appendix to “parity funding of health care contributions ... · appendix to “parity funding of...
TRANSCRIPT
Appendix to “Parity Funding of Health Care Contributions in
Germany: A DSGE Perspective”
- INTENDED FOR ONLINE PUBLICATION
Abstract
In this appendix, we provide a detailed description of the formal model and its calibra-tion. We also include additional simulations as robustness analyses.
A.1. Model
The household sector is partitioned into two types of representative households:optimising and non-Ricardian “rule-of-thumb” (RoT) households, indexed by x = o, rfor optimisers and RoTs. They differ in that RoTs do neither save nor borrow but con-sume all their labour income each period (see Gali et al., 2007). In each region i, RoThouseholds make up a share µi ∈ [0, 1) of total population, while the remaining share(1− µi) behaves Ricardian. As in Gali et al. (2011), household members are representedby the unit square and indexed by a pair (hx, jx) ∈ [0, 1]× [0, 1]. Household membersdiffer in the type of labour service they are specialized in, hx ∈ [0, 1], and by their per-
sonal disutility of work, jx ∈ [0, 1]. If employed, the latter is given by κwi · eǫ
Nit · j
ϕix . It is
zero otherwise. κwa > 0 is an exogenous labour disutility scaling parameter and ǫNa
t isan AR(1) labour disutility shock process. ϕa > 0 determines the shape of the distribu-tion of work disutilities across individual household members. Values not indexed by xare common across household types. Assuming that the utility of household memberspositively depends on consumption and that there is full risk sharing of consumptionwithin a household as in Merz (1995) or Andolfatto (1996), the utility of household-typex can be written as
Et
∞
∑s=0
βsi U(
Cix,t+s, Ni
x,t+s (hx))
(A.1)
= Et
∞
∑s=0
βsi e
ǫβit+s
(
Cix,t+s − hiC
ix,t+s−1
)1−σi− 1
1 − σi− κw
i eǫNit+s
∫ 1
0
Nix,t+s (hx)
1+ϕi
1 + ϕidhx
,
where 0 < βi < 1 is a subjective discount factor, ǫβit depicts an AR(1) pref-
erence shock process, Cix,t is household type x-specific private consumption, and
Preprint submitted to German Economic Review August 16, 2018
hi ∈ [0, 1] is an external habit persistence parameter based on type-specific aggre-gate consumption of the previous period, Ci
x,t−1. σi governs the elasticity of in-
tertemporal substitution. Nix,t (hx) ∈ [0, 1] denotes the household type x-specific
employment rate in period t among workers specialized in labour-type hx. Con-sumption of private goods, Ci
x,t, is a composite of goods produced at home andabroad. In region a, the household type-x consumption aggregator is given by
Cax,t =
[
(naa)
1ηa
(
Ca,ax,t
) ηa−1ηa +
(
nab · eǫb,a
t
) 1ηa(
Ca,bx,t
) ηa−1ηa + (na
c)1
ηa
(
Ca,cx,t
) ηa−1ηa
] ηaηa−1
, where nai ,
with i = a, b, c, are the weights of goods in the consumption bundle according to theirorigin, implying na
a + nab + na
c = 1, and ηa is the elasticity of substitution between thesegoods. The aggregator is analogous for region b. Region c is assumed to be given by
a VAR process for simplicity. ǫb,at is an AR(1) trade preference shock process and C
i,jx,t,
with i, j = a, b, c, is a good consumed by households of type x in region i which isproduced in region j. The consumer price index (CPI) of region a, Pa
t , is derived by
Pat Ca
x,t = Pa,at Ca,a
x,t + Pa,bt Ca,b
x,t + Pa,ct Ca,c
x,t , where Pi,jt is the producer price index (PPI) of
goods produced in country j purchased in i.Households’ consumption expenditures amount to (1 + τc,a
t ) Pat Ca
x,t, where τc,at is
the consumption tax rate. Income is given by net wage income from employment in the
private and the public sector, NP,at and NG,a
t , paying nominal gross wages Wat and WG,a
twhich are both taxed by the rate τw,a
t . Real wages are, hence, given by wat = Wa
t /Pat
and wG,at = WG,a
t /Pat . Note that neither employment nor wages are indexed by x as
we assume that wage bargaining and employment distribution are undertaken by aunion and the government, who both distribute labour and wages uniformly acrosshousehold types (explained in more detail in the next sections). Unemployed house-hold members receive nominal unemployment benefits Pa
t · UBa. Those members whodecided to participate in the labour market, La
x,t, but who did not find a job are unem-ployed, ie Ua
x,t = Lax,t − Na
t . Here, it is important to note that, while employment ratesand wages are independent of the household type, the number of household membersparticipating in the labour market can differ across types. Furthermore, householdsreceive a type-specific lump-sum transfer TRa
x,t. For optimising households, we have
to take into account that they can save and borrow. Bi,jo,t are private bonds purchased in
country i issued by country j, BG,ao,t is a government bond issued by the fiscal authority in
country a, which is held by domestic households only, and Iao,t are purchases of invest-
ment goods, which is an aggregator analogously to private consumption. πat = Pa
t /Pat−1
is CPI inflation. Hence, optimisers also receive interest on their bond holdings, at rates
ia,jt for private and iG,a
t for government bonds. Furthermore, optimisers receive a return,ra
k,t, on their capital, kao,t and pay lump-sum taxes Ta
o,t. Capital depreciates at rate δa and
the government taxes capital gains net of depreciation at rate τk,at . Da
o,t are the profits
of firms and eǫRP,EAt and eǫRP,RoW
t are exogenous “risk premium” shock processes for theEuro Area as a whole and for the rest of the world similar to Christoffel et al. (2008)
2
and Coenen et al. (2013). They can also be interpreted as uncovered interest rate parity(UIP) shocks reflecting the degree of divergence between countries (or regions) in line
with Rabanal and Tuesta (2010). The nominal exchange rate Sj,at is defined as country j
currency per unit of country-a currency. Clearly, Sj,at is one within the monetary union.
Note that, for RoT households, it holds that Bi,jr,t = BG,a
r,t = Iao,t = ka
r,t = Tar,t = Da
r,t = 0for all t. Summarising, households face the following CPI-deflated budget constraint:
(1 + τc,at ) Ca
x,t + Iax,t + Ba,a
x,t + ∑j=b,c
Sa,jt B
a,jx,t + BG,a
x,t
= (1 − τw,at )
(
wat NP,a
t + wG,at NG,a
t
)
+ UBa(
Lax,t − Na
t
)
+ TRax,t +
(1 + ia
t−1
)eǫRP,EA
t−1
πat
Ba,ax,t−1 +
(
1 + ia,bt−1
)
eǫRP,EAt−1
πat
Ba,bx,t−1
+ Sa,ct
(1 + ia,c
t−1
)eǫRP,RoW
t−1
πat
Ba,cx,t−1 (A.2)
+
(
1 + iG,at−1
)
πat
BG,ax,t−1 +
(
1 − τk,at
)
rak,t ua
t kax,t−1 + τk,a
t δa kax,t−1 + Da
x,t − Tax,t.
The law-of-motion for capital is given by
kao,t = (1 − δa) ka
o,t−1 +
Iao,t − Ia
o,tψi
a
2
(
Iao,t
Iao,t−1
− 1
)2
eǫIat (A.3)
which states that today’s capital stock equals yesterday’s capital stock net of deprecia-
tion plus new investments net of investment adjustment costs, ψia/2
(
Iao,t/Ia
o,t−1 − 1)2
,
and ǫIat is an exogenous AR(1) investment technology shock process (see Christiano et
al., 2005, 2011, for a discussion). The parameter ψia determines the costs of investment
adjustment. The maximisation problem of the households yields the standard equa-tions.
Turning to labour demand and supply, we have to differentiate between private andpublic sector demand. As in Forni et al. (2009), we assume that labour demand in bothsectors gets uniformly allocated among household types and that public sector labour
demand, NG,at , and wages, wG,a
t , follow exogenous autoregressive processes describedin the fiscal policy section in the appendix. Consistent with OECD data we assume that,in steady state, public sector wages include a markup, mga, on private sector wages.In the private sector, a perfectly competitive agency buys the differentiated individuallabour services supplied by households, transforms them into a homogenous composite
of labour input, NP,at , and sells that to intermediate goods producers. Hence, labour
3
agencies solve for each variety of labour service, h,
maxNP,a
t (h):h∈[0,1]NP,a
t =
(∫ 1
0
(
NP,at (h)
)(θwa,t−1)/θw
a,tdh
)θwa,t/(θ
wa,t−1)
,
subject to a given level of the wage bill∫ 1
0 Wat (h) NP,a
t (h)dh = WBat . The solution
of this problem is the private-sector labour demand for each variety h, NP,at (h) =
(Wa
t (h)Wa
t
)−θwa,t
NP,at , where Wa
t is the average nominal wage paid in the private sector. To-
tal employment is an aggregate of public and private employment, Nat = NP,a
t + NG,at .
θwa,t is the time-varying elasticity of substitution between different types of labour and
follows θwa,t/
(θw
a,t − 1)= ρθw
aθw
a,t−1/(
θwa,t−1 − 1
)
+(1 − ρθw
a
)θw
a /(θw
a − 1)+ ν
θwa
t , where
νθw
at is an i.i.d. shock with mean zero and variance σθw
a ; νθw
at can be interpreted as a wage
markup shock.In order to derive a labour market equilibrium, we will have to determine labour
supply and demand as well as wage setting. Let us, first, turn to the labour supplydecision of households. Taking labour market conditions (ie wages and employment)as given, any household member specialized in type hx labour will find it optimal toparticipate in the labour market if and only if utility from working exceeds his or herdisutility. When defining the marginal member for which this condition holds withequality as La
x,t and noting that jx ∈ [0, 1], Lax,t can be seen as the labour supply of
household-type x; see Gali et al. (2011) for a more detailed discussion. Hence, thelabour supply decision of households can be summarized as
λax,t
[
(1 − τw,at )
(
wat NP,a
t + wG,at NG,a
t
)
+ UBa(
Lar,t − Na
t
)]
= Nat κw
a eǫNat+s(
Lax,t
)ϕa ,
(A.4)where λa
x,t is the marginal utility of consumption.
To determine wages in the private sector, we assume that there are utilitarian unionsfor each labour type hx, representing optimising and RoT households according to theirshares in population. Unions maximize income of its members by optimally choos-ing nominal wages Wa
t (h), taking into account the disutility of work and the effects onlabour supply and demand. Furthermore, wage setting is due to Rotemberg adjustment
4
costs, indicated by the parameter γwa . Formally, each union maximizes
Et
∞
∑s=0
βsa eǫ
βta
t+s
µa
[
λr,at+s
(
(1 − τw,a
t+s
)
(
Wat+s(h)
Pat+s
NP,at+s(h) +
WG,at+s(h)
Pat+s
NG,at+s(h)
)
+UBa(
Lr,at+s(h)− Na
t+s(h))− adjW,a
t
)
− κwa eǫNa
t+sNa
t+s(h)1+ϕa
1 + ϕa
]
+(1 − µa)
[
λo,at+s
(
(1 − τw,a
t+s
)
(
Wat+s(h)
Pat+s
NP,at+s(h) +
WG,at+s(h)
Pat+s
NG,at+s(h)
)
+UBa(
Lo,at+s(h)− Na
t+s(h))− adjW,a
t
)
− κwa eǫNa
t+sNa
t+s(h)1+ϕa
1 + ϕa
]
,
with respect to
Wat+s(h), NP,a
t+s(h), Lr,at+s(h), Lo,a
t+s(h) : h ∈ [0, 1]
subject to (A.4) for each
household type x, the private-sector labour demand curve and Nat = NP,a
t + NG,at .
Rotemberg wage adjustment costs, adjW,at , are defined as
adjW,at =
υwa
2
(
Wat+s(h)
(πa
w,t+s−1
)ξwa (πa)1−ξw
a Wat+s−1(h)
− 1
)2Wa
t+s
Pat+s
.
The solution is symmetric, so that Wat (h) = Wa
t , Lo,at (h) = Lo,a
t , Lr,at (h) = Lr,a
t and
NP,at (h) = NP,a
t for all h in equilibrium. Defining Lat = (1 − µa)Lo,a
t + µaLr,at as the
total labour force, we can then define the unemployment rate as URat = (La
t − Nat ) /La
t .We also allow for potential indexation on past wage inflation, πa
w,t−1, and steady-state
wage inflation, indicated by the parameter ξwa ∈ [0, 1]. The first-order conditions of this
problem then determine wages in the private sector.
We assume that, in each country, there is a measure-P i continuum of firms in the fi-nal goods sector, with i = a, b, c as the country index, and P i being country-size. Firmsare owned by optimising households. Each final goods producer purchases a variety ofdifferentiated intermediate goods, bundles these and sells them to the final consumerunder perfect competition. The producer price index (PPI) of goods produced in coun-
try i and sold in j is defined as Pi,jt . We assume that the law of one price holds across re-
gions, so firms in country a set their price Pa,at for all markets. Multiplying with the nom-
inal exchange rate, then, yields the price of country-a goods charged in the other coun-
tries, ie Pb,at = Sb,a
t Pa,at and Pc,a
t = Sc,at Pa,a
t , where the nominal exchange rate Sj,at is de-
fined as country j currency per unit of country-a currency. Clearly, Sj,at is one within the
monetary union (ie, for the euro area, Sb,at = Sa,b
t = 1 ∀ t). The maximisation problem
of the representative final goods firm reads maxyat (z):z∈[0,1] Pa,a
t Yat −
∫ 10 Pa,a
t (z)yat (z)dz,
5
where Yat =
(∫ 10 ya
t (z)(θa,t−1)/θa,tdz
)θa,t/(θa,t−1)is the production function, ya
t (z) the de-
mand for each differentiated input good z, and Pa,at (z) the price of each input. θa,t is
the time-varying elasticity of substitution between differentiated goods and follows
a process θat/ (θat − 1) = ρθaθat−1/ (θat−1 − 1) + (1 − ρθa) θa/
(θa − 1
)+ νθa
t , where
νθat is an i.i.d. shock with mean zero and variance σθa . It can be interpreted as a
price markup shock. The first-order condition of the maximisation problem yields
yat (z) = (Pa,a
t (z)/Pa,at )
−θa,t Yat , which implies that the PPI of country a is given by
Pa,at =
(∫ 10 Pa,a
t (z)1−θa,t dz)1/(1−θa,t)
.
Private intermediate goods firms on the continuum z ∈ [0, 1] operate as mo-nopolistic competitors in the product market. Each firm produces its intermedi-ate good variety with the following Cobb-Douglas production function ya
t (z) =
eǫAat eǫ
Agt
(
ζa
(
KG,at
)ηKG,a (
NG,at
)ηNG,a)
[Ka
t−1(z)]αa[
NP,at (z)
]1−αa
−Ωa, where ǫAat is an
AR(1) productivity shock process, identical across firms in country a, ǫAg
t is an analo-gous shock on global productivity, identical across firms in all regions, and Ωa is a fixedcost yielding steady-state profits to be zero. The parameter 0 < αa < 1 gives the share
of private capital, Kat , in production. NP,a
t denotes private-sector employment. Public
investment, IG,at , which determines the public-sector capital stock, KG,a
t , and public em-
ployment, NG,at , both provided by the government, affect private-sector productivity as
stated in the firms’ production function (see D’Auria, 2015, Leeper et al., 2009, 2010,
and Pappa, 2009, for a discussion): ηKG,a determines the relevance of public capital in
the private-sector productivity function and ηNG,a the relevance of public employment
(for ηKG,a = ηNG,a = 0, there is no effect), while ζa > 0 is a scaling parameter. Both areoutside the firms’ control.
With rak,t being the consumer price index (CPI)-deflated rental rate of capital and
(1 + τsc,at ) wa
t being gross labour costs, including CPI-deflated private-sector wages, wat ,
and the firms’ social security contributions at rate τsc,at , firm z’s cost minimisation prob-
lem yields the following capital-to-labour ratio
rak,t
wat (1 + τsc,a
t )=
NP,at (z)
Kat−1(z)
·αa
1 − αa,
which is common to all firms. Real CPI-deflated marginal costs are hence given by
mcat =
(
rak,t
)αa
(wat (1 + τsc,a
t ))1−αa
eǫAat eǫ
Agt
(
ζa
(
KG,at
)ηKG,a (
NG,at
)ηNG ,a)
ααaa (1 − αa)1−αa
6
and are also common across firms. Each intermediate goods producer sets its own pricePa,a
t (z) to maximise intertemporal profits: the difference between revenues and produc-tion as well as Rotemberg price adjustment costs, the latter indicated by a cost parameterγa. The maximisation problem in CPI-terms can be stated as
maxPa,a
t+s(z):z∈[0,1]Et
∞
∑s=0
βsa
λao,t+s
λao,t
[(Pa,a
t+s(z)
Pat+s
− mcat+s
)
yat+s(z) (A.5)
−γa
2
(
Pa,at+s(z)
(πa,at+s−1)
ξa (πa,a)1−ξa Pa,at+s−1(z)
− 1
)2Pa,a
t+s
Pat+s
︸ ︷︷ ︸
=adjP,at
Yat+s
,
subject to yat (z) = ya
t (z) = (Pa,at (z)/Pa,a
t )−θa Ya
t . The parameter ξa ∈ [0, 1] determinesthe magnitude of price indexation on past inflation, πa,a
t−1, or steady-state inflation,πa,a (see Ascari et al. 2011). As optimisers own firms the intertemporal discount factorof a firm includes only the marginal utility of optimising households, λa
o,t, determinedby household maximisation described in the main text.
Fiscal policy making is included as follows. The real (CPI-deflated) per capita valueof end-of-period government debt evolves according to a standard debt accumulationequation,
BG,it =
(1 + iG,it−1)
πit
BG,it−1 + PDi
t, (A.6)
where PDit = Gi
t − Revit denotes the real per capita primary deficit, being defined as
total primary expenditures (excluding interest payments iG,it−1 on outstanding debt),
Git = Ri,i
t
(
CG,it + IG,i
t
)
+ UBi(
µi Ur,it +
(
1 − µi)
Uo,it
)
(A.7)
+(
1 + τsc,it
)
NG,it wG,i
t + TRit (A.8)
minus primary revenues,
Revit =
(
τw,it + τsc,i
t
) (
wit NP,i
t + wG,it NG,i
t
)
+ τk,it
(
rk,it − δi
)
Kit−1
+τc,it Ci
t + Tio,t. (A.9)
CG,it and IG,i
t denote government consumption and investment, respectively. As we as-sume full home bias here, which can be justified by the fact that there is evidence for astrong home bias in government procurement (see, among others, Brulhart and Trion-fetti, 2004), and because the budget constraint is CPI-deflated, we correct for this by the
7
relative price between home-country PPI and home-country CPI, Ri,it = Pi,i
t /Pit .1 Un-
employment benefits UBi are paid to unemployed optimizing households, (1− µi)Uo,it ,
and to unemployed rule-of-thumb households, µi Ur,it , where U
j,it , with j = o, r denotes
the type-specific unemployment rate and µi is the share of rule-of-thumb householdsin total population. Furthermore, the government has to finance public-sector employ-
ment including the corresponding social security contributions, where NG,it is public
employment, wG,it the public-sector and τsc,i
t the social security contribution rate. Thereare also transfers TRi
t which are also to households in a lump-sum way. On the rev-enue side distortionary taxes are introduced on wages τw,i, capital τk,i and consump-tion τc,i. Labor taxes and social security contributions for wage income in the private
sector wit NP,i
t and in the public sector wG,it NG,i
t constitute the first part of governmentincome. Capital tax income is generated by capital taxes on the net return (depreciation
rate deducted from the gross return) rk,it − δi on capital Ki
t−1. Household consumption
Cit is taxed with the consumption tax rate τc,i and ultimately there is a lump-sum tax as
well. All available fiscal instruments follow a rule governed by the following exogenousprocesses:
log
(Xt
X
)
= ρX,i log
(Xt−1
X
)
− ξX,BG,i,i log
(
BG,it−1
ωd Yi
)
− ξX,y,i log
(
Yit−1
Yi
)
+ψX,i νX,it +
(
1 − ψX,i)
νX,it−1, (A.10)
for instruments X ∈ CG,i, IG,i, TRi, wG,i and
Xt − X = ρX,i (Xt−1 − X) + ξX,BG,i,i log
(
BG,it−1
ωd Ya
)
+ ξX,y,i log
(
Yit−1
Yi
)
+ψX,i νX,it +
(
1 − ψX,i)
νX,it−1, (A.11)
for X ∈ τw,i, τsc,i, τk,i, Tio, NG,i. νX,i
t is an i.i.d. (discretionary) fiscal policy shock with
mean zero and variance σX,i, ρX,i is a persistence parameter and ξX,BG,i,i measures theresponsiveness of the corresponding instrument to deviations in the debt-to-GDP ratiofrom its long-run target, ωd. In order to guarantee stability in the debt ratio, for at
least one instrument the coefficient ξX,BG,i,i must be positive (see, among others, Schmitt-Grohe and Uribe, 2007, for a discussion). ξX,y,i can be interpreted as an ad-hoc automaticstabilizing component as in Coenen et al. (2013). As in Leeper et al. (2009), we allowfor anticipation effects of fiscal policy with a weight of
(1 − ψX,a
). Following Coenen
1Given public investment, the public sector capital stock naturally evolves according to KG,it =
(1 − δG
i
)KG,i
t−1 + IG,it with δG
i as the depreciation rate on public capital. We abstract from capital adjust-ment costs here because public investment is assumed to be given by an exogenous stochastic process.
8
et al. (2013), we assume that capital taxes are kept constant while consumption taxesfollow an AR(1) process including anticipation effects but no reaction on debt or outputdeviations. The same holds for public employment and public wages.
We assume that, in the monetary union, there is only one central bank setting thepolicy rate iEA
t . Following Stähler and Thomas (2012), it responds to deviations of area-wide CPI inflation, which is a population share-weighted average of inflation in countrya and region b, from its long-run target, and to the area-wide output gap, according to asimple Taylor-type rule,
log
(1 + iEA
t
1 + iEA
)
= ρai log
(
1 + iEAt−1
1 + iEA
)
+ (1 − ρai ) φEA
π
(
s · log
(πa
t
πa
)
+ (1 − s) · log
(
πbt
πb
))
(A.12)
+ (1 − ρai ) φEA
y
(
s · log
(Ya
t
Ya
)
+ (1 − s) · log
(
Ybt
Yb
))
+ νMEA
t
where s = P a
P a+Pb is the relative population-weight of country a in the monetary union,
ρai is a smoothing parameter, φEA
π and φEAy are the monetary policy’s stance on inflation
and output gap; and νMEA
t denotes an i.i.d. monetary policy shock with mean zero and
variance σMEA.
While there is only one policy rate in the monetary union, namely iEAt , there are
two interest rates governing private savings, iat and ib
t , and, thus, separate foreign as-set holding decisions in each country. This could render foreign asset positions to benon-stationary. A common way to guarantee stationarity of foreign asset trade in open-economy DSGE models is the introduction of a risk premium that depends on the rel-ative net foreign asset position of each country (see, among others, Schmitt-Grohe andUribe, 2003). We will discuss the precise modeling of the risk premium in the sectionon international linkages, but note that different risk premia can imply different interestrates ia
t and ibt . Depending on the net foreign asset position of each country the interest
rate prevailing in the corresponding country may be above or below the policy rate. Therelation between the two rates is given by
log
(1 + iEA
t
1 + iEA
)
= s · log
(1 + ia
t
1 + ia
)
+ (1 − s) · log
(
1 + ibt
1 + ib
)
.
In order to simplify the trading structure of privately traded international bonds,and in order to avoid having to take a stance on the detailed portfolio choice of agents,we assume that residents in country a can sell bonds to region b, but not vice versa. Byallowing residents of region b to sell a-bonds short, b can effectively borrow from a as
9
well. The same logic allows bond trade of region c with country a or region b only totake place via bonds issued by region c.
To determine interest rates paid to or charged from investors abroad, we assumethat the interest rate country-a residents have to pay to region-b residents depends onthe net debt position of a vis--vis b as in Schmitt-Grohe and Uribe (2003), who alsoprovide a discussion of different ways of modeling risk premia. This logic applies toall regions i trading bonds with region j and can, for i, j = a, b, c and i 6= j, formally besummarised by
1 + ii,jt =
(
1 + ijt
)[
1 − φ
(
exp
(
reri,jt B
i,jt
Ri,it Yi
t
−Bi,j
Ri,iYi
)
− 1
)]
,
where Ri,it = Pi,i
t /Pit and rer
i,jt is the real exchange rate between region i and j, de-
termined in detail in the next subsection. Hence, if the term(
reri,jt B
i,jt
)
/(
Ri,it Yi
t
)
−(
Bi,j)
/(
Ri,iYi)
is negative, country i’s indebtedness vis--vis country j increases above
the “normal” steady-state level, Bi,j/(
Ri,iYi), which can be zero, and the interest rate i
i,jt
will contain a markup on the interest rate that region-j residents would have to pay ijt
(the reason why home and foreign rates may differ in the optimisers’ budget constraintof the main text). The opposite is true for the term being positive.
Market clearing implies that total supply must equal total demand. Hence, for coun-try a it holds that the entire production of country-a goods is used either domesticallyor internationally. Hence, taking into account capital utilisation costs, it holds that
Yat = CG,a
t + IG,at + Ca,a
t + Ia,at +
nab
nba
(
Cb,at + Ib,a
t
)
+na
c
nca(Cc,a
t + Ic,at ) + ADJa
t , (A.13)
where CG,at + IG,a
t is domestic public and Ca,at + Ia,a
t domestic private consumption and
investment demand;(
naj /n
ja
) (
Cj,at + I
j,at
)
, for j = b, c, is private foreign consump-
tion and investment demand expressed in per-capita terms; and ADJat = adjP,a
t /Ra,at +
adjW,at /Ra,a
t are total adjustment costs for price adjustments, adjP,at /Ra,a
t , and wage ad-
justments, adjW,at /Ra,a
t . We have to take into account that the cost functions are ex-pressed in CPI-terms, while the rest of equation (A.13) is expressed in PPI-terms. Ananalogous equation holds for region b.
Given that we assume the third region c to be a VAR process, we can simplify the restof the world’s consumption and investment demand of country-j products (j = a, b) to
Cc,jt + I
c,jt = nc
j Rc,jt
(
gc,c + gc,i)
eǫc,jt Yc
t , (A.14)
where Yct is the rest of the world output, described below, gc,c and gc,i are consumption
10
and investment shares of this output, respectively, and ǫc,jt is an exogenous AR(1) shock
process for import preferences of country-j products. An analogous shock is includedfor intra-European trade (see Gadatsch et al., 2016, for details). Region c is representedby the following VAR process:
Yct
πct
ict
ǫAg
t
=
a1,1 a1,2 a1,3 0a2,1 a2,2 a2,3 a2,4
a3,1 a3,2 a3,3 a3,4
0 0 0 a4,4
Yct−1
πct−1
ict−1
ǫAg
t−1
+
1 0 0 0c2,1 1 0 c2,4
c3,1 c3,2 1 c3,4
0 0 0 1
νY,ct
νπ,ct
νi,ct
νAg
t
,
where νY,ct , νi,c
t , νπ,ct , and ν
Ag
t are iid shocks with mean zero and variance σY,c, σi,c, σπ,c,
and σAg .
Given international trade in goods and assets, we have to determine the net for-eign asset position between all regions. Taking into account the bond trading structuredescribed before, country a’s foreign bond position can be expressed in their respectiveCPI-terms as
rera,ct Ba,c
t + Bat
︸ ︷︷ ︸
=n f aat
=(1 + ia,c
t−1
) rera,ct Ba,c
t−1
πct
+(
1 + ib,at−1
) Bat−1
πat
+Ra,at Ya
t − Cat − Ia
t − CG,at − IG,a
t ,
and likewise for country b as
rerb,ct Bb,c
t + rerb,at Bb,a
t︸ ︷︷ ︸
=n f abt
=(
1 + ib,ct−1
) rerb,ct Bb,c
t−1
πct
+(
1 + ib,at−1
) rerb,at Bb,a
t−1
πat
+Rb,bt Yb
t − Cbt − Ib
t − CG,bt − IG,b
t ,
where it holds that Bat = − (P a/P a) Bb,a
t . These equations state that each countrycan only consume as much as the sum of its own production and interest paymentson outstanding asset holdings, or it will have to take up debt. In other words, thecurrent account of country i, cai
t = n f ait − n f ai
t−1, is balanced if and only if countryi consumes its entire production plus interest payments. Otherwise, the current ac-count will, depending on the country’s consumption stance, be positive or negative andcountry i’s net for foreign asset position, n f ai
t, will naturally increase or decrease. Be-cause bond markets also need to clear in equilibrium, it is straightforward to derive
Bct = −
(P a
P c Ba,ct + Pb
P c Bb,ct
)
, where it holds that n f act = Bc
t . For further reference, we
note that, from the perspective of country a, the real exchange rate between regions are
11
related as follows
rerc,at =
1
Ra,ct
, rerc,at =
1
rera,ct
and rerb,ct =
rerb,at
rerc,at
,
And changes in the nominal exchange rate are given by
∆Sa,ct =
πat
(rera,c
t /rera,ct−1
)
πct
in which πa,ct = πc
t ∆Sa,ct holds. Realising that analogous relations hold between all
regions a, b, c and remembering that ∆Sa,bt = ∆Sb,a
t = 1 (because a and b form a monetaryunion) allows us to derive the remaining relations.
B.1. Calibration
In calibrating the model, we strongly rely on the model’s estimation described inmore detail in Gadatsch et al. (2016). To estimate the model, a large data set for Ger-many, the rest of the euro area (an aggregate of the countries Austria, Belgium, Finland,France, Italy, the Netherlands, Portugal, and Spain), and the rest of the world (an aggre-gate of Brazil, Canada, China, India, Japan, Russia, the United Kingdom, and the UnitedStates) was constructed, containing a rich set of quarterly fiscal variables (16 out of 38series are fiscal). A detailed description of the estimation results would go beyond thescope of this paper. Therefore, the reader is referred to Gadatsch et al. (2016) for moredetails, while we restrict ourselves to presenting only the estimation results (ie the pa-rameter calibration) here. Table B.1 summarises the parameters that were not estimated,as well as the targeted steady-state values that the model is supposed to replicate.
Table B.1: Calibrated parameters and targeted steady-state values
Parameter ValueGermany Rest of Euro Area
PreferencesIntertemporal elasticity of substitution , σ 1 1Discount factor, β 0.9985 0.9985Parameter influencing Frisch elasticity, ϕ 10 9Population size, P 1 2.6
TechnologyCapital share, α 0.33 0.33
continued on next page
12
continued from previous page
Parameter ValueGermany Rest of Euro Area
Rate of depreciation (private), δ 0.015 0.015
Rate of depreciation (public), δG 0.015 0.015Public sector productivity shifter, ζ 1.22 1.16Subs. Elasticity: intermediate goods, θ 4 4Subs. Elasticity: different types of labor, θw 5.4 5.1Fixed costs, Ω 0.35 0.28
International sectorRisk premium parameter, φ 0.01
Fiscal policyLabor income taxes, τw 0.304 0.277Capital taxes, τk 0.214 0.316Consumption taxes, τc 0.183 0.196SSC (employers), τsc 0.167 0.246
Public purchases ratio, CG
GDP 0.111 0.1006
Public investment ratio, IG
GDP 0.017 0.028
Public employment ratio, NG
N 0.228 0.231
Transfers (incl. UB benefits) ratio,TR+(L−N)UB
GDP 0.190 0.183
Replacement ratio, UBw(1−τw)
0.351 0.351
Public markup, mg 0.030 0.030
Government debt ratio (quarterly), BG
GDP 2.4 2.4
Monetary policyInflation rate (quarterly), π 0.0045Interest rate (quarterly), i 0.00475
Labor and goods marketUnemployment rate, UR 0.082 0.095Employment rate, N 0.433 0.363Wage markup 0.263 0.287Frisch elasticity 0.192 0.218Price markup 0.333 0.333
International sectorRelative prices and real exchange rates 1 1Net foreign assets 0 0
Import share vis-a-vis Ger or RoE, Ci,j+Ii,j
GDPi 0.130 0.066
Import share vis-a-vis RoW, Ci,c+I I,c
GDPi 0.244 0.244
Table B.2 summarises the parameter values resulting from estimating the model,while Table B.3 depicts the estimated VAR process for the rest of the world.
13
Table B.2: Parameter calibration euro area
Germany Rest of euro area
PreferencesShare of RoT households, µ 0.283 0.209Distribution of transfers, µ 0.476 0.356Habit formation, h 0.494 0.748Subs. elasticity: home and foreign goods, η 0.979 0.893
FrictionsInvestment adj. costs, υ 4.951 4.930Price adj. costs, υp 69.811 67.316Wage adj. costs, υw 61.801 79.885Price indexation, ξ 0.351 0.447Wage indexation, ξw 0.507 0.301Elasticity of pub. inv. w.r.t. output 0.084 0.070Elasticity of pub. emp. w.r.t. output 0.074 0.092
AR coefficients (fiscal rules)Labour taxes, ρτw 0.826 0.842Consumption taxes, ρτc 0.921 0.929SSC (employer), ρτsc 0.925 0.869Public consumption, ρCG 0.822 0.920Public investment, ρIG 0.783 0.857Public employment, ρNG 0.951 0.983Transfers, ρTR 0.844 0.941Lump sum taxes, ρT 0.533 0.808Public wages, ρwG 0.897 0.881
Debt feedback coefficients (fiscal rules)
Labour taxes, ξb,τw-0.005 0.022
SSC (employer), ξb,τsc-0.007 0.007
Public consumption, ξb,g 0.097 0.161
Public investment, ξb,IG0.219 0.197
Transfers, ξb,TR 0.166 0.128
Lump sum taxes, ξb,T 0.163 0.094
Output feedback coefficients (fiscal rules)
Labour taxes, ξy,τw0.073 0.036
SSC (employer), ξy,τsc-0.006 0.016
Public consumption, ξy,g 0.167 0.185
Public investment, ξy,IG0.199 0.189
Transfers, ξy,TR 0.197 0.220Lump sum taxes, ξy,T 0.188 0.194
continued on next page
14
continued from previous page
Germany RoE
Pre-announcement coefficients (fiscal rules)Labour taxes, ψτw 0.605 0.738Consumption taxes, ψτc 0.561 0.664SSC (employer), ψτsc 0.681 0.738Public consumption, ψCG 0.777 0.823Public investment, ψIG 0.786 0.764Public employment, ψNG 0.519 0.656Transfers, ψTR 0.725 0.744Lump sum taxes, ψT 0.802 0.298Public wages, ψwG 0.749 0.830
Monetary PolicyInterest rate smoothing, ρi 0.841 0.841Reaction to inflation, φπ 1.796 1.796Reaction to output, φy 0.054 0.054
AR coefficients (non-fiscal shocks)Technology, ρA 0.899 0.915Investment-specific technology, ρI 0.757 0.721Preference, ρβ 0.565 0.815Labour disutility, ρN 0.971 0.971Risk premium EA, ρRP,EA 0.772 0.772Risk premium RoW, ρRP,RoW 0.529 0.529Price markup, ρθ 0.558 0.582Wage markup, ρθw 0.709 0.512Export preference RoE/GER, ρRoE/ρGER 0.937 0.929Export preference RoW, ρRoW 0.861 0.864
Standard deviationsTechnology, σA 0.009 0.006Investment-specific technology, σI 0.046 0.031Preference, σβ 0.019 0.018Labour disutility, σN 0.026 0.022Risk premium EA, σRP,EA 0.004 0.004Risk premium RoW, σRP, RoW 0.006 0.006Price markup, σθ 0.077 0.053Wage markup, σθw 0.236 0.362Export preference RoE/GER, σRoE/σGER 0.028 0.028Export preference RoW, σRoW 0.028 0.015Interest rate, σi 0.0008 0.0008Labour taxes, στw 0.002 0.002Consumption taxes, στc 0.002 0.002SSC (employer), στsc 0.001 0.002Public consumption, σCG 0.018 0.012
continued on next page
15
continued from previous page
Germany RoE
Public investment, σIG 0.078 0.038Public employment, σNG 0.000 0.0002Public wages, σwG 0.012 0.015Transfers, σTR 0.013 0.011Lump sum tax, σT 0.021 0.016
Table B.3: Parameter calibration rest of the world
RoW
a11 0.776a12 0.208a13 0.348a21 -0.054a22 0.515a23 0.533a24 -0.134c21 0.467c24 0.228a31 0.025a32 -0.003a33 0.926a34 -0.109c31 0.072c32 0.002c34 -0.108a44 0.792Technology, σA 0.007Inflation, σπ 0.006Interest rate, σi 0.001Global technology, σz 0.005Subs. elasticity: home and foreign goods, η 0.478
C.1. Transition paths of alternative fiscal instruments
As mentioned in the main text of the paper, we show the transition paths of al-ternative fiscal instruments here. More precisely, Figure C.1 shows the path of usingpublic consumption as the debt-stabilising instrument. Because of the drop in publicconsumption, the recessionary effects are more pronounced and last longer.
16
Figure C.1: Adjustment paths given changes in the contribution rate using public consumption to stabilise debt
5 10 15 20
Quarters
-0.04
-0.02
0
0.02
GDP (DE)
5 10 15 20
Quarters
-0.1
0
0.1
0.2
0.3
Private consumption (DE)
5 10 15 20
Quarters
-0.2
-0.15
-0.1
-0.05
0
Investment (DE)
5 10 15 20
Quarters
-0.04
-0.02
0
0.02
0.04
Inflation (DE)
5 10 15 20
Quarters
-0.02
0
0.02
0.04
0.06
0.08
Unemployment rate (DE)
5 10 15 20
Quarters
-0.4
-0.3
-0.2
-0.1
0Gross wages (DE)
5 10 15 20
Quarters
-0.4
-0.2
0
0.2
0.4
0.6
Net wages (DE)
5 10 15 20
Quarters
-0.1
0
0.1
0.2
0.3
Labour costs (DE)
5 10 15 20
Quarters
-0.01
0
0.01
0.02
GDP (eurozone)
5 10 15 20
Quarters
-0.01
0
0.01
0.02
Private consumption (eurozone)
5 10 15 20
Quarters
-0.02
0
0.02
0.04
Investment (eurozone)
5 10 15 20
Quarters
0
5
1010-3 Inflation (eurozone)
5 10 15 20
Quarters
-0.02
0
0.02Unemployment rate (eurozone)
5 10 15 20
Quarters
-5
0
5
1010-3 Gross wages (eurozone)
5 10 15 20
Quarters
-0.1
0
0.1Competitiveness (eurozone/DE)
5 10 15 20
Quarters
-2
0
2
4
6
10-3 ECB rate
Increase in employer contributions Decrease in employee contributions Parity funding
17
Figure C.2: Adjustment paths given changes in the contribution rate using general transfers to stabilise debt
5 10 15 20
Quarters
-0.05
0
0.05
GDP (DE)
5 10 15 20
Quarters
-0.05
0
0.05
0.1
Private consumption (DE)
5 10 15 20
Quarters
-0.1
-0.05
0
0.05
0.1
Investment (DE)
5 10 15 20
Quarters
-0.01
0
0.01
0.02
0.03
Inflation (DE)
5 10 15 20
Quarters
-0.05
0
0.05
Unemployment rate (DE)
5 10 15 20
Quarters
-0.4
-0.3
-0.2
-0.1
0Gross wages (DE)
5 10 15 20
Quarters
-0.4
-0.2
0
0.2
0.4
0.6
Net wages (DE)
5 10 15 20
Quarters
0
0.1
0.2
0.3
Labour costs (DE)
5 10 15 20
Quarters
-5
0
5
10
10-3 GDP (eurozone)
5 10 15 20
Quarters
-5
0
5
1010-3Private consumption (eurozone)
5 10 15 20
Quarters
-0.01
-0.005
0
0.005
0.01Investment (eurozone)
5 10 15 20
Quarters
0
2
4
10-3 Inflation (eurozone)
5 10 15 20
Quarters
-0.01
0
0.01Unemployment rate (eurozone)
5 10 15 20
Quarters
-2
0
2
410-3 Gross wages (eurozone)
5 10 15 20
Quarters
-0.06
-0.04
-0.02
0
0.02
0.04
Competitiveness (eurozone/DE)
5 10 15 20
Quarters
0
1
2
3
4
10-3 ECB rate
Increase in employer contributions Decrease in employee contributions Parity funding
18
Figure C.3: Adjustment paths given changes in the contribution rate using consumption taxation to stabilise debt
5 10 15 20
Quarters
-0.05
0
0.05
GDP (DE)
5 10 15 20
Quarters
-0.1
0
0.1
Private consumption (DE)
5 10 15 20
Quarters
-0.15
-0.1
-0.05
0
0.05
Investment (DE)
5 10 15 20
Quarters
-0.01
0
0.01
0.02
0.03Inflation (DE)
5 10 15 20
Quarters
-0.05
0
0.05
0.1Unemployment rate (DE)
5 10 15 20
Quarters
-0.4
-0.3
-0.2
-0.1
0Gross wages (DE)
5 10 15 20
Quarters
-0.4
-0.2
0
0.2
0.4
0.6
Net wages (DE)
5 10 15 20
Quarters
0
0.1
0.2
0.3
Labour costs (DE)
5 10 15 20
Quarters
-5
0
5
1010-3 GDP (eurozone)
5 10 15 20
Quarters
-5
0
5
10-3Private consumption (eurozone)
5 10 15 20
Quarters
-10
-5
0
5
10-3 Investment (eurozone)
5 10 15 20
Quarters
0
2
4
10-3 Inflation (eurozone)
5 10 15 20
Quarters
-15
-10
-5
0
5
10-3 Unemployment rate (eurozone)
5 10 15 20
Quarters
-1
0
1
2
3
10-3 Gross wages (eurozone)
5 10 15 20
Quarters
-0.04
-0.02
0
0.02
0.04Competitiveness (eurozone/DE)
5 10 15 20
Quarters
0
1
2
3
4
10-3 ECB rate
Increase in employer contributions Decrease in employee contributions Parity funding
19
When using transfers to all households, there s no difference in aggregate variables,but household type-specific consumption reacts differently (see Figure C.2). In qualita-tive terms, the same holds for the use of consumption taxation, which further dampensthe positive consumption response (Figure C.3).
D.1. Alternative model design regarding unemployment benefits
In the GEAR model, unemployment benefits are assumed to be given by a fixed pa-rameter (see also the model description in this appendix). However, as we discussed inthe main text, the literature has shown that assumptions regarding unemployment ben-efits are a relevant element driving the results. This especially holds when the focus is onthe financing of unemployment benefits only. While that is not the case for analysing theeffects of reintroducing parity funding in the health care system, it may still affect the re-sults. And, indeed, we see that, when making unemployment benefits wage-dependent(by assuming unemployment benefits to be given by UBa
t = rrsa (1 − τw,at )wa
t , whererrsa = 0.3511 is the replacement rate), the tax shift generates smaller effects (as onewould expect after reading the literature). However, form a qualitative perspective, ef-fects remain basically analogous (see Table D.1 for a comparison of the long-run effects;Table D.2 shows the welfare results).
Table D.1: Long-term effects of changes in the contribution rate
Variable ScenarioWage-dependent UB Baseline
...in GermanyGDP 0.02 0.03Private consumption 0.02 0.04...of optimisers -0.03 -0.01...of liquidity-constrained 0.14 0.15Investment 0.02 0.04Unemployment rate 0.06 -0.02Real gross wages -0.44 -0.44Real net wages 0.28 0.28Real labour costs -0.07 -0.01Employee contribution rate -0.50 -0.50Employer contribution rate 0.50 0.50Financing instrument -0.03 0.11
...in eurozoneGDP 0.000 0.001Private consumption 0.001 0.001
Note: Table shows long-run changes of selected variables relative to initial steady-state values in percent(percentage points for rates and ratios). Financing instrument is lump-sum taxation.
20
Table D.2: Welfare assessment
Financing measure Consumption equivalentsNon-liquidity-constrained households Liquidity-constrained households Economy-wide
Lump-sum tax -0.026 (-0.049) 0.063 (0.119) -0.001 (-0.002)Consumption tax -0.001 (-0.001) -0.000 (-0.001) -0.001 (-0.001)Public consumption 0.022 (0.041) 0.049 (0.094) 0.029 (0.056)Transfers 0.081 (0.073) -0.213 (-0.197) -0.002 (-0.003)
Note: Welfare presented as life-time consumption equivalents taking into account the transition. In brackets, we report a puresteady-state comparison.
21