Math/Physic/Economic/Statistic Problems
There are two effort level for the agent and they are . For the agent the cost for high effort is 500 and the cost for low effort is 0. The reservation utility for the agent is 500. When the agent is working for the principal, there can be three levels of profit and they are . The principal is risk neutral and the agent is risk averse. The agent’s utility from the wage, w, would be . The agent’s total utility would be where . When the effort level is high the probabilities that each result would happen are . When the effort level is low they are .
In the ideal case where the principal can observe the effort level of the agent, what will be the wage system like? In other words, what will be ?
If the principal cannot observe the effort level of the agent, what will be ? And what will be the effort level of the agent in that case? What will be the profit of the principal?
Eric is an owner of a big firm and James applied for that firm. If James is smart, he will make $10 for Eric and Eric will pay him $7 (so that Eric’s profit from James will be $3). If James is not-smart, he will make $4 for Eric and Eric will pay him $3 (so that Eric’s profit from James will be $1). The probability that James is smart is 1/2. Actually, Eric does not know if James is smart or not and he will pay the average wage ($7+$3)/2=$5 and the expected profit for Eric would be $7-$5=$2. Then, professor Hahn opened a “Useless University (=ULU)” and he gives a degree only to the students who pass his difficult examination.
In order to pass the examination, the students should take professor Hahn’s class and the tuition is $1 per month. If the student is smart, it takes only 1 month for him to pass the exam. On the other hand, if the student is not-smart, it takes 5 months to pass the exam. We will assume that Eric can check if James passes the exam or not, but Eric cannot check how many months James took professor Hahn’s class. Find out two Perfect Bayesian equilibria of this game, one separating and one pooling.
Two players play the following game for infinite times.
Cooperate Betray
Cooperate 10, 20 -25, 30
Betray 15, -23 -12, -19
For the player to continue to cooperate what would be the ranges of their discount factor, , respectively?
(professor – TA game) Professor Hahn can give a TA scholarship to Gong Yi for maximum 2 years. At the beginning of each year professor Hahn decides whether he will give a scholarship to Gong Yi or not. Gong Yi can get a scholarship in t=2, only if he gets it in t=1. Basically, professor Hahn and Gong Yi will play the following game twice.
Prof. Hahn
No scholarship Give scholarship
TA Gong Yi
(0,0) Work Don’t work
(3,X) (-2,3)
Gong Yi can be a Hard working type with probably 0.3 and can be a Lazy type with probability 0.7 Professor Hahn does not know Gong Yi’s type. If Gong Yi is hard working, it will be X=5 and Gong Yi will always work if he gets a scholarship. If Gong Yi is lazy, it will be X= 1. There is no time discount for t=2.
Find out a Perfect Bayesian Equilibrium of the game.
Keat and Sunday are neighbors. They want to live in a clean environment and want their street to be clean. The utility from the clean street is same for both at 10 if only one person clean it and the utility will be same for both at 12 if both people clean it. The problem is someone has to clean the street in order to have a clean street and cleaning the street is a public good. Keat’s cost of cleaning the street is ch and Sunday’s cost is cj. Both ch and cj can have two values, either 5 or 12. The probability that the cost will be 5 is 3/4.
The payoffs will be as the following table.
Sunday
Clean Don’t
Keat Clean 12- ch, 12- cj 10- ch,10
Don’t 10, 10- cj 0,0
Find three Bayesian equilibria of the game.
Consider a duopoly with a Cournot competition. The demand of the market is Q=2-p. Both firm 1 and firm 2’s marginal costs can take two values. For firm 1 it can be MC=5/4 with probability 1/3 and MC=3/4 with probability 2/3. For firm 2 it can be MC=5/4 with probability 2/3 and MC=3/4 with probability 1/3. Each firm knows its own MC but does not know the MC of the other firm. (But the probabilities are known to everyone.) What will be he Bayesian equilibrium of the game?
