(A1) Using (1) and (A1), the Lagrangian can be written as L

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Appendix A NOT FOR PUBLICATION Exogenous Monitoring Costs

This Appendix solves for the household optimization problem, the bank optimization problem, the equilibrium condition of the equity market, the condition for eliminating incentives to provide risky loans, and the indirect utility function with risky loans, all under the assumption that monitoring costs are exogenous and relate only to safe loans.

Household Optimization Problem

Consider first the household optimization problem. Substituting for  = [^+1_ −_+1^ κ]^ from (3) in (2) yields the consolidated budget constraint

^_+ ^+1_

^ = (1−κ)+ ^_+1κ

^  (A1)

Using (1) and (A1), the Lagrangian can be written as L^ = ln^_+Λln^+1_ +

½

(1−κ)+_+1^ κ

^ −^_− ^+1_

^

¾



where  is the Lagrange multiplier. The first-order conditions with respect to ^_ and^+1_ are

L^

^_ : 1

^_ −= 0

L^

^+1_ : Λ

^_ − 

^ = 0

Combining these conditions yields the standard Euler equation

^_+1

^_ =Λ^ (A2)

Substituting (A2) for ^_+1 in (A1) yields

^_ = ( 1

1 +Λ)[(1−κ)+^_+1κ

^ ] (A3)

In turn, substituting this result in the Euler equation (A2) yields

^_+1 = ( Λ

1 +Λ)[(1−κ)^+^_+1κ] (A4) Again, substituting (A3) for ^_ in (2) gives optimal deposits as:

= (1−κ)−^_ = Λ

1 +Λ(1−κ)−( 1

1 +Λ)_+1^

^ κ (A5)

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Substituting (A3) and (A4) in (1) yields the indirect utility function for house- holds (given that depositors share consumption with bankers):

_^ = ln

½ ( 1

1 +Λ)(1−κ+^_+1κ

^ )

¾

(A6) +Λln

½ ( Λ

1 +Λ)[(1−κ)^+^_+1κ]

¾

 Bank Optimization Problem

Consider next the bank optimization problem. When^_ = 0, (expected) bank profits are given by

Π+1|^_=0 =^_+1^_ −^−^_+1 (A7) From the balance sheet identity (15) and the regulatory constraint (16),

=^_ +_^−+^^_ = (1−)(^_ +_^) +^_^ (A8) which, for _^= 0, gives

 = (1−)_^+^_^

Substituting this expression forin (A7) and using =^_ from (16) yields Π+1|^_=0 =^_+1^_ −^(1−)^_ −^^^_ −^_+1^_ (A9) The first-order condition with respect to ^_ (which is also the zero-profit condition) is

^_+1−^(1−)−^^−^_+1= 0

which gives

^_+1

^

¯¯

^_=0

= 1−+^+ (^_+1

^ )= 1−[1−(^_+1

^ )]+^ (A10) When^_ =_^, expected bank profits are now, given limited liability (which means that banks earn no return if borrowers default) and deposit guarantee (which means that interest payments on deposits are not state contingent),³²

Π+1|^_0 =^_+1^_ +·0 + (1−)^_+1^_ −^−^_+1 (A11) whereas from (A8), with^_ =_^,

 = (1−)(1 +)^_ +^_^ Substituting this result in (A11) yields

Π+1|^_0 = [^_+1+ (1−)^_+1]^_ −^(1−)(1 +)_^−^^^_ −_+1^ 

32Note that in (A11) payment of equity returns is not state contingent either; if they are, the last term^_+1_would be replaced by(1−)^_+1_.

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that is, using (16) and (17) holding with equality, so that  =(1 +)_^, Π+1|_^0 = [^_+1+ (1−)^_+1]^_ −^[1−+(^_+1

^ )](1 +)_^−^^_^ (A12) Thefirst-order condition with respect to^_ (which is again also the zero-profit condition) gives now

^_+1+ (1−)^_+1−^

½

[1−+(^_+1

^ )](1 +) +^

¾

= 0

Using the result in Proposition 1, _+1^ = (1 +)^_+1. The above expression can thus be rewritten as

^_+1

^

¯¯

^_0

= [1−+(^_+1^)](1 +) +^

1 + (1−)(1 +) =Γ (A13) so that

^_+1

^

¯¯

^_0

= (1 +)Γ (A14)

Now, using the regulatory constraint (16) yields, when _^= 0,

^_ =⁻¹ (A15)

Similarly, when _^=^_ (16) gives =(1 +)^_, so that

_^ = [(1 +)]⁻¹ (A16) In addition, from (15) or (A8),

= (1 +^)_^− when ^_ = 0

= (1 +^+)_^− when _^=^_

that is, using (A15) and (A16) to eliminate _^, yields the demand for equity in proportion of deposits:

= (1 +^

 −1)⁻¹ when ^_ = 0 (A17)

 = [1 +^+

(1 +) −1]⁻¹ when ^_ =^_ (A18) where it can be noted that⁻¹(1 +^)−10and that also

1 +^+

(1 +) −1 = 1 +^+−(1 +)

(1 +) = (1−)(1 +) +^

(1 +) 0

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Now, substituting (A10) into (A7), and (A13) and (A14) into (A11), yields Π+1|^_=0 =^

½

[1−+(^_+1

^ ) +^]^_ −− ^_+1

^ 

¾



Π+1|^_0 =^

½

Γ[1 + (1−)(1 +)]^_ −−^_+1

^ 

¾



Using (A15) and (A16) to substitute out for _^, and (A17) and (A18) for, these expressions yield, given the definition of Γ,

Π+1|^_=0 =^[1−+(^_+1^) +^

 −(1 +^

 −1)− ^_+1

^ ]

Π+1|_^0 =^

½[1−+(^_+1^)](1 +) +^

(1 +)

−[1 +^+

(1 +) −1]−^_+1

^

¾

 that is

Π+1|_^=0 = Π+1|_^0 = 0

Thus, banks make zero (expected) profits in both equilibria.

Equilibrium Condition of the Equity Market

In equilibrium, the demand for, and the supply of, equity must be equal.

Substituting (A5) for the supply of deposits  in (A17) and (A18), the bank demand for equity is given by

 = (1 +^

 −1)⁻¹

½Λ(1−κ)

1 +Λ −( κ

1 +Λ)^_+1

^

¾

 when _^= 0 (A19)

= [1 +^+

(1 +) −1]⁻¹

½Λ(1−κ)

1 +Λ −( κ

1 +Λ)^_+1

^

¾

 when _^=_^ (A20) Equating (A19) and (A20) with equity supply  =κ, yields, dropping 

on both sides, κ = (1 +^

 −1)⁻¹

½Λ(1−κ)

1 +Λ −( κ

1 +Λ)_+1^

^

¾

, when ^_ = 0

κ = [1 +^+

(1 +) −1]⁻¹

½Λ(1−κ)

1 +Λ −( κ

1 +Λ)^_+1

^

¾

, when^_ =_^ which can be rearranged to give

(1− 1 +^

 )κ+ Λ(1−κ)

1 +Λ = ( κ

1 +Λ)^_+1

^ , when _^= 0

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[1− 1 +^ +

(1 +) ]κ+Λ(1−κ)

1 +Λ = ( κ

1 +Λ)^_+1

^ , when ^_ =_^ or equivalently

^ =Φ1^, when ^_ = 0 (A21)

^ =Φ2^, when ^_ =^_ (A22) where

Φ1 = 1 +Λ κ

½

(1− 1 +^

 )κ+Λ(1−κ) 1 +Λ

¾



Φ2 = 1 +Λ κ

½

[1− 1 +^+

(1 +) ]κ+Λ(1−κ) 1 +Λ

¾



The conditions for_+1^  ^are thusΦ1Φ2 1. It can readily be seen that an increase in  raises Φ1 and Φ2 and therefore raises ^_+1. Because, as shown earlier, ⁻¹(1 + ^)−1  0, it can be shown as well that Φ1κ  0; and similarly for Φ2. The reason is that a higherκ increases the supply of equity.

From the above expressions we also have Φ2−Φ1 = (1 +Λ)

½

−1 +^+

(1 +) + 1 +^



¾

 or

Φ2−Φ1 = (1 +Λ)

½−(1 +^ +) + (1 +^)(1 +)

(1 +)

¾

 so that

Φ2−Φ1 = (1 +Λ) ^

(1 +) 0 (A23) Condition to Eliminate Incentives to Provide Risky Loans

To prove Proposition 6 requires showing that the representative bank’s expected profits are negative when providing risky loans. To establish this result, notefirst that for the equilibrium with^_ = 0to hold, we must have zerodemand for risky loans, so that from Proposition 1,

^_+1

^_+1 1 + (A24)

given that, when ^_+1 = (1 +)^_+1 entrepreneurs can choose to use the risky technology and that no equilibrium exists when ^_+1 (1 +)^_+1.

On the supply side of the loan market, for the equilibrium with _^ = 0 to hold, banks must not have any incentive ex ante to oﬀer risky loans. If they do, as noted in the text, the most profitable option is to oﬀer^_ =_^. In that case, their expected excess return ex ante is given by

Π+1|^_0− Π+1|_^=0 ≥0

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that is, from (A9) and (A12), together with (A21) and (A22),

[^_+1+ (1−)^_+1]^_ −^(1−+Φ2)(1 +)_^−^^^_

−{^_+1_^−^(1−+Φ1)^_ −^^^_}≥0

Thus, for the deviation from the equilibrium with ^_ = 0 not to be profitable we must have, after simplifying,

(1−)^_+1^_ −^[(1−)+{Φ2(1 +)−Φ1}]_^ ≤0

or equivalently

½

(1−)^_+1

^ −[(1−)+(1 +)Φ2 −Φ1]

¾

_^ ≤0

When (A24) holds with equality–the only value of ^_+1 for which entrepreneurs will choose to use the risky technology–this condition becomes, using (A14), corresponding to _^=_^, to substitute out for ^_+1 and assuming positive lending in equilibrium,

(1−)(1 +)Γ−[(1−)+(1 +)Φ2−Φ1]≤0

or, given the definition of Γin (A14) and (A22) for ^, (1−)(1 +)[(1−+Φ2)(1 +) +^

1 + (1−)(1 +) ]−[(1−)+(1 +)Φ2−Φ1]≤0

(A25) From the first term, it can be shown that (given thatΦ2 1) an increase in

raises income, because banks increase their lending rate. The second term can be written as

−−[−+ (1 +)Φ2−Φ1] =− −[(Φ2 −1) + (Φ2−Φ1)]

Again, Φ2 1and from (A23) Φ2 Φ1. The direct eﬀect of an increase in is negative. The indirect eﬀect depends on

Φ2

 = Λ(1 +^+)

(1 +)² 0 (Φ2−Φ1)

 =−(1 +Λ)^ (1 +)² 0

so that

Φ2

 + (Φ2−Φ1)

 = [Λ(1 +)−^] (1 +)² 

which is likely positive, given the values ofΛand^ discussed in the text. Thus, an increase in  has an ambiguous eﬀect on condition (A25), because it raises both income and costs. If the cost eﬀect dominates (which occurs in particular, if

is relatively small), a highermakes it more likely that the condition will hold.

Equation (A25) can be solved numerically for a threshold value , as discussed˜ in the text.

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Indirect Utility Function with Risky Loans

Substituting (A22) for ^ in (A6) and using (12) for yields the household indirect utility function in the equilibrium with risky loans as

_^ = ln

½ ( 1

1 +Λ)(1−κ+κΦ2)(1−)

¾

+Λln

½ ( Λ

1 +Λ)(1−κ+κΦ2)^(1−)

¾

 that is

_^ =_^+ (1 +Λ) ln (A26) where

_^ = ln[(1−κ+κΦ2)(1−)

1 +Λ ] +Λln[Λ(1−κ+κΦ2)^(1−)

1 +Λ )]

which is the expression reported in the text.

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Appendix B NOT FOR PUBLICATION Endogenous Monitoring and Default Risk

In this Appendix we generalize the analysis to account for monitoring costs on risky loans and for the case where banks choose the degree of monitoring optimally. We also assume that the probability of default is inversely related to the degree of monitoring, as in Allen et al. (2011) and Dell’Ariccia et al. (2014), for instance, where monitoring eﬀort is directly equal to the success probability of thefirm. For simplicity, we abstract from the monitoring cost of safe loans.

With monitoring on risky loans only (^ = 0), the balance sheet constraint (15) becomes

_^+^_ =+−^__^ (B1) We also assume that the probability of failure,, is negatively related to the intensity of monitoring, ^_:

 =(^_ )

where(^_)is convex (⁰ 0and⁰⁰0), andlim_^

→0 = 1, andlim_^

→∞≤ 1, to ensure that ∈(01). A simple function that satisfies these properties is

 = 1− ^_

1 +^_  (B2)

where  ∈(01).

When^_ = 0, the profit function takes the same form as (A9) with ^ = 0:

Π+1|^_=0 =^_+1^_ −^(1−)^_ −_+1^ ^_ from which it can be established that the loan spread is now

^_+1

^

¯¯

^_=0

= 1−+Φ1 where

Φ1 = 1 +Λ κ

½ (1− 1

)κ+Λ(1−κ) 1 +Λ

¾



When_^=^_, and using (B1), expected profits are now given by, instead of (A12),

Π+1|_^0 = [^_+1+ (1−)^_+1]_^−^[1−+(^_+1

^ )](1 +)^_ −^^__^ Banks now maximize profits with respect to both_^ and^_. The first-order condition with respect to ^_ is given by:

^_+1+ [1−(^_)]_+1^ −^

½

[1−+(^_+1

^ )](1 +) +^_

¾

= 0 (B3)

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and with respect to ^_ as:

−⁰(^_ )^_+1 −^ = 0

that is, using (B2),

^_+1

(1 +^_ )² −^= 0 ⇒ ^_ =

r^_+1

^ −1 (B4) which shows that an increase in the loan spread, by increasing the return to lending (through a fall in the default probability) raises the optimal intensity of monitoring.

Using the result in Proposition 1, ^_+1 = (1 +)^_+1. Equation (B3) can thus be rewritten as

^_+1

^

¯¯

^_ 0

= [1−+(_+1^ ^)](1 +) +^_ 1 + [1−(^_)](1 +) 

In addition, from the equilibrium condition of the equity market (A22) when

_^=_^, we get ^ =Φ2^, where Φ2 is now given by Φ2 = 1 +Λ

κ

½ (1− 1

)κ+Λ(1−κ) 1 +Λ

¾

 so that

^_+1

^

¯¯

_^0

= (1−+Φ2)(1 +) +^_ 1 + [1−(^_ )](1 +) =Γ

which implies that

^_+1

^

¯¯

^_0

= (1 +)Γ (B5)

From this expression, it can be established that an increase in monitoring costs has am ambiguous eﬀect on the loan spread, given that ⁰ 0:

Γ

^_ = [1 + [1−(^_)](1 +)]+ [(1−+Φ2)(1 +) +^_]⁰(^_ ) {1 + [1−(^_)](1 +)}² ≷0

(B6) On the one hand, an increase in ^_ raises the spread, through a direct cost eﬀect. On the other, it lowers the default probability and therefore the loan rate as well, because the expected return to lending increases. From (B2), ⁰() =

−(1 +)², which implies therefore that the net eﬀect is positive (negative) if

, which measures the magnitude of the entrepreneurial moral hazard eﬀect, is suﬃciently small (large).

Substituting (B5) in (B4) yields therefore

^_ =p

(1 +)Γ−1 = s

(1 +)[(1−+Φ2)(1 +) +^_]

1 + [1−(^_ )](1 +) −1 (B7)

(10)

which is an implicit solution for the optimal intensity of monitoring.

To solve for the impact of on^_ requires using the implicit function theo- rem. Writing (B7) as (^ ) yields^=−()(^). Conse- quently, given the assumption thatΦ2 1,



 = (1 +)(Φ2−1)(1 +)

2(1 +^) 0 

^ = 1− (1 +) 2(1 +^)( Γ

^)≷0

which implies therefore from (B6) that ifΓ^0(that is, if the moral hazard eﬀect is weak, or the cost eﬀect dominates) then ^  0; by implication,

^ 0. An increase in the capital adequacy ratioreducesmonitoring eﬀort, which implies that the default probability increases.

By contrast, if instead Γ^  0 (that is, the moral hazard eﬀect is strong), and large enough (greater than unity) to ensure that ^  0, then ^  0. An increase in the capital adequacy ratio now increases monitoring eﬀort, which implies that the default probability falls.