Download targeting financial stability: macroprudential or monetary policy?

targeting financial stability:
macroprudential or monetary policy?
David Aikman†
Julia Giese‡
Sujit Kapadia§
∗
Michael McLeay¶
October 2016
PLEASE DO NOT QUOTE OR DISTRIBUTE WITHOUT PERMISSION
Abstract
This paper examines monetary and macroprudential policy interaction in a calibrated
two-period new-Keynesian model. The model incorporates the possibility of a credit
boom leading to a future financial crisis, and a loss function reflecting financial stability
considerations. Deploying the countercyclical capital buffer (CCyB) improves outcomes
relative to when the interest rate is the only policy tool. The instruments are typically
strategic substitutes, with monetary policy loosening when the CCyB tightens, though
the CCyB should be activated later if monetary policy is constrained at the zero lower
bound. We also examine when the instruments are strategic complements and assess
how market-based finance and a risk-taking channel of monetary policy affect our
results.
JEL CLASSIFICATION: E52, E58, G01, G28.
KEY WORDS: macroprudential policy; monetary policy; financial stability; countercyclical capital buffer; financial crises; credit.
∗ The
views expressed in this paper are those of the authors and not necessarily those of the Bank of
England. Thanks [to be added].
† Financial Stability Strategy and Risk, Bank of England ([email protected]).
‡ Monetary Analysis, Bank of England ([email protected]).
§ Research Hub, Bank of England ([email protected]).
¶ Monetary Analysis, Bank of England ([email protected]).
1.
Introduction
The global financial crisis highlighted deficiencies in macro-financial policy toolkits. Monetary policy focused on price stability; prudential regulation was mostly concerned with
practices at individual firms. Some central banks had a mandate to consider the stability of
the financial system as a whole, but few had tools to ensure the resilience of the system. The
events of 2007-08 demonstrated starkly how price stability was no guarantee of financial
stability.
Some had argued before the crisis that monetary policy should be used to tackle financial
imbalances (Borio and Lowe, 2002 [6]; White, 2006 [33]) while others argued that the costs
outweighed the uncertain benefits (Bernanke and Gertler, 2001 [4]; Kohn, 2006 [25]; Posen,
2006 [28]). Since the crisis, macroprudential instruments have come to the fore as an
important addition to the toolkit of many central banks or regulatory authorities. These are
mainly prudential tools set with the stability of the whole financial sector in mind. Some
tools seek to address cross-sectional risks while others are concerned with time-varying
risks (Aikman, Haldane and Nelson, 2015 [1]). The countercyclical capital buffer, designed
under Basel III, is an example of the latter.
Time-varying macroprudential policies share similarities with monetary policy. Both
look to identify cycles - the financial cycle in the case of macroprudential policies and the
business cycle in the case of monetary policy - and seek to tighten the calibration of tools
in the upswing. While macroprudential policy makers are primarily concerned with the
resilience of the financial system, they should not achieve this at any cost and also need to
take into account the effects of policies on growth and employment. Similarly, monetary
policy seeks to keep inflation close to target, but not necessarily at the expense of output
and employment. They also work through similar channels: monetary policy changes
the price of credit through interest rates and has effects on asset prices, among others.
And macroprudential policy may affect credit provision, either through directly affecting
quantities or by changing banks’ funding costs and thereby lending spreads.
1
Policymakers hold different beliefs over the relative effectiveness of the tools in relation
to financial stability. Some argue that only monetary policy is likely to be effective across
the whole financial system because interest rates ’get in all the cracks’ (Stein, 2013) [31].
Others argue for a combination of policies: Shin (2015) [30] notes that ’both monetary policy
and macroprudential policies have some effect in constraining credit growth and the two
tend to be complements’ And yet others question whether monetary policy should have an
explicit financial stability objective at all since small changes in monetary policy may not
be effective in reducing the crisis probability and ensuring resilience, and monetary policy
may not be powerful enough to clean up after a crisis (eg Kohn, 2015) [26]. According
to some, if anything monetary policy should ’lean with the wind’ rather than against it
(Svensson, 2015) [32].
In this paper, we explore the implications of giving the policymaker an additional tool
to monetary policy, a countercyclical capital buffer (CCyB). Monetary policy can focus
on what it is most effective at, ie keeping inflation stable and the output gap closed.
And macroprudential policy can be used to reduce the probability of a crisis and thereby
increasing resilience, allowing for a separation of objectives and staying true to the Tinbergen
principle of one objective, one tool. But the use of the CCyB is not costless: it affects both
aggregate demand and potential supply, triggering endogenous reactions of monetary
policy. The balance of the two effects determines how optimal monetary policy should
respond and allows us to study cases where the policies are strategic substitutes and
strategic complements, ie they move in the opposite or same direction, respectively.
We use a similar but extended framework to Ajello et al (2015) [2]. They study the
effectiveness of monetary policy in a simple semi-structural new-Keynesian model with a
twist: in addition to an IS and Phillips curve, they have a crisis probability and an equation
governing credit growth. The higher credit growth, the higher the crisis probability, and the
policymaker minimises a two-period loss function, where the second-period can either be a
crisis or a non-crisis outcome. In our set up, the IS and Phillips curve also contain credit
2
spreads which in turn are affected by the setting of the CCyB and the crisis probability
decreases the higher the CCyB. Moreover, the loss function contains an additional term that
reflects the policymakers risk tolerance with regard to crises. While the private sector is
aware that a financial crisis may occur in the second period, it does not take the effects of
its own behaviour on the crisis probability into account in its decisions in the first period.
In the paper, we explore different scenarios around the interaction of monetary and
macroprudential policies. Following Ajello et al (2015), we use a 2 period version of the
model in order to focus on the key trade-offs underlying the optimal policy decision.1
We might typically expect a higher CCyB to reduce demand and have limited effects
on supply. In this case, monetary policy should loosen to offset the effects of the CCyB
on output growth, and the policies are strategic substitutes. This also holds if monetary
policy is constrained by the zero lower bound, although it may then be optimal to wait until
financial stability risks are somewhat more elevated before activating the CCyB, because
monetary policy cannot offset any negative aggregate demand effects.
The policies are strategic complements in two cases: If the CCyB has larger negative
effects on supply than on demand (or indeed the effect on demand is positive) or if the
CCyB becomes less effective, for example if there are leakages of credit to unregulated
sectors, it may be optimal to lean against the wind with monetary policy for financial
stability purposes.
This paper is organised as follows. Section 2 introduces the model, Section 3 discusses
the baseline optimal policy results and Section 4 explores extensions.
2.
Macroprudential policy in a New Keynesian model
In this section, we introduce a parsimonious model for understanding the interaction of
monetary and macroprudential policies. Our model develops Ajello et al (2015): a two1 Svensson
(2015) [32] points out in the context of Ajello et al (2015)’s model that the two-period assumption
is not innocuous, however. We plan to explore how optimal policy changes in an infitnite horizon version of
the model in future work.
3
period New Keynesian model, which features the possibility for a crisis in period 2. A
crisis is an event that causes a discontinuous drop in output, whose likelihood is positively
related to the degree of leverage built up in the economy in period 1.
2.1.
Benchmark model
Aggregate demand and supply are determined by a variant of the canonical two-equation
New Keynesian model, featuring a forward-looking IS curve and Phillips curve.2
ps
ps
y
y1 = E1 y2 − σ(i1 − E1 π2 + ωs1 ) + ξ 1
ps
π1 = E1 π2 + κy1 + νs1 + ξ 1π
(1)
(2)
where y1 is the gap between output from its efficient level, π1 is the deviation of inflation
from target, i1 is the deviation of the central bank nominal interest rate from its steady state
level and s1 , as discussed below, is the deviation of the credit spread from its steady state
y
level. ξ 1 is a demand or consumption preference shock, while ξ 1π is a cost-push or mark-up
shock.
These equations contain two departures from the canoncial model.
First, following Curdia and Woodford (2010) [14], we introduce a role for fluctuations in
credit spreads from their steady state level, s1 , in driving macroeconomic equilibria. Credit
spread shocks in this structure are isomorphic to ’cost push’ shocks in that they introduce a
trade-off between stabilising real activity and inflation.
Higher credit spreads push down on aggregate demand via the IS equation (1) by
increasing the interest rates facing households and firms wishing to borrow for consumption
and investment. The sensitivity of demand to credit spreads may differ from the sensitivity
2 See
Woodford (2003) [34], Galı́ (2008) [17], for example.
4
to the policy rate and ω 6= 1.3 Higher credit spreads also affect aggregate supply. They
increase firms’ marginal costs and act as an endogenous cost-push shock, reducing the
economy’s productive capacity.4
Second, we assume that the private sector’s expectations of period 2 outcomes, denoted
ps
E1 , are myopic and do not take into account the true model of how financial crises occur
(Gennaioli, Schleifer and Vishny, 2015) [18]. In particular, risks are neglected such that
expectations are taken to be a weighted average of crisis and non-crisis states, where the
weights are exogenous and do not respond when financial system vulnerabilities build.
ps
E1 π2 = eπ2,c + (1 − e)π2,nc
ps
E1 y2 = ey2,c + (1 − e)y2,nc
(3)
(4)
where e is an exogenous parameter reflecting the probability agents attach to a financial
crisis occurring between period 1 and period 2, and the subscripts c and nc denote outcomes
in the crisis and non-crisis states respectively.
The financial side of our model consists of a law of motion for real credit growth, a
macroprudential transmission mechanism, and an equation specifiying the probability of a
financial crisis.
B1 = φ0 + φi i1 + φs s1 + ξ 1B
3 There
(5)
are two opposing effects that could lead to this parameter being smaller or larger than 1. All else
equal, the fact that only the subset of agents who are borrowers will be affected by changes in credit spreads
would lead to a smaller effect than changes in risk-free rates, which affect everyone. But other factors could
lead to a larger effect: credit constrained borrowers may have a higher sensitivity to changes in rates than
savers.
4 This channel typically arises in a variety of ways in models of financial frictions. In models featuring
only households, such as Curdia and Woodford (2010), it may come from the effect of higher credit spreads
in distorting labour supply decisions. Or spreads could increase the cost of firms’ working capital, as in
Carlstrom, Fuerst and Paustian (2010) [11]. In models with capital accumulation and binding credit constraints
facing firms, such as Gertler and Karadi (2011) [19], higher credit spreads lead to capital shallowing, reducing
labour productivity.
5
s1 = ψk1
γ1 =
exp(h0 + h1 B1 + h2 k1 )
1 + exp(h0 + h1 B1 + h2 k1 )
(6)
(7)
where B1 is defined as 3 year cumulative real credit growth, k1 is the setting of the CCyB in
period 1, and γ1 is the probability of a financial crisis occurring between periods 1 and 2.
Real credit growth depends (negatively - φs , φi < 0) on interest rates and credit spreads
over the period. A constant term, φ0 , captures the steady state rate of credit growth, while
ξ 1B is a shock to the quantity of credit, which may be correlated with the demand and
y
supply shocks ξ 1 and ξ 1π .
The CCyB influences the likelihood of a financial crisis via two channels. First, all else
equal, activating the CCyB directly reduces the probability of a bad outcome (h2 < 0 in (7)).
This reflects the resilience benefits of higher bank capital, which increases the loss absorbing
capacity of the banking system for a given distribution of shocks, and could also be viewed
as affecting the cost of a crisis. This channel is consistent with the empirical evidence that
banks that entered the crisis with more capital had higher survival probabilities (Berger
and Bouwman, 2013) [3] and were better able to maintain their supply of loans during the
crisis (Carlson, Shan and Warusawitharana, 2013) [10]. It is also consistent with theories
that emphasise the role of bank capital in incentivising banks to monitor their loans over
the cycle (Holmstrom and Tirole, 1997) [22]. We choose to model this channel as part of the
crisis probability. An alternative would be to endogenise the cost of a crisis by modelling it
separately from the probability of a crisis and making it dependent on the level of capital
in the economy. The two options are qualitatively equivalent for our purposes.
Second, activating the CCyB forces banks to rely on a more equity rich funding structure,
which, in turn, pushes up their overall cost of capital. Banks, we assume, are able to pass on
some of these increased costs, resulting in higher credit spreads for real economy borrowers
and hence lower credit demand in the first period. We capture this relationship in equation
6
(6). The CCyB can influence the risk environment facing banks through this ’leaning’
channel: activating the CCyB tempers a built up credit growth and hence reduces the
likelihood of a financial crisis. Note also that the CCyB will also have knock-on effects on
both aggregate demand and supply via this channel, via the IS curve (1) and Phillips Curve
(2).
In the baseline version of our model, monetary policy can also be used to influence the
likelihood of a financial crisis by leaning against the build-up risks associated with credit
growth. Higher interest rates reduce credit growth by lowering credit demand and hence
influence the risk environment facing the banking system. Credit supply, captured by (6), is
independent of monetary policy in our baseline model. In Section 4, we consider relaxing
that assumption. We also explore extensions of the model that emphasise further qualitative
differences in the transmission mechanisms of the two tools. For instance, the scope of
application of the CCyB is much narrower than monetary policy in that it is applied only
to banks.
2.2.
Loss function
We focus in the main on the case of a single policymaker that sets monetary and macroprudential policy jointly to minimise an overall loss function. This allows us to analyse the
normative question of how policy should be set to meet given objectives, free of institutional constraints. It closely approximates the institutional set-up of many central banks
where monetary and financial policy committees have overlapping objectives, overlapping
membership and common information sets.
We assign our policymaker a simple loss function that captures the mandates of central
banks with responsibility for monetary and financial stability policies. The loss function is
an augmented version of that typically used in the monetary policy literature. Overall loss
7
is the sum of period 1 loss and discounted period 2 loss.
L = L1 + βL2
(8)
Period 1 loss features the central bank’s traditional monetary policy objectives, which
typically involve the dual mandate of an inflation target plus a concern for output or
employment volatility. In our formulation, the policymaker seeks to minimise the quadratic
loss resulting from deviations of inflation and output from their respective targets. λ
denotes the weight assigned to output deviations relative to those in inflation.
L1 =
1
[(π12 + λy21 )
2
(9)
This is augmented with endogenous and exogenous financial stability concerns in period
2, where loss is given by:
L2 =
1
2
2
[(1 − γ1 )(π2,nc
+ λy22,nc ) + γ1 (1 + ζ )(π2,c
+ λy22,c )]
2
(10)
The first two terms capture outcomes if there is no crisis, and are similar to the period 1
objectives of the central bank. We capture the central bank’s financial stability concerns via
the second two terms, which capture the output and inflation costs of a crisis event. Since
these costs are larger than those in a non-crisis state, the policymaker has an endogenous
motivation to minimise γ1 .
The final terms are also increasing in ζ, an extra exogenous weight placed on avoiding
the costs associated with a financial crisis. When ζ > 0, the policymaker attempts to
minimise the probability of a financial crisis to a greater degree than quadratic expected
losses alone would imply is optimal. In effect, the presence of the financial stability
objective skews the central banks’ objectives towards crisis prevention. The greater is ζ, the
more the central bank’s objectives are skewed in the direction of crisis avoidance relative
to symmetrically stabilising inflation and output. This formulation of financial stability
8
objectives is novel and qualitatively distinct from simply increasing the weight attached to
future output gaps .
There are two compelling justifications for augmenting the standard loss function in
this way. First, in the aftermath of the financial crisis many central banks are mandated
to achieve financial stability goals by elected governments on behalf of the public. For
example, the Bank of England’s Financial Policy Committee has a mandate to ’take action to
remove or reduce systemic risks’ with a view to ’protect[ing] and enhance[ing] the resilience
of the UKs financial system’. Such financial stability mandates in part reflect a desire by
taxpayers to avoid bearing the bailout costs of future systemic crisis. This idea is explored
in Peek, Rosengren and Tootell (2015) [27], who posit a quadratic financial stability term,
arguing that too much financial stability is undesirable since there are costs in reducing
credit availability to an unacceptably low level. We model the cost of achieving financial
stability directly through its effect on potential output, which provides an endogenous
motivation for avoiding ’too much’ financial stability. Financial stability mandates may
also reflect a preference to avoid the distributional effects of financial crises, which may be
difficult to offset with other policy tools.
A second justification for ζ > 0 arises if there are uncertain welfare costs of crises.
Robust control considerations would suggest avoiding the worst case outcome the output
losses associated with a severe crisis in period 2 rather than targeting expected period
2 output. Crises may also generate additional costs that are not captured in our simple
two-period framework. We show in the appendix, for instance, that if there are hysteresis
effects associated with financial crises such that depressed output persists into future
periods (Blanchard, Cerutti and Summers, 2015) [5], the policymaker optimally places a
higher weight on avoiding crises in the first place.
The policy problem facing the central bank therefore is to choose an interest rate and
CCyB rate in period 1 to minimise the loss function (8) subject to the IS curve (1), the
Phillips curve (2), the dynamics of credit (5) and the crisis probability function (7), taking
9
Table 1: Benchmark parameter values
Parameter Description
Standard Parameters
β
Discount Factor
σ
Interest-rate sensitivity of ouptut
κ
Slope of the Phillips Curve
λ
Weight on output stabilisation
i∗
Long-run natural nominal rate of interest
Effect of the CCyB
ψ
Effect of the CCyB on credit spreads
ω
Effect of spreads relative to policy rate on y
ν
Effect of spreads on the Phillips Curve
Financial conditions equation parameters
φ0
Average real credit growth
φi
Coefficient on interest rates
φs
Coefficient on spreads
Crisis probability equation parameters
h0
Constant
h1
Coefficient on leverage vairable
h2
Coefficient on k1 , (resilience effect of CCyB)
e
Private sector perception of crisis probability
Period 2 parameters
y2,c
Deviation of output from efficient in crisis state
Parameter
Notes
0.99
0.57
1.03
0.05
3%
Matches r*=1%
Burgess et al (2013)
Burgess et al (2013)
Standard welfare-based
Rachel and Smith (2015)
0.2
1
0.41
1pp equity = 20bps - MAG (2010)
Cloyne et al (2015), updated
Franklin, Rostom and Thwaites (2015)
0.21
-1.4
-6.1
Historical average
Cloyne et al (2015), updated
Cloyne et al (2015), updated
-1.7 + 0.11h2
5.18
-27.8
0.0005
All estimated using
dataset constructed in
Bush et al (2015)
Arbritarily small
-0.032
3.2% lost output per year
Brooke et al (2015)
No effect
Steady state
Steady state
Risk-neutral policy
π2,c
y2,nc
π2,nc
ζ
Shocks
y
SD (ξ 1 )
Deviation of inflation from target in crisis state
Deviation of output from efficient in non-crisis state
Deviation of inflation from target in non-crisis state
Extra weight on E(crisis cost)
0
0
0
0
Standard deviation of demand shocks
0.0125
SD (ξ 1π )
Standard deviation of cost-push shocks
0.0011
SD (ξ 1B )
Standard deviation of credit shocks
0.16
Similar to risk premium shock in
Burgess et al (2013)
Similar to mark-up shocks in
Burgess et al (2013)
Set to match historical data
ps
as given private sector expectations, E1 . The appendix contains a formal solution to the
central bank’s problem.
2.3.
Parameter choices
We calibrate the model using the parameters in Table 1. Our approach involved estimating
equation (7) for the probability of a crisis and its determinants on a cross-country dataset.
For our other parameters, governing the effect of our two policy tools on credit, aggregate
demand and supply, we used empirical estimates from the literature for the UK, taken
from a variety of econometric models and estimated DSGE models already in use for policy
analysis at the Bank of England.
We interpret the time periods in our model as 3 years to capture the notion that financial
10
system vulnerabilities build up in response to the prolonged credit booms that appear to
be a feature of the data (Aikman, Haldane and Nelson, 2015 [1]; Borio, Drehmann and
Tsatsaronis, 2012 [15]). And macroprudential policy tools such as the CCyB have longer
implementation lags than monetary policy, such that a longer timeframe captures the
interaction between the tools in a more meaningful way.
For the CCyB, a change in the requirement in period 1 can be interepreted straightforwardly as a change to a new ratio, announced and implemented and reaching its full
effectiveness all within the same three-year period. For monetary policy, there are range of
different policy changes that could be implemented over a three-year horizon. The specific
experiment we consider in our calibration exercise is a change in the policy rate in the
first quarter of period 1, held constant for three years. In the various models we use, we
implement this using a series of unanticipated monetary policy shocks.
The model’s standard macroeconomic parameters, which determine the effect of monetary policy on demand and supply, are calibrated using the Bank of England’s main DSGE
forecasting model: COMPASS, set out in Burgess et al, 2013 [9]. Rather than reading across
directly from the Euler equation and Phillips Curves in that much larger model, we match
σ and κ, respectively the interest rate sensitivity of output and the slope of the Phillips
Curve, to the reduced form estimates from a series of monetary policy shocks in Burgess et
al. For ease of comparison with other papers, all variables are meaured as average annual
values. So σ = 0.57 implies that an increase in the annual policy rate of 1 percentage point,
for a period of 3 years, reduces the level of GDP by 0.57%. While κ = 1.03 implies that a
fall in GDP by 1% over the 3-year period reduces annual inflation by an average of 1.03pp.
The parameters governing the probability of a financial crisis are estimated using the
cross-country panel dataset in Bush et al (forthcoming). Figure 1 illustrates the relationship
between credit growth, the CCyB and the likelihood of a crisis. Evaluated at a CCyB
of zero and the sample average rate of credit growth of 21%, the estimated coefficients
imply that an increase in real credit growth of 1pp per year to 24% would increase the
11
annual probability of a crisis by around 0.4 percentage points. In addition, the ratio of the
coefficients
h2
h1
implies that an increase in the CCyB of 1pp is sufficient to offset the impact
on the crisis probability of just over 5% real credit growth over 3 years, or 1.8% per year.
Figure 2 shows the (in-sample) performance of the estimated equation at predicting the
probability of a crisis in the UK. The line peaks in 2008 with an estimated annual crisis
probability of 12%.
Turning to the transmission mechanism of the CCyB, we set φs , the sensitivity of credit
growth to spreads, equal to -6.1, based on Cloyne et al [13]. This is over 4 times as large
as the effect of interest rates on credit growth taken from the same paper. A 1 percentage
point increase in the CCyB is set to increase spreads by 20 basis points. This number is at
the top end of the estimates reported in the Macroeconomic Assessment Group (2010) [21]
meta study, conducted by the Financial Stability Board and Basel Committee on Banking
Supervision to quantify the potential impact of increasing bank capital requirements on
lending and growth. Combined with the effect of spreads on credit growth, it implies
that a 1pp increase in the CCyB reduces three year credit growth by 1.2 percentage points,
similar to the median estimate reported in the Macroeconomic Assessment Group study
(a reduction of 1.4% after 18 quarters). We set the sensitivity of output to spreads equal
to that on interest rates, ω = 1, broadly consistent with Cloyne et al (2015). Together,
these estimates imply that even ignoring its resilience benefits, the CCyB has an absolute
advantage in dampening credit growth for a given cost to aggregate demand.
We calibrate the parameter governing the impact of spreads on inflation, ν, jointly with
φs and κ, using the estimates reported in Franklin, Rostom and Thwaites (2015) [16]. This
paper finds that each percentage point fall in corporate lending in the recession reduced
UK labour productivity by 0.25%. Combining this with the effect of the CCyB on lending
described above gives its effect on potential supply. Our calibration implies a relatively
large impact of the CCyB on potential supply . It may be an overestimate if one believes
that some of fall in UK labour productivity in recent years has been driven by the scale and
12
nature of the financial crisis rather than being a more general phenomenon caused by lower
corporate lending per se.
In the event of a crisis, output contracts by an average of around 3% per year, in line
with the post-resolution estimates from Brooke et al (2015) [7]. Inflation, however, remains at
target in the crisis state, reflecting the experience internationally in the years following the
2008 crisis (Gilchrist et al, 2015) [20]. As we abstract from further shocks, the policymaker is
able to achieve zero output and inflation gaps in period 2 if a crisis does not occur.
Finally, the standard deviations of the demand and inflation shocks are calibrated close
to the estimated standard deviations of similar shocks in Burgess et al (2013) [9] - a risk
premium shock for demand, and a combination of two mark-up shocks for inflation. The
standard deviation of the credit shock is set to loosely match the standard deviation of
credit in the data.
3.
Results
In this section, we describe our results. We first examine the monetary policy-only case
and show that welfare is higher with active use of the CCyB. We then discuss trade-offs for
monetary and macroprudential policies and explore the interaction between monetary and
macroprudential policy further under different assumptions.
3.1.
Benchmark case: monetary policy-only economy
We begin by examining the benchmark monetary policy-only economy, where the CCyB is
switched off and spreads are exogenous. In this case, the first-order condition governing
the central bank’s interest rate policy in period 1 can be shown to be (see appendix):
σ(κπ1 + λy1 ) =
where
∂γ1
∂i1
∂γ1 ∂L
∂i1 ∂γ1
(11)
is the derivative of the crisis probability with respect to interest rates, and
13
∂L
∂γ1
2
2 + λy2 ), the policymaker’s expected discounted
= − β(π2,nc
+ λy22,nc ) + β(1 + ζ )(π2,c
2,c
loss from a crisis, is a function of the severity of the crisis state and the skewness of the
central bank’s objective function.
This condition is an augmented version of the standard description of optimal monetary
policy, λy1 = −κπ1 (Clarida, Gali, Gertler, 1999) [12]. Intuitively, it collapses to the standard
condition when either the probability of a financial crisis γ1 is insensitive to credit growth,
or when credit growth is insensitive to interest rates, φi = 0. Outside of these special cases,
monetary policy must balance two trade-offs. First, the standard intra-temporal trade-off
between stabilising inflation and output in period 1 in the presence of cost-push shocks.
Second, the inter-temporal trade-off between stabilising output (and inflation) in period 1
and stabilising the probability of a financial crisis, which would depress output in period 2.
Figure 3 presents the intertemporal trade off for monetary policy alone when faced
by a credit shock of different sizes.5 Each point on the trade-off curves is the solution
to the optimal policy problem for a different value of ζ. Points to the northeast of each
line are strictly inferior, while points to the southwest are infeasable. The y-axis shows
the period 1 loss from using monetary policy to counter credit growth if not required for
monetary stability: raising interest rates results in an undershoot of the period 1 inflation
and output targets for certain. The x-axis shows the financial stability benefit of doing so:
lower credit growth reduces the crisis probability, so increases period 2 output in expectation.
The stronger is credit growth, the further this intertemporal trade-off shifts out, and the
more willing the central bank should be to raise interest rates. Period 1 welfare loss is scaled
to be a proportion of the period 2 loss, conditional on a crisis occurring. This means that a
policymaker with symmetric objectives, ζ = 0, has indifference curves with slope -1, through
points (1,1), (2,2), etc. Skewing the policymaker’s objectives towards avoiding financial
crises by increasing ζ has the effect of increasing the slope of the indifference curves,
leading to tighter policy, lower inflation, output and crisis probability. But interestingly,
this quantitative difference is the only effect of a positive ζ, the qualitative nature of the
5 We
assume the shock is independent of any cost-push and demand shocks, which are set to zero.
14
intertemporal trade-off is unchanged to that under symmetric objectives.
3.2.
Two instruments: adding macroprudential policy
With one instrument, balancing monetary and financial stability objectives requires higher
interest rates than if monetary policy was focused exclusively on inflation and output
stabilisation. The stronger is credit growth, the greater the conflict between monetary
and financial stability objectives and the larger the relative losses under the inactive
macroprudential regime. By contrast, under an active macroprudential regime, the CCyB
can be directed at stabilising the crisis probability by matching the resilience of the banking
system to the risk environment it faces, while interest rates can focus on inflation and output
stabilisation. This, of course, is Tinbergen’s famous result: that to achieve n independent
targets there must be at least n effective instruments.
If using the CCyB were costless, there would be perfect separation of instruments. The
green circle in figure 4 shows that in this case, (and with unbounded instruments), then the
first-best outcome of no crises could be achieved. This could be achieved with ν = 0 by
setting capital requirements to an (infinitely) high level. Even if ω > 0, the negative effects
of high capital ratios on demand could be perfectly offset by monetary policy, absent the
zero lower bound on interest rates.
Away from that special case, while losses are unambiguously lower under the active
macroprudential regime, there is no free lunch. This is shown in the blue lines in figure
4. Although much improved relative to monetary policy only, a trade-off continues to
exist between monetary and financial stability objectives. This is because actively using the
macroprudential tool makes intermediation more costly, affecting the supply side of the
economy. This has the side-effect of worsening the traditional monetary policy trade-off
between output and inflation.
To examine these trade-offs in more detail, we first define the natural rates of output
15
and interest as the levels of each consistent with inflation being at target.6 Setting all
expectations terms and the cost-push shock to zero in (2), setting y1 = y1n , and substituting
in (6) gives the the natural level of output, as a deviation from its steady state.
y1n ≡ −
νψ
k1
κ
(12)
Substituting (12) for y1 and (6) for s1 into (1), and again setting expectations to 0 gives
the natural rate of interest:
r1n ≡ (
νψ
y
− ωψ)k1 + σ−1 ξ 1
κσ
(13)
Evidently, the natural rate can rise or fall when the CCyB increases. If the supply effects of
the CCyB are large, the first term will dominate, and the natural rate will be increasing in
the CCyB. Similarly, if increasing the CCyB has only a small negative effect on demand,
then the natural rate will increase. Given that the CCyB introduces a monetary policy
trade-off, tracking the natural rate is not optimal policy, however. But it provides a useful
benchmark to compare optimal policy to below.
For optimal policy, the policymaker sets period 1 interest rates and the CCyB to minimise
the Loss Function (8) subject to the constraints imposed by the private-sector equilibrium
conditions. As we show in the appendix, with two instruments, this gives one first order
condition for each policy instrument, each trading off output, inflation and the crisis
probability. But with two condtions and these three target variables, we show that we can
also derive conditions giving independent trade-offs between any two of target variables.
Any two of these trade-offs give the optimality conditions that completely characterise
optimal policy.
For ease of comparability with the monetary policy only case, we show the optimality
condition governing the intra-temporal monetary trade-off between output and inflation in
6 In
the standard New Keynesian model, the natural rates of output and interest would equate to the levels
of each that would prevail were prices and wages perfectly flexible.
16
period 1. We then discuss the second inter-temporal monetary-financial stability trade-off
between period 1 output and the probability of a crisis in period 2.
Intra-temporal trade-off
The optimal split between output and inflation in our model has a lower weight on output
than the policymaker’s actual weight λ.
λy1 + κπ1 = 0
where λ ≡
h1 ψ(φs −φi ω )+h2
λ
ν
h1 ψ(φs −φi (ω − κσ
)+h2
(14)
< λ.
The intuition underlying λ can be seen by defining χ ≡
∂γ1
∂k
( ∂γ1
1
∂i1
∂y1
∂i1
∂y1
∂k1
− 1) as the comparative
advantage of macroprudential policy at affecting the crisis probability, versus monetary
policy at affecting the IS curve/aggregate demand. This allows us to express λ as λ =
∂y
χ ∂k 1
1
∂yn
1 + χ ∂y1
∂k1
∂k1
λ.
When the instruments become orthogonal, either because interest rates do not affect the
∂y
1
1
crisis probability ( ∂γ
∂i → 0), or the CCyB does not affect aggregate demand ( ∂k → 0), then
1
1
the comparative advantage tends to infinity (χ → ∞). In that case, or if the cost of using
∂yn
the CCyB tends to 0 ( ∂k1 → 0), then (λ → λ) and the optimal monetary policy prescription
1
is the same as in the textbook model. When either the tools have an infinite comparative
advantage, or when macroprudential policy is costless, there is no advantage to monetary
policy adjusting to try to affect the crisis probability. But when using the CCyB is costly,
then at the margin there will be some benefit in using interest rates to take on some of the
responsibility.
Adding the CCyB as a second instrument makes the optimal inflation-output choice
independent of the expected crisis loss, unlike (11), the monetary policy only case, where
both output and inflation are lower if the crisis probability is higher. The financial stability
motive does still affect the equilibrium inflation and output outcomes in two distinct ways,
17
however. First, is that with the CCyB as a an active policy instrument, it imparts an
endogenous cost-push type effect, so that even in the absence of any exogenous cost-push
shocks, it will be impossible to close both output and inflation gaps at once.
Second, is that the effective weight on output stabilisation in (14) is lower than the
weight in the loss function (8). This is because the CCyB imparts positive cost-push effects
that push up on inflation and down on output, so placing a lower weight on output means
setting a higher interest rate for a given level of the CCyB. This effect is present because
the CCyB is costly. Similarly to the monetary policy only case, it is always optimal at
the margin for monetary policy to be somewhat tighter than with no financial stability
concerns, in order for the CCyB to be somewhat lower. The difference is that with two
instruments, tighter monetary policy always means bringing inflation closer to target
Inter-temporal trade-off
The policymaker’s optimality condition trading off period 1 potential output with the
probability of a financial crisis in period 2 is given by:
λy1 (−
νψ
∂γ
∂L
∂γ νψ
) = ( 1 + 1(
− ωψ))(−
)
κ
∂k1
∂i1 κσ
∂γ1
(15)
The intution behind the equation can be seen using the natural rate definitions above:
λy1
∂y1n
∂γ
∂γ ∂r n
∂L
= ( 1 + 1 1 )(−
)
∂k1
∂k1
∂i1 ∂k1
∂γ1
(16)
The optimality condition sets the policymaker’s marginal cost of increasing the CCyB/credit
spreads in terms of lower potential output today, on the LHS, equal to the marginal gain
in terms of a lower crisis probability, multiplied by the perceived cost of a crisis,
∂L
∂γ1 .
The
first term in brackets is the lower risk of a systemic crisis with a higher CCyB, while
the second term is the additional effect from changing interest rates in response to any
macroprudential policy induced change in the natural rate of interest. If the demand
effect of macroprudential policy dominates, this second term will be negative, so the gain
18
from increasing spreads will be somewhat offset by looser monetary policy, for unchanged
inflation.
Policy settings
Figure 5 shows, under our benchmark calibration, the optimal policy settings required to
implement the two conditions above. The charts show the policy functions for the CCyB
and the interest rate in response to a credit shock of varying size. As the size of the credit
shock increases, the policymaker chooses to increase the CCyB, to lean against credit growth
and increase resilience. At the same time, policy reduces the interest rates in order to offset
some of the reduction in demand brought about by the CCyB. To explore the mechanism in
detail, the corresponding equilibrium outturns for four other variables are shown in Figure
6.
Absent policy, the rate of credit growth would increase one for one with the size of the
credit shock. With policy, a higher CCyB and a lower interest rate both affect this increase,
with the CCyB leading to higher spreads, leaning against the credit shock, while the lower
interest rate exacerbates it. In our calibrated model, the first effect dominates, such that a
small amount of the credit shock is offset in the top-left panel. Qualitatively, optimal policy
is not particularly dependent on the exact parameterisation of the credit growth equation
(5), however. Even if we were to set φs = 0, so that only monetary policy is able to lean
against credit growth, policy should still cut interest rates. The reason is that it is the crisis
probability, rather than credit growth itself, that has a direct effect on the policymaker’s
loss. And the CCyB still has a comparative advantage in affecting the crisis probability due
to its resilience benefits, even if it is unable to lean against the credit cycle. Thus in the
top-right panel, a higher CCyB is able to mitigate the effect of the credit shock on the crisis
probability, despite most of it feeding through into higher credit growth.
Instead of leaning against the credit cycle, the optimal role of monetary policy is largely
to best offset the effects of the CCyB on output and inflation. First, the demand effects of the
19
CCyB are larger than the supply effects, so with a higher CCyB the natural rate of interest
falls - to which monetary policy responds by cutting rates. In addtion, the policymaker
places a positive weight on output deviations, so optimally chooses to cut interest rates
even further than the fall in the natural rate, in order to split the cost-push effects of the
CCyB between output and inflation according to (14). As a result, equilibrium output falls
by less than y1n , while inflation rises.
3.3.
The zero-lower bound on interest rates
Until now, we have discussed optimal policy under the assumption that both instruments
are set optimally in an unconstrained fashion. But in practice, monetary policy may face
a zero lower bound (ZLB) constraint. Figure 7 shows how this would affect the trade-off
facing the policymaker. The blue lines show the policymaker’s menu of options in response
to a credit shock with both instruments freely available. The red lines show the equivalent
policy frontier when monetary policy is assumed to be constrained, all else equal. With
monetary policy unable to offset any of the demand effects of a higher CCyB, using it
becomes far more costly. As a result, the trade off facing the policymaker rotates out from
the point on the horizontal axis where the CCyB is set to zero. For our baseline calibration,
the trade-off remains less steep than when monetary policy is the only instrument available,
however. This is due to the comparative advantage of the CCyB in dealing with credit
shocks that affect the crisis probability - for a given reduction in the crisis probability, the
demand cost is much smaller if that is achieved using the CCyB than via monetary policy.
Given the steeper trade-off, for given policymaker preferences, it is optimal to use the
CCyB less when monetary policy is constrained. Figure 8 shows how the CCyB policy
function changes at the zero lower bound. As the credit shock grows, the optimal response
of the CCyB, while still positive, is far more muted. The larger the CCyB response would be
if monetary policy were unconstrained, the harder the constraint bites on macroprudential
policy, and the more that financial stability concerns have to be balanced with monetary
20
stability.
4.
When are the policies complements or substitutes?
In this section, we focus specifically on the interaction between monetary and macroprudential policies in our model. We explore when the two policies are complements and
move in the same direction in response to a shock, or substitutes, and move in opposite
directions. In our benchmark model, the policymaker may choose to move interest rates
and the CCyB in the same direction for two distinct reasons. First, is that although we have
assumed so far that our credit and macro shocks are uncorrelated, in practice the business
cycle and credit cycle are highly intertwined. We first explore how even if the policymaker
largely uses the interest rate to offset macro shocks and the CCyB to offset credit shocks,
the correlation between the two types of shocks may drive a correlation in the instruments.
Second, is that even if shocks are completely independent, the strategic substitutability or complementarity (Bulow, Geanakoplos and Klemperer, 1985 [8]) between the two
instruments means that one instrument will be changed purely in response to the change
in the other. We next explore how our varying the calibration of our model to capture
different beliefs about the structure of the economy can switch the instruments from being
strategic substitutes to complements. This alters the nature of interaction between the
instruments - such that the policymaker tightens and loosens both together even when
faced with independent shocks.
Away from our benchmark model, we consider other views about policy interaction. This
section finally explores how some small changes to our model could give rise to different
types of interaction often cited by policymakers, within our parsimonious framework.
4.1.
Responses to different shocks
The optimal response of period 1 policy to a one standard deviation innovation in each of
the three different shocks is shown in Figure 9. The first 6 bars show the response of each
21
instrument to each shock under the assumption that they are completely independent. The
final 2 bars show the response to a joint shock to credit and aggregate demand, under the
other extreme assumption that these are perfectly correlated.
In response to an isolated positive credit shock, as discussed in the results section, it is
optimal for the CCyB to tighten. It does so partly to dampen credit growth, but even more
so to enhance resilience and reduce the probability that high credit growth will lead to a
systemic financial crisis in period 2. Although the setting of interest rates optimally takes
into account benefit from tightening them in reducing the crisis probability, the main role
for monetary policy is to offset the negative effects of a higher CCyB on aggregate demand.
So monetary policy is loosened slightly.
In response to a positive demand shock, in contrast, the impact of the shock can largely
by offset by a tightening in interest rates, leaving period 1 output little changed. The
optimal response of the CCyB is to loosen slightly, since with higher interest rates (and
lower credit growth), the crisis risk becomes lower, so the marginal benefit of using the
CCyB falls. Finally, following a positive cost-push shock, the optimal response is again for
the interest rate to tighten and the CCyB to loosen. But in this case, the monetary policy
tightening is smaller, partly due to the need to balance the effects shock between higher
inflation and lower output. The CCyB loosens, and by more, since with a positive cost-push
shock already pushing up on inflation and down on output, the marginal cost of using the
CCyB and reducing potential supply becomes larger.
The final two bars show the case of perfectly positively correlated credit and demand
shocks. This could be thought of as the typical policy setting when the credit and business
cycles are closely aligned. Or equivalently, it could represent optimal policy when credit
shocks had large endogenous effects on aggregate demand, which we abstract from in our
model. With correlated positive shocks, both instruments should be tightened. The direct
tightening of monetary policy in response to a demand shock dominates the indirect effect
coming from strategic substitutability between the instruments, which would tend towards
22
Table 2: Optimal policy under different sets of parameters as crisis cost/financial stability
weight rises
Policy interaction
∆k1
∆i1
Parameter case
Intuition
Strategic complements
+ive
+ive
νψ κ2 κ+λ > σωψ
(Supply effect of CCyB)*(policymaker
weight on inflation) > demand effect of
CCyB
νψ κ2 κ+λ < σωψ,
Strategic substitutes
+ive
-ive
>
∂γ1
∂k1
∂γ1
∂i1
Relative effect of the CCyB/interest rates
< (σωψ − νψ κ2 κ+λ )σ−1 on crisis probability < relative effect of the
CCyB/interest rates on demand and supply
(σωψ − νψ κ2 κ+λ )σ−1
Strategic substitutes and -ive
instrument switches
+ive
Demand effect of CCyB is bigger than
(weighted) supply effect, but the CCyB is
still relatively more effective at reducing
crisis probability
∂γ1
∂k1
∂γ1
∂i1
loosening. Similarly, the direct tightening required in the CCyB in response to the positive
credit shock is larger than the indirect loosening needed in response to the rise in interest
rates.
4.2.
Response to a credit shock under different calibrations
Even in response to a credit shock that leaves aggregate demand conditions unchanged,
changing our assumptions with regard to a key parameter in the model changes the way
monetary and macroprudential policies interact. In this sub-section, we examine three
mutually exclusive cases, summarised in Table 2. As in our benchmark case, policies may
be strategic substitutes - a tightening in one policy instrument reduces the marginal benefit
to tightening the other. Or they may be strategic complements - a tightening in one policy
instrument increases the marginal benefit to tightening the other. And if they are strategic
substitutes, there may be circumstances when the objectives of each instrument switch,
such that the CCyB should target inflation and output, and interest rates should aim to
reduce the probability of financial crises.
Table 2 shows that the interaction between the policies depends cruicially on two
features: whether the (weighted) supply cost of macroprudential policy is larger than its
23
demand cost; and which policy has a comparative advantage at acheiving which goal.
On the first, when macroprudential policy tightens, it reduces both actual and potential
output, relative to their effecient level. The policymaker’s desire to smooth volatility in
actual output means they care directly about the reduction in output. But they also care
about any change in actual output relative to potential output, via the inflation term. If
the fall in potential output is large enough, such that the output gap increases, and if the
policymaker’s weight on inflation is large enough, then the policymaker will respond to the
macroprudential policy tightening by tightening monetary policy. This will also lower the
crisis probability somewhat, reducing the required macroprudential tightening. Note that
the result relies on the relative supply and demand effects of the CCyB. If an increase were
to increase aggregate demand, rather than reduce it as in our benchmark case, it would
also be optimal to tighten monetary policy even if the supply effect of the CCyB were to be
small. As the demand effect gets more negative relative to the supply effect, or the relative
weight on inflation falls, the policymaker switches to wishing to reduce interest rates in
response to increases in the CCyB, as in the benchmark case.
The policymaker sets jointly optimal policy, so has no in-built preference to use each
instrument for a specific goal. The decision as to which policy to tighten, when there is
an increased desire to target the financial stability goal, depends entirely on which policy
has a comparative advantage. The definition of comparative advantage is slightly more
general than the χ term defined earlier. Macroprudential policy is tightened to reduce the
crisis probability if and only if it is relatively more effective than monetary policy at doing
so, relative to their effects at traditional monetary policy goals. Effectiveness at monetary
policy goals is given by the weighted average of the marginal effect of the policy on output,
with the policymaker’s output weight, and the output gap, with the policymaker’s inflation
weight.
Figure 10 shows how the monetary policy response varies when parameters are set
such that the policies are strategic complements. The blue lines show the standard case
24
when policies are strategic substitutes. But in the green lines, the CCyB is assumed to boost
aggregate demand, and optimal policy involves monetary policy tightening to offset this. In
the red line, the CCyB negatively impacts supply by 5 times as much as in the benchmark
case. This has two effects, first it makes using macroprudential policy more costly, so less
active as a result. But as with positive demand effects, it also makes the policymaker choose
to tighten monetary policy at the same time as tightening the CCyB.
The other extreme case is when the comparative advantage of the instruments switches.
The parameters in this theoretical example are unlikely in practice, and the mechanism
partly depends in there being no lower bound for the CCyB. It could occur if the (negative)
demand effect of macroprudential policy (ω) became very large. As the demand effect
grows, monetary and macroprudential policy operate on more and more similar margins
- their comparative advantage falls. This means that the policies generally have to both
move further in opposite directions to achieve a similar outcome for output, inflation and
the crisis probability. At the point where the comparative advantage switches, it quickly
becomes optimal for the instruments to switch roles, and monetary policy tightens to
reduce the crisis probability, while macroprudential policy loosens to offset some of the
spillovers to output and inflation.
4.3.
Introducing a market-based finance sector
In this subsection, we return to the calibration of our benchmark model but extend it
to allow for a market-based finance sector that is not subject to macroprudential policy
alongside a banking sector to which macroprudential policies apply. The intuition is that as
the banking system is subject to tighter and costly macroprudential policy, there might be
leakage to the non-banking system, ie non-banks step in and extend credit more cheaply.
This implies that there are limits to the effectiveness of the CCyB in reducing the crisis
probability as it cannot lower the probability of a crisis emanating from the market-based
finance system.
25
Rather than equation (7), let the probability of a crisis be equal to:
γ1 = bγ1BL + (1 − b)γ1SL
(17)
where γ1BL is the probability of a crisis arising in the banking sector and γ1SL is the probability
of a crisis arisisng in the shadow banking sector. The parameter b is the share of lending
done by banks in steady state. Lending growth for each sector is determined by an
analogous equation to (5):
BL1 = φ0B + φi i1 + φsB s1 + ξ 1B
(18)
SL1 = φ0S + φi i1 + φsS s1 + ξ 1S
(19)
We assume that φsB < 0 - higher capital requirements (and bank lending spreads) reduce
the amount of lending done by banks. But we also assume there is some leakage of
macroprudential policy to the unregulated, shadow banking sector, such that φsS > 0. Both
equations have the same parameter governing the effect of interest rates on credit growth,
however, capturing the Stein (2013) argument that monetary policy gets in all the cracks. For
simplicity, we assume that both shocks are perfectly correlated and ξ 1B = ξ 1S
We calibrate φsS = −3φsB , based on Aiyar, Calomiris and Wieladek (2014), and set the
shadow banking sector equal to 20% of the overall stock of lending. If macroprudential
policy reduces bank credit growth by 1pp, shadow bank credit growth increases by 3pp.
But given the smaller stock of shadow bank lending, together, these parameters imply that
around three-quarters of the reduction in regulated bank lending leaks into the shadow
banking sector.
Figure 11 shows how the optimal policy response to a credit shock changes when the
shadow banking channel is switched on, for different policymaker preferences for avoiding
crises. The channel leads the policymaker to attenuate their policy response. This is because
26
the marginal benefit of tighter macroprudential policy will always be smaller, now that
it boosts shadow banking credit (all else equal). But less tight macroprudential policy
also means that monetary policy can set interest rates higher, both because there is less
macroprudential policy induced cost-push shock to effect, and also because with a less
effective macroprudential tool, there is more need to lean against the wind.
4.4.
Introducing a strong risk-taking channel of monetary policy
Similar conclusions can be reached when we allow for non-linear interaction between
the two policies in their effect on credit. In particular, we add such an interaction term
to equation (5). This captures the argument of Shin (2015) [30] that monetary policy
and macroprudential policies are complements, with higher levels of one increasing the
marginal effectiveness of the other. Intuitively, this could be interpreted as a ’risk-taking
channel’ of monetary policy, under which combinations of low interest rates and high
capital requirements encourage banks to lend to riskier borrowers. Under this interpretation,
B1 would proxy for the growth rate of credit risk, which may increase even if the growth
rate of credit quantities remains unchanged.7
Let real credit growth by given by:
B1 = φ0 + φi i1 + φs s1 + φi,s i1 s1 + ξ 1B ,
(20)
where φi,s < 0.
Then lower interest rates make the CCyB less effective at reducing lending growth. This
is illustrated in Figure 12 - although the interaction coefficient has to be large to make
policies strategic complements and therefore give monetary policy a large role in leaning
7 An
alternative way of introducing a risk-taking channel of monetary policy to the model would be to allow
credit spreads to be negatively correlated with the level of interest rates. Given the endogenous cost-push
effect of credit spreads in our Phillips Curve, this would be equivalent to introducing a negative version of a
cost channel of monetary policy (Ravenna and Walsh, 2006). [29] This would make using macroprudential
policy more costly, since monetary policy would be unable to costlessly offset its effects. But it would not
qualitatively change the results presented in Section 3.
27
against the wind.
5.
Conclusion
Our simple framework allows us to explore various assumptions around how monetary
and macroprudential policies affect the economy and interact with each other. We show
that depending on the assumptions made, the two policies may be strategic substitutes or
complements, ie they move in the opposite or the same direction. In particular, while our
benchmark case suggests that they would typically be substitutes, they may be complements
when an increase in capital requirements has particularly large negative supply effects or
when such an increase affects aggregate demand positively. And under certain conditions,
monetary policy can be seen as a last line of defence, for example when the effectiveness
of capital in reducing the probability of crises decreases with its level. But under most
assumptions the implications for monetary policy are relatively small. The findings do,
however, speak to understanding the impact of changes in capital requirements better, in
order to determine the optimal policy mix.
References
[1] David Aikman, Andrew Haldane, and Benjamin Nelson. Curbing the credit cycle. The
Economic Journal, 2014.
[2] Andrea Ajello, Thomas Laubach, David López-Salido, and Taisuke Nakata. Financial
stability and optimal interest-rate policy. 2015.
[3] Allen Berger and Christa Bouwman. How does capital affect bank performance during
financial crises? Journal of Financial Economics, 109(1):146–176, 2013.
[4] Ben S Bernanke and Mark Gertler. Should central banks respond to movements in
asset prices? The American Economic Review, 91(2):253–257, 2001.
28
[5] Olivier Blanchard, Eugenio Cerutti, and Lawrence Summers. Inflation and activity–two
explorations and their monetary policy implications. Technical report, National Bureau
of Economic Research, 2015.
[6] Claudio EV Borio and Philip William Lowe. Asset prices, financial and monetary
stability: exploring the nexus. 2002.
[7] Martin Brooke, Oliver Bush, Robert Edwards, Jas Ellis, Bill Francis, Rashmi Harimohan,
Katharine Neiss, and Caspar Siegert. Measuring the macroeconomic costs and benefits
of higher UK bank capital requirements. Financial Stability Paper, 35, 2015.
[8] Jeremy I Bulow, John D Geanakoplos, and Paul D Klemperer. Multimarket oligopoly:
Strategic substitutes and complements. Journal of Political economy, 93(3):488–511, 1985.
[9] Stephen Burgess, Emilio Fernandez-Corugedo, Charlotta Groth, Richard Harrison,
Francesca Monti, Konstantinos Theodoridis, and Matt Waldron. The Bank of England’s
forecasting platform: COMPASS, MAPS, EASE and the suite of models. 2013.
[10] Mark Carlson, Hui Shan, and Missaka Warusawitharana. Capital ratios and bank
lending: A matched bank approach. Journal of Financial Intermediation, 22(4):663–687,
2013.
[11] Charles Carlstrom, Timothy Fuerst, and Matthias Paustian. Optimal monetary policy
in a model with agency costs. Journal of Money, Credit and Banking, 42(s1):37–70, 2010.
[12] Richard Clarida, Jordi Gali, and Mark Gertler. The science of monetary policy: A New
Keynesian perspective. Journal of Economic Literature, 37:1661–1707, 1999.
[13] James Cloyne, Ryland Thomas, Alex Tuckett, and Samuel Wills. An empirical sectoral
model of unconventional monetary policy: The impact of QE. The Manchester School,
83(S1):51–82, 2015.
29
[14] Vasco Curdia and Michael Woodford. Credit spreads and monetary policy. Journal of
Money, Credit and Banking, 42(s1):3–35, 2010.
[15] Mathias Drehmann, Claudio Borio, and Kostas Tsatsaronis. Characterising the financial
cycle: don’t lose sight of the medium term! 2012.
[16] Jeremy Franklin, May Rostom, and Gregory Thwaites. The banks that said no: banking
relationships, credit supply and productivity in the UK. 2015.
[17] Jordi Galı́. Monetary policy, inflation, and the business cycle: an introduction to the New
Keynesian framework. Princeton University Press, 2008.
[18] Nicola Gennaioli, Andrei Shleifer, and Robert Vishny. Neglected risks: The psychology
of financial crises. Technical report, National Bureau of Economic Research, 2015.
[19] Mark Gertler and Peter Karadi. A model of unconventional monetary policy. Journal of
Monetary Economics, 58(1):17–34, 2011.
[20] Simon Gilchrist, Raphael Schoenle, Jae Sim, and Egon Zakrajsek. Inflation dynamics
during the financial crisis. 2015.
[21] Macroeconomic Assessment Group. Assessing the macroeconomic impact of the
transition to stronger capital and liquidity requirements, 2010.
[22] Bengt Holmstrom and Jean Tirole. Financial intermediation, loanable funds, and the
real sector. the Quarterly Journal of economics, pages 663–691, 1997.
[23] Sujit Kapadia. Optimal monetary policy under hysteresis. Department of Economics,
University of Oxford, 2005.
[24] Mervyn King. Twenty years of inflation targeting, 2012.
[25] Donald Kohn. Monetary policy and asset prices. Technical report, 2006.
30
[26] Donald Kohn. Implementing macroprudential and monetary policies: The case for
two committees, 2015.
[27] Joe Peek, Eric Rosengreen, and Geoffrey Tootell. Should US monetary policy have a
ternary mandate? The Economic Journal, 2014.
[28] Adam S Posen. Why central banks should not burst bubbles. International Finance,
9(1):109–124, 2006.
[29] Federico Ravenna and Carl Walsh. Optimal monetary policy with the cost channel.
Journal of Monetary Economics, 53(2):199–216, March 2006.
[30] Hyun Song Shin. Macroprudential tools, their limits and their connection with
monetary policy, 2015.
[31] Jeremy Stein. Overheating in credit markets: Origins, measurement, and policy
responses, 2013.
[32] Lars Svensson. Discussion of Ajello, Laubach, López-Salido, and Nakata, ’Financial
stability and optimal interest-rate policy’, 2015.
[33] William R White. Is price stability enough? 2006.
[34] Michael Woodford. Interest and prices: foundations of a theory of monetary policy. 2003.
Appendix 1 - Solving the model
Period 1 private sector expectations of period 2
The private sector has all of the same information as the policymaker conditional on the
state, but does not know the probability of a crisis occurring between periods 1 and 2.
Instead, agents assume there is an exogenous probability e of a crisis occurring.
31
So the private sector’s period 1 expectations are equal to:
ps
E1 π2 = eπ2,c + (1 − e)π2,nc = eπ2,c
(21)
ps
E1 y2 = ey2,c + (1 − e)y2,nc = ey2,c
(22)
At time t = 1, the values these expectations are known by the policymaker, and crucially,
independent of period 1 policy. So the period 1 policymaker’s problem reduces to a static
problem.
Period 1
The period 1 policymaker chooses period 1 settings of the CCyB and interest rates to
minimise the Loss function:
L=
1 2
(π + λy21 ) + βγ1 (1 + ζ ) E1 [ L2,c ] + β(1 − γ1 ) E1 [ L2,nc ]
2 1
(23)
Subject to the Phillips Curve:
ps
π1 = κy1 + E1 π2 + νs1 + ξ 1π
(24)
and the IS curve
ps
ps
y
y1 = E1 y2 − σ(i1 − E1 π2 + ωs1 ) + ξ 1
(25)
Since L2,nc = 0, the policymaker’s problem reduces to:
mini1 ,k1 L =
1 2
(π + λy21 ) + βγ1 (1 + ζ ) E1 [ L2,c ]
2 1
(26)
We then substitute (24) in place of inflation in (26), and (25) in place of output. And
ps
ps
noting that E1 [ L2,c ], E1 π2 , and E1 y2 are exogenous parameters from the perspective of
32
the period 1 policymaker, differentiating with respect to i1 and k1 gives the two first order
conditions which hold at the minimised value of L:
∂L
∂γ
= −σ(κπ1 + λy1 ) + 1 β(1 + ζ ) E1 [ L2,c ] = 0
∂i1
∂i1
(27)
∂γ
∂L
= −σωψ(κπ1 + λy1 ) + νψπ1 + 1 β(1 + ζ ) E1 [ L2,c ] = 0
∂k1
∂k1
(28)
The crisis probability is given by:
γ1 =
exp(h0 + h1 B1 + h2 k1 )
1 + exp(h0 + h1 B1 + h2 k1 )
(29)
We can substitute into this the equation determining credit growth, as well as the Phillips
Curve, IS curve and the macroprudential policy effect, to give an equation in terms of
spreads and interest rates. Differentiating this gives the two marginal crisis risk equations:
∂γ1
exp(h0 + h1 B1 + h2 k1 )
=
h (φ )
∂i1
(1 + exp(h0 + h1 B1 + h2 k1 ))2 1 i
(30)
exp(h0 + h1 B1 + h2 k1 )
∂γ1
=
(h (ψφs ) + h2 )
∂k1
(1 + exp(h0 + h1 B1 + h2 k1 ))2 1
(31)
Which means the relative effect of each policy on the crisis risk is a constant parameter,
which we define as A:
A≡
∂γ1
∂k1
∂γ1
∂i1
=
h1 ψφs + h2
h1 φi
(32)
Multiplying through (27) by − A, and adding to (28), gives:
( Aσ − σωψ)(κπ1 + λy1 ) + νψπ1 = 0
33
(33)
This can be be expressed as
λy1 + κπ1 = 0
(34)
where
λ :=
χσωψ
νψ
κ
+ χσωψ
λ
(35)
and χ is a parameter defined as the comparative advantage of macroprudential policy at
affecting the crisis probability, versus monetary policy at affecting the IS curve/aggregate
demand. And
νψ
κ
is the marginal effect of macroprudential policy on the natural rate of
output.
χ≡
∂γ1
∂k
( ∂γ1
1
∂i1
∂y1
∂i1
∂y1
∂k1
− 1)
(36)
Equation (33) can also be used to back out the optimal relationship between the two
policy instruments, interest rates and spreads. This can then be substituted into one of the
two first order conditions, to give an equation determining the optimality condition trading
off changes in the crisis probability with output/inflation losses in period 1:
λy1 (−
where
∂L
∂γ1
∂γ
∂γ νψ
∂L
νψ
) = ( 1 + 1(
− ωψ))(−
)
κ
∂k1
∂i1 κσ
∂γ1
(37)
is the policymaker’s expected discounted cost of a financial crisis, taking into
account the extra weight ζ that they place on the expected cost of a crisis.
∂L
= β((1 + ζ ) E1 [ L2,c ] − E1 [ L2,nc ])
∂γ1
(38)
The intution behind equation (37) can be seen by defining the natural rates of interest
34
and output as those that would obtain with flexible prices (but with credit frictions):
λy1
∂y1n
∂γ
∂γ ∂r n
∂L
= ( 1 + 1 1 )(−
)
∂k1
∂k1
∂i1 ∂k1
∂γ1
(39)
At the optimal setting of policy, the marginal loss from changing output is equated
to that from changing inflation. So the condition can equivalently be expressed as an
optimality condition trading off inflation today with the cost of a financial crisis tomorrow
(with output today unchanged.) This gives:
∂π
∂γ
∂γ
π1 1 = ( 1 − 1
∂k1
∂k1
∂i1
∂y1
∂k1
∂y1
∂i1
)(−
∂L
)
∂γ1
(40)
The LHS is the marginal cost of increasing inflation through the cost-push effect of
higher credit spreads. This is set equal to the marginal gain from a lower crisis probability,
made up of two terms. The first term on the RHS is the marginal gain from a lower crisis
probability of higher spreads. The second is the marginal increase in the crisis probability
from offsetting the demand effects of higher spreads using interest rates.
Appendix 2 - Uncoordinated policies
Until now, we have considered monetary and macroprudential policies being set jointly,
capturing cases when these are set by the same policymaker, or at least in a co-ordinated
fashion. But in some locations the institutional set-up is such that two policymakers with
different objectives have responsibility for setting the policies. In this section we examine
the extent to which this leads to suboptimal outcomes.
Outcomes may of course depend on how objectives are split. We try to capture how
this has often been done in practice, by assigning our monetary policymaker a standard
monetary policy loss function that ignores tail outcomes such as financial crises:
LM =
1 2
(π + λi y21 )
2 1
35
(41)
For the macroprudential policymaker, one option would be to assign what is left from
our joint loss function - its effect on the crisis probability. But this would be somewhat
one sided. In practice, macroprudential seeks to balance its benefits with the costs of
excessive regulation on intermediation and ultimately on the economy’s supply potential.
We therefore also assign the macroprudential policymaker a desire to care about the natural
rate of output, defined by (12). This gives the following macroprudential policy loss
function (setting E1 [ L2,nc ] = 0):
LF =
1
λ (yn )2 + βγ1 (1 + ζ ) E1 [ L2,c ]
2 k 1
(42)
Nash policies
In this subsection we examine how policy differs in the uncoordinated case, under the assumption that the two policymakers take each other’s policy as given in a Nash equilibrium,
with both acting as Stackelberg leaders with the respect to the private sector.
Minimising the monetary policymaker’s assigned section of the loss function:
mini1 L M =
1 2
(π1 + λi y21 )
2
(43)
gives the familiar monetary policy first order condition under discretion (Clarida, Gali,
Gertler, 1999) [12]
σ (κπ1 + λi y1 ) = 0
(44)
compared to the jointly optimal case (also given earlier by (11)):
σ(κπ1 + λy1 ) =
∂γ1 ∂L
∂i1 ∂γ1
(45)
Assuming (λi = λ), meaning that the monetary policymaker is assigned the same weight on
output losses as the joint policymaker, then (44) implies that the under Nash, conditional on
36
a given level of the CCyB k1 , monetary policy is set looser than is optimal. This is because
the monetary policymaker fails to take into account that raising interest rates also creates
a positive externality for financial stability. The rise in interest rates also dampens credit
growth and reduces the probability of a crisis in the subsequent period. Delegation means
that the Nash monetary policymaker suboptimally fails to lean against the credit boom.
The macroprudential policymaker minimises their loss function:
mink1 L F =
1
λ (yn )2 + βγ1 (1 + ζ ) E1 [ L2,c ]
2 k 1
(46)
to give the first order condition:
∂L F
νψλk n ∂γ1 ∂L F
y1 +
=−
=0
∂k1
κ
∂k1 ∂γ1
(47)
or rearranging,
λk y1n (−
∂γ
νψ
∂L
) = 1 (− F )
κ
∂k1
∂γ1
(48)
The equation sets the marginal loss from a higher CCyB lowering the natural rate of output
equal to the marginal benefit from reducing the crisis probability. This compares to the
jointly optimal intertemporal condition (given earlier as (15)):
λy1 (−
νψ
∂γ
∂L
∂γ νψ
) = ( 1 + 1(
− ωψ))(− F )
κ
∂k1
∂i1 κσ
∂γ1
(49)
Again assuming that (λk = λ), there are two differences to the macroprudential policymaker’s decision under Nash. First, on the left hand side of (48), the Nash policymaker is
assigned to care about the natural rate of output, rather than the actual level of output. This
leads to a higher marginal cost of using the CCyB, since the Nash policymaker disregards
that the cost-push effect of using the CCyB can be optimally split between two goals (output
and inflation), rather than borne solely by one (potential output). Given the quadratic
nature of losses, this leads them to overstate the marginal cost of the CCyB.
On the right hand side of (48), however, the Nash macroprudential policymaker is
37
missing the term that depends on the marginal effect of changing the interest rate on the
crisis probability. This second missing effect derives from the fact that under Nash, the
macroprudential policymaker ignores the externality that the CCyB exerts on demand
and inflation via the output gap. If, as under our benchmark calibration, the demand
effect of the CCyB is larger than the supply effect, this is a negative externality. The CCyB
creates a negative output gap which is costly over and above its effect on potential output.
Even though interest rates can be cut to offset the externality - and are, in the Nash case
- such a monetary loosening increases the crisis probability. The Nash macroprudential
policymaker fails to take account of this, since the output gap does not enter their loss
function. Consequently, they overstate the marginal benefits of using the CCyB. If the supply
effect of the CCyB were larger, the opposite would be true, and the Nash policymaker
would understate the policy’s marginal benefits, relative to the jointly optimal case.
Depending on the relative sizes of the two effects, the CCyB may be set suboptimally
high or low by the uncoordinated macroprudential policymaker. Under our benchmark
calibration delegation means that the CCyB is set too low for a given monetary policy
setting. And in response, interest rates loosen by less under Nash policies than if they were
responding to the higher, jointly optimal setting for the CCyB. Overall, the Nash settings
of policy implies too tight a setting for macroprudential policy and too loose a setting for
monetary policy. But for practical policymaking purposes, there may be other gains from
delegating responsibilities. A key question is therefore whether these suboptimal settings
are quantiatively large.
To get a sense of this, Figure 13 plots the policy functions for interest rates and the
CCyB as credit growth varies in response to a credit shock, assumed for simplicity to
be uncorrelated with the other shocks. Relative to the co-ordinated case, delegation has
only a small effect on policy settings. For our benchmark calibration, while there it is not
strictly true that macroprudential policy and monetary policy should only focus on their
respective goals, doing so is nonethelesss a fairly close approximation to jointly optimal
38
policy. If there are any other material gains from splitting objectives not captured in our
simple framework - such as improved accountability, or greater specialised expertise on
committees, for example - then these could easily outweigh any losses.
Stackelberg policies
An alternative way of delegating responsibilities would be to assign one of the policymakers
to act first within period 1, as a Stackelberg leader, with the other policymaker as a
Stackelberg follower.8 Given the longer timeframe that macroprudential policy operates
over, it seems natural to assign the macroprudential policymaker as the first mover, the
Stackelberg leader.
As the Stackelberg follower, the monetary policymaker’s decision is identical to under
Nash equilibirum policy - macroprudential policy is predetermined, so taken as given.
Condition (44) again determines interest rates for a given setting of the CCyB, ignoring the
externality from monetary policy to financial stability. By acting first, the macroprudential
policymaker is able to take advantage of this, however, and take into account the effect that
changes in the CCyB have on interest rates and the knock-on effect to financial stability. The
size of this effect can be seen by substituting the IS curve (1), Phillips Curve (2) and spreads
equation (6) into (44). Rearranging gives the policy function for interest rates, conditional
on k1 :
i1 =
σ (κ 2
κ
1 ps
ps
ps
( E1 π2 + νψk1 + ξ 1π ) + E1 y2 + E1 π2 − ωψk1
σ
+ λi )
(50)
Differentiating this with respect to k1 gives the effect of the CCyB on interest rates:
di1
κ 2 νψ
= 2
− ωψ
dk1
κ + λi σκ
(51)
The Stackelberg macroprudential policymaker takes this effect into account when setting
8 In
both case the private sector is assumed to be a Stackelberg follower to both policymakers, however.
39
the level of the CCyB. With i1 a function of k1 , rather than taken as given, the (rearranged)
first order condtion for the problem
mink1 L F =
1
λ (yn )2 + βγ1 (1 + ζ ) E1 [ L2,c ]
2 k 1
(52)
becomes
λk y1n (−
∂γ
∂γ di
∂L
νψ
) = ( 1 + 1 1 )(− F )
κ
∂k1
∂i1 dk1
∂γ1
(53)
Substituting (51) into this gives the macroprudential policymaker’s optimality condition:
νψ
λk y1n (− )
κ
∂γ1 ∂γ1
∂L
κ 2 νψ
=(
− ωψ))(− F )
+
( 2
∂k1
∂i1 κ + λi σκ
∂γ1
(54)
As with the Nash policymaker, the left hand side of (54) shows that the Stackelberg
policymaker overstates the marginal cost of using the CCyB. But differently, the second term
on the right hand side is unambiguously smaller than the equivalent term in the jointly
optimal policy condition - the Stackelberg policymaker always understates the marginal
benefit. Intuitively, this is because the macroprudential policymaker is forced to not only
internalise all of the the externality the CCyB exerts by creating an output gap, but also an
additional effect, coming from the monetary policymaker choosing to run a positive output
gap in response to the cost-push effects of the CCyB. Put differently, interest rates are cut
to fully offset effects of the CCyB on demand, but only raised enough to partly offset the
effects of the CCyB on supply. This asymmetry means the Stackelberg macroprudential
policymaker is aware that any tightening will be offset by loose monetary policy more than
is jointly optimal.
As a result of higher marginal costs and lower marginal benefits, the CCyB is set
unambiguously lower than under jointly optimal policy. For a given level of the CCyB,
interest rates are set lower, although if the two instruments are substitutes, they may not
be unconditionally lower. Figure 13 also plots the policy functions under Stackelberg
competition. Again, quantitatively, the differences between the three sets of policies are
40
very small.
Appendix 3 - Long-run potential supply effects - motivating different
weights in the loss function
In our benchmark model macroprudential policy and credit spreads have a negative impact
on the economy’s supply potential. This is essentially a static effect, however. Potential
output is lower when macroprudential policy is tight, but the effect reverses the moment
policy is loosened. And the event of a financial crisis has no impact on the economy’s
supply potential.
In practice, one possible cost of macroprudential policy is that it may have persistent
effects on the supply capacity of the economy. Similarly, to the extent running negative
output gaps induces hysteresis, tight monetary policy may have a similar effect. As might
the financial crisis itself.
In principle one could model each of these effects by allowing policy or financial crises
to have effects on level of potential output consistent with stable prices. But as we show
below, doing so is isomorphic to endogenising the weights on different parameters in the
loss function. Specifically:
If macroprudential policy affects period 2 potential output, the policymaker should include
minimising deviations in the macroprudential tool in the loss function, where the weight on the
CCyB depends positively on the potential output effect
If the period 1 output gap affects period 2 potential output, the policymaker should place a larger
weight (λ) on period 1 output in their loss function the larger is the potential output effect
If a financial crisis reduces period 3 potential output, the policymaker should place a larger weight
(ζ) on the expected cost of a financial crisis in the loss function, where the weight depends positively
41
on the potential output effect
We show this in the next three subsections. For simplicity and without loss of generality,
we assume no shocks materialise and that this is known by everyone. So the only uncertainty
is over whether a financial crisis occurs or not. We maintain the assumption that the crisis
is a one off event, which either occurs between periods 1 and 2, or not at all.
Macroprudential policy affects period 2 potential output
The simplest way of introducing such effects would be to make period 2 potential output
n ), a linear function of period 1 credit spreads.9
in the absence of a crisis, (y2,nc
n
y2,nc
= τs1 = τψk1
(55)
In period 2, in the event of no financial crisis, the policymaker would then minimise the
following welfare loss function:
L2,nc =
1 2
(π
+ λy22,nc )
2 2,nc
(56)
subject to the Phillips Curve: (in the absence of shocks and since period 2 is the terminal
period, future expectations play no role)
n
π2,nc = κ (y2,nc − y2,nc
)
(57)
giving the first order condition:
κπ2,nc + λy2,nc = 0
9 One
(58)
could also assume that period 2 potential output is affected if there is a crisis too, although it seems
reasonable to think of crises as being large negative demand shocks that policy is unable to fully offset, such
that a small fall in potential output would not make much difference to the crisis loss, and may actually
reduce it, be preventing inflation falling as far below target.
42
which can be substituted back into (57) for π2,nc to give
(κ 2 + λ)y2 = κ 2 y2n
(59)
and minimised welfare loss of
L2,nc =
1 λκ 2 τ 2 ψ2 2
1 λκ 2
n 2
(
y
)
=
k
2 κ2 + λ 2
2 κ2 + λ 1
(60)
Therefore the expected discounted welfare loss in the event of no crisis can be given as:
1
λk k21
2
(61)
λκ 2 τ 2 ψ2
κ2 + λ
(62)
βL2,nc =
where
λk ≡
This implies that the period 1 policymaker’s loss function,
L=
1 2
(π + λy21 ) + βγ1 (1 + ζ ) E1 [ L2,c ] + β(1 − γ1 ) E1 [ L2,nc ]
2 1
(63)
can then be written
L=
1 2
(π1 + λy21 + (1 − γ1 )λk k21 ) + βγ1 (1 + ζ ) E1 [ L2,c ]
2
(64)
The weight that the policymaker places on minimising deviations in the CCyB from
zero, (1 − γ1 )λk , is positively related to the size of the hysteresis effect, τ. It also depends
endogenously on the probability of a crisis. As the probability of a crisis increases, the
policymaker adjusts optimal policy less due to potential output effects, because the effects
only matter when a crisis does not occur.
43
Hysteresis effects - output gaps affect period 2 potential output
As in Kapadia (2005) [23], we could also let period 2 potential output be affected by
period 1 output, irrespective of whether period 1 ouptut was affected by monetary or
macroprudential policy. This is a true hysteresis effect, rather than a long-run supply effect
of macroprudential policy. Again, we assume that the effect only affects potential when a
crisis does not occur.
n
y2,nc
= f y1
(65)
As in Kapadia (2005), we impose an upper bound on hysteresis by having the period 2
policymaker choose i2,nc or equivalently y2,nc to minimise the loss function:
L2,nc =
1
n
2
(λ(y2,nc − y2,nc
)2 + π2,nc
+ λh (yn − y2,nc )2 )
2
(66)
subject to the Phillips Curve:
n
π2,nc = κ (y2,nc − y2,nc
)
(67)
n
Gives optimal period 2 output as a weighted average of y2,nc
and its maximum value:
y2,nc =
λ + κ2
λh
n
y
+
yn
2,nc
λ + κ 2 + λh
λ + κ 2 + λh
(68)
Substituting this back into the Phillips Curve (67) gives optimal period 2 inflation of
π2,nc =
κλh
n
(yn − y2,nc
)
λ + κ 2 + λh
(69)
So minimised welfare loss is equal to:
L2,nc =
1 (λ + κ 2 )λh n
n
(
(y − y2,nc
)2 )
2
2 λ + κ + λh
44
(70)
or equivalently
L2,nc =
1 (λ + κ 2 )λh 2
yn 2
(
)y ) + t.i.p.
f
(
1
−
2 λ + κ 2 + λh
f y1 1
(71)
where t.i.p stands for the term in yn that is independent of policy.
This implies the period 1 loss function can be rewritten as:
L=
1 2
(π + λy21 ) + βγ1 (1 + ζ ) E1 [ L2,c ] + t.i.p.
2 1
(72)
where
(λ + κ 2 )λh 2
yn
λ ≡ λ + β(1 − γ1 )
)
f
(
1
−
f y1
λ + κ 2 + λh
(73)
All of the optimality condtions from Appendix 1 still hold, but now the weight on
period 1 output in the loss function is an endogenous function of the hysteresis effects,
and of the crisis probability. As long as the limit on the hysteresis channel is large enough
n
that y2,nc
< yn , then the weight placed on output deviations lambda will be strictly greater
than the policymaker’s period preferences, reflecting the additional hysteresis benefit the
following period. A low crisis probability also means a greater weight is placed on period
1 output than in the case with no hysteresis, since there is a high probability that the
policymaker will get to benefit from higher period 2 potential output.
Financial crises have persistent effects on the economy’s supply capacity
A final type of hysteresis effect that could affect optimal policy is if financial crises have
persistent effects on potential supply, but smaller reductions in output do not. We can
model this simply by assuming a third period, where potential output is unchanged from
steady state if there was not a crisis in period 2, but falls if there was. The parameter p
45
represents the size of this hysteresis channel.
y3∗ =



0
if no crisis in period 2


− p
if crisis in period 2
(74)
In the event of no crisis between periods 1 and 2, the policymaker will be able to achieve
0 welfare loss in period 3. If there is a crisis however, the policymaker will select i3 , or
equivalently y3 , to minimise the loss function:
L3,c =
1 2
(π + λy23,c )
2 3,c
(75)
subject to
∗
π3,c = κ (y3,c − y3,c
)
(76)
Doing so gives the following standard first order condition, specifying the optimal
output inflation trade-off:
∂L3,c
= κπ3,c + λy3,c = 0
∂y3,c
(77)
∗ and rearranging gives an equation for y
Substituting in (76) for π3,c , (74) for y3,c
3,c as a
function of the hysteresis effect, p.
y3,c = −
κ2
p
κ2 + λ
(78)
κλ
p
+λ
(79)
And substituting into (76) for y3,c gives
π3,c =
κ2
46
In the event of a crisis, period 3 loss is therefore equal to:
L3,c =
1 2
κ2 λ
[π3,c + λy23,c ] =
p2
2
2(κ 2 + λ )
(80)
We assume that there is only one possible crisis event - between periods 1 and 2. So
period 2 discretionary policy has no effect on period 3 outcomes. The period 1 loss function
is therefore:
L=
1 2
(π + λy21 ) + (1 − γ1 )( βE1 [ L2,nc ] + β2 E1 [ L3,nc ]) + γ1 (1 + ζ )( βE1 [ L2,c ] + β2 E1 [ L3,c ])
2 1
(81)
Using the fact that expected welfare loss in periods 3 is 0 if there is no crisis as well as
equation (80), this can be rewritten as
L=
1 2
β2 κ 2 λ 2
(π1 + λy21 ) + (1 − γ1 ) βE1 [ L2,nc ] + γ1 (1 + ζ )( βE1 [ L2,c ] +
p )
2
2(κ 2 + λ )
(82)
or equivalently as
L=
1 2
(π1 + λy21 ) + (1 − γ1 ) βE1 [ L2,nc ] + γ1 β(1 + ζ ) E1 [ L2,c ]
2
(83)
where
ζ ≡ ζ + (1 + ζ )
κ2 λ
βp2
2(κ 2 + λ) E1 [ L2,c ]
(84)
ζ is therefore equal to the additional weight placed on avoiding crises, plus an additional
term that is an increasing function of the size of those hysteresis effects. This set-up is
equivalent to a model with no perisistent hysteresis effects but a higher value of ζ. Indeed,
such persistent effects could be a justification for the ζ term itself. If ζ = 0, so that the
47
policymaker only cares about mean expected losses, then (84) becomes
ζ≡
κ2 λ
βp2
2(κ 2 + λ) E1 [ L2,c ]
(85)
implying that any additional weight placed on avoiding crises is solely due to their
persistent effects on potential output.
Figures
Annual probability of a crisis (%)
7
CCyB of 0%
CCyB of 1%
CCyB of 2.5%
Steady state
credit growth
6
5
4
3
2
1
0
−2
0
2
4
6
8
10
12
Annual real credit growth (%)
Figure 1: Crisis probability and credit growth for different levels of the CCyB
48
Annual probability of a crisis in the UK (%)
14
12
10
8
6
4
2
0
1985
1990
1995
2000
Year
2005
2010
2015
Figure 2: Model-implied crisis probability for the UK
Period 1 welfare loss (as % of crisis loss)
5
0% cumulative 3 year real credit growth
− monetary policy only
30% cumulative 3 year real credit growth
− monetary policy only
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Period 2 annual crisis probability (%)
Figure 3: Monetary and financial stability trade-off with monetary policy only
49
Period 1 welfare loss (as % of crisis loss)
5
0% cumulative 3 year real credit growth
− monetary policy only
30% cumulative 3 year real credit growth
− monetary policy only
Two instruments (no supply costs of CCyB)
0% cumulative 3 year real credit growth
− two instruments (supply cost)
30% cumulative 3 year real credit growth
− two instruments (supply cost)
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Period 2 annual crisis probability (%)
CCyB setting (%)
Figure 4: Monetary and financial stability trade-off with monetary policy only and with
two instruments
10
5
0
−20
−15
−10
−5
0
5
10
15
20
25
30
15
20
25
30
Annual real credit growth (%)
Policy rate (%)
3
2.8
2.6
2.4
2.2
−20
−15
−10
−5
0
5
10
Annual real credit growth (%)
Figure 5: CCyB and interest rate policy functions as size of credit shock varies
50
Annual crisis probability (%)
After policy annual
real credit growth (%)
20
10
0
−10
−20
−20
−10
0
10
20
Annual real credit growth (%)
30
2
1.5
1
0.5
0
−20
−10
0
10
20
30
Annual real credit growth (%)
Annual inflation (%)
Output (% deviation
from efficient)
0
−0.2
−0.4
−0.6
−0.8
−20
−10
0
10
20
2.03
2.02
2.01
2
−20
30
−10
Annual real credit growth (%)
0
10
20
30
Annual real credit growth (%)
Figure 6: Optimal policy outcomes as size of credit shock varies
Period 1 welfare loss (as % of crisis loss)
5
0% cumulative 3 year real credit growth − zero lower bound
30% cumulative 3 year real credit growth − zero lower bound
0% cumulative 3 year real credit growth − two instruments
30% cumulative 3 year real credit growth − two instruments
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Period 2 annual crisis probability (%)
Figure 7: Optimal policy under different initial credit conditions, with and without binding
zero lower bound
51
10
Both instruments
Zero lower bound
9
CCyB setting (%)
8
7
6
5
4
3
2
1
0
−20
−15
−10
−5
0
5
10
15
20
25
30
Annual real credit growth (%)
Figure 8: CCyB policy function as credit shock varies, with and without binding zero lower
bound
2.5
CCyB
Policy rate
Optimal policy change (pp)
2
1.5
1
0.5
0
−0.5
−1
credit
demand
cost−push
credit and demand
1 standard deviation shock:
Figure 9: Optimal policy response to different shocks
52
3.8
Benchmark calibration
CCyB boosts aggregate demand
CCyB has large potential supply effects (5 x Benchmark)
3.7
Optimal interest rate (%)
3.6
3.5
3.4
3.3
3.2
3.1
3
2.9
2.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Optimal CCyB setting (%)
10
5
−20
0
20
CCyB setting (%)
15
10
8
6
4
2
−20
Annual real credit growth (%)
2.1
2.05
2
1.95
−20
0
0
20
Annual real credit growth (%)
Policy rate (%)
Inflation (Ann. %)
Annual crisis prob. (%)
Figure 10: Optimal CCyB and interest rate under different sets of parameters
20
3
2.8
2.6
2.4
2.2
−20
Annual real credit growth (%)
0
20
Annual real credit growth (%)
No market−based finance
Market−based finance sector has 75% lending share
Figure 11: Optimal policy response to a credit shock under different policymaker preferences, with and without a market-based finance sector
53
CCB setting (%)
1
0.5
0
5
10
15
(1+ζ), FS weight in loss function
2.1
2
1.9
1.8
0
5
10
15
(1+ζ), FS weight in loss function
8
6
4
2
0
20
Policy rate (%)
Annual crisis probability (%)
Inflation (Ann. %)
2
1.5
20
0
3
2.8
2.6
2.4
2.2
0
5
10
15
(1+ζ), FS weight in loss function
5
10
15
20
20
(1+ζ), FS weight in loss function
No risk−taking channel
Large risk−taking channel (φis=−5000)
CCyB setting (%)
Figure 12: Optimal policy response to a credit shock under different policymaker preferences, with and without non-linear interaction term between policies
7
6
5
10
12
14
16
18
20
Annual real credit growth (%)
Policy rate (%)
2.65
2.6
2.55
Jointly optimal policies
Uncoordinated policy (Nash)
Uncoordinated policy (Stackelberg)
2.5
2.45
10
12
14
16
18
20
Annual real credit growth (%)
Figure 13: Optimal policy functions under coordination, Nash equlibrium and Stackelberg
equilibrium
54