-
Banks, their Balance Sheets and Optimal
Monetary Policy
David Aikman a,∗ , Matthias Paustian baBank of England,
Threadneedle Street, London, EC2R 8AH, UK
bCenter for European Integration Studies (ZEI B),
Walter-Flex-Str. 3, 53113Bonn, Germany
Abstract
We incorporate financial constraints in a standard dynamic new
Keynesian model.These constraints are derived endogenously from two
moral hazard problems infinancial markets: Unobservable project
choice of entrepreneurs generates a role forentrepreneurial net
worth in financial contracts, while unobservable monitoring
bycommercial banks gives rise to bank capital as a determinant in
aggregate lending.We study the transmission of shocks to monetary
policy and bank capital and totalfactor productivity. Furthermore,
we analyze optimal monetary policy. It is shownthat interest rate
rules should not respond to asset prices or loan supply.
Optimalmonetary policy does not fully stabilize the inflation rate
in response to shocks.However, the optimal amount of volatility in
the inflation rate is extremely small.
Key words: Banks; moral hazard; financial market imperfections;
RamseyJEL classification: E44; E32; E52
1 Introduction
A number of empirical studies have identified an important role
of banksfor real economic activity. Samolyk (1994) examines the
relationship betweenbanking conditions and economic performance at
the U.S. state level andshows how regional banking conditions can
affect local economic activity byimpacting on a region’s ability to
fund local investments. Peek and Rosengren(2000) focus on real
estate lending of Japanese banks in U.S. states following
∗ Corresponding authorEmail addresses:
[email protected] (David Aikman),
[email protected] (Matthias Paustian).
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the Japanese banking crisis. They find that reduced real estate
lending byJapanese banks was not compensated by increased lending
of domestic banksand had a significant and sizeable impact on real
construction projects. Har-rison et al. (1999) analyze the
relationship between economic growth and costof bank monitoring
using U.S. state level data. They find an inverse, albeitsmall,
relationship between state per-capita income and the cost of
banking- suggesting a feedback between real and financial
development. Gambacortaand Mistrulli (2004) analyze a sample of
Italian banks and show that bankcapital matters in the propagation
of different types of shocks, owing to theexistence of regulatory
capital constraints and imperfections in the market forbank
fund-raising.
Despite the rich empirical literature on the role of banks,
there is only a smallnumber of theoretical macroeconomic models
linking the banking sector tothe macroeconomy. 1 Friedman (1991)
notes: “Traditionally, most economistshave regarded the fact that
banks hold capital as at best a macroeconomic ir-relevance and at
worst a pedagogical inconvenience.”The lack of a
tractablemacroeconomic model that provides a meaningful role for
banks in intermedia-tion has left several important questions
unanswered: Do variations in banks’net worth significantly affect
macroeconomic outcomes? Does incorporatingbanks into a standard
business cycle model change the propagation of busi-ness cycle
shocks? Are there implications for the conduct of monetary
policyarising from an explicit focus on bank loan supply?
The contribution of this paper is to provide a small
quantitative general equi-librium model in which banks serve a
meaningful role in intermediation anduse it for policy analysis.
Its virtue is that it shares essential features with theworkhorse
models for business cycle analysis: The real business cycle
modeland its successor, the new Keynesian sticky price model. We
build on the foun-dation of Chen (2001), who in turn builds on
Holmstrom and Tirole (1997). Inthat model two moral hazard problems
in the market for loans and depositsimply an important role for
firms’ net worth and for bankcapital in incentivecompatible
financial contracts: Unobservable project choice of
entrepreneursgenerates a role for entrepreneurial net worth in
financial contracts, while un-observable monitoring by commercial
banks gives rise to bank capital as adeterminant in aggregate
lending. We add to this moral hazard setup variablelabour supply,
aggregate capital accumulation, concave preferences over
con-sumption, monopolistic competition and sticky prices to be able
to analyzemonetary policy questions.
Two papers are closely related. The first is Meh and Moran
(2004). These
1 See Holmstrom and Tirole (1997), Bolton and Freixas (2001),
Ennis (2001), Chen(2001), Van den Heuvel (2002) and Meh and Moran
(2004) for a non exhaustive listof theoretical models on banks with
some reference to the macroeconomy.
2
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authors also build a general equilibrium model with a banking
sector on thefoundations of Holmstrom and Tirole (1997). However,
their paper does notexplore the implications of credit market
imperfections for the conduct ofoptimal monetary policy as is the
focus of this paper. Furthermore, theirpaper differs in a number of
modeling choices from this one. 2 The secondone is Collard and
Dellas (2005). They analyze optimal monetary policy ina model with
quadratic costs to price adjustment, a monetary friction andtax
distortions. They show that optimal monetary policy tolerates only
smalldepartures from complete price stability, thereby confirming
the case for pricestability made by Goodfriend and King (2001). The
present paper analyzeswhether credit market imperfections give rise
to more substantial deviationsfrom stabilizing inflation.
We first show that incorporating a dual moral hazard problem in
credit exten-sion significantly affects the response of
macroeconomic variables to businesscycle shocks. Credit constraints
bring about an amplified response of macro-economic variables to
technology and monetary shocks relative to the casewhere moral
hazard in financial markets is absent. The amplification is dueto
the fact that the net worth of both entrepreneurs and banks serves
to ame-liorate moral hazard problems and is affected by market
prices. Variationsin the price of physical assets therefore feed
back into the economy via thecollateral effects stressed in
Kiyotaki and Moore (1997). We then show thatthe contribution of
banks’ net worth to amplifying and propagating businesscycle
fluctuations in this model is relatively small. The reason is that
firms’net worth and banks’ own funds are substitutes in the
determinant of aggre-gate credit extension. In any reasonably
calibrated model however, banks networth is likely to be an order
of magnitude smaller than the net worth ofentrepreneurs. Therefore
variations in banks own funds are only a small por-tion of total
net worth and must be large to significantly impact on
aggregateoutcomes.
We then turn to how monetary policy should optimally be
conducted in thismodel. To what extent do distortions in the credit
markets give rise to a de-parture from full price stability? This
question is addressed in two parts. First,it is asked whether
interest rate rules should include a response to financialvariables
such as asset prices or loan supply. It is shown that a
mechanicalresponse to asset prices or loan supply can be quite
costly in welfare terms,leading to welfare losses equivalent to up
to one per cent of period consumptionin per period.
2 First, Meh and Moran (2004) assume intra-period loan contracts
contrary to theinter-period contracts used here. Second, they base
the moral hazard problem in theinvestment good sector whereas our
model assumes moral hazard in intermediategood production. Third,
they utilize a limited participation assumption to introducea role
for monetary policy, whereas we use a sticky price framework.
3
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Second, we search numerically within a broad family of rules for
the optimalmonetary policy rule that maximize household’s expected
utility . Optimalmonetary policy does not fully stabilize the price
level in response to shocksto bank’s net worth. Rather, it takes a
slightly counter-cyclical stance andsmooths some of the inefficient
fluctuations stemming from the exogenousshocks to banks net worth.
Quantitatively, the amount of inflation inducedby optimal monetary
policy is very small and roughly comparable to what isfound by
Collard and Dellas (2005) for the case of tax distortions.
This paper is organized as follows. Section 2 introduces the
model set up.Section 3 explains the structure of financial
contract. Section 4 discusses ag-gregation and equilibrium
conditions. In section 5, we discuss the conditionsthat imply a
role for bank capital and firms net worth in intermediation.
Sec-tion 6 contains the calibration. Section 7 explains the
internal propagationmechanism in this model via impulse responses.
Section 8 analyzes optimalmonetary policy. Finally, section 9
concludes.
2 Structure of the model
There are three agents with distinct preferences and access to
production tech-nologies: Households, bankers and entrepreneurs.
The production structurefor the final output good has two layers.
Intermediate goods production takesplace at the initial layer. Both
households and entrepreneurs can produce in-termediate goods from
capital goods, but have access to different productiontechnologies.
Households marginal product of capital is decreasing, whereasthat
of entrepreneurs is constant. In the steady state of our model, due
tocredit constraints, households hold too much capital relative to
the surplusmaximizing allocation which equates the marginal product
of capital of thetwo agents. Intermediate goods as well as labour
are an input in the productionof a continuum of slightly
differentiated goods at the second layer of produc-tion. Imperfect
competition at this layer allows us to introduce nominal
pricerigidities. A competitive bundler then aggregates the
differentiated goods intofinal output.
The final output good can be either consumed or be used as
investment intoaugmenting the aggregate capital stock. Investment
takes place in a compet-itive sector that uses output as material
input and combines it with rentedexisting capital stock to produce
new capital goods. At the end of a periodafter intermediate good
production has taken place, the owners of existingcapital rent out
their capital stock to the investment good sector and receivetheir
rental rate. 3
3 Note that both production of intermediate goods and subsequent
rental of the
4
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2.1 Households
We utilize a money in the utility function framework. Households
are infi-nitely lived, their period utility is separable in
consumption and leisure. Theirobjective is to maximize
E0
∞∑
t=0
βt
(cht
)1− 1σ
1− 1σ
+χ
1 + ψ(1− nt)1+ψ + H (mt)
. (1)
Here, mt are real money balances. In every period, households
choose theirconsumption cht , how much labour nt to supply, how
much deposits dt to makeand how much capital goods kht to purchase.
The relative price of capitalgoods in terms of the final
consumption goods is qt and the relative price ofintermediate goods
is vt. Households receive real wage income w
rt nt, rental
income Zkt from renting their capital holdings to the investment
good sector,lump sum profits from the monopolistic retailer Πt, the
revenue from thesales of home production vtG
(kht−1
)and of last period’s capital stock as well
as the real gross return from yesterdays deposits rdt−1dt−1. The
central bankadjusts nominal monetary transfers trt to households in
order to support itspolicy instrument rnt , the nominal interest
rate. Households face the followingperiod-by-period budget
constraint in real terms:
qt(kht − (1− δ̃)kht−1
)+ dt + c
ht + mt =
rdt−1dt−1 + wrt nt + vtG
(kht−1
)+ Zkt (1− δ̃)kht−1 + Πt + trt +
mt−1πt
.
Here, δ̃ is the rate of depreciation of the physical capital
used in householdproduction of intermediate goods and πt ≡ Pt/Pt−1,
where Pt is the aggregateprice index. The first order conditions
with respect to kht , dt and nt are:
qt(cht
)− 1σ = βEt
(cht+1
)− 1σ
[(1− δ̃)(qt+1 + Zkt+1) + vt+1G
′ (kht
)], (2)
(cht
)− 1σ = βEt
rdtπt+1
(cht+1
)− 1σ , (3)
χ(1− nt)ψ = wrt(cht
)− 1σ . (4)
Since monetary policy follows an interest rate rule, we omit the
first-ordercondition for money holdings. In this model, the
first-order condition for real
capital stock occur within one period. The same unit of capital
can therefore beused for production of intermediate goods and
subsequent rental to the investmentgood sector.
5
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money balances merely serves to back out the money supply that
supports agiven interest rate.
2.2 Entrepreneurs
There exists a continuum of risk neutral entrepreneurs.
Entrepreneurs have aconstant probability πe of surviving to the
next period. 4 Dying entrepreneursare replaced by new entrepreneurs
as to keep the population size constant.They derive utility from
consuming goods ce and enjoying private benefitsB ∈ {0, b, B} per
unit of capital they purchase. Entrepreneur i maximizes
theintertemporal objective function:
E0
T̃∑
t=0
βt (ceit + Bkeit) . (5)
0 < β < 1 is a constant discount factor, T̃ is the
stochastic time of death andkei denotes entrepreneur i’s holdings
of physical capital.
Entrepreneurs optimize subject to the following period-by-period
budget con-straint:
qtkeit = wit + l
dit − ceit. (6)
where wit is the entrepreneur’s net worth in period t, and ldit
is the entrepre-
neur’s demand for loans from the banking sector in period t.
At the beginning of each period, an entrepreneur can choose
between threeproduction technologies. In case of success, each
technology yields a total netreturn of R intermediate goods per
unit of capital invested as well as (1−δ) ofthe capital stock which
is rented to the investment good sector at rate Zkt andthen sold at
price qt. If the project fails, it yields no output and the
capitalstock is lost as well. The return is verifiable by all
agents in the economyat zero cost. Projects differ both in their
probability of success and in theprivate benefits 5 per unit of
capital they offer to the investing entrepreneur,as described in
the box below:
4 The purpose of this assumption is to preclude the possibility
that the entre-preneurial sector will eventually accumulate
sufficient net worth to become self-financing (see Carlstrom and
Fuerst (1997), and Bernanke et al. (2000) for furtherdiscussion).
An often-used alternative would be to assume that entrepreneurs
dis-count the future at a higher rate than households. See
Carlstrom and Fuerst (2001)for a discussion of the differing
macroeconomic implications of each assumption.5 Private benefits
capture the idea that the entrepreneur gets some kind of
non-monetary return from some projects. A common interpretation is
that they captureeffort. Lower effort is clearly a benefit to the
entrepreneur, but leads to a lowerprobability of success.
6
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project probability of success private benefits
‘good’ pH 0
‘bad’ pL b
‘rotten’ pL B
The “good” project has a high probability of succeeding, pH ,
but offers noprivate benefits; the “bad” project has a low
probability of succeeding, pL(pH − pL ≡ ∆p > 0), and private
benefits, b, per unit of capital invested;finally, the “rotten”
project shares the same low probability of succeeding,pL , but
offers higher private benefits, B, per unit of capital invested (B
> b).Notice that this implies that in the absence of monitoring,
entrepreneurs willalways prefer the rotten project to the bad one.
This simple two outcome-threeaction principal-agent set-up is taken
from Holmstrom and Tirole (1997).
We assume that only the good project has a positive expected net
presentvalue:
pHEt[vt+1R + (1− δ)(qt+1 + Zkt+1)
]− rtqt > 0 (7)
0 > pLEt[vt+1R + (1− δ)(qt+1 + Zkt+1)
]+ B − rtqt
where r is the opportunity cost of funds.
Households are unable to verify the flow of private benefits to
entrepreneurs atany cost. This is the source of the model’s first
moral hazard problem: entre-preneurs may be tempted to deliberately
reduce the probability of a projectsucceeding in order to enjoy
private benefits. Given (7), however, only thegood project can be
chosen in equilibrium. We therefore require an incentiveconstraint
to induce entrepreneurs to implement this outcome:
pHEt(vt+1R + (1− δ) (qt+1 + Zkt+1)− fit)ket ≥ (8)pLEt(vt+1R +
(1− δ)(qt+1 + Zkt+1)− fit)keit + Bkeit
Here, fit is the portion of the per unit project return that is
pledged to anoutside financier. The following constraint limits the
due repayment fit thatany entrepreneur can promise to repay without
destroying incentives
fit ≤ Et[vt+1R + (1− δ)(qt+1 + Zkt+1)
]− B
∆p(9)
Since the per unit repayment is independent of entrepreneur
specific variableswe henceforth use the symbol Ft. This simply
states that the entrepreneur’sexpected return from carrying out the
good project must exceed the expectedreturn from the rotten
project. The entrepreneur cannot credibly commit torepay more than
Ft per unit of capital.
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With linear preferences, the entrepreneur only cares about the
present dis-counted value of her consumption. In the neighbourhood
of the steady state,the agency problem implies that the expected
rate of return from accumulatingcapital exceeds the discount
factor. 6 The entrepreneurs problem has a simplesolution: the net
present value of consumption will be maximized by postpon-ing
consumption and accumulating wealth until the period in which she
re-ceives a signal to exit the economy. She will then consume her
entire resourcesbefore exiting. In every period, the entrepreneur
receives an endowment of ee
units of the final good. 7 Thus, aggregate entrepreneurial
consumption andnet worth, respectively, are given by:
Cet = (1− πe) pH[vtR + (1− δ)(qt + Zkt )− Ft−1
]Ket−1, (10)
Wt =πepH
[vtR + (1− δ)(qt + Zkt )− Ft−1
]Ket−1 + e
e. (11)
2.3 Banks
Banks play a distinct role in the model because they have a
monitoring technol-ogy. By spending a certain amount of resources c
proportional to the projectscale under their supervision, they can
distinguish rotten projects from theothers. Monitoring is assumed
to imply a strong enough punishment on theentrepreneur such that
rotten projects will not be chosen by the entrepreneursin a
monitoring equilibrium. Their incentive constraint can be relaxed
to:
Ft ≤ Et[vt+1R + (1− δ)(qt+1 + Zkt+1)
]− b
∆p(12)
Ceteris paribus, the entrepreneur can now pledge a higher
repayment to bankersand therefore achieve higher leverage.
Banks also face a moral hazard problem in that depositors cannot
observewhether this monitoring has actually taken place. For
monitoring to occur, asecond incentive constraint must be
satisfied, regulating the share of the duerepayment that must be
paid to the bank, 0 < Rbt < 1:
pHRbtFtk
ejt − ckejt ≥ pLRbtFtkejt. (13)
6 To see this define, Ωt ≡ vt+1R+(1−δ)(qt+1 +Zkt+1)−Ft In the
aggregate, rate ofreturn on entrepreneurial capital is EtΩtK
et
πeΩt−1Ket−1+eeWithout special assumptions on
the distribution of the shocks, this expression is not
guaranteed to always exceed thediscount factor. Calibrating πe
-
Solving for Rbt yields:
Rbt ≥c
∆pFt. (14)
The intuition is identical to the case for the entrepreneur: in
order to beinduced to carry out costly monitoring, the bank must be
given a sufficientlyhigh financial stake in the outcome of the
project. This required payment tothe bank lowers the entrepreneur’s
leverage.
For monitoring to exist in equilibrium, it must be the case that
the gains fromintermediary finance in terms of reduced opportunity
costs of not shirkingoutweigh the delegation costs of providing the
bank with the necessary incen-tives to monitor. 8 Depending on the
model’s parameters both a monitoringequilibrium (indirect finance)
and an equilibrium without monitoring (directfinance) are possible.
We calibrate the model such that in the steady banksmonitor and
analyze small perturbations around the steady state.
The formal problem of a banker in a monitoring equilibrium can
be describedas follows. Denote by kej the stock of entrepreneurial
capital under bank j’ssupervision.
Banks are assumed to be risk neutral and face a constant
probability πb ofsurviving to the next period. A typical bank j
maximizes the following lifetimeutility:
E0
T̃∑
t=0
βt(cbjt − ckejt
) . (15)
where cbj is bank j’s consumption.
Bank j uses its net worth net of monitoring costs, ajt − ckejt,
and its depositsfrom households dj to finance loans l
sjt to entrepreneurs:
lsjt = ajt − ckejt + djt. (16)
The agency cost problem again implies that the rate of return
from investingexceeds the discount factor β and the bank will find
it optimal to postponeconsumption and accumulate net worth until
the period in which it exits theeconomy. We again assume that the
banker can consume the accumulated networth just before death.
Banks receive an endowment eb of the final good ineach period. Once
the signal to die realizes, a fraction 1−πb of bankers consume8 We
implicitly make the assumption that project returns within a bank’s
loanportfolio are perfectly correlated. As Diamond (1984) shows,
with perfectly diver-sified loan portfolios (the opposite extreme),
the delegation costs of involving anintermediary asymptotically
fall to zero as the number of projects being monitoredrises and the
bank can be induced to monitor without holding any own capital.
9
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their net worth. Consumption and aggregate net worth at the
beginning ofthe next period are given by:
Cbt =(1− πb
) [pHFt−1Ket−1 − rdt−1Dt−1
], (17)
At = πb[pHFt−1Ket−1 − rdt−1Dt−1
]+ ee. (18)
2.4 The final goods sector and price setting
There is a continuum of imperfectly competitive firms indexed by
z with unitmass, producing slightly differentiated wholesale
products Yt(z). A competitivebundler uses these varieties Yt(z) as
inputs into production of a homogeneousfinal good Y . The
production function of the bundler is given by the Dixit-Stiglitz
index:
Yt =[∫ 1
0Yt(z)
²−1² dz
] ²²−1
. (19)
where ² > 1. Each wholesale firm z uses intermediate goods
Mt(z) and labourNt(z) to produce output according to
Yt(z) = TtMt(z)γNt(z)
1−γ (20)log Tt = ρ log Tt−1 + vt (21)
Cost minimization of firm z gives rise to these first-order
conditions
wrt =Xt (1− γ)Tt(
Mt(z)
Nt(z)
)γ, (22)
vt =Xt γTt
(Mt(z)
Nt(z)
)γ−1. (23)
Here Xt denotes real marginal cost. Since all firms optimize
against the samevector of input prices, marginal cost is equal
across firms.
We introduce price stickiness following the widely used approach
of Calvo(1983). In any given period, there is a constant
probability 1−θ of receiving asignal that allows the firm to reset
its price. Is the random signal not received,the firm carries on
the price posted in the last period and satisfies any demandat that
price. The problem of a firm that receives a signal to change its
price inperiod t is to maximize expected profits as valued by the
households’ marginalutility of consumption in those states of the
world where the price remainsfixed. The firm’s problem can be
written as
maxP ∗t (z)
Et∞∑
i=0
(θβ)i(Cht
)− 1σ
[P ∗t (z)Pt+i
]1−²Yt+i −Xt+i
[P ∗t (z)Pt+i
]−²Yt+i
(24)
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The optimal price P ∗t can then be expressed as
P ∗t =²
²− 1Et
∑∞i=0(θβ)
i(Cht+i
)− 1σ Xt+iP
²t+iYt+i
Et∑∞
i=0(θβ)i(Cht+i
)− 1σ P ²−1t+i Yt+i
(25)
As is well known, with Calvo (1983) pricing, the consumption
based priceindex Pt evolves as
Pt =[θP 1−²t−1 + (1− θ) (P ∗t )1−²
] 11−² . (26)
2.5 The investment good sector
The end of period capital stock K̃t−1 = (1−δ)pHKet−1+(1−δ̃)Kht−1
is combinedwith consumption goods as material input to produce new
capital goods. Theproduction function is concave in material input.
This gives rise to a price ofcapital - henceforth asset price -
that is increasing in the investment to capitalratio, as in Tobin’s
q. Existing capital and newly produced capital are
perfectsubstitutes, so they sell at the same price. The production
function for newphysical capital goods of firm j is
Y kt,j = µIφt,jK̃
1−φt−1,j (27)
Here, 0 < φ < 1, and µ is a constant chosen such that the
steady state price ofnew capital is unity. Capital producing firms
maximize their profits, qtY
kt,j −
Zkt K̃t−1,j − It,j, through choice of It,j and K̃t−1,j. The
first-order conditionsare:
1 = φqtµ
(It,j
K̃t−1,j
)φ−1, (28)
Zkt = (1− φ)qtµt(
It,j
K̃t−1,j
)φ. (29)
Due to the linear homogeneity of the production function, all
firms choose thesame ratio of investment to capital, which allows
to drop the subscript j andmove to aggregate variables. The
evolution of the aggregate capital stock is
Kt = µIφt K̃
1−φt−1 + K̃t−1, (30)
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3 The financial contract and monetary policy
In order to keep the model tractable, we follow Carlstrom and
Fuerst (1997)by assuming that there exists enough anonymity between
agents that only oneperiod contracts are feasible. The contract
specifies the division of the projectreturn amongst the parties
providing finance, contingent on the two possibleproject
outcomes.
As Holmstrom and Tirole (1997) and Chen (2001) demonstrate, one
optimalcontract in this setting will involve the entrepreneur and
bank investing alltheir net worth, w and a respectively, and
households putting up the difference(q + c) ke − w − a. If the
project succeeds, the net per unit return is dividedso that the
incentive constraints (12), (14) bind:
Ft = Et[vt+1R + (1− δ)(qt+1 + Zkt+1)
]− b
∆p(31)
Rbt =c
∆pFt. (32)
If the project fails, no one is paid anything.
We assume that the central bank sets the nominal interest rate
following asimple rule:
rnt =
(1
β
)1−ρr (rnt−1
)ρrπ
(1−ρr)(1+κ)t e
ut . (33)
In loglinear form, rules of this type have been employed in a
number of studies.ρr parameterizes the degree of inertia in the
rule, (1 + κ) gives the long runresponse of the nominal interest
rate to inflation and ut represents a monetarypolicy shock. We link
the nominal interest rate with the real interest rate ondeposits
via the Fisher equation, rnt = Etr
dt πt+1.
9 The reason we specify de-posit contracts in real terms is that
we later analyze to what extent monetarypolicy should depart from
full price stability. Sticky nominal goods prices arethe only
relative price that is affected by inflation in this setup.
Therefore,it is easy to identify the channel through which
inflation affects real activitywhich would be more difficult with
multiple nominal contracts.
9 Specifying the interest rate policy via the Fisher equation is
a shortcut for a moreexplicit modeling of policy manipulating the
nominal rate on other assets such asbonds.
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4 Equilibrium
Aggregation is straightforward here, because entrepreneurs
obtain the sameamount of capital per unit of net worth. Despite the
heterogeneity of net worthacross entrepreneurs and banks, all that
matters is the aggregate quantity ofeach type. We use upper-case
letters to denote economy-wide aggregate quan-tities. The model is
closed with market clearing conditions for consumptiongoods,
capital goods, intermediate goods and credit.
Market clearing for physical capital goods requires:
Ket + Kht = Kt. (34)
The market clearing condition for bank credit is given by the
aggregation of(6) and (16):
Ldt = qtKet −Wt = At + Dt − cKet = Lst . (35)
Market clearing for intermediate goods requires:
Mt = pHRKet−1 + G(K
ht−1). (36)
Market clearing for the final good requires:
Yt + eb + ee − cKet = Cet + Cbt + Cht + It. (37)
Free entry into the market for banks implies that banks earn
only the minimalamount of informational rents inducing them to
monitor. Therefore, the rateof return on Dt real units of deposits
is given by:
rdt Dt = pH(1−Rbt)FtKet (38)
An individual bank may be faced with all their projects under
supervisionfailing. In order to make this consistent with a safe
return on householdsdeposits, we assume as in Carlstrom and Fuerst
(1997) a mutual fund thatpools households savings and diversifies
the idiosyncratic risk. 10 Aggregaterisk cannot be diversified. One
can therefore not fully exclude the possibilitythat due to
unexpectedly low market prices vt+1 and qt+1, even the
successfulentrepreneurs cannot repay their loans. However, the fact
that entrepreneurscan only credibly commit to repay a fraction of
the value of a successfulproject implies that the unpledged portion
of the project return acts as abuffer against aggregate shocks. In
our calibration entrepreneurs can crediblepledge less than half of
the project value, therefore even very large shocks are
10 Recall that each bank faces a perfectly correlated loan
portfolio. Then in caseall projects fail, an individual bank will
not be able to service its obligations todepositors.
13
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unlikely to cause default of successful entrepreneurs and the
deposit returncan indeed be viewed as safe. 11
We end the discussion of equilibrium with an aggregate
production function.As first pointed out by Yun (1996), the
following relation between aggregatefactor inputs Mt, Nt and
aggregate output Yt holds.
Yt ≡[∫ 1
0Yt(z)
²−1² dz
] ²²−1
=Tt
P̃tMγt N
1−γt . (39)
Here P̃t is a price dispersion terms that captures the
efficiency costs of pricedispersion across producers due to sticky
prices. It evolves as:
P̃t = (1− θ)(
P ∗tPt
)−²+ θ
(Pt−1Pt
)−²P̃t−1 (40)
General equilibrium in this model are sequences{Pt, P
∗t , P̃t, Z
kt , wt, qt, vt, r
dt , r
nt
}∞t=0
as well as{Xt, Yt, Nt, Mt, It, C
ht , C
et , C
bt , Dt, R
et , R
bt , Kt, K
et , K
ht , At, Wt
}∞t=0
that
satisfy the following equations:(2) - (4), (10), (11), (17),
(18), (22), (23), (25) -(40), the Fisher equation and a
transversality condition for household’s capitalholding.
We define two key financial variables and analyze its cyclical
variation in laterparts of the analysis: Firms’ leverage qtK
et /Wt and banks’ capital to asset
ratios At/(Lt + cKet ).
5 When are bankcapital and firms’ net worth essential?
Combining (38) with (35) and (31)-(32), we find a relation
linking equilibriumpurchases of capital by entrepreneurs to the sum
of entrepreneurial and banknet worth – the key equation in the
model:
KEt =Wt + At
Zt. (41)
where the variable Zt is defined as follows:
Zt ≡ qt + c− (rdt )−1pHFt(1−Rbt) (42)11 Since households’ labour
income is risky, a state contingent return on households’deposits
that is negatively correlated with the labour income would dominate
thenon-contingent pay-off of deposits used here. Such a contract
would be feasible giventhe large size of entrepreneurial net worth.
We do not pursue it here, since depositcontracts observed in
practice are typically non-contingent.
14
-
Zt must be positive for there to be a role for net worth. To see
this, notethat Zt is the difference between the per-unit cost of
investing – equal to thecurrent price of capital qt plus the
monitoring resource cost c – and the maxi-mum discounted expected
return that can be promised to depositors withoutdestroying
incentives. Were Zt to be negative, depositors funds would be
suffi-cient to finance the entire project without the need for
entrepreneurs’ or banks’capital. When Zt goes up, more resources
must be devoted to deterring moralhazard, implying less investment
and a greater role for net worth.
Two types of equilibria can therefore emerge in this framework.
In the first,total returns generated by the project are high
relative to the overall projectcost and credit constraints do not
bind. High project returns make it possibleto pay the entrepreneur
and bank sufficiently well to ensure that they havethe right
incentives, while maintaining a sufficiently high rate of return
fordepositors to entice them to participate in the funding of the
project. In thissituation, entrepreneurs are able to borrow enough
to buy the entire capitalstock, and the most efficient level of
output will be achieved. In this equilib-rium, the economy will
behave as if all agents had perfect information.
What happens if total project returns are not large enough to
offer depositors asufficient rate of return once the required
payment to entrepreneurs and bankshas been subtracted? In this
case, the credit constraint binds. And depositorswill ask both the
entrepreneur and the bank to invest their own funds in theproject,
thereby sharing the burden of its financing cost. Depositors, in
thiscase, will not in general make deposits that are sufficient to
fund entrepreneurs’desired borrowing needs. And as a result,
entrepreneurs will not own the entirecapital stock. The remaining
capital stock will be owned by the less productivedepositors.
Output is therefore below its optimal level.
This equilibrium provides a clear role for both entrepreneurial
net worth andbank capital. Capital, net worth and deposits can be
thought of as comple-ments for the financing of projects in this
situation: better capitalized bankswill be able to attract more
deposits and higher net worth firms will be ableto borrow more.
6 Functional forms and calibration
The functional form we choose for households production function
is G(kht ) =λε
(kht−1
)ε. We take one period to be a quarter, and set β to 0.99,
consistent
with a gross real interest rate of 1.04. The depreciation rate
of entrepreneurialcapital δ is set to 0.025. Households capital
depreciates at a slightly higher rate
15
-
such that (1− δ)pH = 1− δ̃. 12 Household utility is taken to be
logarithmic inleisure and consumption, implying that ψ = −1 and σ =
1. This assumptiontogether with a steady state labour supply of
1
3as governed by χ yields a
Frisch elasticity of labour supply −1−NψN
= 2. The capital share γ is set at 0.35.ρ is set at the standard
value 0.9. The standard deviation of the innovationto total factor
productivity is also standard, σu = 0.007.
As in Bernanke et al. (2000) we require firms’ leverage to be
roughly 2 inthe steady state. The fraction of total capital held by
entrepreneurs in thesteady state is set to 0.5. Bank’s capital to
asset ratio is taken to be 10 percent. Steady state monitoring
costs per dollar of extended credit amount toroughly 6 cents. This
is in line with the findings of Harrison et al. (1999),who report
average costs of. 5.85 cents for banks in 48 U.S. states for
theperiod 1982-1994. The annualized return to bank capital is
calibrated to be10 per cent. To match these features, we set ε at
0.5, c at 0.03 and λ at 1, theper-unit net return R at 22.5 and b =
0.35 . We follow Carlstrom and Fuerst(1997) and set the quarterly
failure rate at 1 per cent, requiring pH = 0.99.The difference in
success probabilities, ∆p, is taken to be 0.35. We choose φsuch
that the elasticity of the price of physical capital with respect
to theinvestment to capital ratio is 0.75. This is in line with
empirical estimates forOECD countries obtained by Eberly (1997)
ranging from 0.51 to 1.51.
We calibrate ² = 7, implying a markup of 16.66 per cent, roughly
consistentwith the evidence in Basu and Fernald (1993) for U.S.
manufacturing. Theaverage duration of price contract is calibrated
to 2 quarters, i.e. θ = 1
2. This
in line with the evidence in Bills and Klenow (2004), who find
an averagefrequency of price adjustment of roughly 6 months when
excluding temporarysales. Finally, inertia in the monetary policy
rule ρr is set to 0.9 and theinflation response 1 + κ = 1.5.
7 Model analysis: Impulse responses
To illustrate the working of the model, we plot impulse response
functions toshocks to bank capital and monetary policy. To identify
how the moral hazardproblems influence the business cycle dynamics,
we also plot the response ofthe variables in an economy without any
financial frictions. 13 We start witha shock to bank capital.
12 This simplifies the law of motion for capital and as well the
computation of theno moral hazard reference model.13 This
frictionless reference model is sketched in our technical appendix.
All deepparameters are identical to the moral hazard model, we only
assume that monitoringand project choice are observable and
contractable.
16
-
7.1 Bank Capital shock
We consider an exogenous shocks to bank capital, impulse
responses are dis-played in Figure 1. A fall in bank capital that
is not brought about endoge-nously by movements in the economy
could represent international factors.The Japanese banking crisis
for instance had an influence on the loan sup-ply of U.S.-based
Japanese banks as shown in Peek and Rosengren (1997).More
generally, we consider such a shock in order to isolate the role of
thebanking sector for business cycle dynamics. In particular, we
consider a 25 %fall in banks net worth. For the baseline
calibration the capital-asset ratio ofbanks is 10 %, so for
constant loan supply, this corresponds to a decline of
thecapital-asset ratio to 7.5%.
Note that a shock to bank capital would not have any effects on
aggregateactivity in a frictionless benchmark economy as households
would be willingto increase deposits. In the economy with agency
costs in the banking sector,the fall in bank capital requires that
banks cut back their lending as indicatedby (41). With less own
funds at stake, banks can only be induced to monitor ifthey extend
a smaller loan. Given that entrepreneurs will be able to hold
lesscapital, households must be induced to buy extra capital to
clear the market.The required fall in the asset price lowers the
return on the entrepreneursprojects and this effect decreases
entrepreneurial net worth.
Entrepreneurs’ incentive constraints are affected by two
opposing movementsin relative prices: The price of intermediate
goods rises as the redistributionof capital holdings reduces total
supply of intermediate goods. A rise in theprice of intermediate
goods increases the incentive to choose the good project,the fall
in the asset price adversely influences incentives. The former
effectdominates and entrepreneurs can credibly pledge to repay a
larger amountper unit of his capital holdings. That effect
ameliorates the adverse impact ofthe fall in banks net worth on
aggregate activity, but cannot offset it.
The dynamics of output are also affected by the response of
labour. Laboursupply falls as the drop in intermediate good
production decreases labour’smarginal product and therefore the
real wage. Note that a one time fall inbank net worth induces a
considerable amount of endogenous persistence intothe dynamics of a
number of variables. Although the shock only lasts for oneperiod,
it takes about 10 quarters for the effect on output to die
away.
It should be noted that large variations in bank capital are
necessary to gen-erate sizeable fluctuations in other macroeconomic
variables. To understandthis result consider (41). What matters for
aggregate credit extension is onlythe sum of entrepreneurs’ and
bankers’ net worth. Broadly consistent with thedata, we have
calibrated the leverage ratio of entrepreneurs to be 2 and the
17
-
1 20
−0.5
0output
no frictionsfrictions
1 20
−0.2
−0.1
0labor
1 20−1.5
−1
−0.5
0Intermediate goods
1 20−0.2
−0.1
0household consumption
1 20
−1.5
−1
−0.5
0entrepreneurs net worth
1 20
−20
−10
0banks net worth
1 20
−1.5
−1
−0.5
0entrepreneur capital
1 20
−0.15−0.1
−0.05
0asset price
1 200
0.5
due repayment
1 20
−0.5
0return to banker
1 20
−3
−2
−1
0loans
1 20−20
−10
0capital−asset−ratio
1 20−1.5
−1−0.5
0
leverage
1 200
0.01
0.02
nominal interest rate
1 20
−0.1
−0.05
0
real interest rate
Fig. 1. Response to a bankcapital shock
capital-to-asset ratio of banks to be 10 per cent. It follows
that entrepreneurshold 10 times as much as net worth as banks.
Therefore modest shocks tobanks net worth constitute only a small
variation in overall net worth andaccording to (41) have only a
small impact on the capital holdings of thehigh productive
entrepreneurs. See Aikman and Vlieghe (2004) for a
furtherdiscussion of the importance of the bank capital
channel.
7.2 Monetary policy shock
Next, we consider the reaction of key variables to an
expansionary monetarypolicy shock, displayed in Figure 2.
The expansionary monetary policy shock increases nominal
aggregate demand.Given that some firms cannot adjust their prices,
inflation erodes their markupand they increase production. That
strongly boosts labour demand and in turnthe marginal product of
the intermediate goods. The initial increase in theprice of
intermediate goods contributes to an increase in asset prices (as
thephysical capital is used to produce intermediate goods) and
therefore the networth of banks and entrepreneurs. Given increased
net worth, larger loans can
18
-
1 20
1
2
3
output
no frictionsfrictions
1 200
2
4
labor
1 200
2
4
Intermediate goods
1 20
0.51
1.52
2.5household consumption
1 200
5
entrepreneurs net worth
1 200
5
banks net worth
1 200
5
entrepreneur capital
1 200
1
2
asset price
1 200
0.5
1
due repayment
1 20
−1
−0.5
0
return to banker
1 200
2
4
6
loans
1 20
−6
−4
−2
0
capital−asset−ratio
1 20
−0.4
−0.2
0leverage
1 20
−0.3
−0.2
−0.1
nominal interest rate
1 20
−1.5
−1
−0.5
real interest rate
Fig. 2. Response to a monetary policy shock
be supplied while still satisfying incentive constraints. The
increased extensionof loans to the high productive entrepreneurs
implies that a the supply ofintermediate goods increases strongly.
The output of final goods thereforeincreases much more than in the
world where capital market imperfectionsare absent. There is a
strong supply side of the monetary expansion.
The output effect of the monetary shock is both amplified and
more persistent(as measured by the half life) in the economy with
financial frictions relativeto the economy without moral hazard.
Half of the initial impact on output hasceased one period after the
shock in the moral hazard world, whereas it takestwo periods in the
economy with moral hazard. Part of the stronger outputresponse is
explained through the increase in labour. Clearly, the
amplificationof shocks due to credit market imperfections hinges
partly on the labour supplyelasticity.
7.3 Technology shock
We end the exposition of the working of the model by discussing
the impulseresponses to an adverse shock to total factor
productivity Tt, displayed in Fig-
19
-
ure 3. For ease of exposition, we shut off price stickiness by
specifying completeprice stability as an implicit monetary policy
rule. The internal mechanism for
1 20−1.4−1.2
−1−0.8−0.6−0.4−0.2
output
no frictionsfrictions
1 20−0.08
−0.06
−0.04
−0.02
labor
1 20
−1
−0.5
0Intermediate goods
1 20
−1.2−1
−0.8−0.6−0.4−0.2
household consumption
1 20
−1.5
−1
−0.5
0entrepreneurs net worth
1 20
−1.5
−1
−0.5
0banks net worth
1 20
−1.5
−1
−0.5
0entrepreneur capital
1 20−1.2
−1−0.8−0.6−0.4−0.2
asset price
1 20−2
−1
0due repayment
1 200
1
2
return to banker
1 20
−3
−2
−1
loans
1 200
2
capital−asset−ratio
1 20−1
−0.5
0leverage
1 20−0.4
−0.2
0
nominal interest rate
1 20−1
0
1inflation
Fig. 3. Response to a technology shock
propagation of technology shocks is similar as for the
aforementioned shocks.The decrease in technology lowers directly
the demand for intermediate goods,as their marginal product becomes
smaller. This lowers the return from theentrepreneurial projects
and therefore their net worth. In turn, the decreasednet worth of
firms leads to smaller credit extension. A larger fraction of
thecapital stock is therefore held by less productive households
and as a secondround effect intermediate good output falls sharply.
That effect accounts forthe additional fall in output from period 2
onwards. Again credit constraintsserve to propagate the effects of
exogenous shocks via their effect on net worthof banks and
entrepreneurs.
8 How should monetary policy be conducted?
It has been shown that introducing moral hazard in credit
extension can sig-nificantly change the transmission and
propagation of business cycle shocks.Large and variable deviations
between the allocation under moral hazard andthe efficient
allocation absent moral hazard give rise to a potential role
for
20
-
monetary policy. 14 Due to sticky prices, monetary policy can
influence realeconomic activity in this model. It may be beneficial
(in terms of a modelconsistent welfare measure) for monetary policy
to smooth some of the ineffi-cient fluctuations arising from credit
constraints. While monetary authoritiescannot directly influence
the origins of the moral hazard problems it may ad-dress them
indirectly. For instance, when negative technology shocks hit
theeconomy, asset prices and net worth are low and credit
constrained agents canobtain less outside finance. This further
contributes to the fall in aggregateoutput above what would happen
in a first best economy. Monetary policyhas the ability to
stimulate aggregate demand and may to some extent offsetthe fall in
asset prices thereby ameliorate the second round effects of
creditconstraints on aggregate activity. However, stimulating
demand comes at acost, as inflation implies that the bundler in
(19) demands the varieties in away that is socially inefficient.
The welfare cost of this inefficient allocation ofvarieties in the
bundler is captured by the price dispersion term P̃tin (39).
The literature on welfare based monetary policy in models with
sticky priceshas made a case for price stability. Khan et al.
(2003) analyze a monetarymodel in a cash-credit good set-up,
staggered price setting and no capitalaccumulation. They show that
optimal monetary policy tolerates only verysmall departures from
full price stability in an environment with several dis-tortions.
Collard and Dellas (2005) have analyzed a monetary model with
taxdistortions and capital accumulation and confirmed the case for
price stabilityin their model. Here, we analyze whether large
cyclical variations in the inef-ficiency gap - the gap between
actual and efficient levels of real activity - thatare induced by
moral hazard in financial contracts give rise to more
significantdepartures from price stability than found in the
aforementioned studies.
To study this question we proceed in two steps. First, we
analyze the welfareeffects of simple monetary policy rules that
respond to financial variables inaddition to inflation. Second, we
find the optimal interest rate rules within aparticular family of
rules. 15
We compute welfare using the numerical solution techniques
described inSchmitt-Grohé and Uribe (2004b) and Collard and
Juillard (2001). That tech-nique is based on a second-order Taylor
approximation of the models equationaround the deterministic steady
state and has been shown to give rise to ac-curate welfare
comparisons. 16 Welfare effects of policy rules are compared
14 Our monetary policy analysis focuses on interest rate policy.
We believe thereis little role for governmental regulation of
capital-asset ratios in this model. Themarket required
capital-asset ratios present a second best solution with little
roomfor government regulation.15 We do not consider here, whether
random monetary policy as in Dupor (2003)could improve upon price
stability.16 The technical appendix briefly describes the method
and some accuracy checks
21
-
as percentage points of equivalent variation in the consumption
process. Letmonetary policy rule A yield welfare of V A and let the
sequences of consump-
tion and labour supply under that rule be denoted with{CAt ,
N
At
}∞t=0
. The
required percentage variation ξ of the consumption sequence
under monetarypolicy rule A that makes the household as well off as
some alternative mone-tary policy rule yielding welfare of V B is
defined by the relation
V B = E0∞∑
j=0
[(1 + ξ)CAt+j
]1− 1σ
1− 1σ
+χ
1 + ψ
(1−NAt+j
)1+ψ(43)
For the log utility specification in this paper, we have V B =
log(1+ξ)1−β + V
A It
follows that ξ = exp((1− β)(V B − V A))− 1.
8.1 Simple targeting rules
In this section we analyze the welfare effects of simple
monetary policy rulesof the following loglinear form
r̂nt = (1− α0)(α1π̂t +J∑
j=2
αj ĝjt) + α0r̂nt−1 (44)
Here, gjt are additional variables that the central bank
responds to on top ofinflation. A prominent question in the
literature is whether monetary policyshould respond to financial
variables such as asset prices or credit aggregates.While Bernanke
and Gertler (2001) or Gilchrist and Leahy (2002) argue thata
response to asset prices in an interest rate rule is not warranted,
no formalanalysis of the welfare performance of such rules has been
conducted. In ourmodel movements in asset prices feed back to the
real economy, because theyinfluence the net worth of credit
constraint agents and therefore their abilityto obtain loans. We
therefore include the growth of asset price and the growthrate of
loan supply into the reaction function. Working with levels instead
ofgrowth rates gave similar conclusions as the ones presented
below.
Figure 4 plots −100ξ as a function of the response of the
interest rate tocredit growth and asset price growth, i.e it
expresses welfare as the percentagegain over full price stability.
That is, our reference level V B is welfare undercomplete price
stability.
Given the choice of the other parameters of the policy rule,
responding tocredit growth and or asset price growth does not
increase welfare relative to a
applied to our model. We use the log transformation as our
change of basis function.That is we express the log deviations of
the model’s endogenous variables as aquadratic function of the log
deviations of the state variables.
22
-
−5
0
5
−5
0
5
−1.2
−1
−0.8
−0.6
−0.4
−0.2
asset price growth
Welfare effects
credit growth
wel
fare
Fig. 4. Welfare effects of responding to credit and asset price
growth
policy of full price stability. On the contrary, mechanically
responding to thesefinancial variables has a considerable adverse
effect on welfare. The welfare lossstems from the dispersion of
output across producers of differentiated goods asindicated by
(39). Our formal welfare analysis therefore supports the
positiontaken by Gilchrist and Leahy (2002) that central banks
should not augmentsimple interest rate rules with a response to
asset prices. We next move to amore systematic analysis of optimal
monetary policy rules.
8.2 Optimal interest rate rules
We next search numerically for the optimal rule within a simple
family of mon-etary policy rules. We consider the family of
interest rate rules that respondlog-linearly to all state
variables. Since in linearized rational expectations mod-els all
endogenous variables are linear functions of the states, our
approach isquite general and similar to Collard and Dellas (2005).
17
17 These authors also allow to react to crossproducts of the
states. We have alsoexperimented with crossproducts in our model
and further lags of states. Only avery small gain in welfare was
achieved by additionally including crossproducts orfurther lags. To
save on computational time we chose to exclude them. In our
setup,we optimize for each shock a new rule, whereas Collard and
Dellas (2005) lookedfor one rule that maximizes welfare in face of
several shocks. In that case, allowingfor crossproducts becomes
crucial.
23
-
The optimal rule maximizes the welfare measure numerically
through of choiceof coefficients in an interest rate rule of the
following log-linear form:
r̂nt = c0π̂t +J∑
j=1
cjŜj,t−1 + cJ+1et (45)
According to this rule the nominal interest rate reacts to
current period infla-tion, to all endogenous state variables Sj,t−1
and to the current period exoge-nous shock et.
18 The response to inflation is added to ensure determinacy
andfor numerical stability. While the choice of the family of rules
to consider issomewhat arbitrary, a restriction must be placed to
limit the function space.
For all computed welfare measure we study a decomposition into
mean andvariance based on the following second-order Taylor
approximation to ex-pected utility of households:
E {U(Ct, Nt)} = a1E(C)− a2VAR(C)− a3E(N)− a4VAR(N)
+O(||ξ||3)
Here aj, j = 1, ..., 4 are coefficients and O(||ξ||3) denotes
constants and termsof higher than second order in the amplitude of
the exogenous shocks. Wedecompose welfare into means and variances
of consumption and labour, as inCollard and Dellas (2005) and
compute welfare for 3 monetary policy rules:Full price stability
(henceforth FPS), our baseline monetary rule (BMR) -equation (33)
without any monetary policy shocks - and the numerically op-timized
monetary rule (OMR).
The following table depicts the welfare effects of shocks to
banks net worthand to technology under the three aforementioned
monetary policy rules. 19
We also plot the standard deviation (SDV) of inflation in order
to measurethe departure from full price stability. The last column
expresses the welfaregain relative to a policy of full price
stability and as a fraction of one per centof period
consumption.
The first conclusion emerging from the table is that the simple
baseline rule(BMR) yields a very similar level of welfare as the
rule that completely stabi-lizes inflation (FPS). This is in line
with findings of Schmitt-Grohé and Uribe
18 We do not let the optimization routine choose c0, but rather
impose c0 = 10in order to avoid determinacy problems and to ensure
that the optimization isinitialized in a promising region of the
parameter space. This is not a restrictionas the optimization
routine is free to undo a too strong response to inflation
bychoosing higher reactions to the states.19 Bank capital shocks
are modeled as an innovation to (18) with standard deviation0.007.
This calibration implies that output is largely driven by
technology shocks,only 5 per cent of the forecast error variance
decomposition is due to bank networth shocks. Financial variables
such as are more strongly affected by this shock.Roughly 12 per
cent of forecast error variance is due to the net worth shock.
24
-
case SDV(C) SDV(N) E(C) E(N) SDV(π) −100ξbank net worth
shock
BMR 0.00229 0.00060766 1.0442545 0.3228945 0.00086 -0.0007
FPS 0.00304 0.00071266 1.0442657 0.32289514 0 0
OMR 0.00283 0.00065809 1.0442736 0.32289759 0.00035 0.0002
technology shock
BMR 0.022147 0.00063566 1.0444055 0.32289436 0.0036 -0.0143
FPS 0.024036 0.00034722 1.0445947 0.32289365 0 0
OMR 0.024051 0.00045403 1.0446017 0.32289759 0.00025 0.0002
(2004a) who report that targeting inflation strongly or weakly
has small ef-fects on welfare as long as the equilibrium remains
determinate. Consideringthe optimized monetary policy rule (OMR)
shows that one can improve uponfull price stability but that the
gain from doing so is small. The above tableshows that it is
crucial to account for the effects of increases in average levelsof
consumption E(C) and labour E(N) to capture the welfare effects of
mon-etary policy rules. For shocks to bank capital, both BMR and
OMR inducea smaller standard deviation of consumption and labour
relative to full pricestability. However, it is the fact the
optimal rules increases the average level ofconsumption and the
baseline rule decreases the average level of consumptionthat
accounts for the welfare ranking.
To further characterize optimal policy, we plot impulse response
functions toa shock to banks’ net worth under the optimized rule.
Figure 5 shows that theoptimal monetary policy response to the bank
capital is to induce a very mod-est amount of inflation on impact.
Due to sticky prices, that boosts demandand implies a significantly
smaller fall in employment relative to full pricestability. Output,
consumption and asset prices also fall by slightly less. Thesmaller
fall in asset prices implies that the entrepreneur can credibly
pledgea larger due repayment to outside parties. The fact that
inflation eventuallyundershoots its steady state level after an
initial expansion is a feature of op-timal policy under commitment
that is often reported in the literature, seeFigure 1 in Steinsson
(2003, p. 1443) and Woodford (2003) for a discussionof the history
dependence of optimal policy. Such a policy exploits the for-ward
looking nature of price setting which improves the inflation output
gaptrade-off.
We therefore conclude that the credit market imperfections
introduced in thismodel due not warrant a significant departure
from the objective of pricestability. Only a small amount of
inflation is tolerated under the optimal policy.
25
-
1 20−0.6
−0.4
−0.2
0output
optimalprice stability
1 20−0.2
−0.1
0
labor
1 20
−1
−0.5
0Intermediate goods
1 20
−0.2
−0.1
0
household consumption
1 20−1.5
−1
−0.5
0
entrepreneurs net worth
1 20
−15
−10
−5
banks net worth
1 20−1.5
−1
−0.5
0entrepreneur capital
1 20
−0.2
−0.1
0
asset price
1 200
0.05
0.1
due repayment
1 20−0.12
−0.1−0.08−0.06−0.04−0.02
return to banker
1 20−3
−2
−1
0loans
1 20−15
−10
−5
capital−asset−ratio
1 20−1.5
−1
−0.5
0
leverage
1 20−0.3−0.2−0.1
0
nominal interest rate
1 20
0
0.02
0.04
inflation
Fig. 5. Response to bank net worth shock under the optimized
rule
9 Conclusion
This paper has evaluated a monetary general equilibrium model
featuringcredit frictions that arise from a dual moral hazard
problem in financial con-tracts. A dual moral hazard problem as
outlined in Chen (2001) is embedded ina New Keynesian model with
sticky prices. Incentive compatibility constraintsin financial
contracts imply a role for net worth of entrepreneurs and banks
inthe extension of credit. The model is shown to significantly
alter business cy-cles dynamics relative to the standard sticky
price model. In particular, creditconstraints work to amplify and
propagate exogenous shocks via their effecton net worth of both
banks and entrepreneurs. The role of banks in affectingbusiness
cycle dynamics is shown to be small. The reasons lies in the
factthat only the sum of entrepreneurs’ and banks’ net worth
matters for capitalholding of entrepreneurs in this model. Banks
net worth however is roughlyan order of magnitude smaller that of
entrepreneurs. Therefore, it takes largevariations in banks’ net
worth to significantly affect macroeconomic variables.
The model is then used for policy analysis. It is first shown
that standardinterest rate rules should not respond to asset price
growth or loan in growthon top of responding to inflation. Reacting
strongly to these financial variables
26
-
may decrease households expected utility by up to one per cent
of periodconsumption.
Furthermore, it is shown that optimal monetary that maximizes
expected util-ity of households does not fully stabilize inflation
in response to technologyshocks or bank net worth shocks. However,
the amount of inflation variabilitytolerated in response to either
technology shocks or bank net worth shocks isextremely small. This
result can been seen as supporting the case for pricestability in a
model with multiple credit frictions. One may wonder to whatextent
this result is due to the particular parameterization of this
model. Wenote that the near optimality of price stability arises in
this model despite acalibration that probably overstates the
adverse effect of credit constraints inthis model. For instance, we
have calibrated the model such that in the steadystate half of the
aggregate capital stock is held by households. That allocationis
far from surplus maximizing. It implies that household’s marginal
productis roughly 20 times smaller that of entrepreneurs. It is
this relatively large dif-ference in relative productivity which
gives the model a strong amplificationand propagation mechanism. It
appears that despite strong inefficient fluctua-tions arising from
credit constraints, smoothing of business cycle fluctuationsis not
an important objective for policy in monetary models with
staggeredprice setting.
We conclude with some caveats to our policy conclusion. First we
have usedCalvo (1983) contracts to introduced staggered price
setting. As shown inPaustian (2004), that device may overstate the
welfare losses from inflation.Second, we have not allowed for
nominal contracts. Inflation can therefore noterode the real debt
burden of agents in our model and we have therefore shutoff a
channel through which central banks can influence real activity in
thismodel. Preliminary extension of the present analysis in this
direction showthat these caveats do not significantly affect our
results.
Acknowledgments
We would like to thank in advance our discussant Reint Gropp.
Special thanksto Jan Vlieghe for valuable work in earlier stages of
this paper. Thanks toJürgen von Hagen and Haiping Zhang for
helpful comments. We thank con-ference and seminar participants at
the European Winter Meeting of theEconometric Society in Stockholm,
the Bank of England, the Bundesbank,the University of Munich, the
University of Würzburg and the Center for Eu-ropean Integration
Studies at the University of Bonn. The views expressedhere are
those of the authors and do not necessarily reflect those of the
Bankof England.
27
-
Technical Appendix - not for publication
The no moral hazard reference model
Here, we document the modifications to the model that are
required whenthere is no moral hazard at the level of the bank and
the firm. Specifically,the modification we conduct makes each banks
decision to monitor perfectlyobservable at zero cost to all other
agents in the economy. The surplus max-imizing allocation of
capital is the one that equates the marginal productof capital of
household production with the marginal product of the
en-trepreneurs. Making monitoring public information permits us to
drop thebanks incentive compatibility constraint from the list of
equilibrium condi-tions. Banks in this case will no longer be able
to command a share of thesurplus from projects they monitor. Like
households, therefore, banks will re-ceive an expected return just
sufficient to satisfy their participation constraint.The incentive
to continually postpone consumption and accumulate net worthis
therefore eliminated and, given linear utility, the exact time path
of con-sumption and savings will be indeterminate. An interesting
extreme case toanalyze involves the bank accumulating zero net
worth and simply consum-ing its endowment period by period.
Similarly, we assume that the choice ofthe project is fully
observable at no cost and therefore the entrepreneurs nolonger
command a share of the project. As was the case for bankers, there
con-sumption path is indeterminate and we analyze the case where
they consumetheir endowment in any period. The economy then
collapses a version of thestandard dynamic new Keynesian model. The
equations characterizing equi-librium in this case are summarized
below. Equilibrium in this model are se-quences
{Cht , Nt, K
et , K
ht , K̃t, Kt, Z
kt , It, Xt,Mt, qt, p
∗t , P̃t, πt, vt, r
nt , r
dt
}that sat-
isfy the following equilibrium conditions, as well as the Fisher
equation and a
28
-
transversality condition for households capital holdings:
qt(cht
)− 1σ = βEt
(cht+1
)− 1σ
[(1− δ̃)(qt+1 + Zkt+1) + vt+1G
′ (kht
)]
λ(Kht
)²−1= phR
(Cht
)− 1σ = βEtr
dt
(Cht+1
)− 1σ
χ(1−Nt)ψ = XtTt(1− γ)(
MtNt
)γ (Cht
)− 1σ
vt = TtXtγ(
NtMt
)1−γ
1 = θπ²−1t + (1− θ) (p∗t )1−²P̃t = (1− θ) (p∗t )−² + θπ²t
P̃t−1
TtMγt N
1−γt = C
ht + It
Mt =λ
²
(Kht−1
)²+ pHrK
et−1
Kt = Kht + K
et
rnt =
(1
β
)1−ρr (rnt−1
)ρπ
(1−ρr)(1+κ)t e
ut
1 = φqtµ
(It
K̃t−1
)φ−1,
K̃t = (1− δ)pHKet + (1− δ̃)KhtZkt = (1− φ)qtµt
(It
K̃t−1
)φ
Kt = µIφt K̃
1−φt−1 + K̃t−1
p∗t =²
²− 1Et
∑∞i=0(θβ)
i(Cht+i
)− 1σ Xt+iπ
²t,t+iTt+iM
γt+iN
1−γt+i
Et∑∞
i=0(θβ)i(Cht+i
)− 1σ π²−1t,t+iTt+iM
γt+iN
1−γt+i
Here, p∗t is defined asP ∗tPt
and πt,t+j ≡ Πji=1πt+i, i.e the cumulative inflationrates
between period t and t + j.
9.1 Solution method and computation of welfare measure
We solve the model using the second-order solution method
outlined in Schmitt-Grohé and Uribe (2004b). The accuracy of the
obtained solution is assessed byuse of Euler equation residuals in
the appendix. To fix notation, consider thegeneric representation
for rational expectations models used by Schmitt-Grohé
29
-
and Uribe (2004b)
Et f(yt+1, yt,xt+1,xt) = 0. (46)
f is a known function describing the equilibrium conditions of
the modeleconomy, yt is a vector of co-state variables and xt a
vector of state variablespartitioned as xt = [x1,t; x2,t]. x1,t is
a vector of endogenous state variablesand x2,t a vector of state
variables following an exogenous stochastic process
x2,t+1 = Lx2,t + Ñσ²t. (47)
L and Ñ are known coefficient matrices, ²t is a vector of
innovations withbounded support, independently and identically
distributed with mean zeroand identity covariance matrix I. σ is a
parameter scaling the standard devi-ation of the innovations. The
solution to the model described by (??) is of theform
yt = g(xt, σ), (48)
xt+1 = h(xt, σ) + σN²t+1, with: N =
0
Ñ
. (49)
Schmitt-Grohé and Uribe (2004b) derive the second-order Taylor
approxima-tion to the policy functions g(·) and h(·) and provide
MATLAB codes for thenumerical implementation. The approximate model
dynamics obtained fromtheir second-order approximation can be
compactly expressed as
yt = Gxt +1
2G∗(xt ⊗ xt) + 1
2gσ2, (50)
xt+1 = Hxt +1
2H∗(xt ⊗ xt) + 1
2hσ2 + σN²t+1. (51)
yt and xt are expressed as deviations from the deterministic
steady state.G and H are coefficient matrices representing the
linear part of the Taylorapproximation. The matrices G∗ and H∗ form
the second-order part jointlywith the vectors g and h.
Unconditional means of the model’s variables can be constructed
as follows.Let µy,Σy denote unconditional mean and covariance
matrix of y, respec-tively. To construct first and second moments
of the co-state variables assumecovariance stationarity and take
expectation of (50) and (51)
µy = Gµx +1
2G∗vec(Σx + µxµ
′x) +
1
2gσ2, (52)
µx = Hµx +1
2H∗vec(Σx + µxµ
′x) +
1
2hσ2. (53)
30
-
Since variances can be computed accurately up to second-order
from the linearpart of the policy function, it is sufficient to
approximate vec(Σx + µxµ
′x) ≈
vec(Σx) and vec(Σy + µyµ′y) ≈ vec(Σy). It is then possible to
construct the
variances using the formulas for second moments of stationary
vector autore-gressive processes given in Hamilton (1994,
p.265)
vec(Σx) = σ2(I −H ⊗H)−1(N ⊗N)vec(I), (54)
vec(Σy) = (G⊗G)vec(Σx) (55)
Given these approximations for the variances, the means can be
computedfrom (52) and (53) as
µx =1
2(I −H)−1
(H∗vec(Σx) + hσ2
)(56)
µy = Gµx +1
2G∗vec(Σy) +
1
2gσ2 (57)
Given an expression for the unconditional mean of the models
variables wecan easily compute welfare.
Euler equation residuals
Judd (1998) advocates the inspection of Euler equations
residuals to assess theaccuracy of the obtained solution. One plots
the residual in the Euler equationas a function of the state
variables of the system. Let xt+1 = h
s(xt) denotetransition function for the state variables obtained
under solution method sWe denote the dependence of a generic
co-state variable y on the state vectorxt by y(xt). The residual
arising from the Euler equation for capital holdingsis:
Rs(xt) = 1−
{βEtC (hs(xt))
− 1σ
[q (hs(xt)) +
v(hs(xt))β
(λKe(xt)
K̄
) 1²
]}−σ
C(xt)(58)
The residual gives the error from following the approximated
policy rule asa fraction of current period consumption. 20 Under
certain conditions the ap-proximation error of the policy function
is of the same order of magnitudeas the Euler equation residual as
pointed out by Santos (2000). Figures (6)and (7) plot the absolute
value of the consumption Euler equation residualobtained from the
second-order accurate solution method over a range of de-viations
of the 2 state variables from −10% to +10% from their steady
statelevels 21 The residual is expressed in base 10 logarithms for
the ease of visu-alization.
20 The monetary policy rule that is used to obtain the solution
is the baseline rule
31
-
−0.1
−0.05
0
0.05
−0.1
−0.05
0
0.05
−4.8
−4.6
−4.4
−4.2
−4
−3.8
−3.6
deviation state 2
Euler residual using approximation of order 1
deviation state 1
log1
0 E
uler
res
idua
l
Fig. 6. Euler residuals: First-order approximiation
−0.1
−0.05
0
0.05
−0.1
−0.05
0
0.05
−8.5
−8
−7.5
−7
−6.5
−6
−5.5
−5
deviation state 2
Euler residual using approximation of order 2
deviation state 1
log1
0 E
uler
res
idua
l
Fig. 7. Euler residuals: Second-order approximiation
One can see that the second-order approximation yields smaller
Euler equa-tion residual over the entire state space. The average
22 Euler equation resid-ual from the second-order approximation is
more than order of magnitudesmaller than the one from the
first-order approximation. When we plot theEuler equation residuals
for other selected state variables, similar plots
emerge.Nevertheless, it would be good to compute an average Euler
equation error
with an inflation inertia coefficient of 0.9 and a long-run
response to inflation of 1.521 The two state variables are the
capital stock and the return to the banker, theother states are
fixed at their deterministic steady state values. The expectation
iscomputed using Gaussian quadrature with 10 nodes.22 We do not
weight the points in the state space with their probabilities, as
thesein turn can only accurately be computed using the unknown
exact solution.
32
-
over all points in the state space.
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