Second-best considerations in correcting cognitive biases.

1. Introduction

Suppose you know someone who is overly confident about his prowess in some realm of endeavor. Time after time, your friend has overestimated his ability on exams or his likelihood of success in business ventures. Surely, this kind of error reduces well-being, and you, as a good friend, should try to teach your friend to be less optimistic.

The theory of second best suggests that such an intervention would not necessarily help if your friend has other cognitive biases. In a system of interacting biases, the correction of any single one has ambiguous welfare implications. In their classic article on the theory of second best, Lipsey and Lancaster (1956, p. 12) write, "it is not true that a situation in which more, but not all, of the optimum conditions are fulfilled is necessarily, or is even likely to be, superior to a situation in which fewer are fulfilled." If your friend's decision-making is biased in other ways, then an intervention to reduce overconfidence may well reduce his welfare. Furthermore, the excessive confidence of your friend does not imply that there are ways to make him better off or even that his decision-making is suboptimal.

Studies in psychology and behavioral economics have identified several situations in which correction of a bias may lead to worse decisions. Camerer, Loewenstein, and Weber (1989) discuss the "curse of knowledge." In predicting the behavior of others, agents are unable to ignore the information they have. As a result, agents may become better off when they have less information. In a later article, Loewenstein, O'Donoghue, and Rabin (2003) find that projection bias may be counteracted by another bias in making choices about future consumption. In this setting, working to correct projection bias could lead people to make worse choices. Rabin (1999) offers an example in the context of saving, suggesting that overly high projections of consumption needs in retirement offset the undersaving problem. If so, the consumption-saving decision, though affected by cognitive biases, may still be correct. Kahneman and Lovallo (1993) consider biases and their effects within the firm. In the article most closely related to this one, Benabou and Tirole (2002) consider a formal model in which, among other effects, overconfidence ameliorates a lack of will power.

Though the previous literature has discussed interactions among cognitive biases, it has given little attention to their correction. Absent the knowledge of correct decisions or consideration of all relevant biases, interventions may reduce welfare. If there is selection on the quality of decisions made, and there are reasons to think that such selection exists, then one might even expect the decisions that emerge from the admittedly biased system to be in some sense "good." While these concerns are standard in second-best situations, their implications in a cognitive bias setting have not been addressed.

The results are formally demonstrated in the context of an optimization problem with no uncertainty. A person chooses the level of effort to expend in a project today that yields benefits at some near tomorrow. In the model presented, the individual differs from the standard neoclassical agent in that he has cognitive biases. In addition to overestimating the effects of his action, the individual is time-inconsistent and directly feels regret (or, equivalently, self-satisfaction) based on the level of effort provided. Changes in the strengths of these effects lead to changes in the agent's effort provision. Issues in the theory of second best arise when attempts are made to improve decision-making by ameliorating some, but not all, of the biases.

Any discussion of the second best requires a notion of the first best. For reasons given below, the first-best decision is taken to be the effort provided by an agent who is free from cognitive biases. (The parameterization of preferences admits such an agent as a special case.) Among the many biases discussed in the literature, the ones presented here were chosen not just because they are well established but also because of the insights they yield regarding the difficulty of welfare analysis in the presence of cognitive biases. Cognitive biases can range from problems of statistical inference and systematically incorrect expectations to preference-related issues such as reference points, framing, and regret. Thus, the argument that the agent's true preferences should be considered those of the unbiased agent is strongest for the case of overconfidence: The individual is just wrong about the result of a given level of effort on the outcome. For hyperbolic discounting, one can appeal to the fact that an agent who could commit to an action some time before having to bear the costs of it would choose to act in a time-consistent manner, even though it is arbitrary to claim that the individual's preferences at one time better represent his interests than those at another time. Even more questionable is ignoring the psychic cost of regret. Nonetheless, if one follows the standard formulation of utility as a function of consumption, then the full consequences of bad decisions are borne through their effects on consumption. Any other effect on utility, as through regret, is superfluous for purposes of maximizing consumption in the absence of other distortions. Considering consumption the ultimate aim of effort allows disregarding psychic payoffs and adopting the unbiased agent's utility as the true utility. Because this article aims to demonstrate interaction effects among biases, the choice of specific ones modeled is not crucial. Further, this article investigates the effects of correcting biases. If the cognitive biases represent the true preferences of the agent, there is no reason to change them.

Studying the interaction of cognitive biases sheds light on one of the vexing questions in the field of behavioral economics: Why don't individuals correct their biases? This article gives three related but separate answers to that question. First, it may be that individuals have limited knowledge of the system of interacting biases. In such a situation, the welfare consequences of correcting some subset of them are ambiguous. Second, when correction of biases is costless, if the dimensionality of the biases is greater than that of the decision (as in the model presented in this article), there may be a set of biases that result in the efficient level of action. An agent whose true welfare is that of an unbiased agent will be indifferent among any elements of the bias space that maximize utility. To an outside observer who can measure biases but does not know the efficient decision, the optimality of the agent's choice will be unrecognized. Third, even an individual with full information about the nature of the biases may rationally choose to correct them only partially when correction is costly. A researcher could observe an individual with partially corrected biases and not be aware that the decision-maker had already made the optimal correction.

Some authors have claimed that a particular bias exists in order to improve a decision. In such a vein, Lea and Webley (1997) discuss pride as a mechanism that ensures the consistency of decision. For example, they consider that a young girl who may find it in her short-term interest to become a prostitute will not do so because of pride. Thus, they conclude that pride may be considered adaptive for the long term. Prelec (1991) considers the pain of payment to have a functional role in counteracting biases that would otherwise lead to chronic overspending. Any such conclusion requires serious and careful study. The degree of simplicity with which biases are considered in this article precludes similar conclusions. Again, this article should not be interpreted as saying that the biases under study are adaptive in the sense that bias x exists because it counteracts bias y.

The next section reviews the relevant literature on the existence of the cognitive biases used in the article. The third section models the agent's decision-making as though the agent were trying to maximize a perceived utility function that incorporates cognitive biases, when, in actuality, the utility function of the agent is that of the standard, unbiased agent. The fourth section studies optimal bias correction. The fifth section makes functional form assumptions allowing an analytic solution to illustrate the correction of offsetting biases. Conclusions follow.

2. The Biases

A burgeoning literature describes the departures of human decision-makers from rationality. Time-inconsistency, regret, and overconfidence are singled out in this article. Endowment effects, loss aversion, failures in Bayesian updating, projection and hindsight biases, the availability and representativeness heuristics, and a host of others have also been documented but are not treated here (Earl 1990; Camerer 1995; Rabin 1998). Such departures are not surprising if the human mind is, as biological and psychological research suggests, a "large and heterogeneous network of functionally specialized computational devices" rather than a "general purpose computer" (Cosmides and Tooby 1994, p. 329). Although evolutionary considerations may be important in directing research, for the purpose of this article, the cause of the biases is not germane.

Hyperbolic Discounting

The notion that individuals value the present more strongly than the future in ways that can lead to inconsistent action plans seems clear in informal observation. Strotz (1955) observes that there are many examples of precommitment consistent with such preferences, including the piquant examples of furlough before beginning armed service and the customary period of engagement before marriage. Experiments have found that the discount rates of humans and animals for rewards decline quasi-hyperbolically. (Frederick, Loewenstein, and O'Donoghue 2002 offer a survey of departures from exponential discounting.)

One metaphor that has been offered for hyperbolic discounting is that of an individual at different times as different selves. The present self overweights current benefits and costs relative to the future. The specific functional form of discounting adopted follows Laibson (1997) in using the ([beta], [delta]) formulation of Phelps and Pollak (1968). Specifically, the utility of a consumption stream at time t can be written as follows:

[U.sup.t]([u.sub.t], [u.sub.t+1], ..., [u.sub.T]) = [[delta].sup.t][u.sub.t] + [beta] [T.summation over [tau]=t+1] [[delta].sup.[tau]][u.sub.[tau]],

with [beta], [delta] [member of] (0, 1]. In a study of the short term, an exponential discount factor ([delta]) near one is necessary for the medium or long term to have any effect on the agent's decision-making.

Regret

In the simplest sense, bad decisions are bad decisions because they have bad consequences. Not spending enough time with one's spouse may lead to divorce; not studying hard enough for a test may lead one to do poorly; having an additional drink may leave one with a headache the next day. People make these kinds of decisions, consider them mistakes, and regret them. But if the decisions are bad because of bad consequences, why should one suffer an additional utility loss via regret? Utility will be lower in the future because of the decision. Not only is regret superfluous in a standard model of decision-making under certainty and complete information, it could only be distortionary. Thus, for regret to have a useful instrumental role, some type of second-best effect is necessary.

Nonetheless, regret is a common and powerful emotion that has been shown to affect well-being. Ritov and Baron (1992) find that people prefer a bad outcome when it is not the result of an action they have taken, even when the outcome is the same. Individuals apparently prefer errors of omission than commission, evidently because of the psychic costs imposed by the accountability for active decision-making. In the model, regret is considered a disutility from not working hard, that is, as a level of self-satisfaction that increases in the level of effort provided.

Overconfidence

The overconfidence of individuals in their own efforts has long been recognized. Smith (1776) wrote in The Wealth of Nations that, "[t]he overweening conceit which the greater part of men have in their own abilities is an ancient evil remarked by the philosophers and moralists of all ages" (I.10.29). Studies that demonstrate overconfidence are legion. As Camerer (1997, p. 173) writes in a survey, "[d]ozens of studies show that people generally overrate the chance of good events, underrate the chance of bad events and are generally overconfident about their relative skill or prospects." In another survey, Babcock and Loewenstein (1997, p. 111) write, "well over half of survey respondents rate themselves in the top 50 percent of drivers, ethics, managerial prowess, productivity, and a variety of desirable skills." Fischoff, Slovic, and Lichtenstein (1977) find that people had sufficient confidence in their judgment of these kinds of issues to stake money on their validity. Overconfidence is considered in the model a systematic misperception of the effects of effort.

3. The Model

In the situation under study, an agent takes an action that is costly in the present and provides benefits in the future. Because the temporal scope of this problem is assumed to be short, discounting of the traditional, exponential sort is assumed negligible. The agent's perceived preferences over levels of effort can be described by a function that depends on a scalar level of effort, e, and that is parameterized by the biases. The vector [theta] [equivalent to] ([alpha], [beta], [gamma]) [member of] R x (0,1] x R represents overconfidence, present bias, and regret. Perceived utility is expressed by

U(e, [theta]) = -c(e) + [beta][[upsilon](e, [alpha]) + r(e, [gamma])],

where c(*) is the cost of effort, [upsilon](*,*) is the perceived benefit of effort, and r(*,*) is satisfaction with the choice or the absence of regret. The subutility functions are assumed to be twice continuously differentiable. For all 0, the cost of effort, c(*), is quasi-convex and the perceived benefit of effort, [upsilon](*,*), and the satisfaction with choice, r(*,*), are quasi-concave. An additional condition is that the curvature of at least one of the three subutility functions must be strict. Furthermore, the second cross-partial derivatives are assumed to be nonnegative. Intuitively, this means that the marginal effect, or perceived marginal benefit, of additional effort is monotonically increasing in each parameter. (1) The parameters are anchored, so that when [alpha] = [beta] = [gamma] = 1, the individual is equivalent to a standard, unbiased agent. For convenience, define [[delta].sup.1] = (1, 1, 1). Thus, for all e, [upsilon] (e, 1) is the actual benefit and, because the unbiased agent experiences no regret, r(e, 1) = 0.

The agent's problem is to maximize perceived utility with effort assumed to be taken from some admissible range. One can think of effort as, say, the amount of time in a given period spent on the task. The agent's problem is

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The question of interest is how the agent's decision depends on the biases. The following proposition establishes a few properties of the solution to the agent's perceived problem.

PROPOSITION 1. The maximized perceived utility function, [U.sup.*]([theta]), is continuous and optimal effort [e.sup.*]([theta]) is a continuous, nondecreasing function.

PROOF. The curvature assumptions on the subutility functions imply that the utility function is strictly quasi-concave in e for all [theta]. Further, the choice set is convex valued. By the Maximum Theorem, the maximized value function is continuous on [theta] and the optimal effort [e.sup.*]([theta]) is a continuous function on [theta] (e.g., Sundaram 1996). The comparative static result follows directly from Milgrom and Shannon's (1994) theorem 5. The real line is a lattice, [R.sup.3] is partially ordered, and the positive cross-partials satisfy increasing differences. The theorem's requirements are met and [e.sup.*]([theta]) is nondecreasing in [theta].

The unbiased agent's solution is denoted [e.sup.*]([[theta].sup.1]). For the masons given in the Introduction, this level of effort is considered optimal for all agents, even those with biases. The agent's true utility is then U([e.sup.*]([theta]), [[theta].sup.1]). Because in this model the optimal effort correspondence maps from [R.sup.3] to R, there is generically a large set of parameters [theta] [member of] [THETA], albeit of measure zero, for which effort level [e.sup.*]([[theta].sup.1]) is chosen and U([e.sup.*]([theta]),[[theta].sup.1]) = U([e.sup.*]([[theta].sup.1]),[[theta].sup.1]). Thus, the efficient action and maximal utility can be obtained even in the presence of the biases. The effects of changes in the cognitive biases on the true welfare of the agent are described by the behavior of U([e.sup.*]([theta]),[[theta].sup.1]). (The cause of such changes is ignored temporarily.) Whether a change in the biases 0 increases or decreases effort depends on their interaction. Depending on the initial values of the parameters, effort can be either under- or overprovided. If effort is underprovided, then marginally exacerbating a bias that results in overprovision as well as marginally ameliorating a bias that results in underprovision are both welfare enhancing. The opposite holds if effort is underprovided.

PROPOSITION 2. If [e.sup.*]([theta]) < [>][e.sup.*]([[theta].sup.1]), welfare is weakly enhanced by local increases (decreases) in overconfidence or regret or by local decreases (increases) in present bias.

PROOF. The parametric monotonicity of [e.sup.*] implies that increases in any parameter raise the agent's effort choice. By definition of a quasi-concave function, for all x, y [member of] ([e.bar], [bar.e] with x [not equal to] y and for all [lambda] [member of] (0, 1), U([lambda]x + (1 - [lambda])y, [theta] [sup.1]) > min{U(x, [[theta].sup.1]), U(y, [[theta].sup.1])}. Consider the cases in which [e.sup.*]([theta]) [not member of] {[e.bar], [bar.e]} and [e.sup.*]([theta]) < [e.sup.*]([[theta].sup.1]). Then U(e) > U([e.sup.*]([theta])) for any e [member of] ([e.sup.*]([theta]), [e.sup.*]([[theta].sup.1])). Any parametric increase that causes [e.sup.*] to increase without exceeding [e.sup.*]([[theta].sup.1]) will thus increase welfare. Above, we assume that [e.sup.*]([theta]) [not member of] {[e.bar],[bar.e]}. The proposition holds weakly because the effort choice may be a boundary value both before and after a parametric change, in which case the true utility would not be changed. The same argument holds if [e.sup.*]([theta]) > [e.sup.*]([[theta].sup.1]) for decreases in the parameters.

There is no general relation between [e.sup.*]([[theta].sup.1]) and [e.sup.*]([theta]) for some [theta] as it becomes closer to or farther from [[theta].sup.1]. There are conditions, though, under which statements can be made.

DEFINITION 1. Biases offset each other if they have opposite effects on the maximized value of the choice variable.

In this model, hyperbolic discounting offsets both overconfidence and regret. When multiple offsetting biases are increased, the resulting change in effort is ambiguous with respect to the optimal level. Only when offsetting biases are corrected in opposite directions can there be a statement about the effect on effort. An example of offsetting biases is offered in section 5.

4. Bias Manipulation

Suppose that there is a perfectly informed, benevolent educator who cares about the welfare of the decision-maker and the cost of correcting biases. The objective function for the educator is assumed to correctly characterize the real resource costs of the biases and their corrections in the following manner:

W([theta], [[theta].sup.0]) = U([e.sup.*]([theta]), [[theta].sup.1]) - C(d([theta], [[theta].sup.0])),

where [[theta].sup.0] represents the decision-maker's original biases, d(*,*) is distance, and C(*) is continuous and monotonically increasing. The other restrictions placed on C(*) are that C(0) = 0 and [lim.sub.d([theta],[[theta].sup.0) [right arrow] [infinity]] C(*) = [infinity].

The educator's problem is to choose the parameters efficiently, that is,

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PROPOSITION 3. A solution to the educator's problem exists, and it weakly increases the agent's welfare.

PROOF. Recognize first that the objective function is continuous in the choice variable because both the benefit function and cost function are continuous by assumption. Artificially compactify the choice set in the following way. Consider the choice [[theta].sup.1] that results in the value W([[theta].sup.1], [[theta].sup.0]). Because this choice is feasible, it is a lower bound on the maximum value attainable by the educator. The cost of changing the biases to the unbiased level is C([[theta].sup.1], [[theta].sup.0]). The educator's choice can be constrained to the following admissible set: [[THETA].sup.a] = {[theta] [member of] [THETA] | C([theta], [[theta].sup.0]) [less than or equal to] C([[theta].sup.1], [[theta].sup.0])}. This set is bounded beca the restrictions on C(*,*), and is closed, so it is compact. Thus, a solution, possibly set valued, to the problem exists. Because the educator can always maintain the status quo at zero cost, any change that entails costs must increase the agent's welfare by more than those costs. QED.

The framework above assumes that the problem is known with certainty to the educator. The educator might not know the full extent of cognitive biases or the functional form of the decision-making process. One way of describing this uncertainty would be with a distribution over different combinations of biases, but doing so requires knowledge of what the biases are.

The sophisticated agent could act as her own educator in the following two-stage game. In the first stage, the agent would choose the bias corrections to maximize her true utility given the decision that she will make in the second stage. This might correspond to reading self-help books or taking a class in decision-making. In the second stage, the agent would choose effort. When the biases affect preferences, this implies that the agent would be expending resources to change them, which is not rational. However, the true utility function can be considered meta-preferences, and the resource use as part of a plan to maximize over the meta-preferences.

5. An Example

Assumption of functional forms for utility allows the illustration of two main points of this article, namely, the second-best nature of the bias correction problem and the potential observational equivalence of agents with uncorrected or optimally corrected biases.

The agent's perceived problem is assumed to take the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

From the first-order condition of the agent's maximization, an interior solution is characterized by the following equation:

(1) [e.sup.*]([theta]) = [beta]([alpha] + [gamma] - 1).

A boundary solution is also possible. Notice that the solution for the traditional agent ([e.sup.*]([[theta].sup.1])) is equal to one.

Consider an arbitrary case where [[theta].sup.0] = ([[alpha].sup.0], [[beta].sup.0], [[gamma].sup.0]) = (1.5, 0.5, 1.3). Equation 1 reveals that the agent's effort choice equals 0.9 < 1. In this case, even though the agent is overconfident and regrets low provision of effort, the present bias involved with hyperbolic discounting results in an effort provision below optimal. The true utility equals 0.495, which is less than the optimum of 0.5. The second-best nature of the problem can be seen in the context of the discussion in the Introduction. Even though your friend is overconfident, reducing the overconfidence will reduce his already too-low effort provision and make him even worse off than he would otherwise have been.

The second issue is the observational obscurity of optimality. Optimal correction, either with or without correction costs, can be consistent with cognitive biases. In other words, an individual may have cognitive biases and yet still be optimizing. This can be seen in the context of the educator's problem,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

where d(*,*) denotes Euclidean distance. Given the initial values of the parameters, numerical search yields a solution of [[theta].sup.*] = (1.51, 0.55, 1.31). This results in effort provision e([[theta].sup.*]) = 0.97. Note that this is still below the effort of the unbiased agent. An outside observer unaware of the costs of correction would not know that the biases had been optimally corrected. Further, even the observer aware of the costs of correction but ignorant of the form of the agent's preferences would have no way of knowing whether the correction was optimal.

The final point is that, because the biases are offsetting, an observer might not recognize that an agent was making the optimal effort provision even if the agent were doing so. Suppose the observer could measure each of the biases and knew that the true utility was characterized by [[theta].sup.1] = (1, 1, 1) but could not calculate the optimal effort. If the observer then assayed the biases and found that [theta] = (1.52, 0.55, 1.31), then he or she would think that the agent was diverging from the optimum. But [e.sup.*]([theta]) = 1 (and is, in fact, the closest point in the parameter space to [[theta].sup.0] that achieves the optimum), so the agent is, in fact, exerting the optimal level of effort. If we define a sophisticated agent as one that knows her true utility function, then any vector in the cognitive bias space that maps to the optimal effort is as good as the unbiased one. In this example, there is, in the positive orthant of the bias space, a two-dimensional manifold, every point of which maps to the efficient effort level. Generally, a sophisticated agent, one that knows her true utility function, is indifferent between any parameter that yields the efficient solution.

6. Conclusion

The biases studied in this article were selected for the strength of evidence of their existence and also because the interactions among them could be represented simply. Not only are the effects of these cognitive biases far more complicated than modeled, but there may also be other offsetting biases in effort provision. Additional real-world complexity only strengthens the importance of explicit second-best considerations.

The biases were also chosen to illustrate the subtleties involved in saying that a biased agent chooses suboptimally. As discussed in the Introduction, such arguments are forthright when the agent is simply mistaken, reasonable when there are temporary preference changes, and strained when they involve true psychic costs. For the purpose of illustrating cognitive biases, one could select among pure statistical inference problems such as the misweighting of polar probabilities, the conjunctive fallacy, etc.

The system of interacting biases studied in this article gave rise to three main points. First, in the presence of other biases, the correction of any single one has ambiguous welfare properties. Second, when correction of biases is costly, full correction will generally not occur, so that the observation of biases does not imply potential welfare gains. Third, if agents can costlessly change their biases and know their true utility function, there is no reason for them to choose to be unbiased if there are other sets of biases that lead to the efficient effort choice. Both of the latter points suggest that an outside observer must know a great deal about cognitive biases, their effect on decision-making, and their cost of correction in order to make statements about whether a particular bias should be corrected.

Of these three points, both the first and second would continue to hold in a setting where an agent had a multidimensional effort decision. While the interactions among particular biases would no longer be characterized generally, it should still be possible to define biases as offsetting with respect to specific decisions. With more dimensions of choice, the effect on welfare from changing a bias would only become more difficult to determine. Previous discussions of cognitive biases have lamented the tendency for individuals to correct a particular decision without learning a general role to eliminate the relevant biases. The second-best perspective suggests that improving decisions one at a time may prove a useful strategy.

My thanks to Jonathan Baron, Brent Goldfarb, Avner Greif, Jonathan Hamilton, Thomas Nechyba. Dina Older-Aguilar, Matthew Rabin, participants at the 1999 RSF Conference on Behavioral Economics, and three anonymous reviewers for helpful conversations and comments.

(1) Regret is modeled as entering the utility function through positive self-satisfaction so that the cross-partial derivatives have the same sign. Because the perceived utility function does not reflect the agent's welfare, its properties matter only insofar as they affect the argument maximum of the expression. The effects of changes in the parameters on the value of the function itself are immaterial.

References

Babcock, Linda, and George Loewenstein. 1997. Explaining bargaining impasse: The role of self-serving biases. Journal of Economic Perspectives 11:109-26.

Benabou, Roland, and Jean Tirole. 2002. Self-confidence and personal motivation. Quarterly Journal of Economics 117: 871-915.

Camerer, Colin. 1995. Individual decision making. In The handbook of experimental economies, edited by John H. Kagel and Alvin E. Roth. Princeton: Princeton University Press, pp. 587-703.

Camerer, Colin. 1997. Progress in behavioral game theory. Journal of Economic Perspectives 11:167-88.

Camerer, Colin, George Loewenstein, and Martin Weber. 1989. The curse of knowledge in economic settings: An experimental analysis. Journal of Political Economy 97:1232-54.

Cosmides, Leda, and John Tooby. 1994. Better than rational: Evolutionary psychology and the invisible hand. American Economic Review 84:327-32.

Earl, Peter E. 1990. Economics and psychology: A survey. The Economic Journal 100:718-55.

Fischoff, Baruch, Paul Slovic, and Sarah Lichtenstein. 1977. Knowing with certainty: The appropriateness of extreme confidence. Journal of Experimental Psychology: Human Perception and Performance 3:552-64.

Frederick, Shane, George Loewenstein, and Ted O'Donoghue. 2002. Time discounting and time preference: A critical review. Journal of Economic Literature 40:351-401.

Kahneman, Daniel, and Dan Lovallo. 1993. Timid choices and bold forecasts: A cognitive perspective on risk taking. Management Science 39:17-31.

Laibson, David. 1997. Golden eggs and hyperbolic discounting. Quarterly Journal of Economics 62:443-77.

Lea, Stephen, and Paul Webley. 1997. Pride in economic psychology. Journal of Economic Psychology 18:323-40.

Lipsey, Richard, and Kelvin Lancaster. 1956. A general theory of second best. Review of Economic Studies 24:11-32.

Loewenstein, George, Ted O'Donoghue, and Matthew Rabin. 2003. Projection bias in predicting future utility. Quarterly Journal of Economics 118:1209-48.

Milgrom, Paul, and Chris Shannon. 1994. Monotone comparative statics. Econometrica 62:157-80.

Phelps, Edmund, and Robert Pollak. 1968. On second-best national saving and game theoretic equilibrium growth. Review of Economic Studies 35:185-97.

Prelec, Drazen. 1991. Values and principles: Some limitations on traditional economic analysis. In Socio-economics: Toward a new synthesis, edited by Amitai Etzioni and Paul Lawrence. New York: M.E. Sharpe, pp. 131-45.

Rabin, Matthew. 1998. Psychology and economics. Journal of Economic Literature 36:11-46.

Rabin, Matthew. 1999. Comment. In Behavioral dimensions of retirement economics, edited by Henry Aaron. Washington, DC: Brookings Institution Press and Russell Sage Foundation, pp. 247-51.

Ritov, Ilana, and Jonathan Baron. 1992. Status-quo and omission biases. Journal of Risk and Uncertainty 5:49-61.

Smith, Adam. 1776. The Wealth of Nations. London: Strahan and Cadell.

Strotz, Robert. 1955. Myopia and inconsistency in dynamic utility maximization. Review of Economic Studies 23:165-80.

Sundaram, Rangarajan. 1996. A first course in optimization theory. New York: Cambridge University Press.

Gregory Besharov, Department of Economics, Duke University, 305 Social Sciences Building, Durham, NC 27708, USA; E-mail [e-mail].

Received November 2002; Accepted October 2003.