Question 3 (25 points total, each part worth 5 points): Ralph, an expected-value decision maker, is having trouble with his income tax return.He must decide whether or not to take a deduction for certain expenses that the Internal Revenue Service (IRS) might disallow if they audit his return.He must pay $15,000 in taxes without the deduction, and he would pay $10,000 with the deduction. Ralph believes that there is probability of 50% that the deduction will be allowed if his return is audited.He also thinks that the probability that his return will be audited is P. The Problem is that Ralph plans to run for political office and he feels that if the government audits his return and disallows the deduction this will hurt his chances of winning the election in addition to being forced to pay the extra $5000 in taxes and another $5000 in penalties.To avoid the public embarrassment, Ralph would be willing to pay $x (as a cover up payment in addition to the extra tax and penalty).Ralph must decide whether or not to take the deduction and risk an audit. Answer the following questions.You can represent the payoffs as losses (negative values) in which case higher is better, or you can represent the payoffs as costs (positive values) in which case lower is better.Either option is fine but be consistent!

a. Draw a decision tree representing Ralph's decision problem

b. For P=0.1 and x=$100,000, what should Ralph work on assuming he is risk neutral?

c. For x=$100,000, Draw the expected value of Ralph's alternatives as a function of P and determine the threshold value of P above which Ralph should not take the deduction.

d. Calculate the value of x as a function of P that will make Ralph indifferent between taking and not taking the deduction.

e. Ralph can ask the advice of H&R Blockhead & Co. who, for a $500 fee, will provide Ralph perfect information on whether the deduction would pass an audit.Assuming that P=0.1, and x=$100,000, should Ralph hire H&R Blockhead?

Question 4 (25 points): Professor Harris is putting together his tax return and is considering whether or not to purchase audit insurance.The way the audit insurance works is that if he is audited and fined, then the insurance covers the cost of the fine regardless of the fine amount.Professor Harris draws the following decision tree and determines that he should purchase the insurance.Note that this decision tree depicts the payoffs as costs, so smaller is better.You can depict the payoffs as losses (negative values) in which case higher is better if you want.

I = Purchase Audit Insurance or not

A = Audited and Results of Audit

Professor Harris does a little more research and sees that Professor Gamble gives audit predictions for $500.Looking at Professor Gamble's historical performance, Professor Harris sees that

When the audit result is not audited, Professor Gamble

• correctly predicted "Not Audited" 70% of the time
• incorrectly predicted "Audited: 2000 fine" 20% of the time
• incorrectly predicted "Audited: 4000 fine" 10% of the time

When the audit result is audited: 2000 fine, Professor Gamble

• incorrectly predicted "Not Audited" 20% of the time
• correctly predicted "Audited: 2000 fine" 60% of the time
• incorrectly predicted "Audited: 4000 fine" 20% of the time

When the audit result is audited: 4000 fine, Professor Gamble

• incorrectly predicted "Not Audited" 5% of the time
• incorrectly predicted "Audited: 2000 fine" 15% of the time
• correctly predicted "Audited: 4000 fine" 80% of the time