Homework 1 (due in class February 18)

ON FEBRUARY 25 WE WILL HAVE TEST 1 (COVERING THE MATERIAL COVERED BY THIS HOMEWORK)

Problem 1. A set of alternatives D consists of three objects: x, y, z. Four people stated their preferences on D:

Peter: x™y, x-z , y-z

Tom: x™y, x™z , y-z

Jane: x™y, x™z , y™z

Mary: x™y, x™z , z™y

Which of these preferences is rational?

Problem 2. Given a choice between receiving $100 and receiving nothing, Mary decided that she would take nothing. Does this information alone allow you to conclude that Mary’s preference relation violates axioms of the expected utility theory?

Problem 3. Suppose that you have asked your friend Peter to tell you whether he prefers a lottery in which he gets $20 with probability 1 (for sure) or a lottery in which he gets $15 with probability 0.5 and $10 with probability 0.5. Given the rationality conditions, as specified by the von Neumann-Morgenstern expected utility theory, is it possible for Peter to prefer the lottery over the sure payment? Is it possible for him to prefer the sure payment over the lottery? Can he be indifferent between the sure payment and the lottery?

Problem 4. Given the choice between the following lotteries,

A: get X with probability 0.5 and get Y with probability of 0.5

B: get Y with probability 1

C: get X with probability 0.25 and Y with probability 0.75

D: get X with probability 1

Jon prefers B over D, and A over C. Do Jon’s preferences violate the axioms of expected utility theory? Explain your algebraically showing why Jon’s preferences did or did not violate the axioms of the expected utility theory.

Problem 5. John, like most people, prefers to get more money than less. Suppose now that you have asked John to tell you whether he prefers a lottery in which he gets $30 with probability 0.9 and 0 (nothing) with probability 0.1 or a sure payment of $20. Given the rationality conditions, as specified by the von Neumann-Morgenstern expected utility theory, can John prefer the lottery over the sure payment? Can he prefer the sure payment over the lottery? Can he be indifferent between the sure payment and the lottery?

Problem 6. Jon, when asked about his preferences among various alternatives involving objects A and B, stated that he prefers I over II and prefers IV over III. The four alternatives are as follows:

I: get A with probability 1 (for sure);

II: get B with probability 1 (for sure);

III: get B with probability 0.5 and A with probability 0.5;

IV: get B with probability 0.8 and A with probability 0.2.

By preferring I over II and IV over III did Jon violate the axioms of the expected utility theory?

Explain your algebraically showing why Jon’s preferences did or did not violate the axioms of the expected utility theory.