1. Consider the model of corruption explored by Shleifer and Vishni’s where there is one government-produced good X. There is a demand for that good described by the inverse  demand equation Qd = 10 – P. The official government price for the good is Pg=3. The government pays the cost of producing the good. A bureaucrat can restrict the supply of X. Since there is no risk of detection, the public official has incentives to ask for a bribe to supply the good.  

Consider the model of “no theft” where the consumer pays the official government price plus a bribe in order to obtain X. Assume that the official marginal revenue for selling the  good in this context is given by Qc=(7/2) – (1/2)P.  

a) In the model of “no theft” what is the amount of the bribe that the corrupt official will charge?  

b) In the same model of corruption with no theft, what is the total cost that the consumer will have to pay in order to obtain the good X?  

c) Now consider the “model with theft” where consumers only pay a bribe but not the official government price. In this context, what is the total amount they will pay the corrupt official in order to obtain good X?  

2. Suppose that economic outcomes can be classified as either good or bad. Governments  differ in ability and this affects the likelihood of good outcomes. There are two types of governments: high ability or low ability. The prior probability that a government is high ability is 1/2. The probability that the economy is good given that the government is high ability is 3/4 while the probability that the economy is good given that the government is low ability is 1/4.  

In this case, the incumbent government can manipulate the economy and the electorate will learn (update) their beliefs about the ability of the incumbent government based on the observed state of the economy.  

Use Bayes Theorem to find the answer to all the following questions.  

a) What is the probability that the government is high ability given that the economy is good? 

 b) What is the probability that the government is high ability given that the economy is bad?  

Suppose that the opposition is a high type with probability 1/2. Voters vote for the government with the highest probability of being of a high type.  

c) What is the probability that the incumbent government will win an election against the opposition if the economy is good?  

d) What is the probability the incumbent government will win if the economy is bad?   

3. The government of a certain country has installed a new device that monitors the activity of public officials working for an agency prone to corruption. There are only two types of public officials: corrupt and honest. A device called “Fire Alarm” indicates 

whether officials are corrupt or honest. Using the device, the probability that an official is charged with corruption is 1/2. The probability that an official is charged with corruption given that the official is actually corrupt is 2/5. The probability that an official is honest is 7/9.  

Based on the results provided by the device, an official has been charged with corruption. Use Bayes’ Theorem to find the probability that the official is actually honest given that the device indicates that the official is corrupt. You may want to first find the probability that the official is corrupt given the corruption charges in order to find the probability that the official is honest given the corruption charges.