代写ECON30001 Problem Set 1代做留学生SQL 程序

ECON30001 Problem Set 1

Semester 1 2024

Question 1. When considering a possible Raphael painting, the National Gallery lists five possible attributions that could be made:

• By Raphael (B): almost certainly painted by the master himself.

• Attributed to Raphael (A): possibly painted by the master, some doubt.

• by the Studio of Raphael (S): painted by a pupil, probably under the master’s di-rection.

• a Follower of Raphael (F): painted by someone at the time, influenced by Raphael.

• an Imitator of Raphael (I): painted by someone highly influenced by Raphael, per- haps at a much later date.

John is an art collector, looking to find Raphael paintings.  He can place an Imitator, but cannot distinguish between other attributions.

Peter and Robert have some expertise in art identification.  Peter can place any painting into either “By or Attributed or Studio” or “Follower or Imitator” .  Robert can place any painting into either “By or Attributed”, “Studio or Follower”, or “Imitator” .

(a) Write down each person’s information structure. Sketch them in a state-space diagram.

(b) Does Robert have better information than John?

(c) Does Peter have better information than John?

(d) Does asymmetric information exist between Peter and Robert?

(e) Suppose that John really wants a “By Raphael” . Since Peter cannot accurately identify this attribution, is it worth listening to him at all?

Question 2. There is a virus, disease Y, that is a serious concern for human health. Fortu- nately, a very accurate test for disease Y has been developed.  The test is 99.9% accurate. Someone with disease Y is correctly diagnosed with probability 0.999, and someone not infected is correctly diagnosed with probability 0.999.  It has been estimated that 50,000 people in the population of 100 million are infected.

Suppose that Bob is selected at random and tested.  He receives the diagnosis that he is infected. What is the probability that Bob is infected?

Question 3. Consider two expected utility representations U = Eu and V = Ev .  Prove that U and V are ordinally equivalent if u and v are cardinally equivalent.

Question 4. In the lecture, we used the axioms of mixture monotonicity, reduction, sub- stitution, and continuity.  There are many other axiomatisations in the literature.  A well- known axiom is the following:

(The Sure Thing Principle)

For all lotteries p1 ,… , pJ, q1 ,…, qK, r and s we have:

where α1 , …, αJ and β1 ,… , βK are such that the compound lotteries above are well-defined.

(a) Represent the lotteries above diagrammatically and explain the intuition of the sure- thing principle.

(b) Substitute the expected utility functional form to show that the sure-thing principle is a necessary condition for expected utility maximisation.

Question 5. Consider the following choices:

and,

This pair of choices was first proposed by Nobel Laureate Maurice Allais in the early 1950s, shortly after the publication of von Neumann and Morgenstern’s work. Allais’ prediction, later confirmed by a large body of work in experimental economics, was that the modal pattern of preferences would be:2

A ≻ B and A' ≺ B' .

(a) Show, by substituting the expected utility functional form, that the modal pattern of preferences are incompatible with expected utility maximisation.

(b) Show that it is the sure thing principle in particular that is violated in Allais’ example (see Question 4).