EC9A1, Problem Set 2
G. Trigilia, February 2016
Exercise 1
In the example from the lecture notes about Costly State Verification (CSV),
find the optimal contract.
Solution: From the lecture notes, we know that the optimal contract is
debt, characterized by a face value D and verification in bankruptcy states.
From the break even condition for the financier we get (in thousands of $):
Z
D
[x − 40]f (x)dx + (1 − F (D))D
200 =
80
because income is uniformly distributed between 80 and 420. Further, from
the formula for the CDF of a uniform:
1 − F (D) = (420 − D)/340
and from the PDF: f (x) = (340)−1 .
Plugging inside and solving the integral we get:
200 = (680)−1 [760D − D2 ]
The quadratic equation has a unique feasible solution (the negative root),
that is: D∗ = 288.4.
Exercise 2
A corporate raider considers raising K > 0 funds to finance a takeover bid.
He knows that either the bid is successful, in which case he will get RS back
from his investment (with probability p), or it is unsuccessful and he will
get RF ∈ (0, RS ). Investment is subject to costly state verification, and the
audit cost is µ > 0.
1. First, show that the pledgeable income is maximized, and the audit
cost minimized, if the lender gets the full income in case (i) a failure is
reported; and (ii) no audit takes place;
2. Second, show that if the face value of debt D < RS , a random audit
policy dominates deterministic audit.
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Solution: (1) Denote the probability of audit in case of reported failure
as yF , and the repayment by SF . For a given D ∈ (K, RS ), the break even
condition for the financier reads:
pD + (1 − p)[SF − yF µ] = K
The pledgeable income equals the LHS of the break even condition above. It
is increasing in SF and decreasing in yF .
The incentive compatibility constraint reads:
RS − D ≥ (1 − yF )[RS − SF ]
and notice that it is must be binding. To see why, argue by contradiction.
Suppose it is not binding. First, this could depend on the fact that we
are verifying too often (yF is too high). Now, because this has an adverse
effect on the pledgeable income, we could decrease yF and achieve higher
pledgeable income levels without violating incentive compatibility. This is
in contradiction with our current goal of maximizing the pledgeable income,
hence it could not be.
The second possibility is that we are choosing a too low SF . Again, a
higher SF would increase the pledgeable income satisfying incentive compatibility, hence reaching a contradiction again.
As a result, the incentive constraint must be binding.
A final observation is the following: suppose that SF < RF , and the
associated audit probability is yF . Then, there exists a pair (yF0 , SF0 ) such
that (i) SF0 > SF and yF0 < yF , and (ii) it makes the incentive constraint
binding as well.
From the break even condition for the financier, it becomes obvious that
the prime contract increases the pledgeable income relative to the original
contract we started with.
(2) As a result of part (1), it must be that SF = RF , and the optimal
audit probability yF is the one that makes the incentive constraint binding:
D − RF
RS − RF
which is a well defined probability because RS > D > RF > 0.
Furthermore, yF < 1 implies that random audit is optimal. Could you
argue intuitively why this must be the case?
RS − D = (1 − yF )[RS − RF ] ⇐⇒ yF =
Exercise 3
For ease of notation, in this exercise assume that yS and yF denote the probabilities of NO liquidation, conditional on the reported state of the project.
As before, assume that D < RS .
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The incentive constraint for the successful entrepreneur not to claim failure reads:
RS − D + yS RC ≥ RS + yF RC z; ⇐⇒
(yS − yF )RC ≥ D
The optimal contract solves the following problem:
S
max
p R − D + yS RC + (1 − p)yF RC s.t.
{D,yS ,yF }
(yS − yF )RC ≥ D, and
p D + (1 − yS )L + (1 − p)(1 − yF )L ≥ I
A few observation help us in solving the problem:
1. The latter constraint (zero profit condition) must be binding. To see
why, note that if it does not bind one can lower D and this (i) still
satisfies the incentive constraint; (ii) increases the objective function –
hence contradicting the optimality of a contract with non-binding zero
profit condition;
2. At the optimal contract we have: yS = 1. To see why, argue by contradiction. Start with 1 > yS . We know that D > I > 0 because the
project is risky. Therefore, by incentive compatibility it must be that
yS > yF . Now, consider an alternative contract with yS0 = yS + , for
some small > 0. To keep the zero profit condition binding, we have to
set also D0 = D + L. The incentive constraint holds because RC > L,
and the objective function increases by p[RC − L] > 0, contradicting
the optimality of the original contract;
3. Finally, the incentive constraint must be binding. Again we argue by
contradiction. First, notice that we must have yF < 1. Suppose though
that the incentive constraint is slack, and consider an alternative contract such that yF0 = yF + . To make the zero profits binding, we need
to set D0 = D + L(1 − p)/p. It is again easy to check that the objective function increases by (1 − p)(RC − L) > 0, reaching the desired
contradiction.
As a result of these observations we can use the substitution method and
solve for the optimal contract. I leave the last step to you.
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