Can
we estimate the value of a life?
Government programs that try to keep people alive longer, such as policies curtailing pollution and accidents, must somehow value the ensuing benefits in order to know how much to spend. If we have only $500,000 to spend on either of two programs, with one worth $800,000 and the other only $300,000, then the choice is clear. But since the usual benefit is an extension of life, we cannot avoid considering the economic value of a life.
Assume that a life is worth an infinite number
of dollars. It’s certainly
reasonable. Now consider two programs:
Plan A would save 50 million lives, and Plan B would save 1 life. Both would cost the same amount. Which would we adopt?
Surely, this is a no-brainer: Plan A. But look again. Since both programs have the same cost, we
would prefer the plan with the higher value.
Apparently, Plan A is worth 50 million times infinity, and Plan B is
worth 1 times infinity. But these
estimates are absurd, since infinity is not a cardinal number. We cannot conclude that 50 million times
infinity is greater than infinity, because infinity is incomparable. If we are to choose between the programs,
then we must assume a finite value of life.
The problem does have a practical solution
although it is not fully satisfying.
Rather than consider the value of a life per se, think about the value
of a slight reduction in the risk to life.
The value of a 1% reduction may be $10,000; of a 2% reduction, $30,000,
and so on. Each of these decreases of
risk has a finite value. And yet life
itself has an infinite value, because the value of going from a 100% risk of
death to a 99% risk is infinite. The
value of a life is roughly the sum of the values of all the slight reductions,
including the one from 100% to 99%, the one from 99% to 98%, and so forth, down to the last one, from 1% to 0%. This sum is infinite since part of it is infinite.
No way out?
No way out?
This approach is workable, because most
programs do attempt small reductions of risk when life is already pretty safe –
for example, the avoidance of fatal fires, by requiring a fire detector in each
apartment. It’s sensible to think that
such a small decrease of risk has a finite value since you are not willing to
pay your entire income for a fire detector; $10 or $15 should do it. This is like going from a 2% risk of death to
a 1% risk.
Another problem with the marginal approach is that it implies a vastly disproportional change in the value of a reduction of risk. A decrease in the risk of death from 100% to 99% is worth far more than 100 times the value of a decrease in risk from 2% to 1%. In fact, the first risk is worth any number of the second risk or even of most larger risks. But are we truly willing to devote all our time and money to cutting the risk of death to 99% for one advanced cancer patient rather than to reduce risks from 90% to near zero for a million patients in earlier stages of cancer? Perhaps we must either assume a finite value of life or compare programs without using values.
One possibility is to prefer the program that saves more lives at the same total cost as other programs. This rule is enticing but arbitrary -- and too specialized to be of much use, since it applies only to the case of equal costs. If costs differ between programs, then we must compare them to the various values, which is the dilemma that we were trying to avoid. –Leon Taylor tayloralmaty@gmail.com
Notes
Define p as the risk of an early death in percentage terms, varying from 0 through 100. Define MV(p) as the marginal value of avoiding an early death – that is, of continuing a life – for given p. The variables p and MV are continuous. Integrating MV(p) over the full range of p gives us the total value of a life, which is an improper integral.
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