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Government Expenditures and Business Cycles
- Policy Reaction and Surprise Shocks
Christian Glocker1
University of Basel
This Draft: November 2010
Abstract
This paper analyzes the effects of discretionary fiscal policy on German business cycle
fluctuations by means of an estimated DSGE model based on low frequency oscillations. The
results highlight that fiscal policy has a strong impact on the amplitude of fluctuations while
hardly any on the duration of business cycles. To the extent that fiscal policy shocks are an
important source for triggering economy wide fluctuations, fiscal policy also strongly reacts
to shocks originating elsewhere. Thus standard fiscal policy specifications in terms of simple
AR(1) processes for government expenditures ignore the fact that there can indeed be a strong
endogenous dependence of governmental variables on economic fluctuations.
JEL Classifications: C13, C32, E3, E62Key Words: Fiscal Policy, Business Cycles, DSGE Models, Estimation
1Universitat Basel, Wirtschaftswissenschaftliches Zentrum (WWZ) - Department for Quantitative Methods, Pe-ter Merian-Weg 6, 4002 Basel, Switzerland, Phone: +41 (0) 61 267 33 68, Fax: +41 (0) 61 267 23 97, E-mail:[email protected] and [email protected]
1 Introduction
This paper seeks to understand the impact of fiscal policy on business cycles once it is characterizedas a responder to observed fluctuations versus a scenario where it acts as a source of fluctuations.To this end a dynamic stochastic general equilibrium (DSGE) model is formulated with an extendedfiscal policy apparatus. There are two different types of policy mechanisms at work. On the onehand, the model specifies fiscal policy as reacting to changes in the economic stance. On the otherhand, fiscal policy is extended with an exogenous stochastic term in order to elaborate on surprisesin fiscal policy. The model is used to judge the importance of surprise fiscal policy shocks as wellas to analyze the impact of its endogenous component. I do so by estimating the model using onlylow frequency oscillations of German time series data.
As far as the profoundness of the analysis of fiscal policy in DSGE models is concerned,up to now the literature surrounding this issue has received considerably less attention than theliterature on monetary policy reaction functions. In DSGE models the specification of the fiscalauthority typically involves some type of fiscal closure rule. Except for a few studies (Gal, Lopez-Salido and Valles (2007), Perez and Hiebert (2004)) this rule is still completely exogenous and thefiscal authority is not allowed to react to various shocks affecting the model economy. This is ofcourse a rather limited description of the fiscal authorities reaction to shocks other their own ones.In addition to this, the government is usually described in a very limited way. Rather than settinga profound government sector, it is usually preferred to stick to lump sum taxes in a frameworkwhere there is no public debt.
The model that I construct has two key features. First it nests the now standard newKeynesian model, as for instance in Smets and Wouters (2007), and the Diamond-Mortensen-Pissarides model. This allows to disentangle the adjustment in the labour market into its extensiveand intensive margins. Second, I model a tax and an unemployment payments structure to actas so called automatic stabilizers in order to separate fiscal policy fluctuations which are due toautomatic stabilizers as opposed to those which are due to their discretionary component.
The key findings are as follows. First, fiscal policy has a strong impact on the amplitudeof fluctuations, however, there is no effect on the duration of business cycles. In particular, fiscalpolicy can smooth cyclical fluctuations but there is no possibility of fundamentally disturbingcyclicalities in their lengths.
Second, relative to the other shocks, surprise fiscal policy shocks are a major source offluctuations. Fiscal policy shocks explain a large amount of the fluctuations in inflation andunemployment for short horizons (up to 28%), but for longer forecasting horizons this magnitudedeclines strongly. For output, fiscal policy shocks explain around 20-25% of the fluctuations ratherindependently of the specific horizon considered.
Third, as the estimates for the fiscal policy rules indicate, government expenditures sig-nificantly contribute to dampening cyclical oscillations. Considering a fiscal authority’s aim ofdampening cyclical fluctuations indicates that especially for unemployment and output, counter-cyclical fiscal policy is a powerful tool, whereas for inflation its impact is limited due to the ratherrigid fluctuations therein.
I pursue a particular limited information econometric strategy to estimate and evaluatethe model. The basic idea behind this approach can be described as follows. DSGE models
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and especially new Keynesian and Real Business Cycle models are theoretical models aiming atexplaining business cycles. However, usually the data used for estimation are a composition ofcycles of different periodicities; there are long periodic cycles describing growth patterns, seasonalcyclicalities, purely random components and among them, business cycle patterns. Since the modelis only aimed at explaining low cyclicalities and especially business cycle frequencies, I hence makeonly use of this part of information in the data. Carrying out this estimation procedure hencerequires a frequency domain approach. I proceed by using a procedure proposed by Smith (1993)and Gourieroux, Monfort and Renault (1993) called Indirect Inference. I minimize a measure ofdistance between the model and empirical spectral density functions as proposed by Diebold et al.(1998) using only those fluctuations which correspond to business cycle fluctuations.
The paper is structured as follows. Section 2 describes the model. Section 3 addresses theeconometric methodology used to appropriately adjust the model to the data; section 4 discussesthe impact of the various fiscal policy rules and analyzes the effects of different specifications onthe fluctuations triggered by fiscal policy. Section 5 concludes.
2 The model
The present model has the key features that many authors have found useful for capturing the data;these include habit formation, investment adjustment costs, variable capital utilization, nominalprice and real wage rigidities.
The key changes in this model are in the labour market. Rather than considering aggregatehours worked irrespective of its dependence on the intensive and extensive margin, I introducevariation occurring at both margins. I do so by extending the new Keynesian Model by theMortensen-Pissarides model of search and matching.
The economy consists of two types of households, a continuum of firms, called the wholesalefirms who produce differentiated intermediate goods, the retail firms, a fiscal authority and a centralbank in charge of monetary policy. I use a representative family construct for each householdtype, similar as in Merz (1995), in order to introduce complete consumption insurance. Themodel features two important non-Ricardian elements: liquidity constraint agents and distortionarylabour and capital taxation.
2.1 Households
I assume a continuum of infinitely lived households, indexed by j ∈ [0, 1]. Following Gal, Lopez-Salido and Valles (2004) and Gal, Lopez-Salido and Valles (2007), a fraction 1 − ς of the totalpopulation has access to capital markets where they are able to trade a full set of contingentsecurities and buy and sell physical capital which they accumulate over time and rent out to firms.These type of consumers are called Ricardian consumers. The remaining fraction ς of householdsdo not own any assets nor do they have access to capital markets and hence they just consume theircurrent labour income. These types of households are referred to as non-Ricardian consumers.
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2.1.1 Ricardian households
The starting point for the Ricardian consumers is based on the now conventional monetary DSGEmodel developed by Christiano, Eichenbaum and Evans (2005), Smets and Wouters (2007) andothers. There is a representative household for the Ricardian consumers. The number of familymembers currently employed is No
t = (1 − ς)Nt, where o refers to optimizing, i.e.: Ricardianconsumers, and Nt is the amount of people currently employed. Employment is determined througha search and matching process described in section 2.2. Individuals gain utility from consumptionand leisure. Each family member offers Ho,s
t hours of work. Individuals not currently workingare searching for a job. Hours worked will be determined jointly within the bargaining over thereal wage between firms and households. Accordingly, conditional on No
t and Ho,st , the household
chooses consumption Cot , government bonds Bot , capital utilization zt, investment Iot and physicalcapital Ko
t to maximize the intertemporal utility function
Et
[ ∞∑s=0
βs · 1ξut+s
· u(Cot+s − h · Cot+s−1, H
o,st+s
)](1)
where h is the degree of habit persistence in consumption of Ricardian consumers and ξut isa preference shock with mean unity which obeys log ξut = ξu + εξ
u
t where εξu
t ∼ WN(0, σ2ξu) and
the instantaneous utility function satisfies u(Cot −hCot−1, Ho,st ) = 1
1−σc (Cot −hCot−1)1−σc − v(Ho,st )
with σc being the coefficient of relative risk aversion and v(Ho,st ) = χ
(Ho,st )1+φ
1+φ .Let Do
t be lump sum profits, Bot the quantity of nominally riskless one period governmentbonds carried over from period t−1 and paying one unit of the numeraire in period t, Rt the grossnominal rate of return on bonds purchased in period t− 1, rKt the rental rate on physical capital,Γ the amount of unemployment benefits an unemployed household member receives, Wt is the realmarket wage rate and {τK , τN} the tax rates on capital and labour income, respectively; then thehouseholds’ intertemporal budget constraint is
Cot + Iot +R−1t
BotPt
=Pt−1
Pt
Bot−1
Pt−1+ (1− τN )WtN
ot H
o,st + (2)
(1− τK)rKt ztKot − ψ(zt)Ko
t−1 +Dot + ΓUot
Households of the Ricardian consumers type own capital and choose the capital utilizationrate zt, which transforms physical capital Ko
t into effective capital Kot according to: Ko
t = ztKot−1.
The cost of capital utilization per unit of physical capital is ψ(zt) and I assume that in the steadystate ψ(1) = 0 and z = 1.
The physical capital accumulation dynamics are given by
Kot = (1− δ)Ko
t−1 + ξIt
(1− f
(IotIot−1
))Iot (3)
where ξIt is an investment specific technology shock with mean unity, δ is the depreciationrate and the function f(·) satisfies the following steady state conditions: f(1) = f ′(1) = 0 andf ′′(1) =: 1
κI> 0.
Based on this set-up, the first order conditions for the Ricardian consumers’ problem can
4
be written as(Cot )
λot =1
(Cot − hCot−1)σc− hβEt
[ξut
ξut+1(Cot+1 − hCot )σc
](4)
(Bot )
1 = βEt
[Λt,t+1 ·
PtPt+1
]·Rt (5)
(zt)(1− τK) · rKt = ψ′(zt) (6)
(Iot )
1 = ξIt qKt
(1− f
(IotIot−1
)− f ′
(IotIot−1
)· I
ot
Iot−1
)+ (7)
βEt
[Λt,t+1q
Kt+1 · f ′
(Iot+1
Iot
)·
(Iot+1)2ξIt+1
(Iot )2
](Ko
t )qKt = βEt
[Λt,t+1
((1− τK)rKt+1zt+1 − ψ(zt+1) + (1− δ)qKt+1
)](8)
where
Λt,t+1 :=ξut λ
ot+1
ξut+1λot
(9)
is the stochastic discount factor. qKt is the real shadow value of physical capital in place, i.e.:Tobin’s Q. Except for the treatment of the labour supply condition, the household’s optimizationproblem is conventional. Among the first order conditions, I have not listed an intratemporaloptimality condition linking the real wage to the consumer’s marginal rate of substitution betweenconsumption and labour. The reason is that wages and hours worked are determined within abargaining process outlined in section 2.4.
2.1.2 Non-ricardian households
The Non-Ricardian consumers do neither borrow nor save due to a lack of access to capital markets.Hence they are not able to smooth their consumption path as a reaction to fluctuations in labourincome or substitute intertemporally in response to interest rate changes. The number of familymembers currently employed is Nr
t = ςNt, where an r refers to Non-Ricardian (i.e.: rule ofthumb) consumers. Each family member employed offers Hr,s
t hours of work. Each period thesehouseholds solve a static optimization problem; let their instantaneous utility function be given byu(Crt , H
r,st ) = 1
1−σc (Crt )1−σc − v(Hr,st ) with v(Hr,s
t ) = χ(Hr,st )1+φ
1+φ , which is maximized given thefollowing static budget constraint
Crt = (1− τN )WtNrt H
r,st + ΓUrt (10)
Consumption is determined simply by after tax labour income. As it was the case for theRicardian consumers, hours worked Hr,s
t is determined within the households’ and firms’ bargaining
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over hours worked and the real wage. It is assumed that the tax rate imposed on the labour incomeof the two types of consumers is the same.
2.1.3 Aggregation
Aggregate consumption (Ct) is specified by a weighted average of the relevant magnitudes for eachhousehold type: Ct = ςCrt + (1− ς)Cot , the amount of employed and unemployed family membersof both household types is given by Nt = (1 − ς)No
t + ςNrt and Ut = (1 − ς)Uot + ςUrt . Similarly,
aggregate investment and the aggregate capital stock (both for the effective and physical capitalstock) are defined by: It = (1− ς)Iot and Kt = (1− ς)Ko
t .For the further analysis I assume that, first of all, firms hire workers independently of
their household type and that, once at work, the productivity of workers is independent of theirhousehold type, so that
Hst = Hr,s
t = Ho,st (11)
This simplification allows consumption of the Ricardian households to be expressed in termsof aggregate consumption
Cot =1
1− ς(Ct − ς2(1− τN )WtNtH
st − ς2ΓUt
)(12)
2.2 Matching of vacancies and unemployment
There is a continuum of intermediate goods producing firms, called the wholesale firms, indexedby i ∈ [0, 1]. Firms post vacancies Vt(i) in order to attract unemployed workers and employs nt(i)workers. The fixed labour force consists of nt =
∫ 1
0nt(i)di workers and aggregate vacancies are
given by Vt =∫ 1
0Vt(i)di. Each worker who has a job supplies Ht units of hours worked, which are
determined jointly with the real wage. Effort of each worker is constant. All unemployed workers attime t search for jobs. The timing assumption is such that in case an unemployed worker matcheswith a vacancy, he starts working there immediatelly within that period. This implies that thepool of unemployed people Ut searching for a job at time t is given by the difference between thetotal labour force, which is normalized to unity and the amount of employed people at the end ofperiod t− 1, that is nt−1
Ut = 1− nt−1 (13)
The function matching unemployed workers Ut and firms with a vacancy Vt is
mt = ξmUζmt V 1−ζmt (14)
where ξm is the exogenous part in the matching process. ζm is the elasticity to unemploymentand mt is the total amount of matches in period t. Let qt denote the probability of a vacancybeing filled in period t by defining
qt =mt
Vt= ξm ·
(VtUt
)−ζm= ξmϑ−ζmt (15)
6
where ϑt := VtUt
measures the degree of labour market tightness. Similarly, the probabilitypt a searching worker finds a job is given by
pt =mt
Ut= ξmϑ1−ζm
t = ϑt · qt (16)
Firms and workers take qt and pt as given. The definitions of qt and pt in (15) and (16)show that there is an intricate connection between the process linking workers to jobs and the onelinking jobs to workers. The total flow of hires for an individual firm (i) in period t+ 1 is qtVt(i).
Finally, each period firms separate from a fraction s of their current workforce nt−1(i). Oncea worker loses his job, he is not allowed to search until the next period. This restriction impliesthat fluctuations are triggered by cyclical fluctuations in hiring rather than due to fluctuationsin separations. Gertler and Triggari (2006) use a similar timing assumption based on empiricalevidence in support for this phenomenon outlined by Hall (2005) and Shimer (2005, 2007).
2.3 Wholesale firms
Production takes place at the wholesale firms. These firms are competitive. They hire workers andnegotiate wage contracts with them. Wholesale firms and workers bargain over the real wage andthe amount of hours worked based on an efficient Nash bargaining solution. The production for atypical wholesale goods firm depends on the amount of workers hired. Each output good Yt(i) isproduced by a firm i at time t using capital and labour as inputs. The firm considers the workersas identical and independent of the corresponding household type and the output of job j at firmi at time t is
eξAt kt(i)αHt(j, i)1−α (17)
where ξAt is an aggregate AR(1) random disturbance term with mean zero. Firms rent aunit of physical capital kt(i) = ztkt−1(i) from the households at the capital rental rate rKt . Ht(j, i)denotes the amount of hours worked of worker j at firm i. Assuming identical workers implies thatHt(j1, i) = Ht(j2, i) = Ht(i). Total output at firm i is given by a production function including theaggregate disturbance ξAt , the amount of employment relationships nt(i) and average per workercapital kt(i), so that
Yt(i) = eξAt kt(i)α
∫ nt(i)
0
g(Ht(i)1−α, n
)dn (18)
Following Pissarides (2000), the identical worker assumption implies that g(H,n) = g(H)f(n);since the mass of workers is uniformly distributed over [0, 1], f(n) = 1 ∀n ∈ [0, 1] and for g(H) Iassume: g(H) = H, which implies that the production function simplifies to
Yt(i) = eξAt kt(i)αnt(i)Ht(i)1−α (19)
Wholesale firms solve a profit maximization problem based on the following expected streamof revenues net of expenses2 for an individual firm i at time t
2The total real wage bill for firm i is the product of the mass of employment relationships at time t times thenumber of hours worked by each employee times the hourly real wage rate: nt(i)Ht(i)Wt. Similar reasoning applies
7
Et
[ ∞∑s=0
βsΛt,t+s(nt+s(i) ·
(µte
ξAt+s kt+s(i)αHt+s(i)1−α− (20)
rKt+skt+s(i)−Wt+sHt+s(i))− cvVt+s(i)
)]where Λt,t+1 is the households’ stochastic discount factor defined in equation (9), since
households are the firms’ owners, profits are evaluated in terms of value attached to them. Theemployment flow equation for firm i is given by
nt(i) = (1− s)nt−1(i) + qt−1Vt−1(i) (21)
This implies that the adjustment cost for labour is 1qt
(nt+1(i) − (1 − s)nt−1(i)). Thereforesearch theory is an alternative way of introducing adjustment costs which are linear in nt(i) andthe vacancy cost.
At each point in time, the firm maximizes its value choosing its capital stock, the work forceand the amount of vacancies posted. The first order conditions for the firms optimization problemare
(kt(i))
αµteξAt
(Ht(i)kt(i)
)1−α
= rKt (22)
(nt(i))
κt(i) = µteξAt kt(i)αHt(i)1−α − rKt kt(i)−
WtHt(i) + (1− s)βEt [Λt,t+1κt+1(i)] (23)
(Vt(i))
cvqt
= βEt [Λt,t+1κt+1(i)] (24)
The Lagrange multiplier κt is the shadow value of employment. Equation (22) states thatthe real rental rate on capital equals the marginal product of capital.
Combining equations (23) and (24) yields the following job creation condition
cvqt
= βEt
[Λt,t+1
(µt+1e
ξAt+1 kt+1(i)αHt+1(i)1−α − rkt+1kt+1(i)− (25)
Wt+1Ht+1(i) + (1− s) cvqt+1
)]The job creation condition equates the expected return of creating a new job, expressed by
the right hand side, to the cost of a vacancy creation, which is given by the expected duration ittakes to fill a vacancy 1/qt times the associated costs of a vacancy, given here by the vacancy cost
to the capital costs.
8
parameter cv.
2.4 Wage bargaining
The matching friction gives rise to a bilateral monopoly context. When workers and firms match,there are rents to be split: if one party - either the worker or the firm - turns down a wage offer,finding another potential partner is costly. I proceed by following Pissarides (2000) and Mortensenand Pissarides (1994) and specifically Christoffel and Linzert (2005) by applying the efficient Nashbargaining solution. In this framework, the bargained real wage (wt) and hours worked (Ht) aredetermined as the outcome of a Nash bargaining between workers and firms - bargaining over thenominal wage would lead the the same wage schedule to be derived in what follows.
2.4.1 Bellman equations
I assume that each firm-worker pairing is equally productive so that the wage rate is the sameeverywhere. Firms consider workers independent of their corresponding household type. However,a worker’s valuation of his employment and unemployment status depends on his household typein terms of the marginal rate of substitution between consumption and leisure. Consider workeri of either household type. This worker’s value function in the employment and unemploymentstates satisfy
Y E,ht (i) = (1− τN )wtHt −v(Ht)λht
+
βEt
[Λt,t+1
((1− s)Y E,ht+1 (i) + sY U,ht+1 (i)
)]∀ h ∈ {o, r} (26)
Y U,ht (i) = Γ + βEt
[Λt,t+1
(pt+1Y
E,ht+1 (i)+ (27)
(1− pt+1)Y U,ht+1 (i))]∀ h ∈ {o, r}
where v(Ht) and λot are defined as in section 2.1 and λrt = (Crt )−σc .Note that in fact only the value function for employment depends on the consumer type due
to a different valuation of marginal consumption and Γ is the amount of unemployment benefitsgiven to an unemployed worker by the fiscal authority. The value of unemployment hence dependson the current flow value Γ and the likelihood of being employed versus unemployed in the nextperiod.
The value function of firm i for job j is (compare equation (23))
Jt(i, j) = µtyt(i, j)− rkt kt(i)− wtHt(i) + βEt [Λt,t+1(1− s)Jt+1(i, j)] (28)
Finally the free entry condition implies that (compare equation (24))
cvqt
= βEt [Λt,t+1Jt+1(i, j)] (29)
The stochastic discount factor Λt,t+1 is as defined in equation (9) and the output in firm i
of job j - yt(i, j) - is as defined in equation (17). Bargaining takes place over the real wage wt and
9
hours worked Ht taking the aggregate price level Pt as given.
2.4.2 Wage setting
Due to the presence of Ricardian and non-Ricardian consumers, the Nash product is extended totake both kind of consumers into account. In particular, I assume that both groups bargain jointlywith the firms where each group’s weight is given by its corresponding share in the population.
The equilibrium real wage and hours worked are derived from the maximization of thefollowing Nash product
maxw,H
((1− ς)
[Y E,o − Y U,o
]+ ς
[Y E,r − Y U,r
])η · J1−η (30)
where η ∈ (0, 1) measures the bargaining power of firms and households.The first order condition with respect to the the real wage implies the following expression
for the target wage wt
wtHt =1− η
1− τN
(Γ +
mrstHt
1 + φ
)+ η
(µtyt(i, j)− rkt kt(i) + cvϑt
)(31)
where the marginal rate of substitution is given by
mrst = χHφt
((1− ς)λot
+ς
λrt
)(32)
The result is intuitive. The real wage workers get is increasing in the unemployment benefit(Γ), the relative bargaining strength of the worker (1− η), the tightness in the labour market andthe tax rate on households’ labour income. Workers get a weighted average of the unemploymentbenefits and the surplus, which consists of the output of job j minus the costs on capital neededfor job j to be productive.
Finally, the first order condition with respect to hours worked implies the following twooptimality conditions
wt = (1− α)µteξAt
(kt(i)Ht(i)
)αand (1− τN )wt = mrst (33)
These two equations close the labour market of the model economy.
2.4.3 Real wage rigidity
Due to some un-modeled friction, I assume that not all labour supply reaches the market; specifi-cally, the steady state supply of labour is fixed at some exogenously imposed level below the levelof the frictionless economy. Locally around the steady state, households are therefore willing tosupply labour at the going real market wage Wt, assumed to be governed by
Wt = (Wt−1)ν · (wt)1−ν · eεWt (34)
10
where εWt ∼WN(0, σ2W ). This is a way of introducing real wage rigidity as recently postu-
lated by e.g. Hall (2005) and Shimer (2005). The particular specification here follows Blanchardand Gal (2007).
2.5 Retailers
Retailers are monopolistic competitors and set prices for their goods in a staggered fashion. Theybuy intermediate goods from the wholesale firms, transform them and sell them as either invest-ment/capital or as a consumption good.
Let Yt(r) be the output sold by retailer r and let pt(r) be its corresponding sale price in thegoods market. Final goods denoted with Yt are a Dixit-Stiglitz type aggregator of the individual
goods Yt(r) according to: Yt =(∫ 1
0Yt(r)
1− 1ξRt dr
) ξRtξRt −1
where ξRt is the elasticity of substitution
between two individual goods and evolves according to log ξRt = ξR+εξR
t where εξR
t ∼WN(0, σ2ξR).
As in Calvo (1983) firms are not allowed to change prices unless they receive a randomprice-change-signal. The likeliness that firm r can re-optimize its price in any period is constantand equal to 1− θ. Let Yt+k|t(r) be the output of retail firm r at time t+ k that could adjust itsprice pt+k|t(r) optimally in period t the last time. Cost minimization on the final goods sector can
be written as: Yt+k|t(r) =(pt+k|t(r)
Pt+k
)−ξRt+k· Yt+k where pt+k|t(r) is the price of intermediate good
r. Firms that are allowed to adjust their prices in period t, set the price to maximize expecteddiscounted profits. In the periods between price re-optimization firms adjust their prices accordingto the following indexation rule:
pt+k|t =
{pt+k−1|t (Πt+k−1)ωp Π1−ωp for k = 1, 2, 3, ...pt|t = p∗t for k = 0
(35)
where pt+k|t denotes the price effective in period t + k of a firm that last re-optimized itsprice in period t. Πt := Pt
Pt−1is the aggregate gross inflation rate and ωp ∈ (0, 1) is an exogenous
parameter that measures the degree of inflation indexation. The term Π1−ωp is an adjustment fortrend inflation. The aggregate price level Pt can be expressed in the following way:
Pt =[(1− θ)(p∗t )1−ξRt + θ
(Pt−1Π1−ωpΠωp
t−1
)1−ξRt ] 11−ξRt (36)
The probability that a firm cannot re-optimize its price for k periods is given by θk. Profitmaximization by retailer r who is allowed to re-optimize his price at time t chooses a target pricep∗t to maximize the following stream of future profits
maxp∗t
Et
∞∑k=0
(θβ)kΛt,t+k
p∗tPt+k
· Yt+k|t(r) ·Πk·(1−ωp) ·k∏j=1
Πωpt+j − Cot+k
(Yt+k|t(r)
) (37)
where Cot+k(Yt+k|t(r)) is the associated cost function expressed in real terms and βkΛt,t+kis the associated stochastic discount factor. The first order condition of the retailers’ optimization
11
(37) problem is
Et
∞∑k=0
(θβ)kΛt,t+kYt+k|t(r)
p∗tPt+k
(ξRt+k − 1)Πk(1−ωp)k∏j=1
Πωpt+j − ξ
Rt+kµt+k|t(r)
= 0 (38)
where Co′t+k|t(Yt+k|t(r)) = µt+k|t(r) is the marginal cost function given by equation (26).
2.6 Fiscal and monetary policy
Monetary policy is modeled by a simple Taylor type rule
RtR
=(Rt−1
R
)ρr [(πtπ
)aπ (YtY
)aY ]1−ρr· eε
Mt (39)
where εMt ∼ WN(0, σ2M ) and R, π, Y are the corresponding steady state variables of the
nominal interest rate (or nominal bond yield), the steady state inflation rate and steady stateoutput level. The parameter ρr captures the degree of interest rate smoothing.
The fiscal authority’s budget restriction satisfies
Tt +R−1t B∗t =
B∗t−1
Πt+Gt + UtΓ (40)
where real government bonds satisfy B∗t = Bt/Pt and Πt = Pt/Pt−1. Γ is the amount ofunemployment benefits given to an unemployed worker as outlined in section 2.4.1. Hence thetotal amount of expenditures due to unemployment payments is UtΓ. The total amount of taxescollected satisfies
Tt = τNWtNtHt + τKrKt ztKt−1 (41)
The government budget constraint implies a stable difference equation in nominal debt aslong as Rt < 1 and in real debt as long as Rt/Πt < 1. Hence the fiscal policy sector has to bemodified such as to control for the public debt in case the previous stability conditions are notsatisfied.
2.7 The fiscal policy rule
In the present model the fiscal authority controls government consumption by making use of thefollowing fiscal policy rule (expressed in log-deviations from the corresponding steady state)
Gt = φGGGt−1 + (1− φGG)(φGBB
∗t + φGY Yt + φGU Ut
)+ εGt (42)
where εGt ∼WN(0, σ2G)
where it is assumed that φGG ∈ [0, 1), φGB < 0, φGY < 0, φGU > 0. The policy rule extends the
12
standard model set up in terms of a higher degree of flexibility for the fiscal authority.The rule for Gt is defined such that increases in debt trigger declines in government expen-
ditures in order to control for the deviation of public debt from its long run level and hence toguarantee stability. Based on the log-linearized equilibrium equations outlined in appendix C thecondition for the government budget constraint to be a stable difference equation in B∗t requiresthe following to be satisfied: by(R−1 −Π−1) < γG(1− φGG)φGB where by = B∗/Y and γG = G/Y .
The current specification of the fiscal sector implies that there are two different types offiscal policy mechanisms at work. On the one hand, the tax specification and the unemploymentspecification are two types of the so called automatic stabilizers. These automatic stabilizerschange the fiscal policy stance automatically to altered economic situations. These mechanismsfunction without the government taking any specific actions. In the current model set up, recessionstriggering lower income and higher unemployment automatically imply lower tax revenues andhigher expenditures due to higher unemployment payments which finally both lead to expansionaryfiscal policy effects.
However, the model also has room for discretionary fiscal policy of two types. The fiscalpolicy rule implies that the government actively intervenes to changes in the economic stance bothby relating its actions to current economic conditions and to an exogenous part. The reaction ofthe government to changes in unemployment and output determines that part of the rules whichdependent on the current economic stance, while εG is a purely random intervention, i.e. surprisesin the fiscal authority’s actions.
Throughout the analysis, fiscal policy is analyzed taking monetary policy as given.
2.8 Equilibrium and solution of the model
Assuming representative firms next to representative households in equilibrium the aggregate cap-ital supply (Kt) in period t is equal to total capital demand so that: Kt = ntkt−1. Moreover,labour supply (Nt) in period t is equal to labour demand (nt): Nt = nt. The aggregate resourceconstraint satisfies
Ct + It +Gt + cvVt = Yt − ψ(ztξzt )Kt−1 (43)
and, by using equation (19) and the labour market equilibrium conditions, total outputsupplied can be rewritten in the following way
Yt = eξAt (ztKt−1)α (NtHt)
1−α (44)
In order to solve the model, I first log-linearize the model around the non-stochastic steadystate, provided in appendix B, where it is assumed that all variables except for the aggregateprice level Pt are stationary. The model has a well defined steady state. Appendix C contains thecomplete log-linear model.
13
3 Econometric methodology
Recently, various different methodologies have been developed to estimate the underlying struc-tural parameters of DSGE models given its reduced form representation. Here I proceed by using aprocedure proposed by Smith (1993) and Gourieroux, Monfort and Renault (1993) called IndirectInference. I minimize a measure of distance between the model and empirical spectral densityfunctions as proposed by Diebold et al. (1998). Working with spectral densities enables a decom-position of variation across frequencies which is often useful and the multivariate focus facilitatesthe examination of cross-variable correlations and lead-lag relationships at those frequencies ofinterest. Specifically I only use those frequencies in the data which correspond to low cyclicalitiesincluding business cycle periodicities. In particular, the estimation methodology carried out heredoes not use all the information being contained in the data. This is intentional for the followingreason. DSGE models and especially New Keynesian and Real Business Cycle models are theoreti-cal models aiming at explaining business cycles. However, usually the data used for estimation area composition of cycles of different periodicities; there are long periodic cycles describing growthpatterns, seasonal cyclicalities, purely random components and among them, business cycle pat-terns. Since the model is only aimed at explaining low cyclicalities and especially business cyclefrequencies, I hence make only use of this part of information in the data3. Carrying out thisestimation procedure hence requires a frequency domain approach.
More formally, the model is solved using the algorithm of Uhlig (1999) which yields thefollowing state space representation:
[I(x) −Q(θ)0 I(z)
][xtzt
]=
[P(θ) 0
0 N(θ)
][xt−1
zt−1
]+
[0εt
](45)
where εt ∼WN (0,Σ(θ))
yt =[
R(θ) S(θ)] [ xt
zt
](46)
where xt is a vector of endogenous state variables, yt is a vector of control variables, zt isa vector of exogenous state variables and θ is a vector containing the structural parameters andvariances of the exogenous shock processes. Σ(θ) is a diagonal matrix with the variances of theexogenous shock processes on its main diagonal and zeros everywhere else since it is assumed thatthe structural shocks are all orthogonal to each other.
Let H(ω)∀ω ∈ [0, π] denote the n× n matrix of spectral densities of observed time series (ndenotes the number of variables). Denote by H(θ, ω) the synthetic counterpart of H(ω) of artificialspectral densities generated by the DSGE model. Specifically, for the solution of the DSGE modelgiven by equations (45)-(46), the spectral densities of the variables are given by
3The alternative to this approach is to filter the data before using them. However, the problems with filteringare outlined well in Canova (1998).
14
1990 200013.5
14
14.5
cons
umpt
ion
Time Series
0 2−15
−10
−5(log) Spectrum
1990 20000
0.05
0.1
nom
inal
rat
e
Time Series
0 2−20
−10
0(log) Spectrum
1990 2000−0.2
0
0.2
infla
tion
0 2−15
−10
−5
1990 200012.5
13
13.5
inve
stm
ent
0 2−10
−5
0
1990 200014
14.5
15
outp
ut
0 2−15
−10
−5
1990 20000.9
0.95
1
empl
oym
ent r
ate
0 2−20
−10
0
1990 20006
6.5
7
hour
s w
orke
d
0 2−15
−10
−5
1990 20005
6
7
vaca
ncie
s
0 2−10
−5
0
1990 200013.5
14
14.5
real
wag
e
Time (quarters)0 2
−15
−10
−5
ω1990 2000
−0.2
0
0.2go
vern
men
t
Time (quarters)0 2
−15
−10
−5
ω
Figure 1: Plot of the data (all in logs except the inflation, nominal interest and employment ratewhich are in levels) and their spectral densities (in logs) over all frequencies (ω ∈ [0, π]).
gz (ω, θ) =1
2π(I−N(θ)e−iω
)−1Σ(θ)
(I−N′(θ)eiω
)−1(47)
gx (ω, θ) =(I−P(θ)e−iω
)−1Q(θ) · gz (ω, θ) ·Q′(θ)
(I−P′(θ)eiω
)−1(48)
gy (ω, θ) = R(θ) · gx (ω, θ) ·R′(θ) + S(θ) · gz (ω, θ) · S′(θ)ω ∈ [0, π] (49)
where ω ∈ R, however, it is sufficient to consider ω ∈ [0, π]. Then, the Indirect Inferenceestimator of θ, call it θ, is given by a vector which solves
θ = arg minθ
∫ ωU
ωL
f(H(ω),H(θ, ω)
)�V(ω)dω (50)
where � is the Hadamard-product. In practice the integral is replaced with a sum overfrequencies ω ∈ [ωL, ωU ] ⊂ [0, π]. For what follows, I specify a quadratic loss function withuniform weighting over all frequencies4.
3.1 Estimation
I use data of Germany from the OECD data base on ten variables for the estimation, these are:(1) real GDP, (2) real personal consumption of non-durables, (3) real private investment, (4) hoursworked per employee, (5) inflation, measured by the percentage change of the GDP deflator, (6)
4The computations are carried out in Matlab (Version 6.0.0) using the Differential Evolution Algorithm (seeStorn et al. (2005)).
15
the employment rate, (7) the nominal interest rate on 3-month government bonds, (8) the realwage rate which is defined here as the compensation per hour worked in the non-farm industry,(9) real government consumption and (10) the number of vacancies posted. The GDP-Deflatoris used in all cases where nominal variables need to be converted to real ones. The sample is:1990q1:2006q4.
The empirical spectral densities are estimated based on a parametric estimation. In orderto compute them, I estimate a stationary Bayesian Vector Autoregression (BVAR) using a non-informative prior density, four lags, a time trend and a constant term (see Uhlig (1994) andCanova (2007) for further details). The reason for sticking to a Bayesian VAR is due to the factthat this procedure facilitates the computation of standard errors of the estimated spectra as wellas confidence intervals. All variables enter the BVAR in logarithmic terms except the nominalinterest rate, inflation and the employment rate which are used in levels. A plot of the data usedin the BVAR is given in figure 1. Given these estimates, I transform the BVAR to get an estimatorfor the spectrum of the variables (see for instance Hamilton (1994), chapter 10). Specifically, letthe VMA of the corresponding VAR be given by
yt = µ(t) + Ψ(L)et, et ∼ N(0,Σ) ∀ t = 1, ..., T (51)
An estimator for H(ω) is then given by
H(ω) =1
2π
(Ψ(e−iω)
)Σ(Ψ(eiω)
)′, ∀ ω ∈ [0, π] (52)
I proceed by numerically drawing 10,000 Monte Carlo replications of the posterior densityof the BVAR coefficients and taking the median across all draws. Figure 1 shows the spectra ofall variables next to the data used in the BVAR. The graphs indicate that there are indeed lots offluctuations which DSGE models are not able to account for. Especially hours worked, governmentexpenditures and vacancies have many cyclicalities outside the typical business cycle interval. Totake into account the failure of the DSGE model to explain fluctuations outside the business cycleinterval, I henceforth use only fluctuations which correspond to 2-year up to 15-year periodicities;this implies that the frequencies considered are: ω ∈ [2π/60, 2π/8]. In order to evaluate H(ω) andH(θ, ω) I use a grid of size N for the frequencies: {ωi}Ni=1 ∈ [2π/60, 2π/8].
Note that H(ω) as well as gx (ω, θ), gy (ω, θ) and gz (ω, θ) are matrices with real numberson the main diagonal and complex numbers on the off-diagonal elements. Due to the fact thatwithin the estimation algorithm the same element in H(ω) and H(θ, ω) need not necessarily havethe same form of number (either real or complex) a problem which can emerge henceforth is thatthe estimates of the structural parameters are complex numbers too, which does not make anysense at all. Diebold et al. (1998) proceed by using only the spectra as well as the coherence andthe phase shift functions in their matching exercise. In what follows I proceed by making use onlyof the absolute value of each element in the spectral matrices.
A further important point concerns the level of the spectrum which is matched. Since eachvariable has a different level in its spectrum, which expresses the differences in the variances, thestandard question of how to adjust the model to the empirical data also applies here. However, inthe current procedure, this problem is solved rather easily. Here I proceed by matching normalizedspectra; that is, as can be seen in figure 2 each spectral density function is normalized to unity at
16
Table 1: Calibrated parameters
Parameter Value
β discount factor 0.98δ depreciation rate 0.025p probability a searching worker finds a job 0.5α capital share in output 0.33ζm elasticity to unemployment in matching function 0.5τK capital tax rate 0.35τN labour tax rate 0.40η bargaining power of workers 0.5ξR elasticity of substitution between two final goods 6σc coefficient of relative risk aversion 1by steady state fiscal deficit 0.50cy consumption share in output 0.55γΓ unemployment payments share in output 0.01γG government expenditures share in output 0.20
the lowest frequency. I do so for the following two reasons:First, to the extent that different variances are expressed by different levels of the spec-
tral density function, the estimation procedure gives higher weight to those variables within thematching process whose spectral density function are at a higher level, independently of using anoptimal weighting matrix in the distance function (50) or not. This implies that the fit of thespectral density functions of those variables with rather small variance is indeed poor. Matchingthe spectral density functions based on normalized spectra gives equal weight to all variables.
Second, the empirical spectral density functions are based on the level of the variables whilethose of the theoretical ones only on the deviation of the variables from their corresponding steadystate. This does not affect the shape of the density functions, however, it matters for the levelsince the area under the spectrum is the unconditional variance and (log) linear transformations,as it is done within log-linearization, transforms the variables by a constant term which cruciallyinfluences the level of the variance and hence the level of the spectrum. Since the transformation ofa time series with a constant term effects its variability across all frequencies by the same amount,the spectral shape is henceforth unaffected. To this extent a normalization of the empirical andtheoretical spectral density functions wipes out this discrepancy.
There are a couple of parameters which are not estimated but calibrated instead which aregiven in table 1. These values are based partly on long run statistics as in Trabandt and Uhlig(2006) as well as on estimates obtained in Pytlarczyk (2005) and Christoffel et al. (2007).
3.2 Estimation results
The estimation results are presented in table 2. In order to elaborate on the research questionsstated in the introduction, I estimate three different versions of the previously specified DSGEmodel. The distinctions among them only refer to different fiscal policy specifications. Model 1 intable 2 refers to a fiscal policy structure where fiscal policy is governed as outlined in section 2.7.The second model defines fiscal policy as being completely exogenous to fluctuations other than itsown ones. Specifically, I estimate σ2
G, φGG and φBG only. The third model is as model 1 but omitsσ2G implying that no surprises are allowed in fiscal policy.
In order elaborate on the way the current estimation procedure works and how well the
17
Tab
le2:
Est
imat
edpa
ram
eter
s
Model
1M
odel
2M
odel
3
Para
mete
rE
stim
ate
t-valu
eE
stim
ate
t-valu
eE
stim
ate
t-valu
e
ρA
pro
ducti
vit
ysh
ock
0.7
41
0.2
80.6
97
3.6
20.7
10
2.9
5σ
pro
ducti
vit
ysh
ock
0.0
942
1.1
50.1
078
1.0
80.1
088
0.6
6σ
dis
count
facto
rsh
ock
0.0
932
1.0
60.1
129
0.5
80.2
236
1.1
0σ
govern
ment
exp
endit
ure
s0.2
236
1.4
20.1
960
1.0
6-
-σ
moneta
ryp
olicy
shock
0.1
941
0.7
80.2
129
0.9
40.2
352
1.4
4σ
invest
ment
shock
0.0
928
0.8
20.1
366
1.2
30.1
475
0.5
4σ
mark
up
shock
0.1
032
0.2
90.1
125
0.8
80.1
310
0.8
0σ
wage
shock
0.0
600
11.2
0.0
805
0.4
80.1
132
0.9
2
job
separa
tion
rate
(s)
0.0
36
2.6
60.0
54
3.6
20.0
46
3.6
3re
al
wage
rigid
ity
(ν)
0.7
41
6.5
60.6
91
2.1
70.7
19
4.3
6vacancy
post
ing
cost
(cv)
8.4
90
3.3
49.0
45
2.7
87.9
91
3.0
3
non-r
icard
ian
share
(ς)
0.2
01
0.5
70.2
12
0.7
90.2
86
1.0
7habit
pers
iste
nce
(h)
0.7
34
2.2
60.7
37
4.4
70.7
80
15.8
8la
bour
supply
(φ)
4.1
20
1.9
64.1
97
2.6
74.3
10
0.7
6capit
al
uti
lizati
on
cost
(ψ2/ψ
1)
0.8
72
0.5
60.9
31
0.4
70.9
02
0.2
3in
vest
ment
adju
stm
ent
cost
(κI)
3.2
49
1.8
52.9
25
1.1
12.7
50
0.9
2
Calv
opri
ces
(θp)
0.9
02
4.6
00.8
86
3.5
10.8
72
8.5
7pri
ce
indexati
on
(ωp)
0.9
81
5.0
40.9
69
4.5
40.9
86
4.4
4
govern
ment
exp
endit
ure
ssm
ooth
ing
(φG G
)0.5
67
2.3
60.6
20
3.7
10.5
98
6.1
5
govern
ment
debt
(φG B
)-0
.579
-1.8
8-1
.293
-1.5
9-3
.171
-1.9
5
outp
ut
(φG Y
)-2
.047
-0.9
1-
--2
.773
-0.4
1
unem
plo
ym
ent
(φG U
)2.7
62
1.3
8-
--2
.973
0.3
1
inte
rest
rate
smooth
ing
(ρR
)0.7
59
2.9
10.7
75
1.4
70.7
51
2.3
4in
flati
on
inm
oneta
ryp
olicy
rule
(aπ
)1.5
67
3.7
31.5
30
3.7
41.4
98
4.4
9outp
ut
inm
oneta
ryp
olicy
rule
(aY
)0.0
39
1.0
6-0
.080
-0.0
9-0
.058
-0.5
9
SSR
196.1
51665.8
9437.0
7B
Ksa
tisfi
ed
yes
no
yes
pro
b(B
Ksa
tisfi
ed)
0.6
90.4
00.3
8
Notes:
SSR
denote
sth
esu
mofsq
uare
dre
siduals
,B
Ksa
tisfi
ed=
yes
ifth
eB
lanch
ard
-Kahn
condit
ions
are
sati
sfied
for
the
poin
test
imate
sand
pro
b(B
Ksa
tisfi
ed)
giv
es
the
pro
babilit
yof
these
condit
ions
bein
gsa
tisfi
ed
once
havin
gsi
mula
ted
the
model5,0
00
tim
es
base
don
the
ass
um
pti
on
that
the
stru
ctu
ral
para
mete
rsfo
llow
anorm
al
dis
trib
uti
on.
18
0.2 0.4 0.60
1
2
3
4
5
6
7
8consumption
ω0.2 0.4 0.6
0
1
2
3
4
5
6
7
8nominal rate
ω0.2 0.4 0.6
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5inflation
ω0.2 0.4 0.6
0
2
4
6
8
10
12
14
16investment
ω0.2 0.4 0.6
0
2
4
6
8
10
12
14
16output
ω
0.2 0.4 0.60
2
4
6
8
10
12employment
ω0.2 0.4 0.6
0
1
2
3
4
5
6hours worked
ω0.2 0.4 0.6
0
2
4
6
8
10
12
14
16
18vacancies
ω0.2 0.4 0.6
0
1
2
3
4
5
6
7
8
9real wage
ω0.2 0.4 0.6
0
1
2
3
4
5
6gov. expenditures
ω
Figure 2: Estimated spectral density functions of the data (black solid line) with confidence inter-vals (16th and 84th percentile of the posterior density function). The red dashed lines representthe spectra of the DSGE model.
model spectra fit the empirical ones, I use model 1 to display some interesting patterns. Theanalysis here only focuses on spectra and ignores cross spectra for simplicity. Figure 2 shows theempirical spectra with confidence intervals5 next to the model spectra. As can be seen, the DSGEmodel appropriately mimics the empirical spectra - the theoretical spectra are always inside theconfidence intervals of the empirical ones. Moreover, the DSGE model also adequately describesthe typical hump-shaped empirical pattern.
As far as the robustness of the results is concerned, the estimates of the variances of theexogenous shock processes are highly sensitive to the chosen frequency band, which is, however,rather intuitive. Due to the fact that the area under the spectrum of a time series equals its variance,using only a part of the spectrum, as is done in the current estimation exercise, while ignoring therest crucially affects the estimates of the variances of the exogenous shock processes. Moreover,this induces a discrepancy between commonly estimated values and the ones here. Apart fromthis, however, the current estimation procedure is invariant to the stochastic singularity problemcommon in the estimation of DSGE models.
The estimates of the structural parameters other than those of the fiscal policy rules, aresimilar in size with commonly estimated values; specifically, the estimates of the parameters cap-turing the degree of price rigidity are similar in size as those obtained by Pytlarczyk (2005) andChristoffel, Kuester and Linzert (2007) using Bayesian estimation techniques. Moreover, the es-timates of the interest rate smoothing parameter and the inflation reagibility parameter in themonetary policy rule closely reflect those of Pytlarczyk (2005) and Christoffel, Kuester and Linz-ert (2007). The estimates of the exogenous job separation rate highlight that there is a rather highexogenous degree of inertia in the labour market, especially along the extensive margin. This high
5I compute confidence bands for the spectral density functions by numerically drawing 10,000 Monte Carloreplications of the posterior density, ordering them and extracting the 68% confidence band (from the 16th to the84th quantile) as suggested by Canova (2007).
19
−1 −0.8 −0.6 −0.4 −0.20
0.01
0.02Standard Deviation of π
φGB
−1 −0.8 −0.6 −0.4 −0.20
0.5
1Standard Deviation of U
φGB
−1 −0.8 −0.6 −0.4 −0.20
0.2
0.4Standard Deviation of Y
φGB
0 1 20
0.01
0.02
φGU
0 1 20
0.5
1
1.5
φGU
0 1 20
0.2
0.4
φGU
−2 0 24
6
8
10x 10
−3
φGY
−2 0 20
0.2
0.4
φGY
−2 0 20
0.1
0.2
φGY
Figure 3: Standard deviations of π, U and Y where the effects of the policy rules coefficients arealways analyzed with the remaining parameters being set equal to zero.
degree of rigidity in the labour market transmits into a rather sticky real wage which can be seenwell by the high value of ν.
Prior to commenting on the estimation results specific to the fiscal policy rules I analyzethem in detail regarding their impact on the variability of key macroeconomic variables.
4 Analysis
Using the previously estimated structural parameters, specifically those of model 1, this sectionfocuses on an analysis of the impact of the discretionary fiscal policy rule specified in section2.7 where the impact of each variable and coefficient is explored in isolation of the remainingparameters. The analysis focuses, among others, on second moments, impulse response functions,forecast error variance decompositions and spectral densities. It is important to realize that thesestatistics are, given the variances of the stochastic processes, not stochastic and hence their valuescan be directly obtained from the structural parameters without simulating the model. Hencethere is no need for specifying a density for the stochastic terms in the processes for the exogenousstate variables 6.
4.1 Variability
Fluctuations are at the core for governmental intervention to achieve less variability and smalleramplitudes of fluctuations. Figure 3 plots the standard deviations of inflation, unemployment andoutput as functions of the policy parameters of the fiscal policy rule for government expenditures
6 The second moments for the exogenous state, the endogenous state and the control variables can be computedin the following way
vec (Γz(θ)) = [I−N(θ)⊗N(θ)]−1 vec (Σ(θ)) (53)
vec (Γx(θ)) = [I−P(θ)⊗P(θ)]−1 (Q(θ)⊗Q(θ)) vec (Γz(θ)) (54)
Γy(θ) = R(θ)Γx(θ)R′(θ) + S(θ)Γz(θ)S′(θ) (55)
20
(equation (42)). While letting one specific parameter vary to investigate its impact on the standarddeviations, the remaining coefficients are fixed at their estimated values, such that the mechanismat work in creating changes in the standard deviations can be assigned uniquely to the one analyzed.The results highlight interesting patterns.
The effect of the coefficient specifying the sensitivity of government expenditures to changesin debt (φGB) clearly shows that reactions to changes in debt imply a continuous decline in thestandard deviation of inflation, unemployment and output for larger values of φGB . The intuitionbehind this is the following. A strong reaction of the government to deviations of public debtfrom its steady state value imply also large impacts in size on output, inflation and employmentthrough the aggregate resource constraint. Hence the high variability of government expendituresdue to debt fluctuations amounts into higher instability of those variables due to the nature ofgovernment expenditures as interventions on the demand side.
The effect of φGU also shows rather homogeneous effects on the standard deviations of thethree variables under consideration. In all cases a declining pattern of the variability can beobserved with increasing values of φGU . This is in line with countercyclical fiscal policy. It isapparent that only procyclical fiscal policy significantly increases instability, whereas a fiscal policyrule reacting positively to unemployment deviations or not at all does not imply large differencesin the variability of inflation, unemployment and output.
The effect of stabilization in terms of reacting to output fluctuations shows a rather clearpattern for all three variables. It is interesting to note that here, fiscal stabilization policy in termsof a reaction to output indeed is a powerful tool for not only dampening fluctuations in output,but in unemployment and inflation too.
From a qualitative point of view, changes in the structural variance of surprise fiscal policyshocks as well as its degree of persistency described by the parameter φGG, do not imply any specificchanges in the pattern of the standard deviations of unemployment, inflation and output outlinedpreviously. What changes, however, is the size of the overall variance of unemployment, outputand inflation in absolute terms. The calibrated model replicates empirical variances for these threevariables appropriately. It is interesting to note that changes in the various policy parameters ofthe fiscal policy reaction function cause only small changes in the value of the standard deviationof inflation in absolute terms, whereas, the range of variation is much larger for the other twovariables. This implies that fiscal policy as such has a much stronger impact on output andunemployment than on inflation; more on this in section 4.4.
4.2 Cyclical interdependences
Since the various previously outlined fiscal policy specifications imply rather different patterns forthe variability of the model economy, it is likely that also the cyclical pattern of fluctuations isinfluenced. Due to the fact that the area under the spectrum is the variance, the current sectionextends the previous one in elaborating on the changes in the variability of inflation, unemploymentand output. Specifically the question is whether these changes are induced by changes in theamplitude or frequency of fluctuations. The important thing to note is that a change in thespectrum does not necessarily imply a change in the duration of cyclicalities since it can equallywell affect the amplitude of fluctuations. To this extent a movement of the spectrum form one
21
−1
−0.5
0 0.20.4
0.6
0
1
2
ω
Spectrum of π
φBG
−1
−0.5
0 0.20.4
0.6
0
2
4
ω
Spectrum of U
φBG
−1
−0.5
0 0.20.4
0.6
0
0.2
0.4
ω
Spectrum of Y
φBG
01
2 0.20.4
0.6
0
0.2
0.4
ωφUG
01
2 0.20.4
0.6
0
1
2
ωφUG
01
2 0.20.4
0.6
0
0.1
0.2
ωφUG
−2
0
2 0.20.4
0.6
0
0.5
ωφYG
−2
0
2 0.20.4
0.6
0
0.5
1
ωφYG
−2
0
2 0.20.4
0.6
0
0.1
0.2
ωφYG
Figure 4: Spectrum of π, U , and Y ; ω ∈ [0, π] is the frequency, however, only those frequenciescorresponding the (quarterly) business cycles are plotted (ω ∈ [ 2π
60 ,2π8 ] which corresponds to cycles
of length 8 quarters to 60 quarters).
frequency band onwards to another amounts into a change in the duration of cycles while a general,not necessarily, proportional upward or downward movement would be in favour of a change in theamplitude of fluctuations.
Figure 4 shows the spectrum of inflation, unemployment and output for frequencies corre-sponding to business cycles of two to fifteen years length (ω ∈ [2π/60 ∼= 0.10, 2π/8 ∼= 0.78], forquarterly data). The figures, similarly as before, show the impact of one parameter of the fiscalpolicy rule for government expenditures in isolation of the remaining ones. The parameter spaceover which the coefficients vary is the same as before. The graphs show some interesting patterns.
For the three variables at focus, the cyclicalities are affected rather homogeneously by variousfiscal policy specifications. The graphs of the spectral density functions highlight that for anychosen parameter of the fiscal policy rule for government expenditures, the impact on the spectraldensities is such that the new policy induces proportional changes in the spectra for a rathernarrow frequency band only, while leaves the form of the spectral density functions outside thisband completely unchanged. Across the parameter space, the spectrum does not change its shape,it only indicates a higher or lower value for specific calibrations for a rather narrow band acrossthe frequency space. This implies that different fiscal policies primarily affect the amplitude offluctuations rather than the duration of cyclicalities.
Hence in terms of the variability of the variables induced by the government only the ampli-tude of fluctuations is affected, however, hardly the duration of cycles. To this extent the effect ofthe reagibility of government expenditures to inflation, unemployment and output affects the over-all fluctuations in a similar way as surprise fiscal policy shocks which also only lead to proportionalupward or downward movements of the spectral density functions.
22
0 5 10 15 20 25 30 35−5
0
5
10
15x 10
−3 Impulse Response of π
0 5 10 15 20 25 30 35−0.4
−0.2
0
0.2
0.4Impulse Response of U
0 5 10 15 20 25 30 35−0.2
0
0.2
0.4
0.6
Horizon (in quarters)
Impulse Response of Y
Figure 5: Impulse response functions of inflation (π), unemployment (U) and output (Y ) to agovernment expenditures shock.
4.3 The adjustment paths
In general, the impulse response functions due to a shock in government expenditures (shown infigure 5 with an approximate 68% confidence interval for the response functions7) are in line withgeneral wisdom - inflation and output increase while unemployment declines, which is supported byvarious other empirical studies (see for instance Mountford and Uhlig (2008), Coenen and Straub(2005) and Blanchard and Perotti (2002) among others).
Both output and unemployment display a rather strong effect initially which, however,dies out very quickly. The initial reaction of these two variables to this surprise shock is ratherheterogeneous. Unemployment drops by around 0.2% while output increases by around 0.2%points. Eventhough the shock as such is rather persistent (φGG = 0.55) its long-lasting effects onthese two variables is negligible. Inflation displays a hump-shaped adjustment pattern with bothpositive and negative deviations. To the extent that these deviations are of rather equal length,their size is negligibly small.
Within the DSGE model, the source of the fiscal policy shock is uniquely defined. From anempirical point of view, however, this is everything else but clear. To this extent, there are threerealistic sources: (a) changes in the relative weights defined by the fiscal authority when reactingto fluctuations in output, unemployment, etc., (b) imperfect information on the part of the fiscalauthority about the current or future economic stance and (c) changes in the fiscal policy stanceunrelated to the current or future economic stance as well as unrelated to the prevailing policyreaction function.
The first source of fiscal policy shocks refers to the decision making process within thefiscal authority. Different fiscal policy positions on how to set government expenditures are likelydescribed by different preferences concerning the relative weights determining the reagibility ofthose with respect to fluctuations in output, unemployment, etc. As a result, the decision makingprocess itself can be random. In this case the random component in the fiscal policy reaction
7The confidence intervals are obtained by assuming an (approximate) normal distribution for the estimatedparameters, simulating the model and extracting the 16th and 84th percentile.
23
function corresponds to random fluctuations in the preferences of the fiscal authority.The second source of fiscal policy shocks refers to measurement errors caused by lags in the
collection of the data of those variables which are essential within the fiscal authority’s decisionmaking process. The fiscal authority can observe the actual economic stance and reverse policyactions due to measurement errors only after the final data has become available. Hence, witha fiscal reaction function based on revised data due to previous misperceptions of the economicstance, all previous policy actions show up in the model as deviations from the rule, which canthen be interpreted as unexpected fiscal policy shocks.
The third source refers to genuine surprise shocks which are indeed unrelated to the currentfiscal policy reaction function as well as to the current or future economic stance.
In order to quantify the importance of these surprise shocks, the next section presents someevidence using forecast error variance decompositions.
4.4 Surprise fiscal policy shocks
Surprise fiscal policy shocks are an important source of fluctuations. This can be best seen intable 2 from the estimates of model 1. In model 1 all fiscal policy parameters are estimatedincluding those of the variances (σ2
G). The estimates for the fiscal policy rule for governmentexpenditures clearly indicate countercyclical fiscal policy with especially a rather strong reactionto output deviations. The estimate for the surprise in fiscal policy (σ2
G) has a rather large valuecompared to the other shocks. Now, the crucial thing to notice is that when σ2
G is omitted in theestimation (consider model 3), specifically its value is set to zero, higher variability is introducedby a strong procyclical fiscal policy of government expenditures in terms of a procyclical behaviourwith respect to unemployment. This implies that when ignoring surprise fiscal policy shocks, animportant source of fluctuation is discarded such that a different fiscal policy stance has to accountfor this lack of variability.
Moreover, despite in-significant effects of unemployment and output on government expen-ditures (consider the estimates of model 1 in table 2), ignoring those variables leads to a changein the variance of surprise fiscal policy shocks which henceforth captures the omitted effects ofthose variables. Specifically, the variance of surprise fiscal policy shocks is much larger in model 1where countercyclical policy is allowed in terms of a reaction to output and unemployment, whilein model 2 this damping factor is not prevalent and hence also the standard deviation of surprisefiscal shocks is much smaller. As it turns out, the drop in the standard deviation is rather severe:from 0.22 down to 0.09. Moreover, for the point estimates, model 1 satisfies the Blanchard-Kahnconditions (Blanchard and Kahn (1980)) while model 2 does not. As Lubik and Schorfheide (2004)argue one can expect a richer pattern of the autocovariance functions which cannot be reproducedwith parameters from determinate models. Hence model 1 offers a richer environment as comparedto model 2 such as to adjust the model to the data given that determinacy is satisfied.
When focusing on the estimates of model 1, it is apparent that fiscal policy is both animportant force in contributing to fluctuations, due to its strong dependence on output and un-employment fluctuations, as well as an important source of fluctuations. To the extent that gov-ernment expenditures cause rather different persistences across various variables, a surprise shocktriggers important fluctuations in many of them. This is confirmed by the forecast error variance
24
Table 3: Forecast error variance decomposition
shock σA σu σG σM σI σR σW
Horizon: 4 quarters
Inflation 14.29 1.63 15.49 21.09 1.44 10.32 25.74Unemployment 22.55 14.93 27.83 15.82 4.07 2.57 12.23Output 38.44 2.74 23.09 11.87 9.95 3.12 10.79
Horizon: 16 quarters
Inflation 13.41 8.67 6.34 25.04 11.46 14.33 20.75Unemployment 16.99 13.24 22.15 16.21 12.47 3.52 15.42Output 24.16 6.11 26.00 13.22 8.22 8.01 14.28
Horizon: 32 quarters
Inflation 15.40 13.67 3.35 15.03 15.46 13.33 23.76Unemployment 17.66 12.77 9.90 16.79 14.18 8.83 19.86Output 13.79 6.70 25.59 16.66 11.27 9.21 16.78
Notes: σA denotes the technology shock, σu the preference shock, σG the gov-ernment expenditures shock, σM the monetary policy shock, σI the investmentshock, σR the price markup shock and σW the real wage shock.
decomposition8 outlined in table 3. As can be seen from the table, fiscal policy shocks explaina large amount of the fluctuations in inflation and unemployment for short horizons, but thenthis magnitude declines strongly. For output, fiscal policy shocks explain around 20-25% of thefluctuations rather independently of the specific horizon considered. Relative to the other shocks,fiscal policy shocks are a major source of fluctuations.
5 Conclusion
This study analyzed the influence of discretionary fiscal policy based on an estimated DSGE modelfor Germany. Discretionary fiscal policy was designed to have government expenditures react tooutput and unemployment with an additional stochastic error term. In particular, the focus was to
8For the model’s solution given by equation (45) and (46), the forecast error variance decomposition for thevariables in the endogenous state vector xt is given by
γm,n(h) =
Ph−1j=0
“1′m,(x)
Aj(θ)Σ(θ)1/21n,(z)
”2
Ph−1j=0 1
′m,(x)
Aj(θ)Σ(θ)Aj(θ)′1m,(x)
(56)
where the Aj(θ) matrices satisfy A(L) = (I − P(θ)L)−1Q(θ)(I − N(θ)L)−1 so that A0(θ) = Q(θ), A1(θ) =Q(θ)N(θ) + P(θ)Q(θ) and Aj(θ) = Aj−1(θ)N(θ) + P(θ)jQ(θ) ∀ j > 1. 1
m,(x)is a column vector of zeros of the
same size as xt with a one in the mth row and 1m,(z)
is a column vector of zeros of the same size as zt with a one
in the mth row.For the control variables yt the forecast error variance decomposition is given by
γm,n(h) =
Ph−1j=0
“1′m,(y)
Bj(θ)Σ(θ)1/21n,(z)
”2
Ph−1j=0 1
′m,(y)
Bj(θ)Σ(θ)Bj(θ)′1m,(y)
(57)
where the Bj(θ) matrices satisfy B(L) = R(θ)A(L) + S(θ)(I − N(θ)L)−1 so that Bj(θ) = R(θ)Aj(θ) +S(θ)N(θ)j ∀ j ≥ 0 and the Aj(θ) matrices are as defined above. 1
m,(y)is a column vector of zeros of the same
dimension as yt with a one in the mth row.γm,n(h) then denotes the fractions of the forecast error variance until horizon h of variable m which is accounted
for by the nth shock.
25
investigate, for output, unemployment and inflation, in how far fluctuations therein can be affectedby fiscal policy. The important point here, among others, was to figure out the effect of fiscal policyonce it is characterized as a force in contributing to the changing characteristics of cycles versus ascenario where it acts as a cause of fluctuations.
The following conclusions can be drawn from the analysis. Discretionary fiscal policy hasa strong impact on the amplitude of fluctuations while basically no on the duration of businesscycles. The results highlight that, similar as for monetary policy, fiscal policy can smooth cyclicalfluctuations but there is no possibility of fundamentally disturbing cyclicalities in their lengths.
Apart from the fact that countercyclical discretionary fiscal policy can even be destabilizing(Friedman (1953)) the results obtained here clearly indicate that countercyclical fiscal policy indeedcontributes strongly to diminish the amplitude of fluctuations.
Fiscal policy shocks originating in government expenditures are an important source ofeconomic fluctuations as especially the forecast error variance decomposition displayed. To theextent that this technique indicates the importance of surprises in fiscal policy for explainingeconomic fluctuations, the applied econometric methodology strengthens this result since oncethe variance of the policy rule is omitted from the model, implying that there are no surprisesin fiscal policy to be allowed, the discretionary fiscal policy stance changes significantly froma countercyclical to a strongly procyclical one so as to account for the fluctuations triggered bysurprise shocks. Relative to the other shocks, fiscal policy shocks are a major source of fluctuations.
But besides the fact that fiscal policy is an important source for causing fluctuations, itis also an important responder such that, as the estimates for the fiscal policy rules indicate,government expenditures significantly contribute to dampening cyclical oscillations. To the extentthat government expenditures play the most important role of governmental intervention to theeconomy, the prevalence of countercyclical fiscal policy can clearly be stated.
Considering a fiscal authority’s aim of dampening cyclical fluctuations indicates that forunemployment and output, countercyclical fiscal policy is a powerful tool, whereas for inflation itsimpact is limited due to the rather rigid fluctuations therein.
Moreover, the results highlight that once a researcher’s aim is to describe the fluctuationsobserved empirically by means of a DSGE model, a standard set up of the fiscal authority in termsof simple AR(1) processes ignores that there can indeed be a strong endogenous dependence ofgovernmental variables on economic fluctuations.
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29
Appendix
A Data and variables
The Bayesian VAR analysis applied to compute the empirical spectral density functions uses quar-terly German data on the following variables: (1) real GDP (Yt), (2) real personal consumption ofnon-durables (Ct), (3) real private investment (It), (4) hours worked per employee (ht), (5) infla-tion, measured by the percentage change of the GDP deflator (Pt), (6) the unemployment rate (Ut)from which the employment rate can be obtained (Nt = 1− Ut), (7) the nominal interest rate onlong term government bonds (it), (8) the real wage rate which is defined here as the compensationper hour worked in the non-farm industry (Wt), (9) real government consumption (Gt) and (10)the number of vacancies posted (Vt). All data series are taken from the OECD database and putinto the BVAR without further transformations. The series range from 1990q1 until 2007q1 andthe data are not seasonally adjusted. All variables entering the BVAR are expressed in logarithmicterms except for the employment rate, the nominal interest rate and the inflation rate.
The data vector in the BVAR is defined as follows
yt = [log(Yt), log(Ct), log(It), log(ht), ...
...πt, Nt, it, log(Wt), log(Gt), log(Vt)]′ (58)
Table 4: Data: definitions and sources
Vt Vacancies; OECD, MEI: DEU Unfilled job vacancies
It Investment; OECD, MEI: DEU Gross fixed capital formation,
constant prices
Ct Consumption; OECD, MEI: DEU Private final consumption expenditure,
constant prices
ht Hours worked; OECD, EO: DEU Hours worked per employee -
total economy
Ut Unemployment rate; OECD, MEI: DEU Unemployment rate: survey-based
all persons
Yt Gross Domestic Product; OECD, MEI: DEU Gross domestic product,
constant prices
Pt Price level; OECD, EO: DEU Gross dometsic product, deflator,
market prices
Wt Real wage rate; OECD, MEI: DEU Hourly wage rate: manufacturing
Gt Real government consumption; OECD, MEI: DEU Government final consumption expenditure,
constant prices
it Nominal interest rate; OECD, EO: DEU Long term interest rate on government bonds.
Notes: MEI stands for Main Economic Indicators (URL:http://oecd-stats.ingenta.com/OECD/TableViewer/DimView.aspx?TableName=6jmeicd.ivt&IF Language=eng)and EO for Economic Outlook (URL:http://oecd-stats.ingenta.com/OECD/TableViewer/DimView.aspx?TableName=4deo.ivt&IF Language=eng).All data are freely accessible.
30
B Steady state
The derivations here are part of the solution of the model as outlined in section 2.8.
• First obtainN =
p
s+ p
U = 1−N
MC = µ =ξR − 1ξR
rK =1− β + βδ
β(1− τK)
• Then get
k = H(αµrK
) 11−α
K = k ·N
Y = Kα(HN)1−α
I = δK
• Note thatz = qK = 1
Cr = Co = C
• FinallyC = cy · Y
V = Up1
1−ζm
ϑ =V
U
w = W = (1− α)µ(k
H
)α
mrs = (1− τN )µ(1− α)(k
H
)αχ =
mrs
(1− ς)Hφλo + ς Hφ
λr
C Log-linearized equilibrium
The following equations describe the equilibrium of the model in log-linearized form as referred toin section 2.8.
31
• Physical capital accumulation dynamics
Kt = (1− δ)Kt−1 + δ(It + εIt
)(59)
• Consumption-saving
−(Rt − Et[πt+1]
)= εξut − λot + Et
[λot+1 − ε
ξut+1
](60)
• Capital utilization
rkt =ψ2
ψ1zt (61)
where ψ1 := ψ′(1) = 1/β − 1 + δ and ψ2 := ψ′′(1).
• Investment
It =κI
1 + β
(qKt + εIt
)+
11 + β
It−1 +β
1 + βEt
[It+1
](62)
• Tobin’s Q
qKt = Et
[−(Rt − πt+1
)+ (1− β(1− δ))rKt+1+ (63)
β(1− δ)qKt+1
]• Marginal consumption of Ricardian consumers
(1− hβ) · λot =σch
1− hCot−1 +
σchβ
1− hEt
[Cot+1
]+
hβ(Et
[εξut+1
]− εξut
)− σc(1 + h2β)
1− hCot (64)
• Marginal consumption of Non-Ricardian consumers
λrt = −σcCrt (65)
• Consumption of Non-Ricardian consumers
Crt = c1
(Wt + Nt + Ht
)+ c2 · Ut (66)
where c1 = ς(1−τN )(1−α)µcy
and c2 = ςΓUC .
• Aggregate ConsumptionCt = ςCrt + (1− ς)Cot (67)
• Marginal rate of substitution
(b3 + ς)mrst = φ(b3 + ς)Ht − φb3λot − φςλrt (68)
where b3 = (1−h)(1−ς)1−hβ
32
• Capital rentingrkt = µt + ξAt + (1− α)
(Ht − kt−1 − zt
)(69)
• UnemploymentUt = −p
s· Nt−1 (70)
• Matchingmt = ζm · Ut + (1− ζm) · Vt (71)
• Transition probabilities
qt = mt − Vt (72)
pt = mt − Ut (73)
• Market tightnessϑt = Vt − Ut (74)
• Employment flowNt = (1− s) · Nt−1 + s · mt−1 (75)
• Aggregate wage
wt =1− η1 + φ
[mrst − φHt
]+
η
1− α
[µt + ξAt − α · rKt +
cvγνp
µs· ϑt]
(76)
where γν = V/Y
• Hours workedmrst = ξAt + µt + α
(kt−1 + zt − Ht
)(77)
• Real market wageWt = ν · Wt−1 + (1− ν) · wt + εwt (78)
• Phillips curve
πt =ωp
1 + βωpπt−1 +
(1− θ)(1− θβ)θ(1 + βωp)
(µt +
εξR
t
ξR − 1
)+
β
1 + βωpEt [πt+1] (79)
• Employment
b1 · κt = µt + ξAt − (1− α)Wt − αrKt +
(1− s)b1βEt[κt+1 − Rt + πt+1
](80)
where b1 = cvqβ ·
NY µ
• Vacancies− qt = Et
[πt+1 − Rt + κt+1
](81)
33
• Monetary policy rule
Rt = ρr · Rt−1 + (1− ρr)(aππt + aY Yt
)+ εMt (82)
• Government budget constraint
Tt + byR−1[B∗t − Rt
]= byΠ−1
[B∗t−1 − πt
]+ γG · Gt + γΓ · Ut (83)
where γG = G/Y , R−1 = βΠ , γΓ = ΓU
Y and by = B∗/Y
• Tax revenues
Tt = τN (1− α)µ[Wt + Nt + Ht
]+ τKαµ
[rKt + zt + Kt−1
](84)
• Fiscal policy rule
Gt = φGGGt−1 + (1− φGG)(φGBB
∗t + φGY Yt + φGU Ut
)+ εGt (85)
• Aggregate capital stockKt = Nt + kt−1 (86)
• TechnologyYt = ξAt + (1− α) ·
(Ht + Nt
)+ α ·
(Kt−1 + zt
)(87)
• Resource constraint
Yt = cyCt + iy It + γGGt +ψ1K
Yzt + cvγvVt (88)
where iy = I/Y
34
