代做Ec 992 Spring 2024 Advanced Microeconomics Problem set 1代写Web开发

日期：2024-08-27 04:13

Problem set 1

Advanced Microeconomics,

Ec 992, Spring 2024

Decision-making under risk and uncertainty

You should attempt all the problems. Please hand in your solutions to problems 1,3, 5 at the beginning of the class in week 17. Your can (and are encouraged to) work in groups, but should write up the solutions on your own.

1. Exercise 6.B.1 of MWG textbook.

2. Exercise 6.B.2 of MWG textbook.

3. Exercise 6.B.5 of MWG textbook.

4. X = {0, 4, 6} is a set of monetary payoffs. A lottery L = (ℓ1, ℓ2, ℓ3) over X specifies probability ℓ1 of 0, ℓ2 of 4, and ℓ3 of 6 respectively. Suppose C expresses the following ranking over lotteries: (0, 0, 1) ≻ (0, 1, 0) ≻ (0.1, 0, 0.9) ≻ (0.8, 0, 0.2) ≻ (0.75, 0.25, 0) ≻ (1, 0, 0), where ≻ denotes a strict preference. Can C’s preferences over lotteries be represented by an expected utility function where the Bernoulli utility u(·) over monetary payoffs satisfies u(6) > u(4) > u(0) = 0? Explain your answer.

5. Exercise 6.C.17 of MWG textbook.

6. Exercise 6.C.18 of MWG textbook.

7. Exercise 6.D.4 of MWG textbook

8. Exercise 6.F.1 of MWG textbook.

9. There is an urn with 300 balls. 100 of these are red (R) and the rest are either blue (B) or yellow (Y). Consider the following two choice situations.

Situation I.

Act A: win ↔100 if a ball drawn from the urn is R and nothing otherwise.

Act B: win ↔100 if a ball drawn from the urn is B and nothing otherwise.

Situation II.

Act A’: win ↔100 if a ball drawn from the urn is R or Y and nothing otherwise.

Act B’: win ↔100 if a ball drawn from the urn is B or Y and nothing otherwise.

Suppose an individual strictly prefers A to B and strictly prefers B’ to A’.

(a) Formulate the above in terms of a decision problem under uncertainty by defining the set of states S, set of outcomes X, appropriate acts (‘horse-race’ lotteries) as maps from the set of states to lotteries over sets of outcomes.

(b) Show that the individual’s preferences can’t be represented by a subjective expected utility function, where his beliefs are represented by a probability measure over the states.

(c) Identify the axiom that is violated, preventing the representation of beliefs by a probability measure over states.

(d) The probability that the red ball is drawn is 1/3. Suppose the individual has a set of probabilities P over the set of states S and makes decisions using the following max-min decision rule. Letting u : X → R denote the Bernoulli or von-Neumann Morgenstern utility over outcomes, the utility of any act H is given by

What property must the set P satisfy in order to generate the observed preferences? Prove your answer.