Constant Relative Risk Aversion

There is a game of chance called "St Petersburg," which is the simplest thing possible. Take a fair coin and flip it. You have bet 2 rubles on the outcome. If the coin comes up heads then you win 2 rubles, but if it comes up tails you play again, this time for 4 rubles. Each time, the stake is doubled, so n plays yields a prize of 2ⁿ rubles. Each flip of the coin is called a "trial" and the string of trials with their outcomes that concludes the game, is called a "consequence."

The probability of a consequence of n flips ('P(n)') is 1 divided by 2n, and the "expected payoff" of each consequence is the prize times its probability. The ‘expected value’ of the game is the sum of the expected payoffs of all the consequences. Since the expected payoff of each possible consequence is 1 ruble, and there are an infinite number of them, this sum is an infinite number of rubles. This became known as the St. Petersburg Paradox.

Bernoulli, the Swiss philosopher and mathematician, suggested the problem lay in rewarding people with money rather than utility. At the back of the paradox is the assumption that (2ⁿ)(2^-n) = 1 for all values of n; as n becomes (or could become) infinitely large, the sum of probable outcomes reaches ∞. In fact, that's not true for utility, and Bernoulli proposed that the expected utility--as opposed to expected payout in rubles--was necessarily finite.¹

Economists were increasingly interested in risk² because it applies to virtually all decisions, particularly those related to savings. Suppose you have a temporary employee working at a longterm assignment. The temp could be dismissed at any second; temps are almost never given any notice, and an abrupt dismissal typically causes the temp a lot of hardship. Because of this, the temp faces a risk if she accrues any debt; savings are vital to surviving periods between assignments. Yet there is also potential benefit in taking night school courses--say, in accounting. She must therefore weigh the risk (weighted for consequences) of dismissal, against the probability of getting a permanent job with benefits (weighted for the benefits of doing so).

Putting this another way, let U be the utility experienced by the temp. U is a function of consumption C, which of course varies over time; U = U(C(t)). The temp may prefer to take risks in order to enhance her estimated future consumption: an increase in income caused by risky investment of scarce money in tuition. For small values of θ, marginal utility diminishes more slowly--i.e., U˝(C) is smaller--than for larger values. That's the crucial significance of θ.

In the equation above, a high value of θ signifies that the consumer is quickly sated by increasing consumption. Hence, both high values of consumption all at one time, and a high payoff from a high risk bear less gratification, than would be the case if θ were low. Hence, another term for "constant relative risk aversion" is "constant intertemporal elasticity of substitution" (CIES). On average, the tendency to accept risk (in exchange for a payoff) and the tendency to accept a major belt-tightening (in exchange for a future payout) are comparable.

In the graph below, the horizontal axis C(z) refers to a random outcome; the probability that z₁ happens is p, and the probability that z₂ happens is (1-p). In other words, either z₁ or z₂ can happen. So the expected outcome E(z) is pz₁ + (1-p)z₂. Now, please notice someone has drawn a chord between points A and B. Notice that the expected utility E(U) is substantially lower than the utility of the expected outcome u[E(z)]; or just notice D and E. The position of E on the chord is dependent on the ratio of p:(p-1).

The behavioral inference drawn from this chart is that the utility of expected income U[E(z)] is greater than the expected utiliy E(U), i.e.,

U[pz₁ + (1-p)z₂] > Upz₁ + U(1-p)z₂
This is just a complicated way of saying that risk aversion inflicts a severe hit on the utility of bundle of benefits.

The function above was developed by Milton Friedman and Leonard Savage in 1948. Friedman & Savage also speculated on other shapes of the risk-utility function, but the curve above has a certain usefulness for the economics profession. You see, if a person has a curve very much unlike the one shown above, then one can be presented with a series of risks, each of which one finds acceptable, that lead one into any position; the other party--say, the casino management--can always make a profit, and essentially "pump" money out of players. While some people undoubtedly are like that, the population in the aggregate cannot be, or the economy would grind to a halt forever.

If we are looking at the function as a CIES graph, then the horizontal access merely represents increasing values of consumption. If, however, we are looking at the function as a CRRA graph, then it makes sense to regard the horizontal axis as a series of equally likely payouts. A segment between z_i and z_j with a length of 1% of the entire horizontal axis, would have a 1% possibility of happening.
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NOTES

1 For those of you unfamiliar with calculus: some algebraic functions, like f(x) = x^-2 can be graphed from 0 to infinity, and the total area under their curve is finite. This seems impossible, but it's true.

2 Risk and uncertainty are (usually) regarded as distinct topics in economics. Risk is quantifiable; uncertainty is not. Or, in the words of Frank L. Knight,

The essential fact is that "risk" means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character; and there are far-reaching and crucial differences in the bearings of the phenomenon depending on which of the two is really present and operating. There are other ambiguities in the term "risk" as well, which will be pointed out; but this is the most important. It will appear that a measurable uncertainty, or "risk" proper, as we shall use the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all. We shall accordingly restrict the term "uncertainty" to cases of the non-quantitative type. It is this "true" uncertainty, and not risk, as has been argued, which forms the basis of a valid theory of profit and accounts for the divergence between actual and theoretical competition.
[Risk, Uncertainty, and Profit, 1921]