VE =
r(EMV)-
pa(α)
r (EMV )
* pk1 + φ
(5)
Turning to debt contracts, the bank’s expected outcome is r[zpD+ (1 - z)npk2]. Using the
lender’s participation constraint, this is equivalent to:
D = [pa (α )- r (1- z )npk 2 ]
(6)
rzp
With probability p , the firm is successful, repays the loan and retain control over output
production Φ . With probability (1 - p), the firm fails, goes bankrupt and produces a
subsistence amount of home production φ. The firm’s expected net income can thus be
defined as VD = rp(k1 -D)+ pΦ + (1 - p)φ. By substitution, we have:
VD = [r (em )- pa {g )] + p φ + (1 - p φ
(7)
zt
The optimal choice of contract for a type-1 firm can then be characterized by the function
V* = VE - VD ; that is, from (5) and (7):
(1 - p)(1 -φ)- [r (emv - pa(α))](1- z)npk2 = 0
(8)
V ʌ 7 z (EMV )
An equity contract is chosen if VE- VD > 0, while a debt contract is chosen if VE- VD < 0 .
For each level of efficiency α ∈ (0,1) , it can be shown that V * is a concave function in
z ∈ (0,1) , with limz→0 V* = +∞ and limz→1 V* = (1 - p)(Φ -φ) > 0 .