Let (x1; x2) (0; 0) denote Juneís state-contingent wealth,where x1 0 is her wealth in the state in which the truck does notcrash and x2 0 is her wealth in the state in which the truck doescrash. Draw a graph with the horizontal axis measuring the quantityx1 and the vertical axis measuring the quantity x2 and plot Juneísstate-contingent wealth if she does not take out any insurance.Suppose Juneís preferences over state-contingent wealth bundles (x1; x2)conform to the theory of Subjective Expected Utility
Question
Let (x1; x2) (0; 0) denote Juneís state-contingent wealth,where x1 0 is her wealth in the state in which the truck does notcrash and x2 0 is her wealth in the state in which the truck doescrash. Draw a graph with the horizontal axis measuring the quantityx1 and the vertical axis measuring the quantity x2 and plot Juneísstate-contingent wealth if she does not take out any insurance.Suppose Juneís preferences over state-contingent wealth bundles (x1; x2)conform to the theory of Subjective Expected Utility
Solution
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To plot June's state-contingent wealth, we first need to know her initial wealth and the potential loss from the truck crash. Let's assume for simplicity that her initial wealth is W and the potential loss is L.
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If June does not take out any insurance, her wealth in the state where the truck does not crash (x1) will be W, and her wealth in the state where the truck does crash (x2) will be W - L.
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On a graph with x1 on the horizontal axis and x2 on the vertical axis, June's state-contingent wealth without insurance would be represented by a point at coordinates (W, W - L).
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Now, suppose June's preferences over state-contingent wealth bundles (x1, x2) conform to the theory of Subjective Expected Utility. This means that she evaluates each bundle based on its expected utility, which is a weighted average of the utilities of the wealth levels in each state, with the weights being her subjective probabilities of the states.
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If June is risk averse, her utility function will be concave, meaning that she prefers a certain outcome to a risky one with the same expected wealth. In this case, she would prefer to take out insurance to reduce the risk of a large loss.
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On the graph, the set of bundles that give June the same expected utility as her current wealth without insurance would be represented by an indifference curve. This curve would be downward sloping and convex to the origin, reflecting June's risk aversion.
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If June takes out insurance, she pays a premium in both states, reducing her wealth in the state where the truck does not crash, but increasing her wealth in the state where the truck does crash (because the insurance covers part or all of the loss). On the graph, this would be represented by a movement from the point without insurance towards the 45-degree line, which represents equal wealth in both states.
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The optimal insurance for June would be the one that gets her on the highest possible indifference curve, given the cost of the insurance. This would typically involve full insurance if the insurance is actuarially fair (i.e., the premium equals the expected loss), and partial insurance if the insurance is actuarially unfair (i.e., the premium is greater than the expected loss).
Similar Questions
Illustrate on your diagram from part (a) her state-contingentwealth bundles corresponding to the set of choices going from zero cov-erage (that is, C = 0) to full coverage (that is, C = L). [Hint: eachadditional dollar of coverage reduces Juneís wealth by q dollars in theevent the truck does not crash and increases it by q + 1 in the eventthe truck crashes.
Without needing to know anything more about Juneís at-titude toward risk (except that she is strictly risk averse), explain whyif June believes the insurance is actuarially unfair, that is q is strictlygreater than her subjective probability that the truck will crash, thenJune will not fully insure, that is, she will choose C < L. Illustrateyour your answer by drawing appropriate indi§erence curves for Juneon your diagram from part (d)
Consider an individual with preferences defined over two goods, X1 and X2. This consumer has preferences such that she must have 1/3 of a unit of X1 with every 1/2 unit of X2. In addition, let P1 = 1, P2 = 1, and suppose this individual has an income of $45,000. (a) Draw indifference curves with X1 on the horizontal axis to depict this consumer’s 1 preferences. Comment.
Suppose there are only two goods in the economy: Yo-yos and Marbles. Bob only cares about his consumption of Yo-yos; the more Yo-yos he has the happier he is, regardless of his consumption of Marbles. In contrast, Lucy only cares about his consumption of Marbles; the more Marbles she has the happier she is, regardless of her consumption of Yo-yos. (a) Draw Bob’s indifference curves, putting Yo-yos in the x-axis and Marbles in the y-axis. (b) Draw Lucy’s indifference curves, putting Yo-yos in the x-axis and Marbles in the y-axis. (c) Do Bob’s and Lucy’s preferences comply with the property of diminishing marginal rate of substitution? (d) Are Bob’s and Lucy’s preferences transitive?
Explain what it means for June to be deemed strictly riskaverse and what this implies for the utility function from your answerto part (b). Illustrate in your diagram from part (a) what this meansfor her indi§erence curves.Suppose now June can purchase insurance for the transport of her householde§ects at a price of q 2 (0; 1) per dollar of coverage. That is, if she chooses apolicy that covers a maximum of C in the event of a loss, she pays qC to theinsurance company and they agree to pay her in the event of a loss, the valueof her loss up to the agreed maximum cover of C. So, in particular, if she3takes out a policy with the maximum coverage of L, then she will be fullyinsured in the event of the truck crashing since her state-contingent wealthwill be M qL in the event the truck does not crash and M qL L + L =M qL in the event the truck crashes.
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