A brief explanation of a certain economic concept: the value of a year of life (VYL). This numerical value relates the value of additional consumption to the value of living longer/reducing risk of dying.

Formal Derivation

We start by imagining an agent living for some time $T$. He has some flow of consumption $c$ he gets throughout his life. This is a flow so it is measured in units of dollars per second. For simplicity I assume it’s constant throughout his life. One way to imagine it is that he has $c$ square meters of farmland he uses to farm and create crops. He doesn’t get $c$ per year or something, just $c$. So his lifetime utility $U$ is $\int_{0}^{T}u(c)\,dt$ which is simply equal to $T u(c)$.

Now we want to see how he would trade off living a bit longer or shorter to having more consumption. The deal is that he would:

Lose a fraction $a$ of his life, he’d live only $T(1-a)$ years instead of $T$.
But he gains a fraction on consumption to compensate, instead of consuming $c$ he gets to consume $(1+\ell) c$.

We want to know for what values of $\ell$ and $a$ our agent will be indifferent between the shorter richer or the longer poorer life, that is when does $u\big(c (1+ \ell)\big) T(1-a) = u(c) T$ hold.

Now it turns out that it makes sense to think of $\ell$ as relating linearly to $a$. If we live $1\%$ shorter life we want some % longer life. So we express $\ell$ in terms of $a$ as $\ell=am$. And we want to solve for $v$, we want to know what the multiplicative factor is, e.g. if $v$ is 2 the agent wants 2% more consumption to accept a 1% shorter life.

Substituting $a v$ for $\ell$ we get $u(c (1+am)) T(1-a) = u(c) T$ we want to know what $v$ makes this equation hold, makes the agent indifferent.

Notice we can cancel the $T$ on both sides, the absolute length of the agent’s life is irrelevant in this setup. Whether you live 1000 years or 10, your value $v$ is unchanged. Cancelling gives us

\[u\big(c + c v a\big) (1-a) = u(c). \tag{$\ast$}\]

Now what we really care about is the marginal value of $v$, that is for very small $a$ what is your factor. For finite $a$ the solution depends slightly on $a$ but in practice $v$ will be very similar for small $a$, if we let $a$ be 1% or 0.1% it will lead to almost identical results, but for analytic simplicity it makes sense to take the limit for $a$ going to $0$. The factor that makes $(\ast)$ hold for a vanishingly small sacrifice $a$. Working in this limit also lets us replace $u$ by its linearization exactly rather than approximately.

The way we will do this is to write the utility function as a Taylor series and then take the limit with $a$ going to zero. For the Taylor series we expand $u$ around $c$ to get $u\big(c + x\big) = u(c) + u'(c) x + \tfrac12 u''(c) x^2 + \cdots$

Now, we only care about the function at the point $c + cma$ so we substitute $cma$ in for $x$ to get $u\big(c + c v a\big) = u(c) + u'(c) (c v a) + \tfrac12 u''(c) (c v a)^2 + \cdots$ Grouping all terms past the first two into an $O(a^2)$ term we get

\[u\big(c + c v a\big) = \underbrace{u(c)}_{k=0} + \underbrace{u'(c) c v a}_{k=1} + \underbrace{O(a^2)}_{k\ge 2}.\]

Substituting this expansion into $(\ast)$ and then simplifying gives us

$[u(c) + u'(c) c v a + O(a^2) ] (1-a) = u(c)$ Next I multiply out $(1-a)$, cancel the $u(c)$ on each side, divide through by $a$, and finally bring one $u(c)$ term over on the other side. $\begin{aligned} u(c) + u'(c) c v a + O(a^2) - a u(c) - a^2 u'(c) c v - O(a^3) &= u(c) \\ u'(c) c v a + O(a^2) - a u(c) - a^2 u'(c) c v - O(a^3) &= 0 \\ u'(c) c v + O(a) - u(c) - a u'(c) c v - O(a^2) &= 0 \\ u'(c) c v + O(a) - a u'(c) c v - O(a^2) &= u(c). \end{aligned}$ So far we have only written out our original indifference equation in a different way, not changed anything substantive. Now after these preparations we are finally ready to take the limit wrt. $a$ $\begin{aligned} u(c) &= \lim_{a \to 0} \Big(\, u'(c) c v + O(a) - a u'(c) c v - O(a^2) \,\Big) \\ &= u'(c) c v \end{aligned}$ Only the constant $u(c)$ and the linear $u'(c) c v a$ survive the limit; the rest are higher order in $a$.

Now we simply divide by $u'(c) c$ to solve for $v$ and get

\[\boxed{ v = \dfrac{u(c)}{u'(c) c} \text{ or } v(c) }\]

This finishes the derivation, given any utility function we have a clear way to determine $v$ or $v$. But this doesn’t give us any great intuition yet, the next section is meant for this.

Intuition

Numerical example

I will start by going through the same steps as in the formal derivation with some made up numbers to get some intuition.

I use the following numbers:

\[a = 2\%\]
\[c = 4\ \$/\text{hour}\]
\[u(c) = 140\ \text{util}\]
\[u'(c) = 2.5\ \text{util}/\$\]

as $v$ is the number we want to derive we don’t assign any value to it, we also don’t need to assign anything for $T$ as it cancels out as we saw above. With these numbers our indifference condition is

\[u(4 (1+2\%\times v)\$ /\text{h}) (1-2\%) = 140 \text{util}\]

we can simplify this to get $u( 4\$ /\text{h} + 4 \times 2\% \times v \$ /\text{h})98\% = 140 \text{util}$

So our original life with consumption of $4\$ /\text{h}$ is worth $140$ utils. But we could also live 2% shorter but get 2% times $v$ more consumption instead, there is some $v$ that makes these two sides equivalent.

Writing the utility function as a Taylor series and assuming the $O(a^2)$ term is negligible we get

$\big[u(4\$ /\text{h}) + u'(4\$ /\text{h}) \times 2\% \times 4 v \$ /\text{h}\big]98\% = 140 \text{util}$ this simplifies into $u'(4\$ /\text{h}) \times 98\% \times 4 v \$ /\text{h} = 140 \text{util}$ solving for $v$ and cancelling terms we get $v = \frac{140 \text{util}}{u'(4\$ /\text{h}) \times 0.98 \times 4\$ /\text{h}}$ substituting $2.5\ \text{util}/\$$ for $u'(c)$ we get $v = \frac{140 \text{util}} {2.5\frac{\text{util}}{\$} \times 4 \$ }$ this reduces to $v = \frac{140 \text{util}}{10 \text{util}} = 14.$ If you want to shorten my life by some small amount say 2%, I would only agree if I got $14 \times 2 = 28\%$ more money.

The extremes

A useful intuition can be had by comparing the VYL people at the extremes of consumption, a standard first world citizen vs an ultra rich person, e.g. Jeff Bezos and a very poor person, e.g. a medieval peasant.

Bezos has much more money than me, this means that more consumption increases his utility relatively less than mine. I might accept a deal where I lose one year of my life as long as I get 5 years of consumption for doing this.

But for Bezos his utility barely increases at all if he gets 5 years of his own consumption, even though its absolute value is much higher than mine.

His VYL will be much higher than mine, he will be more willing to give up consumption for safety.

The other extreme case is comparing me vs a medieval farmer. She barely has enough to survive, her life is barely positive. In the most extreme case we could imagine her current utility to be something like 0.1, increasing her consumption by just 20% could mean she stops being malnourished, maybe her children will survive the next famine. It could improve her utility to 0.4. An increase of times 4, her VYL is very low. She is willing to give up almost no consumption for a longer life or more safety.

Comparing different utility functions

To understand how $v$ changes with more consumption it is useful to compare a log and a square root agent.

For $u(c) = \log(c)$ what is the VYL? It’s very simple to calculate the derivative of $\log$ is $1/c$ so the two $c$’s cancel and we get $v(c) = \log(c)$ it happens to be the same as the utility function, it’s increasing just as we might expect. The wealthier the agent the more safety conscious he is.

Now the square root agent has the following utility function $u(c)=c^{0.5}$ it is risk averse, we might expect this to imply that its VYL is also increasing. But calculating it we get $v(c) = c^{0.5}/(0.5 c^{-0.5} c) = 2$ the VYL is constant, it doesn’t change with more consumption. This is a bit counterintuitive. We can understand it by looking at how the three different components of $v$ change as we increase consumption. We can compare the square root agent’s $v$ at two points, 5 and 30. we get

\[v(5) = 2.23/(5 \times 0.223) = 2\]
\[v(30) = 5.48/(30 \times 0.091) = 2\]

His utility has approximately doubled, his consumption grew by a factor of 6 and his marginal utility decreased by around $2/3$ so we get $2/(6 \times 1/3) = 2$. The function isn’t convex enough to make the VYL increase with more consumption.