Taxpayer's pre-tax income is given by y and the objective is to maximize the expected income. The tax rate is given by t=0.2. Given the amount of tax evasion E, the tax paid if evasion is successful is given by t(y-E). If cheating is not successful, the tax paid is given by 't.y' and the person has to pay a penalty that is proportional to the total amount of tax evaded 'f.t.E'.

(a) If the penalty rate is f =2, how high should the probability of detecting evasion be to discourage tax evasion?

(b) Suppose that 90% of taxpayer's income is from wages and salaries that are subject to third-party reporting, so that misreporting of that type of income is automatically detected. The remaining 10% is self-employment income. Misreporting of self-employment income can be only detected during an audit, but audits are successful at discovering cheating only 50% of the time. The audit rate is p=20%. What fraction of income will the person hide when penalty rate is f=5? What will happen when f=10?

(c) Consider again just one type of income and the tax rate of t=0.2. Suppose that the probability of detecting evasion is not constant but rather varies with the level of tax evasion. More specifically, the probability of discovering tax evasion is p(E)=E/y. (i.e. it is equal to the fraction of income evaded). How much tax evasion would you expect to see if f=1?