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On the Limits of Macroprudential Policy

Marcin Kolasa

Abstract

This paper studies how macroprudential policy tools applied to the housing market can complement the interest rate-based monetary policy in achieving one additional stabilization objective, defined as keeping either economic activity or credit at some exogenous (and possibly time-varying) levels. We show analytically in a canonical New Keynesian model with housing and collateral constraints that using the loan-to-value (LTV) ratio, tax on credit or tax on property as additional policy instruments does not resolve the inflation-output volatility tradeoff. Perfect targeting of inflation and credit with monetary and macroprudential policy is possible only if the role of housing debt in the economy is sufficiently small. The identified limits to the considered policies are related to their predominantly intertemporal impact on decisions made by financially constrained agents, making them poor complements to monetary policy, which also operates at an intertemporal margin. These limits can be overcome if macroprudential policy is instead designed such that it sufficiently redistributes income between savers and borrowers.

JEL: E32; E58; E63; G21; G28

Corresponding author: Marcin Kolasa, SGH Warsaw School of Economics and Narodowy Bank Polski, Warszawa, Poland, E-mail:

Article note: Much of this paper was written while the author was visiting the Columbia University. The paper greatly benefited from useful comments by two anonymous referees, the author’s discussions with Bianca De Paoli, Gauti Eggertsson, Matteo Iacoviello, Jonathan Kreamer, Jennifer La’O, Anna Lipinska, Matthias Paustian, Martin Uribe and Michael Woodford, as well as from comments received from the participants to a seminar at the Federal Reserve Board, Annual SED Meeting in Warsaw, CEF Conference in Bordeaux, EEA-ESEM in Geneva, Dynare Conference in Rome, and WIEM in Warsaw. The views expressed herein are those of the author and not necessarily those of Narodowy Bank Polski.


Appendix

A.1 Model equations

In this section of the Appendix we present a full list of equations making up the benchmark NK macrofinancial model. The variables without time subscripts denote their steady state values.

Households

Euler equation for patient households

(A.1)cP,t1=βPEt{cP,t+11πt+11}Rt

Impatient households’ budget constraint

(A.2)cI,t+pχ,t(χI,tχI,t1)+Rt1lt1πt1=lt+wtnI,t+tt

Collateral constraint

(A.3)lt=mtpχ,tχI,t

Euler equations for impatient households

(A.4)(1τl,t)cI,t1=βIEt{cI,t+11πt+11}Rt+Θt

Rigid housing demand of patient households

(A.5)χP,t=χP

Housing Euler equation for impatient households

(A.6)cI,t1pχ,t(1+τχ,t)=AχχI,tσχ+βIEt{cI,t+11pχ,t+1(1+τχ,t+1)}+Θtmtpχ,t

Labor supply (for i = {I, P})

(A.7)wtci,t1=Anni,tφ

Labor aggregate

(A.8)nt=ωnI,t+(1ω)nP,t

Firms

Marginal cost

(A.9)mct=wtεt

Optimal price set by reoptimizing firms

(A.10)p˜t=μΩtϒt

Auxiliary functions for optimal price

(A.11)Ωt=cP,t1mctyt+βPθEt{πt+1μμ1Ωt+1}
(A.12)ϒt=cP,t1yt+βPθEt{πt+11μ1ϒt+1}

Price indexes

(A.13)1=θ(ππt)11μ+(1θ)p~t1μ1

Market Clearing

Goods market

(A.14)yt=ωcI,t+(1ω)cP,t

Aggregate output

(A.15)ytΔt=εtnt

Price dispersion index

(A.16)Δt=θΔt1πtμμ1+(1θ)p˜H,tμ1μ

Housing market

(A.17)χ=ωIχI,t+(1ωI)χP,t

A.2 Inflation and Output Targeting (Proof of Proposition 1)

We prove Proposition 1 by showing that stabilization of inflation and output at some exogenously moving targets using the short-term interest rate and any of the three macroprudential instruments (LTV ratio, tax on credit, and tax on new property) is inconsistent with the existence of stable rational expectations equilibrium.

Let us denote any linear function of current and past exogenous variables up to time t as exot. Formally, exot=A(L)ϵt, where A(L) = a0 + a1L + a2L2 + … is any lag polynomial in L (with Lϵt=ϵt1), ϵt denotes a k × 1 vector of exogenous shocks, while a0, a1, … are 1 × k vectors of numbers, possibly dependent on the structural parameters of the model.

First note that if output is at its target at all times, the market clearing condition (19) implies

(A.18)ωcc^I,t+(1ωc)c^P,t=exot

If additionally inflation is at the target at all times, the Phillips curve implies w^t=ε^t and hence Eq. (20) can be rewritten as

(A.19)ωnc^I,t+(1ωn)c^P,t=exot

These two equations can be solved for c^P,t and c^I,t as functions of exogenous shocks only, which together with the Euler equation for patient households (10) leads to the same result for R^t. From the above and the intratemporal conditions (15) and (16), it follows that also n^P,t and n^I,t depends only on shocks.

Using these results, one can show that the budget constraint of impatient households (12) can be reduced to

(A.20)1βPl^t1l^ttlt^t=exot

It is clear from this equation that credit is explosive in response to shocks if t^t=0, independent on whether macroprudential policy uses as its instrument the LTV ratio m^t, tax on credit τl,t or property purchase tax τχ,t, or even all of them simultaneously.

Let us now assume instead that only transfers are used as an additional instrument to the interest rate. Then the Euler equation for impatient households (11) implies that Θ^t is a function of exogenous shocks only while the collateral constraint (13) is simply reduced to

(A.21)l^t=p^χ,t

Plugging these results into the optimality condition for housing (14) yields

(A.22)l^t=βPEt{l^t+1}+exot

which uniquely determines a non-explosive path of credit l^t while equation (A.20) determines the value of transfers consistent with this path.

A.3 Inflation and Credit Targeting (Proof of Proposition 2)

We prove Proposition 2 by showing that perfect targeting of both inflation and credit using the short-term interest rate and any of the three macroprudential instruments (LTV ratio, tax on credit, and tax on new property) is consistent with the existence of stable rational expectations equilibrium only if the model parameters satisfy a certain restriction.

If inflation is at the target at all times, Eq. (20) implies

(A.23)φ(ωcc^I,t+(1ωc)c^P,t)+ωnc^I,t+(1ωn)c^P,t=exot

which allows us to express patient households’ consumption as a function of that of impatient ones and exogenous variables

(A.24)c^P,t=ac^I,t+exot

where aφωc+ωnφ(1ωc)+1ωn>0 and exot denotes any linear function of exogenous variables up to time t, as defined in Appendix A.2.

Plugging this condition into the model equations and restricting credit to be at its exogenous target at all times leads to the following system of equations

(A.25)ac^I,t+aEt{c^I,t+1}+R^t=exot
(A.26)scc^I,t+1βP+R^t1tlt^t=exot
(A.27)βIβPp^χ,tτχ,t+βIEt{p^χ,t+1+τχ,t+1}+c^I,tβIEt{c^I,t+1}+(1βIβP)(Θ^t+m^t)=exot
(A.28)c^I,t+τl,tβIβPEt{c^I,t+1}+βIβPR^t+(1βIβP)Θ^t=exot
(A.29)m^t+p^χ,t=exot

where sccIl+wnIφl>0. If we use equation (A.26) to eliminate c^I,t and equation (A.28) to eliminate Θ^t, and equation (A.29) to eliminate p^χ,t, the system further reduces to

(A.30)ascβPR^t1asctlt^t+(1ascβP)R^t+asctlEt{t^t+1}=exot
(A.31)βPβI(τχ,tm^t)βPEt{τχ,t+1m^t+1}1βPsctlEt{t^t+1}βPβIτl,t+(1+1βPscβP)R^t=exot

Let us first assume that redistributive taxation is not available. Then, since both a and sc are strictly positive, equation (A.30) implies that, for a stable equilibrium to exist, we must have 2aβPsc. This restriction obtains irrespective of which macroprudential instrument is used.

Let us now work more on the formula for sc. First note that the steady state Euler equations (A.1) and (A.4), evaluated at the steady state, imply

(A.32)Θ=cI1(1βIβP)

This formula can be used to substitute for Θ in the housing Euler equation (A.6), which, after using the collateral constraint (A.3) to substitute for pχχI, gives

(A.33)cIl=βI(1βP)AχβP

Next note that the budget constraint of impatient households (A.2) evaluated at the steady state implies

(A.34)wnIl=cIl+βP11

Now we are ready to derive the formula for sc as a function of deep model parameters

(A.35)sc=cIl+φ1wnIl=(1+φ1)cIl+φ1(βP11)=1βPφβP(1+βIφ+1Aχ)

Plugging this equation into condition 2aβPsc gives

(A.36)φωc+ωnφ(1ωc)+1ωn1βP2φ(1+βIφ+1Aχ)

which is restriction (22) in the main text.

The condition derived above ensures that the policy rate R^t is stable in response to shocks. To verify whether this also guarantees stability of a macroprudential instrument chosen to complement monetary policy, we need to examine equation (A.31). The stability of τl,t follows immediately, but also that of m^t and τχ,t since 0<βI<1.

Finally, suppose that, instead of the three macroprudential instruments, we use redistributive taxes t^t. Then, system (A.30), (A.31) can be cast in the following matrix notation

(A.37)Γ0[R^tEtt^t+1]=Γ1[R^t1t^t]+Exot

where

(A.38)1βPsctlEt{t^t+1}+(1+1βPscβP)R^t=exot
Γ0=[1ascβPasctl1+1βPscβP1βPsctl]Γ1=[ascβPasctl00]

Note that the determinant of Γ0 is (1+aβP)tlsc1, which is non-zero for t > 0, except for very special parametrization, so we can evaluate Γ=Γ01Γ1. One of its eigenvalues is 0, i.e. lies inside the unit circle, so there exist at least one stable solution to system (A.30), (A.31), see Blanchard and Kahn (1980).

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Received: 2018-08-28
Accepted: 2020-05-13
Published Online: 2020-07-08

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