Consider a risk-averse individual with fixed income $10,000 and utility function U = ln(W),

where W is his wealth or income. He has to report his income to a tax authority. A constant

proportion, t, of income is payable in taxes, 0 < t < 1. The individual can choose to hide some of

his income by underreporting his income. Let x be the amount reported, where 0 x 10,000. If

x = $10000, then he truthfully reports his income. If x < $10,000, then he under-reports his

income. The probability that the tax authority will audit his tax return is p, where 0 < p < 1.

When the audit is carried out, his true income, $10,000, will be revealed. So the probability that

he will be found guilty of tax evasion, if he has indeed under-reported his income, is p. If he is

found guilty (caught) of tax evasion, he has to pay (a) the remaining taxes owed to the

government, and (b) pay a fine, F, for the crime of tax evasion. The fine is equal to a constant

number, , times the remaining taxes owed to the government. Therefore, the more taxes that he

evades, the bigger is the fine.

(i) Write down the individual's expected utility function. Call it (x).

(ii) Suppose that = 0, how much income will he report? Prove your answer.

(iii) What is the intuition for your answer in (ii)? Explain carefully.

(iv) How is your answer in (iii) related to the design of fines to deter unlawful behavior? Be sure

to relate your answer to the practice of, for example, asking public officials who have embezzled

public funds to refund only the amount they have embezzled.

(v) Suppose p = 0.5 and > 0, how much income, x*, will he report? Assume that is such that

0 < x* < 10,000.

(vi) What is the effect of on x*? What is the intuition?