Simon Fraser University
Spring 2013
Econ 302 Final Exam
Instructor: Songzi Du
Section D200
Wednesday April 24, 2013, 8:30 – 11:30 AM
This exam has two parts. Write your name, SFU ID number, and tutorial section number
on each part.
Part II (Questions 5 – 9):
• Name:
• SFU ID number:
• Tutorial section number:
1. This is a closed-book exam.
2. You may use a non-graphing calculator.
3. If you use decimals in calculation, keep two decimal places.
4. There is no separate exam booklet. Write your solution in the space following each
question.
5. Show your work! Partial credits are given. Answers without proper explanation/calculation
will be penalized.
6. We will accept a request for regrade only if the solution is written with a pen.
7. The final exam has 135 points (45% of the class grade).
8. Stay in your seat if less than 20 minutes remain in the exam. You may leave early if
you finish 20 minutes before the exam is over.
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5. (15 points) You have an utility over money u(m) = m and start with $10000.
Consider a risky project with probability 0.5 of gaining $5000, probability 0.3 of losing
$1000, and probability 0.2 of losing $4000. Find the expected value of this project (on the
total wealth). Find your expected utility if you take this project, and find the certainty
equivalent and risk premium for the this project. Would you take the project, and why? In
your calculations keep two decimal places.
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6. (15 points) The principal is considering hiring a salesman to make to an important
sale. If the sale is successful, it will generate $1,000,000 to the principal. The salesman can
exert effort (e = 1) or no effort (e = 0) in the sales job; if he exert effort (e = 1), the sale will
succeed with probability p; if he does not exert effort (e = 0), the sale will fail for sure. The
salesman is risk averse and has utility (w)1/3 − 10e, where w is the wage he receives from
the principal, and e is either 0 or 1. The principal is risk neutral and has an utility which
is the money from the sale (if any) minus the wage of the salesman. Both the principal and
the salesman maximizes their expected utilities.
Suppose that the salesman has an outside option of $1000 (from another source of employment), which translates to an utility of 10001/3 = 10; the principal has an outside option
of $0.
(i) Suppose that p = 0.2. Solve the principal’s first best problem when he can observe the
effort of the salesman and offers a flat wage, and solve the principal’s second best problem
when he cannot observe the effort of the salesman and offers wages contingent on the success
or failure of the sale. Your solution must specify the optimal wage(s) and whether the
salesman is hired. For the second-best problem you may assume that the (IR) and (IC)
constraints bind for the optimal solution. Exhibit your final answers with two decimal
places.
(ii) Suppose that p = 0.1. Solve the principal’s first best and second best problems.
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7. (15 points) (i) The worker has two types: skilled (s = 1) or unskilled (s = 0); the
probability that that the worker is skilled is 30%. A worker of either type can choose to get
educated (e = 1) or not (e = 0); suppose that a skilled worker gets educated with probability
10%, and a unskilled worker gets educated with probability 80%. Calculate the conditional
probability that a worker is skilled given that he has an education.
(ii) The Census Bureau has estimated the following survival probabilities for men:
• probability that a man lives at least 50 years: 95%;
• probability that a man lives at least 60 years: 85%;
• probability that a man lives at least 70 years: 65%.
What is the conditional probability that a man lives at least 70 years given that he has
just celebrated his 60th birthday?
In your calculations keep two decimal places or use fraction.
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8. (15 points) Consider a third -price auction of a single, indivisible object: the highest
bidder gets the object and pays the third highest bid; the other bidders do not get the object
and do not pay.1 Suppose that there are three bidders.
(i) Give an example such that bidder 1 with value v1 does not have an incentive to bid
truthfully given bids b2 and b3 (i.e., specify the values of v1 , b2 and b3 ). Explain.
(ii) Give an example such that bidder 1 with value v1 does have an incentive to bid
truthfully given bids b2 and b3 (i.e., specify the values of v1 , b2 and b3 ). Explain.
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If there are ties for the highest bid, each of the highest bidders gets the object and pays the third highest
bid with equal probabilities. That is, ties are resolved by a fair randomization.
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9.
(15 points) Consider an average-price auction of a single, indivisible object: the
highest bidder gets the object and pays the average of all bids; the other bidders do not
get the object and do not pay.2 For example, if b1 = 0.9 and b2 = 0.8, then bidder 1 wins
the object and pays (0.9 + 0.8)/2 = 0.85. (The auctioneer might (naively) use this kind of
auction because he wants the winning bidder to pay the “fair” price.)
Suppose that there are two bidders. The private value vi of each bidder is uniformly
distributed on the interval [0, 1] and is independent of one another.
(i) Construct the symmetric equilibrium bidding strategy b(vi ). Hint #1: b(vi ) = Avi so
your job is just to calculate the constant A. Hint #2: if bidder 1 wins by bidding b(x), then
his expected payment conditional on winning is
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(b(x) + E[b(v2 ) | b(v2 ) ≤ b(x)]) = (Ax + E[Av2 | v2 ≤ x]),
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and
E[v2 | v2 ≤ x] =
x
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by our assumption of uniform distribution.
(ii) Calculate the expected payment of bidder i of value vi (the overall expected payment,
weighted by the probabilities of winning and losing).
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If there are ties for the highest bid, each of the highest bidders gets the object and pays the average bid
with equal probabilities. Hint: don’t worry about ties.
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