fiscal consolidation in a disinflationary environment: price...
TRANSCRIPT
Fiscal Consolidation in a Disinflationary
Environment: Price- vs. Quantity-Based Measures
⇤
Evi Pappa† Rana Sajedi‡ Eugenia Vella§
January 2016
PRELIMINARY DRAFT
Abstract
An important feature of the current economic conditions in the EU, whichchallenges the design and implementation of macroeconomic policy, is inflationuncertainty. With monetary policy at the zero lower bound, and inflation wellbelow its target, a key issue for policy makers is the e↵ect this has on thetransmission of fiscal policy. We aim to address this question, in particularcomparing the e↵ects of price-based and quantity-based fiscal instruments.In this paper we focus on the public wage bill, and consider a DSGE modelin which the government can consolidate their debt through reductions in thepublic wage or public employment. We find that the low-inflation environmenteliminates the expansionary e↵ects of the reduction in the public wage bill inboth cases, with increased debt-to GDP levels during the consolidation process.Our results also indicate that consolidation through cuts in public wages takesless time than through cuts in public employment.
JEL classification: E32, E62
Keywords: fiscal consolidation, public wage bill, zero lower bound, un-employment
⇤Acknowledgements. The views expressed here in no way reflect those of the Bank of England.†Corresponding author: Department of Economics, European University Institute, Via della
Piazzuola 43, 50133 Florence, Italy, Tel: (+39) 055 4685 908, e-mail: [email protected]
‡
European University Institute and Bank of England. e-mail: [email protected]
§
Department of Economics, University of She�eld. e-mail:e.vella@she�eld.ac.uk
1
1 Introduction
An important feature of the current economic conditions in the EU, which challenges
the design and implementation of macroeconomic policy, is inflation uncertainty.
With monetary policy constrained by the zero lower bound (ZLB henceforth), in-
flation in the euro area has remained below the ECB’s medium-run objective for
some time. While some recent studies have looked at the impact of the ZLB on
fiscal policy, research on the di↵erential impact of inflation on di↵erent budgetary
items is limited. In this context, the aim of this paper is to examine the e↵ects
of alternative fiscal consolidation strategies to reduce the public wage bill, specifi-
cally comparing price-based measures and quantity-based measures, under di↵erent
inflation environments.
As seen in Figure 1, since 2012, the inflation rate across the euro area has been
trending downwards and still remains below the ECB’s 2% target. At the same
time, monetary policy easing has been subdued, with nominal interest rates at the
ZLB, and the e↵ects of unconventional measures, such as the recent asset purchases,
remaining uncertain.
This environment has important implications for fiscal policy. Firstly, low in-
flation is generally considered to make fiscal consolidation more di�cult. Indeed,
historically, periods of high inflation have been used to reduce debt-to-GDP ratios,
for example in many western countries following both the First and Second World
War (see Reinhart et al. 2015). From a theoretical point of view, low inflation re-
duces the growth in nominal GDP and, all else equal, raises deficit- and debt-to-GDP
ratios. Debt dynamics would be left unchanged if nominal interest rates fall by the
same magnitude as inflation, thus leaving real rates unchanged. Instead, when nom-
inal rates have hit the ZLB, falling inflation leads to rising real interest rates, making
it more di�cult to reduce government debt-to-GDP ratios.
Moreover, much of the literature, both theoretical and empirical, has found that
2
Figure 1: Inflation and Interest Rates in the Euro Area Source: ECB, Eurostat
fiscal multipliers are higher when monetary policy is constrained. In particular,
Eggertsson (2011) found that the government spending multiplier goes from below
0.5, to around 2.3 at the ZLB, and that tax multipliers even change sign and become
negative at the ZLB. Similar results are found in the studies of Christiano et al.
(2011), Coenen et al. (2012) and De Long and Summers (2012). Empirically, Ilzetzki
et al. (2013) corroborate these results, finding that government spending multipliers
are substantially higher in countries operating under fixed exchange rates. Nakamura
and Steinsson (2014) draw similar conclusions regarding the multiplier of military
spending, although their analysis is not a direct comparison of di↵erent monetary
regimes. Based on these principles, several papers discuss the potential role of fiscal
stimulus in alleviating a ZLB crisis: Correia et al. (2013) suggest an alternative
stimulus strategy to the use of government spending, based on consumption taxation,
and Rendahl (2015) focuses on amplification e↵ects in the labour market due to the
ZLB and how an expansionary fiscal policy can best exploit these. The converse of
these arguments is that attempting to carry out fiscal consolidation in a liquidity
trap can be very costly, and even self-defeating.
Another important way in which low inflation a↵ects fiscal policy is the fact that
3
inflation shocks can be expected to have a di↵erent impact, both in terms of size
and timing, across di↵erent government revenue and expenditure categories. In line
with the research highlighted above, Jalil (2012) finds that the di↵erences between
the estimated multipliers of government spending and taxation can be explained
by the di↵erential response of monetary policy. Erceg and Linde (2013) find that
the magnitude of the output contraction induced by spending-based consolidation
is roughly three times larger when monetary policy is constrained by the ZLB than
when it is unconstrained. They also find that a tax-based consolidation is less costly
in the short-run than a spending-based consolidation, while the opposite is true
when monetary policy is unconstrained. McManus et al. (2014) find that the ZLB
has di↵erent e↵ects on di↵erent fiscal consolidation instruments, and should therefore
be considered when designing austerity packages.
One aspect of this comparison which has been overlooked is that the e↵ectiveness
of consolidation packages that focus on quantity-based measures instead of price-
based measures may be di↵erent depending on the inflation environment. Recent
austerity packages implemented in many European countries, like Greece and Spain,
have placed special emphasis on the reduction of the public wage bill. In that con-
text, reducing the wage bill via cutting wages (price-based measure) or reducing
public employees (quantity-based measure) may have a di↵erent budgetary impact
depending on the inflation environment. This paper aims to uncover the potential
e↵ect of a low-inflation environment on these alternative consolidation strategies.
To this end, we develop a DSGE model through which we can study the di↵eren-
tial e↵ects of quantity-based and price-based consolidation measures. In particular,
we consider a New-Keynesian model with nominal rigidities in the form of monopolis-
tic retailers facing price-stickiness. In order to build a complete model of the labour
market, we incorporate both search and matching frictions, leading to involuntary
unemployment, and an endogenous labour force participation decision, leading to
4
voluntary unemployment. Finally, to study the e↵ects of the public wage bill, we
allow the government to hire public employees to produce a public good that is used
by private firms.
Following Erceg and Linde (2013) and Pappa et al. (2015), fiscal policy responds
to the deviation of the debt-to-GDP ratio from a target value, and fiscal consolidation
occurs when this target is hit by a negative shock. We focus attention on two fiscal
consolidation instruments on the part of the government: public wage cuts and public
vacancy cuts. We consider each instrument separately, assuming that if one is active,
the other remains fixed at its steady state value. We then repeat this experiment
when the economy faces low inflation due to a liquidity trap. This setup allows us to
compare, for a given consolidation volume, the e↵ects of the alternative consolidation
strategies in di↵erent environments.
In a low inflation environment, induced by a positive shock to the discount rate,
a much larger cut in the public wage bill is required to bring the debt-to-GDP ratio
to the desired level. The rise in the real interest rate when the ZLB constraint is
binding leads to a rise in public debt and, as a result, makes consolidation more
costly. The fall in demand following the discount rate shock creates a drag on the
private sector, meaning that the consolidation in this environment has large negative
e↵ects.
The remainder of the paper is organised follows. In Section 2, we provide the de-
tails of the model. Section 3 discusses the results of the di↵erent policy experiments.
Section 4 concludes.
2 The Model
We consider a DSGE model with search and matching frictions, endogenous labour
participation choice, and sticky prices in the short run. There are three types of firms
in the economy: (i) a public firm that produces a good that is used for productive
5
and utility enhancing purposes, (ii) private competitive firms that use private inputs
and the public good to produce an inermediate good, and (iii) monopolistic retailers
that use the intermediate good to produce a final good. Price rigidities arise at the
retail level, while labour market frictions occur in the intermediate goods sector.
The representative household consists of private and public employees, unemployed,
and labour force non-participants. The government collects taxes and uses revenues
to finance the wages of public employees, the costs of opening new vacancies in the
public sector and the provision of unemployment benefits.
2.1 Labour markets
We consider search and matching frictions in both the private and public labour mar-
kets. In each period, jobs in each sector, j = p, g, are destroyed at a constant fraction
�
j and a measure m
j of new matches are formed. The evolution of employment in
each sector is thus given by:
n
jt+1 = (1� �
j)njt +m
jt (1)
We assume that �p> �
g in order to capture the fact that, in general, public employ-
ment is more permanent than private employment.
The new matches are given by:
m
jt = ⇢
jm(�
jt )↵(uj
t)1�↵ (2)
where the matching e�ciency, ⇢jm, can di↵er in the two sectors. From the match-
ing functions specified above we can define, for each sector j, the probability of a
jobseeker being hired, hjt , and of a vacancy being filled, fj
t :
hjt ⌘
m
jt
u
jt
(3)
6
fjt ⌘
m
jt
�
jt
(4)
2.2 Households
The representative household consists of a continuum of infinitely lived agents. The
members of the household derive utility from leisure, which corresponds to the frac-
tion of members that are out of the labour force, lt, and a consumption bundle, cct,
defined as:
cct = [↵1(ct)↵2 + (1� ↵1)(y
gt )↵2 ]
1↵2
where ygt denotes a public good, taken as exogenous by the household, and ct is private
consumption. Following Neiss and Pappa (2005), we also allow for variable labour
e↵ort, et, which leads to separable disutility. The instantaneous utility function is
thus given by:
U(cct, lt) =cc
1�⌘t
1� ⌘
+ �l
1�'t
1� '
� �2e
1+'2t
1 + '2
where ⌘ is the inverse of the intertemporal elasticity of substitution, � > 0 is the
relative preference for leisure, and ' is the inverse of the Frisch elasticity of labour
supply. The elasticity of substitution between the private and public goods is given
by ⌘1�↵2
.1
At any point in time, a fraction n
pt (ng
t ) of the household members are private
(public) employees. Campolmi and Gnocchi (2014) and Bruckner and Pappa (2012)
have added a labour force participation choice in New Keynesian models of equilib-
rium unemployment. Following Ravn (2008), the participation choice is modelled as
a trade-o↵ between the cost of giving up leisure and the prospect of finding a job. In
particular, the household chooses the fraction of the unemployed actively searching
for a job, ut, and the fraction which are out of the labour force and enjoying leisure,
1When this elasticity is greater than one, ct and ygt are substitutes, and when it is below onethey are complements. The Cobb-Douglas specification is obtained when it is equal to zero.
7
lt, so that:
n
pt + n
gt + ut + lt = 1 (5)
The household chooses the fraction of jobseekers searching in each sector: a share st
of jobseekers look for a job in the public sector, while the remainder, (1 � st), seek
employment in the private sector. That is, ugt ⌘ stut and u
pt ⌘ (1� st)ut.2
The household owns the private capital stock, which evolves according to:
k
pt+1 = i
pt + (1� �
p)kpt �
!
2
✓
k
pt+1
k
pt
� 1
◆2
k
pt (6)
where ipt is private investment, �p is a constant depreciation rate and !2
⇣
kpt+1
kpt� 1⌘2
k
pt
are adjustment costs.
The intertemporal budget constraint is given by:
(1+⌧c)ct+i
pt+
Bt+1⇡t+1
Rt
[rpt�⌧k(rpt��
p)]kpt+(1�⌧n)(w
ptn
pt et+w
gtn
gt )+$ut+Bt+⇧
pt�Tt
(7)
where ⇡t ⌘ pt/pt�1 is the gross inflation rate, wjt , j = p, g, are the real wages in the
two sectors, rpt is the real return on capital, $ denotes unemployment benefits, Bt is
the real government bond holdings, Rt is the gross nominal interest rate, ⇧pt are the
profits of the monopolistic retailers, discussed below, and ⌧c,⌧k, ⌧n, and T represent
taxes on private consumption, private capital, labour income and lump-sum taxes
respectively.
Thus the problem of the household is to choose ct, ut, st, npt+1, n
gt+1, k
pt+1and Bt+1
to maximise lifetime utility subject to the budget constraint, (7), the law of motion of
employment in each sector, (1), the law of motion of capital, (6), and the composition
of the household, (5). Details of the household’s optimisation, and the resulting first
order conditions, is provided in Appendix A.1. For use below, we define the marginal
2For simplicity, we will abstract from variable labour e↵ort in the public sector.
8
value of an additional private sector employee as:
V
Hnpt = �ctw
pt et(1� ⌧n)� �l
�'t + (1� �
p)�npt (8)
= �ctwpt et(1� ⌧n)� �l
�'t + (1� �
p)�Et(VHnpt+1)
where �ct and �npt are the lagrange multipliers on the budget constraint and the law
of motion of private employment respectively.
2.3 Production
2.3.1 Intermediate goods firms
Intermediate goods are produced with a Cobb-Douglas technology:
y
pt = (Atn
pt et)
1��(kpt )�(ygt )
⌫ (9)
where At is a labour augmenting productivity factor, kpt and n
pt are private capital
and labour inputs, et is the e↵ort intensity of labour, and y
gt is the public good used
in productive activities, taken as exogenous by the firms. The parameter ⌫ regulates
how the public input a↵ects private production: when ⌫ is zero, the government good
is unproductive.
Since current hires give future value to intermediate firms, the optimization prob-
lem is dynamic and hence firms maximize the discounted value of future profits. The
number of workers currently employed, npt , is taken as given and the employment
decision concerns the number of vacancies posted in the current period, �pt , so as
to employ the desired number of workers next period, n
pt+1.
3 Firms also decide
the amount of the private capital, kpt , to be rented from the household at rate r
pt .
3Firms adjust employment by varying the number of workers (extensive margin) rather than thenumber of hours per worker. According to Hansen (1985), most of the employment fluctuationsarise from movements in this margin.
9
The problem of an intermediate firm with n
pt currently employed workers consists of
choosing k
pt and �pt to maximize:
Q
p(npt ) = max
kpt ,�pt
�
xt(Atnpt )
1��(kpt )�(ygt )
⌫� w
ptn
pt et � r
pt k
pt � �
pt + Et
⇥
⇤t,t+1Qp(np
t+1)⇤
(10)
where xt is the relative price of intermediate goods, is a utility cost associated with
posting a new vacancy, and ⇤t,t+1 = �
�ct+1
�ctis the discount factor. The maximization
takes place subject to the private employment transition equation, where the firm
takes the probability of the vacancy being filled as given:
n
pt+1 = (1� �
p)npt +
fpt �
pt (11)
The first-order conditions are:
xt�y
pt
k
pt
= r
pt (12)
fpt
= Et⇤t,t+1[xt+1(1� �)y
pt+1
n
pt+1
� w
pt+1et+1 + (1� �
p)
fpt+1
] (13)
According to (12) and (13) the value of the marginal product of private capital should
equal the real rental rate and the marginal cost of opening a vacancy should equal the
expected marginal benefit. The latter includes the marginal productivity of labour
minus the wage plus the continuation value, knowing that with probability �p the
match can be destroyed.
The expected value of the marginal job for the intermediate firm, V Fnpt is:
V
Fnpt ⌘
@Q
p(npt )
@n
pt
= xt(1� �)y
pt
n
pt
� w
pt et +
(1� �
p)
fpt
(14)
10
2.3.2 Retailers
There is a continuum of monopolistically competitive retailers indexed by i on the
unit interval. Retailers buy intermediate goods and di↵erentiate them with a tech-
nology that transforms one unit of intermediate goods into one unit of retail goods,
and thus the relative price of intermediate goods, xt, coincides with the real marginal
cost faced by the retailers. Let yit be the quantity of output sold by retailer i. The
final consumption good can be expressed as:
yt =
ˆ 1
0
(yit)✏�1✏di
�
✏✏�1
where ✏ > 1 is the constant elasticity of demand for each variety of retail goods. The
final good is sold at a price pt =h´ 1
0 p
1�✏it di
i
11�✏
. The demand for each intermediate
good depends on its relative price and on aggregate demand:
yit =
✓
pit
pt
◆�✏
yt (15)
Following Calvo (1983), we assume that in any given period each retailer can reset
its price with a fixed probability (1 � �). Firms that are able to reset their price
choose p
⇤it so as to maximize expected nominal profits given by:
Et
1X
s=0
�
s⇤t,t+s(p⇤it � pt+sxt+s)yit+s
subject to the demand schedule, (15), in each period. Since all firms are ex-ante
identical, p⇤it = p
⇤t for all i. The resulting expression for p⇤t is:4
p
⇤t =
✏
✏� 1
Et
P1s=0 �
s⇤t,t+syt+sxt+sp✏t+s
Et
P1s=0 �
s⇤t,t+syt+sp✏�1t+s
(16)
4See Appendix A.3 for the derivation of this condition.
11
Under the assumption of Calvo pricing, the price index is given by:
pt =⇥
(1� �)(p⇤t )1�✏ + �(pt�1)
1�✏⇤ 11�✏ (17)
2.4 Government
The government sector produces the public good using public capital and labour:
y
gt = (Atn
gt )
1�µ(kgt )
µ (18)
where we assume that productivity shocks are not sector specific and µ is the share
of public capital. The public good, which is provided for free, provides productivity
and utility enhancing services.
The government holds the public capital stock. Similar to the case of private
capital, the government capital stock evolves according to:
k
gt+1 = i
gt + (1� �
g)kgt �
!
2
✓
k
gt+1
k
gt
� 1
◆2
k
gt (19)
Government expenditure consists of public investment, public wages, public vacancy
costs and unemployment benefits, while revenues come from the consumption, capital
income, labour income and lump-sum taxes. The government deficit is therefore
defined by:
DFt = i
gt + w
gtn
gt + v
gt +$ut � TRt
where TRt ⌘ ⌧n(wptn
pt et + w
gtn
gt ) + ⌧k(r
pt � �
p)kpt + T + ⌧cct denotes tax revenues.
The government budget constraint is given by:
Bt +DFt = R
�1t Bt+1⇡t+1 (20)
We assume that tax rates are constant and fixed at their steady state levels, and
12
we do not consider them as active instruments for fiscal consolidation. Similarly we
assume that government investment is held fixed at it’s steady state value, ig = �
gk
g,
keeping the public capital stock constant. Thus the government has two potential
fiscal instruments, vg and wg. We consider each instrument separately, assuming that
if one is active, the other remains fixed at its steady state value. For 2 {vg,wg},
we assume fiscal rules of the form, following Erceg and Linde (2013) and Pappa et
al. (2015):
t = (1�� 0) � 0
t�1
"
✓
bt
b
⇤t
◆� 1✓
�bt+1
�b
⇤t+1
◆� 2#(1�� 0)
where bt =BtYt
is the debt-to-GDP ratio and b
⇤t is the target debt-to-GDP ratio, given
by the AR(2) process:
log b⇤t � log b⇤t�1 = µb + ⇢1(log b⇤t�1 � log b⇤t�2)� ⇢2 log b
⇤t�1 � "
bt
where "bt is a white noise shock representing a fiscal consolidation.5
2.5 Closing the model
2.5.1 Monetary policy
There is an independent monetary authority that sets the nominal interest rate to
target zero net inflation, subject to the ZLB:
Rt = max(1, R⇡⇣⇡t ) (21)
5Notice that public wage cuts reduce the wage bill in the public sector in the same period, whilepublic vacancy cuts reduce it with a lag from next period.
13
2.5.2 Resource constraint
Private output must equal private and public demand. The resource constraint is
given by:
y
pt = ct + i
pt + i
gt + (�pt + �
gt ) (22)
Following Gomes (2015), we define total output, Yt, as private output plus the public
wage bill: Yt = y
pt + w
gtn
gt .
2.5.3 Wage bargaining
Private sector wages are determined by ex post (after matching) Nash bargaining.
Workers and firms split rents and the part of the surplus they receive depends on
their bargaining power. If we denote by # 2 (0, 1) the firms’ bargaining power, the
Nash bargaining problem is to maximize the weighted sum of log surpluses:
maxwp
t
�
(1� #) lnV Hnpt + # lnV F
npt
where V Hnpt and V
Fnpt have been defined above. The optimization problem leads to the
following solution for wpt :
6
w
pt et = (1� #)xt(1� �)
y
pt
n
pt
+#
(1� ⌧n)�c,t�l�'t (23)
Hence, the equilibrium wage is a weighted average of the marginal product of em-
ployment and the disutility from labour, with the weights given by the firm and
household’s bargaining power respectively.
6See Appendix A.2 for the full derivation.
14
Table 1: Calibration of Parameters and Steady-State ValuesParameter/Variable Description Value
Preferences:
� Household discount factor 0.99↵1 Share of public good in consumption 0.85↵2 Private/public elasticity of substitution �1.25
Labour Market:
(1� l) Labour force participation 65%u/(1�l) Unemployment rate 10%ng/n Share of public employment 18%
/wp Vacancy costs as a share of wages 4.5%
Production:
⌫ Productivity of public good 0.15�, µ Share of capital in production 0.36kg/kp Public-private capital ratio 0.31� Price-stickiness 0.75
Policy Parameters:
⇣⇡ Taylor-rule inflation targeting parameter 1.1⇢1, ⇢2 Debt-target law of motion 0.85, 0.0001b Steady-state debt-to-GDP ratios 103%
2.6 Model Solution and Calibration
We solve the model by linearising the equilibrium conditions around a non-stochastic
steady state in which all prices are flexible, the price of the private good is normalized
to unity, and inflation is zero. When considering the ZLB, which is a non-linear
constraint, we use the Occbin toolkit provided by Guerrieri and Iacoviello (2015).
Table 1 shows some of the key parameters and steady-state values targeted in
our calibration. In particular, note that we allow the public good to be both utility
enhancing, as a complementary good to the private consumption good, and produc-
tive in the private production function. Full details of the calibration strategy are
provided in Appendix C.
15
3 Results
We begin by considering a shock to the debt-to-GDP target which drives this ratio
around 5% below its steady state after 3 years. We simulate the response to this
shock under the two alternative policy instruments, �gand w
g. We then consider the
same shock in a low inflation environment. Following the literature, this environment
is induced by assuming a positive shock to the household’s discount rate, �, which
causes inflation to fall, driving the nominal interest rate to its lower bound.7 For
ease of exposition, we consider the cases of the two policy instruments in turn, and
compare the simulations with and without the shock to the discount rate in each
case.
3.1 The E↵ects of Quantity-Based Measures: A Cut in Pub-
lic Vacancies
Figure 2 shows the responses when public vacancies are the active consolidation
instrument. The blue line shows the baseline simulations in response to the fiscal
consolidation shock.8 We see that the cut in public vacancies causes a fall in public
employment, and hence both the public wage bill and public output, with a lag. Some
of the jobseekers leaving the public sector move towards the private sector, causing
a rise in private employment. At the same time, the reduction in expenditure on
the public wage bill creates a positive wealth e↵ect for the household, causing a rise
in private consumption and investment, which leads to a rise in capital. The rise in
both private employment and capital lead to a rise in private output, despite the fall
in public output, which also serves as an input in private production.
The green line depicts the responses in a low inflation environment induced by
the shock to the household’s discount rate, without yet imposing the ZLB constraint.
7We assume that the shock decays with auto-regressive parameter 0.5.8The dotted line in the debt-to-GDP ratio graph shows the path of the exogenous debt target.
16
Figure
2:FiscalCon
solidationwithPublicVacan
cyCuts
24
68
1012
-0.50
0.5
PAR
TIC
IPA
TIO
N R
ATE
24
68
1012
-4-20PRIV
ATE
CO
NSU
MPT
ION
24
68
1012
123PRIV
ATE
INVE
STM
ENT
24
68
1012
0
0.51
1.5
GR
OSS
REA
L IN
TER
EST
RA
TE
24
68
1012
0
0.050.1
PRIV
ATE
CA
PITA
L
24
68
1012
-15
-10-505
PRIV
ATE
VA
CA
NC
IES
24
68
1012
-10-50
PRIV
ATE
WA
GES
24
68
1012
-50510PR
IVA
TE J
OB
SEEK
ERS
24
68
1012
012PRIV
ATE
EM
PLO
YMEN
T
24
68
1012
-3-2-1
PRIV
ATE
OU
TPU
T
24
68
1012
-80
-60
-40
-20
PUB
LIC
VA
CA
NC
IES
24
68
1012
-15
-10-50
PUB
LIC
WA
GE
BIL
L
24
68
1012
-2002040
PUB
LIC
JO
BSE
EKER
S
24
68
1012
-15
-10-50PU
BLI
C E
MPL
OYM
ENT
24
68
1012
-10-50
PUB
LIC
OU
TPU
T
24
68
1012
-2-10G
RO
SS N
OM
INA
L IN
TER
EST
RA
TE
24
68
1012
-2-10IN
FLA
TIO
N
24
68
1012
246UN
EMPL
OYM
ENT
RA
TE
BA
SELI
NE
UN
CO
NST
RA
INED
CO
NST
RA
INED
24
68
1012
-4-2024D
EBT-
TO-G
DP
RA
TIO
17
Here we see that the nominal and the real interest rates fall sharply. Since agents
become more patient, we observe a fall in private consumption and a larger increase
in private investment compared to the baseline case. The latter leads to a higher
accumulation of private capital. However, despite the rise in capital, the demand
contraction leads to a fall in private labour demand and hence private employment.
Notice that the contraction in the private sector leads to a rise in public debt despite
the consolidation shock. This means that public vacancies fall by much more than
the baseline case, contraction public employment and output by more. This further
reinforces the fall in private output.
We next impose the ZLB constraint on the nominal interest rate and examine
again the responses to the combination of the consolidation and discount factor
shocks, as shown by the red lines. As we can see, the nominal interest rate now
cannot fall to the necessary level. The fall in inflation therefore translates to a large
rise in the real interest rate, putting upwards pressure on the debt-to-GDP ratio,
which rises by even more in the short run. In this case, a much larger cut in public
vacancies is required. The e↵ects of the shock are amplified accordingly, with a larger
fall in public, and hence private, output.
3.2 The E↵ects of Price-Based Measures: A Cut in Public
Wages
Figure 3 shows the responses when public wages are the active consolidation instru-
ment. Again, the blue lines depict the baseline simulations with the consolidation
shock only. We see that the public wage cut causes a fall in the fraction of jobseek-
ers in the public sector, and hence public employment and public output are again
reduced. As before, this causes a movement of jobseekers towards the private sector,
and boosts private employment. Despite the fall in income, we see that again the
consolidation causes a positive wealth e↵ect for the household, raising consumption
18
Figure
3:FiscalCon
solidationwithPublicWageCuts
24
68
1012
-0.4
-0.200.2
PAR
TIC
IPA
TIO
N R
ATE
24
68
1012
-4-20PRIV
ATE
CO
NSU
MPT
ION
24
68
1012
0123PRIV
ATE
INVE
STM
ENT
24
68
1012
012G
RO
SS R
EAL
INTE
RES
T R
ATE
24
68
1012
0
0.050.1
PRIV
ATE
CA
PITA
L
24
68
1012
-202468PR
IVA
TE V
AC
AN
CIE
S
24
68
1012
-10-50
PRIV
ATE
WA
GES
24
68
1012
051015
PRIV
ATE
JO
BSE
EKER
S
24
68
1012
0
0.51
1.5PR
IVA
TE E
MPL
OYM
ENT
24
68
1012
-3-2-10PR
IVA
TE O
UTP
UT
24
68
1012
-12
-10-8-6-4-2
PUB
LIC
WA
GES
24
68
1012
-15
-10-5
PUB
LIC
WA
GE
BIL
L
24
68
1012
-60
-40
-200
PUB
LIC
JO
BSE
EKER
S
24
68
1012
-6-4-20PUB
LIC
EM
PLO
YMEN
T
24
68
1012
-4-20PU
BLI
C O
UTP
UT
24
68
1012
-2-10G
RO
SS N
OM
INA
L IN
TER
EST
RA
TE
24
68
1012
-3-2-10IN
FLA
TIO
N
24
68
1012
-4-202UN
EMPL
OYM
ENT
RA
TE
BA
SELI
NE
UN
CO
NST
RA
INED
CO
NST
RA
INED
24
68
1012
-4-2024D
EBT-
TO-G
DP
RA
TIO
19
and investment, and reducing labour force participation. Hence, despite the fall in
public output, we again see a rise in private output. It is also important to note that
the consolidation is much more successfull in the case of public wage cuts, with the
debt-to-GDP ratio falling to its new target after 3 years.
We turn next to the e↵ects of fiscal consolidation in the presence of the positive
shock to the household’s discount rate. The fact that inflation remains below the
steady state level so persistently implies that the government debt-to-GDP rises
dramatically above the steady state level, despite the deep fall in public wages. In
other words, in order for debt to be lowered to the new target level, a much deeper
cut in the public wage bill is required in the presence of the ZLB. This depresses
public employment and public output by far more than the baseline case. Hence,
despite the rise in private employment and capital, we see a fall in private output.
When the ZLB is imposed, the rise in the real rate again puts upward pressure on
public debt, further amplifying the required fall in public wages and hence the impact
of the shock.
3.3 The Productivity of the Public Sector Output
The results we present are, of course, very sensitive to the assumed value for the
productivity of the public good (⌫), as this is crucial in determining the e↵ects of
cuts in public wages or vacancies even in the baseline model when the ZLB does not
bind. Take, for example, the e↵ect of the vacancy cut: the fall in public employ-
ment generates a fall in public output and at the same time an increase in private
employment given that workers shift their supply away from public jobs. Whether
private output increases or not, and whether the consolidation is actually successful
or not, depends crucially on the productivity of the public good. If the public good
is productive, the cut in public vacancies reduces public output and, as a result,
besides the increase in private employment, for some parameterizations of ⌫, private
20
output could decreases after the consolidation shock.
In the case of wage cuts, the subsequent decrease of the private wage decreases
marginal costs of firms in the private sector and this increases the demand for labour
and boosts private employment. Due to the fall in public wages and the increase in
demand in the private sector, unemployed shift their supply of labour to the private
sector and, hence, public employment is also decreasing, as in the case of vacancy
cuts, but for di↵erent reasons. Again, the productivity of the public good is crucial.
The more productive is the public good, the more private firms will increase vacancies
and the smaller the impact of the consolidation in unemployment will be, generating
a smaller impact of the consolidation in the private sector.
Now in a low inflation environment the demand e↵ect from the consolidation
shock coupled with the negative demand e↵ect from the discount factor shock will
crucially interplay with the e↵ects coming from the fall in public output. Whether
public output is productive or not will crucially a↵ect the transmission of consolida-
tion shocks in a low inflation environment. Figures 4 and 5 compare the responses
using the previous calibration, in which public output is assumed to be somehow
productive, ⌫ = 0.15, with the case in which ⌫ is reduced by half. As it is clear from
the figures, making the public sector less productive implies a need for a smaller
fiscal consolidation after the discount factor shock, and a smaller and less persistent
fall in private output.
3.4 The Taylor Rule Coe�cient on Inflation
Given the focus of the paper on the e↵ects of fiscal consolidation when monetary pol-
icy is constrained, the fiscal and monetary policy interactions become very important
in our context. We therefore turn next to examining the e↵ects of fiscal consolidation
in an economy constrained by the ZLB when the response of the nominal interest
rate to inflation through the Taylor rule is higher. In Figures 6 and 7 the blue lines
21
Figure
4:FiscalCon
solidationwithPublicVacan
cyCuts
under
theZLB
-Di↵erentProductivityof
PublicGoo
d
05
10
-0.50
0.5
PAR
TIC
IPA
TIO
N R
ATE
05
10-4-20PR
IVA
TE C
ON
SUM
PTIO
N
05
100.51
1.5PR
IVA
TE IN
VEST
MEN
T
05
100
0.51
1.5
GR
OSS
REA
L IN
TER
EST
RA
TE
05
100
0.1
0.2
PRIV
ATE
CA
PITA
L
05
10
-15
-10-505
PRIV
ATE
VA
CA
NC
IES
05
10
-10-8-6-4-2
PRIV
ATE
WA
GES
05
10
0510PR
IVA
TE J
OB
SEEK
ERS
05
10
012PRIV
ATE
EM
PLO
YMEN
T
05
10-3-2-10
PRIV
ATE
OU
TPU
T
05
10-80
-60
-40
-20
PUB
LIC
VA
CA
NC
IES
05
10
-15
-10-50
PUB
LIC
WA
GE
BIL
L
05
10
-2002040
PUB
LIC
JO
BSE
EKER
S
05
10
-15
-10-50PU
BLI
C E
MPL
OYM
ENT
05
10
-10-50
PUB
LIC
OU
TPU
T
05
10-1
-0.8
-0.6
-0.4
GR
OSS
NO
MIN
AL
INTE
RES
T R
ATE
05
10
-2.5-2
-1.5-1
-0.5
INFL
ATI
ON
05
10246U
NEM
PLO
YMEN
T R
ATE
BA
SELI
NE
CA
LIB
RA
TIO
N
LOW
ER P
RO
DU
CTI
VITY
YG
05
10
-4-2024D
EBT-
TO-G
DP
RA
TIO
22
Figure
5:FiscalCon
solidationwithPublicWageCuts
under
theZLB
-Di↵erentProductivityof
PublicGoo
d
05
10-0.6
-0.4
-0.200.2
PAR
TIC
IPA
TIO
N R
ATE
05
10
-4-3-2-1PRIV
ATE
CO
NSU
MPT
ION
05
10
0
0.51PR
IVA
TE IN
VEST
MEN
T
05
10012
GR
OSS
REA
L IN
TER
EST
RA
TE
05
100
0.050.1
PRIV
ATE
CA
PITA
L
05
10-4-20246
PRIV
ATE
VA
CA
NC
IES
05
10-12
-10-8-6-4-2
PRIV
ATE
WA
GES
05
1001020PR
IVA
TE J
OB
SEEK
ERS
05
100
0.51
1.5PR
IVA
TE E
MPL
OYM
ENT
05
10
-3-2-10PR
IVA
TE O
UTP
UT
05
10
-12
-10-8-6-4-2
PUB
LIC
WA
GES
05
10-20
-15
-10-5
PUB
LIC
WA
GE
BIL
L
05
10-80
-60
-40
-20020
PUB
LIC
JO
BSE
EKER
S
05
10-8-6-4-20PU
BLI
C E
MPL
OYM
ENT
05
10-5-4-3-2-1
PUB
LIC
OU
TPU
T
05
10-1
-0.50
GR
OSS
NO
MIN
AL
INTE
RES
T R
ATE
05
10
-3-2-10IN
FLA
TIO
N
05
10
-4-202UN
EMPL
OYM
ENT
RA
TE
BA
SELI
NE
CA
LIB
RA
TIO
N
LOW
ER P
RO
DU
CTI
VITY
YG
05
10
-4-2024D
EBT-
TO-G
DP
RA
TIO
23
correspond to our baseline calibration (⇣⇡ = 1.1), while the red lines correspond to
the case in which the inflation coe�cient in the Taylor rule is higher (⇣⇡ = 1.5).
As we see in both figures, the tighter is monetary policy the less significant are
the e↵ects of the fiscal consolidation. The drop in private wages and inflation is
smaller. Consequently, the fall in the nominal interest rate is smaller, while the real
interest rate falls by more. The debt-to-GDP ratio increases by much less and as a
result the required cut in the fiscal instrument for debt consolidation is smaller. In
the labour market the e↵ects appear generally mitigated.
4 Conclusions
In this paper, we have set up a DSGE model with search and matching frictions,
nominal rigidities, and public employment. This rich model allows us to study non-
trivial reallocation of agents in and out of the labour force, and between the public
and private sector. In the baseline case, a fiscal consolidation through a cut in public
wages is able to reduce the public debt-to-GDP ratio faster than public vacancy
costs, although both have similar e↵ects on private output. Hence, public wage cuts
are a preferable consolidation strategy.
In a low inflation environment, induced by a positive shock to the discount rate,
a much larger cut in the public wage bill is required to bring the debt-to-GDP ratio
to the desired level. The rise in the real interest rate when the ZLB constraint is
binding leads to a rise in public debt and, as a result, makes consolidation more
costly. The fall in demand following the discount rate shock creates a drag on the
private sector, meaning that the consolidation in this environment has large negative
e↵ects.
24
Figure
6:FiscalCon
solidationwithPublicWageCuts
under
theZLB
-Di↵erentTaylorRule
Coe�cient
05
10
-0.4
-0.200.2
0.4
PAR
TIC
IPA
TIO
N R
ATE
05
10
-4-3-2-1PRIV
ATE
CO
NSU
MPT
ION
05
10012PR
IVA
TE IN
VEST
MEN
T
05
10
012G
RO
SS R
EAL
INTE
RES
T R
ATE
05
100
0.050.1
0.15
PRIV
ATE
CA
PITA
L
05
10
-2024
PRIV
ATE
VA
CA
NC
IES
05
10
-10-8-6-4-2
PRIV
ATE
WA
GES
05
10051015
PRIV
ATE
JO
BSE
EKER
S
05
100
0.51
1.5PR
IVA
TE E
MPL
OYM
ENT
05
10-3-2-10
PRIV
ATE
OU
TPU
T
05
10-12
-10-8-6-4-2
PUB
LIC
WA
GES
05
10
-15
-10-5
PUB
LIC
WA
GE
BIL
L
05
10
-60
-40
-200
PUB
LIC
JO
BSE
EKER
S
05
10
-6-4-20PUB
LIC
EM
PLO
YMEN
T
05
10
-4-20PU
BLI
C O
UTP
UT
05
10-1
-0.50
GR
OSS
NO
MIN
AL
INTE
RES
T R
ATE
05
10-3-2-10
INFL
ATI
ON
05
10-4-202U
NEM
PLO
YMEN
T R
ATE
BA
SELI
NE
CA
LIB
RA
TIO
N
HIG
HER
TA
YLO
R C
OEF
FIEC
IEN
T0
510
-4-2024D
EBT-
TO-G
DP
RA
TIO
25
Figure
7:FiscalCon
solidationwithPublicVacan
cyCuts
under
theZLB
-Di↵erentTaylorRule
Coe�cient
05
10
-0.50
0.5
PAR
TIC
IPA
TIO
N R
ATE
05
10-4-3-2-1PR
IVA
TE C
ON
SUM
PTIO
N
05
100.51
1.52PR
IVA
TE IN
VEST
MEN
T
05
10
0
0.51
1.5
GR
OSS
REA
L IN
TER
EST
RA
TE
05
100
0.1
0.2
PRIV
ATE
CA
PITA
L
05
10
-15
-10-505
PRIV
ATE
VA
CA
NC
IES
05
10
-10-8-6-4-2
PRIV
ATE
WA
GES
05
10
0510PR
IVA
TE J
OB
SEEK
ERS
05
10
012PRIV
ATE
EM
PLO
YMEN
T
05
10-3-2-10
PRIV
ATE
OU
TPU
T
05
10-80
-60
-40
-20
PUB
LIC
VA
CA
NC
IES
05
10
-15
-10-50
PUB
LIC
WA
GE
BIL
L
05
10
-2002040
PUB
LIC
JO
BSE
EKER
S
05
10
-15
-10-50PU
BLI
C E
MPL
OYM
ENT
05
10
-10-50
PUB
LIC
OU
TPU
T
05
10-1
-0.50
GR
OSS
NO
MIN
AL
INTE
RES
T R
ATE
05
10
-2-10IN
FLA
TIO
N
05
10
246UN
EMPL
OYM
ENT
RA
TE
BA
SELI
NE
CA
LIB
RA
TIO
N
HIG
HER
TA
YLO
R C
OEF
FIEC
IEN
T0
510
-4-2024D
EBT-
TO-G
DP
RA
TIO
26
References
Bruckner, M. and Pappa, E.: 2012, Fiscal expansions, unemployment, and labor force
participation: theory and evidence, International Economic Review 53(4), 1205–
1228. 2.2, C.2
Calvo, G.: 1983, Staggered prices in a utility maximizing framework, Journal of
Monetary Economics 12(3), 383–398. 2.3.2
Campolmi, A. and Gnocchi, S.: 2014, Labor market participation, unemployment
and monetary policy, Bank of Canada Working Paper No. 2014/9 . 2.2
Christiano, L., Eichenbaum, M. and Rebelo, S.: 2011, When Is the Government
Spending Multiplier Large?, Journal of Political Economy 119(1), 78–121. 1
Coenen, G., Erceg, C. J., Freedman, C., Furceri, D., Kumhof, M., Lalonde, R., ...
and Trabandt, M.: 2012, E↵ects of Fiscal Stimulus in Structural Models, American
Economic Journal: Macroeconomics 4(1), 22–68. 1
Correia, I., Farhi, E., Nicolini, J. and Teles, P.: 2013, Unconventional Fiscal Policy
at the Zero Bound, American Economic Review 103(4), 1172–1211. 1
De Long, J. and Summers, L.: 2012, Fiscal Policy in a Depressed Economy, Brookings
Papers on Economic Activity pp. 233–297. 1
Eggertsson, G.: 2011, What Fiscal Policy is E↵ective at Zero Interest Rates?, NBER
Macroeconomics Annual 2010 25, 59–112. 1
Erceg, C. and Linde, J.: 2013, Fiscal Consolidation in a Currency Union: Spending
Cuts vs. Tax Hikes, Journal of Economic Dynamics and Control 37(2), 422–445.
1
Gomes, P.: 2015, Heterogeneity and the public sector wage policy, Manuscript . 2.5.2
27
Guerrieri, L. and Iacoviello, M.: 2015, OccBin: A toolkit for solving dynamic mod-
els with occasionally binding constraints easily, Journal of Monetary Economics
70(C), 22–38. 2.6
Ilzetzki, E., Mendoza, E. and Vegh, C.: 2013, How Big (Small?) Are Fiscal Multi-
pliers?, Journal of Monetary Economics 60(2), 239–254. 1
Jalil, A.: 2012, Comparing Tax and Spending Multipliers: It’s All About Controlling
for Monetary Policy, Available at SSRN 2139855 . 1
McManus, R., Ozkan, F. G. and Trzeciakiewicz, D.: 2014, Self-defeating austerity at
the zero lower bound, Department of Economics, University of York . 1
Nakamura, E. and Steinsson, J.: 2014, Fiscal Stimulus in a Monetary Union: Evi-
dence from US Regions, American Economic Review 104(3), 753–792. 1
Neiss, K. S. and Pappa, E.: 2005, Persistence without too much price stickiness: the
role of variable factor utilization, Review of Economic Dynamics 8(1), 231–255.
2.2, C.3
Pappa, E., Sajedi, R. and Vella, E.: 2015, Fiscal Consolidation with Tax Evasion
and Corruption, Journal of International Economics 96(S1), 56–75. 1
Ravn, M.: 2008, The consumption-tightness puzzle, NBER International Seminar
on Macroeconomics 2006 pp. 9–63. 2.2
Reinhart, C., Reinhart, V. and Rogo↵, K.: 2015, Dealing with Debt, Journal of
International Economics 96(S1), 43–55. 1
Rendahl, P.: 2015, Fiscal Policy in an Unemployment Crisis, Review of Economic
Studies (forthcoming). 1
28
Appendices
A Derivations
A.1 Household’s maximisation problem
We can write in full the Lagrangean for the representative household’s maximisation
problem. Firstly, we can incorporate the composition of the household, as well as
the definition of the total consumption bundle, directly into the utility function of
the household. Then, we can plug the definition of the matches m
jt =
hjt u
jt into
the law of motion of employment in each sector, and also replace i
pt in the budget
constraint using the law of motion of private capital. We are left with 3 constraints,
and the following Lagrangean:
L = E0
1X
t=0
�
t{
[↵1(ct)↵2 + (1� ↵1)(ygt )↵2 ]
1�⌘↵2
1� ⌘
+ �[1� ut � n
pt � n
gt ]
1�'
1� '
� �2e
1+'2t
1 + '2
��ct[(1 + ⌧
ct )ct +
p
gt
pt
y
gt + k
pt+1 � (1� �
p)kpt +
!
2
✓
k
pt+1
k
pt
� 1
◆2
k
pt
+Bt+1⇡t+1
Rt
� [rpt � ⌧
k(rpt � �
p)]kpt � [rpt � ⌧
k(rpt � �
p)]kpt �
(1� ⌧
nt )(w
ptn
pt et + w
gtn
gt )�$ut � Bt � ⇧
pt + Tt]
��npt
h
n
pt+1 � (1� �
p)npt �
hpt (1� st)ut
i
��ngt
h
n
gt+1 � (1� �
g)ngt �
hgt stut
i
}
The controls are ct, ut, st, npt+1, n
gt+1, k
pt+1and Bt+1. The first order conditions are:
[wrt ct]
cc
(1�⌘�↵2)t ↵1c
(↵2�1)t � �ct(1 + ⌧
ct ) = 0 (24)
29
[wrt kpt+1]
�ct
1 + !
✓
k
pt+1
k
pt
� 1
◆�
��Et�ct+1[1��p+r
pt+1�⌧k(r
pt+1��
p)+!
2(
✓
k
pt+2
k
pt+1
◆2
�1)] = 0
(25)
[wrt Bt+1]
� �ct1
Rt
+ �Et�ct+11
⇡t+1= 0 (26)
[wrt n
jt+1]
� �njt � �Et[�l�'t+1 � �ct+1(1� ⌧n)w
jt+1e
jt+1 � �njt+1(1� �
j)] = 0 (27)
[wrt st]
�npt hpt = �ngt
hgt (28)
[wrt ut]
� �l�'t + �ct$ + �npt hpt (1� st) + �ngt
hgt st = 0 (29)
[wrt et]
� �2e'2t � �ctw
ptn
pt = 0 (30)
Equations (24)-(26)are standard and include the arbitrage conditions for the returns
to private consumption, private capital and bonds. Equation (27) relates the ex-
pected marginal value from being employed to the after-tax wage, the utility loss
from the reduction in leisure, and the continuation value, which depends on the
separation probability. Equation (29)states that the marginal utility of the unem-
ployment benefit, minus the marginal utility from leisure should equal the expected
marginal values of being employed, given te share of unemployed searching in each
sector. Equation (28) is an arbitrage condition according to which the choice of the
share, st, is such that the expected marginal values of being employed are equal
across the two sectors.
30
We can define the marginal value to the household of having an additional member
employed in the private sector, as follows:
V
hnpt ⌘
@L
@n
pt
= �ctwpt et(1� ⌧n)� �l
�'t + (1� �
p)�npt (31)
= �ctwpt et(1� ⌧n)� �l
�'t + (1� �
p)�Et(Vhnpt+1)
where the second equalities come from equation (27) for each j respectively.
A.2 Derivation of the private wage
The Nash bargaining problem is to maximize the weighted sum of log surpluses:
maxwp
t
n
(1� #) lnV hnpt + # lnV f
npt
o
where V
hnjt and V
fnjt
are defined as:
V
hnpt ⌘
@L
@n
pt
= �ctwpt et(1� ⌧
nt )� �l
�'t + (1� �
p)�npt (32)
V
Fnpt ⌘
@Q
p
@n
pt
= xt(1� �)y
pt
n
pt
� w
pt et +
(1� �
p)
fpt
(33)
The first order conditions of this optimization problem is:
#V
hnpt = (1� #)�ct(1� ⌧
nt )V
fnpt (34)
Plugging the expressions for the value functions into the FOC, we can rearrange to
find the expression for the private wage. Using (32),(33) and (34) we obtain:
w
pt et = (1� #)[xt(1� �)
y
pt
n
pt
+(1� �
p)
fpt
] +#
(1� ⌧n)�c,t(�l�'t � (1� �
p)�npt) (35)
Finally, taking the time t expectation of34 evaluated at time t+1, and using the
31
FOCs of the household and firm, we obtain
#�npt = (1� #)�ct(1� ⌧
nt )
fpt
which allows us to simplify 35 to obtain the final expression for the private wage
w
pt et = (1� #)xt(1� �)
y
pt
n
pt
+#
(1� ⌧n)�c,t�l�'t (36)
A.3 Derivation of the optimal price condition
An optimising retailer will choose the optimal price, p⇤it to maximise:
Et
1X
s=0
�
s⇤t,t+s(p
⇤it
pt+s
� xt+s)yit+s
subject to
yit+s =
✓
p
⇤it
pt+s
◆�✏
yt+s
We can rewrite the objective function, plugging in this constraint, as follows:
Et
1X
s=0
�
s⇤t,t+syt+s
✓
p
⇤it
pt+s
◆�✏✓p
⇤it
pt+s
� xt+s
◆
= Et
1X
s=0
�
s⇤t,t+syt+s
✓
p
⇤it
pt+s
◆1�✏
�
✓
p
⇤it
pt+s
◆�✏
xt+s
!
For simplicity, let ✓s ⌘ �
s⇤t,t+syt+s. We can multiply out the terms in the
objective function to get:
Et
1X
s=0
✓s
✓
p
⇤it
pt+s
◆1�✏
� Et
1X
s=0
✓sxt+s
✓
p
⇤it
pt+s
◆�✏
= (p⇤it)1�✏
Et
1X
s=0
✓sp✏�1t+s � (p⇤it)
�✏Et
1X
s=0
✓sxt+sp✏t+s
32
Taking the derivative with respect to p
⇤it, we have the following first order condi-
tion:
(1� ✏)(p⇤it)�✏Et
1X
s=0
✓sp✏�1t+s + ✏(p⇤it)
�✏�1Et
1X
s=0
✓sxt+sp✏t+s = 0
(1� ✏)(p⇤it)Et
1X
s=0
✓sp✏�1t+s = �✏(p⇤it)
�✏�1Et
1X
s=0
✓sxt+sp✏t+s
(p⇤it)�✏
(p⇤it)�✏�1
=�✏
(1� ✏)
Et
P1s=0 ✓sxt+sp
✏t+s
Et
P1s=0 ✓sp
✏�1t+s
(p⇤it)�✏+✏+1 =
✏
✏� 1
Et
P1s=0 ✓sxt+sp
✏t+s
Et
P1s=0 ✓sp
✏�1t+s
p
⇤it =
✏
✏� 1
Et
P1s=0 ✓sxt+sp
✏t+s
Et
P1s=0 ✓sp
✏�1t+s
Note that the right hand side no longer depends on i, so that we have p
⇤it = p
⇤t .
Reinserting ✓s ⌘ �
s⇤t,t+syt+s, we have the expression for the optimal price level
p
⇤t =
✏
✏� 1
Et
P1s=0 �
s⇤t,t+syt+sxt+sp✏t+s
Et
P1s=0 �
s⇤t,t+syt+sp✏�1t+s
B Model Equations
B.1 Dynamic Equations
We write the model as 38 equations in 38 unknowns:
nt + ut + lt = 1 (37)
n
pt+1 = (1� �
p)npt +m
pt (38)
33
n
gt+1 = (1� �
g)ngt +m
gt (39)
nt = n
pt + n
gt (40)
m
pt = ⇢
pm(�
pt )↵ (up
t )1�↵ (41)
m
gt = ⇢
gm(�
gt )↵(ug
t )1�↵ (42)
ut = u
pt + u
gt (43)
fpt =
m
pt
�
pt
(44)
hpt =
m
pt
u
pt
(45)
hgt =
m
gt
u
gt
(46)
fgt =
m
pt
�
gt
(47)
k
pt+1 = i
pt + (1� �
p)kpt �
!
2
✓
k
pt+1
k
pt
� 1
◆2
k
pt (48)
cct = [↵1(ct)↵2 + (1� ↵1)(y
gt )↵2 ]
1↵2 (49)
cc
(1�⌘�↵2)t ↵1c
(↵2�1)t = �ct(1 + ⌧
ct ) (50)
� �ct1
Rt
+ �Et�ct+11
⇡t+1= 0 (51)
34
�ct
1 + !
✓
k
pt+1
k
pt
� 1
◆�
��Et�ct+1[1��p+r
pt+1�⌧k(r
pt+1��
p)+!
2(
✓
k
pt+2
k
pt+1
◆2
�1)] = 0
(52)
� �npt � �Et[�l�'t+1 � �ct+1(1� ⌧n)w
pt+1 � �npt+1(1� �
p)] = 0 (53)
� �ngt � �Et[�l�'t+1 � �ct+1(1� ⌧n)w
gt+1 � �ngt+1(1� �
g)] = 0 (54)
� �l�'t + �ct$ + �npt hpt = 0 (55)
�npt hpt = �ngt
hgt (56)
�2e'2t = ��ctw
ptn
pt (57)
y
pt = (Atn
pt )
1��(kpt )�(ygt )
⌫ (58)
xt�y
pt
k
pt
= r
pt (59)
fpt
= �Et�c,t+1
�c,t
"
xt+1(1� �)y
pt+1
n
pt+1
� w
pt+1 + (1� �
p)
fpt+1
#
(60)
w
pt = (1� #)[xt(1� �)
y
pt
n
pt
+(1� �
p)
fpt
] +#
(1� ⌧n)�c,t(�l�'t � (1� �
p)�np,t) (61)
y
pt = ct + i
pt + i
gt + (�pt + �
gt ) (62)
y
gt = (Atn
gt )
1�µ(kgt )
µ (63)
k
gt+1 = i
g + (1� �
g)kgt �
!
2
✓
k
gt+1
k
gt
� 1
◆2
k
gt (64)
yt = y
pt + w
gtn
gt (65)
35
TGt = i
g + w
gtn
gt + v
gt +$ut (66)
TRt = ⌧n(wptn
pt + w
gtn
gt ) + ⌧k(r
pt � �
p)kpt + ⌧cct (67)
DFt = TGt + T � TRt (68)
Bt +DFt = R
�1t Bt+1⇡t+1
Rt = R⇡
⇣⇡t (69)
t = (1�� 0) � 0
t�1
"
✓
bt
b
⇤t
◆� 1✓
�bt+1
�b
⇤t+1
◆� 2#(1�� 0)
(70)
log b⇤t � log b⇤t�1 = µb + ⇢1(log b⇤t�1 � log b⇤t�2)� ⇢2 log b
⇤t�1 � "
bt (71)
1 = (1� �) (p⇤t )1�✏ + �⇡
✏�1t (72)
⌦1,t = �ctytxt + �Et⇡✏t+1⌦1,t+1 (73)
⌦2,t = �ctyt + �Et⇡✏�1t+1⌦2,t+1 (74)
p
⇤t =
✏
✏� 1
⌦1,t
⌦2,t(75)
C Calibration Strategy
C.1 Labour market variables
We calibrate e = 1, such that it does not e↵ect the rest of the steady state. We
calibrate the labour-force participation rate, the unemployment rate, and the share
36
of public employment in total employment to match the observed average values
from the Italian data (1� l = 0.65, urate = u1�l
= 0.1, ng
n= 0.18). Then we get u, n,
np
n, n
p, n
g as follows:
u = u
rate(1� l)
n = 1� l � u
n
p
n
= 1�n
g
n
n
j =n
j
n
n
We set the following values for the separation rates, �p = 0.063 and �g = 0.06. Then
we get mj from (1) at the steady state:
m
j = �
jn
j
We calibrate the ratio of unemployed searching in two sectors as up/ug = 4. Then, it
holds by definition:
u
p =u
1 + up/ug
u
g = u� u
p
hj =m
j
u
j
Since there is no exact estimate for the value of the private vacancy-filling prob-
ability, fp, in the literature, we use what is considered as standard by setting it
equal to 0.1 and then we assume that fp =
fg. Hence, we get:
�
j =m
j
fj
The elasticity in the matching functions, ↵, is set equal to 5. Then the e�ciency
37
parameter for private matches, ⇢pm, is given by inverting the matching function:
⇢
jm =
m
j
(�j)↵ (uj)1�↵
C.2 Production
We set the capital depreciation rates, �j, equal to 0.025. Following the literature, we
set the discount factor, �, equal to 0.99. The tax rates on capital and labour income
are calibrated to 30%. Next, we get rp and R from (25) and (26), respectively:
r
p =1
(1� ⌧k)
✓
1
�
� 1
◆
+ �
p
R =1
�
The elasticity of demand for intermediate goods, ", is set equal to 10, which implies
a gross steady-state markup, ""�1 , equal to 1.11. The price of the final good is
normalized to one, meaning that x is determined from (16):
x ="� 1
"
We set the capital share in the production function of the private good equal to 0.36.
Then we obtain yp
kpfrom (12):
y
p
k
p=
r
p
�x
We set the shares of public capital in public production, µ, equal to 0.36, of the
public good in private production, ⌫, equal to 0.12. Further, using data from Kamps
(2006) we set kg
kp= 0.31, close to the mean value for 1970-2002. Since we restrict
our case to a deterministic steady state, we normalize At to one. Then from (9) and
38
(18), kp is determined by:
k
p =
"
y
p
k
p(np)
�(1��)
(ng)µ⌫�⌫✓
k
g
k
p
◆�µ⌫#
1�+µ⌫�1
and then we get ypand k
gby definition:
y
p =y
p
k
pk
p, kg =k
g
k
pk
p
and i
p and i
g from (6) and (19) at steady state:
i
p = �
pk
p, i
g = �
gk
g
and y
g from (18):
y
g = (ng)1�µ(kg)µ
Following Hagedorn and Manovskii (2008), Galı (2011), and Bruckner and Pappa
(2012), we calibrate the cost of posting a vacancy, , by targeting vacancy costs per
filled job as a fraction of the real private wage, wp , choosing 0.045 as a target as in
Galı (2011). Also, we set the replacement rate, $wp , equal to 0.4 (in accordance with
the range [0.2, 0.4] in Petrongolo and Pissarides, 2001). Then, we can get wp from
(13):
w
p = x(1� �)y
p
n
p
✓
1 +�
p
fp
w
p
◆�1
and it follows that and b are given by:
=
w
pw
p
b =b
w
pw
p
39
C.3 Households
We derive private consumption from the resource constraint (22)
c = y
p� i
p� i
g� (�p + �
g)
We set the consumption tax rate to 15%, the intertemporal elasticity of substi-
tution, 1⌘, equal to 1, the Frisch elasticity of labour supply, 1
, equal to 0.25 (in the
range of Domeij and Floden, 2006). We set the share of private consumption in the
consumption bundle, ↵1 = 0.85, and the elasticity of substitution between the two
goods, ↵2 = �1.25, such that the two goods are complements. This gives us the
total consumption bundle by definition:
cc = [↵1(c)↵2 + (1� ↵1)(y
g)↵2 ]1↵2
We derive the two lagrange multipliers from the household’s first order conditions,
(24) and (27) for j = p, respectively:
�c = (1 + ⌧
c)�1↵1cc
(1�⌘�↵2)c
(↵2�1)
�np =��c(wp(1� ⌧n)�$)
1� �(1� �
p) + �
hp
This allows us to derive � from (29), after substituting in (28):
� = l
'(�c$ +
hp�np)
and the firm’s bargaining power from (23):
40
# =x(1� �)(yp/np) + (1� �
p)/ fp� w
p
x(1� �)(yp/np) + (1� �
p)/ fp� (�l�' � (1� �
p)�np)/�c(1� ⌧n)
Following Neiss and Pappa (2005)we set '2 = 0.5, and we use (30) to calibrate
�2 such that e = 1:
�2 = ��cwpn
p
Finally, we derive the lagrange multiplier from (28), and the public wage from
(27) for j = g:
�ng =�np
hp
hg
w
g =�l�' + �ng(R� 1 + �
g)
�c(1� ⌧n)
This also allows us to define total output, Y = y
p + w
gn
g.
C.4 Fiscal Policy
We set the steady-state debt to GDP ratio, b, equal to 103%, so that by definition:
B = bY
and using the government’s budget constraint, (20), in steady state, we have:
DF = (� � 1)B
Next, we calibrate the steady state value for lump-sum transfers, T , from the
41
definition of the deficit:
T = i
g + w
gn
g +$u+ �
g� ⌧k(r
p� �
p)kp� ⌧n(w
pn
p + w
gn
g)� ⌧cc�DF
C.5 Other parameters
Finally, the model’s steady state is independent of the degree of price rigidities, of
the monetary policy rule, the debt-targeting rule for lump-sum taxes, and of the size
of the capital adjustment costs. We set the probability that a firm does not change
its price within a given period, �, equal to 0.75, the Taylor rule coe�cient, ⇣⇡, equal
to 1.1, and the adjustment costs parameter, !, equal to 150. Finally, we set the
parameters for the persistence of the debt-target shock,⇢1 and ⇢2, equal to 0.85 and
0.0001, respectively.
42