Uncertainty in Microeconomic - Essay Example

An individual is said to be risk averse if he prefers a certain income over a fair gamble that is associated with an expected income that is equal to the certain income. The degree of risk aversion is then the extent to which the individual prefers the certain income over the uncertain income. In terms of a utility function, this is measured by the distance between the utility generated by the certain income and the utility generated by the gamble which has an expected income equal to the certain income. To better understand this, consider the diagram below. Figure 1: Risk Aversion and the concavity of the utility function Consider a rational individual with a typical upward rising concave (implying diminishing Marginal Utility) utility function U(.) defined over money incomes X. Suppose the individual can either play a lottery where two possible outcomes are possible: X1 a low income and X2 a high income. Further assume that both are equally likely. That is, both occur with probability = ?. Thus if X1 is realized the individual gets U(X1) and if X2 realizes, the individual derives U(X2). Then, the expected income from the lottery is ?[X1+X2] and the expected utility is ? [U(X1) +U(X2)]. The scenario is depicted in the diagram above. Now, observe that the utility derived by the individual from a certain income of ?[X1+X2] which is equal to U?[X1+X2] lies above ? [U(X1) +U(X2)], the expected utility from the lottery with an expected earning of ?[X1+X2]. This shows that the individual prefers a certain income over an above a lottery with an expected income that is equal to certain income. Thus, the individual is risk averse. Further, note that the further U?[X1+X2] lies above ? [U(X1) +U(X2)], the more risk averse is the individual, since the preference for the certain income is even greater in that case. This implies that the more concave the utility function the greater will be the risk aversion of the individual. Now, consider Mr. D’s utility function. It is shown in the diagram below. Figure 2:Mr. D's utility function Mr. D’s utility function is U=which is a straight line from the origin with a slope of. Thus, the second derivative of the function is zero everywhere implying zero risk aversion. Thus, we can conclude that Mr. D is a risk neutral agent. A risk neutral agent in contrast to a risk-averse agent is indifferent between picking a certain income and playing a lottery which has the equal expected income. Therefore, Mr. D is not influenced by the fact that the income from the lottery is attached to some risk bearing which is not true in case of the certain income. The indifference implies that in making decisions, Mr. D would compare expected incomes between choices and pick the option associated with the highest expected income. 2. We shall assume =2 and =3. The resulting utility functions are shown in the diagram below (figure 3). We are concerned with the rational decision in Mr. D’s case. Now, Mr. D is rational if he picks the option that generates the greatest utility for him. In the present context, when =2, if Mr. D tries to earn $60000 the expected income is. The resulting expected utility therefore is $70000. This is point A on the diagram below (figure 3). Alternatively, if Mr. D decides to stay within speed limits, the certain income=expected income=$30000. Therefore, the utility obtained in this case=expected utility obtained = $60000.This is point C in figure 3. Therefore, if =2, pursuing the income of $60000 generates a higher utility. Thus, taking the risk of breaking the speed limit to earn $60000 is the rational choice. Again if =3, and Mr. D tries to earn $60000 the expected income remains equal to. However, the resulting expected utility now is 3(35000) = $105000. This is shown as point C in the diagram below (figure 3). The utility derived from the certain income now is = $90000. This is point D in figure 3. Therefore, the expected utility of pursuing the yearly income of $60000 yields is higher compared to that of pursuing the income of $30000. Thus, when =3, the rational choice still remains the pursuit of an annual income of $60000. Figure 3: Point A - expected utility if alpha = 2, Point B expected utility if alpha=3 when Mr. D tries to earn 60000 and Point C and D are expected utilities when he earns 30000 with alpha = 2 and 3 respectively. Thus, it turns out that when alpha takes any value greater than unity, speeding and trying to earn 60000 is the rational decision. This is evident from the fact that point A lies above point C on the same utility function and point B lies above point D on the other utility function. And this is particularly true in this case since Mr. D is risk neutral. Thus as long as the expected income is higher than the certain income, it generates higher utility to pursue the risky alternative. Another way to state this is that the compensation for taking on the risk makes taking the risk the worthwhile choice. 3. Now, alpha takes values less than unity. We assume 0.5 and 0.75 as the two values for alpha. The resulting utility functions are shown in the diagram (figure 4) below. Figure 4: Same as figure 1, only with Alpha less than unity. As in the case of problem 2, the expected incomes remain the same. However, the resulting expected utilities are different. When, the expected utility for Mr D if he chooses to pursue the $60000 a year earning opportunity is = 0.5(35000)= $17500. This is represented as point A in the diagram (figure 4). The utility from pursuing the riskless $30000 a year option is = $15000. This is depicted by Point C in the diagram. In this case as well, since Mr. D is assumed to be rational, he shall prefer to go for the $60000 annual earning. If, on the other hand , the expected utility from pursuing the risky option of $60000 generates an expected utility of $26250 whereas going for the risk free option of $30000 results in an expected (=certain) utility of $22500. Therefore, going for the risky option is again the rational choice. Thus, even in this case, where alpha takes values less than unity, Mr.D will prefer to try to earn $60000 as well. Although this is associated with greater risk, this is the rational choice since Mr. D is risk neutral. Observe again as in problem 2, inspite of alpha taking values smaller than unity, point A lies above and to the right of point C on the lower utility function and point B lies above and to the right of point D on the higher utility function. Thus, as long as the expected earning is higher than the certain earning of $30000, he gains higher utility. 4. Although the utility functions undergo no change, the expected income from trying to earn $60000 falls. This is the result of the fact that due to additional preventive measures the probability of getting caught while over speeding has shot up to 0.9 implying the complementary probability of not getting caught while over speeding has fallen to 0.1. Now, the expected income from trying to earn $60000 is = 0.9(10000) +0.1(60000) = 9000+6000 = 15000. The certain income still is 30000. In the context of problem 2, if, the expected utility from the risky option is $30000 while that from the risk free option is $60000. Therefore, now it is rational to go for the risk free option of staying within the speed limits. Similarly if, the expected utility from the risky option is $45000 while the (certain=expected) utility from the risk free option is $90000. Thus, the risk free option is rational in this case as well. In the context of problem 4, when, the expected utility from the risky option = $7500 while it is = $15000 from the certain option. Again if , the expected utility from the risky option of crossing the speed limit to pursue the higher annual income is $11250 while that from the safe option of staying within speed limits is $$22500. Thus, the rational choice would be to go for the risk free option of earning $30000 annually since if generates a higher expected utility. Thus, in both situations of problems 2 and 3, Mr. D will go for the certain income by staying within the speed limits. Therefore, the fundamental conclusion from these problems is that given an individual is risk neutral, the rational choice will always be to pursue a gamble that has a higher expected income compared to any given certain income. In problems 2 and 3, pursuing the risky option generated higher expected utility compared to the certain utility of pursuing the risk free option. This was the result of the fact that the chances of getting caught were equal to the chances of getting away while over speeding. As a result, regardless of whether alpha took values greater or lesser than unity, going for the risky option generated a higher expected utility compared to the safe option. However, in case of problem 4, because of additional precautions taken by authorities, the risk of getting caught increased substantially. Consequently, the expected utility from going for the risky option fell considerably. As we found out, the expected utility from the risky option was smaller than the utility of going for the safe option regardless of whether alpha was greater or less than unity. Thus, the rational choice became going for the risk less option of staying within speed limits and earning $30000 a year. Reference Varian H (2003), Intermediate Microeconomics: a modern approach, London:Norton. Appendix The Excel sheet: Y U (alpha=0.5) U(alpha=0.75) U(alpha = 2) U(alpha = 3) 1 0.5 0.75 2 3 1.5 0.75 1.125 3 4.5 2 1 1.5 4 6 2.5 1.25 1.875 5 7.5 3 1.5 2.25 6 9 3.5 1.75 2.625 7 10.5 4 2 3 8 12 4.5 2.25 3.375 9 13.5 5 2.5 3.75 10 15 5.5 2.75 4.125 11 16.5 6 3 4.5 12 18 6.5 3.25 4.875 13 19.5 7 3.5 5.25 14 21 Read More