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 Q_{d} = 10 - 2P. The official government price for the good is P_{g}=3. The government pays the cost of producing the good. A bureaucrat can restrict the supply of X. The fact that there are no risks of detection gives this public official 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 Q_{c}=8-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?