We're fresh from a small round of blog discussions about the value of human life and the policy implications of executing murderers and executing 'vermiscripters' (computer virus writers). Steven Landsburg here and Ravikiran here assert that people who claim that a value cannot be attached to human life are making "invalid" and "wrong" statements. The crux of their argument is that we make choices about our lives and our safety all the time. So if I drive a bike at a 100 kmph for half an hour today, and that increases the chances of my death by .01%, then the fact that I am making this tradeoff means that utility (.01 % increased safety of life) = utility (driving a bike at 100 kmph for half an hour). It is easy to see that the RHS of this equation can be replaced in money terms - for e.g if I am willing to pay 1000 Rs to drive a bike for half an hour at 100 kmph, then utility (1000 rs) = utility (driving....half an hour) and by substitution, utility (.01% increased safety of life) = utility (1000 Rs). So far so good - the argument is flawless until here. But truth is a little more nuanced after this.

The flawed leap in the argument comes when people extend utility (1000 rs) = utility (.01% increased safety) to , say, utility (100000 rs) = utility (1% increased saftey). There are two assumptions underlying this extrapolation, both wrong, one more wrong than the other. The first assumtion is that utility (100000 rs) = 100 * utility (1000 rs). The second, and more incorrect assumption, is that utility (1% increased safety) = 100 * utility (.01% increased safety). (Of course, mathematically it is possible to come at the same conclusion without either fo the two assumptions, using a third more integrated one. But the decomposition into its parts will help illustrate the folly in logic better, and hence I take this route).

Let's deal with the first assumption. In classical analysis, goods are supposed to have diminishing marginal utility. Money is the one good that is exempted from this consideration - an extra buck is always as good as any other extra buck. It makes some amount of intuitive sense, plus it is necessary to exempt money from the concept of diminishing marginal utility if one is to compare the utility (and hence prices) of other goods. However, in decision analysis under uncertainity, things change a bit.

Rational people are risk averse. If I was to offer you a game where you could win 150000 rupees with 50% chance and win 50000 Rs with 50% chance, you would typically not accept this gamble at a price of Rs 100,000. The expected value of this game is 100000 - it has an expected return of 0 and it is irrational to take it up. You are bearing some risk, and hence would want a non-zero return. (In fact you would want a return above the risk free rate of interest, but lets not get into that). Mathematically, if U(x) is the utility function of money, U(50000) + U(150000) < 2 * U(100000). What kind of a function would U be? Concave to the x axis. Consider the simple example of U(x) = sqrt (x). It is easy to check that the inequality holds (in fact, it reduces to the RMS-AM inequality, or equivalently, to the statement that variance is non zero). Now, it's easy to see that with U(x) = sqrt (X), U(100000) = 10 U(1000) < 1000 U(1000). This is, of course, a cooked-up example that makes the inequality very strong, but it is easy to see that for any concave function, U(kx) < k*U(x). Hence, U(100000) < 100 * U (1000).

(Through all that not-so-fancy math, all that we're saying is that a 100 times more money will not make you a 100 times more happy. Which is something we all intuitively know. But since we are trying to counter some 'rational' arguments, we will stick with the math. )

The second assumption, and this is where the extrapolation gets hit very badly, can be invalidated by a simple thought experiment at the extreme. You make me two promises - one in which you promise to save me from near certain death, and another in which you promise to provide me with an extra .01% of safety. What is the worth of your first promise vis-a-vis your second promise to me? Is it worth 10,000 times more? Far from it. It is worth a whole lot more. I want to consume, I want to ride bikes, I want to do various things, but most of all, I want to live. I'd give you my entire wealth (everything above subsistence) if you saved me from near certain death. In fact, if someone was willing to take a leveraged position on my life, I'd be willing to give you a huge chunk of my future wealth as well. Put mathematically, the marginal utility of extra safety of life is increasing. U(1% safety) > 100 * U(.01% safety).

(Actually, it will not increase monotonically, for reasons that should be obvious. But we shall sacrifice behavioural assertions and mathematical rigour for the purpose of relevant analysis)

Now, if we put the mathematical formulations of our two intuitive results together, what do we get? U(1% safety) > 100 * U(.01% safety) = 100 * U(1000) > U(100000). The extrapolation of the 'value of life' from a single decision has been hit on two counts. Essentially, extrapolations like this will end up severely understating the actual value that we place on our safety and our lives. And therefore, the time frame of the analysis and the absolute numbers becomes of critical importance. It is useless to compare one exceution of a murderer with one execution of a vermiscripter if over my lifetime, many more than one executions take place.

Hence, dear economists and other rational type people - if you are going to estimate the value of my life by extrapolating from the value I place on .01% of my safety, say it like that. Because the answer that you will get will be different from your answer if you were to extrapolate it from the value I place on 1% of my safety. If you directly ask me the value of my life, I will refuse to put a figure to it. Not because I'm being a moralist. But because I am then thinking about the value of my entire life. 1 or 0. Life or death. Not 10000 times .01% or 100 times 1%. And I can genuinely not answer that, except for saying that my life is rather priceless. Not because I am irrational. But because I am rational and hence risk-averse. Because the marginal utility of money is diminishing while the marginal utility of safety is increasing. Because the way you frame a question will lead to a different answer.

The time frame of policy and an approximation of the utility of safety/life are of critical importance beofre one starts asserting things about the value of life and the costs and benefits of deterrence. If you are only going to multiply two numbers, don't bother. I learnt multiplication and pretty much mastered it when I was 7 years old.

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## 2 comments:

Seriously,everytime I read one of these ' lets do the math' rants I am left wondering about the sanity of the person in question. Dude, unless you're talking about partial differntial equations or probability distributions, you're not talking insight ok?

You said that in your earlier post. I'd rather preferred you to puke on the articles like Nilu does :-)

Naah. Lets leave puking to Nilu. I know I said that, but it was necessaery to use some math to illustrate the entire gamut of mathematical flaws in that article.

Of course, you should remember to not hold me accountable to the word for rants. One can be talking insight with utility functions, you know ;-).

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