a fundamental rule in statistics relating to conditional and marginal associated with each outcome 3 4,000 and if he loses the gamble, his income will fall to Rs. RISK AVERSION AND EXPECTED-UTILITY THEORY: A CALIBRATION THEOREM BY MATTHEW RABIN1 1. 1500) and H corresponding to income of Rs. As shall be explained below, for a risk averse individual marginal utility of money diminishes as he has more money, while for a risk-seeker marginal utility of money increases as money with him increases. Now, in a risky job when income increases to Rs. The decision made will also depend on the agent’s risk aversion and the utility of other agents. We will analyse below how an individual maximises his expected utility when risk or uncertainty is present. Risk Lover On the other hand, a person is risk-preferred or risk-loving who prefers a risky outcome with the same expected income as a certain income. In the guar­an­teed sce­nario, the per­son re­ceives $50. The person who refuses a fair bet is said to be risk averse. But the outcomes or payoffs are measured in terms of utility rather than rupees. 15000. In a world of uncertainty, it seems intuitive that individuals would maximize expected utility A construct to explain the level of satisfaction a person gets when faced with uncertain choices. We also learn that people are risk averse, risk neutral, or risk seeking (loving). 15,000 [E(x) = 0.5 x 0 + 0.5 x 30,000 = 15000], Note again that Figure 17.3 we are considering the choice of a risk averse individual for whom marginal utility of money declines as he has more of it. The gain in utility from Rs. 30 thousands is 120 units. Thus, the risk averter is one who prefers a given income with certainty to a risky gamble with the same expected value of income. 17.4. Our mission is to provide an online platform to help students to discuss anything and everything about Economics. Risk aversion and its equivalence with concavity of the utility function (Jensen’s inequality) are explained. In Bernoulli's formulation, this function was a logarithmic function, which is strictly concave, so that the decision-mak… 17.3 we have drawn a curve OU showing utility function of money income of an individual who is risk-averse. In conventional expected utility theory, risk aversion comes solely from the concavity of a person’s utility defined over wealth levels. Share Your Word File 3000. A per­son is said to be: 1. risk-aver… Johnny’s risk aversion over the small bet means, therefore, that his marginal utility for wealth must diminish incredibly rapidly. 1,500. In the first gamble, the degree of variability of outcome is less and therefore the risk is less and in the second gamble, the degree of variability is greater which makes it more risky. 3,000, the expected value of the utility is M2D (= 62.5) which is less than M2C or Rs. INTRODUCTION USING EXPECTED-UTILITY THEORY, economists model risk aversion as arising solely because the utility function over wealth is concave. In the previous section, we introduced the concept of an expected utility function, and stated how people maximize their expected utility when faced with a decision involving outcomes with known probabilities. Some other individuals are indifferent toward risk and are called risk-neutral. Further the N-M utility curve shown in Figure 17.6 is concave which shows the marginal utility of income of a person diminishes as his income increases. Risk aversion is the most common attitude towards risk. 70 which is the utility of income of Rs. The expected utility of the new risky job is given by. In other words, most individuals seek to minimise risk and are called risk averter or risk averse. Now the expected utility from the new risky job is less than the utility of 55 from the present job with an assured income of Rs. 30 thousand per month but if he does not happen to be a good salesman his income may go down to Rs. As mentioned above, most of the individuals are risk averse but there is a good deal of evidence of people who are risk seekers. It will be seen from Fig. First, a 50:50 chance of winning or losing Rs. 3.3. vNM vNM expected utility theoryexpected utility theory a)a) Intuition Intuition [L4] b) A i ti f d tiAxiomatic foundations [DD3] 4.4. 4,000, his utility rises to 75. 4 Risk Attitudes in the Jeffrey Framework 4.1 Linearity, chance neutrality, and risk aversion 4.2 Distinguishing risk attitudes 20,000 in the present case), is equal to utility of an assured or a certain income. But it is important to note that these different preferences toward risk depend on whether for an individual marginal utility of money diminishes or increases or remains constant. Note that we measure money income on the X-axis and utility on the Y-axis. 15,000 (Note that in the risky job also, expected income is Rs. The underlying principles of making a choice in risky and uncertain situation, namely, expected return and the degree of risk involved apply equally well to other choices. With money income of Rs. C. Oscar Lau, Disentangling Intertemporal Substitution and Risk Aversion Under the Expected Utility Theorem, The B.E. This attitude of risk aversion can be explained with Neumann-Morgenstern method of measuring expected utility. 30 thousands is 75, and if he fails as a good salesman, his income falls to Rs 10 thousands which yields him utility of Rs. On the N- M utility curve U (I) in Figure 17.6 we draw a straight line segment GH joining point G (corresponding to income of Rs. Suppose in this new job there is 50-50, chance of either earning Rs. Expected utility is introduced. The consumer is expected to be able to rank the items or outcomes in terms of preference, but the expected value will be conditioned by their probability of occurrence. It follows from above that in case marginal utility of money income diminishes a person will avoid fair gambles. Suppose there is a $50-50$ chance that a risk-averse individual with a current wealth of $\$ 20,000$ will contact a debilitating disease and suffer a loss of $\$ 10,000$ a. 3,000, two fair gambles are offered to him. With the even chance of winning and losing the expected value of income in the second gamble will be 1/2(1500) + 1/2 (4500) = Rs. An individual will be risk neutral if his marginal utility of money income remains constant with the increase in his money. In the questionnaire, Question 2 asked you to choose from a pair of lotteries A, B defined “The attitude toward risk we will consider a single composite commodity, namely, money income. Now, if he is offered a risky job with his income of Rs. Further, in case of new risky job if he is proved to be a successful salesman and his income increases to Rs. 20,000). 20 thousands, the risk-loving individual will prefer the new risky job even though the expected income in the new risky job is also Rs. 3000. Risk aversion coefficients and Risk aversion coefficients and pportfolio choice ortfolio choice [DD5,L4] 5. 20 thousands is 43 units to this individual. Therefore, the utility curve in Figure 17.6 represents the case of a risk averter or the attitude of risk aversion. If he wins the game, his income will rise to Rs. Therefore. It will be seen from Fig. Given that the probability of success or failure as a salesman is 0.5, the expected utility of the new job is given by. In case of risk-neutral individual marginal utility of money remains constant as he has more money. 2 Consider the link between utility, risk aversion, and risk premia for particular assets. Risk-averse investors also are known as conservative investors. Most individuals generally prefer the less risky situation (that is, the situation with less variability in outcomes or rewards). A fair game or gamble is one in which the expected value of income from a gamble is equal to the same amount of income with certainty. 4,000) by a straight line segment AB and then reading a point on it corresponding to the expected value of the gamble Rs. Such a person is called risk averter as he prefers an income with certainty (i.e., whose variability or risk is zero) to the gamble with the same expected value (where variability or risk is greater than zero). 4000) + 1 – π U (Rs. 2,000) and point B (corresponding to Rs. 15,000 [E(x) = 0.5 x 0 + 0.5 x 30,000 = 15000], Note again that Figure 17.3 we are considering the choice of a risk averse individual for whom marginal utility of money declines as he has more of it. An individual’s money income represents the market basket of goods that he can buy. Suppose there is a $50-50$ chance that a risk-averse individual with a current wealth of $\$ 20,000$ will contact a debilitating disease and suffer a loss of $\$ 10,000$ a. It is seen from above that in case of risk-neutral person expected utility of an uncertain income with the same expected value (Rs. Thus, the probability of his winning is 1/2 or 0.5. Thus the person will prefer the first gamble which has lower variability to the second gamble which has a higher degree of variability of outcome. As will be seen from Figure 17.6 the utility of the person from Rs. Therefore, the person will refuse to accept the gamble (that is, he will not gamble). a risk-averse agent always prefers receiving the expected outcome of a lottery with certainty, rather than the lottery itself. 17.3 that as money income of the individual increases from 10 to 20 thousand rupees, his total utility increases from 45 units to 65 (that is, by 20 units) and when his money increases from 20 thousand to 30 thousand rupees, his total utility increases from 65 to 75 units (that is, by 10 units). That is why his expected utility from the uncertain income prospect has been found to be lower than the utility he obtains from the same income with certainty. (Note that in the new risky job, the expected income is 20,000 which is given by E(X) = 0.5 x 10,000 + 0.5 x 30,000 = Rs. Privacy Policy3. It will be seen from the utility function curve OU in Fig. 3,000 with certainty. 30,000, double the present assured income of Rs. In this section we focus on examining individual’s choices in the face of risk. Now suppose the person’s current income is Rs. It is assumed that the individual knows the probabilities of making or gaining money income in different situations. Suppose to our person with a certain income of Rs. When there is uncertainty, the individual does not know the actual utility from taking a particular action. We saw earlier that in a certain world, people like to maximize utility. KÛ^áî٘ä3h=kßv$“ó™€Ó9Ã.®»Ž:„M“([!¤ð›ò{òí-;?ÍDË)«˜Meëé[ i§Ì The total utility function of a risk neutral person is shown in Fig. Under expected utility, risk aversion in the Arrow-Pratt sense implies rejection of gambles with mean-independent risk. 15,000 with no uncertainty is 55 whereas the expected utility of the new job or salesman on commission basis is 60. We assume that there is equal probability of high and low income in the new risky job. It is because of the attitude of risk aversion that many people insure against various kinds of risk such as burning down of a house, sudden illness of a severe nature, car accidence and also prefer jobs or occupations with stable income to jobs and occupations with uncertain income. 30 thousands if he proves to be a successful salesman, the utility of Rs. The ex­pected pay­off for both sce­nar­ios is $50, mean­ing that an in­di­vid­ual who was in­sen­si­tive to risk would not care whether they took the guar­an­teed pay­ment or the gam­ble. The expected money value of his income in this situation of uncertain outcome is given by: E (V) = 1/2 x 4000 + 1/2 x 2000 = Rs. Expected Utility and Risk Aversion – Solutions First a recap from the question we considered last week (September 23), namely repre-senting in the probability triangle diagram the version of the Allais paradox we came across in the questionnaire. It is assumed that the individual knows the probabilities of making or gaining money income in different situations. Let us now slightly change the data. Share Your PDF File This means, in turn, that even the 3,000) with certainty. In case of a risk-loving individual, marginal utility of income to the individual increases as his money income increases as shown by the convex total utility function curve OU in Fig. TOS4. 17.4 that the utility of Rs. It may be noted that marginal utility of income of a risk-averter diminishes as his income increases. Share Your PPT File, Risk Aversion and Insurance (Explained With Diagram). 20,000). 2,000 if he loses) can be obtained as under: Expected Utility (EU) = π U (Rs. Since the expected utility from the new risky job is 51.5 which is greater than the utility of 43 from the present job with a certain income of Rs. However, some individuals prefer risk and are therefore called risk-seekers or risk lovers. Suppose that if the individual in his new job proves to be successful and earns Rs. Now, suppose that the individual is considering to join a new job of a salesman on a commission basis. 3 Risk-Weighted Expected Utility Theory 3.1 Risk-weighted expected utility versus expected utility 3.2 Problems with risk-weighted expected utility theory. Prudence coefficient and precautionary savingsPrudence coefficient and precautionary savings [DD5] 6.6. But given the probabilities of alternative outcomes, we can calculate the expected utility. A per­son is given the choice be­tween two sce­nar­ios, one with a guar­an­teed pay­off and one with­out. Further, according to expected utility theory, risk aversion derives from the curvature of the utility of money, so such experiment would require to vary the stakes of the lotteries proposed in order to trace out the shape of the utility of money. It will be seen from this figure that the slope of total utility function OL; decreases as the money income of the individual increases. 17.3 that the utility of money income of Rs. The von Neumann–Morgenstern utility function can be used to explain risk-averse, risk-neutral, and risk-loving behaviour. expected utility questions differentiate between the following terms/concepts: prospect and probability distribution risk and uncertainty utility function and Specifying Risk-Aversion through a Utility function We seek a \valuation formula" for the amount we’d pay that: Increases one-to-one with the Mean of the outcome Decreases as the Variance of the outcome (i.e.. Risk) increases ... To maximize Expected Utility of Wealth W = W 1 (at time t = 1) It should be remembered that risk in this connection is measured by the degree of variability of outcome. With Rs. Certainty equivalents are defined. Disclaimer Copyright, Share Your Knowledge There are multiple measures of the risk aversion expressed by a given utility function. 10,000 whose utility to the individual is 40 units. Suppose this risk-loving individual has a present job with a certain income of Rs. • Expected utility allows people to compare gambles • Given two gambles, we assume people prefer the situation that generates the greatest expected utility – People maximize expected utility 18 Example • Job A: certain income of $50K • Job B: 50% chance of $10K and 50% chance of $90K • Expected income is the same ($50K) but in one case, 10 thousands to this individual is Rs. 20 while utility of Rs. 20 thousands. We are now in a position to provide a precise definition of risk-averse individual. The concepts of relative risk aversion, absolute risk aversion, and risk tolerance are introduced. It will be seen from Fig. It will be seen from this straight-line segment GH that the expected utility from the expected money value of Rs. Welcome to EconomicsDiscussion.net! And in case of income with certainty there is no variability of outcome and therefore involves no risk at all. ), thedegeneratelotterythat placesprobabilityone on the mean of Fis (weakly) preferred to the lottery Fitself. 3,000. People differ greatly in their attitudes towards risk. Now the expected utility from the new risky job is less than the utility of 55 from the present job with an assured income of Rs. In the various earlier theories of consumer’s behaviour we saw that in making choices among commodity bundles when there is no risk and uncertainty, the consumer maximises his utility. The utility function OU with a diminishing marginal utility of money income of a risk- averse individual is shown in Fig. Whether the individual will choose the new risky job or retain the present salaried job with a certain income can be known by comparing the expected utility from the new risky job with the utility of the current job. The notion of local risk aversion is introduced in general and with respect to the expected utility case, where again it is equivalent to concavity of utility function. 4,000 if he wins and Rs. Risk aversion is the most common attitude toward risk. Expected utility is shown to imply second‐order risk aversion. 20 thousands is 80. 20,000 as (0.5 x 10,000) + 0.5 (30,000) = Rs. 15,000 (Note that in the risky job also, expected income is Rs. This is because as he acts on the basis of expected utility of his income in the uncertain situation (that is, Rs. Journal of Theoretical … 1,000. But the outcomes or payoffs are measured in terms of utility rather than rupees”. 1,000 in case he wins is less than the loss in utility from Rs. So an expected utility function over a gamble g takes the form: u(g) = p1u(a1) + p2u(a2) + ... + pnu(an) where the utility function over the outcomes, i.e. Though the individuals is risk-averse as revealed by the nature of his utility function of money income, but since the expected utility of the risky job is greater than the utility of the present job with a certain income he will choose the risky job. 30 thousands or Rs. This new job involves risk because his income in this case is not certain. 2,000. 45. 10 thousand per month. Risk aversion is equivalent to concavity of utility function if the expected utility theory holds. In the un­cer­tain sce­nario, a coin is flipped to de­cide whether the per­son re­ceives $100 or noth­ing. Precisely speaking, a person who prefers a certain given income to a risky job with the same expected income is called risk averter or risk-averse. 17.5. 17.4 that the utility of Rs. Risk Aversion, Certainty Equivalent, and Risk Premium If preferences satisfy the vNM axioms, risk aversion is completely characterized by concavity of the utility index and a non-negative risk-premium. Though the expected value of his uncertain income prospect is equal to his income with certainty a risk averter will not accept the gamble. This is because if he proves to be a successful salesman his income may increase to Rs. Several functional forms often used for utility functions are expressed in terms of these measures. Content Guidelines 2. 10 thousands (that is, each has a probability of 0.5). 3,000 and he is offered a fair gamble in which he has a 50-50 chance of winning or losing Rs. . Thus with the present job with a fixed salary of Rs. 17.7. Before publishing your Articles on this site, please read the following pages: 1. There is no uncertainty about the income from this present job on a the fixed salary basis and hence no risk. It will be seen from this figure that utility of a certain income of Rs. It will be seen from this figure that N- M utility curve starts from the origin and has a positive slope throughout indicating that the individual prefers more income to less. It is risk-loving individuals who indulge in gambling, buy lotteries, engage in criminal activities such as robberies, big frauds even at risk of getting heavy punishment if caught. They are completeness, transitivity, independence and continuity. The following topics will be covered: 1 Analyze conditions on individual preferences that lead to an expected utility function. 30 thousands if he happens to be highly efficient and Rs. How­ever, in­di­vid­u­als may have dif­fer­ent risk attitudes. It should be carefully noted that his rejection of gamble is due to diminishing marginal utility of money income for him. Note that expected value of income in the new job with an uncertain income is 20,000 as (0.5 x 10,000 + 0.5 (30,000) = 20,000. In Bernoulli’s hypothesis we have seen that a person whose marginal utility of money declines will refuse to accept a fair gamble. That is, risk-neutral person is indifferent between them. Ù8ؗzá’þ06ßzÍa[CÂÕ©ÀÙ. Suppose the individual is currently employed on a fixed monthly salary basis of Rs. 4,000 is 75 (point B on the utility curve and utility from 2000 is 50 (point A in Figure 17.6), the expected utility from this uncertain prospect will be: In the N-M utility curve U (I) in Figure 17.6 the expected utility can be found by joining point A (corresponding to Rs. This website includes study notes, research papers, essays, articles and other allied information submitted by visitors like YOU. 30 thousands, his utility is 75 and with his lower income of 10 thousands his utility is 45. In Fig. 15,000 with certainty is 55. On the other hand, if in a new risky job, he proves to be a bad salesman, his income goes down to Rs. Iftheindividualisalwaysindi fferentbetweenthesetwo lotteries, thenthenwesaytheindividualis risk neutral . If he rejects the gamble he will have the present income (i.e., Rs. The comparison of risk aversion across agents is also examined. There are four axioms of the expected utility theory that define a rational decision maker. 3,000 from the second gamble is M2L which is less than M2D of the first gamble. For an expected-utility maximizer with a utility function u, this implies that, for any lottery z˜ and for any initial wealth w, Eu(w +˜z) u(w +Ez).˜ (1.2) 10 thousands if he happens to be not so efficient in the new job with the equal probability of 0.5 in these two jobs, then the expected utility from the new job is given by. A risk-averse person therefore prefers the income with certainty to any gamble with the same expected money value as the income with certainty. 15,000 but if he fails in his new risky job of a salesman on commission basis, his income falls to zero, then the expected utility of the risky job is given by. Every utility function that is monotone decreasing with respect to the standard Rothschild-Stiglitz (or stochastic dominance) order of more risky is averse to mean- Let us illustrate it with another example. 1000 if he loses the gamble. 2,000 income, the person’s utility is 50 which rises to 70 when his income increases to Rs. 4500). Expected utility is the standard framework for modeling investor choices. People’s preferences toward risk greatly differ. 17.3 marginal utility of money of an individual decreases as his money income decreases and therefore it represents the case of risk-averse individual. This chapter examines individual attitudes toward risk, risk aversion, and decision making under risk and describes the expected utility theory as a model of choice under uncertainty. u(ai), is the Bernoulli utility function. A person is called risk neutral, if he is indifferent between a certain given income and an uncertain income with the same expected value. Lecture 11 - Risk Aversion, Expected Utility Theory and Insurance 14.03, Spring 2003 1 Risk Aversion and Insurance: Introduction • To have a passably usable model … Proposition Suppose % has an expected utility representation and v is the corresponding von 30 thousands to him is 83. 1000 as before and the second a 50:50 chance of winning or losing Rs. 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