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 1 / 22
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 2 / 22
Introduction Overview Expected Utility Theory Problems with EUT Prospect Theory Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 3. Risk Preferences I 3 / 22
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? 1 Taken from Holt, Charles (2007): Markets, Games, and Strategic Behavior , Addison Wesley. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 3. Risk Preferences I 4 / 22
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 2 U ( 1 . 000 . 000 ) + 1 1 2 U ( 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 5 / 22
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 option (since 516.000 > 500.000). In this case, we say that the individual is “risk neutral.” If U ( x ) = p x , then the individual would choose the safe option: p p p 1 1 . 000 . 000 + 1 32 . 000 � 590 < 707 � 500 . 000 . 2 2 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 6 / 22
Expected Utility Theory Finally, if U ( x ) = x 2 , 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 outcome. 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 7 / 22
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 2 n 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 8 / 22
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 outcome CE . For example, if U ( x ) = p x , then the certainty equivalent for the lottery from above is de…ned by p p p 1 1 . 000 . 000 + 1 32 . 000 = U ( CE ) = CE ! CE � 347 . 443 . 2 2 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 9 / 22
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. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 3. Risk Preferences I 10 / 22
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 ? Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 3. Risk Preferences I 11 / 22
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) Incentives and Behavior 3. Risk Preferences I 12 / 22
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. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 3. Risk Preferences I 13 / 22
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. 2 Rabin, Matthew (2000): “Risk Aversion and Expected-Utility Theory: A Calibration Theorem,” Econometrica 68(5), 1281 - 1292. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 3. Risk Preferences I 14 / 22
Problems of EUT 100 X=101 X=105 X=110 X=125 L g L , 101 g L , 105 g L , 110 g L , 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 ; g L , X EUR ; 0 , 5 ) . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 3. Risk Preferences I 15 / 22
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