2- Complementarities arise in all sorts of situations. Here is a tax evasion problem. Suppose that each of N citizens in a country needs to pay a tax of T every year to the government. Each citizen may decide to pay or to evade the tax. If an evader is nabbed, the law of the country stipulates payment of a fine of amount F, where F > T. However, the government's vigilance is not perfect, because it has limited resources to detect evaders. Assume that out of all the people who evade taxes, the government has the capacity to catch only one, and this person is chosen randomly. Distinguish between two cases: In Case 1, one person is randomly chosen from the population to be audited. If he is an evader, then conditional on being audited, he will be caught for sure. In Case 2, each evader has an anomaly on his tax return, and so only one of the evaders is randomly audited.

(a) Write down the expected payoffs in each case from evasion, and show that there are no complementarities in Case 1, while there are complementarities in Case 2.

(b) Contrast Case 1 and Case 2. In Case 1 show that there is typically a unique equilibrium. In Case 2, show that it is always an equilibrium for nobody in society to evade taxes. Is there another equilibrium as well? Find it and describe when it will exist.