Problem
Consider a person who is buying a car. He is looking at a variety of cars, and his key metric for valuation is the number of miles on the car. His value function for a car is as follows
Value = 40000 - miles^.9 (the last bit of notation is "miles to the .9 power") so for a new car he would be willing to pay 40000.
i. What would he be willing to pay for a car with 20,000 miles?
ii. What is the maximum number of miles a car can have that he would be willing to pay a positive amount of money for?
iii. Now consider what happens if he is inattentive. In particular, he has left digit bias such that he fully values the leftmost digit, but underweights all numbers to the right with theta = .6. Like the example from the lecture for a true mileage of 49,830, Perceived Miles = 40,000 + (1 - theta)*9,830
How much does he perceive a car with 89,652 miles? What about 91,252?
iv. He has the same value function as in the beginning of this problem. How much would he be willing to pay for a car that had 20,000 miles? What about 19,950?
v. Now consider the more complex left digit bias model where an inattentive person's perceived mileage for 49,830 is Perceived Miles = 40,000 + (1-theta)*9,000 + (1-theta)^2 *800 + (1-theta)^3 *30 How many miles would 89,652 be perceived as? How much would he be willing to pay for a car with 19,950 miles?