1 λ of the assets of the financial intermediary hence

1 - λ of the assets of the financial intermediary. Hence lenders will only supply funds if the gains
from stealing are lower than the continuation value of the financial intermediary. This gives rise to
the following constraint:
V
j,t
≥ λ ( q
k
t
s
k
j,t
+ q
b
t
s
b
j,t
) ⇒ ν
k
t
q
k
t
s
k
j,t
+ ν
b
t
q
b
t
s
b
j,t
+ η
t
n
j,t
≥ λ ( q
k
t
s
k
j,t
+ q
b
t
s
b
j,t
) (5)
The optimization problem can now be formulated in the following way:
max
{ q
k
t
s
k
j,t
,q
b
t
s
b
j,t
}
V
j,t
, s.t. V
j,t
≥ λ ( q
k
t
s
k
j,t
+ q
b
t
s
b
j,t
)
From the first order conditions we find that ν
b
t
= ν
k
t
. Hence the leverage constraint (5) can be
rewritten in the following way:
ν
k
t
( q
k
t
s
k
j,t
+ q
b
t
s
b
j,t
) + η
t
n
j,t
≥ λ ( q
k
t
s
k
j,t
+ q
b
t
s
b
j,t
) ⇒ q
k
t
s
k
j,t
+ q
b
t
s
b
j,t
≤ φ
t
n
j,t
, φ
t
=
η
t
λ - ν
k
t
(6)
where φ
t
denotes the ratio of assets to net worth, which can be seen as the leverage constraint of
the financial intermediary. The intuition for the leverage constraint is straightforward: a higher
shadow value of assets ν
k
t
implies a higher value from an additional unit of assets, which raises the
continuation value of the financial intermediary, thereby making it less likely that the banker will
steal. A higher shadow value of net worth η
t
implies a higher expected profit from an additional
unit of net worth, while a higher fraction λ implies that the banker can steal a larger fraction
of assets, which induces the household to provide less funds to the banker, resulting in a lower
leverage ratio everything else equal. Substitution of the conjectured solution into the right hand
side of the Bellman equation gives the following expression for the continuation value of the financial
intermediary:
V
j,t
= E
t
h
Ω
t +1
n
j,t +1
i
,
Ω
t +1
= β Λ
t,t +1
(1 - θ ) + θ [ η
t +1
+ ν
k
t +1
φ
t +1
]
Ω
t +1
can be thought of as a stochastic discount factor that incorporates the financial friction. Now
9

substitute the expression for next period’s net worth into the expression above:
V
j,t
= E
t
h
Ω
t +1
n
j,t +1
i
= E
t
h
Ω
t +1
(1 + r
k
t +1
) q
k
t
s
k
j,t
+ (1 + r
b
t +1
) q
b
t
s
b
j,t
- (1 + r
d
t +1
) d
j,t
+ n
g
j,t +1
- ˜ n
g
j,t +1
i
= E
t
h
Ω
t +1
(
r
k
t +1
- r
d
t +1
)
q
k
t
s
k
j,t
+
(
r
b
t +1
- r
d
t +1
)
q
b
t
s
b
j,t
+
(
1 + r
d
t +1
+ τ
n
t +1
- ˜ τ
n
t +1
)
n
j,t
i
(7)
After combining the conjectured solution with (4), we find the following first order conditions:
η
t
= E
t
h
Ω
t +1
(
1 + r
d
t +1
+ τ
n
t +1
- ˜ τ
n
t +1
)
i
(8)
ν
k
t
= E
t
h
Ω
t +1
(
r
k
t +1
- r
d
t +1
)
i
(9)
ν
b
t
= ν
k
t
= E
t
h
Ω
t +1
(
r
b
t +1
- r
d
t +1
)
i
(10)
Ω
t +1
= β Λ
t,t +1
(1 - θ ) + θ [ η
t +1
+ ν
k
t +1
φ
t +1
]
2.2.1 Financial sector support
We assume that support provided to an individual intermediary, if provided, will be proportional
to the intermediary’s net worth in the previous period. Hence individual financial support is given
by:
n
g
j,t
= τ
n
t
n
j,t - 1
, ζ ≤ 0 , l ≥ 0
τ
n
t
= ζ ( ξ
t - l
- ξ )
Repayment of the support is parametrized proportionally to the sector’s net worth in the period
preceding the pay back period:
˜ n
g
j,t
= ˜ τ
n
t
n
j,t - 1
where ˜ τ
n
t
is a scaling factor that is obviously time dependent and incorporates the return paid by
the sector to the government over the support funds.
2.2.2 Aggregation of financial variables
Integrating the individual balance sheets of the financial intermediaries yields the aggregate balance
sheet of the financial sector:
p
t
= n
t
+ d
t
(11)
Aggregation over the asset side of the balance sheet gives the composition of the aggregated financial
system:
p
t
= q
k
t
s
k
t
+ q
b
t
s
b
t
(12)
φ
t
does not depend on firm specific factors, so we can aggregate the leverage constraint (6) across
financial intermediaries to link sector wide assets and net worth:
p
t
= q
k
t
s
k
t
+ q
b
t
s
b
t
= φ
t
n
t
(13)
10