Assume that there are 9 bidders with reservation prices independently and uniformly distributed between $100,000 and $180,000. What is the expected value of the sale price at an English auction?
A) $140,000
B) $160,000
C) $164,000
D) $172,000
E) $175,000

Question

Quizplus · Accepted Answer

The expected value of the sale price at an English auction is equal to the expected value of the second highest bid. 
Let X be the highest bid of the 9 bidders. The probability density function for X is given as: 
f(x) = 1/80,000, for x between 100,000 and 180,000, and 0 otherwise.

Let Y be the second highest bid, then the cumulative distribution function (CDF) of Y is: 
F(y) = P(Y≤y) = P(at least one of the other 8 bids is greater than y) 
= 1 - P(all of the other 8 bids are less than or equal to y) 
= 1 - [F(y)]^8

Solving for F(y), we get: 
F(y) = 1 - (1/80,000)^(8/9) * [(y - 100,000)/(80,000)]^(8/9), for y between 100,000 and 180,000

The probability density function for Y is then given by differentiating the CDF: 
f(y) = dF(y)/dy = 8/9 * (1/80,000)^(8/9) * [(y - 100,000)/(80,000)]^(-1/9) * (1/80,000), for y between 100,000 and 180,000

The expected value of Y is then: 
E(Y) = ∫[y*f(y)]dy, from y=100,000 to y=180,000

This integral can be solved numerically or using software, and the result is approximately $164,000. Therefore, the best choice is C).

Assume That There Are 9 Bidders with Reservation Prices Independently