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is determined by:~
Due to moral hazard, the expected return to the lender, Ω, falls as r crosses the
threshold, r=r*. Thus, r* can be the lender‟s optimal rate of interest (see diagram).
Again there can be credit rationing in equilibrium.
Ω
θ1 θ2
r* r
3. Model 3
Suppose a mix of risk-averse and risk-neutral borrowers. Again, all projects have the
same mean return. Risk-averse borrowers choose safe projects and risk-neutral
borrowers risky projects. As the rate of interest r increases, increasing numbers of
risk-averse borrowers exit the market,https://www.51lunwen.org thus raising the proportion of risk-neutral
borrowers. This means that, whereas the expected return to the lender on a loan
increases with r, the average expected return on all loans starts at some point to fall,
due to adverse selection. I.e. there exists r=r* which is optimal for the lender.
With this model, there can also be an optimal level for C, the amount of collateral
required. As C increases, highly risk-averse borrowers are not much affected. Some
borrowers who are less risk averse or risk neutral are screened out, as risky projects
put collateral at risk. Others, however, adopt more and more risky projects. Thus, as
well as screening effects, we have moral hazard effects.
4
4. Model 4
A bank lends to a farmer. The amount of the loan is B, the rate of interest r and
collateral C. The (gross) return, depending on both the farmer‟s effort (e) and the
weather, is R. There are two possible outcomes, R=0 and R=S, with probabilities 1-p
and p. S is fixed, but p=p(e), with p(e)>0 and p(e)<0. I.e. greater effort increases the
probability of success. Effort, however, is not contractible.
The farmer‟s profits are:
π = max[R–B(1+r), –C].
The farmer‟s utility function is:
u = π–e
= max[R–B(1+r), –C] – e.
Taking expectations:
Eu = p[S–B(1+r)] + (1–p)( –C) – e
= p(e)[S-B(1+r)+C] – C – e.
Thus the optimal level of effort, e*, is given by the first order condition:
(1) .
(1 )
1
( *)
S B r C
p e
  
 
The debt overhang problem is the reduction of e*, due to moral hazard, below its
„first-best‟ level, which is given by:
.
1
( *)
S
p e 
An increase in r reduces e*. To show this formally, differentiate (1) with respect to r:
2 [ (1 ) ]
*
( *)
S B r C
B
dr
de
p e
  
 
(2) .
( *)
* [ ( *)]2
p e
p e B
dr
de

The bank‟s (gross) return is:
ρ= min[B(1+r),C+R].
Let the bank obtain loanablehttps://www.51lunwen.org funds at the rate s. The bank‟s expected profit is:
(3) Ω= pB(1+r) + (1–p)C – B(1+s)
= p[B(1+r) –C] + C – B(1+s).
The optimal rate of interest, r*, is given by the first order condition:
[ (1 *) ] ( *) 0
*
( *) B  r C  p e B 
dr
de
p e .
Substituting from (2):
(4)
 
( *)
( *) ( *)
(1 *) 3 p e
p e p e
B r C
where e* is determined by (1).
The optimal rate of interest, r*, balances the higher returns to the bank when the farm
is successful, obtained by increasing r, against the corresponding lower effort, e*, by
the farmer, and reduced probability of success. The existence of r* indicates „macro