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 $$Q_d = 10 – P.$$ 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 $X$. 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 $$Q_c=\frac{7-P}{2}.$$
a) In the model of “no theft” what is the amount of the bribe that the corrupt official will charge?
Hint: The official wants to maximize his income, so will set the price so the marginal income is zero. The bribe is the excess of the price over $3$.