Assume, rst, that the contract in period 1 speci es α in addition to the principal (K), the interest rate (r), and the strategy (Si). When α is endogenous, then α, K and r will
always be set at such levels that the debtor prefers the socially optimal strategy SQ in
state L because this maximizes his expected pro t. The debtor's maximization problem then is max K≥0, r≥0, 0≤α≤1 p[V + B ln (K + 1) − (1 + r)K]+ +(1 − p) max {V + γK − (1 + r)K; α(V + γK)} (3.4) s.t. ICC: max {V + γK − (1 + r)K; α(V + γK)} ≥ ≥ π[V + B ln(K + 1) − (1 + r)K] + (1 − π)αV, (3.5) PC: p(1 + r)K + (1 − p) min {(1 + r)K; (1 − α)(V + γK)} − K ≥ 0. (3.6) When designing the contract, the debtor maximizes her expected payo formulated in the maximand. With probability p, state H will occur in period 1 and the project will continue till period 2. Then the debtor retains the value of the rm's assets (independent of the project) and the project's upside payo and is able to repay the whole debt.15
With probability (1 − p), state L occurs and the debtor quits the project. Depending on what yields her higher payo, she chooses between out-of-bankruptcy liquidation, which gets her full value of the rm minus the value of the debt, and ling for a bankruptcy
14The main justi cation for the law's existence is muting creditors' incentives to race to be the rst
to collect. As White (2007) puts it: "When creditors realize that a debtor rm might be insolvent, they have an incentive to race against each other to be rst to collect. This is because, as in a bank run, the earliest creditors to collect will be paid in full, but later creditors will receive nothing. The race to be rst is ine cient, since the rst creditor to collect may seize assets that the rm needs for its operations and, as a result, may force the rm to shut down. Early shutdown wastes resources because the piecemeal value of the rm's assets may be less than their value if the assets are kept together and the rm sold as a going concern."
15Note that if the occurrence of state H had not implied full repayment in period 2, the creditor would
reorganization, which frees her from the full debt repayment and gets her fraction α of the total remaining rm's value.
Obviously, the debtor's payo after quitting the project, both from out-of-bankruptcy liquidation and from bankruptcy reorganization, must be higher than her expected pro t from continuation. The incentive compatibility constraint (3.5) assures that the debtor will never gamble on resurrection in state L. The gamble would have got him the upside payo with probability π and fraction α of the rm's assets with probability (1 − π). Since α is endogenous, the contract sets it at such a level that incentivizes the debtor to choose the rst best strategy, SQ, in state L. Such α can always be found as the
creditor is willing to accept higher α when compensated by a higher interest rate. From the creditor's participation constraint (3.6) we can see that in the extreme case of α = 1, the interest rate would reach r = 1−p
p .
Let us rst consider what the rst best solution looks like. The maximization problem in this case is
max
K≥0 V + pB ln(K + 1) + (1 − p)γK − K (3.7)
and the rst best level of K is
KF B = pB
1 − (1 − p)γ − 1. (3.8)
KF B is the level of K that generates the highest surplus. Because the debtor has all
the bargaining power and captures all the ex ante surplus, she would like to set K = KF B.
This will, therefore, be the optimal level of K with α and r adjusted to satisfy the ICC (3.5) and PC (3.6). Because the debtor's maximization problem does not lead to a unique solution for α and r higher α implies higher r, but the optimal level of K and the ex ante expected pro ts of the debtor and the creditor remain the same we assume that they are both set to the minimum level still satisfying the constraints. If α = 0 and
r = 0 satisfy the ICC for K = KF B, then these are the optimal values. Whether this is
possible depends on the model parameters. In particular, consider parameter B, which can be thought of as the project's upside or pro tability, and denote the maximum value for which α = 0 and r = 0 is compatible with the ICC as B1. With these assumptions
and notation the solution to the debtor's maximization problem can be shown to take on the values stated in the following proposition.
determination of α is Ken= pB 1 − (1 − p)γ − 1 = K F B, ren= 0 if B ≤ B1, 1−p p h 1 − (1 − α∗)V +γKF B KF B i otherwise, α∗ = 0 if B ≤ B1, V +pB ln(KF B+1)−[1−(1−p)γ]KF B V +p+(1−p)ππ γKF B otherwise, S = SQ. (3.9)
The main point of the previous proposition is that when the degree of softness can be determined freely in the contract, the rst best solution (level of investment and the optimal strategy choice) can always be attained. When the project is not too pro table (B < B1), a completely tough law (α = 0) will produce the rst best solution. The debtor
is never tempted to continue the project in the bad state and the creditor is always repaid in full, which means he is willing to accept r = 0. When, on the other hand, the project's pro tability exceeds a certain threshold (B ≥ B1), the debtor needs to be incentivized
to liquidate the project in the bad state by receiving a fraction of the rm's residual value. The creditor is not always repaid in full and needs to receive positive r in order to satisfy his participation constraint. The threshold B1 depends positively on the rm's
value V and the degree to which the project assets can be re-deployed elsewhere (γ). It depends negatively on the probability of the good state (p) and on the probability of project success in the bad state (π). The negative dependence on p results from the dependence of KF B on p: higher p leads to higher KF B, which increases the value of the
project and makes it more tempting for the debtor to continue in the bad state.
In Appendix 3.A.2 we demonstrate the dependence of K, α and r on the model's parameters. The optimal investment K is linearly increasing in the project upside pa- rameter B. α and r are discontinuous functions of B. They both equal zero as long as the ICC (3.5) can be satis ed for K = KF B, r = 0, α = 0; they both jump up discontinuously