John manages Rachel’s used CD music store. To provide John with the incentive to sell CDs, Rachel offers him 50% of the store’s profit. John has the opportunity to misrepresent sales by fraudulently recording sales that actually did not take place. Let t represent his fraudulent profit. John’s expected earnings from reporting the fraudulent profit is 0.5t.

Rachel tries to detect such fraud and either detects all or none of it. The probability that Rachel detects the entire fraud is t/(1 + t) and the probability that Rachel does not detect the fraud is 1 - t/(1 + t).

Hence, Rachel’s probability of detecting fraud is zero if John reports no fraudulent profit, increases with the amount of fraudulent profit he reports, and approaches 1 as the amount of fraud approaches infinity. If Rachel detects the fraud, then x 70.5 is the fine that John pays Rachel per dollar of fraud.

John’s expected fine of reporting fraudulent profit t is t2x/(1 + t). In choosing the level of fraud, John’s objective is to maximize his expected earnings from the fraud, 0.5t, less his expected fine, t2x /(1 + t).

As a function of x, what is John’s optimal fraudulent profit? (Hint: check the second-order condition.) Show that 0t /0x 6 0. Also show that as x S , John’s optimal reported fraudulent profit goes to zero. (Hint: See Solved Problem 19.1.) M