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Federal Reserve Bank of Chicago

The Tradeoffs in Leaning
Against the Wind
François Gourio, Anil K Kashyap,
and Jae Sim

August 2017
WP 2017-21
Working papers are not edited, and all opinions and errors are the
responsibility of the author(s). The views expressed do not necessarily
reflect the views of the Federal Reserve Bank of Chicago or the Federal
Reserve System.

NBER WORKING PAPER SERIES

THE TRADEOFFS IN LEANING AGAINST THE WIND
François Gourio
Anil K Kashyap
Jae Sim
Working Paper 23658
http://www.nber.org/papers/w23658

NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
August 2017

The views expressed in this paper do not necessarily reflect the views of the Federal Reserve
System, the Federal Reserve Board, the Federal Reserve Bank of Chicago, the Bank of England,
or the National Bureau of Economic Research. Kashyap acknowledges research support from the
Houblon/Norman/George fellowship at the Bank of England, a grant from the National Science
Foundation administered through the NBER and the Initiative on Global Markets at Chicago
Booth. We thank participants at the IMF Annual Research conference, seminar participants at the
Federal Reserve Board, the Federal Reserve Bank of Chicago, Andrew Filardo, Galina Hale,and
Lars Svensson for helpful comments. We are responsible for all errors.
At least one co-author has disclosed a financial relationship of potential relevance for this
research. Further information is available online at http://www.nber.org/papers/w23658.ack
NBER working papers are circulated for discussion and comment purposes. They have not been
peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies
official NBER publications.
© 2017 by François Gourio, Anil K Kashyap, and Jae Sim. All rights reserved. Short sections of
text, not to exceed two paragraphs, may be quoted without explicit permission provided that full
credit, including © notice, is given to the source.

The Tradeoffs in Leaning Against the Wind
François Gourio, Anil K Kashyap, and Jae Sim
NBER Working Paper No. 23658
August 2017
JEL No. E52,E58,G28
ABSTRACT
Credit booms sometimes lead to financial crises which are accompanied with severe and
persistent economic slumps. Does this imply that monetary policy should “lean against the wind”
and counteract excess credit growth, even at the cost of higher output and inflation volatility? We
study this issue quantitatively in a standard small New Keynesian dynamic stochastic general
equilibrium model which includes a risk of financial crisis that depends on “excess credit”. We
compare monetary policy rules that respond to the output gap with rules that respond to excess
credit. We find that leaning against the wind may be attractive, depending on several factors,
including (1) the severity of financial crises;(2) the sensitivity of crisis probability to excess
credit; (3) the volatility of excess credit; (4) the level of risk aversion.

François Gourio
Economic Research
Federal Reserve Bank of Chicago
230 South LaSalle St
Chicago, IL 60604
francois.gourio@chi.frb.org
Anil K Kashyap
Booth School of Business
University of Chicago
5807 S. Woodlawn Avenue
Chicago, IL 60637
and NBER
anil.kashyap@chicagobooth.edu

Jae Sim
Division of Research & Statistics
Federal Reserve Board
20th Street & Constitution Avenue, NW
Washington, D.C. 20551
jae.w.sim@frb.gov

Introduction

Following the Global Financial Crisis of 2008, policymakers around the world have made it a priority
to reduce the risk of future crises. New prudential and regulatory policies have been developed to
promote financial stability. But the question of whether financial stability concerns should play
a role in the setting of monetary policy remains controversial. In this paper we investigate the
wisdom of what has come to be known as “leaning against the wind” (LAW), that is having
monetary policy react to perceived financial imbalances such as excess credit growth, which has
been found empirically to predict financial crises.1
One argument against LAW is that financial stability could be better delivered by an appropriate
set of macroprudential policies, that is making prudential and regulatory policies respond to the
state of the economy, which would leave monetary policy free to focus on its usual inflation and
output stability objectives.2 While this separation is attractive in principle, we believe it is often
difficult to implement practically (as recently explained by Dudley (2015)). Many countries such
as the United States have a limited set of macro-prudential tools, and suffer from a dispersion of
regulatory authorities. The tools are difficult and slow to adjust, and their effects remain fairly
uncertain. Monetary policy has broad effects (it “gets in all the cracks” as Stein (2013) famously
noted) while macro-prudential tools are perhaps too narrow (e.g. they lead to a migration of
activities from the regulated banking system to the unregulated shadow banking system). These
considerations motivate our focus on monetary policy.
The second main argument against LAW is that under inflation targeting, stabilizing inflation
is sufficient to stabilize the macroeconomy, as argued by Bernanke and Gertler (1999).3 Even if
there is a trade-off, monetary policy has likely small effects on the likelihood of financial crises, so
that having a meaningful effect on this likelihood would require a large interest rate change, at the
cost of a large deviation of output and inflation relative to what could be achieved. This argument
1

See Schularick and Taylor (2012).
For instance, see Korinek and Simsek (2016) and Farhi and Werning (2016).
3
A distinct argument states that it is preferable to “mop up after the crash”, but this argument seems less
compelling now in light of the difficulties in stabilizing the economy in the aftermath of the most recent financial
crises - for instance, the zero lower bound and reduced potency of monetary policy when agents want to deleverage.
2

has been made most clearly by Svensson (2016).
Our starting point is that financial crises are very costly. As Reinhart and Rogoff (2009) and
Cerra and Saxena (2008) emphasized even before the most recent crisis, recovery from crises are
typically slow so that the hangover from a crisis is different than from a regular recession. Figure
1 shows GDP since the crisis and suggests that there has been a permanent drop in the level of
GDP amounting to about 13 percent. More studies since then have documented that financial
crises have very persistent effects.4 Preventing a crisis may, therefore, bring different benefits than
those associated with smoothing out inefficient business cycle fluctuations (Barro (2009)). This
consideration features prominently in our analysis.
Our main contribution is to propose a stylized dynamic stochastic general equilibrium (DSGE)
model to assess the efficacy of LAW. We depart from the usual model in two ways. First, we follow
Gourio (2013) and introduce a standard capital structure choice in which debt has a tax subsidy,
but creates the risk of costly bankruptcy (that is avoided for an all equity financed firm). Capital
is accumulated by firms that face costs of issuing equity. We introduce a “financial shock” in this
economy by assuming that the tax benefit varies over time. As we explain below, this is a shorthand
for various forms of inefficient credit use. We view the tradeoff theory as providing a compact way
to introduce variation in the use of debt financing (that could in fact arise for many reasons).
Our second modification is to introduce the possibility of a large financial crisis that can hit the
economy. This is similar to the rare disasters that have received much attention recently in the asset
pricing literature.5 We assume that the financial crisis leads to a significant, permanent reduction
in total factor productivity and a one-off shock to the capital stock. We view this modeling choice
as a convenient device to capture that financial crises lead to large and highly persistent declines
in output and consumption. Given that there is not much of a consensus as to why crises are so
costly, this simplification is a reasonable way to model them without taking a stand on a particular
reason why losses seem to be so persistent. In particular, as Figure 1 suggests, in the aftermath of
the most recent crisis U.S. real GDP dropped permanently below the pre-crisis trend, and shows
no sign of returning to the trend. As of 2016, real GDP was about 13 percent below the pre-crisis
trend. Given this consideration, associating a permanent reduction in the production capacity with
4
5

See Blanchard, Cerutti, and Summers (2015) and Martin, Munyan, and Wilson (2015).
See Barro (2006), Gabaix (2012), Tsai and Wachter (2015) and Gourio (2012).

Figure 1: Real GDP and Its Pre-Crisis Trend

disaster-type crises is a convenient shortcut to replicate the crisis dynamics of real GDP.
We study two types of collapses. The first kind of financial crisis, as in Gourio (2012) and
Gourio (2013), occurs exogenously. The second supposes that the probability of the crisis depends
on the amount of inefficient credit. By comparing the two alternatives, we can understand how the
policy consequences may differ when leaning against the wind can change the likelihood of a crisis.
The model allows for the usual productivity and demand shocks in addition to the financial
shocks. The centerpiece of the analysis is a comparison of different monetary policy rules that
vary with respect to the signals on which the central bank’s policy rate is set. In our baseline, we
compare policies that rely on perfectly measured variables and then in some extensions analyze
what happens when the central bank must rely on imperfectly measured proxies. In this respect we
follow in the long line of papers starting with Bernanke and Gertler (1999) and Gilchrist and Leahy
(2002) that ask whether monetary policy should take account of asset price movements. A common
conclusion in that literature is that after accounting for movements in inflation, and possibly output,
there is no need to respond to asset prices. We explore whether the same conclusion holds in our
environment.
Our main finding is that gains from responding to financial shocks depend importantly on the

relative importance of the various shocks hitting the economy and the nature of the financial crisis
risk. In some versions of the model, for instance when only productivity and demand shocks are
present, the possibility of a crisis (endogenous or not) makes little difference for policy. In this
environment, stabilizing inflation is optimal. Loosely speaking, once the central bank eliminates
demand shocks and accommodates productivity shocks, it can stabilize inflation and simultaneously
control crisis risk to the extent possible. In this setup, even if financial crises are endogenous it
will make little difference for the policy choices because the central bank’s control of demand will
also control credit and limit crisis risk. This result is consistent with the previous literature, in
particular Bernanke and Gertler (1999).
On the other hand, when there are also financial shocks, then failing to respond to credit build
ups leads to larger crisis risk. Because crises are very costly, the optimal policy trades off leaning
against the wind to reduce crisis risk against the costs of larger fluctuations in output and inflation.
We emphasize that this result arises even though monetary policy is not a particularly powerful tool
for managing the risk of financial crisis. The agents in our model are also not terribly risk averse.
Nonetheless, by lowering the probability of financial crisis, the central bank generates welfare gains
because of the large cost of financial crises. In general, the more risk averse are households and the
larger is the size of the crisis, the stronger is the case for LAW. We describe these mechanisms in
more detail below and provide a preliminary quantitative assessment.
The remainder of the paper proceeds as follows. In the next section, we provide a brief literature
review. In section 3, we introduce the model. Most elements are very standard and are common to
many New Keynesian models. In presenting the model, therefore, we concentrate on the two novel
aspects mentioned above. Section 4 discusses the parameters used and examines basic properties of
the model economy. Finally, in section 5, we compare the performance of a number of policy rules
for different versions of the model, and illustrate how several key parameters affect our results.
Section 6 concludes.

Literature Review

Smets (2014) provides an excellent survey of most of the research on leaning against the wind
through 2014, so we summarize his main conclusions and then focus on a few notable papers written

since his survey. Smets notes that the case for using monetary policy to promote financial stability
depends in part on the availability and effectiveness of other tools. The paper then reviews a number
of analyses, most notably Lim, Costa, Columba, Kongsamut, Otani, Saiyid, Wezel, and Wu (2011),
that study the experience using macroprudential tools and reaches two important conclusions:
that “the empirical literature tentatively supports the effectiveness of macroprudential tools in
dampening procyclicality” and “to what extent such measures are effective enough to significantly
reduce systemic risk is, however, as yet unclear.”
Given the ambiguity over whether financial stability can be delivered without appealing to
monetary policy, the paper then turns to the question of what the evidence says regarding the
effectiveness of monetary policy in limiting the build up of financial vulnerabilities. Here again
the evidence is mixed. On the one hand, there are a variety of studies that link higher risk-taking
by banks with looser monetary policy. Smets stresses that the risk-taking can occur on both the
asset-side of the banks’ balance sheet as they reach for yield and through funding choices that
entail extra reliance on short-term financing. He argues that although there is ample evidence of
risk-taking, the question of whether actively using monetary policy to head it off creates too much
collateral damage remains open. He cites several articles that suggest, for instance, that using
monetary policy to forestall property price booms would have created a recession. Overall we read
his paper as suggesting that there may be scope for leaning against the wind, but doing so would
entail non-trivial risks.6
Perhaps the most prominent paper written after the Smets survey is Svensson (2016). This
paper provides a simple and transparent framework for evaluating LAW policies. It starts with
empirical estimates of the effects of higher interest rates on the likelihood of a crisis (obtained by
combining estimates of the effect of interest rates on credit, and of credit growth on the likelihood
of crisis (Schularick and Taylor (2012))) and on inflation and output in the short run as well as
the cost of a financial crisis (a sharp, temporary recession). Svensson emphasizes that on the one
hand, tighter policy reduces the risk of financial crisis in the short-run but increases it later on since
the effect of tighter policy works through the growth rate of credit (and the long-term level of real
credit is assumed to be independent of monetary policy because of long-run neutrality). On the
6

Smets also stresses that if the central bank is given responsibility for financial stability and fails to achieve it,
then the bank’s monetary independence could be compromised. Though as Peek, Rosengren, and Tootell (2015)
mention, central banks that are simply acting as a lender of last resort can also face this kind of pressure.

other hand, tighter policy reliably reduces growth and inflation in the short-run. Overall, the costs
of slowing down the economy are much higher than the gains from only marginally reducing the
risk of a crisis. Indeed, if one accounts for the fact that crises are to a certain extent inevitable and
unavoidable, then a policy that steers the economy to be above potential during non-crisis periods
is optimal. Hence, Svensson argues that a careful treatment of this problem calls for leaning with
the wind.7
The IMF 2015 staff study (IMF (2015)) reaches similar conclusions to Svensson. On their
reading of the empirical literature, a 100 basis point increase in the central bank policy rate for
one year is needed to reduce the probability of a crisis by only 0.02 percentage point per quarter.
There is obviously much uncertainty around this estimate, but they argue that even using the
largest reported estimates of a 0.3 percentage point reduction per quarter in crisis risk, the costs of
a slowdown are likely to exceed the gains from preventing a crisis. Ajello, Laubach, Lopez-Salido,
and Nakata (2016) similarly argue that the optimal response is small for the median estimate of
the effect of monetary policy on risk of crisis, but may be significant if the policymaker takes into
account the uncertainty surrounding the estimate, and focuses on the worst-case scenario.8
Filardo and Rungcharoenkitkul (2016), in contrast, reach the opposite conclusion studying
optimal monetary policy in an environment of recurring, endogenous financial booms and busts. In
their environment leaning systematically over the whole cycle is justified because leaning not only
influences the probability of a crisis, but also smooths the financial cycle, resulting in less virulent
boom and bust episodes. The optimal monetary policy in this setting calls for progressively stronger
leaning as financial imbalances grow but reducing the degree of leaning against the wind as a crisis
becomes imminent. The persistence of the financial cycle and the degree to which monetary policy
influences the amplitude and duration of booms and busts are key distinguishing features of the
modelling approach.9
Our approach cannot be easily mapped into the Svensson style calculation. There are several
critical differences. First, in terms of methodology we optimize a policy rule in a DSGE model
7

Juselius, Borio, Disyatat, and Drehmann (2017) argue that one cost of low interest rates is an exacerbation of
the financial cycle.
8
See also Gerdrup, Hansen, Krogh, and Maih (2016) and Bauer and Granziera (2016) for recent studies of the
effectiveness of monetary policy in LAW.
9
Filardo and Rungcharoenkitkul (2016) solve for the optimal nonlinear policy rule using collocation method. This
allows the intensity of leaning to change with the level of financial imbalances. Linear rules do not permit this
possibility.

while Svensson conducts a one-time cost/benefit analysis. Second, our objective function is utility
while he bases his analysis on a quadratic loss function. Third, we model crises as permanent
effects on output while he considers them a temporary “gap” in unemployment or output. For
instance, in our benchmark calculation the level of output drops by 10 percentage points in a crisis.
Svensson assumes a five percentage points increase in unemployment for two years followed by
a return to normal. The total loss in output in his baseline is, therefore, much smaller than in
ours. Below we show that with much smaller crises a LAW policy is not welfare improving. Finally,
there is a difference between the way the models approach long-run monetary neutrality. In our
model, monetary policy shocks have only transient effects on credit and other variables, similar to
Svensson. But Svensson’s specification implies that lower credit growth reduces the probability of
crisis in the short-run before increasing it in the medium run. In our model, LAW can deliver a
lower probability throughout.
IMF (2015), like Smets, questions whether monetary policy is the right tool to address these
problems and proposes a three part test that should be considered before monetary policy should
be used to lean against the wind. First, are financial risks in the economy excessive? If they are
not, then adjusting monetary policy is unnecessary. Second, can other tools be used, particularly
macroprudential ones, be used instead of monetary policy? Finally, will monetary policy, if set in
a conventional fashion based on inflation and output developments, take care of financial stability
concerns?
Our model allows us to partially address two of the three considerations. We suppose that
monitoring financial risks is challenging. Inefficient credit movements may not be observable, so
we can study policies that can only be based on noisy indicators of financial risk. Our model has
multiple shocks, so we can also study which ones give rise to scenarios where there is a genuine
tradeoff between managing the near term inflation and output fluctuations and preventing crises; as
will be clear, there are some shocks where a standard inflation targeting central bank will contain
financial risks just as a by-product of following its mandate.
We do not discuss macroprudential tools. Partly, this is a tractability issue. There is no
consensus model that integrates macroprudential policy levers in a standard monetary model. As
Smets (2014) emphasizes even the empirical evidence how this might work is mixed. Developing
that kind of framework is beyond the scope of our paper.
7

More importantly, in many countries the scope for deploying macroprudential tools is limited.
The case study developed by Adrian, de Fontnouvelle, Yang, and Zlate (2015) highlights some of the
challenges in the U.S. context. In their hypothetical scenario that they dub a “tabletop exercise”,
the Federal Reserve is facing a situation where commercial real estate prices are rising sharply, while
its inflation and employment objectives are close to being met. Most of the funding fueling the boom
are coming from small banks and through capital markets (via securitization).10 When confronted
with this scenario, the four Federal Reserve Bank Presidents who were attempting to implement
policies to manage the situation concluded that “from among the various tools considered, tabletop
participants found many of the prudential tools less attractive due to implementation lags and
limited scope of application. Among the prudential tools, participants favored those deemed to
pose fewer implementation challenges, in particular stress testing, margins on repo funding, and
supervisory guidance. Nonetheless, monetary policy came more quickly to the fore as a financial
stability tool than might have been thought before the exercise.”

Model

The model economy consists of a representative household, a continuum of monopolistic competitors, a representative investment good producer, and a continuum of financial intermediaries that
hold capital financed by debt and equity. All firms, including the intermediaries are owned by
the household and therefore discount future cash flow using the stochastic discount factor of the
representative household.

3.1

Households

The representative household has preferences

∞
X

β s−t U (Cs , Ns )

s=t
10

This funding constellation matters because in the U.S. the central bank can use some tools, such as stress tests,
to steer decisions for very large banks. Restricting the behavior of small banks and stopping securitization is more
difficult.

where
U (Ct , Nt ) =

Nt 1+υ
Ct 1−τ
−
.
1−τ
1+υ

(1)

The household consumption bundle is made up of differentiated products,
Z

Ct =

Ct (i)

1
1−η

1−η
di
.

The dual problem of cost minimization gives rise to a good-specific demand,

Ct (i) =

where Pt ≡

hR
1
0

Pt (i)
Pt

−η
Ct

i1/(1−η)
Pt (i)1−η di
.

The representative household earns wage income (wt Nt ), the profits of intermediate-goods firms
(ΠFt ) and the profits of financial intermediaries (ΠIt ). (The investment good producer makes zero
profits due to perfect competition and constant return to scale.) The household saves by holding
securities issued by financial intermediaries and government bonds (BtG ), which are zero in netsupply. The bonds issued by the intermediaries are unsecured risky bonds. We denote the price of a
bond by qt . If the bond issuer avoids default, the bond returns one unit of consumption tomorrow.
In default, the household receives a partial payment. Since there is a continuum of issuers, the
law of large number applies and the household can form rational expectations about how many
bonds fail and how many deliver the promised payment. We denote the probability of default by
Ht and the average recovery rate conditional upon default by RtD . We can then express the budget
constraint of the household as

G
Ct = wt Nt + ΠFt + ΠIt − qt Bt + [(1 − Ht ) + Ht RtD ]Bt−1 + Rt−1 Bt−1
− BtG .

(2)

We denote the Lagrangian multiplier associated with the budget constraint by Λt . The household’s
efficiency conditions are summarized as:

Ct : Λt = UC (Ct , Nt ),

(3)

Nt : Λt wt = −UN (Ct , Nt ),

BtG

: 1 = βEt

Λt+1
Rt+1
Λt

(4)

(5)

and

Bt : qt = βEt

Λt+1
D
(1 − Ht+1 ) + Ht+1 Rt .
Λt

(6)

A few remarks are in order. First, the two static FOCs together also imply the following efficiency
condition.
wt = −

UN (Ct , Nt )
.
UC (Ct , Nt )

(7)

Second, we assume that the economy is subject to the risk premium shock of Smets and Wouters
(2007). Following Fisher (2015), we interpret this as the shock to the demand for safe asset.11 We
denote the shock by Ξt and modify the FOC as

1 = βEt

Λt+1
Ξt Rt+1 .
Λt

(8)

These shocks do not affect the flexible economy and hence are an inefficient source of business cycle
fluctuations.
Third, the FOC for intermediary bond holding plays the role of the pricing equation for intermediary problem. We will provide more details on this, including the determinants of the recovery
rate RtD , when we discuss the intermediary problem. For later purposes, we define the stochastic
discount factor of the household as
Mt,t+1 ≡ β

3.2

Λt+1
.
Λt

(9)

Investment Goods Producers

We assume that there exists a continuum of competitive firms indexed by k ∈ [0, 1]. These firms
produce an identical composite good It using a linear technology subject to an adjustment cost
related to changing the level of investment. We parameterize the costs to be κ/2 (It /It−1 − 1)2 It−1 .
11
In this interpretation, the shock Ξt can be viewed as a disturbance to demand for money and hence can also be
thought of as shifting nominal aggregate demand.

The composite good It is sold at a price Qt to be used in the production of capital. Production
of the composite good requires the use of all varieties of intermediate goods. Since the industry is
competitive, the size of an individual firm is indeterminate. Hence we assume a representative firm
that is a price taker. The profit maximization problem of the investment goods producers can be
cast as choosing the input level given the cost of adjusting investment level, i.e.,

max Et
Is

∞
X

Mt,s

s=t

Is
κ
− 1 Is−1 .
Qs Is − Is +
2 Is−1

The FOC of the problem is given by

It
κ It+1 2
Qt = 1 + κ
− 1 − Et Mt,t+1
−1 .
It−1
2
It

3.3

(10)

Retailers

There exists a continuum of monopolistic competitors indexed by i ∈ [0, 1]. These retailing firms
combine labor and capital using a Cobb-Douglas production technology

Yt (i) = Zt Kt (i)α Nt (i)1−α

where Zt is the aggregate technology. Following Rotemberg (1982), we assume that the retailers
are subject to quadratic costs of adjusting prices

2
ϕ
pt (i)
ϕ Pt (i)
− 1 Yt =
Πt
− 1 Yt ,
2 Pt−1 (i)
2
pt−1 (i)
where pt (i) ≡ Pt (i)/Pt and Πt ≡ Pt /Pt−1 is aggregate inflation. Hence, the firm’s static profit is
given by
Πt (i) = pt (i)Yt (i) − wt Nt (i) −

rtK Kt (i)

2
ϕ pt (i)
−
Πt − 1 Yt .
2 pt−1 (i)

where wt ≡ Wt /Pt is the real wage. The retailers are owned by the representative household,
and hence discount future cash flow using the stochastic discount factor of the household. Pricing

maximizes the present value of expected profits

L = Et

∞
X

Mt,s {Πs (i) + µs (i)[Zs Ks (i)α Ns (i)1−α − Ys (i)] + ν s (i)[ps (i)−η Ys − Ys (i)]}

s=t

where ν s (i) and µs (i) are the shadow values of the demand constraint and technological constraints.
The efficiency conditions in a symmetric equilibrium (where all firms choose an identical relative
price) are:
wt = (1 − α)µt

rtK = αµt

Yt
,
Nt

Yt
,
Kt

ν t = 1 − µt

(11)

(12)

(13)

and

Yt+1
0 = 1 − ϕΠt (Πt − 1) − ην t + ϕEt Mt,t+1 Πt+1 (Πt+1 − 1)
.
Yt

3.4

(14)

Financial Intermediaries

This part of the model follows the setup in Gourio (2013). We assume that there exists a continuum
of financial intermediaries indexed by s ∈ [0, 1]. The financial intermediaries combine debt and
equity capital to invest in physical capital. From now on we omit the intermediary index.
If intermediary invests Qt Kt+1 at time t, then at time t + 1 its return on the asset will be
K
εt+1 Rt+1
= εt+1

K + (1 − δ)Q
rt+1
t+1
,
Qt

where εt+1 is an idiosyncratic risk associated with the intermediary. The shocks are iid across time
and producers, have a cdf H(·), and a pdf h(·). (In practice we assume that εt+1 follows a lognormal
destribution, log εt+1 ∼ N (−0.5σ 2 , σ 2 )).
The intermediary here can be thought of integrating a set of financially unconstrained borrowers
with a banking system. In a more complete set up where even borrowers are subject to financial
constraints, we could have richer financial accelerator mechanism that comes both from the bor12

rowers and the lenders. Here we collapse the actors together so that when the banks expand, they
directly create more physical capital (as in Gertler and Karadi (2011)).
The choice of debt vs. equity is driven by the standard trade-off model from corporate finance.
For now, we assume that debt is set in real terms12 and has a tax advantage χt > 1. This means that
for each unit of debt issued at time t, the corporation receives a subsidy equal to χt − 1 > 0. This
subsidy is a stand-in for many considerations that make debt issuance attractive. For instance, it is
commonly argued that the presence of debt is beneficial as it gives stronger incentives on managers
to maximize profits, and to avoid engaging in empire building. One can view χt as a shortcut for
such an “agency benefit” to debt. On the other hand, if there are no benefits to debt but simply
issuance costs, χt could be less than unity. Critical for our purpose is the assumption that χ varies
over time. One could think of χ varying because of unmodeled changes in the ease of placing debt
issues. The intermediary’s problem is to choose capital and debt (and hence equity) to maximize
its expected present discounted value.
We also assume that the issuance of equity is costly and that the cost per-unit of equity issuance
is an increasing function of the equity share relative to the size of the project:

St
,
γt = γ
Qt Kt+1

γ(0) = 0, γ 0 (·) ≥ 0 and γ 00 (·) ≥ 0

where St is equity issuance today.13 The maximization problem of the intermediary can then be
expressed as

max

Bt+1 ,St ,Qt Kt+1

Et [Mt,t+1 max (Vt+1 − Bt+1 , 0)] − 1 + γ

St
Qt Kt+1

St .

(15)

K Q K
where Vt+1 = εt+1 Rt+1
t t+1 is the value at time t+1. The maximization is subject to the funding

constraint:
Qt Kt+1 = χt qt Bt+1 + St ,
12

(16)

This is rather innocuous since our financial crises will not have deflation, so changing this assumption would not
materially affect the results.
13
This equation assumes that the producer only maximizes its one-period ahead value. It is easy to see that this
corresponds to maximizing its long-term value because the present value of rents is zero due to free entry.

where qt is the price of the bonds and the debt pricing equation is given by

qt = Et Mt,t+1

Vt+1
1Vt+1 <Bt+1 ζ
+ 1Vt+1 ≥Bt+1
Bt+1

(17)

1Vt+1 <Bt+1 is a dummy indicating default, and ζ is the recovery rate. The intermediary decides on
debt and capital, recognizing that higher leverage will lead to lower bond prices.
To derive the efficiency conditions of the problem, first, we rewrite the bond pricing function as
K
qt Bt+1 = Et Mt+1 [Ω(ε∗t+1 )ζRt+1
Qt Kt+1 + (1 − H(ε∗t+1 ))Bt+1 ],

where Ω(x) ≡

Rx
0

εdH(ε) = xh (x), and ε∗t+1 ≡

Bt+1
,
K Q K
Rt+1
t it+1

(18)

i.e., the default threshold.14 Substitut-

ing (16) and (18) into (15), we reexpress the objective function as

max

K
Et Mt+1 [(1 − (1 − χt ζ)Ω(ε∗t+1 )) − (1 − χt )(1 − H(ε∗t+1 ))ε∗t+1 ]Rt+1
Qt Kt+1

St
St
− Qt Kt+1 1 + γ
Qt Kt+1 Qt Kt+1

Bt+1 ,Kt+1

(19)

Dividing (18) by the size of the balance sheet Qt Kt+1 , we define

qt Bt+1
Bt+1
K
K
≡
= Et Mt+1 [Ω(ε∗t+1 )ζRt+1
+ (1 − H(ε∗t+1 ))ε∗t+1 Rt+1
].
L
Qt Kt+1
Qt Kt+1

(20)

Using (16) and (20), we transform the second line of (19) into an expression that does not depend
on the amount of equity:

St
St
Qt Kt+1 1 + γ
Qt Kt+1 Qt Kt+1

Bt+1
Bt+1
= Qt Kt+1 1 + γ 1 − χt L
1 − χt L
Qt Kt+1
Qt Kt+1

Bt+1
.
≡ Qt Kt+1 Γ
Qt Kt+1
Importantly, Γ(Bt+1 /Qt Kt+1 ) depends only on leverage and not separately on Qt Kt+1 . Hence,
14

D
Note that the recovery rate that appears in the household problem can be expressed as Rt+1
=

K
Ω(ε∗
t+1 )ζRt+1 Qt Kt+1
.
H(ε∗
t+1 )Bt+1

the FOC for capital can be expressed as

Bt+1
1=Γ
Qt Kt+1

−1

K
Et Mt+1 Rt+1
λt+1

(21)

where
λt+1 = 1 + (χt − 1) ε∗t+1 1 − H ε∗t+1

− (1 − ζχt ) Ω ε∗t+1 .

(22)

The efficient level of leverage is determined by

0 = Et Mt+1 (χt − 1)(1 − H(ε∗t+1 )) − (1 − χt ζ)ε∗t+1 h(ε∗t+1 ) − (χ − 1)ε∗t+1 h(ε∗t+1 ) − Γ0

Bt+1
.
Qt Kt+1

This expression can be shown equivalent to

χt − 1
St
St
Et Mt+1 1 − H ε∗t+1
+γ
+ γ0
χt
Qt Kt+1
Qt Kt+1

St
St
∗
∗
0
= (1 − ζ t ) Et Mt+1 εt+1 h εt+1 1 + γ
+γ
.
Qt Kt+1
Qt Kt+1

3.5

(23)

Financial Crises

We now describe how we introduce financial crises into the model. We assume that aggregate
technology Zt evolves over time as the sum of a standard AR(1) shock and a unit root process
affected by rare downward jumps:
Zt = Ztr Ztp ,

r
log Ztr = ρZ log Zt−1
+ σ Z eZ,t ,

p
Zt+1
= e−Xt+1 bc ,
Ztp

(24)

where Xt+1 is the “financial crisis” shock; specifically Xt+1 = 0 with probability 1 − pt an Xt+1 = 1
with probability pt . When a crisis occurs, the level of technology drops by bc percent. We assume
the following reduced-form law of motion for the probability of a crisis:
log pt = b0 + b1 log(Bt /Btf )

(25)

where Btf is the efficient level of credit that prevails in an economy without price distortions.
The reduced-form assumes that the probability of a crisis is an increasing function of the level of

inefficient credit. This framework directly implies an “externality” since higher debt increases the
probability of a crisis, which is not internalized by financial intermediaries.15 We refer to this as
“inefficient credit” and hence implicitly assume that the steady-state distortion that favors debt
(that, is the steady-state tax subsidy χ > 1) does not create a risk of financial crisis.
We also assume that the capital accumulation process is affected by the financial crisis in the
same fashion: financial intermediaries invest It and “expect” to obtain
Ktw = (1 − δ)Kt + It ,

but their capital stock that is realized at beginning of time t + 1 is actually
Kt+1 = Ktw e−Xt+1 bc .

That is, in the (unlikely) event of a financial crisis, the capital stock is not what the intermediaries
expected it to be. This amounts to assuming a “capital quality” shock that is perfectly correlated
with the productivity shock. This assumption is made largely for technical reasons in our case: it
allows using a simpler solution method as we explain below.
Finally we further assume that the utility function is affected by a crisis realization. We do
this because the preferences we use are not compatible with balanced growth, so that a one-time
decline in productivity may lead to a change in hours. For tractability, we assume that

U (Ct , Nt ) =

N 1+υ
Ct1−τ
− Jt1−τ t
1−τ
1+υ

where Jt is the cumulative disaster effect,
Jt = e−Xt bc Jt−1 .

We then redefine variables by detrending by Zt , e.g. Yet = Yt /Zt , etc. Under the assumptions above,
15

While this is a convenient short-cut, Cairo and Sim (2016) provides a structural model that delivers the same
prediction. In order to study the relationship between price stability and financial stability, Cairo and Sim (2016)
endogenize the production and income distribution in the financial crisis model of Kumhof, Ranciere, and Winant
(2015). Cairo and Sim (2016) also allows for nominal rigidities and labor market frictions. Cairo and Sim (2016)
shows that in this structural model of financial crisis, the correlation between debt and the probability of financial
crisis is as high as 0.92. This is one way to justify our reduced form specification for the crisis risk.

Table 1: Structural Parameters
Parameters
β
γ̄
α
τ
κ
δ
η
χ̄
ζ
υ
Π̄
ϕ
σ
pss
b1
bc
ρZ
ρχ
ρΞ
σZ
σχ
σΞ

Interpretation
Discount factor
Quadratic cost of equity issuance
Capital share
Constant relative risk aversion
Investment adjustment cost
Depreciation rate
Elasticity of substitution between goods
Steady state tax benefit
Recovery rate
Inverse of Frisch elasticity of labor supply
Gross inflation rate target
Price adjustment cost
Idiosyncratic volatility
Average probability of financial crisis
Sensitivity of log prob to credit deviation
Size of output drop if financial crisis
Persistence of the technology shock
Persistence of the financial shock
Persistence of the demand shock
Volatility of the technology shock
Volatility of the financial shock
Volatility of the demand shock

Value
0.99
0.167
0.36
2
5
0.025
2.0
1.005
0.50
1/3
1
130
0.2007
0.005
5
0.10
0.90
0.90
0.90
0.01
0.0097
0.0035

the system of equations of detrended variables does not depend on Xt . That is, the detrended
system has no jumps. This implies that it can be solved using standard perturbation techniques.
For the details of transforming the original system of equations into the detrended system, see the
appendix. Also see Gourio (2012), Isoré and Szczerbowicz (2015) and Gabaix (2011) for detailed
detrending methodology for this kind of model.16

Basic model properties

We first discuss the parameters used for our model, then illustrate the model dynamics using
impulse response analysis.

4.1

Calibration

Table 1 summarizes the calibration of the model parameters. We set the time discount factor
β = 0.99 simplying annual real rate of 4 percent. Capital share of production α is set equal to
16
Note that we also need to assume that financial crisis affects the investment goods production function so that
the producer’s adjustment cost is not affected by the disaster realization.

0.36 as is standard in the literature. The depreciation rate δ is calibrated equal to 0.025. We set
investment adjustment cost κ equal to 5 to produce reasonable investment volatility.
We assume risk aversion of 2 and an inverse Frisch elasticity of labor supply of 1/3. Regarding
the elasticity of substitution between goods, we choose η = 2, consistent with the results in Broda
and Weinstein (2006). With this choice, we set the price adjustment cost ϕ = 130 to match the
fact that micro studies suggest that prices adjust about once a year.
We set the “tax benefit” parameter to a relatively low value of χ̄ = 1.005, to take into account
that debt incurs issuance costs as well as tax benefits, as discussed above. We choose the bankruptcy
cost of default to be ζ = 0.5. This value allows us to match the recovery rate on U.S. corporate
bonds. Regarding the functional form of the equity issuance cost, we assume a quadratic form:

St
Qt Kt+1

= γ̄

St
Qt Kt+1

2
.

We next calibrate the two parameters σ and γ̄ (the volatility of idiosyncratic shocks to firms and
the equity issuance cost) to match a probability of default of 15% per year and average leverage of
around 0.65. This implies that σ = 0.2007 and γ̄ = 0.167.
Steady state probability of a financial crisis is set to 2% per year or 0.5% per quarter, corresponding to two crises per century. The size of the output drop is set to 10%. This number
is significantly smaller than the values typically used in the asset pricing literature on disasters.
This number is also smaller than the recent US experience, as discussed in the introduction. The
sensitivity of the financial crisis probability to excess credit is 5, so that a 20% increase in inefficient
credit doubles the probability of financial crisis. We study extensively the sensitivity of our results
to these parameters below.
Regarding the aggregate shock processes, we take an agnostic approach and set all the persistence parameters equal to 0.9. We calibrate the standard deviation of technology shock to equal to
0.01. We then choose the other shock volatilities so that the variance decomposition of output can
be allocated to technology shock, demand shock and financial shock with 42.5,42.5, and 15 shares,
respectively. The 15% share for financial shocks is at the lower end of the estimates implied by
Christiano, Ilut, Motto, and Rostagno (2010) and Fuentes-Albero (2014). We study the sensitivity
of the results to the importance of the financial shocks as well.

Figure 2: Impulse Response to Productivity Shock: Baseline
(a) Output (%)

0.5

(b) Investment (%)

0.4

1.5

0.3

0.2

0.5

0.1

-0.05

-0.1

-0.5
0

20
(d) Debt (%)

0.4

-0.15
0

(e) Prob. of Crisis (Ppt)

-0.01

0.35

(f) Policy Rate (Ppt)

0
-0.05

-0.02
0.3

-0.1
-0.03

0.25

-0.15

-0.04

0.2

-0.05
0

-0.2
0

Quarters

4.2

Quarters

Model Properties With a Standard Policy Rule

As a first step, we illustrate how our model economy behaves in response to the three fundamental
impulses that we consider - a productivity shock, an aggregate demand shock, and the financial
shock. As a further diagnostic we also report the effect of a monetary policy shock. To solve the
model, we assume a standard inertial Taylor (1999) rule:
Rt = 0.85 × Rt−1 + 0.15 × (R∗ + 1.5 × (π t − π ∗ ) + yet )

(26)

where yet is the output gap,17 and π t is the year-over-year inflation rate. We summarize the main
mechanisms in the model by explaining what happens to output, investment, inflation, debt, the
policy rate, and the probability of a crisis.
A productivity shock, shown in Figure 2, leads to higher output and lower inflation as is common
in New Keynesian models. The policy rule leads the central bank to cut the policy rate but not
sufficiently to stabilize inflation or to allow output to rise in line with potential. Put differently,
17

We define this gap to be the difference between the level of output and the one that would prevail in an economy
without nominal rigidities and without financial shocks.

Figure 3: Impulse Response to Demand Shock: Baseline
(a) Output (%)

(b) Investment (%)

0.5

-0.2

-0.4

-0.5

-0.1

-0.6

-1

-0.2

-0.8

-1.5

-0.3

-1

-2
0

20
(d) Debt (%)

-0.4
0

(e) Prob. of Crisis (Ppt)

-0.05

-0.2

-1

-0.1

-0.4

-0.15
0

Quarters

(f) Policy Rate (Ppt)

-0.5

-1.5

0.1

-0.6
0

Quarters

lower inflation reflects the decline in current and future marginal costs that arise from higher
productivity and the fact that monetary policy does not bring demand in line with this higher
supply.
The output surge leads to higher borrowing to finance investment, but because output does not
keep up with the growth in potential, credit actually rises less than in the frictionless benchmark. As
a result, the annualized probability of crisis falls modestly (by 4 basis points, so that the probability
drops from 2% per year to 1.96%).
The response to a negative demand shock is shown in Figure 3. A negative demand shock leads
to lower output and inflation; the shock also leads to a lower policy rate, but the assumed policy
rule does not respond enough to offset completely the effects of the shock. Lower output in turn
leads to lower debt and lower risk of financial crisis. Since the shock does not affect the flexible
economy and the credit thereof, probability reduction due to deleveraging is sizable in this case.
Next, in Figure 4, we show the effect of a financial shock, which reflects an inefficient shock to
credit supply. This type of shock leads to a large expansion of credit which reduces the user cost of
capital and leads to a boom in investment and, to a lesser extent, also in output. The lower user

Figure 4: Impulse Response to Financial Shock: Baseline
(a) Output (%)

0.3

(b) Investment (%)

0.2

0.1

-0.1

-0.2

-0.1

-0.2

-1
0

20
(d) Debt (%)

-0.3
0

0.4

0.3

0.2

(e) Prob. of Crisis (Ppt)

(f) Policy Rate (Ppt)

-0.02
-0.04
-0.06
-0.08

0.1

-0.1

0
0

Quarters

-0.12
0

Quarters

cost feeds through to lower inflation. The spike in debt (that is permitted with this policy rule)
significantly increases the risk of financial crisis, from 2% per year to 2.37% per year.
Finally we illustrate how a “monetary shock” affects this model economy. Although we are
most interested in optimal monetary policy rules, showing the impact of a deviation from the rule
is informative about certain aspects of the model. Figure 5 displays the responses of our main
variables to a 100 basis point (1%) increase in the policy rate. One important takeaway from the
figure is that the shock leads to a decline in output and inflation.18 The output drop leads to a
decline of credit and hence the probability of crisis.
An important conclusion from this exercise is that the sensitivity of the risk of crisis to an
increase in the policy rate is by no means extreme - this fairly large monetary shock only generates
on impact a reduction of 8 basis points in the annual probability of crisis, i.e. moving it from 2%
to 1.92%. This is magnitude of the change is consistent with the empirical estimates reviewed by
IMF (2015). We share the view of IMF (2015) that these estimates are quite uncertain, but it
18

Our model does not generate hump-shapes in response to this shock because it lacks some of the propagation
mechanisms introduced by Christiano Eichenbaum and Evans (2005) or Smets and Wouters (2007) such as inflation
indexation or consumer habits. We believe this is not critical for our results.

Figure 5: Impulse Response to Monetary Policy Shock: Baseline
(a) Output (%)

0.2

(b) Investment (%)

0.1
0

0.05
0

-0.1
-0.2

-0.05
-0.2

-0.4

-0.1

-0.3

-0.6

-0.4
0

(d) Debt (%)

-0.15
0

-0.2

-0.02

-0.4

-0.04

(e) Prob. of Crisis (Ppt)

(f) Policy Rate (Ppt)

1
0.8
0.6
0.4

-0.6

-0.06

-0.8

0.2

-0.08
0

Quarters

0
0

Quarters

is important to note that our subsequent conclusions about the desirability of leaning against the
wind are not driven by a presumption that monetary policy has powerful effects on the risk of a
crisis.

Optimal simple rules

Having established the basic model properties, we consider policy rules that specify the interest
rate as a function of last period’s interest rate, inflation, the output gap and/or the “credit gap”,
that is, Bt /Btf , the deviation of credit from the level that would prevail with only productivity
shocks and flexible prices. Previous research shows that such rules typically perform well in models
like ours. Because real time measurement of the output and credit gaps is difficult, we also study
rules that rely on imperfectly measured version of these variables, namely deviations of output and
credit from their steady state values.19 Our goal is to establish the conditions when responding
to credit may be beneficial. The benchmark for comparisons is the welfare of a representative
consumer. This consumer cares not only about the usual fluctuations in output and inflation, but
19

See, among many others, Orphanides and Williams (2002) and Edge and Meisenzahl (2011).

also about risks that bring large persistent drops in output and consumption. As we will see, in
some configurations of the model, the central bank finds it preferable to respond to credit gap
rather than the output gap, even though this leads to higher output and inflation volatility.
Our main result is that leaning against the wind can be beneficial provided that three conditions
are met: (1) financial crises have important output effects; (2) financial shocks are important, i.e.
the variance of the financial shocks and the associated swing in inefficient credit are large enough,
and (3) financial crises are endogenous, i.e. they are caused in part by inefficient credit. In
contrast, if there are no financial shocks, even with other financial imperfections present, we obtain
the standard result that stabilizing inflation is a sufficient condition for maximizing welfare. In
this latter case, a simple Taylor rule that puts enough weight on the output gap can maximize
welfare.20 If there are financial shocks, but financial crises are exogenous, a simple rule that puts
weight on the output gap still outperforms credit-based rules, because targeting the output gap is
a more direct way to eliminate undesirable fluctuations in output and inflation.
Obviously, these results depend on parameter choices. For instance, it is clear that if financial
crises have small effects, or the variance of financial shocks is small, responding to output may still
be preferable to responding to credit. In the results that follow we have calibrated the financial
shocks so that they account for 15% of the variance of output, and demand and productivity shocks
equally account for the remainder (i.e., 42.5% each). We discuss some robustness exercises after
we introduce our main findings. However, because we have not estimated the model, we view these
results as being indicative rather than dispositive. Put differently, rather than giving a definitive
answer to the question of whether leaning against the wind is desirable, we think our framework is
useful precisely because it permits us to understand, within a fairly standard DSGE model, which
parameters and model features govern whether responding to credit conditions is beneficial.

5.1

Methodology

We consider policy rules of the following form:
Rt = ρRt−1 + (1 − ρ)(R∗ + φπ (π t − π ∗ ) + φy ỹt + φb bt )
20

This result is sometimes called “divine coincidence”. The same outcome can be achieved by maximizing the
inflation coefficient. Of course, in the presence of price markup shocks, this result breaks down.

Table 2: Benchmark Model
Welfare
Consumption equivalent (%)
Coefficient φy
Coefficient φb
400×SD(Π)
100×SD(Y )
400×E(P)
400×SD(P)

Output gap only
-143.35
0
100
–
1.45
2.20
2.06
0.83

Credit gap only
-143.15
0.177
–
1.90
2.41
4.57
1.98
0.29

Both gaps
-143.14
0.185
80.09
100.0
2.36
4.30
1.99
0.29

where π t is again the year-over-year inflation rate, yet is the output gap and bt is the credit gap, i.e.

log Bt /Btf . Note that bt is the variable which determines the probability of a financial crisis, as
given by equation (25). Throughout this exercise we set ρ = 0.85 and φπ = 1.5. Our motivation
for imposing these restrictions is to make analysis transparent, and to require that the policy rule
resembles the kind that broadly describes actual central bank decisions. We then consider the
welfare consequences of policy rules with different coefficients for φy or φb . Specifically, we rank
rules according to the utility they provide to the representative consumer and find the value of φy
and/or φb that maximizes this expected utility.21,22 We first consider the simple case where the
central bank responds to only one gap so that φb = 0 or φy = 0. We then discuss results when we
optimize over φb and φy jointly.

5.2

Main Result

Table 2 summarizes our main finding. When we select the best rule that depends solely on a
correctly measured output gap, the optimal sensitivity is very high,23 around 100, so that monetary
policy eliminates all inefficient fluctuations of output. As can be seen, this monetary policy rule
generates also a relatively small volatility of inflation. The standard deviation of the probability
21
In contrast, many papers maximize a quadratic loss function of inflation and unemployment. In our case this
approach would not capture the cost of financial crises, which permanently lower productivity. It is also a priori
attractive to use a micro-founded welfare criterion.
22
In practice, we first rewrite the system of equations that determines the equilibrium around the stochastic
trend induced by disaster. This system can then be solved using standard perturbation methods since it has no
jumps. We then use a second-order approximation of the utility to obtain conditional welfare, that is the utility
obtained by the agent if the state variables are at their nonstochastic steady-state values. The result with unconditional welfare (the average utility in the new steady-state) are quite similar however. See appendix for details, and
https://sites.google.com/site/fgourio/ for the code used to solve the paper.
23
We set an upper bound of 100, and a lower bound of 0, to ensure that the optimization problem is well-posed.
Allowing for values higher than 100 does not materially alter the results.

of crises is 0.83 percent. Crises occur about 2.06 percent of the time, which differs from 2 percent
owing to Jensen’s inequality.24 As a result, households face the risk that crises can be more frequent
than that. When we select the best rule that depends solely on the correctly measured credit gap,
we obtain a coefficient of 1.90 on the credit gap. This rule generates significantly greater volatility
of output and inflation than the one based on the output gap.25 Yet, the credit-gap based rule
outperforms the output-gap based rule in terms of welfare. The difference in utility is equivalent
to a permanent increase of consumption of 0.18%, a significant number. For comparison, if one
were to follow Lucas (1987) and compute the welfare gain of exogenously removing all business
cycle volatility of consumption, the benefits amount to 0.058%.26 In contrast, the same Lucas-style
calculation yields a benefit of 5.50% of exogenously removing all disasters.27
In all of the comparisons that follow, we report the consumption equivalent change between a
rule based only on the output gap and those that depend on the credit gap or both gaps; by this
convention, the consumption equivalent for the rule that focusing on output gap only is always
zero.
The gain in welfare occurs because the LAW policy is sacrificing cyclical volatility in order to
limit the financial crisis risk: the probability of a financial crisis is now both smaller and substantially less volatile. The reduction in the mean probability of crisis is driven, in part, by the
functional form we use to insure that the crisis probability lies between zero and one.28 While
this effect may seem at first mechanical, it reflects the reality that the financial crisis probability is
bounded below (by zero). As such, decreasing the volatility of financial crisis leads to lower mean
because the mean is driven by the occasional upswings.
Figures 6, 7 and 8 depict the response of macroeconomic aggregates to the three fundamental
shocks under the standard Taylor (the solid blue line), the rule that responds only to the output gap
(the dotted green line), and the rule that responds only to the credit gap (the dashed red line). Our
24
The level of crisis probability is given by pt = exp[b0 + (Bt /Btf )b1 ]. As a result, the average value of crisis
probability is affected by the volatility of excess credit.
25
The output volatility measure does not take into account financial crises.
26
This calculation is based on the benefits of removing consumption volatility starting from the standard Taylor
(1999) rule.
27
The benefit obtained from reducing the disaster probability exogenously from 2.06ppt to 1.98ppt is 0.23%. Hence
our result is in line with the Lucas calculation. We obtain smaller gains because our gains come at a cost of higher
business cycle volatility. Moreover, our model incorporates other costs of volatility, including inflation and labor.
And, as discussed in Lester, Pries, and Sims (2014), there may be gains from higher volatility as well.
28
We specify a process for the log of the probability and that implies that lower volatility also brings a lower mean.

Figure 6: Impulse Response to Financial Shock: Optimal Simple Rule
(a) Output (%)

(b) Investment (%)

0
-0.5

-0.2

-0.4

-0.6

-1

-0.8

-1
-1.5

-1

-2
10

(d) Debt (%)

(e) Prob. of Crisis (Ppt)

(f) Policy Rate (Ppt)

0.5

0.3
0

0.2

0.1

-0.5

0
10

Quarters

-1
10

Quarters

Note: Solid blue, doted green and dashed red lines indicate the cases of the baseline
monetary policy rule, eq. (26), optimized output-gap rule and optimized creditgap rule (LAW), respectively.

main conclusion is best understood by comparing what the different rules imply for policy in the
aftermath of a financial shock in figure 6. The credit-gap rule tightens policy, which leads to a much
lower debt expansion and consequently a lower risk of a crisis. The cost of this policy is large in
terms of the deviation of inflation and output from target. This policy, nevertheless, delivers higher
welfare because it meaningfully lowers the probability of a financial crisis. In contrast, the output
gap based policy cuts interest rates because inflation is low and lower rates help keep output close
to its target. The cost of this choice is a rise in the financial crisis risk. Finally, following a standard
Taylor rule leads the central bank to gradually cut rates as it trades off inflation undershooting
against a modest output boom. In this case, debt also accumulates so that the crisis probability
rises even more substantially.
Figures 7 and 8 show the performance of the different rules in face of demand and productivity
shocks. Another cost of the credit-gap policy rule is that it does less well than the output-gap
based rule in response to standard demand and productivity shocks. While the output-gap based
policy offsets completely the demand shock and accommodates almost perfectly the productivity
26

Figure 7: Impulse Response to Demand Shock: Optimal Simple Rule
(a) Output (%)

(b) Investment (%)

0
0

0
-0.2

-0.1

-0.5
-0.4
-1

-0.2

-0.6
-1.5

-0.3

-0.8
10

(d) Debt (%)

(e) Prob. of Crisis (Ppt)

(f) Policy Rate (Ppt)

-0.02
-0.04
-0.5

-0.5

-0.06
-0.08
-1

-0.1

-1

-0.12
-1.5
10

Quarters

Note: Solid blue, doted green and dashed red lines indicate the cases of the baseline
monetary policy rule, eq. (26), optimized output-gap rule and optimized creditgap rule (LAW), respectively.

shock, the credit-gap based policy responds less aggressively to both of these shocks. The relatively
passive response implied by the optimized credit rule for these shocks is because if it were more
aggressive in these cases, it would also be even more responsive to financial shocks: i.e., a higher
coefficient on the credit gap would help in responding to these shocks, but would exaggerate even
more the output and inflation deviations in response to the financial shock.

5.3

Understanding the results

To confirm the interpretation that we have offered for the main findings, it is instructive to shutdown
various features of the model to see how doing so changes the results. A particularly helpful
experiment is to turn off the financial shocks (i.e. set σ χ = 0) and make the financial crises
exogenous events (e.g. b1 = 0). The environment then amounts to a standard New Keynesian
model that includes a debt-equity tradeoff in capital structure and exogenous crises. The main
findings are summarized in Table 3. In this environment, the optimal policy is one that responds
enough to either the output or credit gap, and essentially perfectly stabilizes inflation. After
27

Figure 8: Impulse Response to Technology Shock: Optimal Simple Rule
(b) Investment (%)

(a) Output (%)

0.6

0.4

1.5

-0.02

-0.04
-0.06

0.5

0.2

-0.08
0

-0.1

0
10

(d) Debt (%)

(e) Prob. of Crisis (Ppt)

(f) Policy Rate (Ppt)

0.6
-0.1

-0.01
0.4

-0.2

-0.02

-0.3

-0.03

0.2

-0.4
-0.04
0

-0.5
10

Quarters

Note: Solid blue, doted green and dashed red lines indicate the cases of the baseline
monetary policy rule, eq. (26), optimized output-gap rule and optimized creditgap rule (LAW), respectively.

a demand shock, monetary policy offsets the shock to fully stabilize output and inflation. On
the other hand, when a productivity shock occurs, the policy keeps inflation on target and lets
output respond fully to the shock. This result is standard in New Keynesian models - the divine
coincidence property (Blanchard and Gali (2007)) applies and so there is no trade-off between
output and inflation volatility, and this optimal policy can be (approximately) implemented by
either simple rule provided they are sufficiently aggressive.29
To further build intuition, we now relax the assumption of exogenous financial crises. The
main results are reported in Table 4. The findings are nearly identical to the prior case. This
is because there is no reason to offset the credit fluctuations driven by the productivity shock,
which are efficient and do not contribute to financial risk. As for demand shocks, the credit
fluctuations they create are actually eliminated once output volatility is eliminated. Hence, there
is no trade-off between credit stabilization and output/inflation stabilization, and the same policies
29

Note that there is no intrinsic reason as to why one simple rule should perform better than the other in terms
of welfare in this case; indeed the welfare difference we find is extremely small, about 0.2 basis point. Also note that
output and inflation volatility as well as mean and standard deviation of financial crisis probability are quite close.

Table 3: No Financial Shocks, Exogenous Financial Crises
Welfare
Consumption equivalent (%)
Coefficient φy
Coefficient φb
400×SD(Π)
100×SD(Y )
400×E(P)
400×SD(P)

Output gap only
-142.98
0
100
–
0.01
2.20
2
0

Credit gap only
-142.99
-0.002
–
96.89
0.01
2.19
2
0

Both gaps
-142.98
0
100
0
0.01
2.20
2
0

Table 4: No Financial Shocks, Endogenous Financial Crises
Welfare
Consumption equivalent (%)
Coefficient φy
Coefficient φb
400×SD(Π)
100×SD(Y )
400×E(P)
400×SD(P)

Output gap only
-142.98
0.00
100
–
0.01
2.19
2.00
0.01

Credit gap only
-142.98
-0.00
–
97.28
0.01
1.66
2.00
0.01

Both gaps
-142.98
0.00
100
100
0.01
2.20
2.00
0.01

as in the previous case can implement an efficient allocation without creating any inefficient credit
movements.
As a third point of comparison, we now reintroduce financial shocks, though unlike in the
benchmark we suppose that crises are exogenous. In this version of the model the output gap rule
does slightly better than LAW. The main findings are summarized in Table 5. The novelty compared
to the previous cases is that the response to the output gap is diminished. This occurs because
the response that would be required to offset demand and productivity shocks is not consistent
with the response needed to respond to the financial shock. But the credit gap rule suffers from
the same issue and has to trade off the response against the different shocks.30 Overall, the LAW
policy underperforms because the volatility it induces by stabilizing credit shocks does not lower
the crisis risk. However, combining the credit and output gap allows a slightly better outcome.
30
In some environments where the financial frictions are sufficiently severe, LAW can dominate an output gap rule
even if financial crises are exogenous. For such an example, see Kiley and Sim (2017). In our model, this result also
seems to be possible.

Table 5: Financial shocks, Exogenous Financial Crises
Welfare
Consumption equivalent (%)
Coefficient φy
Coefficient φb
400×SD(Π)
100×SD(Y )
400×E(P)
400×SD(P)

Output gap only
-143.08
0
4.11
–
1.28
2.21
2
0

Credit gap only
-143.14
-0.050
–
0.47
1.97
3.20
2
0

Both gaps
-143.07
0.017
2.69
0.47
1.60
2.32
2
0

Table 6: Effect of Financial Crisis Size on Optimal Credit Policy
Financial crisis size ( bc )
Optimal coeff. on credit φb
Consumption equivalent (%)
SD(Y ) under LAW
SD(Π) under LAW
Mean(P) under LAW
Mean(P) under output gap rule
SD(P) under LAW
SD(P) under output gap rule

5.4

1.20
0.06
4.06
2.29
1.993
2.058
0.39
0.82

1.58
0.115
4.37
2.37
1.987
2.059
0.32
0.83

10%
(benchmark)
1.90
0.177
4.57
2.41
1.980
2.060
0.29
0.83

12%

14%

2.14
0.247
4.69
2.44
1.983
2.060
0.26
0.83

2.32
0.324
4.78
2.46
1.982
2.061
0.25
0.83

When is leaning against the wind optimal?

We next ask how certain parameters affect the desirability of leaning against the wind. For simplicity, in these comparisons we focus here on rules that depend either only on the (correctly measured)
output gap or credit gap.

5.4.1

The cost of financial crises

Our benchmark model assumes that a financial crisis leads to a permanent decline in the level of
GDP of 10%. Table 6 illustrates how our results change as we vary this cost from 6% to 14% with
all other parameters kept constant. Several points emerge. First, the welfare benefit of targeting the
credit gap rather than the output gap increases monotonically with the size of the financial crisis.
Second, the bigger is the crisis, the stronger is the response to credit, with the coefficient rising
from 1.20 to 2.32. Third, the volatility of inflation and output rise modestly as the responsiveness
to credit rises, though the probability of the crisis is hardly moving across the different scenarios.

Table 7: Effect of Sensitivity of Crisis to Excess Credit on Optimal Policies
Sensitivity of crisis to excess credit ( b1 )
Optimal coeff. on credit φb
Consumption equivalent (%)
SD(Y ) under LAW
SD(Π) under LAW
Mean(P) under LAW
Mean(P) under output gap rule
SD(P) under LAW
SD(P) under output gap rule
5.4.2

0.42
-0.05
3.15
1.94
1.987
1.987
0.27
0.34

1.13
0.04
4.00
2.28
1.989
2.019
0.32
0.66

5
(benchmark)
1.90
0.17
4.57
2.41
1.984
2.060
0.29
0.83

2.69
0.37
4.89
2.49
1.982
2.115
0.27
0.99

4.06
0.91
5.22
2.55
1.979
2.271
0.26
1.33

The sensitivity of crises to excess credit

Another key parameter for our results is b1 , which measures how much excess credit affects the
likelihood of financial crises. A large value of b1 means that excess credit has a strong effect on the
risk of crisis and hence on welfare. This naturally gives rise to a stronger motive to lean against
excess credit. Table 7 confirms this intuition. First, for low values of b1 , LAW is outperformed
by the output gap rule. Second, the optimal coefficient on credit rises with b1 . This change in the
coefficient partially offsets the increase in the volatility of financial crisis probability that would
otherwise occur mechanically. Third, this policy is chosen despite a clear cost in terms of higher
output and inflation volatility.

5.4.3

The importance of financial shocks

Perhaps most basically, the magnitude of the (inefficient) financial shocks is critical for our results.
We already illustrated that if there are no financial shocks, leaning against the wind brings no
benefits relative to standard policies. Table 8 provides more details on the importance of this
consideration. Here too, we see that the welfare difference between the best credit gap policy and
the best output gap policy is increasing in the variance of financial shocks. The effects on output
and inflation volatility as well as the financial crisis probability are more subtle because they result
both from (i) the higher variance of financial shocks and (ii) the change in policy rule in response to
this higher variance. Nevertheless, when the financial shocks are more important, the LAW policy
delivers more volatility for output and inflation than the output gap rule and a lower probability
of a crisis.

Table 8: Effect of Standard Deviation of Financial Shocks on Optimal Policy
Standard dev. of financial shocks
(relative to benchmark)
Optimal coeff. on credit φb
Consumption equivalent (%)
SD(Y ) under LAW
SD(Π) under LAW
Mean(P) under LAW
Mean(P) under output gap rule
SD(P) under LAW
SD(P) under output gap rule

33%

66%

4.97
0.01
2.71
0.87
1.999
2.007
0.09
0.28

2.35
0.07
3.58
1.65
1.992
2.027
0.19
0.55

100%
(benchmark)
1.90
0.18
4.57
2.41
1.980
2.060
0.29
0.83

133%

166%

1.74
0.32
5.63
3.18
1.974
2.106
0.38
1.01

1.67
0.51
6.75
4.96
1.962
2.165
0.48
1.38

Table 9: Risk Aversion and Leaning Against the Wind
CRRA
Optimal coeff. on credit φb
Consumption equivalent (%)
SD(Y ) under LAW
SD(Π) under LAW
Mean(P) under LAW
Mean(P) under output gap rule
SD(P) under LAW
SD(P) under output gap rule
5.4.4

0.5

1.5

0.74
-0.30
7.81
2.60
1.92
2.084
0.49
0.89

1.35
0.12
4.98
2.41
1.98
2.066
0.36
0.84

2
(benchmark)
1.90
0.18
4.57
2.41
1.98
2.060
0.29
0.83

3.23
0.27
4.00
2.40
1.99
2.049
0.20
0.80

4.76
0.35
3.63
2.39
1.99
2.041
0.15
0.78

The role of risk aversion

We next explore how the willingness of households to bear macroeconomic risk affects our results.
On one side, higher risk aversion makes agents more fearful of financial crises. On the other hand,
higher risk aversion also makes agents less willing to tolerate the higher business cycle volatility
implied by LAW. Moreover, with our assumed preferences, a higher risk aversion implies a lower
elasticity of substitution, which affects the response of the economy to monetary policy (as well as
the dynamics of the model more generally). Table 9 reveals that the first effect seems to dominate
- the higher the risk aversion, the larger the benefits from leaning against the wind. With a risk
aversion of 0.5, an output-gap rule outperforms a credit-gap rule, but the benefits of using the
credit gap rule rise with risk aversion. The optimal policy largely stabilizes fluctuations in financial
crisis risk.
Figure 9 summarizes many of the central findings of the paper. On the horizontal axis, we
vary the size of the financial crisis. On the vertical axis we vary risk aversion. The lines that are
drawn trace out isoquants in units of equivalent consumption between the best LAW policy (that
32

Figure 9: Welfare Gain from LAW: the Effects of Risk Aversion and the Size of Financial Crisis
4
1

0.5

0.2

3.5

0.2

2.5
0

Risk Aversion

1.5

0.2

0
0

0.5
2

Financial crisis size (%)
responds only to the credit gap and inflation) and the best monetary policy rule that responds
only to the output gap (and inflation).31 The zero consumption equivalence curve traces out all
the combinations of the size of the crisis and the representative household’s level of risk aversion
where the two policies deliver equivalent welfare. Points to the right and above the zero curve show
the regions where LAW delivers higher welfare and below and to the left show combinations where
the output gap rule performs better. The outcome for the benchmark model, described in Table 2
with risk aversion of two and a crisis that brings a permanent ten percent output loss, is indicated
by the red dot. The results from Table 6 described how welfare varied when we fixed risk aversion
at two and varied the size of the crisis. This figure fills in the rest of the parameter space. Not
surprisingly, as risk aversion rises, LAW’s relative performance improves. For most combinations,
LAW is advantageous. However, if risk aversion is lower, say one, then a crisis that drops output
by 10 percent is not enough to justify a LAW policy.
31

Each policy is optimized with respect to the coefficient on the credit gap or output gap, as in the exercises above.

Figure 10: Tradeoff between Financial Stability and Traditional Mandates

5.5

Trading off financial stability vs. macroeconomic stability

Our model results demonstrate a significant trade-off between the traditional mandates of monetary
policy – output and inflation stability – and financial stability - stabilizing, and if possible reducing
the probability of financial crisis.
To illustrate this trade-off, we present a “policy frontier” in Figure 10. The policy frontier
depicts the range of outcomes that can be implemented by a LAW policy. The frontier is obtained
by solving the model for many possible values of φb . In particular, in the left panel of Figure 10, we
change the LAW coefficient from 0 to 100 to see what happens to the mean probability of financial
crises (on the vertical axis) and the standard deviation of the inflation rate (on the horizontal axis).
In the right panel, we show the relationship between the mean probability of financial crisis and
the stability of economic activity as measured by the standard deviation of output.32 In a standard
New Keynesian DSGE model, it is common to represent the policy frontier as the pairs of volatility
of output and inflation that can be obtained. Here, we show how these measures vary with the
average probability of a financial crisis.
The two panels indicate that for low values of the LAW coefficient, there can be a region
where the central bank can improve upon both financial stability and traditional monetary policy
objectives. This is possible because a rule that sets interest rates based only on inflation is suboptimal and putting a little weight on the credit gap unambiguously improves outcomes. The
32

The figure is nearly identical when one uses the output gap instead of output.

panels also show that after a certain point, the central bank can reduce the probability of financial
crises only by sacrificing the traditional mandates. The LAW coefficient where the tradeoff begins
differs for output and inflation. This is not surprising since the distortionary effects of responding
to financial shocks differs for inflation and output. Nonetheless, the cost of a crisis is large enough
that utility is maximized (at φb = 1.90) by driving down the probability of a crisis even though
doing so substantially raises inflation and output volatility. This choice reflects the improvement in
the distribution of outcomes due to the lower risk of crisis. The welfare maximization challenge is
to balance these gains against the losses from the increased volatility of the economy. The analysis
suggests that this is an important potential consideration that is often omitted from stabilization
discussions.

5.6

Mismeasurement

An important practical consideration is that neither the output gap nor the credit gap is actually
observable. In practice efficient movements in credit cannot easily be separated from inefficient
ones. In our model, inefficient movements come from demand or financial shocks while efficient
ones come from technology shocks. In reality deregulation, changes in property rights and many
other factors could also lead to a benign surge in credit and a central bank would need to be able
to separate those swings from the inefficient ones.
To quantify this, we search again for the best policy rules in our baseline specification where the
central bank is restricted to just observing actual output and credit - that is, it uses the deviation
from the steady-state rather than the deviation from the efficient benchmark. The results are
shown in Table 10 (and these should be compared to the findings in Table 2 ). The mismeasured
output gap rule now eliminates all fluctuations in output, including the efficient ones. This leads
to higher inflation volatility and noticeably lower welfare. Relying on the mismeasured credit gap
still leads to a similar tradeoff as in the baseline model. The central bank delivers less frequent and
less volatile crises, in exchange for higher inflation and output volatility. The relative performance
of the rule based on mismeasured credit is bigger in this scenario than in the one with both gaps
are perfectly measured. In fact, the best rule when both gaps are considered puts almost no weight
on the output gap (and the welfare is about the same as when only the credit gap is used).
The welfare level is actually slightly higher when the mismeasured credit gap is used instead
35

Table 10: Optimal Policy Rules with Mismeasured Gaps
Welfare
Consumption equivalent (%)
Coefficient φy
Coefficient φb
400×SD(Π)
100×SD(Y )
400×E(P)
400×SD(P)

Output gap only
-143.54
-0.17
100
0
1.77
0.09
2.10
0.90

Credit gap only
-143.25
0.09
0.00
1.61
2.50
3.98
2.01
0.42

Both gaps
-143.25
0.09
0.00
1.61
2.50
3.98
2.01
0.42

of the perfectly measured output gap; this conclusion depends on all the foregoing factors that
have been shown to determine the relative attractiveness of leaning against the wind. For instance,
in parameter configurations where the gains from leaning against the wind are low to begin with,
then tying the policy rate to mismeasured credit gap would not necessarily lead to higher welfare
than a rule that can be set based on a perfectly measured output gap. In these cases, however, the
mismeasured credit gap rule would still outperform the mismeasured output gap rule. One caveat
is that in our simple model, most of the credit variation is inefficient, so that the mismeasurement
problem is not very significant.

5.7

Robustness

A concise way to summarize the findings from our baseline specification is as follows. The advantage of LAW depends on whether financial shocks are responsible for some non-trivial amount of
economic fluctuations and whether they influence the probability of a financial crisis that delivers a
long-lived slowdown. When these shocks are not important for macro outcomes, then LAW delivers
similar outcomes to a conventional Taylor style monetary policy rule. When they are present, but
they do not influence the probability of a crisis, a policy rule that reacts to them delivers worse
outcomes. This deterioration comes because off-setting the credit shocks worsens inflation and
output stabilization without any corresponding benefit. The case when LAW is preferred arises
because dampening the financial fluctuations reduces the probability of a crisis enough to overcome
the higher level of output and inflation variability. The success of LAW in this case depends neither on perfectly observing how much of the credit fluctuations are inefficient nor on agents being
extremely risk averse.

Our results all assume that the monetary policy rule has a fixed coefficient on inflation. If
instead that coefficient is also optimized, the same broad conclusions hold; that is, a monetary rule
that responds to inflation and credit gap optimally outperforms one that responds to inflation and
output gap.
These basic conclusions are present in several other variants of the model where we altered the
nature of the financial fluctuations and the way that they influence a crisis. One set of alternatives
that we considered is the possibility that a crisis depends on the level of debt relative to either output
or capital (rather than the level of debt itself), again compared to the efficient level of the ratio. That
kind of a specification is closer to some of the empirical literature that uses scaled debt movements
to predict crises. When we change the model in this direction, and recalibrate appropriately, our
main conclusions are unaffected. One issue that arises in the alternative specifications is that
leaning against the wind now depends on both movements in the numerator and the denominator
of the financial variable. This complicates the interpretation of what one would expect to some
shocks that move the numerator and denominator in different directions or at different horizons.
That is the main reason for presenting the specification that we featured.
We also experimented with more extreme variants of the financial frictions. As mentioned
earlier, we think of χ as standing in for fluctuations in firms’ ability to place debt. Our baseline
calibration supposes that there are benefits to debt finance, say through tax subsides or reductions
in agency problems with managers that prevent the squandering of funds. The quantitative results
are, however, very similar if we suppose that each dollar of debt issued returns less than a dollar
in available funds. This would be the case if we assumed that χ was standing in for the floatation
costs of issuing debt and these costs were fluctuating.

Conclusion

Conventional discussions about the links between monetary policy and financial stability typically
start by saying that one can appeal to different tools for different jobs. Macro-prudential regulation
can address stability concerns, while monetary policy can attend to managing inflation. We agree
that this would be the ideal arrangement, however, in practice in many countries this is easier said
than done. Macro-prudential policymaking is in its infancy and for some countries the tools barely

exist. These practical concerns motivate our analysis.
On the question of whether central banks should alter monetary policy to contain financial
stability risks IMF staff study (IMF (2015)) says “Based on our current knowledge, and in present
circumstances, the answer is generally no.” We believe this conclusion is premature.
The model we have presented is highly stylized and the parameters are not estimated. Nonetheless, we believe it does capture the ingredients that many of the advocates, and opponents, of leaning
against the wind accept as important. In particular, the model presumes that financial crises are
very costly, and are partly driven by credit conditions which monetary policy can affect, but at
the cost of missing on its traditional inflation and output objectives. The model can easily uncover
circumstances where leaning against the wind is welfare improving.
The model points to a number of factors that will determine the efficacy of leaning against the
wind. Our main hesitations in endorsing the conclusion of IMF staff study (IMF (2015)) are that
many of these factors are difficult to measure and that existing empirical work still do not provide
much guidance about how to calibrate certain of these key elasticities. Perhaps subsequent work
will confirm the IMF conclusion but for now we believe it is too early to say that the question is
settled.
One powerful conclusion from the model is that the case for leaning against the wind likely rests
on accepting higher volatility of inflation and output, in exchange for reducing the risk of crises. If
central banks are going to embrace this policy, they will need to invest substantially in explaining
this tradeoff to the public and to legislatures.

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Appendices
The system has 31 variables:

ν t λK
t
 N t wt
Yt = 
 qt B t
Γt


µt Ct Lt Πt lt Ωt Mt−1,t Λt
Ξt Rt Qt Kt It Yt
RtK
StK 

ε∗t χt Ktw rtK Zt ht
zt∗
Ht 

The corresponding system of equations are:
ν t = 1 − µt

0 = 1 − ϕΠt (Πt − 1) − ην t + ϕEt
Mt−1,t = β

(.27)

Yt+1
Mt,t+1 Πt+1 (Πt+1 − 1)
Yt

Λt
Λt−1

(.28)
(.29)

Λt = Ct −τ

(.30)

Λt wt = Ntυ

(.31)

Rt
1 = Et Mt,t+1 Ξt
Πt+1

(.32)

Ktw = (1 − δ)Kt + It

(.33)

Kt+1 = eXt+1 b Ktw

(.34)

Ct + It = Yt −

2
ψ
Πt − Π Yt
2

(.35)

Yt
Nt

(.36)

wt = (1 − α)µt
rtK = αµt

Yt
Kt

Yt = Zt Ktα Nt1−α
RtK =

(1 − δ) Qt + rtK
Qt−1

StK = Qt Ktw − χt qt Bt+1
ε∗t =

Bt
K
Rt Qt−1 Kt
42

(.37)
(.38)

(.39)
(.40)
(.41)

qt = Et

zt∗ = σ −1 (log ε∗t + 0.5σ 2 )

(.42)

Ht = Φ(zt∗ )

(.43)

ht = φ(zt∗ )

(.44)

Ωt = Φ(zt∗ − σ t )

(.45)

ζ K
∗
∗
Mt+1 1 − H(εt+1 ) +
R Qt Kt+1 Ω(εt+1 )
Bt t+1
K
Γt = Et Mt+1 Rt+1
λK
t+1

(.46)
(.47)

∗
∗
∗
λK
t = 1 + (χt − 1) εt (1 − H (εt )) − (1 − ζχt ) Ω (εt )

(.48)

Γt = 1 + γ (1 − χt L (lt )) (1 − χt L (lt ))

(.49)

K
K
L (lt ) = Et Mt+1 [Ω(ε∗t+1 )ζRt+1
+ (1 − H(ε∗t+1 ))ε∗t+1 Rt+1
]

(.50)

χt − 1
StK
StK
0
Et Mt+1 1 − H
+γ
+γ
χt
Qt Kt+1
Qt Kt+1

St
StK
∗
∗
0
= (1 − ζ) Et Mt+1 εt+1 h εt+1 1 + γ
+γ
Qt Kt+1
Qt Kt+1

ε∗t+1

Qt = 1 + κ

It
It−1

(

− 1 − Et
lt =

κ
Mt,t+1
2

It+1
It

(.51)

−1

Bt
Qt Ktw

(.52)
(.53)

log Rt = (1 − ρR ) log Rt−1 + ρR [log R + ρΠ log(Πt /Π̄) + ρY log(Yt /YtF )]

(.54)

log χt = (1 − ρχ ) log χ + ρχ log χt−1 + σ χ χ,t

(.55)

Zt+1 = eXt+1 b eξt+1 Zt

(.56)

log Ξt = (1 − ρΞ ) log Ξ + ρΞ log Ξt−1 + σ Ξ Ξ,t

(.57)

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