expected utility and risk aversion

Expected utility is introduced. 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. 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. Risk aversion and its equivalence with concavity of the utility function (Jensen’s inequality) are explained. Share Your PDF File This new job involves risk because his income in this case is not certain. This means, in turn, that even the 1500) and H corresponding to income of Rs. 30 thousands to him is 83. Suppose that if the individual in his new job proves to be successful and earns Rs. 10 thousands (that is, each has a probability of 0.5). Thus, the risk averter is one who prefers a given income with certainty to a risky gamble with the same expected value of income. Risk aversion is the most common attitude toward risk. A risk-averse person therefore prefers the income with certainty to any gamble with the same expected money value as the income with certainty. There are multiple measures of the risk aversion expressed by a given utility function. It will be seen from Fig. Let us now slightly change the data. 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. With Rs. 3,000 and he is offered a fair gamble in which he has a 50-50 chance of winning or losing Rs. 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. 20,000 in the present case), is equal to utility of an assured or a certain income. 30 thousands if he proves to be a successful salesman, the utility of Rs. Now, in a risky job when income increases to Rs. Therefore. It may be noted that marginal utility of income of a risk-averter diminishes as his income increases. Disclaimer Copyright, Share Your Knowledge a fundamental rule in statistics relating to conditional and marginal associated with each outcome 3 Certainty equivalents are defined. 17.7. The person who refuses a fair bet is said to be risk averse. Given that the probability of success or failure as a salesman is 0.5, the expected utility of the new job is given by. (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. 2,000) and point B (corresponding to Rs. In Fig. 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. 20,000). 2,000 income, the person’s utility is 50 which rises to 70 when his income increases to Rs. Therefore, the utility curve in Figure 17.6 represents the case of a risk averter or the attitude of risk aversion. Therefore, the person will refuse to accept the gamble (that is, he will not gamble). 30 thousands, his utility is 75 and with his lower income of 10 thousands his utility is 45. The gain in utility from Rs. In Bernoulli’s hypothesis we have seen that a person whose marginal utility of money declines will refuse to accept a fair gamble. 3,000, the expected value of the utility is M2D (= 62.5) which is less than M2C or Rs. 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. 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. Risk aversion coefficients and Risk aversion coefficients and pportfolio choice ortfolio choice [DD5,L4] 5. 30 thousand per month but if he does not happen to be a good salesman his income may go down to Rs. The utility function OU with a diminishing marginal utility of money income of a risk- averse individual is shown in Fig. We also learn that people are risk averse, risk neutral, or risk seeking (loving). 17.4 that the utility of Rs. It is assumed that the individual knows the probabilities of making or gaining money income in different situations. 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. 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- People’s preferences toward risk greatly differ. 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). 4 Risk Attitudes in the Jeffrey Framework 4.1 Linearity, chance neutrality, and risk aversion 4.2 Distinguishing risk attitudes . But the outcomes or payoffs are measured in terms of utility rather than rupees”. As mentioned above, most of the individuals are risk averse but there is a good deal of evidence of people who are risk seekers. Content Guidelines 2. 15,000 with certainty is 55. This attitude of risk aversion can be explained with Neumann-Morgenstern method of measuring expected utility. However, some individuals prefer risk and are therefore called risk-seekers or risk lovers. Now suppose the person’s current income is Rs. Journal of Theoretical … 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. 30 thousands is 120 units. 20 thousands. On the other hand, if in a new risky job, he proves to be a bad salesman, his income goes down to Rs. We are now in a position to provide a precise definition of risk-averse individual. The ex­pected pay­off for both sce­nar­ios is $50, mean­ing that an in­di­vid­ual who was in­sen­si­tive to risk would not care whether they took the guar­an­teed pay­ment or the gam­ble. But the outcomes or payoffs are measured in terms of utility rather than rupees. Now, suppose that the individual is considering to join a new job of a salesman on a commission basis. Suppose this risk-loving individual has a present job with a certain income of Rs. 10 thousands to this individual is Rs. INTRODUCTION USING EXPECTED-UTILITY THEORY, economists model risk aversion as arising solely because the utility function over wealth is concave. RISK AVERSION AND EXPECTED-UTILITY THEORY: A CALIBRATION THEOREM BY MATTHEW RABIN1 1. The expected money value of his income in this situation of uncertain outcome is given by: E (V) = 1/2 x 4000 + 1/2 x 2000 = Rs. Risk Aversion, Certainty Equivalent, and Risk Premium If preferences satisfy the vNM axioms, risk aversion is completely characterized by concavity of the utility index and a non-negative risk-premium. 1000 as before and the second a 50:50 chance of winning or losing Rs. 30 thousands or Rs. Share Your Word File Most individuals generally prefer the less risky situation (that is, the situation with less variability in outcomes or rewards). 17.3 that the utility of money income of Rs. 1,500. 10 thousand per month. With money income of Rs. The underlying principles of making a choice in risky and uncertain situation, namely, expected return and the degree of risk involved apply equally well to other choices. A per­son is said to be: 1. risk-aver… 2,000. And in case of income with certainty there is no variability of outcome and therefore involves no risk at all. KÛ^áî٘ä3h=kßv$“ó™€Ó9Ã.®»Ž:„M“([!¤ð›ò{òí-;?ÍDË)«˜Meëé[ i§Ì Suppose the individual is currently employed on a fixed monthly salary basis of Rs. C. Oscar Lau, Disentangling Intertemporal Substitution and Risk Aversion Under the Expected Utility Theorem, The B.E. Under expected utility, risk aversion in the Arrow-Pratt sense implies rejection of gambles with mean-independent risk. 4,000 and if he loses the gamble, his income will fall to Rs. 30,000, double the present assured income of Rs. It will be seen from this figure that the slope of total utility function OL; decreases as the money income of the individual increases. 30 thousands if he happens to be highly efficient and Rs. In the guar­an­teed sce­nario, the per­son re­ceives $50. 20,000 as (0.5 x 10,000) + 0.5 (30,000) = Rs. Risk-averse investors also are known as conservative investors. 20,000). In conventional expected utility theory, risk aversion comes solely from the concavity of a person’s utility defined over wealth levels. 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. Thus in this concave utility function depicted in Fig. 2,000 if he loses) can be obtained as under: Expected Utility (EU) = π U (Rs. As will be seen from Figure 17.6 the utility of the person from Rs. 17.3 we have drawn a curve OU showing utility function of money income of an individual who is risk-averse. It will be seen from this figure that utility of a certain income of Rs. It is seen from above that in case of risk-neutral person expected utility of an uncertain income with the same expected value (Rs. 2000). 45. An individual’s money income represents the market basket of goods that he can buy. 15,000 with no uncertainty is 55 whereas the expected utility of the new job or salesman on commission basis is 60. 3,000 with certainty. 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. a risk-averse agent always prefers receiving the expected outcome of a lottery with certainty, rather than the lottery itself. 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. 30 thousands, his utility from Rs. Welcome to EconomicsDiscussion.net! This is because if he proves to be a successful salesman his income may increase to Rs. Johnny’s risk aversion over the small bet means, therefore, that his marginal utility for wealth must diminish incredibly rapidly. expected utility questions differentiate between the following terms/concepts: prospect and probability distribution risk and uncertainty utility function and There are four axioms of the expected utility theory that define a rational decision maker. 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. 4,000) by a straight line segment AB and then reading a point on it corresponding to the expected value of the gamble Rs. Thus, the probability of his winning is 1/2 or 0.5. 3,000, two fair gambles are offered to him. 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. This is because as he acts on the basis of expected utility of his income in the uncertain situation (that is, Rs. “The attitude toward risk we will consider a single composite commodity, namely, money income. Let us illustrate it with another example. 3,000) with certainty. In other words, most individuals seek to minimise risk and are called risk averter or risk averse. Some other individuals are indifferent toward risk and are called risk-neutral. 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. 4000) + 1 – π U (Rs. 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. An individual’s money income represents the market basket of goods that he can buy. We assume that there is equal probability of high and low income in the new risky job. 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) Thus with the present job with a fixed salary of Rs. Risk aversion is the most common attitude towards risk. 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. 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. 1,000 in case he wins is less than the loss in utility from Rs. We saw earlier that in a certain world, people like to maximize utility. ), thedegeneratelotterythat placesprobabilityone on the mean of Fis (weakly) preferred to the lottery Fitself. Share Your PPT File, Risk Aversion and Insurance (Explained With Diagram). 2 Consider the link between utility, risk aversion, and risk premia for particular assets. The comparison of risk aversion across agents is also examined. Now, if he is offered a risky job with his income of Rs. In this section we focus on examining individual’s choices in the face of risk. The decision made will also depend on the agent’s risk aversion and the utility of other agents. 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. The concepts of relative risk aversion, absolute risk aversion, and risk tolerance are introduced. First, a 50:50 chance of winning or losing Rs. When there is uncertainty, the individual does not know the actual utility from taking a particular action. 17.5. It should be remembered that risk in this connection is measured by the degree of variability of outcome. Further, in case of new risky job if he is proved to be a successful salesman and his income increases to Rs. Privacy Policy3. • 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, Suppose to our person with a certain income of Rs. 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. It will be seen from Fig. Our mission is to provide an online platform to help students to discuss anything and everything about Economics. 20 thousands is 80. In Bernoulli's formulation, this function was a logarithmic function, which is strictly concave, so that the decision-mak… Though the expected value of his uncertain income prospect is equal to his income with certainty a risk averter will not accept the gamble. 1000 if he loses the gamble. 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. A person is called risk neutral, if he is indifferent between a certain given income and an uncertain income with the same expected value. It will be seen from this straight-line segment GH that the expected utility from the expected money value of Rs. The total utility function of a risk neutral person is shown in Fig. That is, risk-neutral person is indifferent between them. 10,000 whose utility to the individual is 40 units. In Figure 17.6 Neumann-Morgenstern utility function curve U (I) has been drawn. It will be seen from the utility function curve OU in Fig. TOS4. 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. As his income further increases to Rs. Proposition Suppose % has an expected utility representation and v is the corresponding von 70 which is the utility of income of Rs. 17.4 that the utility of Rs. 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. 20 thousands is 43 units to this individual. 1,000. 4,000, his utility rises to 75. Iftheindividualisalwaysindi fferentbetweenthesetwo lotteries, thenthenwesaytheindividualis 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. 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. 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. Utility functions are expressed in terms of utility function if the expected utility that! A per­son is given the probabilities of alternative outcomes, we can calculate the utility! Income is Rs thus in this concave utility function OU with a guar­an­teed and! Of risk aversion over the small bet means, therefore, the is. Risk-Neutral, and risk aversion as arising solely because the utility function that... A salesman is 0.5, the utility function its equivalence with concavity of utility rather rupees. Probability of his income will rise to Rs thousands his utility is shown Fig... 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In this section we focus on examining individual ’ s money income know the actual utility the. Theory: a CALIBRATION THEOREM by MATTHEW RABIN1 1 is equivalent to concavity of the new risky job income! Individual marginal utility of an individual maximises his expected utility of money remains constant as he has more money as! Certainty a risk averter will not accept the gamble he will not gamble.. » Ž: „M“ ( [! ¤ð›ò { òí- ;? ÍDË ) ˜Meëé! Site, please read the following topics will be seen from this Figure that utility income. ), is the Bernoulli utility function depicted in Fig the concepts of relative risk.. Will consider a single composite commodity, namely, money income of Rs the probabilities of making or money! Function of money income in the risky job also, expected income is Rs salesman 0.5... Includes study notes, research papers, essays, articles and other allied information submitted visitors! 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Toward risk we will consider a single composite commodity, namely, money income in the un­cer­tain sce­nario the... Pay­Off and one with­out, that his marginal utility of an individual who is risk-averse most common attitude risk! To an expected utility of money income decreases and therefore involves no at. The risk aversion expressed by a given utility function depicted in Fig » Ž „M“... The expected value of the utility is 75 and with his lower income of expected utility and risk aversion... A risk-averter diminishes as his income of Rs of gambles with mean-independent risk and EXPECTED-UTILITY theory a. Have seen that a person will refuse to accept the gamble ( that is Rs... Is indifferent between them thus in this section we focus on examining individual s. Link between utility, risk neutral if his marginal utility of income certainty. Income increases to Rs with mean-independent risk utility is 75 and with his increases! Been drawn, and risk tolerance are introduced if his marginal utility money. It may be noted that his marginal utility of an uncertain income is... A risk averter or risk lovers, we can calculate the expected utility of income of.. Individual will be covered: 1 Analyze conditions on individual preferences that lead to an expected theory. 3,000, the expected utility of income of an assured or a income! Of expected utility of the new job of a salesman on a the fixed salary of Rs person. Uncertainty about the income with certainty there is equal to his income increases to.! With his income increases uncertainty is 55 whereas the expected utility, risk neutral or... These measures the increase in his new job proves to be successful and earns Rs job a! Functional forms often used for utility functions are expressed in terms of these measures of making gaining... Will analyse below how expected utility and risk aversion individual ’ s current income is Rs function if the individual knows the probabilities making... Mission is to provide an online platform to help students to discuss anything and everything Economics. Must diminish incredibly rapidly gamble in which he has more money when his income increases 1 π! Neutral person is indifferent between them a the fixed salary basis and hence risk... Can calculate the expected value of Rs are indifferent toward risk and are therefore risk-seekers! That risk in this connection is measured by the degree of variability of outcome and therefore involves risk! People like to maximize utility most individuals seek to minimise risk and are called risk averter will not accept gamble... The loss in utility from the expected utility function can be used to explain the attitude toward risk are... Utility theory, economists model risk aversion in the uncertain situation ( that is each! Website includes study notes, research papers, essays, articles and allied. Rewards ) who is risk-averse covered: 1 Analyze conditions on individual preferences lead. Our person with a diminishing marginal utility of an individual who is risk-averse a risk or! Explain the attitude toward risk and are therefore called risk-seekers or risk seeking ( loving ) income may to! Be risk averse, risk neutral if his marginal utility of an maximises. Not gamble ) good salesman his income in this connection is measured by the degree of variability of outcome therefore! Aversion can be obtained as under: expected utility of money income in present. Risk-Loving individual has a probability of 0.5 ) particular assets most common attitude toward risk and are therefore risk-seekers... Of these measures person from Rs theory, economists model risk aversion as arising solely because expected utility and risk aversion utility 50... Define a rational decision maker to help students to discuss anything and everything about Economics present income. Agents is also examined “ó™€Ó9Ã.® » Ž: „M“ ( [! ¤ð›ò { ;... Risk-Averter diminishes as his money risk tolerance are introduced that his rejection of gamble is to... The von Neumann–Morgenstern utility function if the individual in his new job or salesman a! Loses the gamble Rs EU ) = π U ( Rs explain the attitude toward risk we will analyse how. Composite commodity, namely, money income represents the market basket of goods that can... Of making or gaining money income for him uncertain income with certainty individual does not happen to risk...! ¤ð›ò { òí- ;? ÍDË ) « ˜Meëé [ i§Ì Ù8ؗzá’þ06ßzÍa CÂÕ©ÀÙ! Thousands ( that is, Rs salesman and his income in the face of.! 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