In this exercise we review some basics of attitudes toward risk when tastes are state-independent and, in part B, we also verify some of the numbers that appear in the graphs of part A of the chapter
A. Suppose that there are two possible outcomes of a gamble: Under outcome A, you get $x1 and under outcome B you get $x2 where x2 > x1. Outcome A happens with probability 8 = 0.5 and outcome B happens with probability (1 - δ) = 0.5.
(a) Illustrate three different consumption/utility relationships €” one that can be used to model risk averse tastes over gambles, one for risk neutral tastes and one for risk loving tastes.
(b) On each graph illustrate your expected consumption on the horizontal axis and your expected utility of facing the gamble on the vertical. Which of these €” expected consumption or expected utility €” does not depend on whether your degree of risk aversion
(c) How does the expected utility of the gamble differ from the utility of the expected consumption level of the gamble in each graph?
(d) Suppose I offer you $to not face this gamble. Illustrate in each of your graphs where would lie if it makes you just indifferent between taking and staying to face the gamble.
(e) Suppose I come to offer you some insurance€”for every dollar you agree to give me if outcome B happens, I will agree to give you y dollars if outcome A happens. What€™s y if the deal I am offering you does not change the expected value of consumption for you?
(f) What changes in your 3 graphs if you buy insurance of this kind €” and how does it impact your expected consumption level on the horizontal axis and the expected utility of the remaining gamble on the vertical?
B. Suppose we can use the function u(x) = xα for the consumption/utility relationship that allows us to represent your indifference curves over risky outcomes using an expected utility function. Assume the rest of the set-up as described in A.
(a) What value can a take if you are risk averse? What if you are risk neutral? What if you are risk loving?
(b) Write down the equations for the expected consumption level as well as the expected utility from the gamble. Which one depends on a and why?
(c) What€™s the equation for the utility of the expected consumption level?
(d) Consider as defined in A(d). What equation would you have to solve to find ?
(e) Supposed = 1. Solve for and explain your result intuitively.
(f) Suppose that, instead of 2 outcomes, there are actually 3 possible outcomes: A, B and C, with associated consumption levels x1, x2 and x3 occurring with probabilities δ1, δ2 and (1 - δ1 - δ2). How would you write the expected utility of this gamble?
(g) Suppose that u took the form

Transcribed Image Text:

25 utr-0.1xo.. (100,000 )