old, the entrepreneur will be faced with negative profits and will declare bankruptcy,
leaving the proceeds of the project—net of bankruptcy costs—to the lender.
The default productivity threshold in the debt contract ωit is defined by
R'B- 1 = ωR Qt-1 Kit
and implicitly determines the contractual interest rate Ribt :
Rit = ωit
1+
Bit-1
Nit-1
k
Rt .
(1)
Thus, under the incentive-compatible debt contract, the entrepreneur’s expected
profit will be given by ψ (ωit ) Rk Qt-1 Kit, where
∞
(2)
ψ ( ωit )=/ ( ωit - ωit ) f ( ω ) dω.
Jitit
Note that if ωit < ωit, the entrepreneur declares bankruptcy and simply walks away
from the project with nothing. The expected return to the lender under such a
contract can be expressed as ξ (ωit ; μ) RkQt-1 Kit, where
ξ ( ωit ; μ )
ωiit
(1 - μ ) /
0
ωitf ( ω ) dω + ωit
∞ f ( ω ) dω.
Jitit
(3)
In equilibrium, therefore, the optimal debt contract negotiated at the end of period
t - 1 specifies the amount Bit-1 that the entrepreneur can borrow along with the
default productivity threshold ωit, so as to maximize the expected return of the
investment project, subject to the constraint that the lender earns the risk-free rate
Rt:6
max
Bit-1 ,tit
s.t.
ψ (ωit ) Rk (Bit-1 + Nit-1)
ξ(ωit; μ)Rk (Bit-1 + Nit-1) = RtBit-1.
6 See BGG for additional technical details that guarantee a solution without credit rationing and
that ensure entrepreneur’s participation in the project. We checked our numerical results to make
sure that they are consistent with all the assumptions of the BGG framework.
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