O’Donoghue and Somerville (2018) on Risk Aversion

Ted O’Donoghue and Jason Somerville, “Modeling Risk Aversion in Economics.” Journal of Economic Perspectives 32(2): 91-114, Spring, 2018.

• As Rabin and Thaler (2001) indicate, expected utility (EU) maximization seems incapable of explaining people’s risk preferences – even though it does suggest some nice measures of the degree of risk aversion. 

• Other models of risk aversion, however, might prove more empirically sound, while maintaining tractability. That is, we might not need expected utility to analyze problems involving risk aversion, as alternative models could replicate current standard, EU-based results, while offering still more or avoiding the shortcomings associated with the assumption of EU maximization. 

• Consider standard findings associated with insurance: (1) A more risk averse person is willing to pay more for insurance (than is a less risk averse person); and (2) at a fixed price per dollar of insurance (fixed in excess of the actuarially fair price), a more risk averse person will purchase more insurance (than will a less risk averse person). 

• Consider standard findings associated with financial investments: (1) In a world with one safe (riskless) and one risky asset, more risk averse people invest less in the risky asset; and (2) if the population as a whole becomes more risk averse, the price of the risky asset must fall (equivalently, the expected return from holding the risky asset must rise). 

• Consider standard findings of principal/agent analysis, say, when a risk neutral principal hires a risk averse agent: (1) if the agent’s effort is not observable, then to encourage effort, the agent will have to bear some risk (so that lower output leads to less pay); and (2) the unobservability of effort is costly to the principal, who would prefer to contract on effort directly. 

• The various claims made concerning risk aversion in the three previous bullet points do require risk aversion – but they do not require expected utility maximization. That is, many of the ideas that have been developed around the concept of risk aversion – developed in the context of expected utility maximization – remain valid even when expected utility maximization is not descriptively accurate.

• Consider loss aversion as an alternative approach, one where outcomes are judged against a reference point and “losses loom larger than gains.” For prospects with some loss and some gain outcomes, loss aversion can generate risk averse behavior. (This style of loss aversion does not require "diminished sensitivity," the feature of prospect theory that leads to risk averse behavior in the gains domain and risk seeking behavior in the losses domain.)

• A second alternative, also featured in prospect theory, is probability weighting. The general notion is that, in practice, decision weights might not equal objective probabilities. Specifically, probability weighting typically involves the overweighting of low probability events and the underweighting of high probability events. This type of probability weighting can generate, depending on the options, either risk seeking or risk averse behavior. Lotteries, for instance, might be attractive (induce risk seeking behavior) due to the overweighting of the low-probability outcome of a large win. 

• Finally, consider contextual features and salience. Extreme or vivid outcomes (like deaths in terrorist attacks) might garner intense attention, leading to higher decision weights on those outcomes. The contextual feature of the available (although unchosen) options can exert influence by shifting the salience of other outcomes. Again, choices displaying risk aversion can arise from these factors. Expected utility maximization is neither necessary nor sufficient for explaining risk-averse behavior.

Vis and Kuijpers (2018) on Prospect Theory and Foreign Policy

Barbara Vis and Dieuwertje Kuijpers, “Prospect Theory and Foreign Policy Decision-Making: Underexposed Issues, Advancements, and Ways Forward.” Contemporary Security Policy 39(4): 575-589, 2018.

• Risk, in both prospect theory and in useful foreign policy applications, involves outcome uncertainty—so risk considerations are important in the gains domain as well as the loss domain. 

• Probability weighting often is ignored in applications of prospect theory, but it can override the usual “risk averse for gains, risk loving for losses” result. 

• In particular, low probability gains might see risk loving behavior, and low probability losses might be met with risk averse behavior. 

• And in foreign policy applications, low probabilities for unusual events are common. 

• Multiple dimensions are relevant in foreign policy decisions, so there can be multiple reference points, and outcomes might involve gains with respect to some reference points but losses with respect to other reference points. 

• When (to whom and for what decisions) might prospect theory apply? What decisions are better described by expected utility theory? 

• Oddly, ambiguity goes unmentioned in this article.

Professor Thaler’s American Economic Association Presidential Address

Richard H. Thaler, “Behavioral Economics: Past, Present, and Future.” American Economic Review 106(7): 1577–1600, 2016 (working paper pdf available here).

• Economics provides an approach to optimal decision making -- and that is well and good. But we should not let that model distract us from how people actually make decisions.

• When people make decisions, they make them as fallible Humans, not as textbook Econs. They remain fallible Humans irrespective of how often we are told that: (1) their decisions will look "as if" they are Econs; (2) their departures from the full Econ will be unsystematic; (3) when the stakes are high they will convert into Econs; (4) with time they will learn to be Econ; and, (5) the special magic of market settings will see to it that only Econs survive.

• Does the market "get prices right"? Consider the closed-end mutual fund with ticket symbol CUBA. Typically, CUBA is priced at about a 10-to-15 percent discount relative to its underlying assets. But after December 18, 2014, CUBA started to trade at a 70% premium over the value of its underlying securities, and premium pricing continued for about a year.

• Why? On December 18, 2014, President Obama announced that the US would normalize diplomatic relations with Cuba. The CUBA mutual fund has nothing to do with the country of Cuba.

• When Humans make decisions under uncertainty, the sort of preferences they display are not those of expected utility theory. Rather, many decisions seem to involve "prospect theory"-style preferences: (1) utility is based on changes in wealth from some reference point; (2) people are loss averse; and (3) people do not weight potential outcomes according to the objective probabilities.

• For intertemporal preferences, people often display a present bias, a taste for instantaneous gratification, and in many ways, do not exhibit exponential discounting.

• As with preferences, people also do not seem to hold fully rational beliefs. In particular, people display excessive optimism and excessive confidence in their beliefs.

• Actual choices are influenced by "supposedly irrelevant factors [p. 1595]," where the supposition of irrelevance is made within standard economic models. For instance, default settings tend to influence ultimate choices, even in high-stakes situations (such as retirement planning) where the defaults are less-than-optimal and easy to override.

• In the future, economic models will incorporate those behavioral features that best improve their predictive accuracy without imposing high costs in terms of complexity; "behavioral" will disappear as an adjective for a subset of economics, as all economics will be as behavioral as necessary.

The Ellsberg Paradox and Ambiguity Aversion

• OK, this is not really an outline of an article, but at least there is an urn involved. The urn has a total of 90 balls inside of it. Thirty of the balls are black, and the other 60 balls are either red or blue. (That is, anywhere between 0 and 60 of those balls are red, and the remainder of the non-black, non-red balls are blue.) A single ball will be pulled at random from the urn. 

• Situation A: You can choose Bet 1A, which pays $100 if the ball that is chosen is black. Alternatively, you can choose Bet 2A, which pays $100 if the chosen ball is red. Which bet do you prefer? [Spoiler alert: most folks prefer Bet 1A.]

• Situation B: You can choose Bet 1B, which pays $100 if the ball that is chosen is either black or blue. Alternatively, you can choose Bet 2B, which pays $100 if the chosen ball is either red or blue. Which bet do you prefer? [Spoiler alert: most folks prefer Bet 2B.]

• The modal choices in these hypothetical urn-related decision problems, already spoiled for you, are to choose Bet 1A and Bet 2B. 

• These modal choices are inconsistent with expected utility maximization. A person who (strictly) prefers Bet 1A to Bet 2A, and is an expected utility maximizer, must believe that the probability of choosing a black ball (here, precisely one-third) exceeds the probability of choosing a red ball. A person who (strictly) prefers Bet 2B to Bet 2A, and is an EU maximizer, must believe that the probability of choosing a black ball is smaller than the probability of choosing a red ball (because the probability of winning via the blue ball is the same in either alternative, 2A or 2B).

• The disposition that (presumably) leads to these modal choices is termed ambiguity aversion. In Situation A, the subject knows precisely the probability of winning Bet 1A, but is unsure of the probability of winning Bet 2A. In Situation B, the situation is reversed, with Bet 2B being the option with the known probability (precisely 2/3) of winning.

• The modal choices, inconsistent with expected utility maximization, are an example of what has become known as the Ellsberg Paradox, after the analysis given by Daniel Ellsberg in "Risk, Ambiguity, and the Savage Axioms," Quarterly Journal of Economics 75(4): 643-669, 1961 [pdf here]; Ellsberg's version is on pages 654-655. The version in this post follows closely the presentation in the Introduction (pages 3-4) by Adam Oliver in Behavioural Public Policy, Adam Oliver, ed., Cambridge University Press, 2013.

People are Not Exponential Discounters, and That’s OK

Some Notions Drawn, as I Recall, from Rabin (2002) and Frederick, Loewenstein, and O'Donoghue (2002)

• Would you rather have $20 now or $21 one week from now? If you choose the immediate $20 – a perfectly reasonable choice – then you discount monetary rewards by at least 5 percent per week. Would you rather have $20 now, or $250 one year from now? If you are an exponential discounter, and you preferred the immediate $20 in the initial situation, then you must prefer the immediate $20 to $250 one year hence, as 1.05 to the 52nd power is more than 12.6. If the question concerned two years from now, you would turn down $3100 in two years’ time for an immediate $20.

• Would you rather have $20 now or $22 one week from now? If you chose the immediate $20, then you discount by at least 10 percent per week. Would you rather have $20 now, or $2800 one year from now? If you are an exponential discounter, you must still want the $20, as 1.10 to the 52nd power is more than 140. In two years’ time, you’d turn down $400,000 (1.1 to the 104th power is more than 20,000) for an immediate $20. 

• Rabin showed that the sort of risk aversion over small-stakes gambles that most people display is inconsistent with expected utility theory, because such behavior would necessitate crazy choices for higher stakes gambles. What is demonstrated above is rather analogous, that the sort of time preference that people display for small stakes, short time-frame situations is not consistent with exponential discounting, because it would necessitate crazy choices for longer time-frame choices.

Abeler et al. (2011) on Reference Points and Effort Provision

Johannes Abeler, Armin Falk, Lorenz Goette, and David Huffman, “Reference Points and Effort Provision.” American Economic Review 101: 470-492, April 2011.

• An experiment is conducted in which people engage in a tedious and pointless task, but one that requires some attention. 

• The participants do not know with certainty how much they will be paid. They know that they will either receive a fixed fee (of which they are informed), or their accumulated, piece-rate earnings, each with equal probability. 

• The experiment varied only the fixed fee, which is either low (3 euros) or high (7 euros). The relevant choice for the worker is how long to work. 

• For an expected utility maximizer, the size of the fixed fee will not influence the decision about how long to work. (This claim requires the assumption that utility is separable in money and effort.) Even for a prospect theory decider, if the reference point is the status quo prior to the experiment, the size of the fixed fee will not influence the decision about how long to work. 

• If the worker is a prospect theory decider whose reference point is determined by the fixed fee – perhaps by fixing expectations of earnings – then the fixed fee size will influence the amount of work, as losses relative to the fixed fee will be quite aversive. 

• Sure enough, the participants worked longer when the fixed fee was higher. Further, the most common stopping point occurred when the accumulated earnings equaled the fixed fee, so the payment involved no uncertainty at all. 

• Workers whose responses to a series of questions suggest that they are particularly loss averse are also relatively more likely to stop working at the no-risk point.

Some Material Connected to Machina (1987)

Mark Machina, “Choice under Uncertainty: Problems Solved and Unsolved.” Journal of Economic Perspectives 1(1): 121-154, Summer 1987.

• A short review of choice under uncertainty: The general prospect (or lottery) is (x1, x2, …, xn; p1, p2, …, pn), where the xi’s are monetary outcomes and the pi’s are the associated probabilities.


• The expected value of the prospect (x1, x2, …, xn; p1, p2, …, pn) is p1x1+p2x2+…+pnxn = Σ pixi. 


• The expected utility of the prospect (x1, x2, …, xn; p1, p2, …, pn) is p1 U(x1) + p2 U(x2) +…+ pn U(xn) = Σ pi U(xi), where U(x) is the von Neumann-Morgenstern utility function defined over monetary outcomes xi. The standard model of choice under uncertainty is that a person will choose among prospects in such a manner as to maximize her expected utility.


• A person is risk averse if, when endowed with a riskless prospect, she always declines fair bets (bets that offer her the same expected value as her riskless prospect) – and this is equivalent to diminishing marginal utility of income.


• Machina (1987) and the Triangle Diagram: Fix the (three) dollar outcomes at x1, x2, x3; let x1<x2<x3. 


• With outcomes fixed but probabilities variable, every prospect (x1, x2, x3; p1, p2, p3) can be represented by a point in the unit simplex, which can be graphed as a triangle on p1-p3 axes (because p2 must equal 1-p1-p3, we only need a two-dimensional graph to indicate every prospect).


• Indifference curves for an expected utility maximizer will be linear in this space. Iso-expected value lines also will be linear. 


• We can use the diagram to speak about stochastic dominance; mean-preserving spreads; and risk preferences.


• Expected utility maximization requires that individual choices adhere to the "independence axiom": If the prospect P* is preferred to the prospect P, then the compound prospect aP* + (1-a)P’ is preferred to aP + (1-a)P’, for all prospects P’ and for all 0<a<1. 


• The independence axiom implies indifference curves that are linear in the probabilities, and hence, are straight, parallel lines within the triangle diagram. Nonetheless, many different choices seem to indicate that people have indifference curves that “fan out,” as opposed to being parallel lines.


• One common departure from the independence axiom (and hence from expected utility maximization) is the Allais Paradox (which can be neatly illustrated within the Triangle Diagram). Here's the setting:
  
  
                       Alternative 1                 Alternative 2
  
  
  Situation A:         ($1M;1)                           ($5M, $1M, $0; .1,.89,.01)
  
  
  Situation B:         ($1M,$0; .11,.89)          ($5M,$0; .1,.9)
  
  
  The "M" indicates that all dollar payoffs above involve millions of dollars. People typically choose alternative 1 in situation A, and alternative 2 in situation B. These two choices are inconsistent with expected utility maximization. [Why? To prefer alternative A1 to alternative A2, as an expected utility maximizer, you must have EU(A1) > EU(A2). This inequality can be rewritten as 1u($1) > .1u($5) + .89u($1) + .01u($0), and this inequality can be further simplified to .11u($1) > .1u($5) + .01u($0)*. If you also prefer  alternative B2 to alternative B1, then, as an expected utility maximizer, EU(B2) > EU(B1). This inequality can be rewritten as .1u($5) + .9 u($0) > .11u($1) + .89u($0), and this inequality can be further simplified to .1u($5) + .01u($0) > .11u($1), or equivalently, .11u($1) < .1u($5) + .01u($0). Compare this with inequality *; they contradict each other. Therefore, you cannot be an expected utility maximizer: there are no values for u($5), u($1), and u($0) such that your choices could be consistent with expected utility maximization.]

Kőszegi and Rabin (2006) on Reference-Dependent Preferences

Botond Kőszegi and Matthew Rabin, “A Model of Reference-Dependent Preferences.” Quarterly Journal of Economics 121(4): 1133-1165, 2006. 

·  Utility depends on the consumption bundle and on a reference bundle (u(cǀr) for riskless bundles). Consumption utility, here but one component of overall utility, is the standard utility of microeconomic theory. The other component, “gain-loss utility,” reflects the reference bundle. 

·  Gain-loss utility is assumed to be related to the difference in consumption utility between the bundle chosen and the reference bundle. The model allows for uncertainty both in consumption bundles and in reference bundles. 

·  The reference bundle is not the status quo; rather, it reflects recent beliefs (probabilistic) about outcomes. Fixing the chosen bundle, a “lower” reference bundle leads to higher utility. The endowment effect follows from loss aversion, since the disutility from loss (for an owner) exceeds the utility from gain for a non-owner. People who acquire items expecting to trade will not suffer from the endowment effect, because they have a different reference bundle. 

·  A “preferred personal equilibrium” (PPE) is that (unique) consistent equilibrium with the highest expected utility. PPE reduces to the standard model when there is no uncertainty; reference dependence plays no role in a deterministic environment, as there are no surprises. 

·  Willingness-to-pay for a good depends on the probability that you expect to buy the good and the price you expect to pay. An increase in the likelihood of buying makes for a greater reference “loss” in the event you don’t buy – the “attachment effect.” The more you expect prices to be so low that they will induce you to buy, the more willing you are to buy when the price is higher than expected!

·  The “comparison effect” holds the expected likelihood of purchase constant. In this case, a decrease in the price you expect to pay means that a medium price feels like more of a loss, lowering the willingness to pay the medium price. (You would not buy at high prices in any case). This effect involves a contradiction of sorts with the law of demand: lower expected prices can lead to diminished interest in purchasing. 

·  Taxi drivers and target wages: a driver learns her afternoon wage after she completes her morning shift. If the driver had unexpectedly high morning earnings, she is less likely to drive in the afternoon. Higher expected wages increase the likelihood of working and of staying through the afternoon. As in many other dimensions of economics, whether an event is anticipated or unanticipated leads to large effects on behavior.

Rabin and Thaler (2001) on Risk Aversion and the Failures of Expected Utility

Matthew Rabin and Richard H. Thaler, “Risk Aversion,” Journal of Economic Perspectives 15(1): 219-232, Winter 2001.

·  In the expected utility (EU) model, risk aversion is equivalent to diminishing marginal utility of income.

·  Any meaningful risk aversion over small stakes is inconsistent with EU maximization, as it requires a crazy unwillingness to take on risk at larger stakes. In other words, EU maximizers must be effectively risk neutral for small stakes, such as those in laboratory experiments. Nonetheless, people display risk aversion at small stakes.

·  Extended warranties and rental car insurance are purchased by many people, though the small stakes (relative to lifetime wealth) and high prices involved imply that they should be unattractive to expected utility maximizers.

·  Small-scale risk averse behavior is consistent with loss aversion, where the status quo is the reference point, and with mental accounting (narrow bracketing), a failure to look at the situation in the bigger picture.

·  Even the money pump argument offers more support for loss aversion and narrow bracketing than for expected utility maximization, as people are not reliably turned into money pumps. They are more likely to purchase those ill-advised warranties, for instance, when the warranties are tied to the purchase of the good itself, thereby promoting an isolated view of the transaction; for most decisions, consumers will not be so misled. An expected utility maximizer who bought such a warranty, however, would then necessarily agree to all sorts of ridiculous purchases.

·  The rate-of-return on stocks (as opposed to bonds, say) seems to be excessive, even though there should be some premium for holding stocks because stocks are riskier than bonds. This “equity premium puzzle” might be due to loss aversion and narrow bracketing. The day-to-day fluctuations in stock prices cause many short-term losses for shareholders. The equity premium comes from the fact that loss-averse investors require compensation to put up with all of the short term (mental accounting) losses.