ECON30001 Advanced Microeconomics Coursework, Semester 1 2022-23.

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ECON30001 Advanced Microeconomics

Coursework, Semester 1 2022-23.

Question 1. (15 marks)

Consider a market in which workers choose between working as an academic or as an entrepreneur. A worker’s own productivity as an academic (the revenue generated for the

university),  , is private information.  Universities view the productivity of workers as academicsasarandomvariablethatisuniformlydistributedontheinterval[0, 1]. Thegood academics are bad entrepreneurs, and vice versa. Specifically, a worker with productivity

as an academic  would earn f() as an entrepreneur, where f() = 1 −  . Universities

pay academics a wage w and the market is competitive. Universities and workers are risk neutral expected utility maximisers.

Explain and derive the competitive equilibrium of this market. Discuss the equilibrium’s properties and what policy intervention, if any, might be appropriate.

Question 2. (15 marks)

Bob has wealth of £100 and wants to invest it.  There are two assets: a safe asset and a risky asset.  If Bob invests his £100 in the safe asset, he will end up with £100.  If Bob

invests his £100 in the risky asset, he will end up with £100r.  Bob believes that r is a uniform random variable with support [0, 4].  Bob is an expected utility maximiser with von Neumann-Morgenstern utility u(x) = ^x .

Charlie is a financial expert who can provide reliable information on the risky investment. Charlie will send Bob one of the following messages:

Message

Information

Probability of Message

1

0 ⩽ r ⩽ 3

0.25

2

0 ⩽ r ⩽ 2

0.25

3

1 ⩽ r ⩽ 2

0.25

4

0.5 ⩽ r ⩽ 1.5

0.25

Explain and calculate the value of Charlie’s information to Bob.

Question 3. (20 marks)

a) The following axioms are central in Subjective Expected Utility (SEU) theory:

Axiom 1: For all events A and B , and all outcomes x ≻ y and x!  ≻ y! , we have:

[(A, x); ( ¬A, y)] ≿ [(B , x); ( ¬B , y)]      ⇔      [(A, x! ); ( ¬A, y! )] ≿ [(B , x! ); ( ¬B , y! )].

Axiom 2: For all events C and all outcomes z, z!  we have:

 

[(A1 , x1 ); … ; (Aj , xj ); (C , z)] ≿ [(B1 , y1 ); … ; (Bk , yk ); (C , z)]

⇔         [(A1 , x1 ); … ; (Aj , xj ); (C , z! )] ≿ [(B1 , y1 ); … ; (Bk , yk ); (C , z! )].

Explain these axioms and show that they are necessary for SEU maximisation.

b) Define an order ≿*  over events so that A  ≿*   B if and only if, for some x and y, we have [(A, x); ( ¬A, y)] ≿ [(B , x); ( ¬B , y)]. Explain how such an order can be interpreted if axioms A1 and A2 hold. Assuming these axioms hold, show that the following must be

true:

ForalleventsA, B , C withA∩ C = B ∩ C = ∅wehaveA ≿*  B ifandonlyifA∪ C ≿*  B ∪ C .

c)Anurncontains200balls, eachofwhichareeitherred, black, green, oryellow(R, B , G, Y). There are 50 green and 50 yellow balls. The remaining 100 balls are either red or black,   but nothing more is known. One ball is drawn and bets are placed on its colour. Consider   the following acts:

a = [(R, £100); (B , £100); (G, £0); (Y , £0)]

b = [(R, £100); (B , £0); (G, £100); (Y , £0)]

c = [(R, £0); (B , £100); (G, £0); (Y , £100)]

d = [(R, £0); (B , £0); (G, £100); (Y , £100)]

The modal pattern of preferences is a ≻ b and c  < d . Discuss why this might be the case and explain whether or not such a pattern is compatible with SEU.


2022-11-11