Answers to Problem Set 8
Economics 703
Spring 2016
1. (a) For the first–best contract, the principal will either pay√the agent 0 and have him
put in low effort on both tasks or will pay him g 2 (so that w − g = g − g = 0) and
have him put in high effort on one of the tasks. The payoff to high effort on task 1 is
(A/2) + C − g 2 , the payoff to high effort on task 2 is B − g 2 and the payoff to low effort
on both is C. The assumptions on payoffs ensure that the best option is high effort on
task 1, the next best is high effort on task 2, and the worst is low effort on both tasks.
(b) Clearly, a constant wage of 0 is the cheapest way to induce the agent to choose
low effort on both tasks. So the profit to this is still C. Also, because effort on task 2 is
effectively observable, the cheapest way to induce high effort on this task still costs g 2 :
the principal tells the agent he will receive 0 (or worse) if π2 6= B and g 2 otherwise. This
again makes the payoff to this option equal to B − g 2 . Since B − C > g 2 by assumption,
this is better than low effort on both tasks.
Inducing high effort on task 1 is more complex. Here we have to minimize costs
subject to the individual rationality constraint
1√
1√
wA +
w0 − g ≥ 0
2
2
and the incentive constraint
√
1√
1√
wA +
w0 − g ≥ w0 .
2
2
Explanation: The principal should insist on π2 = C (otherwise, he knows that the agent
put low effort into task 1). Conditional on this, he has to decide how much to pay when
π1 = A and how much to pay when π1 = 0. I’m using wA to denote the wage in the first
case and w0 for the second.
We know that both constraints are binding at the optimum. Hence
√
1√
1√
wA +
w0 − g = 0 = w0 .
2
2
1
The second equality tells us that w0 = 0. Plugging this into the first equality, we get
√
wA = 2g or wA = 4g 2 . The principal’s expected payoff from this is
1
[A − 4g 2 ] + C.
2
For his decision in (b) to be different from that in (a), the payoff to inducing high effort
on task 2 must be larger than this or
1
A − (B − C) < g 2 .
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2. (a) The cheapest way to induce the low effort is to pay a flat wage of 0. This gives
the principal profits of πL . So consider the minimum cost way to induce the high effort.
Let wH denote the wage when profits are πH and wL the wage when profits are πL . The
principal’s problem is to minimize pwH + (1 − p)wL subject to individual rationality
√
√
p wH + (1 − p) wL − c ≥ 0
and the incentive constraint
√
√
√
p wH + (1 − p) wL − c ≥ wL .
Both constraints must bind. (If the second doesn’t bind, we’ll have wH = wL , but this
will lead the agent to choose low effort, a contradiction. If the first doesn’t bind, we can
lower wL , making the second easier to satisfy and raising the principal’s profits.) Hence
√
√
√
p wH + (1 − p) wL − c = 0 = wL .
√
So wL = 0. Therefore, p wH = c, so wH = (c/p)2 . The principal’s expected payoff from
inducing high effort is then
pπH + (1 − p)πL − p(c/p)2 .
This exceeds πL if pπH − pπL − p(c/p)2 > 0 or
πH − π L >
c2
,
p2
which holds by assumption.
(b) Let wS denote the wage when the principal sees high effort and wN the wage
when he does not. Just as before, if the principal wants to induce low effort, he can do
so by setting wS = wN = 0, yielding him profits of πL . The principal’s cost minimization
problem for high effort is to minimize qwS + (1 − q)wN subject to
√
√
q wS + (1 − q) wN − c ≥ 0
2
√
√
√
q wS + (1 − q) wN − c ≥ wN .
This is the same as the problem from (b) with q replacing p, wS replacing wH , and
wN replacing wL . Hence we know that the cost minimizing contract is wN = 0 and
wS = c2 /q 2 . The principal’s payoff to inducing high effort this way is
pπH + (1 − p)πL − qwS − (1 − q)wN = pπH + (1 − p)πL − q(c2 /q 2 ).
This exceeds the payoff to inducing low effort iff
π H − πL >
c2
.
pq
Hence if this inequality holds, the optimal contract is wS = c2 /q 2 and wN = 0. Otherwise,
it is wS = wN = 0.
Comparing the principal’s payoff in (a) to his payoff in (b), we have two cases. First,
suppose it is optimal for the principal to induce low effort in (b). Since this was an option
in (a) and he chose high effort, in this case, the principal is better off in (a). So suppose
it is optimal for the principal to induce high effort in (b). Then his profits in (b) are
higher if his expected wage cost is lower — that is, if
c2
c2
<
q
p
or q > p. If q > p, the principal will induce high effort in (b). In short, the principal is
better off in (b) if and only if q > p. Intuitively, this says that the principal wants to
base the contract on the best possible signal of high effort.
3. The objective function is now
λH [π(eH ) − wH ] + λM [π(eM ) − wM ] + λL [π(eL ) − wL ].
I won’t write out all the constraints. There are three individual rationality constraints
and six incentive compatibility constraints (each of three types has to prefer the truth to
each of two possible lies).
Essentially the same argument as the one in class shows that the individual rationality
constraints for the middle and high types can’t bind. We see that incentive compatibility
implies
wi − g(ei , θi ) ≥ wL − g(eL , θi ), i = M, H
so
wi − gi (ei , θi ) ≥ wL − g(eL , θL ) + g(eL , θL ) − g(eL , θi ).
Since g(eL , θL ) ≥ g(eL , θi ), this implies
wi − gi (ei , θi ) ≥ wL − g(eL , θL ) ≥ v −1 (ū),
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where the last inequality comes from individual rationality for the low type. Hence
wi − gi (ei , θi ) ≥ v −1 (ū) for i = M, H. Therefore, we can ignore the IR constraints for the
middle and high type since they will hold automatically.
This implies that individual rationality for the low type must bind. Otherwise, we
could lower all three wages by ε, not affecting incentive compatibility, and improve the
principal’s payoff. So wL = g(eL , θL ) + v −1 (ū).
A similar argument shows that the constraint that θH not take the contract for θL
cannot bind. To see this, note that the constraint that θH not take the θM contract says
wH − g(eH , θH ) ≥ wM − g(eM , θH ) = wM − g(eM , θM ) + g(eM , θM ) − g(eM , θH ).
Using the constraint that θM not take the contract for θL , we get
wM − g(eM , θM )+g(eM , θM ) − g(eM , θH ) ≥ wL − g(eL , θM ) + g(eM , θM ) − g(eM , θH )
= wL − g(eL , θH ) + [g(eM , θM ) − g(eL , θM )] − [g(eM , θH ) − g(eL , θH )] .
The first term in brackets on the right is the marginal cost to the θM type of increasing
effort from eL to eM . The term in brackets we’re subtracting from that is the marginal
cost to the θH type of the same increase in effort. Since the marginal cost is decreasing
in type, this difference must be positive. Hence the last term is at least wL − g(eL , θH ).
Hence
wH − g(eH , θH ) ≥ wL − g(eL , θH ).
So we can ignore the constraint that θH not take the contract for θL and be sure this will
hold anyway.
This tells us that the constraint that θH not take the θM contract must bind. This is
now the only constraint left which says wH must be bigger than something, so if it holds
with a strict inequality, we can lower wH , improve the principal’s payoff, and not mess
up any of the other constraints. Hence
wH = g(eH , θH ) + wM − g(eM , θH ).
Next, consider the constraint
wM − g(eM , θM ) ≥ wH − g(eH , θM ).
Substituting from the above for wH , we see that this holds iff
wM − g(eM , θM ) ≥ g(eH , θH ) + wM − g(eM , θH ) − g(eH , θM )
or
g(eH , θM ) − g(eM , θM ) ≥ g(eH , θH ) − g(eM , θH )
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which holds iff eH ≥ eM . As in class, it is natural to conjecture that this is not binding
and to use “ignore and verify” on it. So I’ll ignore this constraint for now, verifying it
holds at the end.
This means the constraint that θM not take the θL contract must bind since this is
the only constraint left which says wM must be greater than something. Hence
wM = g(eM , θM ) + wL − g(eL , θM ).
Just as in class, one can show that the incentive compatibility conditions that θL not
take one of the wrong contracts reduce to eH ≥ eL and eM ≥ eL . Again, let use “ignore
and verify” on these (although we’ll be a bit lax on the “verify” part).
So we’ve now eliminated all the constraints and determined that
wL =g(eL , θL ) + v −1 (ū)
wM =g(eM , θM ) + wL − g(eL , θM )
=g(eM , θM ) + g(eL , θL ) − g(eL , θM ) + v −1 (ū)
wH =g(eH , θH ) + wM − g(eM , θH )
=g(eH , θH ) + g(eM , θM ) − g(eM , θH ) + g(eL , θL ) − g(eL , θM ) + v −1 (ū)
Substituting into the objective function, we get
λH [π(eH )−g(eH , θH )] + λM [π(eM ) − g(eM , θM )] + λL [π(eL ) − g(eL , θL )]
− λH [g(eM , θM ) − g(eM , θH ) + g(eL , θL ) − g(eL , θM )]
− λM [g(eL , θL ) − g(eL , θM )] − v −1 (ū).
The first line gives the objective function from the first–best. The second line gives the
cost of the rents we have to give the high type. Note that if he imitates the middle type,
he gets, in effect, the rents the middle type would get plus more due to his own cost
advantage. Finally, the last line (ignoring the v −1 (ū)) gives the rents for the middle type.
The first order condition for eH is π 0 (eH ) = ge (eH , θH ). So, just as in the simpler
model from class, the effort choice for the high type is first best.
The first order condition for eM also looks like what we saw in class for the low type,
namely,
λM [π 0 (eM ) − ge (eM , θM )] = λH [ge (eM , θM ) − ge (eM , θH )].
Thus, as in class, we see that the second–best effort choice for the middle type is below
the first–best. Since the first–best effort choice for the middle type is below that of the
high type, we see that our “ignore and verify” worked on the constraint that eM ≤ eH .
Finally, the first order condition for eL is slightly messier. We have
λL [π 0 (eL ) − ge (eL , θL )] = (λH + λM )[ge (eL , θL ) − ge (eL , θM )].
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Again, this implies that the low type’s second–best effort is below the first–best and
hence is below that of the high type.
The last constraint we should verify is that eL ≤ eM . This is messier and so I will
omit it. To see the issue, think of λM as very small. Intuitively, as it converges to zero,
we’re converging to the two–type case we discussed in class. But the first–order condition
for eM above says that our choice of eM will be getting small — converging to zero in
general. Thus if λM is too small, we will end up with eM below eL and will have to
explicitly impose the constraint that this does not occur. That is, the constraint will
bind and we’ll have to set eL = eM , changing the first–order conditions.
4. (a) This change gives the owner more options but under the assumptions given, these
extra options have no value. To see this, first note that the principal can now offer only
one contract, say (wH , eH ), chosen for only for the high type and “shut down” if it’s the
low type. That is, he can set this so that
wH − g(eH , θH ) ≥ 0 ≥ wH − g(eH , θL ).
Note that he cannot set the contract to induce the low type to work but the high type
to quit since the high type earns more from any given contract than the low type. The
best contract of this form would maximize
λ[π(eH ) − wH ] + (1 − λ)(0)
subject to the constraint above. But note that this gives the principal the same payoff
as in the original formulation of the model but with eL = wL = 0. Since ū = 0 and
v(0) = 0, this contract is actually available to the principal in the original model. Thus
even though we seemed to constrain the principal to deal with both types, he could deal
with the low type only in the trivial sense of offering him no money in exchange for no
work.
(b) Now the argument of (a) breaks down. If the principal does try to get the low
type to take the contract but not take effort, he still has to pay that agent g(0, θL ) > 0
which is worse than what happens if he induces him to turn down the contract instead.
Essentially the same thing happens if g(0, θL ) = 0 but v −1 (ū) > 0. Again, the principal
has to pay the low type a strictly positive amount which is worse than what happens if
the principal instead induces the low type to reject the contract.
I’ll focus my comments on the fixed cost case, setting v −1 (ū) = 0, but it’s easy to
rewrite for the case where g(0, θL ) = 0 but v −1 (ū) > 0.
It is easy to see that the best contract which excludes the low type is to set eH = e∗H
and wH = g(e∗H , θH ). Hence this is optimal whenever λ[π(e∗H ) − g(e∗H , θH )] exceeds the
profits from the usual second–best. The latter is
∗∗
∗∗
∗∗
λ[π(e∗H ) − g(e∗H , θH )] − λ[g(e∗∗
L , θL ) − g(eL , θH )] + (1 − λ)[π(eL ) − g(eL , θL )].
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Hence the new approach is better iff
∗∗
∗∗
∗∗
λ[g(e∗∗
L , θL ) − g(eL , θH )] ≥ (1 − λ)[π(eL ) − g(eL , θL )].
In other words, it’s better to “get rid of” the low type when the expected rents he forces
the owner to pay the high type exceeds the profits the owner earns from him. Intuitively,
this is likely to hold as the low type becomes more and more inefficient. The more
inefficient is the low type, the higher the rents the high type earns from imitating him
and the lower the surplus the owner gets from dealing with him. For example, if the low
type is so inefficient that maxe π(e) − g(e, θL ) is negative (which is possible with the fixed
cost), then certainly this new option would be better.
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