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Risk aversion and money

This discussion from Keynes (from Robert Skidelsky, linked from Steve Hsu) reminds me of a frustrating conversation I’ve sometimes had with economists regarding the concept of “risk aversion.”

Risk aversion means many things, but in particular it is associated with attiitudes such as preferring a certain $30 to a 50/50 chance of having either $20 or $40. The standard model for this set of attitudes is to assume a nonlinear function for money. It is well known that reasonable nonlinear utility functions do not explain this sort of $20/30/40 attitude (see section 5 of this little article from 1998, for example); nonetheless the curving utility function always comes up in discussion, requiring me to waste a few minutes before going on, explaining why it doesn’t explain the phenomenon.

It would be as if any discussion of intercontinental navigation required a preliminary discussion of why the evidence shows that the earth is not flat.

So then there are various patches to utility theory such as prospect theory etc. Myself, I’m partial to Dave Krantz’s goal-based model of decision making (see herehere, and lå–like many models in this field, it’s both descriptive and prescriptive.

In any case, one thing I’ve long thought was funny about classical models of risk aversion was that they traditionally are applied to money, not to goods. So, if you’re risk averse you’d rather have some money now, instead of holding out for the possibility of more money later. A bird in the hand and all that. But classical economics is ultimately about goods, not money: money is only useful to the extent it can buy things. Anyway, the Keynes quotes above reminded me of this, that it’s so usual of us when working in decision analysis to speak of money as if it is a goal in itself.


  1. Kevembuangga says:

    So, if you're risk averse you'd rather have some money now, instead of holding out for the possibility of more money later.

    The now/later distinction is likely the key to understanding apparently irrational behaviors about assessments of value.
    See hyperbolic discounting.
    As for the differences in individual preferences about the same choice case it could come from different "coefficients" of discounting (hyperbole slope) among individuals.

  2. Ted Dunning says:

    I think that the these experiments purporting to show non-linear value of money are inherently broken.

    The problem (in my own Bayesian opinion) is that these experiments start with an assumption that the experimental subject has a simple model of the experimental setup. I don't that assumption is valid.

    Instead, I think that the subjects are themselves Bayesian inference machines and they take the experimental instructions as evidence of the truth rather than the truth. When somebody tells me that I can have a one in a thousand chance of getting $10,000 or a certain chance of getting $1, I am likely to adjust their p = 0.001 statement by my own estimate of how likely they are to be lying. The larger the amount that they are promising me, the more likely I will think that they will lie. Anybody promising me a billion dollars is probably lying, so I will take their dollar.

    This criticism may even apply to the experiments involving much smaller amounts of money, not so much in the form that the subject things that the experimenter won't pay up on a $40 promise, but more in terms of a general cynicism about whether probabilistic things really turn out they way they are predicted.

    This dovetails very nicely with the fact that people can't form 90% confidence intervals accurately from their own experience. An acute observer is likely to have internalized this general failure and view any statements of a probabilistic nature in a highly jaundiced way thus leading to the same sort of apparently non-linear value due simply to incredulity.

  3. Aaron says:

    To economists, risk aversion (for agents whose preferences can be represented as utility functions) means that utility functions are concave. Indeed, concave utility functions basically define risk aversion, as the condition of concavity is equivalent to the definition of risk aversion (at least according to Mas Colell et al., our loveable microeconomics bible). So whatever else risk aversion means to non-economists, it doesn't mean those things to economists. Basically, risk aversion to us is completely characterized by concave utility functions.
    Also, most of classical economics proceeds under the assumption that there are no pesky transactions costs associated with trading money for goods, so having money is equivalent to having goods; they are perfectly (i.e., costlessly) interchangeable.

  4. Andrew Gelman says:

    Kev: I can believe that the time effect can explain a lot, but I'm skeptical of attempts to treat temporal discounting as an automatic solvent to solve all decision paradoxes.

    Ted: Could be. But my impression is that the heuristics-and-biases crowd have done their experiments under enough different conditions that your story doesn't _directly_ explain what's going on. Although your story could work as more of an evolutionary explanation. It's like those experiments on anchoring and adjustment where even random numbers–known to the participants to be random–are used as anchors. They have an effect, possibly because in the wild, so to speak, our anchors aren't random.

    Aaron: I konw that people say this sometimes but I don't buy it. Risk aversion is a phenomenon that it is standard to _model_ using a curving utility function. Reality is not the same as a model. And the well-known point (see, for example, my 1998 paper linked to above) is that you can't seriously consider a utility function that has serious curvature in the $20/$30/$40 range, or it won't work at higher values of money.

    Regarding your second point, of money being assumed to be costlessly interchangeable with goods: exactly, that's what Keynes was pointing out as a flaw in the standard model.

  5. Aaron: I think Andrew's point about "we think in terms of money too often" is exactly what's going wrong if economists really define risk aversion in terms of concave utility functions. Imagine a situation where we're not talking about money at all, but I'm just imagining three different futures (say, spending time writing a paper that gets rejected, spending time writing a paper that gets accepted, and going to the beach). If the sum of the utilities of the first two futures is at least twice the utility of the third, then preferring the third to a 50/50 gamble between the first two looks like a clear case of risk aversion. But there's no way to model this as a concave utility function, unless you somehow want to talk about "decreasing marginal utility of utility" or something crazy like that.

    (Of course, this just raises the question of how we estimate the utilities of these various situations by any means other than looking at a subject's preferences among gambles between them. If you just define utilities in terms of preferences between gambles, then there's no room for the phenomenon I and most other philosophers would consider true risk aversion, and so I guess the term is available as a re-description of the phenomenon of decreasing marginal utility of money, but it seems like an odd use of the term to me.)

  6. marcel says:

    I think that 3 different concepts are being mixed up here, 2 of which economists try to keep distinct, and one which most ignore. They are:

    1) time preference
    2) risk aversion
    3) uncertainty

    The "bird in the hand" example confuses (1) and (2).

    1) Time Preference is based on 2 quantities (of any good, not just money). How many widgets next year/period would it take you to give up exactly 1 widget right now? You know for certain that you will receive those widgets in the future. There is no possibility that you won't. Does the world work this way? Of course not. But for (economists') purposes of defining terms, we imagine that it does.

    2) Risk Aversion relates to a situation in which, at a specified moment, you can either have 1 quantity of a good with certainty, or 1 of (say) 2 quantities, and each quantity has a probability associated with it, the probabilities summing to 1. Again, you know with certainty that you will receive either the first quantity or 1 from among the 2nd group of quantities. There is no possibility, whichever situation is chosen, that it will not happen. You know that if you select the 1 item with certainty, you will receive it, and that if you select the set of quantities, you actually will receive one of them with the specified probability. (The probabilities are frequency based, so if you faced this situation over and over, and repeatedly chose "1 from the set of quantities", the distribution of outcomes would converge on the probabilities.)

    Return for the moment to the phrase "a specified moment." This qualification is important to keep risk aversion from becoming entangled with time preference. The moment can be immediate, or later in the day, week, month, year, etc. The point is that the selection you choose has no effect on when you receive the good.

    3) Uncertainty in economics is usually traced back to Knight and Keynes. It emphasizes that the simplifications made above, that we can know anything about the future (I am tempted to say historical future, oxymoron though that may be) with any kind of certainty, even that weaker one implied by frequentist notions of probability, is silly, and that for decisions that are important and non-reversible, abstracting from this distorts analysis and clouds understanding.

  7. Greg B Davies says:

    It's true that in economics (and most decision theory) risk attitude is solely a function of curvature of the utility function – and this is as true of Knightian uncertainty as it is of risk, and is also true of non-monetary outcomes.

    Why risk aversion should be constituted out of nothing more than diminishing marginal utility for certain outcomes (where risk plays no part whatsoever) is seldom discussed. However, utility functions elicited using preferences for certain amounts (or goods) seldom imply levels of risk aversion that match with those elicited from choices between lotteries where risk is involved.

    I have long believed (see Davies 2006 – Rethinking risk attitude: Aspiration as pure risk, Theory and Decision 61, 159-190) that observed risk attitudes may in part be explained by curvature of the value function which governs marginal (psychophysical) value attached to certain outcomes in the absence of risk, and which only incidentally induce risk attitudes; but also in part by attitudes to pure risk itself.

    Using this approach you can decompose risk attitude into its psychophysical component, and pure risk attitude.

  8. @Greg Davies: Is this article available, and can you provide a URL?