## Black Death – once more with feeling

02/04/2012 § Leave a comment

The post on the Black Death and the VoSL attracted some comments. I thought I’d respond in a new post because, well, because I can.

Seamus Hogan brought in some mathematical notation, so I’ve been trying to figure out the notation for what I was thinking. Here goes:

VoSLs can be estimated by finding examples of people paying to reduce their risks of death. For example, if consumers pay $5,000 more for a car with a safety record that gives them a 1% lower probability of dying in an accident, that suggests a VoSL of $500,000. The equation is:

VoSL = Value / Probability.

We can use this to think about how much to spend now to reduce future impacts of climate change. We multiply the number of lives saved by the VoSL, and then use discounting to compare the future value to present spending:

(VoSL * lives) / (1 + discount rate)^years = spending.

But here’s the problem: VoSL in the future is affected by the quality of life in the future, which is affected by spending on climate change. That is:

Value = f(population),

because when lots of people die, life gets cheaper — this was the quote about the Black Death in the earlier post.

Population = g(spending), so

Value = h(spending).

Putting this back into the estimate of the right level of spending, we get:

((h(spending)/probability)*lives / (1 + discount rate)^ years = spending.

Instead of being able to look at future potential losses and calculating the ‘correct’ level of spending now, we find that spending is on both sides of the equation. Rearrangement gives us:

lives / (1 + discount rate)^ years = spending/(h(spending)/probability).

You can think of the LHS as a constant — the per-life discounting to be applied to future spending. Once you decide the timeframe (100 years?) and the discount rate (good luck), this is just a number. The RHS is a ratio — the ratio of current spending to future VoSL as a function of current spending. Since VoSL increases with spending, we aren’t guaranteed a unique solution. It all depends on *h(spending)*, on that function. This was Seamus’s point — with concave functions for population and quality of life, there are potentially multiple equilibria.

Basically, the reasoning above says that two positions are as economically rational as each other:

- We put a high value on life, and we should preserve future lives by spending now to address climate change.
- It’s all going to descend into brutal chaos, anyway, so we don’t need to bother (insert gratuitous reference to the frightening cultural phenomenon that is The Hunger Games).

More basically, I don’t think economics gives us the answer (provides a unique solution). Instead, it is an ethical problem. Once we sort out the ethical goals, economics can help with the means to achieve them efficiently.

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