A Model of Financial Intermediation
Jesús Fernández-Villaverde
University of Pennsylvania
December 25, 2012
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
December 25, 2012
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A Model with Financial Intermediation
Previous models have a very streamlined …nancial intermediation
structure.
Many of the events of the 2007-2010 recession were about breakdowns
in intermediation.
Kiyotaki and Gertler (2011) incorporate a richer …nancial intermediation
sector.
In particular, we will deal with liquidity.
Di¤erent concepts of liquidity.
Help us to think about unconventional monetary policy.
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A Model of Financial Intermediation
December 25, 2012
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Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
December 25, 2012
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A Model of Financial Intermediation
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A Model of Financial Intermediation
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Representative Household
Preferences:
E0
∞
∑ βt
log (ct
γct
1)
t =0
l 1 +ϑ
χ t
1+ϑ
!
Household can save in:
1
Deposits at the …nancial intermediary, dt , that pay an uncontingent
nominal gross interest rate Rt .
2
Public debt, dgt , that pay an uncontingent nominal gross interest rate Rt .
3
Arrow securities (net zero supply in equilibrium).
The budget constraint is then:
ct + dh,t = wt lt + Rt
1 dh,t 1
+ Tt + zt
where dh,t = dt + dgt .
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
December 25, 2012
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Representative Household
Continuum of members of measure one with perfect consumption
insurance within the family.
A fraction 1
f are workers and f are bankers.
Workers work and send wages back to the family.
Bankers run a bank that sends (non-negative) dividends back to the
family.
In each period, a fraction (1 σ) of bankers become workers and a
fraction (1 σ) 1 f f of workers become bankers. Why?
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
December 25, 2012
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Optimality Conditions
The …rst-order conditions for the household are:
ct
1
γct
βEt
γ
ct +1 γct
λt = βEt λt +1 Rt
= λt
1
χltϑ = λt wt
Asset pricing kernel:
SDFt = β
λt
λt 1
and standard non-arbitrage conditions.
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
December 25, 2012
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Technology
Island model: continuum of islands of measure 1.
In each island, there is a …rm that produces the …nal good with capital
(not mobile) and labor (mobile across islands) and a Cobb-Douglas
production function.
Then, by equating the capital-labor ratio across islands, aggregate
output is:
yt = At ktα lt1 α
where At is a random variable.
Wages satisfy:
wt = (1
α)
yt
lt
Pro…ts per unit of capital:
zt =
Jesús Fernández-Villaverde (PENN)
yt
wt lt
kt
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Liquidity Risk
Each period, investment opportunities arrive randomly to a fraction π i
of islands.
There is no opportunity in π n = 1
πi .
Investment opportunities are i.i.d. across time and islands.
Only …rms in islands with investment opportunities can accumulate
capital.
Then:
kt +1 = ψt +1 π i (1
= ψt +1 [(1
δ) kt + it + ψt +1 π n (1
δ) kt
δ) kt + it ]
where ψt +1 is a shock to productivity of capital.
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
December 25, 2012
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Capital Good Producers
Adjustment costs in investment:
it
f
with f (1) = f 0 (1) = 0 and f 00
it
it
it 1
1
> 0.
Relative price of capital: qti .
Capital good producers:
max E0
∞
λt
∑ βt λ0
qti
it
it
1+f
t =0
it
it
it
1
Optimality condition:
qti
= 1+f
it
it
+
1
Jesús Fernández-Villaverde (PENN)
it
f
1
0
it
1
λ t +1
Et β
λt
A Model of Financial Intermediation
it + 1
it
2
f0
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it + 1
it
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Aggregate Resource Constraint
Government consumption gt .
Then:
yt = ct + 1 + f
it
it
it + gt
1
We will also have, later on, a wide set of policies, that will imply a
relatively involved government budget constraint.
I will skip details because it is mere accounting.
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
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Banks I
Banks are born with a small initial transfer from the family.
Initial equity is increased with retained earnings.
Dividends are only distributed when the bank dies.
Banks are attached to a particular island, which in this period may be
h = fi, n g .
Thus, ex post, they may not be able to lend)wholesale market.
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A Model of Financial Intermediation
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Banks II
Banks move across islands over time to equate expected rate of return:
1
Before moving, they sell their loans.
2
This allows us to forget about distributions: ratio of total …nancial
intermediary net worth to total capital is the same in each island.
Discussion: speci…city in bank relational capital?
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A Model of Financial Intermediation
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Balance Sheet
Net worth: nth .
Besides equity, banks raise funds in a national …nancial market:
1
2
Retail market: from the households, dt at cost Rt . Before investment
shock is realized.
Wholesale market: from each other, bth at cost Rbt . After investment
shock is realized.
Then, bank lend to non-…nancial …rms in their island sth . No
enforcement problem (we can think about sth as equity).
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A Model of Financial Intermediation
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Flow-of-Funds I
Balance sheet constraint (where qth is the price of a loan):
qth sth = nth + bth + dt
Evolution of net worth:
h
i
nth = zt + (1 δ) qth ψt sth
1
Rt
1 dt 1
Rbt
1 i
λ t +i h
n
λ t t +i
h
1 bt 1
Objective function of bank:
Vt = Et
∞
∑ (1
σ ) σi
β
i =1
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A Model of Financial Intermediation
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Flow-of-Funds II
Value function: maximized objective function:
Vt sth , bth , dt
= Et βt
λ t +i
λt
∑ πh
h
Jesús Fernández-Villaverde (PENN)
= max Vt
(1 σ) nth+1
+σ maxdt +1 maxsth ,bth Vt sth+1 , bth+1 , dt +1
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Financial Friction I
We need some …nancial friction to make the intermediation problem
interesting.
Simply costly-enforcement mechanism.
Diversion of funds to family:
θ qth sth
ωbth
default, and close down.
Interpretation of ω.
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A Model of Financial Intermediation
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Financial Friction II
Three cases:
1
ω = 1 (frictionless interbank market). The interbank and loan rates are
the same. They are bigger than the deposit rate if banks are constrained
(only one aggregate constrain holds).
2
ω = 0 (symmetric frictions). The interbank and deposit rate are the
same. The returns on loans if banks on non-investing islands are
constrained.
3
ω 2 (0, 1). The interbank rate lies between the return on loans and the
deposit rate.
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
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Incentive Constraint
Then, the incentive constraint (IC) is:
Vt sth , bth , dt
θ qth sth
ωbth
The Lagrangian associated with the IC is λht and:
h
λt = π i λit + π n λnt
Problem:
h
max Vt sth , bth , dt + λt Vt sth , bth , dt
h
= max 1 + λt Vt sth , bth , dt
Jesús Fernández-Villaverde (PENN)
θ qth sth
h
λt θ qth sth
A Model of Financial Intermediation
ωbth
ωbth
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Guess of Value Functions
We guess that value function is linear in states:
V sth , bth , dt
= νst sth
νbt bth
νdt dt
Interpretation of coe¢ cients:
1
νst : marginal value of assets.
2
νbt : marginal cost of interbank borrowing.
3
νdt : marginal cost of deposits.
Then:
h
1 + λt
Jesús Fernández-Villaverde (PENN)
νst sth
νbt bth
νdt dt
A Model of Financial Intermediation
h
λt θ qth sth
ωbth
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Optimality Conditions
Remember that, from the balance sheet constraint:
bth = qth sth
nth
dt
The FOC are (note the bank takes nth as given and use chain rule to
take derivatives of bth ):
dt :
h
sth :
λht : νdt nth
Jesús Fernández-Villaverde (PENN)
h
1 + λt
θ
h
νdt ) = λt θω
1 + λt (νbt
νst
qth
νst
qth
νdt
νbt
h
= λt θ (1
qth sth
A Model of Financial Intermediation
(θω
ω)
(νbt
νdt )) bth
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Reading the Optimality Conditions
Interpretation:
1
Marginal cost of interbank borrowing is higher than cost of deposits i¤
h
λt > 0 and ω > 0.
2
Marginal value of assets is higher than marginal cost of interbank
borrowing if λht > 0 and ω < 1.
3
Balance sheet e¤ect: equity in bank must be su¢ ciently high in relation
with assets and interbank borrowing.
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
December 25, 2012
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Case A: Frictionless Wholesale Financial Market I
ω = 1.
Arbitrage across asset markets:
qtb = qtl = qt
Marginal value of asset must be equal to the marginal cost of borrowing
on the interbank market
νst
= νbt
qt
The incentive constraint is simply:
qt st
bt = φ t nt
where
φt =
Jesús Fernández-Villaverde (PENN)
νt
θ
µt
A Model of Financial Intermediation
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Case A: Frictionless Wholesale Financial Market II
With a bit of work (which I skip), and by matching coe¢ cients
λ t +1
Ωt +1 Rtk+1 Rt +1
λt
λ t +1
νt = βEt
Ωt +1 Rt +1
λt
µt = βEt
where
Ω t +1 = 1
σ + σ ν t +1 + φ t +1 µ t +1
Rtk+1 = ψt
zt +1 + (1 δ) qt +1
qt
Aggregating:
qt st = φt nt
In this economy, a crisis increases the excess returns for banks of all
types.
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
December 25, 2012
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Case B: Symmetric Frictions I
ω = 0.
Deposits and interbank loans become perfect substitutes:
νt = νbt
Thus, in general, there will be di¤erences in prices of assets across
islands:
qtn > qti
and
µit > µnt
Jesús Fernández-Villaverde (PENN)
0
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Case B: Symmetric Frictions II
The leverage ratio:
qti sti
νt
= φit =
i
θ
µit
nt
n
n
qt s t
νt
φnt =
n
nt
θ µnt
qtn stn
φnt µnt = 0
ntn
With a bit of work (which I skip), and by matching coe¢ cients
0
λ t +1 h 0
Ωt +1 Rthh
Rt +1
+1
λt
λ t +1 h 0
νt = βEt
Ωt +1 Rt +1
λt
µt = βEt
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
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Case B: Symmetric Frictions III
where
0
Ωht +1 = 1
0
Rthh
+1
0
0
σ + σ νt +1 + φht +1 µht +1
0
zt +1 + (1 δ) qth+1
= ψt
qth
Note that know we need to index also by the type of the island in next
period and integrate over it.
Aggregating:
qti sti = φit nti
qtn stn
(qtn stn
Jesús Fernández-Villaverde (PENN)
φnt ntn
φnt ntn ) µnt = 0
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Case B: Symmetric Frictions IV
In this economy, a crisis increases the excess returns for banks of all
types.
Also:
λ t +1 h 0
ih 0
Ωt +1 Rkt
+1
λt
λ t +1 h 0
nh 0
Ωt +1 Rkt
> Et
+1
λt
λ t +1 h 0
Et
Ωt +1 Rbt +1
λt
λ t +1 h 0
= Et
Ωt +1 Rt +1
λt
Et
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
December 25, 2012
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Aggregation
Total bank net worth:
h
h
nth = not
+ nyt
Total net worth of old banks:
nh
h
not
= σπ h zt + (1
i
δ) qth ψt st
1
Rt
where we have net out the interbank loans.
Total net worth of new banks:
h
h
nyt
= ξ zt + (1
Aggregate balance sheet constraint:
dt =
∑
i
δ) qth ψt st
qth sth
1 dt
1
o
1
nth
h =i ,n
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
December 25, 2012
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Market Clearing
Market for securities:
sti = it + (1
stn = (1
δ) π i kt
δ) π n kt
Labor market
χltϑ = λt wt
Debt market:
dht = dt + dg
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A Model of Financial Intermediation
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Policy Experiments
Unconventional monetary policy:
1
Lending facilities.
2
Liquidity facilities.
3
Equity injections.
Classic discussion from Sargent and Wallace (1983): real bills doctrine.
How do we decide between these di¤erent policies?
E¤ect on government budget position.
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A Model of Financial Intermediation
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A Model of Financial Intermediation
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A Model of Financial Intermediation
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Lending Facilities
The central bank lends directly to banks that are constrained.
Central bank is not constrained by its balance sheet (this is more subtle
than it seems, but let us assume it for a moment).
But additional cost τ of underwriting a loan (monitoring, politics...).
The central bank does not subsidize loans...
...but, by increasing funds available, it has an impact on equilibrium
prices and allocations.
New equilibrium condition:
h
h
qth sth = qth spt
+ sgt
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
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Liquidity Facilities
Central bank discounts loans from the interbank lending market.
Banks can divert less funds from the central bank than from the regular
interbank market:
θ (1 ω g )
with ω g > 0.
Then:
qth sth = nth + bth + mth + dt
Penalty rate for discount: di¤erence in the default.
Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
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Equity Injections
Treasury transfers wealth to banks.
Government takes direct ownership position.
Then:
qth sth = nth + ntg + bth + dt
Government must pay a premium.
Why? Problems of issuing equity for banks in a crisis.
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A Model of Financial Intermediation
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Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
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Jesús Fernández-Villaverde (PENN)
A Model of Financial Intermediation
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A Model of Financial Intermediation
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A Model of Financial Intermediation
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A Model of Financial Intermediation
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Issues Ahead
More detailed structure of bank capital.
Di¤erent wholesale markets.
Heterogeneity.
Non-linearities.
Optimal policy.
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A Model of Financial Intermediation
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