Leverage, investment, and optimal monetary policy

Filippo Occhino; Andrea Pescatori

doi:10.1515/bejm-2013-0113

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Leverage, investment, and optimal monetary policy

Published/Copyright: June 21, 2014

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From the journal The B.E. Journal of Macroeconomics Volume 14 Issue 1

Abstract:

We study optimal monetary policy in an economy where firms’ debt overhangs lead to under-investment and under-production. The magnitude of this debt-induced distortion varies over the business cycle, rising significantly during recessions. When debt is contracted in nominal terms, this distortion gives rise to a balance sheet channel for monetary policy. In the presence of real and financial shocks, the monetary authority faces a trade-off between inflation and output gap stabilization. The optimal monetary policy rule prescribes that the anticipated component of inflation should be set equal to a target level, while the unanticipated component should rise in response to adverse shocks, smoothing the debt overhang distortion and the output gap.

Keywords: balance sheet channel of monetary policy; debt overhang; Fisher debt-deflation

JEL Classification:: E32; E52

Corresponding author: Filippo Occhino, Research Department, Federal Reserve Bank of Cleveland, 1455 East 6th Street, Cleveland, OH 44114, USA, e-mail: filippo.occhino@clev.frb.org

aThe views expressed herein are those of the authors and do not necessarily reflect those of the Federal Reserve Bank of Cleveland or of the Board of Governors of the Federal Reserve System.bThe views expressed herein are those of the authors and should not be attributed to the IMF, its Executive Board, or its management.

Acknowledgments

We would like to thank the Editor, an anonymous reviewer, and seminar participants at the 2012 Central Bank Macroeconomic Modeling Workshop in Warsaw, the 2011 Joint Central Bank Conference in Zurich, and Texas A&M University.

Appendices

A Derivation of equations (5) and (6)

This appendix derives equations (5) and (6) from the system of equations (1) through (4), which describes the equilibrium of the model.

Using the definition y≡ωθAk^α, and the expressions for k and b from equations (1) and (3), equation (2) becomes:

ν=ln(E{ω}θAkα)−ln(b)σ+12σν=1σ{ln(E{ω}θA)+αln(E{ω}θAαΦ(ν))1−α−ln(b¯e−(π−E0π))}+12σν=1σ{ln(E{ω}θA)1−α+αln(αΦ(ν))1−α−ln(b¯)+(π−E0π)}+12σ

which is equation (5).

Taking a first-order Taylor expansion of equation (5) with respect to ν, ln(θ), (π–E₀π) and σ, we obtain:

ν^=1σ¯{11−αθ˜+α1−αΦ′(ν¯)Φ(ν¯)ν^+(π−E0π)−(ν¯−σ¯)σ˜}ν^=11−χ¯1σ¯{11−αθ˜+(π−E0π)−(ν¯−σ¯)σ^}

where χ¯≡1σ¯α1−αΦ′(ν¯)Φ(ν¯)∈(0,1); bars over variables denote steady state values derived in the absence of aggregate shocks; while hats and tildes over variables denote, respectively, deviations and log-deviations from steady state values.^[11]

Finally, taking a first-order Taylor expansion of equation (4), and using the previous result, we obtain:

x^=α1−αΦ′(ν¯)Φ(ν¯)ν^x^=α1−αΦ′(ν¯)Φ(ν¯)11−χ¯1σ¯{11−αθ˜+(π−E0π)−(ν¯−σ¯)σ^}x^=χ¯1−χ¯{11−αθ˜+(π−E0π)−(ν¯−σ¯)σ^}

which is equation (6).

B Solution of the optimal policy problem (9)

This appendix solves the optimal monetary policy problem (9).

Notice that monetary policy rules that differ by a scale factor (P′≡κP, or equivalently π′≡ln(κ)+π) induce the same real allocation. If you double up the price level, the face value of debt doubles and there is no change in the real allocation. However, the second term in the objective function, equal to E0(π−π¯)2=E0(π−E0π)2+(E0π−π¯)2, is minimized by the rule such that E0π=π¯. Then, without loss of generality, we can restrict attention to rules where the expected value of inflation is the target inflation rate, adding the constraint

(11)

E0π=π¯ (11)

to the optimal problem (9). Other rules have the same output gap but a larger inflation gap, so they are not optimal.

The optimal problem (9), then, becomes:

minπ(s) (8) subject to: (11), (4), and (5).

After substituting the expression E0π=π¯ in the constraint (5),

minπ(s)12E0{γx2+(1−γ)(π−π¯)2}subject to: E0π=π¯,x=α1−αln(Φ(ν)),ν=1σ{ln(E{ω}θA)1−α+αln(αΦ(ν))1−α−ln(b¯)+(π−π¯)}+12σ.

Equivalently,

minπ(s)12E0{γx2+(1−γ)(π−π¯)2}subject to: E0π=π¯,where: x≡α1−αln(Φ(ν)),ν≡1σ{ln(E{ω}θA)1−α+αln(αΦ(ν))1−α−ln(b¯)+(π−π¯)}+12σ.

After defining π^≡π−π¯,

minπ^(s)12E0{γx2+(1−γ)π^2}subject to: E0π^=0,where: x≡α1−αln(Φ(ν)),ν≡1σ{ln(E{ω}θA)1−α+αln(αΦ(ν))1−α−ln(b¯)+π^}+12σ.

Letting q(s) be the probability of the aggregate state,

minπ^(s)12∑s{γx2+(1−γ)π^2}q(s)subject to: ∑sπ^q(s)=0,where: x≡α1−αln(Φ(ν)),ν≡1σ{ln(E{ω}θA)1−α+αln(αΦ(ν))1−α−ln(b¯)+π^}+12σ.

Letting λ be the Lagrange multiplier (constant across states), the Lagrangian is

ℒ=∑s{12γx2+12(1−γ)π^2+λπ^}q(s)where: x≡α1−αln(Φ(ν)),ν≡1σ{ln(E{ω}θA)1−α+αln(αΦ(ν))1−α−ln(b¯)+π^}+12σ.

Taking the first-order conditions with respect to π^(s), for all states s,

γx∂x∂π^+(1−γ)π^+λ=0where: ∂x∂π^=α1−αΦ′(ν)Φ(ν)∂ν∂π^,∂ν∂π^=1σ{α1−αΦ′(ν)Φ(ν)∂ν∂π^+1}.

Letting χ≡1σα1−αΦ′(ν)Φ(ν), we obtain

∂ν∂π^≡χ∂ν∂π^+1σ∂ν∂π^≡1σ11−χ

and then

∂x∂π^=α1−αΦ′(ν)Φ(ν)1σ11−χ∂x∂π^=χ1−χ

so the previous first-order conditions become:

γxχ1−χ+(1−γ)π^+λ=0.

To obtain an expression for λ, we take the expectation of both sides and use E0π^=0:

E0{γxχ1−χ+(1−γ)π^+λ}=0λ=−γE0{xχ1−χ}.

Substituting this expression for λ into the previous first-order conditions, and using π^≡π−π¯, we obtain

γ{xχ1−χ −E0{xχ1−χ}}+(1−γ)(π−π¯)=0

where χ≡1σα1−αΦ′(ν)Φ(ν). This is solution (10).

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Published Online: 2014-6-21

Published in Print: 2014-1-1

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https://doi.org/10.1515/bejm-2013-0113

Keywords for this article

balance sheet channel of monetary policy; debt overhang; Fisher debt-deflation