Incentives and Behavior Prof. Dr. Heiner Schumacher KU Leuven 3. - - PowerPoint PPT Presentation

Incentives and Behavior

• Prof. Dr. Heiner Schumacher

KU Leuven

• 3. Risk Preferences I
• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 3. Risk Preferences I

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Introduction

In this lecture, we focus on risk preferences, i.e., how people deal with uncertainty. Risk preferences are central for many important decisions: …nancial decision making (Should I invest in stocks?), labor market decisions (What subject should I study?), environmental decisions (Should we invest into climate protection?). The challenge is to …nd a uni…ed explanation for several phenomena that seem to be at odds with rational choice.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 3. Risk Preferences I

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Introduction

Overview Expected Utility Theory Problems with EUT Prospect Theory

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 3. Risk Preferences I

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Expected Utility Theory

Expected utility theory is the standard model in economics that describes how individuals deal with risky situations. As an example, consider the following situation:1 Suppose you are contestant in the game-show “Who Wants to Be a Millionaire?”. You have made it until the 500.000 EUR question, but now you have no clue about the topic. You have saved the 50-50 option that rules out two answers, leaving two that turn out to be unfamiliar. You have two options: you can take the sure 500.000 EUR, or you can guess an answer so that you win 1.000.000 EUR with 50% chance and 32.000 EUR with the other 50%. What would you do?

1Taken from Holt, Charles (2007): Markets, Games, and Strategic Behavior, Addison

Wesley.

• Prof. Dr. Heiner Schumacher (KU Leuven)

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• 3. Risk Preferences I

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Expected Utility Theory

In EUT, risk preferences are given by a utility function U(x) that assigns to each outcome x a value of “utility”, i.e., the utility from the 1.000.000 EUR gain is U(1.000.000). The expected utility from a lottery (like the ones in the example above) is derived from the sum of possible utilities multiplied by the respective probabilities. For example, the expected utility from guessing an option is 1 2U(1.000.000) + 1 2U(32.000), and the expected utility from taking the sure 500.000 EUR is U(500.000). The lottery with the highest expected utility is the preferred lottery.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 3. Risk Preferences I

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Expected Utility Theory

Consider the expected monetary payo¤ from guessing an option: 1 21.000.000 + 1 232.000 = 516.000. If U(x) = x, then the individual would take the risk and take the

• ption (since 516.000 > 500.000). In this case, we say that the

individual is “risk neutral.” If U(x) = px, then the individual would choose the safe option: 1 2 p 1.000.000 + 1 2 p 32.000 590 < 707 p 500.000. Such an individual is called “risk averse.” In particular, an individual is risk averse if her utility function is concave, i.e., it exhibits “diminishing marginal utility.” What is the intuition behind such utility function?

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 3. Risk Preferences I

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Expected Utility Theory

Finally, if U(x) = x2, the individual would prefer a risky lottery to any safe lottery with the same expected monetary payo¤. Such an individual is called “risk loving” or “risk seeking”. Obviously, she prefers guessing an option in the game-show above to the safe

• utcome.

In particular, an individual is risk loving if her utility function is convex. Most individuals are risk averse, especially if stakes are large as the next example shows.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 3. Risk Preferences I

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Expected Utility Theory

Consider the following lottery (the “St.-Petersburg-Paradox”): A fair coin is ‡ipped repeatedly. If heads shows up for the …rst time at the n-th toss, you win 2n EUR. What is the expected payo¤ from this lottery? What would you be willing to pay for this lottery?

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 3. Risk Preferences I

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Expected Utility Theory

We introduce two terms that frequently appear in EUT. The “certainty equivalent” of a lottery is the amount of money CE so that the individual is indi¤erent between the lottery and the safe

• utcome CE. For example, if U(x) = px, then the certainty

equivalent for the lottery from above is de…ned by 1 2 p 1.000.000 + 1 2 p 32.000 = U(CE) = p CE ! CE 347.443. The “risk premium” RP of a lottery is the di¤erence between the lottery’s expected payo¤ and the certainty equivalent. In the example, we have RP = 1 21.000.000 + 1 232.000 CE 516.000 347.443 = 168.557. It is the amount of money the individual is willing to pay in order to get the lottery’s expected payo¤ for sure instead of the lottery.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 3. Risk Preferences I

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Expected Utility Theory

Expected Utility Theory is a simple model that describes how individuals make decisions under uncertainty. Its core features are that (i) the risk attitude only depends on the …nal outcome and the probability distribution over …nal outcomes, (ii) expected utility is linear in probabilities, and (iii) losses are treated in the same way as gains. It is used in all branches of economics. However, on the next slides we will see that EUT also has many problems.

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Problems of EUT

The Allais Paradox Consider the following two lotteries. Lottery A pays with 33% probability a gain of 2500 EUR, with 66% probability a gain of 2400 EUR, and with 1% probability a gain of 0 EUR. Lottery B pays with certainty a gain of 2400 EUR. Which lottery would you choose, A or B?

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Problems of EUT

Now consider the following two lotteries. Lottery A pays with 33% probability a gain of 2500 EUR, and with 67% probability a gain of 0 EUR. Lottery B pays with 34% probability a gain of 2400 EUR, and with 66% probability a gain of 0 EUR. Which lottery would you choose, A or B?

• Prof. Dr. Heiner Schumacher (KU Leuven)

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Problems of EUT

Most individuals prefer B to A and A to B. However, this behavior is inconsistent with EUT. We can show this formally. Many individuals stick to their behavior even if one explains their “inconsistency” to them. Could you imagine an intuitive explanation for this? The Allais-Paradox shows that people do not treat probabilities

• linearly. Consequently, expected utility theory does not make precise

predictions for all situations.

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Problems of EUT

An even more severe problem of EUT is that for small and intermediate lotteries it predicts risk neutral behavior. We know from experiments and …eld data that the typical decision maker is adverse to risk even when the lottery is small (that is, she rejects small lotteries with substantial positive expected payo¤). Rabin (2000) contains a theorem that describes the behavior of an agent with EUT-preferences under small and large stake lotteries: if she rejects a small lottery with positive expected payo¤, then this implies that she rejects a large lottery that is very advantageous for her.2 The table on the next slide (copied from Rabin 2000) contains some surprising numerical examples.

2Rabin, Matthew (2000): “Risk Aversion and Expected-Utility Theory: A Calibration

Theorem,” Econometrica 68(5), 1281 - 1292.

• Prof. Dr. Heiner Schumacher (KU Leuven)

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Problems of EUT

100 X=101 X=105 X=110 X=125 L gL,101 gL,105 gL,110 gL,125 400 400 420 550 1.250 600 600 730 990 36 Bil. 800 800 1.050 2.090 90 Bil. 1000 1.010 1.570 718.190 160 Bil. 2000 2.320 69.930 12.210.880 850 Bil. 4000 5.750 625.670 60.528.930 9,4 Tril. 6000 11.510 1.557.360 180 Mil. 89 Tril. 8000 19.290 3.058.540 510 Mil. 830 Tril. 10000 27.780 5.503.790 1,3 Bil. 7,7 Tril. 20000 85.750 71.799.110 160 Bil. 540 Quint. Interpretation: An agent with EUT-preferences who owns 300.000 EUR and who rejects the lottery (100 EUR; 0, 5; X EUR; 0, 5), also rejects the lottery (L EUR; 0, 5; gL,X EUR; 0, 5).

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 3. Risk Preferences I

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Prospect Theory

Many alternative risk preference models have been developed in order to avoid the problems of EUT. The most convincing one was created by two psychologists, Daniel Kahneman and Amos Tversky. It is called “Prospect Theory” (Kahneman and Tversky 1979, 1992).3 In 2002, Daniel Kahneman received the Nobel Price for this model. Today, more and more economists try to use Prospect Theory in their models (“try” because it is certainly less tractable than EUT). On the following slides, we introduce Prospect Theory.

3Kahneman, Daniel, and Amos Tversky (1979): “Prospect Theory: An Analysis of

Decision under Risk,” Econometrica 47(2), 263-292; Tversky, Amos, and Daniel Kahneman (1992): “Advances in Prospect Theory: Cumulative Representation of Uncertainty,” Journal of Risk and Uncertainty 5(4), 297-323.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 3. Risk Preferences I

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Prospect Theory

A motivating example Consider a student who can spend 20.000 EUR per year. After graduation, she gets a job where she earns enough to spend 60.000 EUR per year. Most likely, she will experience the di¤erence of 40.000 EUR as a gain. Now imagine the same person after some years. In the meantime, she works as a manager and earns enough to spend 100.000 EUR per

• year. However, her company (unexpectedly) goes bankrupt. In her

next job, she only earns enough to spend 60.000 EUR a year. How will she perceive the di¤erence of 40.000 EUR? Most likely, she will consider it as a loss. According to EUT, she should get the same utility from 60.000 EUR in both situations. However, we would expect that she is relatively happy in the …rst situation (after a gain), and relatively unhappy in the second situation (after a loss).

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• 3. Risk Preferences I

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Prospect Theory

According to Kahneman and Tversky’s (1979, 1992) prospect theory, utility is de…ned over gains and losses (relative to a neutral reference point) rather than …nal outcomes (as in EUT). In terms of the example, the student experiences the “prospected utility” v(40.000) in the …rst situation, and the prospected utility v(40.000) in the second situation. The reference point in the …rst situation is 20.000 EUR; and in the second situation, it is 100.000

• EUR. Clearly, we have v(40.000) > v(40.000).

In contrast utility would be u(60.000) under EUT-preferences in both situations.

• Prof. Dr. Heiner Schumacher (KU Leuven)

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• 3. Risk Preferences I

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Prospect Theory

Furthermore, prospect theory assumes that losses weight heavier than gains. In terms of the example above, this means that the individual is bene…ting less from spending 40.000 EUR more, than she su¤ers from cutting back her expenses by 40.000 EUR. As a consequence, the utility (or value) function has a kink in the

• rigin. Here is an example:

v(x) = xα if x 0 λ(x)β if x < 0 , where λ is the coe¢cient of “loss aversion” (its value has been estimated to be around 2). The utility function is concave in the domain of gains, and convex in the domain of losses.

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Prospect Theory

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Incentives and Behavior

• 3. Risk Preferences I

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Prospect Theory

Finally, utility is not linear in probabilities (as in EUT), but is weighted according to some decision weight. This captures the fact that many individuals overweight small probabilities and underweight intermediate probabilities. For example, the Allais paradox shows that the di¤erence between 0.99 and 1.00 has more behavioral impact than the di¤erence between 0.33 and 0.34.

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Prospect Theory

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