Hugh's income is currently 5,000 .$Hisutilityfunctionisshownintheaccompanyingtable.a.CalculateHug{h}^{\prime }smarginalutilityofincome.Whatishisattitudetowardrisk?b.Hughisthinkingaboutgamblinginacasino.Withaprobabilityof0.5hewilllose$\$3,000,$ and with a probability of 0.5 he will win 5,000 .$WhatistheexpectedvalueofHug{h}^{\prime }sincome?WhatisHug{h}^{\prime }sexpectedutility?Willhedecidetogamble?(Supposethathegetsnoextrautilityfromgoingtothecasino.)c.Supposethatthe"spread"(howmuchhecanwinversushowmuchhecanlose)ofthegamblenarrows,sothatwithaprobabilityof0.5Hughwilllose$\$1,000,$ and with a probability of 0.5 he will win \\\ 3,000 .$WhatistheexpectedvalueofHug{h}^{\prime }sincome?Whatishisexpectedutility?Isthisgamblebetterforhimthanthegambleinpart$\mathrm{b}$ ? Will he decide to gamble?

Key Concepts: Marginal Utility Of Income, Attitude Towards Risk, Expected Value, Expected Utility, Risk Assessment Explanation: The problem involves calculating marginal utility of income, then using it to determine attitude towards risk. Then, we need to calculate the expected value of Hugh's income and expected utility in case he gambles (casino) with two different possibilities. Finally, we have to compare if gambling would be better in a narrower spread scenario. Step by Step Solution: Step 1. Calculate marginal utility of income. Formally, the marginal utility of income is defined as the additional utility derived from a one-dollar increase in income. Therefore,\n$MU=\frac{\Delta U}{\Delta m}$ Step 2. From the table, we can see that the first marginal utility of income is $1000,$ and the second is $800.$ Therefore,\n$MU=\frac{800-1000}{5000-4000}=0.4$ utils per dollar. Step 3. Since the computed value is less than one, we can say that Hugh has a decreasing marginal utility of income, i.e., he is risk-averse. Step 4. To calculate the expected value of Hugh's income in case of gambling, we compute the weighted average of the two possible outcomes. Thus,\n$EV(\text{gambling})=0.5(5000-3000)+0.5(5000+5000)=3500$ dollars. Step 5. Now, we need to calculate Hugh's expected utility, and for that, we use the formula $E(U)=U(5000)+pU(2000)+(1-p)U(8000),$ where $p$ is the probability of winning. Using the given utility function, we get,\n$E(U)=0+0.5\ast (10)+0.5\ast (12)=11$ utils. Step 6. Since Hugh is risk-averse, he would not gamble since the expected value of Hugh's income in case of gambling is less than his current income, i.e., $3500<5000.$ Step 7. For the narrowed spread scenario, the expected value of Hugh's income will be as follows:\n$EV(\text{gambling})=0.5(5000-1000)+0.5(5000+3000)=4000$ dollars. Step 8. We can also calculate Hugh's expected utility using the formula as before, and we get,\n$E(U)=0+0.5\ast (9)+0.5\ast (10)=9.5$ utils. Step 9. Overall, we can say that, in this case, the gamble is better for him than the gamble in part (b), and therefore, he will decide to gamble for this scenario.