathbb{E}[u_t] = \frac{\text{VSLY}_t}{\text{VSLY}_{t-1}}\mathbb{E}\Big[u_{t-1}\frac{u'_t}{u'_{t-1}}\Big],\)

and, since investors at t-1 know u_t-1, our estimate of u′_t/u′_t-1 from the observed nominal risk-free rate at t-1 is independent of our estimate of u_t-1 on the basis of the evolution of interest rates and VSLYs from earlier periods. In other words, though in general

\(\mathbb{E}\Big[u_{t-1}\frac{u'_t}{u'_{t-1}}\Big] = \mathbb{E}[u_{t-1}]\mathbb{E}\Big[\frac{u'_t}{u'_{t-1}}\Big] + \text{Cov}\Big(u_{t-1},\frac{u'_t}{u'_{t-1}}\Big),\)

here the covariance term is zero: for any realization of u_t-1, our conditional expectation of u′_t/u′_t-1 is simply the E_t-1[u′_t/u′_t-1] implied by the observed interest rate at t-1. So by the law of iterated expectations, for any t,

\(E[u_t] = u_0\frac{\text{VSLY}_t}{\text{VSLY}_0}\mathbb{E}\Big[\frac{u'_t}{u'_{t-1}}\Big]\mathbb{E}\Big[\frac{u'_{t-1}}{u'_{t-2}}\Big] \cdot\cdot \cdot \mathbb{E}\Big[\frac{u'_1}{u'_0}\Big]=u_0\frac{\text{VSLY}_t}{\text{VSLY}_0}\frac{e^{\delta t}}{R_t}.\)

Finally, again, note that this algebra likewise gives us an estimate of expected growth in “the utility of life” if applied to the VSL instead of the VSLY.

Calculation

Attempting a consistent methodology for measuring the VSL over time, Costa and Kahn (2003) estimate it for American men aged 18-30 with at least some high school education in 1940, 1950, 1960, 1970, and 1980. Converting their estimates to nominal terms, the figures are in the table below.² (I also consider an alternate VSL series in the Robustness section.)

The authors argue that their trend offers a lower bound for the rise in the VSL for various reasons, e.g. because earlier estimates of the VSL are more likely to also be picking up some compensation for risks of non-fatal injury. Also, among the two preferred series they report in Table 6, I’m using the one in which the VSL rose more slowly.

For nominal gross risk-free returns, I compound the returns on 3-month Treasuries. Implied utility, as a multiple of its level in 1940, is calculated for utility discount rates of 0% and 2%. Higher discount rates would of course produce even faster implied utility growth.

Finally, I compare the implied growth in the utility of life (under each discount rate) with that given by a more conventional inference from “real consumption growth”. As summarized just below, I define utility from real consumption in a way that lets it grow especially quickly, making the gap between the implied growth rates in the utility of life all the more striking.

To make the “utility from real consumption” series I assume u(c) = log(c/1000), so that marginal utility in consumption declines more slowly than typically supposed (i.e. only logarithmically), and so that it takes fully $1000/yr in “2025 USD” to have a life worth living (around the global poverty line). I make no adjustments for habit formation, external (“keeping up with the Joneses”) or internal. All these assumptions strike me as conservative. I use the BEA’s accounts for real consumption per capita; the calculations using real GDP per capita are similar.

The “utility from real consumption” series captures growth in the utility of a life-year, not of a life. I plan to update this at some point, but this should not make a huge difference.

• Since consumption for Americans from 1940 to 1980 grew roughly exponentially, log consumption grew roughly linearly. This implies that, on the u(c) = log(c) assumption, flow utility n years into the future from some time t’ (say, 1980) exceeded flow utility n years into the future from some earlier time t (say, 1940) by a smaller proportion than flow utility at t’ exceeded flow utility at t. All else equal, therefore, this approach implies that the utility of a life from t to t’ grew more slowly than the utility of a life-year.

• On the other hand, a 25-year-old man in the US in 1940 had only 42.5 years of life ahead, on average, whereas a 25-year-old man in the US in 1980 had 47.4 (source). If we assume that people don’t discount the future (an upward bias), but ignore the fact that later years came with more consumption than average (a downward bias), this implies that VSL₁₉₈₀/VSL₁₉₄₀ was only 11.5% higher than VSLY₁₉₈₀/VSLY₁₉₄₀.

Extending to today

I worry that not much can be learned by extending the analysis to today using current government figures for the VSL, since this does not come with different numbers by age and sex, and since different ways of estimating the VSL produce such different values. The whole point of Costa and Kahn’s effort was to apply a consistent methodology across time, so that even if they are wrong about the levels, we can say something about the trend: in fact, lower-bound the trend.

That said, the 2024 VSL used by the Department of Transportation is $13.7m, and the nominal gross risk-free return from 1940 to 2024 was 19.65. This implies a utility of life in 2024 of 6.66 using a 0% discount rate—lower than in 1980, as it happens—but 36.45 using a 2% discount rate. By contrast, utility per year from real consumption, as a multiple of its 1940 value, is 1.89.

Robustness

1. 1940 was at the beginning of World War II.

Though the US didn’t enter the war until 1941, perhaps young American men believed that they were almost certain to be drafted and die soon.

Even if they anticipated the war and how deadly it would be, they would have been wrong about this: as noted above, a 25-year-old American man in 1940 lived an average of 42.5 more years, just a few less than in 1980. But regardless, even if we exclude 1940 from the series, and even if we assume they were using a 0% discount rate, we conclude that the utility of life more than tripled from 1950 to 1980.

2. The formula relies on a “representative agent” analysis.

The saving decisions of men aged 18-30 depend on what they expect to happen to their own future marginal utility in consumption, not marginal utility in consumption for the next cohort of 18-30s.

This is true, but since people tend to move up the income distribution from 18-30 to 30+, this seems to imply even faster utility growth. If the true rate at which marginal utility in consumption for a young American man fell over the years was lower than that implied by the interest rate, then utility grew more quickly. (That said, it is possible that young men expect their marginal utility in consumption to be higher in the future, when they have children.)

3) The VSL captures people’s willingness to pay to avoid premature death. But even just from a selfish perspective, the utility gained by avoiding premature death isn’t just a stream of life: it also includes the avoidance of the pain of premature death.

As long as

1. people discount the pain of dying, so that delaying this pain is desirable, and

2. a premature death by an accident on the job tends to be less pleasant than a “planned” death in retirement (or, at least, that the latter is not so much more unpleasant as to make up for the fact that it is being discounted),

then this too implies even faster utility growth. Our calculation tells us that, say, u+1 has been roughly doubling each decade, implying that u itself has grown more quickly.

4. The VSL may have risen more slowly than Costa and Kahn estimate.

The Costa and Kahn estimate implies that the elasticity of the real VSL with respect to real consumption exceeded 1 over the relevant time period (as would be consistent with a model in which u(c) is CRRA with RRA>1), but Viscusi and Aldy (2003) argue that more recent data suggest an elasticity of the real VSL with respect to real consumption of 0.5-0.6.³

Using a Viscusi-and-Aldy-style back-projection of the VSL shrinks the gap between the VSL- and consumption-based estimates of utility growth, but does not come close to eliminating it. The following calculation defines the nominal VSL in 1940 to be 1, without loss of generality, and posits that the elasticity of the real VSL to real consumption was 0.5 from 1940 to 1980. Figures for “real consumption” are taken from the BEA, as above, and re-converted to nominal units using the BEA’s CPI series, also cited above.

5. Maybe the young people who bought bonds expected their marginal utility in consumption to fall especially slowly, or even rise.

This could be either because the people who bought bonds were unusually vulnerable in some way and/or because it was typical to put a lot of weight on the possibility of a large increase in u′, e.g. via a huge recession.

Most young people do have positive savings of some sort, and this was especially true in the mid-1900s. How were they investing instead? Even if we replace the series of nominal gross returns with that of a fantastic hypothetical asset offering risk-free nominal returns of 100% per decade—so about 7% per year—implied utility growth is rapid:

The non-monotonicity of the two series just above highlights just how unrealistically far we have gone to push the numbers down. Still, at least on the Costa and Kahn numbers, even the (absurd!) 0% series implies faster utility growth from 1940 to 1980 (61%), and from 1950 to 1980 (36%), than the 46% and 29% implied by the (optimistic!) calculation based on real consumption.

Equivalently, note that our observation of relatively low interest rates can be squared with slow utility growth if marginal utility in nominal consumption persistently fell much more quickly than expected, i.e. if we had

\(\frac{u'_t}{u'_{t-1}} \ll \mathbb{E}_{t-1}\Big[\frac{u'_t}{u'_{t-1}}\Big]\)

for decades on end. This might not be as surprising as it could seem at first, since to some extent people may be saving as a precaution against a catastrophic drop in consumption which has never materialized (see e.g. Weitzman (2007)), or at least did not materialize in the US from 1940 to 1980. Indeed, the series may be biased by beginning just after the Great Depression, and by being set in the US rather than in one of the many countries in which consumption (and presumably the VSL) has grown less quickly. Though this is a relevant point qualitatively, the calculation above suggests that the bias must be extreme indeed to account for the gap between VSL-based and “real consumption”-based estimates of utility growth. The implied utility of life numbers are those that would follow if bond-buyers expected their marginal utility in consumption not to fall but to double roughly each 10 to 20 years.⁴

Implications

It seems that one or more of the following must be true. At least from 1940 to 1980, at least for young American men,

1. Utility itself grew much more quickly than is commonly believed.

2. The VSL grew much more slowly than is commonly believed.

3. The “utility level assigned to death” fell dramatically. That is, people grew more averse to death for reasons other than improved life, e.g. because they stopped believing in Heaven (or started believing in Hell), or started assigning more altruistic value to staying alive for the sake of others (or stopped assigning as much disvalue to staying alive despite imposing a burden on others).

4. People greatly increased the probability they assigned to various mortality risks, relative to the actual magnitude of the risks. For example, maybe people grew willing to take ever larger pay cuts to get out of coal mining not because their willingness to pay for safety rose, but because they used to think coal mining was less dangerous than it really is, or because they now think it is more dangerous than it really is.
   or

5. Marginal utility in nominal consumption persistently fell much more quickly than expected.

The practical implications of course depend on which of the above hold the most water. I personally take the finding to be a small update on all five points, if only because the gap between the measures of utility growth is so vast that it seems unlikely that any one kind of adjustment could explain it away alone.

As discussed under “robustness check 5”, I think #5 could have some truth to it. If saving is even more “precautionary” than currently appreciated, this lends some support to the idea from Weitzman (2007) and others that we’ve “gotten lucky”, and continued growth is less certain than one would infer from the historical distribution of growth rates or asset returns.

I also think #3 and #4 are somewhat plausible. Culture can shape our behavior in all sorts of arbitrary ways; people may just have gotten more squeamish about the prospect of dying for reasons unrelated to the quality of their lives. And of course people’s beliefs may be systematically biased, especially about low-probability events like workplace fatalities. Insofar as these are important effects, it’s less clear that we should use empirical estimates of willingness to pay for safety when deciding where to set the VSL for policy purposes.

Finally, I take #1 and (to a lesser extent) #2 as evidence for my view, rambled about here, that utility has grown quickly—in particular, that marginal utility in consumption has not fallen quickly, even as consumption and utility in consumption have risen—due to the development of new products for us to spend on. As I note there, when we measure utility growth by (i) estimating the shape of an individual’s utility function in consumption (from risk attitudes) at a given point in time and then (ii) putting a number to how much real consumption has grown, our estimates of utility growth are biased downward, since when a wider range of products is available, marginal utility in consumption is (weakly) higher starting from any given consumption basket. (See also Scanlon (2019).) To the extent that this is right, it has many implications, including:

a. Economic growth is more valuable, at least in the medium term, than commonly supposed.

b. Technological development yielding the creation of new consumption goods is more valuable, relative to that yielding more efficient production of existing consumption goods, than commonly supposed.

c. Even though marginal utility declines quickly at each point in time (so that redistribution is very valuable even if it comes with a one-off deadweight loss), high saving rates and high carbon taxes may also be very valuable, even if we are sure that the future beneficiaries of these sacrifices will be wealthier than we are.

d. Given that u(c,t) can vary a lot with t, we have less reason than commonly supposed to assume that economic growth will indefinitely produce increased concern for health and safety (the premise of Stokey (1998), Hall and Jones (2007), Jones (2016), Jones (2024), Trammell and Aschenbrenner (2025), etc). Though this has tended to happen historically, it is not an inevitable outcome of enrichment. We may even start developing new goods that raise u′ more than u, in which case the VSL will fall.

I thank Chad Jones, Bharat Chandar, Zach Mazlish, Parker Whitfill, and Jordan Rosenthal-Kay for comments.

1. According to GPT-5, the approach seems to be novel, despite its simplicity.

2. I’m using the figures in Costa and Kahn’s Table 6, “logarithmic specification”, and re-inflating by the standard CPI series from the BLS, which they used to deflate their figures.

3. This conclusion would in some ways be theoretically surprising, since even CRRA u(c) with RRA<1 would imply an elasticity of 1, not less. But since we’re throwing out the “u(c)” model in the first place, perhaps we should be more open to this possibility.

4. I.e. precisely every 10 years if risk-free nominal interest rates had been zero.