St Petersburg Paradox (1)
The most primitive way to evaluate a lottery is to calculate its
mathematical expectation, i.e., E[p] = P s∈S p(s)s.
Daniel Bernoulli first doubt this approach in the 18th century when he examined the famous St. Pertersburg paradox.
4. Question 4 (5 points)
Consider a game of election with asymmetric information among voters. Whether candidate A or candidate B is elected depends on the votes of two citizens (denoted by 1 and 2). The economy may be in one of two states, α and β. The citizens agree that candidate A is best if the state is α and candidate B is best if the state is β. The payoff for each citizen is symmetric and given as follows: 1 if the best candidate wins, 0 if the other candidate wins, and 1/2 if the candidates tie. Suppose that citizen 1 is informed of the true state, whereas citizen 2 believes it is α with probability 0.9 and β with probability 0.1. Each citizen may either vote for candidate A, vote for candidate B, or not vote.
6. Question 6 (6 points)
Consider the following labor market signaling game. There are two types of worker. Type 1 worker has a marginal value product of 1 and type 2 worker has a marginal value product of 2. The cost of signal z for type 1 is C 1 (z) = z and for type 2 is
Problem Set 2: Posted on November 18
Advanced Microeconomics I (Fall, 1st, 2013)
1. Question 1 (7 points)
A real-valued function f (x) is called homothetic if f (x) = g(h(x)) where g : R → R is a strictly increasing function and h is a real-valued function which is homo- geneous of degree 1. Suppose that preferences can be represented by a homothetic utility function. Then, prove the following statements.
◮ with probability p, a consumer with wealth x will receive a
times of her current wealth x
◮ with probability 1 − p she will receive b times of x.
Thm Assume that the assumptions of Pratt’s Theorem holds. Then, for any proportional risk, the decision maker 1 is more risk
Problem Set 2: Posted on November 4
Advanced Microeconomics I (Fall, 1st, 2014)
1. Question 1 (7 points)
A real-valued function f (x) is called homothetic if f (x) = g(h(x)) where g : R → R is a strictly increasing function and h is a real-valued function which is homo- geneous of degree 1. Suppose that preferences can be represented by a homothetic utility function. Then, prove the following statements.
5. Mixed Strategy (20 points)
Consider a patent race game in which a “weak” firm is given an endowment of 4 and a “strong” firm is given an endowment of 5, and any integral amount of the endowment could be invested in a project. That is, the weak firm has five pure strategies (invest 0, 1, 2, 3 or 4) and the strong firm has six (0, 1, 2, 3, 4 or 5). The winner of the patent race receives the return of 10. Both players are instructed that whichever player invests the most will win the race and if there is a tie, both lose: neither gets the return of 10.
simultaneously chooses a strategy, and the combination of strategies determines a payoff for each player.. Each chooses her own action without knowing others’ choices.[r]