3.3 Contracting Problem
3.3.2 Full-Scale Competition
I now turn to the case of full-scale competition. When this case holds, all firms apply for loans and will accept the bank’s loan offer so that ρF(E¯,ξL) =ρF(E¯,ξH) =
1. The first order conditions of the bank’s problem yields, in addition to equation (3.19) discussed above, the following equation
ΠB(E¯) = 2t 1−(η−1) 1 λzˆ(E¯,ξ) 1+λ1zˆ(E¯,ξ) ! (3.24)
The above equation links the expected default probability of a contract ˆz(E¯,ξ)
to the total expected profit of the bank. This must hold for both ξL and ξH and thus the default probability remains constant regardless of the realisation ofξ, and ˆ
z(E¯,ξL) = zˆ(E¯,ξH). Using the definition for ΠB(E¯) set out in equation (3.13) yields a system of equations along with equation (3.19) which pins down the de- fault probabilities.
While there does not exist a closed form solution for ˆz(E¯,ξ), the properties of
the equilibrium can be found using the implicit function theorem. I will focus on the impact of a change in the parametert, which governs the market power of banks and hence the level of competition on the banking sector. An increase intwill decrease competition in the banking sector. The relationship betweentand ˆz(E¯,ξ)is set out
in the proposition below.
Proposition 3.4. When a full-scale equilibrium occurs and ρF(E¯,ξL) = ρF(E¯,ξH) =1, an increase in t increases the default probability on loanszˆ(E¯,ξ)
for allzˆ(E¯,ξ)∈h0,ηλ−1i. Furthermore, ifzˆ(E¯,ξ)> ηλ−1 the bank would make an
expected loss by offering the loan contracts and instead rejects the loan application with probability one.
equation (3.24). From equation (3.13), the derivative ofΠB(E¯)is positive whenever ˆ
z(E¯,ξ)≤ηλ−1 and by applying the implicit-function theorem to equation (3.24) the
first part of the proposition immediately follows. From equation (3.24) it follows that ˆz(E¯,ξ)≤ ηλ−1is a necessary condition for the bank to earn a positive expected profit on the set of contracts, ΠB(E¯)>0, and the second part of the proposition immediately follows.
The last contract term to consider is ρB(E¯,ξ), the probability that the bank rejects a firm’s loan application. As discussed in proposition 3.4, if the bank makes an expected loss from offering the set of loan contracts, banks will always reject the loan applications, settingρB(E¯,ξ) =0. Furthermore, it follows from the bank’s first order conditions that following the realisation ofξ, the bank will always accept the loan application, ρB(E¯,ξ) = 1, whenever πB(E¯,ξ)≥0. Intuitively, a profit maximising bank would never reject a profitable loan application.
IfπB(E¯,ξi)<0 for some i∈ {L,H}andΠB(E¯)>0, the bank may choose to reject the loan application with some probabilityρB(E¯,ξi)∈(0,1). This is set out formally in the proposition below.
Proposition 3.5. When a full-scale equilibrium occurs and i) πB(E¯,ξi)<0, ii) πB E¯,ξj>0for some i,j∈ {L,H}and j6=i, and iii) the following condition is
met −2tπB(E¯,ξi)<πB E¯,ξj πF E¯,ξj−2t (3.25)
the bank will reject loan applications following the realisation ofξi with a proba-
following equation ρB(E¯,ξi) =−πB ¯ E,ξj πB(E¯,ξi) πF E¯,ξj−2t πF(E¯,ξi)−2t ! − t 2 πF(E¯,ξi)−2t (3.26)
In the case where
−2tπB(E¯,ξi)≥πB E¯,ξj
πF E¯,ξj−2t
(3.27)
the bank rejects the loan application following the realisation ofξiwith probability
one andρB(E¯,ξi) =0.
Proof. See Appendix 3.7.1.
Proposition 3.5 raises the possibility that banks use cross-subsidisation across states, along with random loan rejections to incentivise firms to apply for loans, while limiting the losses they make following the realisation of ξi. This occurs because firms compete for loan applications rather than on individual contracts. Banks make a positive profit if the realised cost isξjand are willing to make a loss in the case where the realised cost isξiin order to satisfy the participation constraint of the firms and encourage additional loan applications.
The full-scale competition equilibrium occurs if the firms located at the mid- point of the line apply for a loan and accept the loan terms regardless of the realisa- tion ofξ. This implies that the following must hold for bothi∈ {L,H}
t
The right hand side of equation (3.28) is decreasing in ˆz(E¯,ξ)and hence de-
creasing int when the full-scale competition equilibrium occurs. It follows that an increase int, and hence a decrease in competition in the banking sector, means it is less likely that the firms located at the mid-point will apply for and accept a bank loan and that the full-scale competition equilibrium will occur only if competition in the banking sector is high andt is sufficiently low. Another implication of equa- tion (3.28) is that all firms will apply for a loan, either to the left bank if they are located atd∈[0,12)or to the right bank otherwise.