Assignments Are Due at the Beginning of Class on the Due Date

Econ 452, Assign 1, 2006

Economics 452

Assignment 1

Dr. L. Welling Due: Tuesday, Jan. 24

January 13, 2006 Marks: 40 marks

Assignments are due at the beginning of class on the due date.

1.  (5) Show that non-increasing relative risk aversion implies decreasing absolute risk aversion.

Ans: have ; suppose Then Since w>0,

and , the required inequality can hold only if Hence, non-increasing RRA implies decreasing ARA.

(Moral? keep the math as simple as possible.)

2.  (7) Lori is risk averse. She has two pieces of jewelry, each worth $1000, which she wants to send to her sister in Thailand. She is concerned about the safety of sending them. She believes that the probability that the jewelry won't arrive is θ, where 0 < θ < 1. Is her expected utility higher if she sends the articles together or in two separate shipments? Explain.

Ans: if one shipment, only two possible outcomes: both pieces arrive, yielding U(2000), or neither arrives, yielding U(0). Then .

If two shipments, add another possibility: one arrives, one doesn't, yielding U(1000). Calculating the outcomes and the probabilities gives

Comparing the EU's under the two cases, with some manipulation, yields

since U'(0)>0>U"(0).

3.  (8) Let , where b and c are constants. What restrictions on w,b, and c are required to ensure that U(w) is strictly increasing and strictly concave? Show that under those restrictions, U(w) displays increasing absolute risk aversion.

Ans: Interpret x as spending on consumption; then , so can rewrite utility as . Then and . For U(w) strictly increasing and strictly concave, need U'(w)>0>U"(w). To keep signs consistent, need b>w. First inequality requires c>0, while second requires c>1.

Coefficient of absolute risk aversion = -U"(w)/U'(w) = . Then , which signifies increasing absolute risk aversion.

4.  (10) Let John's preferences over goods one and two be described by the utility function , and suppose prices of the goods are . Will John be risk loving, risk neutral, or risk averse when offered lotteries over different amounts of income? Explain.

Ans: with these preferences, John will spend 1/3 of his income on good one, and 2/3 on good two. Let y denote income; then his indirect utility function is where Since utility is concave in y, he will be risk averse with respect to lotteries over income.

5.  (10) Jason's utility function is , where w denotes after-tax income. He faces a tax rate of 50% on earned income; If he reports all his income, his after-tax income will be 100. Jason can choose to report less than his total income, but if he is caught underreporting he will be required to pay the taxes owing plus a fine of F dollars for every dollar of income he failed to report.

a)  (7) Let C denote the amount of income Jason conceals, while the probability of being caught if he conceals income is. Set up his maximization problem, and solve for the optimal level of C.

Ans: i) if reports all income, w=100; at 50% tax rate, that means earned income is 200.

ii) if conceals C, then after tax income is .5(200-C)+C = 100+0.5C if he is not caught; if he is caught, after tax and after fine income is 100-FC. Thus expected utility is . The optimal amount to conceal is found by setting and solving for C=

b)  (3) Show that C is decreasing in both F and .

Ans: clearly true - take partials of expression above.

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