Econ 302
Assignment 7 — Solution
1.
Part (i):
Suppose that the principal offers a flat wage of w. The first best problem is:
max 0.2 × 1000000 − w subject to: (w)1/3 − 10 ≥ 10
w
Clearly, the constraint binds at the optimum: w = (20)3 = $8000. By offering this wage the
principal gets 200000 − 8000 = $192000 > 0. So the principal hires the salesman without
moral hazard.
Suppose that the principal offers a wage of w1 if the sale is successful, and a wage of w0 if
the sale is not successful. The second best problem is:
max 0.2 × (1000000 − w1 ) + 0.8(−w0 ) subject to: 0.2(w1 )1/3 + 0.8(w0 )1/3 − 10 ≥ (w0 )1/3
w1 ,w0
0.2(w1 )1/3 + 0.8(w0 )1/3 − 10 ≥ 10
The two constraints bind at the optimum, which gives two equations:
0.2(w1 )1/3 + 0.8(w0 )1/3 − 10 = (w0 )1/3
0.2(w1 )1/3 + 0.8(w0 )1/3 − 10 = 10
Solving these two equations gives
(w1 )1/3 = 60, (w0 )1/3 = 10,
or equivalently,
w1 = $216000, w0 = $1000.
By offering these wages, the principal gets 0.2 × (1000000 − 216000) + 0.8 × (−1000) =
$156000 > 0. So the principal hires the salesman given moral hazard.
Part (ii):
Suppose that the principal offers a flat wage of w. The first best problem is:
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max 0.1 × 1000000 − w subject to: (w)1/3 − 10 ≥ 10
w
Clearly, the constraint binds at the optimum: w = (20)3 = $8000. By offering this wage
the principal gets 100000 − 8000 = $92000 > 0. So the principal hires the salesman without
moral hazard.
Suppose that the principal offers a wage of w1 if the sale is successful, and a wage of w0 if
the sale is not successful. The second best problem is:
max 0.1 × (1000000 − w1 ) + 0.9(−w0 ) subject to: 0.1(w1 )1/3 + 0.9(w0 )1/3 − 10 ≥ (w0 )1/3
w1 ,w0
0.1(w1 )1/3 + 0.9(w0 )1/3 − 10 ≥ 10
The two constraints bind at the optimum, which gives two equations:
0.1(w1 )1/3 + 0.9(w0 )1/3 − 10 = (w0 )1/3
0.1(w1 )1/3 + 0.9(w0 )1/3 − 10 = 10
Solving these two equations gives
(w1 )1/3 = 110, (w0 )1/3 = 10,
or equivalently,
w1 = $1331000, w0 = $1000.
By offering these wages, the principal gets 0.1 × (1000000 − 1331000) + 0.9 × (−1000) =
−$34000 < 0. So the principal does not hire the salesman given moral hazard.
2.
Part i: Suppose everyone observes the quality of each car.
1. For sales of excellent cars to be feasible, we must have cE ≤ vE , and the feasible price
range for an excellent car is from cE to vE .
2. For sales of good cars to be feasible, we must have cG ≤ vG , and the feasible price
range for a good car is from cG to vG .
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3. For sales of bad cars to be feasible, we must have cB ≤ vB , and the feasible price range
for a bad car is from cB to vB .
Part ii: Suppose no one observes car quality.
For car sales to be feasible, we must have (cE + cG + cB )/3 ≤ (vE + vG + vB )/3, and the
feasible price range for a car is from (cE + cG + cB )/3 to (vE + vG + vB )/3.
Part iii: Suppose only sellers know the quality of each car.
1. For the sales of excellent cars to be feasible, we must have cE ≤ (vE + vG + vB )/3, and
the feasible price range for an excellent car is from cE to (vE + vG + vB )/3. Notice that
when the sales of excellent cars are feasible, the sales of good and bad cars are also
feasible because cB < cG < cE .
2. For the sales of good cars to be feasible, either the sales of all cars are feasible (cE ≤
(vE + vG + vB )/3), or only the sales of good cars and bad cars are feasible and the
excellent cars exit the market (cE > (vE + vG + vB )/3 and cG ≤ (vG + vB )/2). In
the first case (when cE ≤ (vE + vG + vB )/3), the feasible price range for a good car is
from cG to (vE + vG + vB )/3. In the second case (when cE > (vE + vG + vB )/3 and
cG ≤ (vG + vB )/2), the feasible price range for a good car is from cG to (vG + vB )/2.
3. There are three cases when the sales of bad cars are feasible. First, when the sales of
all cars are feasible (cE ≤ (vE + vG + vB )/3), the feasible price range for a bad car is
from cB to (vE + vG + vB )/3. Second, when only the sales of good and bad cars are
feasible (cE > (vE + vG + vB )/3 and cG ≤ (vG + vB )/2), the feasible price range for
a bad car is from cB to (vG + vB )/2. And third, when only the sales of bad cars are
feasible (cE > (vE + vG + vB )/3 and cG > (vG + vB )/2 and cB ≤ vB ), the feasible price
range for a bad car is from cB to vB .
3.
Part (i): see Figure 1.
A perfect Bayesian equilibrium in pure strategy is:
1. Seller of good quality chooses G-Exit.
2. Seller of bad quality chooses B-2000.
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Figure 1: problem 3 part i, seller proposes.
3. Buyer chooses (9K-No, 6K-No, 2K-Yes).
4. Buyer believes that P(G | 9K) ≤ 6/7, P(G | 6K) ≤ 3/7, and P(G | 2K) can be
anything.
(Show steps that lead to this equilibrium.)
Part (ii): see Figure 2. The subgame perfect equilibrium in pure strategy is:
1. Seller of good quality chooses (G-9-Y, G-6-N, G-2-N).
2. Seller of bad quality chooses (B-9-Y, B-6-Y, B-2-Y).
3. Buyer chooses 2000.
4. Since an unskilled worker mixes between acquiring an education (with probability q)
and acquiring no education (with probability 1 − q), we must have
w1 − k = w0 .
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Figure 2: problem 3 part ii, buyer proposes.
But this means that
w1 − c > w0 ,
which means that a skilled worker acquires an education (with probability 1).
Thus, we have w0 = µ(s = 1 | e = 0) = 0. Moreover,
w1 = µ(s = 1 | e = 1) =
Thus,
p
.
(1 − p)q + p
p
− k = 0,
(1 − p)q + p
i.e.,
p = k(1 − p)q + kp,
i.e.,
q=
p(1 − k)
.
k(1 − p)
We must have 0 < q < 1, i.e., p < k < 1.
In summary, when p < k < 1, a semi-separating equilibrium exists in which the unskilled
worker acquires education with probability p(1−k)
(and not acquires education with probak(1−p)
p(1−k)
bility 1 − k(1−p) ), the skilled workers acquires education with probability 1, and the wages
are
w0 = 0, w1 = k.
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