Simon Fraser University
Spring 2016
Econ 302 Assignment 7
Due: in Lecture, Tuesday April 5, 2016.
Show all your work.
1. (3 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.
2. (2 points) Consider a market for used cars with risk-neutral participants, like we did in
the adverse selection lecture. Suppose there are three qualities for cars: Excellent (E), Good
(G) and Bad (B). Let the sellers’ valuations for the three qualities be cE > cG > cB , and let
the buyers’ valuations be vE > vG > vB . Assume that there is an equal number of cars in
each of the three categories.
(i) Suppose everyone observes the quality of each car. What is the condition for sales of
excellent cars to be feasible? Good cars? Bad cars? In each case, what is the feasible price
range?
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(ii) Suppose no one observes car quality. What is the condition for car sales to be feasible?
What is the feasible price range?
(iii) Suppose only sellers know the quality of each car. What is the condition for sales of
excellent cars to be feasible? Good cars? Bad cars? [Caution: Be careful when answering
the latter two cases!] In each case, what prices could you observe?
3.
(3 points) Consider the used car game in the Adverse Selection lecture, with three
possible prices for the car: $9000, $6000, $2000.
(i) Suppose the seller makes the offer (proposing one of the three prices or not selling), draw
the game tree and find a pure-strategy perfect Bayesian equilibrium.
(ii) Suppose the buyer makes the offer (proposing one of the three prices or not buying),
draw the game tree and find a pure-strategy subgame perfect equilibrium.
4.
(2 points) Review the construction of semi-separating equilibrium in the Signaling
Lecture. Construct a semi-separating equilibrium where fraction q of unskilled workers get
education. What is the condition on k and p for this equilibrium to work?
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