How long should I live?

Imagine an individual selfish economic agent. She has a utility function \(u(c)\) where \(c\) is a flow of consumption, in units of something like $/day, for simplicity I’ll assume \(c\) to be constant. Her lifetime utility is \(U = T u(c)\), where \(T\) is for how long she lives. Like any reasonable agent she doesn’t discount her future utility.

Now she faces a trade-off, she has some fixed budget of cash to spend throughout her lifetime, \(B\), and has to decide how long she wants to live. She is unable to earn more money during her lifetime, her total lifetime consumption is constrained to be \(B\). She could live 100 years at \(x\ \$/\text{day}\) or 50 years at \(2x\ \$/\text{day}\).

We can write this as an optimisation problem

\[\max_{T} T u \left(\tfrac{B}{T}\right),\]

using \(c=B/T\) to eliminate both occurrences of \(T\) we get

\[B\cdot \frac{u(c)}{c}.\]

The agent can’t change \(B\) so it drops out in her optimisation problem (alternatively we can normalise it to \(1\)). So this agent is simply maximising \(u(c)/c\), that is her utility flow divided by her consumption flow. Solving for the first-order condition gives us

\[\begin{aligned} \frac{d}{dc}\frac{u(c)}{c} = \frac{u'(c)c - u(c)}{c^2} &= 0 \\ \Rightarrow \ u'(c^*)c^* - u(c^*) &= 0 \\ u(c^*) &= u'(c^*)c^* \\ \frac{u(c^*)}{u'(c^*)c^*} &= 1 \end{aligned}\]

Economists might recognise the term on the left-hand side as the value of a year of life measured in years of consumption, or Value of a Year of Life (VYL) for short. If you don’t know this term I have a post explaning it here.

In brief the VYL is a dimensionless number telling you how many years of consumption you would have to be paid to die one year earlier. Typically the VYL grows with consumption, wealthy agents demand large increases in consumption to give up a year of life, their marginal utility of consumption is low so getting more consumption only increases your utility by a small amount, but dying 1% earlier always reduces your utility by 1%.

So the above equation tells us that the VYL must be equal to 1 at the optimal level of consumption. This might come as a surprise because 1 is quite a small value for the VYL to take on. Current estimates for first-world countries are around 6, wealthy people demand a lot of money to give up a year of their life. Our economic actor would prefer to live a poorer but longer life.

To get some intuition I’ll calibrate the model with some numbers. I make the following assumptions:

  • I standardise \(c = 1\) to mean current US consumption, so \(c = 0.5\) would be living on half of average US consumption.
  • I set the VYL, given \(c=1\), to be \(6\) (in accordance with typical estimates); I will call this value \(V\).
  • Utility has the form \(u(c) = \dfrac{c^{1-\gamma} -1}{1-\gamma} + V\).
  • The coefficient of risk aversion \(\gamma = 2\). The higher this value, the more risk averse the agent is, or alternatively the faster her marginal utility from more money drops off.

Plugging in 1 for the level of consumption we see that \(u(1)=V\), so \(V\) directly determines not only how our agent would trade off consumption for longevity but also how worth living life is at the moment. In setting \(V=6\) I’m assuming that life is worth living, if you disagree you would set \(V\) to be negative and you would get a very different result.

But assuming these parameters for now, if we plug them in, we get an optimal consumption level of about 29% of current US consumption. To determine the length of the agent’s life we simply use \(B/c^* = T^*\). For example with a lifetime budget equal to forty years of US consumption, she will live for \(40/0.29 \approx 138\) years. Instead of spending her budget living well for 40 years she stretches it into 138 years at slightly less than a third of average US consumption.

Technical details for the interested The value of consumption that makes the VYL equal one is $$c^*= \left(\frac{V \left(\gamma-1\right)+1}{\gamma}\right)^{\frac{1}{1-\gamma}}$$ Substituting $$V=6$$ as well as $$\gamma = 2$$ gives us $$ \left(\frac{6+1}{2}\right)^{-1}= 2/7 = 0.29. $$

Her yearly utility is only around 60% of her utility if she lived on US consumption, but by living for more than three times as long she ends up with a lifetime utility around twice as high as otherwise.

Sometimes quantity does beat quality.