Additional appendix relating to the article 'Fiscal sustainability in a new Keynesian model', forthcoming in the Journal of Money, Credit and Banking

Leith, C.

Wren-Lewis, S.

English

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Leith, C., and Wren-Lewis, S. (2012) Fiscal sustainability in a new 
Keynesian model - additional appendix 
 
http://eprints.gla.ac.uk/70226 
 
Deposited on: 3 October 2012 
 
 
Fiscal Sustainability in a New Keynesian Model
- Additional Appendix (Not for Publication)
Campbell Leith∗
University of Glasgow
Simon Wren-Lewis
University of Oxford
This version, September 2012.
Abstract: Recent work on optimal monetary and fiscal policy in New Key-
nesian models suggests that it is optimal to allow steady-state debt to follow
a random walk. Leith and Wren-Lewis (2012) consider the nature of the time-
inconsistency involved in such a policy and its implication for discretionary
policy-making. We show that governments are tempted, given inflationary ex-
pectations, to utilize their monetary and fiscal instruments in the initial period
to change the ultimate debt burden they need to service. We demonstrate that
this temptation is only eliminated if following shocks, the new steady-state debt
is equal to the original (efficient) debt level even though there is no explicit
debt target in the government’s objective function. Analytically and in a se-
ries of numerical simulations we show which instrument is used to stabilize the
debt depends crucially on the degree of nominal inertia and the size of the
debt-stock. We also show that the welfare consequences of introducing debt are
negligible for precommitment policies, but can be significant for discretionary
policy. Finally, we assess the credibility of commitment policy by considering a
quasi-commitment policy which allows for different probabilities of reneging on
past promises. This on-line Appendix extends the results of this paper.
JEL Codes: E62, E63
Keywords: New Keynesian Model; Government Debt; Monetary Policy; Fis-
cal Policy; Credibility.
∗We would like to thank the editor, Paul Evans, three anonymous refer-
ees, Javier Andres, Florin Bilbiie, Tatiana Kirsanova, Luisa Lambertini, Ioana
Moldovan, Raffaele Rossi, Martin Uribe and participants at seminars at the
Universities of Oxford and Valencia and the Bank of Spain for very helpful
comments. We would also like to thank Tatiana Kirsanova and Christoph Him-
mels for providing us with computer code to facilitate the solution of the model
under quasi-commitment. All errors remain our own. We are also grateful to
the ESRC, Grant Nos.RES-156-25-003 and RES-062-23-1436, for financial assis-
tance. Address for correspondence: Campbell Leith, Economics, West Quadran-
gle, Gilbert Scott Building, University of Glasgow, Glasgow G12 8QQ. E-mail
campbell.leith@glasgow.ac.uk.
1
1 Introduction
In Leith and Wren-Lewis (2012) we consider optimal monetary and fiscal policy
in the face of shocks which raise the level of debt, where the policy maker can
either act under full commitment, discretion or an intermediate case of quasi-
commitment. This appendix complements the analysis of the paper, by present-
ing additional results. Firstly, Section 2 explores the implications of treating
government spending as an exogenous stream that needs to be financed, rather
than as a policy instrument which can be chosen optimally. In the main paper
we conduct a robustness analysis by varying the degree of price stickiness and
the initial debt-gdp ratio. Section 3 extends that robustness analysis by varying
the remaining key model parameters, the intertemporal elasticity of substitu-
tion, 1/σ, and the labor supply parameter, ϕ. Across all parameter variants
the key results of the main paper are unaffected. Finally, section 4 provides
the matrix representation of the model considered in the paper in a manner
which enables us to nest the commitment, discretion and quasi-commitment
descriptions of policy considered in the paper. We also outline the numerical
algorithms of Himmels and Kirsanova (2012) which we use to solve the model
under quasi-commitment.
2 Exogenous Government Spending
In the main paper we allowed government spending to be chosen optimally
by the policy maker. We did so to reflect the fact that all the major fiscal
consolidations analyzed by the IMF(2012) relied heavily on spending cuts to
stabilize debt, as it was argued that electorates were increasingly resistant to tax
increases. We found that, contrary to observed behavior, optimal policy would
not imply significant adjustment of the government spending gap in an attempt
to stabilize debt. While, some adjustment through tax revenues was possible,
particularly at low debt-gdp ratios and high degrees of price stickiness, typically
time-consistent fiscal stabilization relied on the reduction of debt service costs
through a relaxation of monetary policy. In this section, for completeness, we
reassess those results when we assume that government spending is not used
as an instrument in the policy maker’s problem. As would be expected from
the earlier results, this does not significantly change the results presented in the
main paper. Here we present the Figures analogous to Figures 1 and 2 in the
main paper.
2
−
0
.4
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.3
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.2
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05
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5 −0.05
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1
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1
1
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
Coefficient G[1,1]
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 1: Evolution of Debt under Discretion as a function of Price Stickiness
and the Debt/GDP Ratio - Exogenous G.
3
1 2 3 4 5 6 7 8 9 10
−0.05
0
0.05
0.1
0.15
 
 
1 2 3 4 5 6 7 8 9 10
−0.5
0
0.5
1
 
 
1 2 3 4 5 6 7 8 9 10
−0.5
0
0.5
1
 
 
1 2 3 4 5 6 7 8 9 10
−0.1
0
0.1
0.2
0.3
 
 
1 2 3 4 5 6 7 8 9 10
−1
−0.5
0
0.5
1
 
 
1 2 3 4 5 6 7 8 9 10
−1
−0.5
0
0.5
 
 
Price Inflation
Price Inflation(C)
Tax 
Tax(C)
Debt
Debt(C)
Output Gap
Output Gap(C)
Government Spending Gap
Government Spending Gap(C)
Interest Rate Gap
Interest Rate Gap(C)
Figure 2: Response to Debt Shock under Commitment (C) and Discretion -
Exogenous G.
4
3 Robustness Analysis
In this section we re-create some of the Figures from the main paper under
alternative parameters. To begin with we allow for alternative values for the
intertemporal elasticity of substitution, 1/σ, specifically σ = 1 and 3, which
covers the ranges of this parameter typically considered in the macro literature.
5
−0.2
−0.2
−0.2
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−0
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−0.1
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0
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−0.1
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−0.1
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−0.05
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5
−0
.0
5
−0
.0
5
−
0
.0
5
−0.05
−0.05 −0.05 −0.05
−0.05
0
0
0
0
0
0.1
0
.1
0
.1
0
.1
0.1
0.1
0.3
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3
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0.3
0.3
0.5
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5
0
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0
.5
0.5
0.7
0.
7
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0.7
0.7
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.7
0.9
0
.9
0
.9
0.
9
0
.9
0
.9
0
.9
0
.9
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 3: Evolution of Debt Under Discretion - σ = 1
6
−
0
.4
−0
.4
−0.
4
−0
.3
−
0
.3
−0.3
−0
.3
−0
.3
−
0.3
−0.3
−0
.3
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.2
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.2
−
0.2
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.2
−0
.2
−0
.2
−
0.2
−0.2
−0.2
−0.2
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−0.1
−0.
1
−0
.1
−0
.1
−
0
.1
−
0.1
−0.1 −0.1
−0.1
−0.1
−0.1
−0.05−0.05−0.05
−0.05
−0
.0
5
−0
.0
5
−0
.0
5
−
0
.0
5
−
0.05
−0.05 −0.05 −0.05 −0.05 −0.05
0
0
0
0
0
0
.1
0
.1
0.1
0
.1
0
.1
0
.1
0.1
0.1
0.3
0
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0
.3
0
.3
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0
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0
.3
0.5
0
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0
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0.7
0
.7
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.7
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7
0.7
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7
0.9
0
.9
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.9
0.
9
0
.9
0
.9
0
.9
0
.9
1 1
1
1
1
1
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 4: Evolution of Debt under Discretion - σ = 3
We now consider the same Figure, but with additional values of the labour
supply parameter ϕ = 2 and 3, having returned to the benchmark calibration
σ = 2.
7
−0.2
−0.2
−0.2
−0
.2
−0
.2
−
0.2
−0.2
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.2
−0
.2
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0
.2
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−0.1
−0
.1
−0
.1
−
0
.1
−
0
.1
−0.1 −0.1 −0.1
−0.1
−0.05−0.05−0.05
−0.05
−0
.0
5
−0
.0
5
−
0
.0
5
−
0
.0
5
−0.05
−0.05 −0.05 −0.05 −0
.05
0
0
0
0
0
0 1
0.
1
0
.1
0
.1
0
.1
0.1
0.3
0.
3
0
.3
0
.3
0.3
0.3
0.5
0.
5
0
.5
0
.5
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0.5
0.7
0
.7
0
.7
0.
7
0.7
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0
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0
.7
0.
7 0.9
0.9
0
.9
0
.9
0.9
0
.9
0
.9
0
.9
0
.9
1
1
1
1
1
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 5: Evolution of Debt under Discretion - ϕ = 2
−0.2
−0.2−0.2
−0
.2
−0
.2
−
0.2
−0.2
−0.2
−0
.2
−0
.2
−0.1−0.1
−0.1
−0.1
−0
.1
−0
.1
−
0
.1
−0.1
−0.1 −0.1 −
0.1
−0.05−0.05
−0.05
−0.
05
−0
.0
5
−0
.0
5
−
0
.0
5
−
0.05
−0.05 −0.05 −0.05 −0.05
0
0
0
0
0
0.1
0.
1
0
.1
0
.1
0.1
0.1
0.1
0.3
0.
3
0
.3
0
.3
0.3
0.3
0.5
0.
5
0
.5
0.5
0.5
0.5
0.7
0.7
0
.7
0.
7
0.
7
0.7
0
.7
0
.7
0
.7
0.
7
0.9
0
.9
0
.9
0.
9
0.9
0
.9
0
.9
0
.9
0
.9
1
1
1
1
1P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 6: Evolution of Debt under Discretion - ϕ = 3
8
We now turn to Figure 4 in the paper which considers the relative contri-
bution to debt stabilization under discretion across the same parameter per-
mutations. Firstly, variations in the inverse of the intertemporal elasticity of
substitution.
−0.1
−0.1 −0
.1
0
0
0
0.2
0.
2
0.2
0.2
0.
5
0.
5
0.5
0.5
0.
8
0.
8
0.
8
0.
8
0.8
0.8
1
1
1 1
1
.1
1.
1
1.1
Contribution of Tax Instruments to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.
05
0.
05
0.05 0.05
0.
2
0.
2
0.2
0.2
0.
5 0.5
0.5
0.
8 0.8
0.8
0 8
1 1 1
1 1 1
Contribution of Surprise Inflation to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−0
.1
−0.1 −0.1
0
0
0 0
0
.2
0.
2
0.2
0.2
0.2 0
.5
0.
5
0.5
0.5
0.8
0.
8
0.8
0.8
1
1
1
Contribution of Monetary Accomodation to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−0
.2
−0
.1
−0
.1
−0.1
−
0.05
−0.05
−0
.0
5
−0.05
−0.02
−0
.02
−0.02
−0.02−0.02
−0.01
−0.01
−0.01 −0.01
Contribution of Government Spending to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 7: Contribution to Debt Stabilisation under Discretion - σ = 1
9
−0.2
−0
.2
−0.2
−0
.2
−0.1
−0
.1
−
0
.1
−0.1
−0.1
0
0
0
0
0
.2
0.
2
0.2
0.2
0
.5
0.
5
0.5
0.5
0
.8
0.8
0
.8
0.
8
0.8 0.8
0.
8
1
1
1 1
1.1
1.
1
1.1 1.1
Contribution of Tax Instruments to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
.0
5
0.05
0.05
0.05
0.
2
0.2
0.2
0.2
0.5
0.5 0.5
0.8
0.8 0.8
1
1
1 1
Contribution of Surprise Inflation to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−0.1
−0.1
−0.1
0
0
0 0
0
.2
0.
2
0.
2
0.2 0.2
0
.5
0.
5
0.5
0.5
0
.8
0.
8
0.8
0.8
1
1
1
1
1.
2
1.2
1.
2
1.2
1
.2
Contribution of Monetary Accomodation to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−
0
.2
−
0
.1
−0
.1
−
0.1
−0
.0
5
−0
.0
5
−0.05
−
0
.0
2
−0.
02
−
0
.0
2
−0.02
−0.01
−0
.0
1
−0.01
−0.01
Contribution of Government Spending to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 8: Contribution to Debt Stabilisation under Discretion - σ = 3
Similarly for the alternative values of the Frisch elasticity.
.
10
−0
.2
−0.2
−0
.2
−0.1
−0.1
−0
.1
−0.1
−0.1
0
0
0
0
0.
2
0.
2
0.2
0.2
0.
5
0.
5
0.5
0.5
0
.8
0.8
0.8
0.
8
0.8
0.8
0.8
1
1
1 1
1
.1 1
.1
1.1
Contribution of Tax Instruments to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.
05
0.
05
0.05
0.05
0.
2
0.
2
0.2
0.2
0.
5
0.5
0.5
0.
8 0.8 0.81 1 11 1
Contribution of Surprise Inflation to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−0
.1
−0.1 −0.1
0 0
0 0
0
.2
0.
2
0.2
0.2
0.2
0.
5
0.
5
0.5
0.5
0.
8
0.
8
0.8
0.8
1
1
1
1
1.
2
1.2
1
.2
1.2
Contribution of Monetary Accomodation to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−
0
.2
−
0
.2
−0
.1
−0
.1
−0.1
−0
.0
5
−0.
05
−0
.0
5
−0.05
−0.02
−0
.0
2
−0.02
−0.02
−0.
02
−0.01
−0.01
−0.01 −0.01
Contribution of Government Spending to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 9: Contribution to Debt Stabilisation under Discretion - ϕ = 2
4 Quasi-Commitment
In this section we we follow Himmels and Kirsanova (2012) in recasting the
quasi-commitment of Schaumburg and Tambalotti (2007) and Debertoli and
Nunes (2010) in a general linear-quadratic form which can be solved using stan-
dard iterative techniques. This also nests the cases of commitment (α = 0) and
discretion (α = 1) which can be solved using the computer codes of Soderlind
(1999). We can write our model in the standard form as,
∙
yt+1
Etxt+1
¸
= A
∙
yt
Etxt
¸
+But +Cξt
where yt =
⎡
⎣
bt
ygt−1
at−1
⎤
⎦, xt =
∙
cgt
πt
¸
, are the vectors of predetermined and
jump variables respectively, while ut =
⎡
⎣
bτ t
ggt
rgt
⎤
⎦ is the vector of controls. The
coefficient matrices are defined as, A = [A0]−1A1, B = [A0]−1B1 and C =
11
−0.2
−0.2
−0
.2
−0.1
−0
.1
−0.1 −0.1
0
0
0
0.2
0.
2
0.2
0.2
0.
5
0.
5
0.5
0.5
0
.8
0.8
0.
8
0.
8
0.
8
0.8
0.8
1 1
1 1
1.
1
1.1 1.1
Contribution of Tax Instruments to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.
05
0
.0
5
0.05
0.05
0.2
0.
2
0.2
0.2
0.
5
0.5
0.5
0.8
0.8 0.8
1 1 11 1
Contribution of Surprise Inflation to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−0
.1
−0.1 −0.1
0
0 0
0
.2
0.
2
0.2
0.2
0.2
0.
5
0.
5
0.5
0.5
0.
8
0.
8
0.8
0.8
1
1
1
1.2
1.
2
1.2
1.2
Contribution of Monetary Accomodation to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−0
.2
−
0.2
−0
.1
−0
.1
−0.1
−
0
.0
5
−0.05
−0
.0
5
−
0.05
−0.05
−0.02−0.02
−
0
.0
2
−
0
.0
2
−0.02
−0.02
−0
.0
2
−0 01
−0.01
−0.01
−0.01
Contribution of Government Spending to Debt Adjustment
P
ri
ce
 S
tic
ki
n
e
ss
 θ p
Debt/GDP ratio (%)
20 40 60 80 100 120 140 160 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 10: Contribution to Debt Stabilisation under Discretion - ϕ = 3
[A0]
−1C1 where,
A0 =
⎡
⎢⎢⎢⎢⎣
β wNτ
b
(1 + ϑ) f −σβ −β
0 1 0 0 0
0 0 1 0 0
0 0 0 1 σ−1
0 γϕ 0 0 β
⎤
⎥⎥⎥⎥⎦
, A1 =
⎡
⎢⎢⎢⎢⎣
1 0 0 σ(β + wNτ
b
) −1
0 0 0 θ 0
0 0 ρa 0 0
0 0 0 1 0
0 0 0 −γσ 1
⎤
⎥⎥⎥⎥⎦
B1 =
⎡
⎢⎢⎢⎢⎣
−wNτ
b
1
1−τ
G
b
0
0 1− θ 0
0 0 0
0 0 σ−1
−γ τ1−τ 0 0
⎤
⎥⎥⎥⎥⎦
The objective function for the policy maker can be written as,
Γ = −N1+ϕ 1
2
E0
∞X
t=0
βt{σθ(bcgt )2 + σ(1− θ)( bGgt )2 + ϕ(bY gt )2 + γ π2t}
= −N1+ϕ 1
2
E0
∞X
t=0
βt{
⎡
⎣
yt
xt
ut
⎤
⎦
0
D0QD
⎡
⎣
yt
xt
ut
⎤
⎦}
12
where
Q =
⎡
⎢⎢⎣
σθ 0 0 0
0 σ(1− θ) 0 0
0 0 ϑ 0
0 0 0 ελ
⎤
⎥⎥⎦ and D =
⎡
⎢⎢⎣
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 θ 0 0 (1− θ) 0
0 0 0 0 1 0 0 0
⎤
⎥⎥⎦
This enables us to write the policy problem for the policy maker operating under
quasi commitment as,
min−N1+ϕ 1
2
E0
∞X
t=0
((1−α)β)t{
⎡
⎣
yt
xt
ut
⎤
⎦
0
D0QD
⎡
⎣
yt
xt
ut
⎤
⎦+αβy0t+1Sdyt+1} (1)
subject to,
yt+1 = A11yt +A12xt +B1ut +Cξt (2)
(1− α)Etxt+1 + αHyt+1 = A21yt +A22xt +B2ut (3)
where y0t+1S
dyt+1 = V (bt+1, at) is the value function of the corresponding
problem under discretion, and H is the link between jump and state variables
for that same problem Therefore as α tends towards zero we move towards
the commitment problem, and as α tends to 1 discretion. More generally, in
the second line of the constraints, we can see that as we increase α the policy
maker is forced to take expectations as given by the solution to the discretionary
problem, while for lower values of α they are able to commit in a manner which
successfully manipulates expectations. In the absence of any endogenous state
variables, this would reduce to the case where expectations are increasingly
taken as given by the policy maker as α increases. The policy problem is then
solved using the computer codes developed in Himmels and Kirsanova (2012),
which corresponds to those developed by Soderlind (1999) in the special cases
of commitment (α = 0) and discretion (α = 1).
Given H and Sd, the combination of the model and focs for the policy
problem can be solved via Schur decomposition to generate a solution in the
following form,
⎡
⎣
ut
xt
ψt
⎤
⎦ = X
∙
yt
ϕt
¸
,
∙
yt+1
ϕt+1
¸
= M
∙
yt
ϕt
¸
, and (4)
W (yt+1,ϕt) =
1
2
∙
yt
ϕt
¸0
U
∙
yt
ϕt
¸
where ψt and ϕt are the vectors of lagrange multipliers associated with the
constraints in (2) and (3), respectively. Himmels and Kirsanova (2012) then
suggest the following solution algorithm.
13
• GuessM , X and U (which implicitly includes guesses of H and Sd).
• Use these guesses to update U by using the guessed solution in (1).
• Obtain the solution to the model and FOCs conditional on the last itera-
tion to updateM and X.
• Continue to iterate until there is no further updating ofM , X and U.
References
[1] Debertoli, D. and R. Nunes (2010). “Fiscal Policy under Loose Commit-
ment”, Journal of Economic Theory 145(3), 1005–1032.
[2] Himmels. C. and T. Kirsanova (2012), “Expectations Traps and Monetary
Policy under Limited Commitment”, Journal of Economic Dynamics and
Control, forthcoming.
[3] IMF (2012): “Balancing Policy Risks,” Fiscal Monitor, April 2011.
[4] Leith, C. and S, Wren-Lewis (2012), “Fiscal Sustainability in a New Keyne-
sian Model”, Journal of Money, Credit and Banking, forthcoming.
[5] Schaumburg, E. and A. Tambalotti (2007), “An Investigation of the Gains
from Commitment in Monetary Policy", Journal of Monetary Economics 54
(2), 302–324
[6] Soderlind, P. (1999), “Solution and Estimation of RE Macromodels with
Optimal Policy”, European Economic Review, 43, pp 813-823.
14


Fiscal sustainability in a new Keynesian model - additional appendix

Enlighten

Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk        Leith, C., and Wren-Lewis, S. (2012) Fiscal sustainability in a new Keynesian model - additional appendix  http://eprints.gla.ac.uk/70226  Deposited on: 3 October 2012   Fiscal Sustainability in a New Keynesian Model- Additional Appendix (Not for Publication)Campbell Leith∗University of GlasgowSimon Wren-LewisUniversity of OxfordThis version, September 2012.Abstract: Recent work on optimal monetary and fiscal policy in New Key-nesian models suggests that it is optimal to allow steady-state debt to followa random walk. Leith and Wren-Lewis (2012) consider the nature of the time-inconsistency involved in such a policy and its implication for discretionarypolicy-making. We show that governments are tempted, given inflationary ex-pectations, to utilize their monetary and fiscal instruments in the initial periodto change the ultimate debt burden they need to service. We demonstrate thatthis temptation is only eliminated if following shocks, the new steady-state debtis equal to the original (eﬃcient) debt level even though there is no explicitdebt target in the government’s objective function. Analytically and in a se-ries of numerical simulations we show which instrument is used to stabilize thedebt depends crucially on the degree of nominal inertia and the size of thedebt-stock. We also show that the welfare consequences of introducing debt arenegligible for precommitment policies, but can be significant for discretionarypolicy. Finally, we assess the credibility of commitment policy by considering aquasi-commitment policy which allows for diﬀerent probabilities of reneging onpast promises. This on-line Appendix extends the results of this paper.JEL Codes: E62, E63Keywords: New Keynesian Model; Government Debt; Monetary Policy; Fis-cal Policy; Credibility.∗We would like to thank the editor, Paul Evans, three anonymous refer-ees, Javier Andres, Florin Bilbiie, Tatiana Kirsanova, Luisa Lambertini, IoanaMoldovan, Raﬀaele Rossi, Martin Uribe and participants at seminars at theUniversities of Oxford and Valencia and the Bank of Spain for very helpfulcomments. We would also like to thank Tatiana Kirsanova and Christoph Him-mels for providing us with computer code to facilitate the solution of the modelunder quasi-commitment. All errors remain our own. We are also grateful tothe ESRC, Grant Nos.RES-156-25-003 and RES-062-23-1436, for financial assis-tance. Address for correspondence: Campbell Leith, Economics, West Quadran-gle, Gilbert Scott Building, University of Glasgow, Glasgow G12 8QQ. E-mailcampbell.leith@glasgow.ac.uk.11 IntroductionIn Leith and Wren-Lewis (2012) we consider optimal monetary and fiscal policyin the face of shocks which raise the level of debt, where the policy maker caneither act under full commitment, discretion or an intermediate case of quasi-commitment. This appendix complements the analysis of the paper, by present-ing additional results. Firstly, Section 2 explores the implications of treatinggovernment spending as an exogenous stream that needs to be financed, ratherthan as a policy instrument which can be chosen optimally. In the main paperwe conduct a robustness analysis by varying the degree of price stickiness andthe initial debt-gdp ratio. Section 3 extends that robustness analysis by varyingthe remaining key model parameters, the intertemporal elasticity of substitu-tion, 1/σ, and the labor supply parameter, ϕ. Across all parameter variantsthe key results of the main paper are unaﬀected. Finally, section 4 providesthe matrix representation of the model considered in the paper in a mannerwhich enables us to nest the commitment, discretion and quasi-commitmentdescriptions of policy considered in the paper. We also outline the numericalalgorithms of Himmels and Kirsanova (2012) which we use to solve the modelunder quasi-commitment.2 Exogenous Government SpendingIn the main paper we allowed government spending to be chosen optimallyby the policy maker. We did so to reflect the fact that all the major fiscalconsolidations analyzed by the IMF(2012) relied heavily on spending cuts tostabilize debt, as it was argued that electorates were increasingly resistant to taxincreases. We found that, contrary to observed behavior, optimal policy wouldnot imply significant adjustment of the government spending gap in an attemptto stabilize debt. While, some adjustment through tax revenues was possible,particularly at low debt-gdp ratios and high degrees of price stickiness, typicallytime-consistent fiscal stabilization relied on the reduction of debt service coststhrough a relaxation of monetary policy. In this section, for completeness, wereassess those results when we assume that government spending is not usedas an instrument in the policy maker’s problem. As would be expected fromthe earlier results, this does not significantly change the results presented in themain paper. Here we present the Figures analogous to Figures 1 and 2 in themain paper.2−0.4−0.3−0.3−0.3−0.3−0.3−0.3−0.3−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.2 −0.2−0.2−0.2−0.2−0.2−0.1−0.1−0.1−0.1−0.1−0.1−0.1−0.1−0.1−0.1 −0.1−0.1−0.1−0.1−0.05−0.05−0.05−0.05−0.05−0.05−0.05−0.05−0.05−0.05 −0.05 −0.05 −0.05 −0.05000000.10.10.10.10.10 10.10.10.30.30.30.30.30.30.30.50.50.50.50.50.50.50.70.70.70.70.70.70.70.70.90.90.90.90.90.90.90.911111Price Stickiness θ pDebt/GDP ratio (%)Coefficient G[1,1]20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9Figure 1: Evolution of Debt under Discretion as a function of Price Stickinessand the Debt/GDP Ratio - Exogenous G.31 2 3 4 5 6 7 8 9 10−0.0500.050.10.15  1 2 3 4 5 6 7 8 9 10−0.500.51  1 2 3 4 5 6 7 8 9 10−0.500.51  1 2 3 4 5 6 7 8 9 10−0.100.10.20.3  1 2 3 4 5 6 7 8 9 10−1−0.500.51  1 2 3 4 5 6 7 8 9 10−1−0.500.5  Price InflationPrice Inflation(C)Tax Tax(C)DebtDebt(C)Output GapOutput Gap(C)Government Spending GapGovernment Spending Gap(C)Interest Rate GapInterest Rate Gap(C)Figure 2: Response to Debt Shock under Commitment (C) and Discretion -Exogenous G.43 Robustness AnalysisIn this section we re-create some of the Figures from the main paper underalternative parameters. To begin with we allow for alternative values for theintertemporal elasticity of substitution, 1/σ, specifically σ = 1 and 3, whichcovers the ranges of this parameter typically considered in the macro literature.5−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.1−0.1−0.1−0.1−0.1−0.1−0.1−0.1 −0.1−0.1−0.1−0.05−0.05−0.05−0.05−0.05−0.05−0.05−0.05−0.05 −0.05 −0.05−0.05000000.10.10.10.10.10.10.30.30.30.30.30.30.50.50.50.50.50.70.70.70.70.70.70.90.90.90.90.90.90.90.9Price Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9Figure 3: Evolution of Debt Under Discretion - σ = 16−0.4−0.4−0.4−0.3−0.3−0.3−0.3−0.3−0.3−0.3−0.3−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.1−0.1−0.1−0.1−0.1−0.1−0.1−0.1−0.1−0.1 −0.1−0.1−0.1−0.1−0.05−0.05−0.05−0.05−0.05−0.05−0.05−0.05−0.05−0.05 −0.05 −0.05 −0.05 −0.05000000.10.10.10.10.10.10.10.10.30.30.30.30.30.30.30.50.50.50.50.50.50.50.70.70.70.70.70.70.70.70.70.90.90.90.90.90.90.90.91 11111Price Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9Figure 4: Evolution of Debt under Discretion - σ = 3We now consider the same Figure, but with additional values of the laboursupply parameter ϕ = 2 and 3, having returned to the benchmark calibrationσ = 2.7−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.1−0.1−0.1−0.1−0.1−0.1−0.1−0.1−0.1 −0.1 −0.1−0.1−0.05−0.05−0.05−0.05−0.05−0.05−0.05−0.05−0.05−0.05 −0.05 −0.05 −0.05000000 10.10.10.10.10.10.30.30.30.30.30.30.50.50.50.50.50.50.70.70.70.70.70.70.70.70.7 0.90.90.90.90.90.90.90.90.911111Price Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9Figure 5: Evolution of Debt under Discretion - ϕ = 2−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.2−0.1−0.1−0.1−0.1−0.1−0.1−0.1−0.1−0.1 −0.1 −0.1−0.05−0.05−0.05−0.05−0.05−0.05−0.05−0.05−0.05 −0.05 −0.05 −0.05000000.10.10.10.10.10.10.10.30.30.30.30.30.30.50.50.50.50.50.50.70.70.70.70.70.70.70.70.70.70.90.90.90.90.90.90.90.90.911111Price Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9Figure 6: Evolution of Debt under Discretion - ϕ = 38We now turn to Figure 4 in the paper which considers the relative contri-bution to debt stabilization under discretion across the same parameter per-mutations. Firstly, variations in the inverse of the intertemporal elasticity ofsubstitution.−0.1−0.1 −0.10000.20.20.20.20.50.50.50.50.80.80.80.80.80.8111 11.11.11.1Contribution of Tax Instruments to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.90.050.050.05 0.050.20.20.20.20.5 0.50.50.8 0.80.80 81 1 11 1 1Contribution of Surprise Inflation to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9−0.1−0.1 −0.1000 00.20.20.20.20.2 0.50.50.50.50.80.80.80.8111Contribution of Monetary Accomodation to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9−0.2−0.1−0.1−0.1−0.05−0.05−0.05−0.05−0.02−0.02−0.02−0.02−0.02−0.01−0.01−0.01 −0.01Contribution of Government Spending to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9Figure 7: Contribution to Debt Stabilisation under Discretion - σ = 19−0.2−0.2−0.2−0.2−0.1−0.1−0.1−0.1−0.100000.20.20.20.20.50.50.50.50.80.80.80.80.8 0.80.8111 11.11.11.1 1.1Contribution of Tax Instruments to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.90.050.050.050.050.20.20.20.20.50.5 0.50.80.8 0.8111 1Contribution of Surprise Inflation to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9−0.1−0.1−0.1000 00.20.20.20.2 0.20.50.50.50.50.80.80.80.811111.21.21.21.21.2Contribution of Monetary Accomodation to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9−0.2−0.1−0.1−0.1−0.05−0.05−0.05−0.02−0.02−0.02−0.02−0.01−0.01−0.01−0.01Contribution of Government Spending to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9Figure 8: Contribution to Debt Stabilisation under Discretion - σ = 3Similarly for the alternative values of the Frisch elasticity..10−0.2−0.2−0.2−0.1−0.1−0.1−0.1−0.100000.20.20.20.20.50.50.50.50.80.80.80.80.80.80.8111 11.1 1.11.1Contribution of Tax Instruments to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.90.050.050.050.050.20.20.20.20.50.50.50.8 0.8 0.81 1 11 1Contribution of Surprise Inflation to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9−0.1−0.1 −0.10 00 00.20.20.20.20.20.50.50.50.50.80.80.80.811111.21.21.21.2Contribution of Monetary Accomodation to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9−0.2−0.2−0.1−0.1−0.1−0.05−0.05−0.05−0.05−0.02−0.02−0.02−0.02−0.02−0.01−0.01−0.01 −0.01Contribution of Government Spending to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9Figure 9: Contribution to Debt Stabilisation under Discretion - ϕ = 24 Quasi-CommitmentIn this section we we follow Himmels and Kirsanova (2012) in recasting thequasi-commitment of Schaumburg and Tambalotti (2007) and Debertoli andNunes (2010) in a general linear-quadratic form which can be solved using stan-dard iterative techniques. This also nests the cases of commitment (α = 0) anddiscretion (α = 1) which can be solved using the computer codes of Soderlind(1999). We can write our model in the standard form as,∙yt+1Etxt+1¸= A∙ytEtxt¸+But +Cξtwhere yt =⎡⎣btygt−1at−1⎤⎦, xt =∙cgtπt¸, are the vectors of predetermined andjump variables respectively, while ut =⎡⎣bτ tggtrgt⎤⎦ is the vector of controls. Thecoeﬃcient matrices are defined as, A = [A0]−1A1, B = [A0]−1B1 and C =11−0.2−0.2−0.2−0.1−0.1−0.1 −0.10000.20.20.20.20.50.50.50.50.80.80.80.80.80.80.81 11 11.11.1 1.1Contribution of Tax Instruments to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.90.050.050.050.050.20.20.20.20.50.50.50.80.8 0.81 1 11 1Contribution of Surprise Inflation to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9−0.1−0.1 −0.100 00.20.20.20.20.20.50.50.50.50.80.80.80.81111.21.21.21.2Contribution of Monetary Accomodation to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9−0.2−0.2−0.1−0.1−0.1−0.05−0.05−0.05−0.05−0.05−0.02−0.02−0.02−0.02−0.02−0.02−0.02−0 01−0.01−0.01−0.01Contribution of Government Spending to Debt AdjustmentPrice Stickiness θ pDebt/GDP ratio (%)20 40 60 80 100 120 140 160 1800.10.20.30.40.50.60.70.80.9Figure 10: Contribution to Debt Stabilisation under Discretion - ϕ = 3[A0]−1C1 where,A0 =⎡⎢⎢⎢⎢⎣β wNτb(1 + ϑ) f −σβ −β0 1 0 0 00 0 1 0 00 0 0 1 σ−10 γϕ 0 0 β⎤⎥⎥⎥⎥⎦, A1 =⎡⎢⎢⎢⎢⎣1 0 0 σ(β + wNτb) −10 0 0 θ 00 0 ρa 0 00 0 0 1 00 0 0 −γσ 1⎤⎥⎥⎥⎥⎦B1 =⎡⎢⎢⎢⎢⎣−wNτb11−τGb00 1− θ 00 0 00 0 σ−1−γ τ1−τ 0 0⎤⎥⎥⎥⎥⎦The objective function for the policy maker can be written as,Γ = −N1+ϕ 12E0∞Xt=0βt{σθ(bcgt )2 + σ(1− θ)( bGgt )2 + ϕ(bY gt )2 + γ π2t}= −N1+ϕ 12E0∞Xt=0βt{⎡⎣ytxtut⎤⎦0D0QD⎡⎣ytxtut⎤⎦}12whereQ =⎡⎢⎢⎣σθ 0 0 00 σ(1− θ) 0 00 0 ϑ 00 0 0 ελ⎤⎥⎥⎦ and D =⎡⎢⎢⎣0 0 0 1 0 0 0 00 0 0 0 0 0 1 00 0 0 θ 0 0 (1− θ) 00 0 0 0 1 0 0 0⎤⎥⎥⎦This enables us to write the policy problem for the policy maker operating underquasi commitment as,min−N1+ϕ 12E0∞Xt=0((1−α)β)t{⎡⎣ytxtut⎤⎦0D0QD⎡⎣ytxtut⎤⎦+αβy0t+1Sdyt+1} (1)subject to,yt+1 = A11yt +A12xt +B1ut +Cξt (2)(1− α)Etxt+1 + αHyt+1 = A21yt +A22xt +B2ut (3)where y0t+1Sdyt+1 = V (bt+1, at) is the value function of the correspondingproblem under discretion, and H is the link between jump and state variablesfor that same problem Therefore as α tends towards zero we move towardsthe commitment problem, and as α tends to 1 discretion. More generally, inthe second line of the constraints, we can see that as we increase α the policymaker is forced to take expectations as given by the solution to the discretionaryproblem, while for lower values of α they are able to commit in a manner whichsuccessfully manipulates expectations. In the absence of any endogenous statevariables, this would reduce to the case where expectations are increasinglytaken as given by the policy maker as α increases. The policy problem is thensolved using the computer codes developed in Himmels and Kirsanova (2012),which corresponds to those developed by Soderlind (1999) in the special casesof commitment (α = 0) and discretion (α = 1).Given H and Sd, the combination of the model and focs for the policyproblem can be solved via Schur decomposition to generate a solution in thefollowing form,⎡⎣utxtψt⎤⎦ = X∙ytϕt¸,∙yt+1ϕt+1¸= M∙ytϕt¸, and (4)W (yt+1,ϕt) =12∙ytϕt¸0U∙ytϕt¸where ψt and ϕt are the vectors of lagrange multipliers associated with theconstraints in (2) and (3), respectively. Himmels and Kirsanova (2012) thensuggest the following solution algorithm.13• GuessM , X and U (which implicitly includes guesses of H and Sd).• Use these guesses to update U by using the guessed solution in (1).• Obtain the solution to the model and FOCs conditional on the last itera-tion to updateM and X.• Continue to iterate until there is no further updating ofM , X and U.References[1] Debertoli, D. and R. Nunes (2010). “Fiscal Policy under Loose Commit-ment”, Journal of Economic Theory 145(3), 1005–1032.[2] Himmels. C. and T. Kirsanova (2012), “Expectations Traps and MonetaryPolicy under Limited Commitment”, Journal of Economic Dynamics andControl, forthcoming.[3] IMF (2012): “Balancing Policy Risks,” Fiscal Monitor, April 2011.[4] Leith, C. and S, Wren-Lewis (2012), “Fiscal Sustainability in a New Keyne-sian Model”, Journal of Money, Credit and Banking, forthcoming.[5] Schaumburg, E. and A. Tambalotti (2007), “An Investigation of the Gainsfrom Commitment in Monetary Policy", Journal of Monetary Economics 54(2), 302–324[6] Soderlind, P. (1999), “Solution and Estimation of RE Macromodels withOptimal Policy”, European Economic Review, 43, pp 813-823.14

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