Credit Frictions and Optimal Monetary Policy∗
Vasco Cúrdia†
Federal Reserve Bank of New York

Michael Woodford‡
Columbia University

June 30, 2008

Abstract
We extend the basic (representative-household) New Keynesian [NK] modelof the monetary transmission mechanism to allow for a spread between the interest rate available to savers and borrowers, that can vary for either exogenousor endogenous reasons. We find that the mere existence of a positive average
spread makes little quantitative difference for the predicted effects of particularpolicies. Variation in spreads over time is of greater significance, with consequences both for the equilibrium relation between the policy rate and aggregate
expenditure and for the relation between real activity and inflation.
Nonetheless, we find that the target criterion – a linear relation that should
be maintained between the inflation rate and changes in the output gap — thatcharacterizes optimal policy in the basic NK model continues to provide a good
approximation to optimal policy, even in the presence of variations in credit
spreads. We also consider a “spread-adjusted Taylor rule,” in which the intercept of the Taylor rule is adjusted in proportion to changes in credit spreads.
We show that while such an adjustment can improve upon an unadjusted Taylor rule, the optimal degree of adjustment is less than 100 percent; and evenwith the correct size of adjustment, such a rule of thumb remains inferior to
the targeting rule.

∗Prepared for presentation at the BIS annual conference, “Whither Monetary Policy?” Lucerne,
Switzerland, June 26-27, 2008. We would like to thank Olivier Blanchard, Bill Brainard, Fiorella
DeFiore, Simon Gilchrist, Marvin Goodfriend, Charles Goodhart, Miles Kimball, Bennett McCallumand Oreste Tristani for helpful comments, and the NSF for research support of the first author
through a grant to the NBER. The views expressed in this paper are those of the authors and do
not necessarily reflect positions of the Federal Reserve Bank of New York or the Federal Reserve
System.
†E-mail : vasco.curdia@ny.frb.org‡
E-mail : michael.woodford@columbia.edu

It is common for theoretical evaluations of alternative monetary policies — mostnotably, the literature that provides theoretical foundations for inflation targeting —to be conducted using models of the monetary transmission mechanism that abstract
altogether from financial frictions.1 There is generally assumed to be a single interest
rate — “the interest rate” — that is at once the policy rate that constitutes theoperating target for the central bank, the rate of return that all households and firms
receive on savings, and the rate at which anyone can borrow against future income.
In models with more complete theoretical foundations, this is justified by assuming
frictionless financial markets, in which all interest rates (of similar maturity) must be
equal in order for arbitrage opportunities not to exist. It is also common to assume arepresentative household, and firms that maximize the value of their earnings streams
to that household, so that there is no need for credit flows in equilibrium in any event;such models imply that a breakdown of credit markets would have no allocative
significance. Many of the quantitative DSGE models recently developed in central
banks and other policy institutions share these features.2
Such models abstract from important complications of actual economies, eventhose that are financially quite sophisticated. Sizeable spreads exist, on average, between different interest rates; moreover, these spreads are not constant over time,
especially in periods of financial stress. And “tighter” financial conditions, indicatedby increases in the size of credit spreads, are commonly associated with lower levelsof real expenditure and employment. This poses obvious questions for the practicalapplication of much work in the theory of monetary policy.3 If a model is to be calibrated or estimated using time series data, which actual interest rate should be taken
to correspond to “the interest rate” in the model? When the model is used to to give
advice about how interest rates should respond to a particular type of shock, which
actual interest rate (if any) should be made to respond in the way that “the interestrate” does in the model? How large an error is likely to be made by abstracting fromcredit frictions, with regard to the model’s predictions for the variables that appear
in it? Moreover, some questions clearly cannot even be addressed using models that
abstract from credit frictions. Most notably, how should a central bank respond to a“financial shock” that increases the size of the spreads resulting from credit frictions?
This paper seeks to address these questions by presenting a simple extension of
1

See, for example, Clarida et al., (1999) or Woodford (2003), among many other references.
The models of Smets and Wouters (2003, 2007) provide an especially influential example.
3
The current generation of DSGE models has been criticized on this ground by Issing (2006) andGoodhart (2007), among others.
2

1

the basic “New Keynesian” model (as developed, for example, in Woodford, 2003)
in which a credit friction is introduced, allowing for a time-varying wedge between
the interest rate available to households on their savings and the interest rate at
which it is possible to borrow. Financial intermediation matters for the allocationof resources due to the introduction of heterogeneity in the spending opportunities
currently available to different households. While the model remains highly stylized,
it has the advantage of nesting the basic New Keynesian model (extensively used innormative monetary policy analysis) as a special case, and of introducing only a small
number of additional parameters, the consequences of which for conclusions about
the monetary transmission mechanism and the character of optimal policy can bethoroughly explored. The approach taken also seeks to develop a tractable model,
with as small a state space as is consistent with an allowance for financial frictions
and heterogeneity, and hence only modestly greater complexity than the basic New
Keynesian model.
Among the questions to be addressed are the following: If the parameters determining the degree of heterogeneity and the size of credit frictions are calibrated so
as to match both the volume of bank credit and the spread between bank deposit
and lending rates in the US economy, how much of a difference does this make (relative to the frictionless baseline) for the model’s predictions for the response of theeconomy to various types of shocks, under a given monetary policy rule? How muchof a difference does it make for the implied responses to real disturbances under an
optimal monetary policy? How much of a difference does it make for the form of
the quadratic stabilization objective that would correspond to the maximization of
average expected utility? How much of a difference does it make for the form of the
optimal target criterion for monetary stabilization policy? And how should policy
optimally respond to a “financial shock”?
The model also provides perspective on “rules of thumb” for policy in times offinancial turmoil proposed in the recent literature. For example, McCulley and Toloui(2008) and Taylor (2008) propose that the intercept term in a “Taylor rule” formonetary policy should be adjusted downward in proportion to observed increases in
spreads.4 Here we use our simple model to ask whether it is correct to say that the4

Similarly, Meyer and Sack (2008) propose, as a possible account of recent U.S. Federal Reserve
policy, a Taylor rule in which the intercept — representing the Fed’s view of “the equilibrium realfunds rate” — has been adjusted downward in response to credit market turmoil, and use the size of2

“natural” or “neutral” rate of interest is lower when credit spreads increase (assuming
unchanged fundamentals otherwise), and to the extent that it is, how the size of thechange in the natural rate compares to the size of the change in credit spreads.
We also ask whether it is approximately correct to say that a proper response toa “financial shock” is to conduct policy according to the same rule as under othercircumstances, except with the operating target for the policy rate adjusted by afactor that is proportional to the increase in credit spreads; and again, to the extent
that such an approximation is used, we ask what proportion of adjustment should bemade.
Other authors have argued that if financial disturbances are an important sourceof macroeconomic instability, a sound approach to monetary policy will have to pay
attention to the balance sheets of financial intermediaries. It is sometimes suggested
that policy should respond to variations in the growth rate of monetary or credit
aggregates, rather than — as in the case of both the Taylor rule and conventionalprescriptions for “flexible inflation targeting” — seeking to determine the appropriate
level of short-term interest rates purely on the basis of observations of or projections
for measures of inflation and real activity. Here we consider two possible interpretations of such proposals: as an argument for targeting monetary and/or credit aggregates, or at least adopting a target criterion that involves such variables alongwith others; or as an argument for their special value as indicators, so that such variables should receive substantial weight in the central bank’s reaction function. We
address the first issue by deriving an optimal target criterion for monetary policy,
under certain simplifying assumptions, and seeing to what extent it involves either
money or credit. We address the second issue, under assumptions that are arguably
more realistic, by computing the optimal responses to shocks, and asking what kindsof indicator variables would allow a simple rule of thumb to bring about equilibrium
responses of this kind.
Of course, we are not the first to investigate ways in which New Keynesian [NK]models can be extended to allow for financial frictions of one type or another. A
number of authors have analyzed DSGE models with financial frictions of one type oranother.5 Many of the best-known contributions introduce obstacles to the willingness
increases in spreads in early 2008 as a basis for a proposed magnitude of the appropriate adjustment.
5
Probably the most influential early example was the model of Bernanke et al., (1999). Morerecent contributions include Christiano et al. (2003, 2007a, 2007b), Gertler et al. (2007), and3

of savers to lend to borrowers, but assume that borrowers directly borrow from the
suppliers of savings. A number of recent papers, however, are like ours in explicitly
introducing intermediaries and allowing for a spread between the interest received bysavers and that paid by borrowers; examples include Hulsewig et al. (2007), Teranishi(2008), Sudo and Teranishi (2008), and Gerali et al. (2008).6
In general, these models have been fairly complex, in the interest of quantitative
realism, and the results obtained are purely numerical. Our aim here is somewhat
different. While the interest of such analyses is clear, especially to policy institutions
seeking quantitative estimates of the effects of particular contemplated actions, we
believe that it is also valuable to seek analytical insights of the kind that can only
be obtained from analyses of simpler, more stylized models. Here we focus on theconsequences for monetary policy analysis of two basic features of economies — heterogeneity of non-financial economic units, of a kind that gives the financial sector a
non-trivial role in the allocation of resources; and costs of financial intermediation,that may be subject to random variation for reasons relating largely to developmentsin the financial sector — in the simplest possible setting, where we do not introduce
other departures from the basic NK model.
Two recent contributions have aims more closely related to ours. Like us, Goodfriend and McCallum (2007) consider a fairly simple NK model, with new model
elements limited to those necessary to allow for multiple interest rates with different
average levels (including, like us, a distinction between bank lending rates and thepolicy rate).7 As in the present paper, a primary goal is to “investigate quantitatively
how much a central bank can be misled by relying on a [NK] model without moneyand banking when managing its interbank-rate policy instrument” (p. xx). De Fiore
and Tristani (2007) also propose a simple generalization of the basic NK model inorder to introduce a distinction between loan rates and the policy; also like us, they
Iacoviello (2005). Faia and Monacelli (2007) consider how two different types of financial frictions
affect welfare-based policy evaluation, though from a perspective somewhat different than the one
taken here; they compare alternative simple rules, rather than computing optimal policy, as we do,
and compute the welfare associated with a particular rule under a complete specification of shocks,
rather than considering what a given simple rule implies about the equilibrium responses to shocks
considered individually. Cúrdia (2008) considers optimal policy in the spirit of the present paper,but in a more complex model with features specific to emerging-market economies.
6
See Gerali et al. (2008, sec. 2) for a more detailed discussion of prior literature.
7
This paper provides a quantitative analysis of type of model first proposed by Goodfriend (2005).

4

consider how financial frictions affect the “natural rate of interest,” and the role ofsuch a concept in inflation determination in an economy with credit frictions.
The approaches taken by these authors nonetheless differ from ours in important
respects. In particular, unlike us, Goodfriend and McCallum assume a representativehousehold model; as a consequence, financial intermediation matters for resource
allocation in their model only because they assume that certain liabilities of banks(transactions balances) play a crucial role in facilitating transactions. We instead
treat the fact that some (but not all) financial intermediaries finance (some) of theirlending by providing accounts that are useful as means of payment as inessential tothe primary function of financial intermediaries in the economy; and in our model,
for the sake of simplicity, we assume that intermediaries finance themselves entirely
by issuing deposits that supply no transactions services at all (so that in equilibrium,
deposits must pay the same interest rate as government debt). De Fiore and Tristani
instead have two types of infinite-lived agents (“households” and “entrepreneurs,”
following Bernanke et al., 1999), one of which saves while the other borrows; but
in their model, unlike ours, agents belong permanently to one of these categories,
and one is tempted to identify the division between them with the division between
households and firms in the flow of funds accounts. This would be desirable, of course,
if one thought that the model did adequately capture the nature of that division, as
the model would yield additional testable predictions. But in fact, there are bothsaving units and borrowing units at a given point in time, both in the household
sector and in the firm sector; and a saving unit at one point in time need not bea saving unit forever. We accordingly prefer not to introduce a distinction between
households and firms (or “households” and “entrepreneurs”) at all, and also not toassume that the identities of our savers and borrowers are permanent.8 In addition,
De Fiore and Tristani, like Goodfriend and McCallum, assume that money must be
used in (some) transactions, while we abstract from transactions frictions of this kind
altogether in order to simplify our analysis.9
8

In fact, De Fiore and Tristani list as an important “undesirable property” of their model the factthat in it, “households and entrepreneurs are radically different agents” (p. 23), as the predictedequilibrium behavior of “households” as a group does not look much like that of the aggregate
household sector in actual economies.
9
Goodfriend and McCallum justify the introduction of a cash-in-advance constraint in theirmodel, stating (footnote 6) that “medium-of-exchange money is implicitly central to our analysis
because it is by managing the aggregate quantity of reserves, which banks hold to facilitate trans-5

We develop our model in section 1, and compare its structure with that of thebasic NK model. We then consider, in section 2, the implications of the model forthe equilibrium effects of a variety of types of exogenous disturbances, under a given
assumption about monetary policy (such as that it conform to a “Taylor rule”), andask to what extent the basic NK model gives incorrect answers to these questions.
Section 3 considers optimal monetary policy in the context of our model, defined tomean a policy that maximizes the average expected utility of households, and again
considers how different the conclusions are from those derived from the basic NK
model. We also consider the extent to which various simple rules of thumb, such asversions of the Taylor rule (Taylor, 1993), can usefully approximate optimal policy.
Section 4 summarizes our conclusions.

1

A New Keynesian Model with Financial Frictions

Here we sketch a model that introduces heterogeneity of the kind needed in order for
financial intermediation to matter for resource allocation, and a limit on the degreeof intermediation that occurs in equilibrium, with a minimum of structure. We stressthe similarity between the model presented here and the basic New Keynesian [NK]model, and show how the standard model is recovered as a special case of the one
developed here. This sets the stage for a quantitative investigation of the degree towhich credit frictions of an empirically realistic magnitude change the predictions of
the model.
actions, that monetary policy affects interest rates.” However, while in their model, banks holdreserves at the central bank only because of a reserve requirement proportional to transactions balances, this need not be true in actual economies, a number of which (such as Canada) have abolishedreserve requirements. Moreover, it is possible in principle for a central bank to control the interest
rate in the interbank market for central-bank deposits without there being any demand for suchreserves other than as a riskless store of value, as discussed in Woodford (2003, chap. 2). Hence
there is no need to introduce a demand for money for transactions purposes in our model, in orderfor it to be possible to suppose that the central bank controls a short-term nominal interest rate,that will correspond to the rate at which banks can fund themselves.

6

1.1

Financial Frictions and Aggregate Demand

We depart from the assumption of a representative household in the standard model,
by supposing that households differ in the utility that they obtain from current expenditure. Each household i seeks to maximize a discounted intertemporal objectiveof the form
·
¸
Z 1
∞
X
t
τ t (i)E0β u(ct (i); ξ t ) −v (ht (j; i) ; ξ t ) dj ,0

t=0

where τ t (i) ∈ {b, s} indicates the household’s “type” in period t. Here ub (c; ξ) andus (c; ξ) are two different period utility functions, each of which may also be shifted by
the vector of aggregate taste shocks ξ t . As in the basic NK model, there is assumed
to be a continuum of differentiated goods, each produced by a monopolistically competitive supplier; ct (i) is a Dixit-Stiglitz aggegator of the household’s purchases ofthese differentiated goods. The household similarly supplies a continuum of different
types of specialized labor, indexed by j, that are hired by firms in different sectors of
the economy; the additively separable disutility of work v(h; ξ) is the same for eachtype of labor, and can be shifted by the taste shock.10
Each agent’s type τ t (i) evolves as an independent two-state Markov chain. Specifically, we assume that each period, with probability 1 − δ (for some 0 ≤ δ < 1) an
event occurs which results in a new type for the household being drawn; otherwise it
remains the same as in the previous period. When a new type is drawn, it is b withprobability π b and s with probability π s , where 0 < π b , π s < 1, π b + π s = 1. (Hencethe population fractions of the two types are constant at all times, and equal to π τ
for each type τ .) We assume moreover thatubc (c; ξ) > usc (c; ξ)for all levels of expenditure c in the range that occur in equilibrium. Hence a change ina household’s type changes its relative impatience to consume,11 given the aggregate
10

As in Woodford (2003), the vector ξ t may contain multiple elements, which may or may not becorrelated with one another, so that the notation makes no assumption about correlation between
disturbances to the utility of consumption and disturbances to the disutility of work.
11
As explained below, all households have the same expectations regarding their marginal utilities
of expenditure far in the future. Hence if type b households have a higher current marginal utility of
expenditure, they also have a higher valuation of current (marginal) expenditure relative to future
expenditure; thus we may say that they are more impatient to consume.

7

state ξ t ; in addition, the current impatience to consume of all households is changedby the aggregate disturbance ξ t .
The coexistence of the two types with differing impatience to consume createsa social function for financial intermediation. In the present model, as in the basic
New Keynesian model, all output is consumed either by households or by the government;12 hence intermediation serves an allocative function only to the extent that
there are reasons for the intertemporal marginal rates of substitution of households
to differ in the absence of financial flows. The present model reduces to the standard
representative-household model in the case that one assumes that ub (c; ξ) = us (c; ξ).
We shall assume that most of the time, households are able to spend an amountdifferent from their current income only by depositing funds with or borrowing fromfinancial intermediaries, and that the same nominal interest rate is available to all
savers, and that a (possibly) different nominal interest is available to all borrowers,13
independent of the quantities that a given household chooses to save or to borrow.
(For simplicity, we shall also assume in the present exposition that only one-periodriskless nominal contracts with the intermediary are possible for either savers or
borrowers.) The assumption that households cannot engage in financial contracting
other than through the intermediary sector introduces the financial friction withwhich the paper is concerned.
Our analysis is simplified (though this may not be immediately apparent!) byallowing for an additional form of financial contracting. We assume that households
are able to sign state-contingent contracts with one another, through which they may
12

The “consumption” variable is therefore to be interpreted as representing all of private expenditure, not only consumer expenditure. In reality, one of the most important reasons for some economic
units to wish to borrow from others is that the former currently have access to profitable investment
opportunities. Here we treat these opportunities as if they were opportunities to consume, in thesense that we suppose that the expenditure opportunities are valuable to the household, but weabstract from any consequences of current expenditure for future productivity. For discussion of
the interpretation of “consumption” in the basic New Keynesian model, see Woodford (2003, pp.
242-243).
13
Here “savers” and “borrowers” identify households according to whether they choose to save or
borrow, and not by their “type”. We assume that at any time, each household is able to save or
borrow (or both at once, though it would never make sense to do so) at market interest rates. In
the equilibrium described below, it turns out that a household i borrows in period t if and only if
τ t (i) = b, but this is a consequence of optimization rather than an implication of a participation
constraint.

8

insure one another against both aggregate risk and the idiosyncratic risk associatedwith a household’s random draw of its type, but that households are only intermittently able to receive transfers from the insurance agency; between the infrequentoccasions when a household has access to the insurance agency, it can only save or
borrow through the financial intermediary sector mentioned in the previous paragraph. The assumption that households are eventually able to make transfers to one
another in accordance with an insurance contract signed earlier means that despite
our assumption of infinite-lived households, households’ respective marginal utilities
of income do not eventually become more and more dispersed as a result of their differing individual type histories. This facilitates aggregation (so that our model still
has a low-dimensional state space), and allows us to obtain stationary equilibrium
fluctuations and to use local methods to characterize them. At the same time, thefact that households may go for years without access to insurance transfers meansthat there remains a non-trivial financial friction for the banking sector to partiallymitigate.14
To simplify the presentation, we assume here that the random dates on whicha given household i has access to the insurance agency are the same dates as thoseon which it draws a new type. Thus with probability δ each period, household i isunable to receive any insurance transfer in the current period, and also retains thesame type as in the previous period. With probability 1−δ, it learns at the beginningof the period that it has access to the insurance agency. In this case, it receives a net
transfer Tt (i) (under the terms of an insurance contract signed far in the past), thatmay depend on the history of aggregate disturbances through the current period, andalso on i’s type history through the previous period (but not on its type in period t,which is not yet known). After receiving the insurance transfer, household i learns its
new type (an independent drawing as explained above), and then makes its spending,saving and borrowing decisions as in any other period, but taking into account its
new type and its post-transfer financial wealth.
Household i’s beginning-of-period (post-transfer) nominal net financial wealth
14

A similar device is commonly used in models of “liquidity,” where access to frictionless financial
intermediation is assumed to be possible only at discrete points in time, and that only a smallerclass of exchanges are possible at interim dates. See, e.g., Lucas and Stokey (1984), Lucas (1990),
Fuerst (1992), or Lagos and Wright (2005). Here we use a similar device to facilitate aggregation,
but without doing so in a way that implies that the allocative consequences of financial frictions are
extremely transitory.

9At (i) is then given by¡¢¡¢
At (i) = [Bt−1 (i)]+ 1 + idt−1 + [Bt−1 (i)]− 1 + ibt−1 + Dtint + Tt (i),(1.1)

where Bt−1 (i) is the household’s nominal net financial wealth at the end of periodt − 1;[B]+ ≡ max (B, 0) ,[B]− ≡ min (B, 0) ;idt is the (one-period, riskless nominal) interest rate that savers receive at the beginning of period t + 1 on their savings deposited with intermediaries at the end of
period t, while ibt is the interest rate at which borrowers are correspondingly able toborrow from intermediaries in period t for repayment at the beginning of period t + 1;
and Dtint represents the distributed profits of the financial intermediary sector. We
assume that each household owns an equal share in the intermediary sector,15 and soreceives an equal share of the distributed profits each period; profits are distributed
each period as soon as the previous period’s loans and depositors are repaid. Notethat the final term Tt (i) is necessarily equal to zero in any period in which household
i does not have access to the insurance agency. A household’s end-of-period nominal
net financial wealth Bt (i) is correspondingly given byZ
Bt (i) = At (i) − Pt ct (i) + Wt (j)ht (j; i)dj + Dt + Ttg ,(1.2)
where Pt is the Dixit-Stiglitz price index in period t (and hence the price of thecomposite consumption good); Wt (j) is the wage of labor of type j in period t; Dt
represents the household’s share in the distributed profits of goods-producing firms;
and Ttg is the net nominal (lump-sum) government transfer received by each householdin period t.
Any pair of identically situated households with access to the insurance agency
will contract with one another so that if, in any state of the world at some futuredate, they again each have access to the insurance agency at the same time, a transfer
will take place between them that equalizes their marginal utilities of income at thattime (if each has behaved optimally in the intervening periods). Given that they15

We do not allow trading in the shares of intermediaries, in order to simplify the discussion ofhouseholds’ saving and borrowing decisions. Euler equations of the form (1.12)–(1.13) below would
still apply, however, even if households could also trade the shares of either banks or goods-producing
firms.

10

have identical continuation problems at that time (before learning their new types),
as functions of their post-transfer financial wealths, such an agreement will ensurethat their post-transfer financial wealths are identical (again, if each has behaved
optimally16 ). If we suppose that at some time in the past, all households originally
started with identical financial wealth and access to the insurance agency, then theyshould have contracted so that in equilibrium, in each period t, all those householdswith access to the insurance agency in period t will obtain identical post-transfer
financial wealth. If we suppose, finally, that transfers through the insurance agency
must aggregate to zero each period (because the agency does not accumulate financial
assets or borrow), then each household with access to the insurance agency at the
beginning of period t must have post-transfer wealth equal toZ
At (i) = At ≡ At (h)dh.
(1.3)
The beginning-of-period wealth of households who do not have access to the insurance
agency is instead given by (1.1), with Tt (i) set equal to zero.
If we let dt denote aggregate real deposits with financial intermediaries at the endof period t,17 and bt aggregate real borrowing from intermediaries, then we must haveZZg
Pt bt = −At (i)di,pt [bt + dt ] =At (i)di,
Bt

St

where Bt is the set of households i for which At (i) < 0, St is the (complementary) setof households for which At (i) ≥ 0, and bgt is real government debt at the end of periodt. We assume that government debt is held directly by savers, rather than by financial
intermediaries, so that the rate of return that must be paid on government debt is
idt , the rate paid on deposits at the intermediaries. (For simplicity, we assume herethat all government debt also consists of riskless, one-period nominal bonds, so that16

It is important to note, however, that the contractual transfer Tt (i) is only contingent on thehistory of aggregate and individual-specific exogenous states, and not on the actual wealth thathousehold i has at the beginning of period t. Thus a spendthrift household is not insured an equal
post-transfer wealth as other households, regardless of how much it has spent in past periods.
17
Here “real” deposits and other real variables are measured in units of the Dixit-Stiglitz composite
consumption good, the price of which is Pt . Deposit contracts, loan contracts, and government debt
are actually all assumed to be non-state-contingent nominal contracts. We introduce real measuresof the volume of financial intermediation because we assume that the intermediation technology
specifies real costs of a given volume of real lending.

11

deposits and government debt are perfect substitutes.) The aggregate beginning-ofperiod assets At of households referred to in (1.3) are then given by
At = [(dt−1 + bgt−1 )(1 + idt−1 ) − bt−1 (1 + ibt−1 )]Pt−1 + Dtint ,(1.4)

integrating (1.1) over all households i.
The supply of government debt evolves in accordance with the government’s flow
budget constraint
bgt = bgt−1 (1 + idt−1 )/Πt + Gt + Ttg /Pt − τ t Yt ,(1.5)

where Πt ≡ Pt /Pt−1 is the gross rate of inflation, Gt is government purchases of thecomposite good, τ t is a proportional tax on sales of goods,18 and Yt is the quantityof the composite good produced by firms. Given the sales tax, the distributed profits
of firms are equal toZ(1.6)
Dt = (1 − τ t )Pt Yt − Wt (j)ht (j)dj,
R
where ht (j) ≡ ht (j; i)di is aggregate labor hired of type j.
We assume an intermediation technology in which real lending in the amount bt
requires an intermediary to obtain real deposits of a quantity
dt = bt + Ξt (bt ),(1.7)

where Ξt (b) is a (possibly time-varying) function satisfying Ξt (0) = 0 and Ξt (b) ≥0, Ξ0t (b) ≥ 0, Ξ00t (b) ≥ 0 for all b ≥ 0. The first term on the right-hand side represents
the funds that the intermediary lends to its borrowers, while Ξt (bt ) represents a real
resource cost of loan origination and monitoring.19 (The quantities in (1.7) should
18

Note that there are two potential sources of government revenue in our model: variation in thesize of the net lump-sum transfers Ttg , and variation in the tax rate τ t . We introduce the process{τ t } as an additional source of time-varying supply-side distortions.
19
This real resource cost can be interpreted in either of two ways: either as a quantity of thecomposite produced good that is used in the activity of banking, or as a quantity of a distinct typeof labor that happens to be a perfect substitute for consumption in the utility of households (sothat the value of this labor requirement in units of the composite good is exogenously given). Theinterpretation that is chosen does not affect the validity of the equations given here, though it affects
the interpretation of variables such as “ct ” in terms of the quantities measured in national income
accounts. See the Appendix for further discussion.

12

be interpreted as referring to the deposits and loans of an individual intermediary;
however, in equilibrium, all intermediaries operate on the same scale, so that in our
eventual characterization of equilibrium, we can identify per-intermediary and aggregate or per-capita quantities.) These costs of intermediation are one of the sourcesof the financial friction in our model. Like Goodfriend and McCallum (2007), we
simply posit an intermediation technology, rather than seeking to provide a behavioral justification for the spread between the interest rate available to savers and theone at which it is possible to borrow. This means that we are unable to consider
possible effects of central-bank policy on the efficiency of the banking system.20 We
can, however, consider the consequences for the effects of monetary policy, and for the
optimal conduct of monetary policy, of the existence of, and of exogenous variationin, obstacles to fully efficient financial intermediation.
Given this technology, a perfectly competitive banking sector will result in an
equilibrium spread ω t between deposit rates and lending rates, such that
1 + ibt = (1 + idt )(1 + ω t ),(1.8)

where ω t = Ξ0t (bt ). We shall allow, however, for additional sources of credit spreads
that are associated with increased resource utilization by intermediaries. Specifically,
we shall assume that the equilibrium spread is given by
1 + ω t = µbt (bt )(1 + Ξ0t (bt ),(1.9)

where µbt ≥ 1 is a (possibly time-varying) markup in the intermediary sector, assumed
here to vary either for exogenous reasons, or perhaps as a consequence of variationin the total volume of lending. (Our allowance for an exogenously varying markup
20

Certainly we do not deny that at least at certain times, central banks do seek to affect theefficiency of the banking system; this is true most obviously in the case of actions taken in acentral bank’s capacity as “lender of last resort” during a financial crisis. However, we regard suchactions as representing a largely independent dimension of policy from monetary policy, by which wemean control of the supply of central-bank balances to the payments system, and of the overnightinterest rate paid for such balances in the interbank market. (Additional lending to intermediaries
through the discount window or similar facilities need not imply any increase in the total supplyof central-bank deposits, as the actions of the Federal Reserve during the most recent crisis havedemonstrated.) Here we are concerned solely with the analysis of the central bank’s monetary
policy decisions, taking as given the evolution of the intermediation frictions (that may reflect otherdimensions of central-bank policy, as well as developments elsewhere in the economy).

13

function is analogous to our allowance for a possibly time-varying “wage markup”
in the treatment below of labor supply.) In allowing for a markup in the loan rates
charged by intermediaries — and in particular, in considering a “financial shock”
in which banking markups increase for reasons treated as exogenous — we follow
Gerali et al. (2008).21 However, we need not view the markup µbt as necessarilyreflecting market power on the part of intermediaries; for example, it might standin for a time-varying risk premium (though we do not explicitly model any sourceof risk22 ), or for variation in the fraction of loans made to fraudulent borrowers.23
What matters is that the sources of spreads between deposit rates and lending rates
may or may not correspond increased real consumption of resources by the activity of
the intermediaries; and we consider the consequences of variations in the efficiency offinancial intermediation of both types. Using this general notation, market-clearing
in the goods market requires thatZ
Yt = ct (i)di + Gt + Ξt (bt )(1.10)
each period, and the distributed profits of intermediaries are given by
int
Dt+1= [(1 + ibt )bt − (1 + idt )dt ]Pt = {[µbt (1 + Ξ0t ) − 1]bt − Ξt (bt )}Pt (1 + idt ).

(1.11)

This completes our description of the flows of both income and goods amonghouseholds, intermediaries, and goods-producing firms. We turn now to the implications of optimal household decisions with regard to consumption, saving, and21

Imperfectly competitive banking is also a feature of the theoretical models of Teranishi (2008)
and Sudo and Teranishi (2008), and the empirical model of Hulsewig et al. (2007).
22
In the technology for financial intermediation specified here, there is no risk, since loans anddeposits are assumed to be perfectly matched, both in maturity and in currency denomination, andthe intermediary’s costs are determined by the value of the loans at origination, not by the realvalue of required repayment.
23
Under the latter interpretation, bt is the real value of loans to legitimate borrowers, who intend
to repay, but χt dollars must be lent for every dollar of legitimate loans, so that total costs ofthe banks are χt bt + Ξt (bt ). This leads to a marginal cost of lending given by χt + Ξ0t (bt ), which
can be written in the same form as the right-hand side of (1.9), under a suitable definition of thefunction µbt (bt ). If these additional costs (χt −1)bt to the banks are windfall income to the fraudulent
borrowers, and all households share equally in the opportunities for income from fraud, then these
are not real resource costs of banking, and the consequences for household budgets are the same as
if the additional charges were pure profits of the banks (and so distributed equally to households as
int
part of Dt+1).

14

borrowing. A household for which At (i) > 0 (i.e., a saver) must satisfy a first-ordercondition
λt (i) = β(1 + idt )Et [λt+1 (i)/Πt+1 ](1.12)
in period t, where λt (i) = uc (ct (i); ξ t ) is the household’s marginal utility of (real)
income in period t, while a household for which At (i) < 0 (a borrower) must instead
satisfyλt (i) = β(1 + ibt )Et [λt+1 (i)/Πt+1 ].
(1.13)
We need not discuss the corresponding first-order condition for a household thatchooses At (i) = 0 exactly (though this is certainly possible, given the kink in households’ budget sets at this point), as no households are in this situation in the equilibria
that we describe here.
Under conditions specified in the Appendix, one can show that there is an equilibrium in which every household of type s has positive savings, while every householdof type b borrows, in every period. Hence the interest rate that is relevant for a given
household’s intertemporal tradeoff turns out to be perfectly correlated with the household’s type (though this is not due to participation constraints). Moreover, because
in equilibrium, households that access the insurance agency in a given period t havethe same marginal utility of income at the beginning of that period (before learning
their new types), regardless of their past histories, it follows that in any period, allhouseholds of a given type have the same marginal utility of income, regardless of
their histories. Hence we can write λτt for the marginal utility of (real) income of any
household of type τ in period, where τ ∈ {b, s}. Thus the equilibrium evolution of themarginal utility of income for all households can be described by just two stochastic
processes, {λbt , λst }.
These two processes satisfy the two Euler equations·¸ª
1 + ibt ©
bbs(1.14)
λt = βEt[δ + (1 − δ) π b ] λt+1 + (1 − δ) π s λt+1 ,
Πt+1
·
¸
ª
1 + idt ©
sbs
λt = βEt(1 − δ) π b λt+1 + [δ + (1 − δ) π s ] λt+1(1.15)
Πt+1
in each period. (These follow from (1.12) – (1.13), taking into account the probabilityof switching type from one period to the next.) It follows that all households of a given
type must also choose the same consumption in any period, and, assuming an interior
15choice for consumption by households of each type, these common consumption levels
must satisfy
λbt = ub0 (cbt ),λst = us0 (cst ),which relations can be inverted to yield demand functions
cbt = cb (λbt ; ξ t ),cst = cs (λst ; ξ t ).

(1.16)

Substituting these into (1.10) yields an equilibrium relation
Yt = π b cb (λbt ; ξ t ) + π s cs (λst ; ξ t ) + Gt + Ξt (bt )(1.17)

linking aggregate demand to the two marginal utilities of income and aggregate borrowing.
The three relations (1.14)–(1.17) generalize the “intertemporal IS relation” of thebasic NK model, which can be expressed by an equation relating aggregate demand
to the marginal utility of income of the representative household (analogous to (1.17))
and a single equation relating that marginal utility of income to the expected real
rate of return implied by the model’s single interest rate. The present model implies
a similar relation between interest rates and the timing of expenditure as in the basicmodel. The main differences are (i) that now there are two different interest rates
that each affect aggregate demand (though with the same sign), by affecting theexpenditure decisions of different economic units, and (ii) that the resources used by
the banking sector can also affect aggregate demand.
The presence of two interest rates relevant to aggregate demand determination
does not mean there are two independent dimensions of monetary policy. Instead,the two rates must be linked by equations (1.8)–(1.9), determining the equilibrium
credit spread.24 If we introduce no further frictions, the policy rate (which is a rate
at which banks are willing to lend short-term funds to one another) corresponds tothe deposit rate idt ;25 and we may suppose that the central bank directly controls
24

Of course, there is an additional, independent dimension of central-bank policy if the central
bank has measures, independent of its control of the policy rate, that can influence the financial
frictions represented by the functions Ξt (bt ) or µbt (bt ). Since we do not here model the underlying
foundations of these frictions, we cannot comment on the nature of such independent dimensions ofpolicy using the present model.
25
We could introduce a distinction between the rate that banks pay depositors and the rate banks
pay one another for overnight funds, by supposing, as Goodfriend and McCallum (2007) do, that16

this rate.26 In the case that banking uses no real resources (so that Ξt (bt ) = 0
regardless of the volume of lending) and the markup µbt is independent of the volume oflending as well, the system consisting of equations (1.8)–(1.9) and (1.14)–(1.17) gives
a complete account of how real aggregate demand is determined by the expected pathof the policy rate idt relative to expected inflation.27 This predicted relation between
aggregate demand and the expected path of future interest rates is of essentially thesame kind as in the basic NK model. Hence the introduction of financial frictions, ofa kind capable of accounting for the observed average size and variability of spreads
between deposit rates and lending rates, need not imply any substantial change in
our understanding of the way in which central-bank control of short-term interest
rates determines aggregate expenditure.
Indeed, the basic NK model remains nested as a special case of the model proposed
here. In the case that both types of households have identical preferences (ub (c; ξ) =us (c; ξ)), and the wedge between the deposit rate and lending rate is always zero(Ξt (b) = 0, µbt (b) = 1, so that ω t = 0 at all times), our model is equivalent to thebasic NK model. For in this case ibt = idt at all times, so that there is a single
interest rate; equations (1.14)–(1.15) then imply that λbt = λst at all times;28 andsince the functions cb (λ; ξ) and cs (λ; ξ) must be identical in this case, equilibrium
must involve cbt = cst at all times. Equation (1.17) then reduces simply to the standard
relation Yt = ct + Gt , while equations (1.14)–(1.15) imply that the common marginalutility of income of all households satisfies the usual Euler equation. Of course, thisparameterization is not the one we regard as most empirically realistic (in particular,
it would not account for observed spreads, as discussed below); but since the model
has exactly the implications of the basic NK model for some parameter values, itbanks must hold unremunerated reserves in proportion to their deposits, while required reserves are
not increased by borrowing funds in the interbank market. We abstract from reserve requirements
here.
26
The issues involved in discussing how the central bank actually controls the policy rate are nodifferent here than in the case of the standard NK model. See, for example, Woodford (2003, chap.
2).
27
To be more precise, the expected path of real interest rates determines only desired current
expenditure relative to expected future expenditure, so that current aggregate demand also depends
on expected long-run output, just as in the basic NK model (see, e.g., Woodford, 2003, chap. 4).
The expected long-run level of output is determined by supply-side factors and by the long-run
inflation target.
28
See the Appendix for demonstration of this.

17becomes merely a quantitative issue to determine how different its predictions are forother parameter values. In fact, our results reported below suggest that for manyquestions, a reasonably parameterized version of this model yields predictions quite
similar to those of an appropriately parameterized version of the basic NK model.

1.2

The Dynamics of Private Indebtedness

We allow in general for the possibility that aggregate real borrowing bt from financial
intermediaries may affect aggregate demand, by affecting the real resources used by
the banking sector (the term Ξt (bt ) in (1.17)), by affecting the equilibrium spread
between the deposit rate and the lending rate (equation (1.9)), or both. Hence ingeneral a complete model of how interest-rate policy affects aggregate demand requiresthat we model the evolution of aggregate bank credit, or alternatively, of aggregate
household indebtedness.
Integrating (1.1) over all those borrowers in period t who did not have access to theinsurance agency in the current period, one finds aggregate net beginning-of-period
assets for these households of−δPt−1 bt−1 (1 + ibt−1 ) + δπ b Dtint .
Adding to this quantity the beginning-of-period assets (At per household) of thosehouseholds who did receive insurance transfers at the beginning period t and thenlearned that they are of type b, one obtains
Z(1.18)At (i)di = (1 − δ)π b At − δPt−1 bt−1 (1 + ibt−1 ) + δπ b Dtint
Bt

for the aggregate beginning-of-period net assets of borrowers in period t. Moreover,integrating (1.2) over all period t borrowers, one obtains
Z
Pt bt = −At (i)di + π b [Pt cbt − wtb − Dt − Ttg ],Btwhere wtτ denotes the real wage income of each household of type τ .29 Finally, using(1.18) to substitute for aggregate beginning-of-period assets, and then using (1.4)
29

The fact that each household of a given type has the same labor supply and same wage income
follows from the fact that in equilibrium each has the same marginal utility of income; see the further
discussion of labor supply below.

18

to substitute for At , using (1.5) to substitute for Ttg , using (1.6) to substitute for
Dt , using (1.7) to substitute for dt , using (1.8) to substitute for ibt , using (1.11) tosubstitute for Dtint , and using (1.17) to substitute for Yt , one obtains
bt = δ[bt−1 + π s ω t−1 (bt−1 )bt−1 + π b Ξt−1 (bt−1 )](1 + idt−1 )/Πt − π b Ξt (bt )+π b [δbgt−1 (1 + idt−1 )/Πt − bgt ] + π b π s [(cbt − cst ) − (wtb − wts )],(1.19)

using the notation ω t (bt ) for the function defined in (1.9).
The dynamics of private indebtedness thus depend, among other things, on the
distribution of wage income across households of the two types. We assume labor
supply behavior of exactly the same kind for both types of households (as a consequence of their identical disutility of working), except for the fact that the marginalutilities of income for the two types of households differ. Any household i, if acting
as a wage-taker in the market for labor of type j, will supply hours ht (j; i) to thepoint at which(1.20)
vh (ht (j; i); ξ t ) = λt (i)Wt (j)/Pt .
Aggregation of the labor supply behavior of the two types is facilitated if, as in
Benigno and Woodford (2005), we assume the isoelastic functional formv (h; ξ t ) ≡1
h1+ν H̄t−ν ,1+ν

where {H̄t } is an exogenous labor-supply disturbance process. Solving (1.20) for thecompetitive labor supply of each type and aggregating, we obtainht (j) = H̄t [λ̃t Wt (j)/Pt ]1/νfor the aggregate supply of labor of type j, where£¤ν
λ̃t ≡ π b (λbt )1/ν + π s (λst )1/ν ,(1.21)or alternatively−1

Wt (j)/Pt = λ̃t (ht (j)/H̄t )ν

(1.22)

for the real wage required if firms are to be able to hire a quantity ht (j) of laborof type j. More generally (and also as in Benigno and Woodford), we allow for the
possibility of imperfect competition in the labor market, and suppose that the real
wage required to hire a given aggregate quantity of labor of type j is given by−1ν
Wt (j)/Pt = µwt λ̃t (ht (j)/H̄t ) ,19(1.23)

where µw
t ≥ 1 is an exogenous, possibly time-varying markup factor, indicating variations in the market power of labor.
The above theory of labor supply implies that households of type τ supply fraction (λτt /λ̃t )1/ν of all labor of each type j, and hence receive that same fraction of
aggregate labor income. However, in order to solve for the dynamics of private indebtedness, we must also determine the distribution of national income between labor andcapital (since profits are distributed equally to all households, unlike wage income).
Once again, aggregation is facilitated by assuming (as in Benigno and Woodford) an
isoelastic production function
yt (i) = At ht (i)1/φ
for each differentiated good i, where φ ≥ 1 and At is an exogenous, possibly timevarying productivity factor, common to all goods. Dixit-Stiglitz preferences (on thepart of government as well as households30 ) imply that the demand for each differentiated good i is given by¶−θµpt (i)
yt (i) = Yt,
Pt
where Yt is demand for the composite good, pt (i) is the price of good i, Pt is the priceof the composite good, and θ > 1 is the elasticity of substitution across differentiatedgoods.
Using these relations to solve for the labor demand ht (i) of each firm i as a functionof its price, integrating over the firms in each industry j to find the total demand forlabor of type j,31 solving for the implied real wage for labor of type j, and finallyintegrating over all types of labor, we obtain a total wage billµ ¶1+ωyZ
Ytw Pt
Wt (j)ht (j)dj = µt∆t ,(1.24)
λ̃t H̄tν At
30

Dixit-Stiglitz preferences imply that household utility depends only on the quantity purchasedof a certain composite good, a CES aggregate of the purchases of the individual goods. We assume
that government purchases quantity Gt of this same composite good, and that the composition of
government purchases minimize the cost of obtaining that quantity of the composite good. We
similarly assume that the resources Ξt used in intermediation are in units of the composite good,
and that intermediaries obtain these resources at minimum cost.
31
Note that we assume, as in Woodford (2003, chap. 3), that all firms in a given industry reevaluate their prices at the same time, so that the price pt (i) is at each time the same for all firms
i in industry j.

20where ω y ≡ φ(1 + ν) − 1 ≥ 0 and
Z µ
∆t ≡

pt (i)Pt¶−θ(1+ωy )di ≥ 1is a measure of the dispersion of goods prices (taking its minimum possible value, 1,
if and only if all prices are identical). In the Calvo model of price adjustment, thisdispersion measure evolves according to a law of motion∆t = h(∆t−1 , Πt ),(1.25)

where the function h(∆, Π) is defined in the Appendix.
Using (1.24) for the total wage bill and our conclusion regarding the distributionof the wage bill between households of the two types, we can solve for the wage
income of households of each type. This solution, together with the consumptionfunctions (1.16), allows us to write the last term in square brackets in (1.19) as afunction of the form B(Yt , λbt , λst , ∆t ; ξ̃ t ), defined in the Appendix, where the vectorξ̃ t of exogenous disturbances includes both the vector of preference shocks ξ t andthe additional exogenous disturbances At and µw
t . The law of motion for private
indebtedness bt can then be written
bt = δ[bt−1 + π s ω t−1 (bt−1 )bt−1 + π b Ξt−1 (bt−1 )](1 + idt−1 )/Πt − π b Ξt (bt )+π b [δbgt−1 (1 + idt−1 )/Πt − bgt ] + π b π s B(Yt , λbt , λst , ∆t ; ξ̃ t ).

(1.26)

This allows us to describe the evolution of real private debt as a function of its
own past level, disturbances to the financial sector (possible exogenous shifts in thefunctions Ξt (b) and ω t (b)), the evolution of the policy rate idt relative to inflation,
the evolution of real government debt bgt , and the additional aggregate variables(Yt , λbt , λst , ∆t ; ξ̃ t ) that determine the relative expenditure and the relative incomesof the two types of households.
The system of equations consisting of (1.8)–(1.9), (1.14)–(1.17), and (1.25)–(1.26),
together with a monetary-policy reaction function (such as a Taylor rule) to specify idt(as a function of variables such as inflation and real activity) and a fiscal rule to specify
the real public debt bgt (also possibly as a function of variables such as inflation andreal activity), then comprise a complete “aggregate demand block” for our model, thatsuffices to determine the evolution of the variables {λbt , λst , ibt , idt , Πt , ∆t , bt } given theevolution of {Yt } and the exogenous disturbances. It remains to specify the model’s21

“aggregate supply block,” that determines aggregate output Yt for any given evolutionof inflation and other variables, in order to have a complete general-equilibrium modelof the monetary transmission mechanism.
A noteworthy property of this system is that when credit frictions matter, Ricardian equivalence generally does not obtain. Even if we consider alternative paths for{bgt } while holding the path of distorting taxes {τ t } constant, so that contemplatedchanges in the path of government debt are achieved entirely through changes in thesize of lump-sum transfers, a change in the path of the public debt will generally
require a different equilibrium evolution of real activity, interest rates and inflation,
contrary to the implication of the basic NK model.32 For (1.26) implies that in theabsence of any offsetting changes in the paths of other endogenous variables, a change
in the path of {bgt } will require an offsetting change in the path of {bt }; essentially,
government borrowing crowds out private borrowing, in the absence of changes inmacroeconomic conditions that increase aggregate private saving.33 In the special
case considered at the end of the previous section (when Ξt = 0 and ω t is independent of the level of private debt), this change in the path of private indebtedness
still has no consequences for the determination of aggregate output, interest rates or
inflation, or for the allocation of consumption or labor effort between the two types
of households, and so Ricardian equivalence still obtains. However, except in thisspecial case, a change in the path of private indebtedness has consequences for aggregate demand determination, by changing the spread between the lending rate andthe deposit rate, by changing the resources used by intermediaries, or both.
32

Crucial to this result is our assumption here that the government can borrow from the private
sector at a rate more favorable than that available to private non-financial borrowers: the rate idt
at which intermediaries are able to obtain funding, rather than the rate ibt paid by households thatmust borrow from intermediaries. In effect, when the public debt is increased the government is(among other things) borrowing at this lower rate on behalf of households that would like to borrowat this rate but are assumed to be unable to do so on their own account. This increases aggregate
demand in somewhat the same way as a reduction in credit spreads does.
33
In the simple case in which ω t (bt ) = Ξt (bt ) = 0, (1.26) determines the evolution of an aggregate
credit variable, bt + π b bgt , in a way that is independent of the composition of that variable, sothat a unit increase in bgt requires a reduction of bt by precisely π b units, so that each borrowing
household must borrow exactly one unit less for each unit that is borrowed (per capita) by thegovernment. The relation between the evolution of the two variables is more complex when private
indebtedness increases credit frictions while government debt does not, but the most important
effect of government borrowing remains the “crowding out” of private borrowing.

22

1.3

Aggregate Supply

It remains to specify the aggregate supply side of the model. This part of the model
remains the same as the basic NK model (as expounded, for example, in Benigno
and Woodford, 2005), except that in modelling the cost of supplying a given quantityof output (and hence the incentives of price-setters) we must take account of thediffering labor supply behavior of savers and borrowers. The model of labor supply
explained above implies that the equilibrium real marginal cost of supplying output
in any industry j is equal tost (j) =φµwtyt (j)ωy.
1+ω
At y H̄tν λ̃tThis differs from the expression in Benigno and Woodford only in that the factor λ̃t
in the denominator is no longer the marginal utility of income of a representative
household, and so is no longer so simply related to aggregate real expenditure.
As in the basic NK model, we assume staggered price adjustment of the kind first
hypothesized by Calvo (1983). This implies an inflation equation of the form
Πt = Π(Zt ),(1.27)

where Zt is a vector of two forward-looking variables, recursively defined by a pair ofrelations of the form
Zt = G(Yt , λbt , λst ; ξ̃ t ) + Et [g(Πt+1 , Zt+1 )],(1.28)

where the vector-valued functions G and g are defined in the Appendix. (Among thearguments of G, the vector of exogenous disturbances ξ̃ t now includes the sales tax
rate τ t , in addition to the disturbances already mentioned; this is relevant to firms’
pricing decisions, as they balance after-tax marginal revenue with the marginal costof supplying more
These relations are of exactly the same form as in the basic NK model, exceptthat two distinct marginal utilities of income are here arguments of G; in the casethat λbt = λst = λt , the relations (1.28) reduce to exactly the ones in Benigno andWoodford (2005). The system (1.27)–(1.28) indicates the nature of the short-run
aggregate-supply trade-off between inflation and real activity at a point in time, given
expectations regarding the future evolution of inflation and of the variables {Zt }.
23

(The precise nature of the implied aggregate-supply relation is discussed further insection 2.1.) Equations (1.8)–(1.9), (1.14)–(1.17), (1.25)–(1.26), and (1.27)–(1.28),
together with equations specifying interest-rate policy and the evolution of the public
debt, then comprise a complete system of equations for determination of the endogenous variables {Yt , λbt , λst , ibt , idt , Zt , Πt , ∆t , bt }, given the evolution of the exogenous
disturbances.

2

Credit Frictions and the Propagation of Disturbances

We have shown that it is possible to generalize the basic NK model in a fairly straightforward way to incorporate credit frictions — more specifically, a spread between thedeposit rate and the lending rate, that may be large or small, constant or variable,
and exogenous or endogenous, depending how we choose to parameterize the model.
This shows, at the very least, that the inherent structure of NK models does not insome way require one to ignore the existence of such frictions. But how much does
this generalization change the implications of the resulting model?
In this section, we consider the effects on the economy of a variety of types of exogenous disturbances, including monetary policy shocks, under simple specificationsof monetary policy. Our goal is not yet to compare monetary policies, but rather tocompare the predictions of alternative model specifications; we wish to determine towhat extent our conclusions about the consequences of a given monetary policy areaffected by the introduction of heterogeneity and credit frictions.

2.1

Log-Linearized Structural Relations

To approach this question, we log-linearize the structural relations of our modelaround steady-state values of the various endogenous variables that represent a perfect foresight equilibrium in the case of no random variation in any of the exogenous
disturbance processes, as discussed further in the Appendix. The solution to theselinear equations under a correspondingly log-linear specification of monetary policy
provides a linear approximation to the equilibrium responses to the various typesof disturbances, in the case that these random variations are small enough. The
linearity of the solution allows us to discuss the equilibrium responses to individual24

shocks independently of whether other exogenous variables change concurrently, andto discuss the size of the responses relative to the size of the shock without caring
about the size of the shock that is considered.
We first summarize the structure of the log-linearized model, as these equations
themselves provide considerable insight into the model’s implications, and the similarities and differences between the predictions of the generalized model and those of
the basic NK model (which is itself most familiar in its log-linearized form). We loglinearize the structural relations of the previous section around a steady state withzero inflation (Π̄ = 1). This means that in our analysis, we shall restrict attention tomonetary policy rules that imply an inflation rate of zero, or one not far from zero, in
the absence of stochastic disturbances. This simplification is familiar in the standard
NK literature, and we follow it here in order to focus solely on the new complications
introduced by heterogeneity and credit frictions. Moreover, we show in section 3 that
according to the present model, optimal monetary policy has this property; hence theapproximation adopted here suffices for the study of monetary policies that are closeenough to optimal policy.
We turn first to the model’s “aggregate demand” block. Log-linearization ofequations (1.14)–(1.15) yields
b

b

s

s

s

b

λ̂t = ı̂bt − Et π t+1 + (δ + (1 − δ)π b )Et λ̂t+1 + (1 − δ)π s Et λ̂t+1 ,λ̂t = ı̂dt − Et π t+1 + (δ + (1 − δ)π s )Et λ̂t+1 + (1 − δ)π b Et λ̂t+1 ,τintroducing the notation λ̂t

(2.1)
(2.2)

τ

m
m
≡ log(λτt /λ̄ ) for τ = b, s; ı̂mt ≡ log(1 + it /1 + ı̄ ) form = b, d; and π t ≡ log Πt for the rate of inflation. (In each case, a variable with abar indicates the steady-state value of the corresponding variable, discussed in the
Appendix.) Log-linearization of (1.8) similarly yields

ı̂bt = ı̂dt + ω̂ t ,(2.3)

where ω̂ t ≡ log(1 + ω t /1 + ω̄). We can similarly log-linearize (1.9) to obtain ω̂ t as alinear function of b̂t , where we define b̂t ≡ log(bt /b̄).
This system can be alternatively expressed in a way that makes the model’s implications more transparent. If we take a weighted average of equations (2.1) and(2.2), weighting the two equations by π b and π s respectively, we obtain
λ̂t = ı̂avg− Et π t+1 + Et λ̂t+1 ,
t
25

(2.4)

b

s

where we now define λ̂t ≡ π b λ̂t + π s λ̂t as the average (log) marginal utility of income
and
ı̂avg
≡ π b ı̂bt + π s ı̂dt
(2.5)
t
as the average of the interest rates that are relevant (at the margin) for all of thesavers and borrowers in the economy. Subtracting (2.2) from (2.1) instead yieldsΩ̂t = ω̂ t + δEt Ω̂t+1 ,
b

(2.6)

s

where Ω̂t ≡ λ̂t − λ̂t is a measure of the inefficiency of financial intermediation, insofar
as the marginal utilities of the two (ex ante identical) types would be equated if
financial markets were frictionless. Equations (2.4) and (2.6) are equivalent to thesystem (2.1)–(2.2), but highlight the fact that the (appropriately defined) averagemarginal utility of income is still related to the expected path of real interest rates— once an appropriate average interest rate is defined — in exactly the same way
as in the basic NK model, while the dispersion of marginal utilities of income acrossdifferently situated members of the population depends only the evolution of thecredit spread {ω t } (which is in turn a function of exogenous conditions affecting thebanking sector, and of the total volume of lending) and not the absolute level of realor nominal interest rates.
Log-linearization of (1.17) yields
Ŷt = sc c̄t − σ̄(λ̂t + sΩ Ω̂t ) + Ĝt + Ξ̂t ,(2.7)

where we define Ŷt ≡ log(Yt /Ȳ ), Ĝt ≡ (Gt − Ḡ)/Ȳ , and Ξ̂t ≡ (Ξt − Ξ̄)/Ȳ ,34 andthe exogenous disturbance c̄t is a weighted average of changes in the impatience toconsume of the two types of households. The coefficient σ̄ measures the (appropriately
weighted) average35 sensitivity of households’ expenditure decisions to variations in
34

We do not define these last two hat variables as log deviations from the steady-state value, sothat we can discuss calibrations in which Ḡ or Ξ̄ may equal zero, though we still consider small
non-zero values of the corresponding disturbances.
35
Definition (2.8) implies that σ̄ is only actually an average of σ b and σ s in the case that Ḡ = Ξ̄ = 0.
More generally, σ̄ is sc times the average household intertemporal elasticity of substitution, wheresc ≡ π b sb + π s ss is the share of private expenditure in aggregate expenditure. In terms of the notation used in Woodford (2003, pp. 80, 243), σ̄ is the coefficient analogous to σ in the representativehousehold model, while the coefficients σ τ are analogous to the intertemporal elasticity of substitution σ C .

26the marginal utility of income (or equivalently, their interest-sensitivity);σ̄ ≡ π b sb σ b + π s ss σ s > 0,(2.8)

where σ τ is each type’s intertemporal elasticity of substitution and sτ ≡ c̄τ /Ȳ is thesteady-state share of each type’s purchases in aggregate national expenditure. The
coefficientsb σ b − ss σ ssΩ ≡ π b π s(2.9)
σ̄
indicates the degree to which aggregate demand is affected by heterogeneity in themarginal utility of income, given the average marginal utility of income; this depends
on the degree to which the expenditure decisions of borrowers are more interestelastic than those of savers. (More precisely, what matters is how much the aggregate
reduction in borrowing by borrowers, per percentage point increase in the interest ratethat they face, exceeds the aggregate increase in saving by savers in response to aninterest-rate increase of the same size.)
Solving (2.7) for λ̂t as a function of aggregate expenditure, and substituting for
λ̂t in (2.4), we obtain an “intertemporal IS relation”
Ŷt = −σ̄(ı̂avg− Et π t+1 ) + Et Ŷt+1 − Et [∆gt+1 + ∆Ξ̂t+1 − σ̄sΩ ∆Ω̂t+1 ],t(2.10)

where gt ≡ Ĝt + sc c̄t is a composite disturbance as in Woodford (2003, pp. 80, 249),
and ∆ indicates a first difference. Note that the first four terms on the right-hand sideof (2.10) are exactly as in the basic NK model (with appropriate generalizations of
the definitions of variables and coefficients to allow for heterogeneity), while the final
two terms exist only in the case of credit frictions. An important difference between
this relation and the standard “IS relation” is that the interest rate appearing in it
is no longer the policy rate. Instead, (2.3) together with (2.5) imply that the policy
rate ı̂dt and the rate that is relevant for the IS relation are linked by the equilibrium
relation(2.11)
ı̂avg= ı̂dt + π b ω̂ t ,tindicating that the spread between them increases when credit spreads increase.
The complete aggregate demand block developed in section 1.1 can then be summarized (in our log-linear approximation) by the intertemporal IS relation (2.10),
together with (2.11) connecting the average interest rate with the policy rate, thelog-linear version of (1.9) for the determination of the credit spread, and (2.6) for27

the determination of the marginal-utility gap Ω̂t . In the case that either ω̂ t or Ξ̂t depends (to first order) on the evolution of b̂t , completion of the system of equilibrium
relations requires a law of motion for aggregate private indebtedness. Log-linearizingbs(1.26), and using the same method as above to substitute for λ̂t and λ̂t , we obtain an
equation for b̂t as a linear function of b̂t−1 , ı̂dt−1 − π t , Ŷt , Ω̂t , b̂gt , b̂gt−1 , and the exogenous
disturbances. (See the Appendix for details.)
Log-linearization of the aggregate-supply block consisting of equations (1.27)–(1.28) yields the log-linear aggregate-supply relationˆ − ν h̄ − (1 + ω )a + µ̂w + τ̂ ] + βE π ,π t = ξ[ω y Ŷt − λ̃
t
t
y t
t
t t+1
t

(2.12)ˆ ≡ log(λ̃ /λ̃),¯through calculations explained in the Appendix, where we define λ̃ttwwh̄t ≡ log(H̄t /H̄), at ≡ log(At /Ā), µ̂wt ≡ log(µt /µ̄ ), τ̂ t ≡ − log(1 − τ t /1 − τ̄ ), andξ≡

1 − α 1 − αβ>0α 1 + ωy θ

(where 0 < α < 1 is the fraction of prices that remain unchanged from one period
to the next) determines the sensitivity of the inflation rate to variation in averagemarginal costs. Note that (2.12) takes exactly the same form as in the basic NKˆ replaces the marginal utility of income of the representativemodel, except that here λ̃thousehold.
ˆ that
It is also important to note that the “average” marginal utility of income λ̃tenters the aggregate-supply relation is in general not exactly the same as the one thatenters the aggregate-demand relation (2.7), or the one that is most directly related
to the expected path of real interest rates in (2.4). The two are related through theidentityˆ = λ̂ + (γ − π )Ω̂ ,
λ̃
t
t
b
t
b
where

Ã b !1/νλ̄.
γ b ≡ πb ¯λ̃ˆ in (2.12), and then using (2.7) to substitute for λ̂ as
Using this to substitute for λ̃ttin the derivation of (2.10), we obtain an aggregate-supply relation
π t = κ(Ŷt − Ŷtn ) + ut + ξ(sΩ + π b − γ b )Ω̂t − ξσ̄ −1 Ξ̂t + βEt π t+1 ,28(2.13)

with a slopeκ ≡ ξ(ω y + σ̄ −1 ) > 0.
Here the composite exogenous disturbance term Ŷtn (the “natural rate of output”)
is a linear combination of the disturbances c̄t , Ĝt , h̄t , and at (sources of variation in
the flexible-price equilibrium level of output that, in the absence of steady-state distortions or financial frictions, correspond to variations in the efficient level of output,
as discussed further in section 3); the additional exogenous term ut (the “cost-pushshock”) is instead a linear combination of the disturbances µ̂wt and τ̂ t (sources ofvariation in the flexible-price equilibrium level of output that do not correspond toany change in the efficient level of output). This is identical to the “new KeynesianPhillips curve” of the basic NK model, with the exception of the terms proportionalto Ω̂t and Ξ̂t , indicating “cost-push” effects of the credit frictions in our extended
model.
Equations (2.10) and (2.13) are thus direct analogs of two of the equations of
the canonical “three-equation model”; the third equation (a central-bank reaction
function, such as a Taylor rule, for the policy rate as a function of inflation andoutput) is unchanged by the existence of credit frictions. In the case that both {ω̂ t }and {Ξ̂t } can be treated as exogenous processes (so that {Ω̂t } is exogenous as well),these same three equations again provide a complete system for the determination ofequilibrium inflation, output and interest rates, except that one must adjoin equation(2.11) to connect the interest rate that appears in the IS relation to the policy rate.
If one substitutes the central-bank reaction function for ı̂dt in (2.11), in order to derive
the implied response of the average interest rate — or if one supposes that the central
bank uses the average interest rate as its operating target, a policy proposal that weanalyze further in section 3 — then one again obtains a three-equation model directly
analogous to the basic NK model. For example, if the central bank follows a Taylor
rule of the formı̂dt = φπ π t + φy Ŷt + ²m
(2.14)t ,then the complete model would consist of (2.10), (2.13), andı̂avg= φπ π t + φy Ŷt + π b ω̂ t + ²mtt .

(2.15)The only differences relative to the basic NK model are that the interest rate appearing in this three-equation system is not the only relevant interest rate and may not29correspond to the policy rate; that the numerical values of the coefficients σ̄ and κ
must take appropriate account of the different degrees of interest-sensitivity of expenditure of different units in the economy; and that time-varying financial frictions
are an additional source of disturbance terms in all three equations.
It follows that at least in this case, inflation and output determination can beunderstood in exactly the same way as in the basic NK model, regardless of theaverage size of credit frictions, or their degree of variability. For example, in the casethat the monetary policy equation involves no response to lagged variables (or toany endogenous variables other than inflation, output, or forecasts of these), and thepolicy implies a determinate rational-expectations equilibrium, that equilibrium will
make inflation, output and interest rates all a function solely of the economy’s current
state — to be precise, the current values of Ŷtn , gt , ut , ω̂ t , Ξ̂t , and the monetary policy
disturbance, and current information about the expected future evolution of thesevariables. (As has often been remarked of the basic NK model, the model is thus
“purely forward-looking,” and cannot explain inflation persistence other than as aconsequence of persistence in the exogenous disturbances just mentioned.)
In the simple case of a monetary policy described by (2.14), the conditions fordeterminacy of equilibrium remain exactly the same as in the basic NK model; in thecase that φπ , φy ≥ 0, these amount simply to the requirement that the policy ruleconform to the “Taylor Principle” (Woodford, 2003, p. 254), i.e., that the response
coefficients satisfy
1−βφπ +φy > 1.
κIf this condition is satisfied, one can solve for inflation as a function of current andexpected future values of the disturbance processes in exactly the same way as is
explained in Woodford (2003, chap. 4, secs. 2.2, 2.4), and the coefficients on currenttmand expected future values of the disturbances c̄t , Ĝt , h̄t , at , µ̂w
t , τ̂ , or ²t at all horizons
are identical to the predictions of the basic NK model, if the latter model is calibrated
to have the same values for the coefficients β, ξ, σ̄, and ω y .36 The only difference in36

Of course, the numerical values of some of these coefficients may be different, owing to theexistence of credit frictions, than they would be in an economy without such frictions, owing, forexample, to an effect of steady-state distortions resulting from credit spreads on the steady-statelevel of output, and hence the point at which various elasticities are evaluated. This would not,however, affect the accuracy of predictions made by the NK model if it were correctly parameterized
to match the elasticities observed in an actual economy.

30

the solution is that shocks to the current or expected future values of the financial
disturbances ω̂ t and Ξ̂t will affect equilibrium inflation as well.
Moreover, not only does the model predict the same numerical responses to nonfinancial disturbances, under given monetary-policy coefficients (φπ , φy ) — so thatone’s conclusions about the desirability of a particular choice of those coefficients,from the point of view of how they effect the economy’s response to non-financial disturbances, will be unchanged — but the predicted responses to financial disturbances
do not really involve any new considerations (beyond the mere fact that such disturbances can occur and ought to be measured in order to properly conduct policy). The
effects of variations in the processes {ω̂ t , Ξ̂t } on inflation and output are predicted tobe the same as the effects that other kinds of disturbances have, when they shift thethree equations to a similar extent and with a similar expected degree of persistence.
Thus the effects of financial shocks on inflation can be decomposed into three types ofeffects that are already present in the basic NK model: the effects of a disturbance tothe “natural rate of interest” (a shift in the real average interest rate consistent witha zero output gap), the effects of a “cost-push shock” (a shift in the size of output gap
required for price stability), and the effects of a “monetary policy shock” (a change
in the average interest rate relative to what would ordinarily follow from the current
inflation rate and output gap, in this case due to a change in the credit spread rather
than a change in the policy rate). Financial disturbances typically have effects of all
three types; but their consequences can be easily understood if the consequences of
those three general types of disturbances are already understood.
The case in which {ω̂ t , Ξ̂t } are both completely exogenous processes is, of course,a fairly special one. If one or both of them depends on the volume of bank lending,as allowed for in our exposition above, a larger system of equations, including the lawof motion for private debt, is needed in order to predict the evolution of inflation,
output and interest rates. We do not seek to present analytical results for this more
complex case, but instead offer some illustrative numerical results.

2.2

Model Calibration

The numerical values for parameters that are used in our calculations are explained in
the Appendix. Many of the model’s parameters are also parameters of the basic NK
model, and in the case of these parameters we assume similar numerical values as in31the numerical analysis of the basic NK model in Woodford (2003). The new parameters that are also needed for the present model are those relating to heterogeneity or
to the specification of the credit frictions. The parameters relating to heterogeneity
are the fraction π b of households that are borrowers, the degree of persistence δ of ahousehold’s “type”, the steady-state expenditure level of borrowers relative to savers,
sb /ss , and the interest-elasticity of expenditure of borrowers relative to that of savers,
σ b /σ s .37
In the calculations reported here, we assume that π b = π s = 0.5, so that there
are an equal number of borrowers and savers. We assume that δ = 0.975, so that the
expected time until a household has access to the insurance agency (and its type isdrawn again) is 10 years. This means that the expected path of the spread between
lending and deposit rates for 10 years or so into the future affects current spending
decisions, but that expectations regarding the spread several decades in the future
are nearly irrelevant.
We calibrate the values of sb and ss to equal 1.1 and 0.3 respectively, so that thesteady-state share of private expenditure in output is 0.7 (the mean of those two values).38 This degree of heterogeneity is chosen arbitrarily for purposes of illustration;
it is important, however, that sb > ss , as this is required in order for the steady state
to involve borrowing by the type b households. The degree of heterogeneity assumed
in our calibrated examples implies steady-state debt b̄ between 0.5 and 0.6 of steadystate output (with the exact fraction depending on the intermediation technology).
Finally, we calibrate the value of σ b /σ s to equal 5. This is again an arbitrarychoice, though the fact that borrowers are assumed to have a greater willingness
to substitute intertemporally is important, as this results in the prediction that anexogenous tightening of monetary policy (a positive value of the residual ²mt in (2.14))
results in a reduction in the equilibrium volume of credit bt (see Figures 1 and 4 below).
37

Another new parameter as a consequence of heterogeneity is the steady-state level of government
debt relative to GDP, b̄g /Ȳ . This parameter need not be specified in the representative-householdbasic NK model, since changes in it simply imply a different steady-state level of net transfers, andthe size of these does not affect any of the equilibrium relations. In the model with heterogeneity,
it does matter, because of the failure of Ricardian equivalence noted above. But in our baseline
calculations, we assume that b̄g = 0.
38
The implied share of government purchases in total output is then 0.3, minus the small part of
output that is consumed by intermediaries. The implied steady-state level of government purchases
therefore varies slightly, depending on our calibration of the intermediation technology.

32

In fact, the degree of asymmetry assumed here results in a predicted contraction of
credit in response to a monetary policy shock of roughly the magnitude indicatedby VAR estimates using recent US data in the case of an exogenous credit spread(Figure 1),39 though the predicted credit contraction is smaller (while still positive)
when we assume that credit spreads increase with the volume of credit. (In the results
reported here, we hold constant our parameterization of preferences when we change
the assumed intermediation technology, in order to clarify how the intermediation
technology affects the results obtained.40 )It is also necessary to specify the steady-state values of the functions ω(b) and Ξ(b)that describe the financial frictions, in addition to making clear what kinds of random
perturbations of these functions we wish to consider when analyzing the effects of“financial shocks.” We here mainly present results for two cases. In each case, we
assume that there is no steady-state financial markup (µ̄b = 1), so that the steadystate credit spread is due entirely to the marginal resource cost of intermediation;
but we do allow for exogenous shocks to the markup µbt (which then becomes slightly
greater than 1), and this is what we mean by the “financial shock” in Figures 15 and
16 below.41 In treating the “financial shock” as involving an increase in markups but
no increase in the real resources used in banking, we follow Gerali et al. (2008).42
The two cases considered differ in the specification of the (time-invariant) intermediation technology Ξ(b). In the case of a linear intermediation technology, we supposethat Ξ(b) = ω̄b, while in the case of a convex intermediation technology, we assume
39

As discussed in the Appendix, our estimated elasticities range from values in the range from -0.2
to -0.4 for samples using only data since the 1990s, though we obtain larger responses, on the orderof -0.9, using samples including data from the 1980s. See also Lown and Morgan (2002) for evidence
similarly supporting the association of credit contraction with a monetary policy tightening.
40
To be precise, we hold constant across specifications of the financial frictions the numericalvalues assumed for the total share of private expenditure in aggregate output, as well as for the
relative shares sb /ss of the two types; and for the aggregate elasticity of intertemporal substitution
σ̄, as well as for the relative elasticities σ b /σ s of the two types. This does not quite mean that theutility functions ub (c; ξ) and us (c; ξ) are invariant; see the Appendix for details.
41
Note that our conclusions regarding both equilibrium and optimal responses to shocks otherthan the “financial shock” are the same as in an economy in which the banking system is perfectlycompetitive (and there are no risk premia), up to the linear approximation used in the numericalresults reported below.
42
These authors cite the Eurosystem’s quarterly Bank Lending Survey as showing that since
October 2007, banks in the euro area had “strongly increased the margins charged on average andriskier loans” (p. 24).

33

that
Ξ(b) = Ξ̃bη

(2.16)

for some η > 1.43 In both cases, in our numerical analyses we assume a steady-state
credit spread ω̄ equal to 2.0 percentage points per annum,44 corresponding to themedian spread between the FRB index of commercial and industrial loan rates andthe federal funds rate, over the period 1986-2007. (Combined with our assumptionthat “types” persist for 10 years on average, this implies a steady-state “marginalbsutility gap” Ω̄ ≡ λ̄ /λ̄ = 1.22, so that there would be a non-trivial welfare gain fromtransferring further resources from savers to borrowers.) In the case of the convex
technology, we set η so that a one-percent increase in the volume of credit increases
the credit spread by 10 basis points (per annum).45 The assumption that η > 1 allows
our model to match the prediction of VAR estimates that an unexpected tighteningof monetary policy is associated with a slight reduction in credit spreads (see, e.g.,
Lown and Morgan, 2002); and the elasticity assumed in our calibration is roughly ofthe magnitude implied by the relative size of our estimates of the respective declines
in the volume of credit and of the average credit spread.46
As a first exercise, we consider the implied equilibrium responses of the model’s
endogenous variables to the various kinds of exogenous disturbances, under the assumption that monetary policy is described by a Taylor rule of the form (2.14). The
coefficients of the monetary policy rule are assigned the values φπ = 2 and φy = 0.25
as in Woodford (2003, chap. 4), allowing comparison between our quantitative results here and those presented there for a calibrated representative-household model.
Among other disturbances, we consider the effects of random disturbances to the error term ²mt in the monetary policy rule. Later, we consider the predicted dynamics
43

One interpretation of this function is in terms of a monitoring technology of the kind assumed
in Goodfriend and McCallum (2007). Suppose that a bank produces monitoring according to a−1
−1
Cobb-Douglas production function, k 1−η Ξηt , where k is a fixed factor (“bank capital”), andmust produce a unit of monitoring for each unit of loans that it manages. Then the produced
goods Ξt required as inputs to the monitoring technology in order to manage a quantity b of loans
will be given by a function of the form (2.16), where Ξ̃ = k 1−η . A sudden impairment of bankcapital, treated as an exogenous disturbance, can then be represented as a random increase in themultiplicative factor Ξ̃. This is another form of “financial shock”, with similar, though not identical,
effects as the financial markup shock considered here; see the Appendix for further discussion.
44
In our quarterly numerical model, this means that we choose a value such that (1 + ω̄)4 = 1.02.
45This requires that η = 6.06.
46
See the Appendix for further details.

34under a variety of other monetary policy specifications as well.

2.3

Numerical Results

We begin by considering the predicted responses to aggregate disturbances of a kindthat also exist in the basic NK model, in order to determine how accurate thatmodel’s answers are about the questions to which it gives answers. We first consider
the case of a linear intermediation technology. In this case, the credit spread ω tevolves exogenously, as assumed in the discussion at the end of section 1.1, but Ξt is
no longer independent of bt . Nonetheless, in this case we continue to find that for areasonable parameterization of the quantity of resources used in intermediation, theexistence of credit frictions makes virtually no difference for the predicted equilibrium
responses to shocks.
This is illustrated in Figures 1-3 for three particular types of exogenous disturbances. In Figure 1 we consider the equilibrium responses to a contractionary monetary policy shock, represented by a unit (one percentage point, annualized) increase
in ²mt . We furthermore assume that the policy disturbance is persistent; specifically,m²t is assumed to follow an AR(1) process with coefficient of autocorrelation ρ = 0.75.
The separate panels of the figure indicate the impulse responses of output, inflation,47
the deposit rate, the lending rate,footnoteIn the present model, the spread between
the deposit rate and the lending rate is exogenously fixed, and so these two variables
necessarily respond by exactly the same amount, except in the case of a shock to theexogenous credit spread itself. However, we include both panels as we use the sameformat for the figures to follow, when inclusion of both is no longer redundant. andaggregate private debt respectively. Figure 2 illustrates the equilibrium responsesof the same variables to a unit positive innovation in the productivity factor, where
the disturbance is again assumed to have an autocorrelation of 0.75, and monetary
policy is conducted in accordance with (2.14) with no random term. Figure 3 shows
the corresponding equilibrium responses in response to an increase in government
purchases by an amount equal to one percent of total output.
In each figure, the predicted impulse responses under our model with financial
47

In the plots, both the inflation rate and the interest rates are reported as annualized rates, sothat 0.10 means an increase in the inflation rate of 10 basis points per annum. In terms of ourquarterly model, what is plotted is not the response of π t , but rather the response of 4π t .

35

frictions (the case labeled ‘FF’ in each figure) are contrasted with those under two
variant parameterizations of the model. The case labeled ‘NoFF’ corresponds toan otherwise identical model in which ω t (b) ≡ 0 and Ξt (b) ≡ 0, but we retain theheterogeneity in preferences, parameterized in the same way as in the ‘FF’ model.
The case labeled ‘RepHH’ is one in which in addition to assuming zero credit frictions, we assume identical preferences for the two types; this model is equivalent toa representative-household model (specifically, to the basic NK model as presented
in Woodford, 2003, chap. 4). Comparison of these three cases allows us to separately consider the degree to which credit frictions as opposed to heterogeneity makea difference.
In each of Figures 1 through 3, we observe that the impulse responses of output,
inflation, and the two interest rates are virtually identical under all three parameterizations of the model. (The same is true for the other aggregate disturbances that haveanalogs in the representative-household model — a common disturbance to the impatience to consume of all households, a disturbance to the disutility of work, a shock
to government purchases, a shock to the tax rate, or a shock to the wage markup —though we do not include these figures here.) We have already explained in section
2.1 why this would be true in the case that the resources used in intermediation are
independent of the volume of lending. Our numerical results indicate that even when
we assume that intermediation uses resources (and indeed that credit spreads are
entirely due to the marginal resource cost of making additional loans), and that therequired resources are proportional to the volume of lending, heterogeneity and theexistence of a steady-state credit spread (of a realistic magnitude) still make only anegligible difference. This is because the contribution of the banking sector to theoverall variation in the aggregate demand for produced goods and services is stillquite small.48
Financial frictions matter somewhat more for equilibrium dynamics if we also assume that credit spreads vary endogenously with the volume of lending. Figures 4-9
show equilibrium responses of the same aggregate variables to a variety of types ofexogenous disturbances, in the case of the “convex intermediation technology” cali48

Note that in each of Figures 1-3, the existence of the credit frictions in the ‘FF’ model makes
a substantial difference for the equilibrium evolution of credit bt relative to the prediction of the‘NoFF’ model. However, this change in the size of the banking sector does not have substantial
consequences for aggregate output, employment, or inflationary pressure.

36

bration discussed in section 2.2. Figures 4, 5 and 7 show responses to the same threekinds of shocks as in Figures 1-3 respectively, but for the alternative intermediation
technology. Figure 6 shows responses to an exogenous 1 percent increase in the wage
markup µw
t ; Figure 8 shows responses to an exogenous increase in the impatience toconsume of savers; and Figure 9 shows responses to an exogenous increase in real government debt by an amount equal to 1 percent of GDP. In each case, the disturbanceis modeled as an AR(1) process with autoregressive coefficient 0.75.
In the case of the monetary policy shock (Figure 4), we again find that the equilibrium responses of output and inflation are nearly the same in all three models,though the ‘FF model’ is no longer quite so indistinguishable from the ‘NoFF’ model.
The most important effect of allowing for endogeneity of the credit spread is on theimplied responses of interest rates to the shock. Because credit contracts in responseto this shock (as noted earlier, though now by less than in Figure 1), the spread
between the lending rate and the deposit rate decreases, in accordance with the empirical finding of Lown and Morgan (2002). This means that the deposit rate needno longer decline as much as does the lending rate. Moreover, because the reduced
spread has an expansionary effect on aggregate demand, both output and inflation
decline slightly less in response to the shock than in the ‘NoFF’ model; this is also a
reason for the deposit rate to decline less, and in fact the reduction in its decline issufficient that the lending rate also declines slightly less than in the ‘NoFF’ model,
despite the reduction in the spread. Thus the most visible effect is on the predicted
response of the deposit rate, which is visibly smaller in the ‘FF model.’ The effectsof financial frictions are similarly mainly on the path of the deposit rate in the caseof a shock to the sales tax rate τ t (not shown).
The effects of financial frictions are somewhat less trivial in the case of the technology shock (Figure 5) and the shock to the wage markup (Figure 6). Though again
the largest effect is on the path of the deposit rate, in these cases the endogeneityof the markup also has non-negligible effects on the equilibrium responses of output
and inflation. (The reason for the difference is that these are now shocks that havelarger effects on the path of credit, and so larger effects on the equilibrium spreadin the case of the convex technology.) Because an increase in productivity leads toan expansion of credit, credit spreads now increase in the ‘FF model’; this is has acontractionary effect on aggregate demand, so that output increases less, while inflation declines slightly more than in the ‘NoFF model’. Similar effects of financial
37

frictions are observed in the case of a disturbance to the disutility of working (anexogenous increase in the multiplicative factor H̄t in (1.22), not shown). The effects
of an increase in the wage markup (Figure 6) are likewise similar, but with opposite
signs.
The effects of financial frictions are even more significant in the case of a shock
to government purchases (Figure 7) or to the consumption demand of savers (Figure8).49 These are both disturbances that crowd out the expenditure of private borrowers(as the most interest-sensitive category of expenditure) to a significant extent, andso substantially reduce equilibrium borrowing and credit spreads. In each case, thereduction in spreads has a further expansionary effect on aggregate demand, so that
output increases by more than in the ‘NoFF’ model, while inflation rises slightly
instead of falling.
Note that the effect would be quite different in the case of a shock to the consumption demand of borrowers rather than savers (not shown). In this case, private
credit would increase rather than decreasing, and by less than in Figure 8, because of
the greater interest-elasticity of the demand of borrowers; this would imply a small
increase in spreads, making the disturbance slightly less expansionary, but with aless dramatic effect than in Figure 8.50 The aggregate effects of financial frictions areeven smaller in the case of a uniform increase in the consumption demand of bothtypes of households, since in this case the effects of the two types of expenditure on
equilibrium credit spreads partially offset one another.
Finally, the consequences of financial frictions are of particular qualitative significance in the case of a disturbance to the path of government debt (Figure 9). Here we
consider a disturbance to fiscal policy that temporarily increases the level of government debt, through a lump-sum transfer to households, which is then gradually takenback over a period of time, so that the path of real government debt is eventually thesame as it would have been in the absence of the shock. In the case of the ‘NoFF
model’, Ricardian equivalence holds, as in the representative household model; andso in these cases, the fiscal shock has no effect on output, inflation, or interest rates.
However, an increase in government borrowing crowds out private borrowing, and49

The shock considered here increases the value of cs (λ) by one percent for each possible value ofλ.
50

The effect of financial frictions in this case is somewhat similar to the case of the technology
shock shown in Figure 4. See the Appendix for the corresponding figure.

38

in the case of the convex intermediation technology, the reduced private borrowing
implies a reduction in spreads. This has an expansionary effect on aggregate demand, with the consequence that both output and inflation increase, as shown in the
figure.51
We obtain similar conclusions in the case of other simple specifications of monetary policy. For example, instead of a Taylor rule, we may assume that the central
bank adheres to a strict inflation target, adjusting the policy rate as necessary in orderto ensure that the inflation rate is unaffected by any exogenous disturbances. Onceagain, we find that neither heterogeneity nor financial frictions make much difference
for the equilibrium responses in the case of the linear intermediation technology (wesuppress these figures). In the case of the convex intermediation technology, equilibrium responses under the ‘FF model’ are slightly different, in the case of shocks
that cause substantial endogenous variation in credit spreads. For example, Figure
10 shows the equilibrium responses under this kind of policy in the case of a shock to
the demand of savers (the same shock as in Figure 8), for the same three alternativemodels as in Figures 4-9. Again, the policy rate is increased substantially more in
the ‘FF model’ than in the ‘NoFF model’, because of the reduction in the spread.
And again the increase in output in response to the shock is somewhat greater in the‘FF model’; however, this effect is much less dramatic than in Figure 8. While thereduction in credit spreads has a small effect on the aggregate-supply relation — theoutput level consistent with price stability is slightly increased — the primary effectis on the IS relation (the average real interest rate consistent with a zero output gap
is increased) and on the relation between the policy rate and the average interest ratethat is relevant for aggregate-demand determination (the lower spread means that ahigher policy rate is needed to achieve a given average interest rate). The latter two
effects make an unadjusted Taylor rule a substantially more expansionary policy inresponse to this kind of shock (as shown in Figure 8), but they are automatically canceled under a (correctly implemented) inflation targeting policy, so that equilibrium
51

Ricardian equivalence does not hold precisely in the ‘FF model’ even in the case of the linear
intermediation technology. However, in this case (not shown) there is no reduction in credit spreads
in response to the shock, and the only consequence for aggregate demand comes from the reductionin the resources used by the banking sector, so that shock is actually (very slightly) contractionary
in this case. (See the Appendix for the corresponding figure.) But there is very little difference inthe predictions of the ‘NoFF’ and ‘FF’ models in the case of that technology, so that we omit thefigure here.

39

outcomes are not so different in the economies with and without financial frictions.
The consequences of financial frictions are even more modest in the case of the otherkinds of disturbances previously discussed (figures not shown here).
To sum up, we find that under an empirically realistic calibration of the average
size of credit spreads, the mere existence of a positive credit spread does not imply
any substantial quantitative difference for our model’s predictions, either about theeffects of a monetary policy shock or about the effects of other kinds of exogenous disturbances under a given systematic monetary policy. What matters somewhat moreis the degree to which there is variation in credit spreads. If spreads vary endogenously (as in our model with a convex intermediation technology), then the effects ofdisturbances are somewhat different, especially in the case of types of disturbances— such as variations in government borrowing, or changes in the relative spending
opportunities available to savers as opposed to borrowers — that particularly affectthe evolution of the equilibrium volume of private credit.
Another important difference of the model with credit frictions is the possibility of exogenous disturbances to the banking sector itself, represented by exogenous
variation in either the intermediation technology Ξt (b) or the intermediation markupµbt (b). Again, these disturbances matter to the determination of aggregate output,
inflation and interest rates primarily to the extent that they imply variation in credit
spreads. The equilibrium effects of disturbances of this kind under alternative monetary policies are considered further below (see Figures 15-16).

3

Optimal Monetary Stabilization Policy

We turn now to the implications of credit frictions for optimal monetary policy. We
shall suppose that the objective of policy is to maximize the average ex ante expectedutility of the households. This implies an objective of the form
E0∞
X

βU (Yt , λbt , λst , ∆t ; ξ̃ t )

(3.1)

t=0

where
U (Yt , λbt , λst , ∆t ; ξ̃ t ) ≡ π b ub (cb (λbt ; ξ t ); ξ t ) + π s us (cs (λst ; ξ t ); ξ t )Ã !− 1+ν
µ ¶1+ω
ν
Yt
1
λ̃t
−ν
H̄t
∆t ,
−
1 + ν Λ̃t
At
40

(3.2)

and

i ν
h ¡ ¢ 1+ν
1+ν 1+νΛ̃t ≡ π b λbt ν + π s (λst ) ν.

Again, the derivation is provided in the Appendix. Note that the final term in (3.2)
represents the average disutility of working, averaging both over the entire continuumof types of labor j and over the two types of households, using the model of equilibrium
labor supply discussed in section 1.2.
While one might reasonably consider the optimal use of fiscal policy for stabilization purposes as well, we shall here consider only the optimal conduct of monetary
policy, taking as given the state-contingent evolution of the fiscal variables {τ t , bgt }.
The problem with which we are concerned is thus the choice of state-contingent paths
for the endogenous variables {Yt , λbt , λst , ibt , idt , Zt , Πt , ∆t , bt }, consistent with the structural relations (1.8)–(1.9), (1.14)–(1.17), (1.25)–(1.26), and (1.27)–(1.28), so as tomaximize (3.1). Note that there is one fewer structural relations per period thanendogenous variables, so that there is one dimension of variation of monetary policy
each period, which may be thought of as the central bank’s choice of the policy rateidt .
We can analyze the solution to this optimization problem, by differentiating a Lagrangian to obtain a system of first-order conditions, that take the form of nonlinear(expectational) stochastic difference equations. (Details are given in the Appendix.)
A first important conclusion, demonstrated in the Appendix, is that w hen there are
no stochastic disturbances (i.e., each of the exogenous variables in ξ̃ t takes a constant value), the first-order conditions for optimality admit a stationary (or steadystate) solution, in which the rate of inflation is zero. Hence under an optimal policy
commitment, in the absence of stochastic disturbances the inflation rate will eventually converge to zero;52 and in the case of small enough stochastic disturbances, the52

Under a once-and-for-all commitment chosen to maximize the objective (3.1) at some initialdate, under no constraints other than those required for a perfect foresight equilibrium from thetime of adoption of the policy commitment onward — the “Ramsey” policy problem — a higher
inflation rate will be chosen initially, in order to exploit the short-run aggregate-supply trade-off
without having to take account of any consequences of anticipation of such inflation in the periodprior to adoption of the policy; but the policy will involve a commitment to eventually reducethe inflation rate to zero, since the long-run inflation commitment internalizes the consequences ofanticipation of the inflation rate in prior periods. (See Woodford, 2003, chap. 7, for discussion ofthis in the context of the basic NK model.) Optimal policy “from a timeless perspective,” in thesense defined in Woodford (2003, chap. 7) and Benigno and Woodford (2005), instead involves zero41

optimal policy commitment will involve an inflation rate that fluctuates asymptotically around zero. To the extent that the disturbances are small enough for a linear
approximation to the equilibrium relations to provide an adequate approximation to
the equilibrium dynamics, optimal policy involves a long-run average inflation rate ofzero, even in the stochastic case.
This result generalizes the one obtained by Benigno and Woodford (2005) for
the representative-household case, and implies that the optimal inflation target isindependent of the (average) severity of the distortions resulting from credit frictions,just as it is independent of the severity of the steady-state distortions resulting frommarket power in the goods or labor markets or from distorting taxes. This indicatesa first important respect in which conclusions about optimal policy derived from amodel that abstracts from financial frictions continue to apply when one allows forsuch frictions. The result also justifies our attention here only to the consequences ofalternative policies that imply a long-run average inflation rate near zero (as in our
analysis of the consequences of a Taylor rule in section 2); since we know that the
optimal policy commitment involves inflation fluctuations around zero, we can restrict
our analysis to policies with that property, both when seeking to further characterize
optimal policy, and when investigating the desirability of simple policy rules that are
intended to approximate optimal policy.

3.1

Linear-Quadratic Analysis

Under certain simplifying assumptions, we can obtain an analytical solution for the
optimal state-contingent policy — or more precisely, for a linear approximation to
optimal policy, in the case of small enough fluctuations around the optimal (zeroinflation) long-run steady state — using the method of linear-quadratic approximation
introduced in Rotemberg and Woodford (1997) and further expounded by Benigno
and Woodford (2007).53 Under these assumptions, we can derive a quadratic loss
inflation at all times, in the absence of stochastic disturbances.
53
These simplifying assumptions are not required in order to compute a valid LQ approximation
to our policy problem, as explained in Benigno and Woodford (2007). However, in general, thealgebraic expression of the solution to the LQ problem will be complex, so we discuss it here only
in a special case allowing a solution of a simple form. The same general method is used, however,to approximately characterize optimal policy in the numerical results presented below, that do not
rely upon the simplifying assumptions introduced in this section.

42

function for monetary stabilization policy with the property that minimization of theloss function is equivalent (in the case of small enough disturbances) to maximization of the utility-based objective (3.1). The linear policy rule that minimizes thequadratic loss function subject to the linear constraints obtained by log-linearizing
the model structural relations (as in section 2.1) then provides a linear approximation
to optimal policy.
We begin by using structural relation (1.17) together with the definition Ωt ≡b
λt /λst to solve for λbt and λst as functions of Yt , Ωt , Ξt , and the exogenous disturbances. Substituting these solutions into (3.2), we obtain a period utility function
Ũ (Yt , Ωt , Ξt , ∆t ; ξ̃ t ) to replace the function U defined in (3.2). We then compute a
quadratic (Taylor-series) approximation to the function Ũ around the values of its
arguments in the zero-inflation steady state, (Ȳ , Ω̄, Ξ̄, 1; 0). This takes an especiallysimple form under the following special assumptions: (i) the steady-state tax distortion τ̄ exactly cancels the distortion resulting from market power in the goods
markets (1 − τ = µp ≡ θ/(θ − 1)), so that in the zero-inflation steady state, priceis equal to marginal cost; (ii) there is no steady-state wage markup (µ̄w = 1); (iii)the steady-state credit spread ω̄ = 0 (which, because of relation (2.6), implies thatΩ̄ = 1 as well); and (iv) there are no resources consumed by the intermediary sector
in steady state (Ξ̄ = 0). Note that the assumed absence of steady-state distortionsof any of these kinds does not mean that we cannot consider the effects of these distortions; as with the other exogenous disturbances, we consider the effects of small
departures from the steady-state values. But in the special case considered in thissection, each of these distortions is assumed to be of order O(||ξ||) in the notationof Woodford (2003, chap. 6). Finally, we also assume (v) that the gradient Ξ̄b = 0
in steady state as well, so that any endogenous variation in the resources used by
the banking sector is of at most second order, though we allow for variations in theintercept Ξt (b̄) that are of order O(||ξ||).54
In this special case, the Taylor series approximation to the period utility function
54

This last assumption is not necessary in order for the approximation (3.3) to average utility to bevalid, but it is necessary in order for the terms in (3.3) that depend on policy to be purely quadratic,i.e., for them to include no linear terms. This last condition is necessary in order for the loss function
to be evaluated to second order under alternative policies using only linear approximations to themodel’s structural relations, as discussed in Woodford (2003, chap. 6) and Benigno and Woodford(2007).

43

takes the simple formo
λ̄Ȳ n −1
ˆ t − λ̄Ξt + t.i.p. + O(||ξ||3 ),(σ̄ + ω y )(Ŷt − Ŷtn )2 + ψ Ω Ω̂2t + (2/1 + ω y )∆2
(3.3)
where the term “t.i.p.” collects terms that are independent of monetary policy (because they depend only on exogenous disturbances) and the residual is of at mostthird order in the amplitude of the disturbances. Here λ is the common steady-statemarginal utility of income of all households (under a calibration in which Ω̄ = 0); σ̄
is the “aggregate” intertemporal elasticity of substitution defined in (2.8) above; thecoefficient ψ Ω > 0 is defined in the Appendix; and hats denote percentage deviations
from the steady-state values of the various variables, as in section 2.1. As in equation(2.13), Ŷtn denotes the percentage deviation of the “natural rate of output” — here
equivalent to both the efficient level of output, given preferences and technology, andto the flexible-price equilibrium level of output in the absence of time-varying tax distortions, wage markups, or financial frictions — from the steady-state level of output.
This variable is a function purely of the exogenous disturbances (described further
in the Appendix); and it is exactly the same function of the exogenous disturbances
as in the basic NK model (as expounded in Woodford, 2003, chap. 3), with the
exception that we must in general take a weighted average of the preference shocksof the two types of households, rather than assuming that their preferences always
shift in exactly the same way, and use σ̄ in place of the elasticity of intertemporal
substitution of the representative household.
Taking a discounted sum of these terms, and using a second-order Taylor series
ˆ t terms in terms of inflation, we obtainapproximation to (1.25) to substitute for the ∆a quadratic objectiveŨt = −

∞
X
t=0

t

β Ũt = −K∞X

β t Lt + t.i.p. + O(||ξ||3 ),

(3.4)

t=0

where K > 0 and the period loss function is of the form
Lt = π 2t + λy (Ŷt − Ŷtn )2 + λΩ Ω̂2t + λΞ Ξbt b̂t ,(3.5)

for certain weights λy , λΩ , λΞ > 0 defined in the Appendix. Maximization of average
expected utility is thus equivalent (to the order of approximation required here) tominimization of the expected discounted value of the loss function (3.5). And because44the terms in (3.5) are purely quadratic, the loss function can be evaluated to second
order using only a log-linear approximation to the equilibrium dynamics of the endogenous variables under a given policy. Hence it is possible to use the log-linearized
structural relations (derived in section 2.1) as the constraints in our (approximate)
optimal policy problem. We thus obtain a linear-quadratic (LQ) problem, the solution
to which provides a log-linear (approximate) characterization of optimal policy.
It is noteworthy that in (3.5), both the form of the output-gap stabilization objective (i.e., the definition of Ŷtn ) and the relative weight λy placed on this objective
are identical (as functions of the model parameters) to those in the corresponding
derivation for the basic NK model, modulo the need to “average” the preferences of
the two types of households, as discussed above. In particular, not only is the output
gap appearing here the same one that appears in the aggregate-supply relation (2.13)— when that relation is written in such a way that the trade-off between inflation
variations and output-gap variations, given inflation expectations, shifts only in response to variations in distorting taxes, wage markups, or financial frictions — just as
in the basic NK model; but in addition the weight is given by λy = κ/θ, where κ is theslope of the inflation/output-gap tradeoff in (2.13), again as in the basic NK model.
Thus in the special case that there are no financial frictions (i.e., ω t = 0, Ξt = 0 at
all times), so that the last two terms in (3.5) vanish, along with the non-standard
terms in (2.13), both the welfare-based loss function and the aggregate-supply relation (that defines the available trade-off between dynamic paths for the two variablesin the loss function) are of exactly the same form as in the basic NK model, once one
defines “average” variables appropriately. The existence of preference heterogeneityof the kind assumed in the present model — heterogeneity both with respect to theinterest-sensitivity of different units’ expenditure decisions and with respect to thetime variation in their opportunities for productive expenditure — does not in itself
require any substantial modification of the theory of optimal monetary stabilization
policy.
If we instead allow for time-varying financial frictions (though no steady-state
distortions, as explained above), but assume that there is no endogenous variationin these frictions — i.e., that {ω t , Ξt } are exogenous processes, independent of theevolution of private indebtedness55 — we continue to obtain a very simple charac55

For example, we may assume that (i) no real resources are used in intermediation, so thatΞt (b) = 0, and the credit spread ω t is a pure financial markup; and (ii) the markup µbt (and hence45terization of optimal policy. In this case, the final term in (3.5) is zero, and thepenultimate term is independent of policy (to second order), since the log-linear approximate structural relation (2.6) implies that if {ω t } is an exogenous process, {Ωt }is also an exogenous process, at least up to a residual that is at most of second order,
so that Ω̂2t is exogenous, at least up to a residual of order O(||ξ||3 ). Hence the loss
function can be written (ignoring terms independent of policy) in the standard New
Keynesian form, which is to say, as
E0∞X

β t [π 2t + λy x2t ],

(3.6)

t=0

where xt ≡ Ŷt − Ŷtn is the welfare-relevant output gap. In this case, the appropriate
objectives of stabilization policy remain as in the basic NK model; the only differences
made by financial frictions relate to the feasible paths for inflation and the output
gap, and to the path for the policy rate required to implement given paths for inflationand the output gap.
Time-variation in financial frictions does matter for the optimal conduct of monetary policy, because they shift both the IS relation (2.10) and the aggregate-supply
relation (2.13)). However, only the latter relation represents a constraint upon theset of achievable outcomes for the target variables, inflation and the output gap. And
the effect of financial frictions on this relation is purely to add an additional additive
disturbance term. The first-order conditions that characterize the solution to the LQ
problem of minimizing (3.6) subject to the constraint that (2.13) hold each period
are unaffected by these disturbances, and so is the optimal target criterion that expresses the linear relation that must exist between the evolution of inflation and of
the output gap in order for the first-order conditions for optimality to be satisfied.
Just as in the basic NK model, the optimal target criterion is of the form56
π t + (λy /κ)(xt − xt−1 ) = 0.

(3.7)

While the state-contingent paths for inflation and the output gap that satisfy thiscriterion are different in the case of a time-varying credit spread, the criterion thatthe spread ω t ) evolves exogenously, and is unaffected by the volume of bank lending.
56
See Woodford (2003, chap. 7) for derivation of this characterization of optimal policy in thecase of the basic NK model, and for further discussion of the implementation of optimal policy using
a target criterion.

46

the central bank should use at each point in time to determine if policy remains on
track is unchanged. Not only can optimal policy still be characterized as “flexible
inflation targeting,” but the optimal target criterion is of an identical form to what
is optimal in the absence of credit frictions.
The implementation of an optimal policy does require the central bank to monitor
the varying size of the credit frictions — more precisely, the varying size of credit
spreads — in order to determine how it must act in order to ensure fulfillment of
the target criterion (3.7). These matter for two reasons. First, the paths of inflationand the output gap consistent with (3.7) depend in general on the evolution of credit
spreads, because of the effect of those spreads on the aggregate-supply trade-off (2.13).
And second, the path of policy rates required to implement given (feasible) paths for
inflation and output depends on the path of credit spreads, because of their effects
on both the relation (2.10) between average interest rates and expenditure and therelation (2.11) between the policy rate and the relevant “average” interest rate.
The required adjustment of the policy rate follows from standard treatments (e.g.,
Clarida et al., 1999; Woodford, 2003, chap. 7) of the optimal interest-rate response
to “cost-push shocks” and variations in the natural rate of interest in the context ofthe basic NK model. Variations in credit spreads result in a total cost-push term in
the aggregate-supply relation equal to57
uFt F = ut + ξsΩ Ω̂t ,where ut ≡ ξ(µ̂w
t + τ̂ t ) is the cost-push term in the absence of financial frictions (buttaking account of preference heterogeneity), and the coefficient sΩ is defined as in(2.9); and they similarly result in a natural rate of interest (understood to mean thereal value of the relevant average rate of interest that would be required to maintaina zero output gap at all times) equal tortn,F F = rtn − sΩ (Ω̂t − Et Ω̂t+1 ),where rtn is similarly the natural rate of interest in the absence of financial frictions(but taking account of preference heterogeneity). The standard theory of how “theinterest rate” should be adjusted in response to exogenous variations in the processes{ut , rtn } continues to apply under this extension of the basic NK model, but it now
57

The cost-push effect of variations in Ω̂t here has a simpler expression than in (2.13), because of
the assumption in this section that Ω̄ = 1, which implies that γ b = π b .

47should be understood to determine the optimal operating target for the average indterest rate iavg
t . The required adjustment of the central bank’s policy rate it is thengiven by (2.11), and this relation is also shifted when credit spreads change.
The precise consequences of these several factors for the proper adjustment of thepolicy rate can be given a simple answer if we make still more special assumptions. In
the “symmetric” case in which sb σ b = ss σ s , uΩ = 0, and as a result variation in credit
spreads has neither a cost-push effect nor any effect on the natural rate of interest.
In this case, the optimal state-contingent evolution of inflation and the output gap
depend only on the evolution of the exogenous disturbances {µ̂wt , τ̂ t }; the implied
optimal evolution of aggregate output depends also on the evolution of preferences
and technology, but is similarly independent of the time variation in credit spreads.
in order to support these
The required evolution of the average interest rate iavgtpaths is also independent of the evolution of credit spreads. Time-variation in credit
spreads matters for the optimal conduct of monetary policy only because the policy
rate associated with a given average interest rate changes, as indicated by (2.11). We
obtain a simple conclusion: an increase in the credit spread ω t reduces the optimal
policy rate idt , by an amount precisely equal to fraction π b of the increase in the credit
spread.
In this simple case, at least, we confirm the appropriateness of the kind of ruleof thumb suggested by McCulley and Toloui (2008) and Taylor (2008): an increasein credit spreads justifies a reduction in the policy rate (i.e., the federal funds rate
operating target, in the case of the U.S. Federal Reserve), below what would otherwise
be justified by current aggregate conditions, by an amount that is proportional to theincrease in credit spreads. However, the appropriate adjustment is not by the fullamount of the increase in credit spreads, as these authors suggest. That would beappropriate only if it were optimal for the rate faced by borrowers to be unaffectedby the increased credit frictions, regardless of the consequences for the interest rate
obtained by savers. In fact, a change in the interest rate obtained by savers creates
distortions, just as a change in the interest rate faced by borrowers does; and when
it is necessary to increase the spread between these two interest rates, necessarily
increasing distortions of one kind or the other, it is optimal to share the adjustment
burden by increasing each kind of distortion to some extent. If there are equal
numbers of savers and borrowers, the optimal rule in the present special case would
be to “split the difference,” and reduce the policy rate by half the increase in the48

spread.
Matters are more complicated, of course, if we assume that either the resources
used in financial intermediation, the size of credit spreads, or both depend on thevolume of bank lending. Rather than seek analytical results in the more general case,
we present some illustrative numerical calculations of optimal policy in calibrated
examples, and investigate the degree to which the insights suggested by the analysisof this simple case continue to apply.

3.2

Numerical Analysis

In order to check the degree to which the results obtained above for a special casecontinue to hold, at least approximately, under more general assumptions, we numerically analyze optimal policy in the calibrated economies for which we have alreadyanalyzed the consequences of two simple policy rule in section 2. Under assumptions
more general than those used in the previous section, we can characterize the optimal
responses to exogenous disturbances of various sorts by deriving the first-order conditions that characterize optimal (Ramsey) policy — the problem of maximizing (3.1)
subject to constraints (1.8)–(1.9), (1.14)–(1.17), (1.25)–(1.26), and (1.27)–(1.28) —and then log-linearizing them around the zero-inflation optimal steady state. The
resulting log-linear equations can be solved for log-linear optimal dynamic responsesof the various endogenous variables to (small enough) random shocks to each of theexogenous disturbance processes.
A useful question about these optimal responses is the degree to which they canbe achieved through one or another rule for the conduct of monetary policy. Given
our results for the special case treated in the previous section, one obvious candidate
for a rule is flexible inflation targeting, here understood to mean a commitment toadjust the policy rate as necessary to ensure that the target criterion (3.7) holds at
all times.58 Other simple proposals that we shall consider include the two policies
58

More precisely, the target criterion that we assume is the one that characterizes optimal policyin the representative-household model. The target criterion discussed in the previous section corresponded to optimal policy in a representative household model in which the steady-state level of
output is efficient, which required that 1 − τ = µp ≡ θ/(θ − 1). More generally, the optimal target
criterion for the representative-household model continues to be of the form (3.7), but the definitionof the output gap xt and the relative weight λy are slightly different, as explained in Benigno andWoodford (2005).

49

considered for illustrative purposes in section 2, namely, the Taylor rule (2.14) withthe coefficients assumed in section 2.2, and strict inflation targeting.
The equilibrium responses of the endogenous variables to a technology shock (ofexactly the same kind previously considered in Figures 2 and 5) under each of thesealternative monetary policies are shown in Figure 11. (The model parameterization
is again the one with a convex intermediation technology, as in Figures 4-10.) The
solid line (labeled ‘Optimal’) indicates the equilibrium responses to the increase inproductivity under the optimal policy. The dashed line (labeled ‘PiStab’) instead
indicates responses under a policy that fully stabilizes inflation (i.e., strict inflation
targeting); the line marked with + signs (labeled ‘Taylor’) indicates responses underpolicy conducted in accordance with the Taylor rule; and the line marked with x’s(labeled ‘FlexTarget’) indicates responses under the flexible inflation targeting policy.
In the case of this kind of shock, the aggregate-supply relation (2.13) implies that
there is little inconsistency between inflation stabilization and output-gap stabilization; and as a consequence strict inflation targeting and flexible inflation targetingare barely distinguishable policies.59 They are also both very close to the optimal
policy, confirming that in this respect the analytical results of the previous section
continue to provide a good approximation. The Taylor rule is instead much less close
to an optimal policy, because it requires policy to be tightened in response to the
output increase, even when this does not represent high output relative to the natural rate (which is increased by the technology shock). Similar results are obtained in
the case of exogenous shocks to the disutility of labor, the demand of borrowers, orto the path of government debt (none of which are shown here): in each case, there
is sufficiently little tension between the requirements of inflation stabilization and of
output-gap stabilization that flexible inflation targeting and strict inflation targetingare similar policies, and both are fairly close to optimal policy (while the Taylor ruleis much less close).
Figure 12 shows the corresponding equilibrium responses in the case of a shock to
the wage markup (of the kind previously assumed in Figure 6), under the same four
alternative policies. This case is interesting because it is one in which strict inflation
targeting is clearly not an optimal policy, even in the representative-household model,
owing to the substantial “cost-push” effect of such a shock. (However, even in this59

In fact, if we did not allow for endogeneity of the credit spread, as in the model with a linear
intermediation technology, they would not differ at all.

50

case, strict inflation targeting is still more similar to optimal policy than is adherence
to the Taylor rule.) The figure shows that in the case of this kind of shock, as with theother “supply shocks” discussed above, flexible inflation targeting is nonetheless quite
a close approximation to optimal policy. A similar result is obtained in the case of ashock to the sales tax rate (not shown): strict and flexible inflation targeting are nolonger nearly identical policies, but flexible inflation targeting closely approximates
optimal policy.
Figure 13 shows the corresponding equilibrium responses to a shock to the level ofgovernment purchases (of the kind previously assumed in Figures 3 and 7); and Figure
14 shows the responses to a shock to the demand of savers (of the kind previously
assumed in Figures 8 and 10). In the case of a shock to government purchases, strict
inflation targeting is again not optimal,60 and in at least some respects the Taylor rule
is in this case closer to the optimal policy than would be strict inflation targeting.
Nonetheless, once again flexible inflation targeting is the closest to optimal policyof the set of simple policy rules considered here, though optimal policy would beslightly tighter in the period of the shock. In the case of a shock to the demand ofsavers as well, flexible inflation targeting is the closest to optimal policy, though again
optimal policy would be slightly tighter in the period of the shock. Flexible inflation
targeting is nonetheless a slightly tighter policy in response to the expansionary shock
than strict inflation targeting would be, and in particular it prevents the persistentincrease in inflation that would be allowed by the Taylor rule.
Finally, Figure 15 shows the corresponding equilibrium responses in the case of ashock to the financial markup µbt , of a size sufficient to increase the credit spread ω t
by one percentage point (annualized). This kind of shock (not discussed in section 2.3
because there is no analogous shock in the models without financial frictions) would
be quite contractionary if monetary policy were conducted in accordance with an
unadjusted Taylor rule.61 But as in the case of a productivity shock (shown in Figure11) or a shock to the disutility of labor supply, either kind of inflation targeting ismuch closer to optimal policy. Yet even in this case, flexible inflation targeting is notquite an equivalent policy to strict inflation targeting, and is closer to the optimal
60

This would be true even in the representative-household model, given the existence of steadystate distortions, as discussed by Benigno and Woodford (2005).
61
This effect could be substantially mitigated in the case of a spread-adjusted Taylor rule, as
discussed in section 3.3 below.

51

policy.
Overall, our conclusion is that while the additional distortions that were omitted in
the special case for which we were able to give an analytical characterization of optimal
policy in section 3.1 are of at least some significance in our calibrated model, theflexible inflation targeting rule (3.7) continues to provide a fairly good approximation
to optimal policy, in response to each of the variety of types of exogenous disturbances
discussed above. Since this is also the rule that characterizes optimal policy in thebasic NK model, in at least this sense the basic NK model remains quite a good guideto policy in the kind of environment that we consider. Of course, it is important toremember that the policy prescription provided by (3.7) is not a complete descriptionof the way in which the policy rate should be adjusted: it is still necessary to use amodel of the transmission mechanism to determine what adjustments of the policy
rate are needed to ensure that the target criterion is satisfied. And the model used
for this latter purpose needs to be one that takes account of the credit frictions — inparticular, that takes account of the variations over time in the size of credit spreads.
But the target criterion around which monetary policy deliberations are structured
may not need to be changed in response to the existence of credit frictions.

3.3

A Spread-Adjusted Taylor Rule

The numerical results of the previous section show that a simple Taylor rule of theform (2.14) does not provide too close an approximation of optimal policy in the caseof a number of different types of possible disturbances. But might an adjustmentof the rule to take account of credit frictions cure some of the problems with theunadjusted Taylor rule? In particular, does the proposal of authors such as McCulley
and Toloui (2008) or Taylor (2008), to adjust the federal funds rate operating target
in response to changes in credit spreads, for any given level of inflation and output,
represent an improvement upon the classic proposal of Taylor (1993)?
Let us consider a generalization of (2.14) of the formı̂dt = φπ π t + φy Ŷt − φω ω̂ t ,(3.8)

for some coefficient 0 ≤ φω ≤ 1. (We now omit the random term ²mt , as there isclearly nothing desirable about unnecessary randomization of policy.) This kind ofrule lowers the intercept of the Taylor rule by an amount equal to fraction φω of the52credit spread. In the case that φω = 1, the rule is equivalent to having a conventional
Taylor rule, but for the lending rate rather than the deposit rate (or interbank fundingrate). The case φω = 1 formalizes the proposal of McCulley and Toloui (2008), andalso captures the idea of proposals that the funds rate should be lowered when credit
spreads increase, so as to prevent the increase in spreads from “effectively tightening
monetary conditions” in the absence of any justification from inflation or high output
relative to potential.
Figure 16 shows the responses of endogenous variables to a “financial shock” ofthe same kind as in Figure 15, but for variant monetary policy rules of the form
(3.8). The response labeled ‘Taylor’ (the dashed line) again corresponds to the standard Taylor rule (the case φω = 0), and is the same as in Figure 14. The responselabeled ‘Taylor+25’ corresponds to the case φω = 0.25, and so on up to ‘Taylor+100,’
corresponding to the case φω = 1. The figure also again plots the responses under
optimal policy, shown by the solid line.
We observe that adjusting the intercept of the Taylor rule in response to changes
in the credit spread can indeed largely remedy the defects of the simple Taylor rule, inthe case of a shock to the economy of this kind. And the optimal degree of adjustment
is close to 100 percent, as proposed by McCulley and Toloui and by Taylor.62 Figure
17 shows that the same is true in the case of a shock to the level of government
debt (of the kind considered in Figure 9). In the case of the “financial shock”, theunadjusted Taylor rule is too contractionary, while in the case of the fiscal shock it is
too expansionary; but in both cases, the spread adjustment corrects the Taylor rulein the right direction and even (for the parameter values assumed here) by almost
exactly the right amount.
This is not true, however, more generally. Figure 18 shows the corresponding
equilibrium responses to a shock to the demand of savers (of the kind considered in
Figures 8, 10, and 14). In the case of this kind of shock, the unadjusted Taylor rule is
too expansionary, and (since the shock decreases credit spreads) a spread adjustment
modifies policy in the right direction, just as in Figure 17. But in this case, a 100
percent spread adjustment would be too contractionary; instead, a 50 percent spreadadjustment most closely approximates optimal policy. Similar conclusions apply inthe case of a shock to government purchases, shown in Figure 19. Again the spread
62

This last conclusion is somewhat sensitive to the parameterization of our model, even in thecase of this particular type of disturbance. See the Appendix for further discussion.

53

adjustment is in the right direction, but a 100 percent adjustment would be far too
much; in this case, the 25 percent spread adjustment leads to responses most similar
to optimal policy.
In other cases, even a 100 percent spread adjustment may not be enough toeliminate the defects of the simple Taylor rule. Figure 20 shows the corresponding
equilibrium responses to a shock to the demand of borrowers. (As with the shockto the demand of savers, we here assume a preference shock that increases cb (λ) by
one percent for each possible value of λ.) In this case, a spread adjustment modifies
policy in the right direction (i.e., it loosens policy), but even a 100 percent spreadadjustment is not nearly enough to produce responses similar to those under optimal
policy.63 The unadjusted Taylor rule tightens policy too much in response to thiskind of disturbance, but this is not primarily a consequence of the increase in credit
spreads resulting from the shock; indeed, the Taylor rule would tighten too much inresponse to this shock even in an economy in which the credit spread is exogenous(as in the case of the linear intermediation technology discussed above). Indeed, theincrease in credit spreads is fairly modest in this case, under our parameterization of
the model, so even a full offset for the increase in the credit spread does not correct
policy by nearly enough.
This is true to an even greater extent in the case of a technology shock, as shown in
Figure 21. Though the spread adjustment again modifies the Taylor rule in the right
direction (making it less excessively contractionary in response to the surge in output
that is justified by increased productivity), it has relatively little effect on equilibriumoutcomes, given the modest increase in spreads. A similar comment applies in thecase of shocks to the disutility of working, to the tax rate, or to the wage markup(not shown).
It is interesting to observe that in all of these cases, the spread adjustment modifies
the simple Taylor rule in the right direction. There is thus some generality to theconclusion that the Taylor rule is too tight in the case of shocks that increase credit
spreads (in most of the above cases, because credit increases) and too loose in thecase of shocks that reduce credit spreads. However, this does not mean that thechange in credit spreads, even if accurately observed by the central bank, would be
63

We do not attempt to calculate the size of spread adjustment that would best approximate
optimal policy in this case; it would clearly be a spread adjustment that would be excessive in thecase of the other types of shocks already discussed, and so not an interesting policy proposal.

54

an ideal indicator of the degree to which a simple Taylor rule is likely to be biased asa guide to welfare-maximizing policy. For the adjustment of the Taylor-rule intercept
in response to the shock that would best approximate optimal policy is not the same
multiple of the change in the credit spread in all cases — indeed, the optimal factor of
proportionality varies widely in the cases considered above.64 Thus simply tracking
credit spreads would not provide enough information to make the correct adjustment:a central bank would also need to determine which kind of disturbance is responsible
for an observed change in credit spreads.
Can one usefully summarize the way in which such information about the particular disturbances currently affecting the economy should be used to determine theappropriate adjustment of the policy rate? Here it is worth noting that the flexible
inflation targeting rule appears superior as a general rule of thumb to any memberof the family of spread-adjusted Taylor rules. The targeting rule implies naturally
implies a certain degree of adjustment for changes in spreads. This is because thetargeting rule requires the central bank to determine the path for its policy rate consistent with projected paths of inflation and output with a certain desirable property;
this requires it to take account of (and to offset, to the extent that it can with a single
instrument) the effects of changes in credit spreads on the structural relations discussed in section 2. As a simple example, an increase in the credit spread increases
the gap between iavg
(the interest rate that is relevant to aggregate demand determit
nation) and the policy rate idt , and given any desired degree of adjustment of iavg
intresponse to an economic disturbance, the policy rate will instead be raised less (orlowered more) in the case of an increase in credit spreads, in order to achieve thedesired effect on iavg
t . This effect alone would justify a credit spread adjustment of
50 percent (in our calibrated model), as a result of equation (2.11). However, thisis not the only adjustment that a forecast-targeting central bank should take into
account; credit spreads also shift the “IS” and aggregate-supply relations as well, as
discussed in section 2. Nor are adjustments for those affects the only ones that differentiate a properly implemented targeting regime from a simple Taylor rule; forecast
64

And of course, no proportional adjustment would fully implement optimal policy in any of thesecases. Ideally, the adjustment of the Taylor-rule intercept would have dynamics somewhat different
from the dynamics of the equilibrium response of the credit spread. For example, one observes in
Figures 18 and 19 that the optimal adjustment factor would be larger in the period of the shock
than in subsequent periods, in each of these cases.

55

targeting also requires that the central bank take account of shifts in the natural rateof output, shifts in the natural rate of interest, and shifts in the aggregate-supply
relation due to “cost-push” disturbances, as discussed in Woodford (2003, chaps. 4,7). Sometimes the adjustments of the policy rate required by considerations of thelatter types will happen to be in the same direction as would be implied by a “spreadadjustment”, but this is fortuitous, and the targeting rule provides a better way of
thinking about the nature and magnitude of the appropriate adjustment of policy.65
While implementation of such a rule requires a greater degree of sophistication (anda greater degree of accurate real-time assessment of current conditions) on the partof the central bank, a rule of this kind provides a better guide to what the central
bank should be seeking to do at each point in time.

3.4

Should Policy Respond to Variations in Aggregate Credit?

Some have suggested that because of imperfections in financial intermediation, it ismore important for central banks to monitor and respond to variations in the volume
of bank lending than would be the case if the “frictionless” financial markets of ArrowDebreu theory were more nearly descriptive of reality. A common recommendation inthis vein is that monetary policy should be used to help to stabilize aggregate private
credit, by tightening policy when credit is observed to grow unusually strongly andloosening policy when credit is observed to contract. For example, Christiano etal. (2007a) propose that a Taylor rule that is adjusted in response to variations inaggregate credit may represent an improvement upon an unadjusted Taylor rule.
In order to consider the possible advantages of such an adjustment, we now propose to replace (2.14) by a rule of the formı̂dt = φπ π t + φy Ŷt + φb b̂t ,(3.9)

for some coefficient φb , the sign of which we shall not prejudge. (Christiano et al.,like most proponents of credit-based policies, argue for the desirability of a positivecoefficient.) Figures 22-24 present the equilibrium responses to three different kinds
65

For example, in the case of the technology shock in Figure 21, the main reasons why the simple
Taylor rule calls for too tight a policy are that it ignores the increase in the natural rate of output
and the reduction in the natural rate of interest; the fact that in the simulations shown in the figure
there is also a modest increase in the credit spread is coincidental.

56

of exogenous disturbances in the case of five different rules of the form (3.9), corresponding to five different possible values of φb . The responses labeled “Taylor” areagain those implied by the unadjusted Taylor rule (the case φb = 0); “Taylor+025”
and “Taylor+050” refer to rules in which φb is equal to .025 or .050 respectively; and“Taylor-025” and “Taylor-050” are rules in which φb is equal to -.025 or -.050 respectively. For purposes of comparison, the responses associated with optimal policy areagain plotted as a solid line in each figure.
In the case of a financial markup shock (Figure 22), the unadjusted Taylor rulemakes policy too tight, as previously shown in Figure 15: both output and inflation
fall significantly in response to the shock, when they should not. Moreover, private
credit contracts in response to this shock, and so one might suppose that a Taylor
rule that responds to credit with a coefficient φb > 0 would represent an improvement. However, Figure 22 shows that in our model, the dynamics of private credit
in response to the shock do not match very well the dynamics of the adjustment of
the Taylor rule needed to implement optimal policy. A value of φb between .025 and.050 will suffice to prevent the undesired initial decline in inflation at the time of theshock, but such a modification of the Taylor rule will then be too inflationary in subsequent quarters. Moreover, such an adjustment does little to prevent the contractionof output in response to the shock; any value of φb large enough to solve that problem
to any substantial extent will be much too inflationary, and will eventually stimulate
output too much in later quarters as well. The problem is that the contraction in
private credit in response to the shock is too persistent for a simple contemporaneous
response to the level of private credit to represent an improvement upon the unadjusted Taylor rule. (Because of the dynamics of private debt implied by (1.26), theeffects upon aggregate credit are substantially more persistent than the shock to thecredit spread may be.)The idea of adjusting the Taylor rule to incorporate a positive response to credit
makes even less sense in the case of other kinds of disturbances. For example, Figure
23 shows the equilibrium responses to a shock to government purchases under thesame set of alternative policies as in Figure 22. In the case of this kind of shock, theunadjusted Taylor rule makes policy slightly too loose, as previously shown in Figure
13. But credit contracts in response to this shock, so a rule of the form (3.9) withφb > 0 would be even worse. One can in fact better approximate optimal policy inresponse to this shock if φb is slightly negative. However, the optimal value of φb is57

quite small, much less negative than -.025 (the smallest negative coefficient for which
we plot the responses), so there is little gain from this kind of adjustment to thesimple Taylor rule.
Similar remarks apply to the case of an increase in the demand of savers (notshown). Responding to credit is even less useful in the case of an increase in the demand of borrowers (also not shown). In the case of this kind of shock, the unadjusted
Taylor rule makes policy too tight (as shown in Figure 20), but credit increases, so
again a negative value of φb would be needed to adjust policy in the right direction.
Moreover, in this case there is a problem similar to the one demonstrated in Figure
22: a negative value of φb that is large enough to prevent the undesired initial decline
in inflation at the time of the shock will then be too stimulative in later periods,
and will at the same time not stimulate output enough in the first year following
the shock (unless it is so large as to be quite inflationary even initially). Again the
problem is that credit responds in too delayed and too persistent a way to be an
ideal indicator of the required adjustment, even when the sign is reversed from theconventional advice.
In other cases, responding to aggregate credit does little to improve the defectsof the simple Taylor rule. Figure 24 shows the equilibrium responses to a technology
shock under the same set of alternative policies. In this case, as in the one just discussed, the unadjusted Taylor rule prescribes policy that is too tight, while private
credit grows strongly in response to the shock. Thus a negative value of φb is needed
to improve upon the unadjusted rule. However, the modest negative values of φb considered in the figure are insufficient to prevent policy from continuing to be much tootight. (Similar results are obtained in the case of other types of “supply” disturbances— a shock to the disutility of working, a shock to the tax rate, or a wage markup
shock — though these figures are not shown here.) We could consider much larger
negative values of φb , but such a policy rule would be highly destabilizing in the caseof any of the “demand” disturbances considered in the previous two paragraphs. Andeven in the case of the technology shock, no value of φb would approximate optimal
policy well, because the dynamics of the required adjustment of policy would notmatch the dynamics of credit, as in Figure 22.
Thus there seems little advantage in modifying the Taylor to include a response
to aggregate credit of the kind proposed in (3.9). To the extent that there is any
advantage to responding to credit, improvement of the equilibrium responses to “non58

financial” disturbances would require a negative value of φb , rather than the positive
value that would correspond to seeking to stabilize aggregate credit.66 Even such
advantages as may be associated with a negative φb (as in the case illustrated in
Figure 23) can be as well achieved by a spread-adjusted Taylor rule (recall Figure19), and the spread-adjusted Taylor rule has the advantage of also prescribing theright direction of adjustment in the case of a “financial shock.” But the flexible
inflation targeting rule better approximates optimal policy than any rule in either ofthese simple families of modified Taylor rules, since in the case of rules of the latter
sorts, the optimal adjustment coefficient is quite different depending on the type of
disturbance considered.

4

Provisional Conclusions

The present analysis represents merely a first attempt at exploration of what is obviously a very rich terrain. Among other obvious limitations of the present analysis, wehave allowed for only one, very simple form of possible endogeneity of credit spreads.67
Further analysis will be necessary before we can judge how robust the conclusionsof the present analysis are to variations in this aspect of our model of the monetarytransmission mechanism. Nonetheless, it may be useful to summarize some of theprovisional conclusions suggested by our results.
66

The fact that in each of these cases a somewhat negative value of φb would be better than any
positive value is not accidental. It is an implication of our observation, in the previous section,that it is quite generally desirable to reduce the Taylor rule intercept in response to an increase incredit spreads. Since in the case of the convex intermediation technology (assumed again in thissection), credit spreads increase if and only if credit increases (in the case of any of the non-financial
disturbances), an adjustment equivalent to a “spread-adjusted Taylor rule” requires that φb < 0.
However, the required sign of the response to credit is reversed in the case of a “financial shock,”since in this case a shock that increases credit spreads reduces credit.
67
For example, we abstract from both from the possible dependence of loan rates on borrowers’
net worth stressed in the “financial accelerator” model of Bernanke et al. (1999) and from the
stickiness of loan rates stressed by Hulsewig et al. (2007), Teranishi (2008), Sudo and Teranishi(2008), and Gerali et al. (2008).

59

4.1

Implications for Models for Policy Analysis

With regard to the most general question raised in our introduction, our results suggest that the basic view of the way in which monetary policy influences aggregate
expenditure and inflation presented in New Keynesian models need not be modified
in any fundamental way as a consequence of the observation that substantial spreads
exist on average between different interest rates in the economy, or that these spreads
are not always constant over time. We have exhibited a simple extension of the basic
NK model in which a time-varying positive spread exists between the interest rateavailable to savers and the interest rate at which borrowers can borrow. Yet in atleast the simplest version of this model, the monetary transmission mechanism isvirtually identical to that of the basic NK model: monetary policy can be viewed
purely as central-bank control of a short-term nominal interest rate (the deposit rateidt ), and the ways in which aggregate expenditure and inflation are determined by the
expected central-bank reaction function for the policy rate (e.g., a Taylor rule specification) are nearly identical to those in the basic NK model. Hence the fundamental
lessons implied by that framework for monetary policy analysis continue to apply (forexample, the conclusion that central banks influence the aggregate economy primarily
by affecting the expected future path of short-run real rates of return, rather than
through their control of the current short rate as such, the conclusion that optimal
policy is history-dependent, and so on).
This does not mean that financial frictions are irrelevant to output and inflation
determination. Even in the simplest case of the present model, time-variation incredit spreads affects both the “IS relation” between the expected path of the realpolicy rate and aggregate expenditure and the “AS relation” between aggregate output and inflation. However, in the simplest version of the model, these spreads simply
contribute additional additive terms to these relations, corresponding to a new formof exogenous disturbance, in addition to the real sources of variation in the naturalrate of interest and natural rate of output, and the “cost-push” disturbances allowed
for in the basic NK model. The occurrence of such additional disturbances matters:
successful monetary policy requires appropriate adjustment of the policy rate in response to disturbances, and this requires monitoring them in real time and correctly
identifying their character as they occur. But the effects of “financial shocks” in thesimplest version of our model with credit frictions are not fundamentally different

60than the effects of a certain linear combination of types of shocks that are alreadyconsidered in the standard NK literature, and the appropriate response to them (fromthe point of view of stabilization of inflation and/or real activity) is the same as would
be appropriate in the case of that combination of familiar shocks.
Among other things, allowing for credit spreads does not in any way require one toreconsider the often-noted de-emphasis of measures of the money supply and sourcesof money demand in standard NK models.68 We have shown that it is possible
to extend the basic NK model to incorporate time-varying credit frictions, without
making any reference to money or introducing a transactions role for money. Thiscan be interpreted as a “cashless” model in which there actually are no transactionsbalances and no government liabilities that are held other than for their pecuniary
returns. But as usual, the model can also be interpreted as one in which money does
supply liquidity services and earns a correspondingly lower return than government
debt, and in which the central bank implements its interest-rate target by adjusting
the supply of money. It remains the case, under this latter interpretation, that whatmatters about monetary policy is the intended reaction function for interest rates, andnot the adjustments of the supply of money that are required in order to achieve theinterest-rate targets, and that, under a plausible calibration of the size of monetary
frictions, the existence of such frictions is of negligible quantitative significance forthe predicted evolution of aggregate activity and inflation.
In particular, there is no intrinsic connection between monetary statistics and thecredit frictions in our model, so that a desire to track time variation in the credit
frictions does not provide any new justification for particular attention to monetary
aggregates. Of course, if we were to assume a mechanical proportionality between
some monetary aggregate and the volume of intermediation through the banking sector — as, for example, in the model of Goodfriend and McCallum (2007), where
deposits are always in strict proportion to base money, owing to an always-binding
reserve requirement, the absence of uses for base money other than satisfaction of
68

Critics of the omission of a fundamental role for monetary aggregates in NK models (e.g.,
Issing, 2006; Goodhart, 2007) often point to the absence of multiple interest rates and of any
role for financial intermediaries as a ground for suspecting the empirical relevance of such models.
Goodhart, in particular, proposes that the absence of financial frictions in such models may accountfor the absence of an essential role for money: “by basing their model on [frictionless financial
markets], the Neo-Keynesians are turning their model into an essentially non-monetary model. So
it is no surprise that monetary variables are inessential in it” (p. 11).

61

the reserve requirement, and the absence of sources of funding for banks that are notsubject to the reserve requirement — then that measure of money could be usefulfor understanding aggregate conditions, to the extent that aggregate lending is an
important determinant of aggregate demand. Even in the world modeled by Goodfriend and McCallum, however, it is unclear why anyone should prefer to monitor
base money rather than bank credit (or aggregate bank liabilities) directly, unless
more direct measures are for some reason unavailable. And in a more realistic modelof banking under current conditions, the connection between the volume of financial intermediation and monetary aggregates is more tenuous. Indeed, there is noconceptual necessity of any connection between money and financial intermediation
at all, as our model shows. Speaking of the importance of monitoring “money andcredit” (in the same breath) because of the importance of financial intermediation to
the economy appears to provide support for the traditional monetarist emphasis on
measures of the money supply, but in fact there is no reason to equate credit withmoney. A proposal (as in the “monetary analysis” of the ECB) to track changes inbank credit to the extent that it belongs among the “components and counterparts”
of a broad measure of the money supply might fortuitously result in attention tosome useful information, but not due to a sound understanding of what one shouldbe trying to measure.
There is a somewhat stronger case for attention to credit aggregates than monetary
aggregates in the conduct of monetary policy. Nonetheless, the mere observation thatcredit frictions are important does not in itself make it obvious that policy should
be conditioned on the observed evolution of the volume of lending. In our model,
it is credit spreads that are of most direct significance as disturbance terms in the
structural equations of our model of the transmission mechanism of monetary policy.
In the simplest case, the part of the model that determines the evolution of aggregate
credit actually decouples from the block of equations that are needed to determine
the evolution of output and inflation. More generally, this is not true, but mainly
because of the possibility that credit spreads are endogenously affected by changes
in the volume of lending.69 In such a case, it would still seem more important tomonitor the endogenous variation in credit spreads in order to judge the appropriate
69

Technically, the volume of financial intermediation can also be relevant to aggregate demand
determination owing to its implications for the resources consumed by the banking sector, but it is
not obvious that this should be quantitatively important.

62

level for the policy rate, rather than to monitor the volume of lending (which would
be one among the several determinants of variation in those spreads).
While our results suggest that in the case of many types of aggregate disturbances (both disturbances that are ordinarily thought of as “demand” disturbances,and others that are considered “supply” disturbances), the effects of shocks on theevolution of output, inflation and interest rates under a conventional specification ofmonetary policy are not too greatly changed by the introduction of credit frictions,this is not uniformly the case. One example is the effects of changes in the path ofgovernment debt owing to shifts in fiscal policy. While fiscal policy has no effects
on inflation, output or interest-rate determination in the basic NK model (if distorting tax rates are assumed not to change, so that the changed path of government
borrowing corresponds only to a change in the time path of net lump-sum transfersfrom the government) — that is, “Ricardian equivalence” obtains — in the presenceof credit frictions of the kind modeled here, this is no longer the case (even under
the special assumption about tax rates). This suggests that the integration of fiscal
variables into models used in central banks for monetary policy analysis may deservea higher place on the research agenda.

4.2

Implications for Monetary Policy

It is more difficult to draw definite conclusions about desirable monetary policies from
our analysis, as conclusions of this kind are likely to be sensitive to details of model
specification. We have nothing to say about the issue of how monetary policy decisions
should take into account financial stability concerns — either possible consequences ofinterest-rate decisions for systemic risks to the financial sector, of the kind emphasized
by the US Federal Reserve in some of its decisions this year, or possible consequencesof interest-rate policy for risk-taking behavior, of the kind hypothesized by Borio
and Zhu (2007) — since we simply abstract from such concerns in our reduced-form
model of the financial sector. In principle, it should be possible to address concernsof these kinds within our framework, by introducing more complex hypotheses about
the nature of endogenous variation in credit spreads, and we hope to pursue such aninvestigation in future work. Here we are able only to shed light on the way in whichmonetary policy should be affected by the magnitude of and time variation in credit
spreads, taking the evolution of credit spreads as a given, that monetary policy (in63the sense of interest-rate policy) is not expected to influence.
In posing this relatively limited question, we do not mean to deny the importanceof central banks’ role as “lender of last resort.” However, contrary to some accounts,
we do not see an intrinsic connection between this dimension of central banking andmonetary policy (in the narrow sense of interest-rate control and control of the supplyof central bank balances for use in payments). It seems to us that the most important
way in which central banks reduce risks to financial stability is by extending credit tofinancial institutions in distress, as in the case of the US Federal Reserve’s traditional
discount-window policy, or the new lending facilities created more recently; and lending of this kind need not imply either a reduction in the central bank’s interest-ratetarget or an expansion of its balance sheet (as has generally been true of the fairly
aggressive lending by the Fed and ECB in recent months). While decisions about this
aspect of central-bank policy are important and sometimes quite subtle, they can beconsidered in relative independence of a bank’s monetary policy objectives, and we
do not seek to say anything about them here. Instead, we take this dimension ofpolicy, with whatever consequences it may have for the evolution of credit spreads, as
given, and consider the proper conduct of monetary policy given the lending capacityof the financial sector (represented by our assumptions about the “intermediation
technology” and “financial markup”).
Of course, even our conclusions about optimal interest-rate policy may well haveto be modified if one takes account of other possible sources of endogenous variation incredit spreads; only replication of our analyses under additional variant assumptions
will be able to determine to what extent this may be true. Nonetheless, we find someof the results from our analyses of alternative policies in the present simple model
quite suggestive. At the very least, these should provide hypotheses deserving offurther study in the context of more complex models. Probably the most interestingof these results relate to the way in which monetary policy should take account ofvariations in credit spreads.
It is often argued that widening credit spreads provide a prima facie case for theappropriateness of lowering the central bank’s policy rate. The argument commonly
given is that when spreads increase, the terms on which it is possible to borrow aretightened, even in the absence of any increase in the policy rate. If it was not desired
to tighten monetary conditions, it is therefore necessary to lower the policy rate, tothe extent required in order to preserve the original cost of borrowing. Authors such64

as McCulley and Toloui (2008) suggest a one-for-one reduction in the policy rate in
response to increases in particular credit spreads.
In our model, it is certainly right that the policy rate is not the only interest ratethat matters in assessing the degree to which financial conditions are expansionary.
We can, to a useful approximation, define an “average” interest rate such that aggregate demand depends on the expected path of this average rate (relative to expected
inflation); and when credit spreads increase, this average rate will be higher relative
to the policy rate than under more ordinary circumstances. Hence it is necessary tolower the policy rate in order to prevent financial conditions from tightening, in therelevant sense. However, this argument alone would not justify lowering the policy
rate by the full amount of the increased spread, nor would it justify lowering it to theextent necessary to keep the rate at which it is possible to borrow from rising. An
increase in credit spreads also causes the lending rate to rise relative to the average
rate, so that it is necessary for the lending rate to rise if financial conditions are not
to be loosened, in the sense that matters for aggregate demand and the generationof inflationary pressure.
In the calibrated numerical examples considered in our paper, it is in some cases
desirable to lower the policy rate by the full amount of the increase in the spread
between the deposit rate and the lending rate; but in a number of cases a much smaller
adjustment would lead to a more nearly optimal policy, while in other cases, even an
adjustment several times as large as the increase in credit spreads would not suffice
to correct the bias associated with a simple Taylor rule. While we find that a spreadadjustment of the kind proposed by McCulley and Toloui would generally represent
the right direction of adjustment of policy, relative to a simple Taylor rule, the simple
adjustment that they suggest is not at all accurate as a guide to the magnitude ofadjustment that would be appropriate.
Still less would our model justify the view, which sometimes appears to be assumed in popular discussions of the appropriate response to strains in credit markets,that the persistence of higher-than-average credit spreads means that interest rates
have not been cut enough. (This is evidently what is assumed when commentators
seek to judge whether policy rates have been cut by the right amount by looking at
whether spreads remain unusually large, rather than asking whether borrowing rates
are actually high.)
It is true that an increase in spreads is a source of deadweight loss in our model,
65

so that a reduction of spreads to normal levels, if this could be done without changing
anything else, would be desirable. But it is not necessarily true that monetary policy
can do anything to undo an increase in credit spreads — this is certainly the casein our simplest model, where the credit spread is determined entirely by exogenous
factors — and lowering interest rates will instead certainly have collateral effects.
Even if monetary policy can influence spreads, it is not obvious that this will bebeneficial. For example, in the case that spreads are endogenous because they dependon the volume of bank lending, monetary policy can lower spreads to the extent thatit can lead to a contraction of bank lending; but that would not necessarily increase
efficiency, even if a reduction in the spread associated with a given volume of lending
would enhance efficiency (by allowing more lending). A better principle may well be
to use monetary policy to mitigate the distortions caused by an increase in credit
spreads, to the extent that this is possible, rather than seeking to use policy to
influence the level of spreads as such.
We have also briefly considered the view that a policy rule that responds toaggregate credit, in addition to inflation and real activity, will better stabilize theeconomy in the case that credit frictions are important. We have found little support
for such a view, at least in the context of our model. In the case of “non-financial”
disturbances of a variety of types, our analysis indicates that a more nearly optimal
response to the disturbance than would result from a simple Taylor rule requires,
if anything, that the intercept of the Taylor rule be reduced when aggregate credit
increases — the opposite of the conventional advice, deriving from the idea thatmonetary policy should seek to stabilize aggregate credit, in addition to inflation andreal activity. But in fact no rule that responds to aggregate credit has very desirable
properties, especially if the same response coefficient is assumed regardless of thenature of the disturbance.
In fact, it is not obvious that one should expect to find any single variable (oreven a small number of variables) that, when added to a standard Taylor rule, will
provide a sufficient guideline for sound policy, regardless of the type of disturbances
to which an economy may be subject. Instead, as has often been argued in othercontexts, a policy prescription that specifies a target criterion that the central bankshould seek to ensure is projected to be satisfied at all time is likely to provide amore robust description of good policy than a prescription for the bank’s reaction
function. For example, our simulations show (consistently with previous literature)66

that a Taylor rule does not always lead to desirable policy responses to disturbances,even in the absence of financial frictions. It can hardly be expected, then, thatmodifying the Taylor rule to properly take account of variation in spreads can solve
all of the problems with such a simple prescription.
A policy rule that directs the central bank to adjust its policy rate as necessary
to satisfy a “flexible inflation target” will automatically require the central bank toadjust the policy rate to take account of changes in the severity of financial frictions,
without any need to modify the target criterion that the bank aims to conform with.
In our model of the transmission mechanism, it is the level and expected future pathof the average interest rate, rather than the policy rate, that determines whether
the evolution of output and inflation should satisfy the target criterion. When therelation between this average rate and the policy rate changes owing to a change
in the size of interest-rate spreads, the level and expected path of the policy raterequired for conformity with the target criterion will obviously change. The required
policy rate will similarly change as a result of the effects of changing credit spreads
on the “IS relation” and aggregate-supply relation discussed in section 2.
When the policy rule is specified in terms of a target criterion, it is not obviousthat the target criterion that should be chosen depends much on the importance ofcredit frictions. In a particularly simple case of a model with time-varying credit
frictions, we have shown that the optimal target criterion remains exactly the sameas in the basic NK model: the central bank should seek to stabilize a weighted averageof the inflation rate and the rate of growth of the output gap (or alternatively, tostabilize an output-gap-adjusted price level). Under more general assumptions, thisexact equivalence does not obtain. Nonetheless, the target criterion that would be
optimal in the case of the model without credit frictions seems still to provide afairly good approximation to optimal policy in the model with credit frictions, whenthese are parameterized to be of an empirically realistic magnitude. Because of theadvantages (for example, from the point of view of communication) of commitment toa simple criterion, and because of the degree to which any attempt to refine the rulewould depend on fine details of the specification of the financial frictions, about which
there is likely to be uncertainty in any event, our results suggest that maintaining
a commitment to the same target criterion as would be optimal in the absence offinancial frictions is not a bad idea. This suggests that a central bank should notchange its target criterion — what it regards as acceptable paths of inflation and real67

activity — at all in times of financial stress, but instead should take account of thosedevelopments only as a result of the changes that they imply for the links between
the policy rate and the evolution of the target variables, and hence for the path ofthe policy rate that is required to satisfy the target criterion.

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[11] De Fiore, Fiorella, and Oreste Tristani, “Credit and the Natural Rate of Interest,” unpublished, European Central Bank, December 2007.
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[21] Issing, Otmar, “The ECB’s Monetary Policy Strategy: Why Did We Choose a
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[22] Lagos, Ricardo, and Randall Wright, “A Unified Framework for Monetary Theory and Policy Analysis,” Journal of Political Economy 113: 463-484 (2005).
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[25] Lucas, Robert E., Jr., and Nancy Stokey, “Money and Interest in a Cash-inAdvance Economy,” Econometrica 55: 491-513 (1987).
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[27] Meyer, Laurence H., and Brian P. Sack, “Updated Monetary Policy Rules:
Why Don’t They Explain Recent Monetary Policy?” Monetary Policy Insights,
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71

[32] Taylor, John B., “Discretion versus Policy Rules in Practice,” CarnegieRochester Conference Series on Public Policy 39: 195-214 (1993).
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72

π

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Figure 1: Impulse responses to a 1 percent (annualized) shock to ²m
t , in three different
models with a linear intermediation technology.

73

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Figure 2: Impulse responses to a 1 percent shock to At , in three different models witha linear intermediation technology.

74

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FF
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Figure 3: Impulse responses to a shock to Gt equal to 1 percent of steady-state output,
in three different models with a linear intermediation technology.

75

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Figure 4: Impulse responses to a 1 percent (annualized) shock to ²m
t , in three different
models with a convex intermediation technology.

76

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Figure 5: Impulse responses to a 1 percent shock to At , in three different models witha convex intermediation technology.

77

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Figure 6: Impulse responses to a 1 percent shock to µw
t , in three different models
with a convex intermediation technology.

78

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Figure 7: Impulse responses to a shock to Gt equal to 1 percent of steady-state output,
in three different models with a convex intermediation technology.

79

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Figure 8: Impulse responses to a 1 percent shock to type s expenditure, in threedifferent models with a convex intermediation technology.

80

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Figure 9: Impulse responses to a shock to bgt equal to 1 percent of steady-state output,
in three different models with a convex intermediation technology.

81

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Figure 10: Impulse responses to the same shock as in Figure 8, under a strict inflation
targeting policy.

82

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Figure 11: Impulse responses to a 1 percent shock to At , under four alternative
monetary policies.

83

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Figure 12: Impulse responses to a 1 percent shock to µw
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monetary policies.

84

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Figure 13: Impulse responses to a shock to Gt equal to 1 percent of steady-state
output, under four alternative monetary policies.

85

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Figure 14: Impulse responses to a 1 percent shock to type s expenditure, under four
alternative monetary policies.

86

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Figure 15: Impulse responses to a shock to µbt that increases ω t (b) by 1 percentage
point (annualized) for each value of b, under four alternative monetary policies.

87

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Figure 16: Impulse responses to the same “financial shock” as in Figure 15, underalternative spread-adjusted Taylor rules.

88

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Figure 17: Impulse responses to a shock to bgt equal to 1 percent of steady-state
output, under alternative spread-adjusted Taylor rules.

89

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Figure 18: Impulse responses to a 1 percent shock to type s expenditure, underalternative spread-adjusted Taylor rules.

90

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Figure 19: Impulse responses to a shock to Gt equal to 1 percent of steady-state
output, under alternative spread-adjusted Taylor rules.

91

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Figure 20: Impulse responses to a 1 percent shock to type b expenditure, underalternative spread-adjusted Taylor rules.

92

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−0.6
0

4

8

12

16

0

4

8

b
OptimalTaylorTaylor+25
Taylor+50
Taylor+75
Taylor+100

1

0.5

0
0

4

8

12

16

Figure 21: Impulse responses to a 1 percent shock to At , under alternative spreadadjusted Taylor rules.

93

π

Y
0.1
0

0

−0.2

−0.1

−0.4

−0.2
−0.3

−0.6
0

4

8

12

16

0

4

d

8

12

16

12

16

b

i

i
0.3

0

0.2
0.1

−0.5

0
−0.1

−1
0

4

8

12

16

0

4

8

b
0

OptimalTaylor−050
Taylor−025
TaylorTaylor+025
Taylor+050

−0.5
−1
−1.5
0

4

8

12

16

Figure 22: Impulse responses to the same kind of “financial shock” as in Figure 15,
under rules that respond to aggregate credit.

94

π

Y
0.2

0.05

0.15
0.1

0

0.05

−0.05

0
0

4

8

12

16

0

4

d

8

12

16

12

16

b

i

i
0.2

0.2

0.15
0.1

0.1

0.05
0

0
0

4

8

12

16

0

4

8

b
0

OptimalTaylor−050
Taylor−025
TaylorTaylor+025
Taylor+050

−0.2
−0.4
−0.6
0

4

8

12

16

Figure 23: Impulse responses to a shock to Gt equal to 1 percent of steady-state
output, under rules that respond to aggregate credit.

95

π

Y
0.01

0.15

0
0.1

−0.01

0.05

−0.02

0

−0.03
0

4

8

12

16

0

4

d

8

12

16

12

16

b

i

i

0.06

0.06

0.04

0.04

0.02

0.02

0

0
0

4

8

12

16

0

4

8

b
OptimalTaylor−050
Taylor−025
TaylorTaylor+025
Taylor+050

0.1
0.05
0
0

4

8

12

16

Figure 24: Impulse responses to a 1 percent shock to type b expenditure, under rulesthat respond to aggregate credit.

96

π

Y
0
2

−0.2
−0.4

1

−0.6
0
0

4

8

12

16

0

4

d

i

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6
8

16

12

16

i
0

4

12

b

0

0

8

12

16

0

4

8

b
OptimalTaylor−050
Taylor−025
TaylorTaylor+025
Taylor+050

1

0.5

0
0

4

8

12

16

Figure 25: Impulse responses to a 1 percent shock to At , under rules that respond toaggregate credit.

97