The Optimal Design of a Fiscal
Union∗
Mikhail Dmitriev†and Jonathan Hoddenbagh‡
First Draft: December 2012
This Version: November 2015
We study cooperative and non-cooperative fiscal policy in a multi-country model
where asset markets are segmented and countries face terms of trade externalities.
We show that the optimal form of fiscal cooperation, or fiscal union, is defined
by the Armington elasticity of substitution between goods from different countries.
We prove that members of a fiscal union should: (1) harmonize tax rates when the
Armington elasticity is low in order to ameliorate terms of trade externalities; and
(2) send fiscal transfers across countries when the Armington elasticity is high in
order to improve risk-sharing. For standard calibrations, the welfare gains from
tax harmonization are as high as 0.3% of permanent consumption for countries
both inside and outside of a currency union. The welfare gains from fiscal transfers
are close to zero for countries outside of a currency union, but rise to between
0.5% (France) and 3.6% (Greece) of permanent consumption for countries inside a
currency union.
Keywords: Fiscal union; Currency union; Monetary union; Optimal fiscal policy.
JEL Classification Numbers: E50, F41, F42.
∗
We thank Fabio Ghironi, Eyal Dvir, Peter Ireland and Susanto Basu for advice and encouragement at each
stage of this project. We also thank Ryan Chahrour, Sanjay Chugh, Christophe Chamley, David Schumacher,
Raúl Razo-Garcia, Laura Bottazzi, and Pedro Gete for helpful comments, as well as seminar participants at
the Bank of Canada, the Bank for International Settlements, Boston College, the Federal Reserve Bank of
Atlanta, the Federal Reserve Bank of Boston, the Federal Reserve Bank of Dallas, Florida State University,
Johns Hopkins SAIS, the Paris School of Economics, Simon Fraser, the University of Adelaide, the University
of Hawaii, Villanova, the BC/BU Green Line Macro Meeting, the Canadian Economic Association, the
Northern Finance Association and the Western Economic Association. Any errors are our own.
†
Florida State. Email: mdmitriev@fsu.edu. Web: http://www.mikhaildmitriev.org
‡
Johns Hopkins. Email: hoddenbagh@jhu.edu. Web: https://www2.bc.edu/jonathan-hoddenbagh/
1 Introduction
“Over the longer-term it would be natural to reflect further on whether we have
done enough in the euro area to preserve at all times the ability to use fiscal policy
counter-cyclically. But it is also clear that. . . this could only take place in the
context of a decisive step towards closer Fiscal Union.” — Mario Draghi, President
of the ECB, Nov. 27, 2014
“Cross-border risk-sharing through the financial system has slid backwards. Europe’s leaders do not currently foresee fiscal union as part of monetary union. Such
timidity has costs.” — Mark Carney, Governor of the Bank of England, Jan. 28,
2015
Currency unions are typically federations comprised of a common central bank, a centralized
fiscal authority and a subset of regional fiscal authorities. The euro area stands in stark contrast:
a currency union with no centralized fiscal authority or fiscal union (Bordo et al (2011)). While
the European Central Bank conducts monetary policy for the euro area as a whole, fiscal policy
is largely carried out by self-interested national authorities. In addition, financial markets in
the euro area play much less of an insurance role than in other federations, primarily due to
lower cross-border ownership of assets. While federations like the U.S., Canada and Germany
smooth over 80 percent of the impact of local shocks through a combination of fiscal transfers
and financial market integration, the euro area only insures half that amount (IMF (2013)).
The relative paucity of cross-border risk-sharing in the euro area combined with the selfinterested behavior of national fiscal authorities has prompted much debate about the need for
a fiscal union between member states. Although the concept of a fiscal union has received a great
deal of attention in policy circles, there is still considerable uncertainty about how to design
such a union. The literature has typically studied this question in an environment of perfect
cross-country risk-sharing, where full consumption insurance is provided by internationally
complete asset markets or by terms of trade movements, and where national fiscal authorities
are not self-interested but cooperative.1 However, the challenge facing the euro area is a direct
consequence of these features missing from the literature: too little cross-country risk-sharing
and too much self-interest on the part of national fiscal authorities.
To address this gap in the literature and the policy realm, we study the optimal design of
a fiscal union in an open economy model where cross-country risk-sharing is imperfect and
fiscal authorities are self-interested. Self-interested national fiscal authorities use fiscal policy
to tilt the terms of trade in their favor, giving rise to terms of trade externalities — a crucial
component of the present crisis in the euro area that has yet to be addressed in the literature.
1
See for example Beetsma and Jensen (2005), Gali and Monacelli (2008), Ferrero (2009) and Farhi and Werning
(2012).
2
We examine the interplay of nominal rigidities with imperfect risk-sharing and terms of trade
externalities and find that the negative welfare impact of these distortions is highly sensitive to
the elasticity of substitution between goods produced in different countries. As such, we find
that the magnitude of this elasticity — the Armington elasticity — governs the optimal design
of a fiscal union.
When the Armington elasticity is equal to one, a common assumption in the literature
following Cole and Obstfeld (1991), terms of trade movements provide complete international
risk-sharing through offsetting income and substitution effects, even in financial autarky. Under
this assumption, there is no need for a fiscal union to improve international risk-sharing. At
the same time, when goods are imperfect substitutes countries are exposed to a relatively high
degree of monopoly power at the export level. Self-interested domestic fiscal policymakers use
this monopoly power to impose a large markup on their exports, giving rise to a large terms
of trade externality. The optimal fiscal union forces domestic fiscal policymakers to internalize
this externality and prevents countries from manipulating their terms of trade. We find that
when the elasticity is close to one countries should cooperate in setting steady state domestic
income tax rates to ameliorate large terms of trade externalities — what we call a tax union.
As the Armington elasticity increases and domestic goods become more substitutable with
foreign goods, the degree of international risk-sharing provided by terms of trade movements
declines, as does each country’s monopoly power, thereby reducing the gains from a tax union.
On the other hand, there is now an increasingly important role for a fiscal union to improve
international risk-sharing. In a currency union with rigid wages, a one percent decline in productivity will raise marginal costs and prices by one percent, and the substitution from domestic
goods to foreign goods will be exactly proportional to the Armington elasticity. For example,
if the Armington elasticity is five a negative one percent productivity shock causes demand for
the home good to drop by five percent. The welfare loss from a negative productivity shock thus
increases in the Armington elasticity. This expenditure switching channel is severely dampened
under a flexible exchange rate, where prices are stabilized by exchange rate depreciation resulting from expansionary monetary policy after a negative productivity shock, regardless of the
value of the elasticity parameter. In a currency union where the common central bank is unable to offset asymmetric shocks, countries should organize a contingent cross-country transfer
scheme to provide international risk-sharing — what we call a transfer union — to compensate
for the decline in aggregate demand.
We compute the welfare gains from a fiscal union for a wide range of elasticities, including
country-specific estimates for European countries from Corbo and Osbat (2013) and Imbs and
Méjean (2010).2 For standard calibrations, the welfare gains from a tax union are as high as
2
Using highly disaggregated data, Eaton and Kortum (2002) estimate the elasticity to be 9.28, Broda and
Weinstein (2006) find an unweighted median of 3.1 and mean of 12.6, while Romalis finds a range of 4 to 13.
Imbs and Méjean (2015) find a mean of 6.7 with a standard deviaion of 4.9, and a median of 5.1. Lai and
3
0.3% of permanent consumption for countries both inside of and outside of currency unions.
The welfare gains from a transfer union are negligible for countries outside of a currency union
(0.01% of permanent consumption), but rise to between 0.5% (France) and 3.6% (Greece)
for countries inside a currency union with incomplete markets, and rise even higher when
countries are cut off from international financial markets.3 The gains from a transfer union are
significantly larger within a currency union because a single monetary policy cannot adequately
offset the negative impact of asymmetric shocks across countries, leading to large gaps in risk
sharing.
In addition to studying optimal fiscal policy, we show that labor mobility can facilitate international risk-sharing and alleviate the negative impact of wage rigidity as workers move
from depressed regions to boom regions. Although labor mobility is enshrined as one of the
four pillars of the European Union, mobility remains quite low within the euro area (see Fig
2). Forming a fiscal union is thus far more important for the euro area than for other federal
currency unions where labor mobility is relatively high and financial markets are more integrated, like the U.S. and Canada. These other federations are much better suited to withstand
asymmetric shocks across regions than the euro area.
Our closed-form model requires two simplifying assumptions: complete openness in consumption for all economies in the model as well as one period in advance nominal rigidities. We
relax both of these assumptions in Section 7 and evaluate the welfare gains from a fiscal union
in a model with consumption home bias and Calvo wage rigidities. As in the closed-form case,
we solve the extended model under non-unitary elasticity so that financial market structure
matters. In the extended model, we consider incomplete markets with cross-border trade in
safe government bonds (incomplete risk-sharing) as well as financial autarky. In this setup,
we find that home bias increases the welfare gains from a transfer union, while home bias
decreases the welfare gains from a tax union. We also find that Calvo rigidities do not change
our results, yielding similar welfare consequences as one period in advance rigidities. We focus
on the closed-form model for the first half of the paper as it generates analytically tractable
and intuitive results, and then shift to the extended model.
Trefler (2002) estimate a range between 5 and 8. More recently, Simonovska and Waugh (2011) find a range
between 3.38 and 5.42, while Feenstra, Obstfeld and Russ (2012) find a median estimate of the elasticity
between foreign countries of 3.1 for the U.S. In a survey of the literature on elasticity estimates, Anderson and
Van Wincoop (2004) conclude a range of five to ten is reasonable. Ruhl (2008) explains why the international
macro and trade literatures have quite different estimates of the Armington elasticity. Corsetti, Dedola and
Leduc (2008) estimate low values of the trade elasticity for the United States. Bussière et al (2013) show
that trade elasticities can vary over the business cycle because of changes in the composition of demand.
3
These gains are larger than Lucas’ (2003) estimates of the welfare cost of business cycle fluctuations (0.05%
of permanent consumption), which we replicate in a flexible wage environment.
4
How We Differ From The Literature
We contribute to the existing literature in three important ways. First, we consider noncooperative equilibria between self-interested fiscal authorities in a currency union, capturing
the current policy environment in the euro area. Second, we relax the assumption of perfect risksharing via complete markets or unitary elasticity. Third, we analyze the welfare implications
of a tax union and of labor mobility. Our ability to tackle these issues is driven by our novel
global closed-form solution that does not restrict the Armington elasticity to one. This provides
a tractable framework to analyze optimal policy and enables us to accurately compare welfare
across a variety of risk-sharing regimes, which is not possible under unitary elasticity.
Early non-microfounded contributions on the conduct of optimal monetary and fiscal policy
among interdependent economies include Canzoneri and Henderson (1990) and Eichengreen and
Ghironi (2002). Another strand of the literature focuses on the division of seignorage within a
currency union. Sims (1999) and Bottazzi and Manasse (2002) examine the interaction between
monetary and fiscal policy when seignorage is distributed by a common central bank. We
abstract from the role of seignorage and focus on the potential for fiscal policy cooperation to
improve welfare. Bottazzi and Manasse (2005) also examine the role of information asymmetry
on the conduct of monetary policy within a currency union. Benigno and De Paoli (2010)
study the ability of a small open economy with a flexible exchange rate to exploit terms of
trade externalities using fiscal policy.
Beetsma and Jensen (2005), Gali and Monacelli (2008) and Ferrero (2009) focus primarily on
the case of cooperative policy in a currency union with internationally complete asset markets.
These papers show that monetary policy should stabilize inflation at the union level and that
cooperative fiscal policy in the form of government spending has a country-specific stabilization
role. Evers (2012) takes a more quantitative approach and estimates the welfare gains from
a variety of “transfer rules”, in essence running a horse race between different types of fiscal
regimes within a currency union.
Farhi and Werning (2012) study cooperative fiscal policy in a transfer scheme under unitary
elasticity, where full international risk sharing is provided via terms of trade movements. They
demonstrate that even when private asset markets are complete internationally, there is a role
for contingent cross-country transfers to provide consumption insurance because the government
can internalize the aggregate demand externality which households and firms do not take into
account. We differ from Farhi and Werning (2012) by focusing on non-unitary elasticity where
private risk-sharing is incomplete, and by quantifying the welfare gains from a transfer union
using elasticities calibrated from the data. The welfare gains from a transfer union under
unitary elasticity are extremely small (0.01%) relative to the gains using elasticities calibrated
from the data (1-3% under incomplete markets).
5
2 The Model in Closed-Form
We consider a continuum of small open economies represented by the unit interval, as popularized in the literature by Gali and Monacelli (2005, 2008). Our model is based on Dmitriev
and Hoddenbagh (2013), although here we consider wage rigidity rather than price rigidity and
extend the closed-form solution for flexible exchange rates to the case of a currency union.
Each economy consists of a representative household and a representative firm. All countries
are identical ex-ante: they have the same preferences, technology, and wage-setting. Ex-post,
economies will differ depending on the realization of their technology shock. Households are
immobile across countries, however goods can move freely across borders. Each economy produces one final good, over which it exercises a degree of monopoly power. This is crucially
important: countries are able to manipulate their terms of trade even though they are measure
zero. As in Corsetti and Pesenti (2001, 2005) and Obstfeld and Rogoff (2000, 2002), we use
one-period-in-advance wage setting to introduce nominal rigidities. Workers set next period’s
nominal wages, in terms of domestic currency, prior to next-period’s production and consumption decisions. Given this preset wage, workers supply as much labor as demanded by firms. We
lay out a general framework below, and then hone in on the specific case of complete markets
and financial autarky. To avoid additional notation, we ignore time subindices unless absolutely
necessary. When time subindices are absent, we are implicitly referring to period t.
Production Each economy i produces a final good, which requires technology, Zi , and aggregated labor, Ni . We assume that technology is independent across time and across countries.
We need not impose any particular distributional requirement on technology at this point. The
production function of each economy will be:
Yi = Zi Ni .
(1)
Households, indexed by h, each have monopoly power over their differentiated labor input,
which will lead to a markup on wages. A perfectly competitive, representative final goods
producer aggregates differentiated labor inputs from households in CES fashion into a final
good for export. Production of the representative final goods firm in a specific country is:
Z
Ni =
1
Ni (h)
ε−1
ε
ε
ε−1
dh
,
0
where ε is the elasticity of substitution between different types of labor, and µε =
ε
ε−1
is the
markup on labor.
The aggregate labor cost index, W , defined as the minimum cost to produce one unit of out1
R
1−ε
1
1−ε
put, will be a function of the nominal wage for household h, W (h): Wi = 0 Wi (h) dh
.
6
Cost minimization by the firm leads to demand for labor from household h:
Ni (h) =
Wi (h)
Wi
−ε
Ni .
In the open economy, monopoly power is exercised at both the household and the country
level: at the household level because of differentiated labor, and at the country level because
each economy produces a unique good. We show in Section 3 that optimizing non-cooperative
policymakers will remove the household markup on labor but will introduce a terms of trade
markup through the income tax rate. Just to be clear, firms have no monopoly power and are
perfectly competitive.
Households In each economy, there is a household, h, with lifetime expected utility
Et−1
(∞
X
βk
k=0
1−σ
Cit+k (h)
1−σ
1+ϕ
−χ
Nit+k (h)
1+ϕ
)
(2)
where β < 1 is the household discount factor, C(h) is the consumption basket or index, N (h) is
household labor effort (think of this as hours worked). Households face a general budget constraint that nests both complete markets and financial autarky; we will discuss the differences
between the two in subsequent sections. For now, it is sufficient to simply write out the most
general form of the budget constraint:
Cit (h) = (1 − τi )
Wit (h)
Pit (h)
Nit (h) + Dit (h) + Tit (h) + Γit (h).
(3)
The distortionary tax rate on household labor income in country i is denoted by τi , while Γit is
a domestic lump-sum tax rebate to households. T refers to lump-sum cross-country transfers.
In the absence of a fiscal union, these cross-country transfers will equal zero (T = 0). Net
taxes equal zero in the model, as any amount of government revenue is rebated lump-sum
to households. The consumer price index corresponds to Pit , while the nominal wage is Wit .
Dit denotes state-contingent portfolio payments expressed in real consumption units, and the
following equation holds in all possible states in all periods:
1
Z
Dit Pit =
Eijt Bijt dj,
0
where Bijt is a state-contingent payment in currency j Eijt is the exchange rate in units of
currency i per one unit of currency j; an increase in Eijt signals a depreciation of currency i
relative to currency j. In a currency union, Eijt = 1 for all i, j, t. When international asset
markets are complete, households perform all cross-border trades in contingent claims in period
7
0, insuring against all possible states in all future periods. The transverality condition simply
states that all period 0 transactions must be balanced: payment for claims issued must equal
payment for claims received. The transversality condition for complete markets is:
E0
(∞
X
)
β t Cit−σ Dit
= 0,
(4)
t=0
while in financial autarky Dit = 0. Intuitively, the transversality condition stipulates that the
present discounted value of future earnings should be equal to the present discounted value of
future consumption flows. Under complete markets, consumers choose a state contingent plan
for consumption, labor supply and portfolio holdings in period 0.
Consumption and Price Indices Households in each country consume a basket of imported
goods. This consumption basket is an aggregate of all of the varieties produced by different
countries. The consumption basket for a representative small open economy i, which is common
across countries, is defined as follows:
Z
1
γ−1
γ
γ
γ−1
cij dj
Ci =
(5)
0
where lower case cij is the consumption by country i of the final good produced by country
j, and γ is the elasticity of substitution between domestic and foreign goods (the Armington
elasticity). Because there is no home bias in consumption, countries will export all of the output
of their unique variety, and import varieties from other countries to assemble the consumption
basket.
Prices are defined as follows: lower case pij denotes the price in country i (in currency i) of
the unique final good produced in country j, while upper case Pi is the aggregate consumer
price index in country i. Given the above consumption index, the consumer price index will be
1
R
1−γ
1
Pi = 0 p1−γ
dj
. Demand by country i for the unique variety produced by country j is:
ij
cij =
pij
Pi
−γ
Ci .
(6)
We assume that producer currency pricing (PCP) holds, and that the law of one price (LOP)
holds, so that the price of the same good is equal across countries when converted into a
common currency. We define the nominal bilateral exchange rate between countries i and j,
Eij , as units of currency i per one unit of currency j. LOP requires that pij = Eij pjj . Given LOP
and identical preferences across countries, PPP also holds for all i, j country pairs: Pi = Eij Pj .
The terms of trade for country j, defined as the home currency price of exports over the home
currency price of imports, is denoted T OTj = pjj /Pj .
8
Take country i’s demand for country j’s unique variety (6), and using LOP and PPP, solve
for world demand for country j’s unique variety:
Z
1
Z
1
cij di =
Yj =
0
0
pij
Pi
−γ
Ci di
LOP+PPP
=
pjj
Pj
−γ Z
1
Ci di = T OTj−γ Cw .
(7)
0
where Cw is defined as the average world consumption across all i economies, Cw =
R1
0
Ci di.
Using goods market clearing and the fact that technology Zit is identically and independently
distributed across time and consumption baskets are identical across countries, we can rewrite
average world consumption as:
1
Z
Cwt =
γ−1
γ
γ
γ−1
Yit
.
(8)
0
Labor Market Clearing Households maximize (2) subject to (3). The first order condition
for labor gives the labor supply condition (which is the optimal preset wage):
Wit =
χµε
1 − τi
Et−1 Nit1+ϕ
n −σ o .
C N
Et−1 itPit it
The representative firm in country i maximizes profit by choosing the appropriate amount
of aggregate labor, leading to the familiar labor demand condition equating the real wage at
time t (Wit /pit ) with the marginal product of labor (Zit ). To obtain the labor market clearing
condition, set labor demand equal to labor supply, and use the fact that the wage is preset at
time t − 1 and that demand for unique variety is (7):4
1=
χµε
1−τ
Et−1 Nit1+ϕ
.
γ−1
1
γ
γ
−σ
Et−1 Cit Yit Cwt
(9)
Taking the expectations operator out of (9) yields the flexible wage equilibrium.
Complete Markets
In complete markets, agents in each economy have access to a full set of domestic and foreign
state-contingent assets. Households in all countries will maximize (2), choosing consumption,
leisure, money holdings, and a complete set of state-contingent nominal bonds, subject to (3).
4
A more general labor market clearing condition, which holds for both producer currency pricing and local
currency pricing is:
n
o
1+ϕ
E
N
t−1
it
χµε
n
o.
1=
pit
−σ
1−τ E
t−1 Cit Yit Pit
9
Complete markets and PPP imply the following risk-sharing condition:
−σ
Cjt
Cit−σ
=
−σ
−σ
Cit+1
Cjt+1
∀i, j
(10)
which states that the ratio of the marginal utility of consumption at time t and t + 1 must be
equal across all countries. Importantly, this condition does not imply that consumption is equal
across countries. Consumption in country i will depend on the initial asset position, fiscal and
monetary policy, the distribution of country-specific shocks, the covariance of global and local
shocks, and other factors.5
When (4), (9), and (10) hold, consumption in country i can be expressed as a function of
world consumption:
Cit =
Et−1
−σ
β s Yit+s Cwt+s
T OTit+s
P
Cwt .
1−σ
Et−1
β s Cwt+s
P
(11)
This defines the optimal consumption allocation for country i in complete markets.6 Using
the fact that Zit is independent across time and across countries, and wages are preset, (11) is
equivalent to
γ−1 Cit = Cw Et−1 Yit γ .
1
γ
(12)
Financial Autarky
The aggregate resource constraint under financial autarky specifies that the nominal value of
output in the home country (exports) must equal the nominal of consumption in the home
country (imports). That is, trade in goods must be balanced. In a model with cross-border
lending, bonds would also show up in this condition, but in financial autarky they are obviously
absent. The primary departure from complete markets lies in the household and economy-wide
budget constraint (i.e. the trade balance), where spending on imports must equal export
revenues: Pi Ci = pii Yi .
Substituting world demand for the unique variety (7) into the trade balance condition, one
can show that demand for country i’s good in financial autarky will be:
1
γ−1
Cit = Cwγ Yit γ .
(13)
Thus, in complete markets consumption is equal to expected domestic output expressed in
consumption baskets; in autarky consumption is equal to realized domestic output expressed
5
A policy change in economy i may lead to a change in consumption. For example, monetary policy affects the
covariance between home production and world consumption, which in turn influences home consumption,
even in complete markets. Fiscal policy can tax consumption and cause a lower level of consumption in
the long-run relative to the rest of the world. In spite of this, it is still possible to characterize an optimal
consumption plan that is robust to changes in monetary and fiscal policy.
6
Details are found in the appendix of Dmitriev and Hoddenbagh (2013).
10
in consumption baskets.
3 Non-Cooperative Policy
In order to study the benefits of international policy cooperation, we must first understand the
non-cooperative Nash equilibrium. What outcomes naturally arise when policymakers do not
cooperate? Our goal in this section is to illuminate the various distortions that are present in the
non-cooperative Nash equilibrium, and to compare and contrast with the global social planner
equilibrium defined in Appendix A. We can then pinpoint specific areas of policy cooperation
that ameliorate welfare decreasing distortions, leading us to the optimal design of a fiscal union.
We begin with the Nash equilibrium under flexible exchange rates and then move to the case
of a currency union.
Flexible Exchange Rates
When exchange rates are flexible, each country has its own central bank and its own fiscal
authority. Before any shocks are realized, national fiscal authorities declare non state-contingent
taxes, and then national central banks declare monetary policy for all states of the world. With
this knowledge in hand, households lay out a state-contingent plan for consumption and labor
as well asset holdings when markets are complete. After that, shocks hit the economy. A
detailed timeline is provided in Figure 1.
Without loss of generality, we assume a cashless limiting economy.7 Central banks set monetary policy in each period by optimally choosing the amount of labor. Although central banks
optimize by choosing labor instead of money or an interest rate, the three are equivalent in
this model.8 We can write down an interest rate rule that gives the exact same allocation.
Domestic fiscal authorities choose the optimal labor tax rate τi . The objective function for
non-cooperative domestic policymakers will be
max max Et−1
Nit
τi
Cit1−σ
Nit1+ϕ
−χ
1−σ
1+ϕ
,
(14)
where the fiscal authority acts first and chooses τi and the central bank then chooses Nit .
We first examine the Nash equilibrium for non-cooperative policymakers when international
asset markets are complete. Policymakers in complete markets will maximize their objective
function subject to production (1), aggregate world consumption (8) and labor market clearing
(9) and goods market clearing (12) constraints.
Proposition 1 Flexible Exchange Rates + Complete Markets When international asset markets are complete and exchange rates are flexible, non-cooperative policymakers will
7
8
Benigno and Benigno (2003) describe a cashless-limiting economy in detail in their appendix, pp.756-758.
We demonstrate the equivalence in Appendix F of Dmitriev and Hoddenbagh (2013).
11
maximize (14) subject to (1), (8), (9), and (12). The solution under commitment for noncooperative policymakers in complete markets is:
Ci =
Ni =
1
χµγ
1
χµγ
1
σ+ϕ
Z
1
(γ−1)(1+ϕ)
1+γϕ
Zi
1+γϕ
(γ−1)(σ+ϕ)
di
,
(15a)
0
1
σ+ϕ
Z
1
(γ−1)(1+ϕ)
1+γϕ
Zi
1−γσ
(γ−1)(σ+ϕ)
di
γ−1
Zi1+γϕ .
(15b)
0
It is optimal for non-cooperative central banks under commitment to mimic the flexible wage
allocation. The optimal tax rate for non-cooperative fiscal authorities is τi = 1 −
µε
.
µγ
Proof See Appendix B. The above allocation replicates the global social planner allocation with the addition of a
terms of trade markup, µγ =
γ
,
γ−1
that lowers consumption and output. It is optimal for central
banks to mimic the flexible wage allocation through a policy of price stability.9 Optimizing fiscal
authorities internalize the negative welfare impact of the domestic markup on differentiated
labor inputs (µε ), and thus choose an income tax rate that cancels out the labor markup.
However, fiscal authorities also want to use their country-level monopoly power. Because
each country in the continuum is measure zero, policymakers do not internalize the impact of
charging a higher markup for their export good on the welfare of other countries. This leads
fiscal authorities to set an income tax rate which reduces hours worked and restricts production,
τi = 1− µµγε , so that exports from each country are subject to a terms of trade markup (µγ ). The
terms of trade externality leads to lower welfare outcomes because households in each country
must pay a higher price on each import good in the consumption basket. Even though asset
markets are complete, the non-cooperative allocation under flexible exchange rates yields lower
welfare than the global social planner allocation due to the imposition of the terms of trade
markup. The need for some sort of international fiscal cooperation that would force domestic
fiscal authorities to internalize this externality is clear.
Now that we’ve examined the complete markets equilibrium when exchange rates are flexible,
we turn our attention to the case of financial autarky. The objective function in financial
autarky will be identical to the complete markets case. Domestic fiscal authorities will first
choose the optimal tax rate, and then central banks will set the optimal monetary policy by
choosing labor. However, there is a slight difference in the constraints faced by policymakers in
complete markets and financial autarky. In complete markets, home consumption is a function
of expected output (12), while in autarky home consumption is a function of actual output
(13). All other constraints are identical in complete markets and financial autarky.
9
In a related paper (Dmitriev and Hoddenbagh 2013), we prove that mimicking the flexible price allocation
is a dominant strategy for small open economy central banks. This result is robust to changes in elasticity
between domestic and foreign goods, the degree of cooperation between policymakers in different countries,
and the degree of financial integration across countries.
12
Proposition 2 Flexible Exchange Rates + Financial Autarky Non-cooperative policymakers in financial autarky will maximize (14) subject to (1), (8), (9), and (13). The solution
under commitment for non-cooperative policymakers in financial autarky is:
Ci =
Ni =
1
χµγ
1
χµγ
1
σ+ϕ
Z
1
(γ−1)(1+ϕ)
1−σ+γ(σ+ϕ)
Zi
(1+ϕ)
(γ−1)(σ+ϕ)
(γ−1)(1+ϕ)
1−σ+γ(σ+ϕ)
Zi
di
,
(16a)
Zi1−σ+γ(ϕ+σ) .
(16b)
0
1
σ+ϕ
Z
1
(γ−1)(1+ϕ)
1−σ+γ(σ+ϕ)
Zi
(1−σ)
(γ−1)(σ+ϕ)
di
(γ−1)(1−σ)
0
It is optimal for non-cooperative central banks to mimic the flexible wage allocation. The optimal
tax rate for non-cooperative fiscal authorities is τi = 1 −
µε
.
µγ
Proof See Appendix B. As in complete markets, central banks find it optimal to mimic the flexible wage equilibrium
through a policy of price stability in financial autarky. On the fiscal side, policymakers again
eliminate the domestic markup µε , but impose a terms of trade markup on their unique export good µγ via the steady state income tax rate. Financial autarky removes cross-country
consumption insurance, as households no longer have the ability to trade in international contingent claims. This can be seen most clearly in (16a), where equilibrium consumption is a
function of idiosyncratic productivity, Zi , and will fluctuate with country-specific shocks to
technology.
Currency Union
Within a currency union, a single central bank sets monetary policy for the union as a whole.
Countries no longer control their domestic monetary policy as they do when exchange rates are
flexible. In the presence of aggregate shocks, the union-wide central bank will stabilize inflation
at the union level, a result shown in Gali and Monacelli (2008). However, for tractability we
assume no aggregate shocks, only asymmetric country-specific shocks. With only one policy
instrument, the union-wide central bank cannot eliminate wage rigidity at the country level in
the presence of asymmetric shocks. As a result, the union-wide central bank does nothing in
our model. None of our results change if we add aggregate shocks: these shocks would simply
be counteracted by the union-wide central bank.
In a currency union each country retains control over it’s own fiscal policy. The objective
function for non-cooperative fiscal policymakers in a currency union is:
max Et−1
τi
Cit1−σ
N 1+ϕ
− χ it
1−σ
1+ϕ
(17)
The constraints faced by policymakers within a currency union are identical to those faced by
policymakers under flexible exchange rates, with the addition of a fifth constraint unique to
13
currency unions. Thus, relative to the optimization problem under flexible exchange rates, we
add one constraint and subtract one FOC. Plugging our expression for the real wage (pii =
Wi
)
Zi
into the demand for country i’s good (7) yields:
Wit
Pit
Yit =
−γ
{z
|
A
Cw Zitγ = AZitγ
}
(18)
where A is a constant. (18) is the additional constraint faced by the policymaker in a currency
union.
Proposition 3 Currency Union + Complete Markets Non-cooperative policymakers in
a currency union will maximize (17) subject to (1), (8), (9), (12), and (18). The solution under
commitment for non-cooperative policymakers within a currency union in complete markets is:
Ci = Cw =
Ni =
1
χµγ
1
χµγ
1
σ+ϕ
1
σ+ϕ
1
 σ+ϕ
γ(1+ϕ)
γ−1



 R
,
 1 (γ−1)(1+ϕ) 
di
Z
i
0

R1
Ziγ−1 di
0
1
 σ+ϕ
γ(1−σ)
γ−1



 R
Ziγ−1 .
 1 (γ−1)(1+ϕ) 
di
Zi
0

(19a)
R1
Ziγ−1 di
0
(19b)
The resulting equilibrium allocation does not replicate the flexible wage equilibrium. The optimal
tax rate for non-cooperative fiscal authorities is τi = 1 −
µε
.
µγ
Proof See Appendix C. Within a currency union, the inability of the union-wide central bank to alleviate asymmetric
shocks across countries leads to the presence of wage rigidity in the optimal allocation. In
addition, non-cooperative fiscal authorities exploit their country-level monopoly power and
impose a terms of trade markup via income tax policy. We thus see the presence of two
distortions in the equilibrium allocation: wage rigidity and a terms of trade markup. As in the
flexible exchange rate allocation, there is no idiosyncratic technology risk in consumption under
complete markets, so consumption will be equalized across countries in equilibrium. However,
welfare will be lower when wages are rigid than when they are flexible, as one can notice by
comparing the above allocation with the Pareto efficient allocation (see Section 6 for details).
Proposition 4 Currency Union + Financial Autarky In financial autarky, non-cooperative
policymakers in a currency union will maximize (17) subject to (1), (8), (9), (13) and (18).
14
The optimal allocation in financial autarky given by a non-contingent policymaker in a currency
union is:
Ci =
Ni =
1
χµγ
1
χµγ
1
σ+ϕ
 R


1
σ+ϕ
1
0
R
1
0
Ziγ−1 di
R1
(γ−1)(1+ϕ)
Zi
di
0
 R


(γ−1)(1−σ)
Zi
di
1
0
(γ−1)(1−σ)
Zi
di
R
1
0
Ziγ−1 di
R1
(γ−1)(1+ϕ)
Zi
di
0
1
 σ+ϕ
1+ϕ
γ−1


Ziγ−1 ,
(20a)
Ziγ−1 .
(20b)
1
 σ+ϕ
1−σ
γ−1


The resulting equilibrium allocation does not replicate the flexible wage allocation. The optimal
tax rate for non-cooperative fiscal authorities is τi = 1 −
µε
.
µγ
Proof See Appendix C. In the autarky Nash equilibrium described in Proposition 4, members of a currency union face
three welfare decreasing distortions: wage rigidity resulting from the absence of country-specific
monetary policy; idiosyncratic consumption risk, caused by lack of access to international
financial markets; and a terms of trade markup, imposed by non-cooperative fiscal authorities
in other countries. The potential for cooperative measures to ameliorate these distortions is
evident, and will be the focus of Section 4.
Contingent Tax Policy
We also consider the role of contingent tax policy in Appendix D. We show that contingent tax
policy within a currency union can play the same role as monetary policy outside of a currency
union, eliminating nominal rigidities and mimicking the flexible-wage allocation. While the
ability of contingent taxes to eliminate nominal rigidities has been shown in other closed and
open economy studies, we are the first to show that the distortionary impact of wage rigidity
and hence the importance of contingent tax adjustments is increasing in the Armington elasticity.10 Our analysis is also the first to demonstrate that international policy cooperation is not
necessary to eliminate the distortionary impact of wage rigidity: non-cooperative contingent
fiscal policy is all that is required.
Empirical evidence from Vegh and Vuletin (2015) shows that tax policy in industrial countries
is acyclical. In line with this evidence, we assume that tax policy is non-contingent for the
remainder of the paper.
10
Two related papers, Adao, Correia and Teles (2009) and Farhi, Gopinath and Itskhoki (2011), show that a
set of state-contingent taxes can mimic the flexible price equilibrium. A non-exhaustive list of other papers
demonstrating the importance of contingent fiscal policy includes Chugh (2006), Chugh and Ghironi (2015),
Correia et al (2013), Kersting (2013), Lewis and Cardoso-Costa (2015), Schmitt-Grohe and Uribe (2004)
and Siu (2004).
15
4 Cooperative Policy in a Fiscal Union
In this section we analyze fiscal policy cooperation. Although the concept of fiscal cooperation
may be quite broad, we focus here on two specific types — a tax union and a transfer union. In
a tax union, fiscal policymakers in each country cooperatively set steady state income tax rates
to maximize the welfare of the union as a whole. A tax union may be viewed as a cross-country
agreement on income tax setting between domestic fiscal authorities, or as a set of tax rates
chosen by a supranational (or federal) fiscal authority. In a transfer union, fiscal policymakers in
each country arrange contingent cross-country transfers to maximize the welfare of the union as
a whole. The transfer scheme may derive from an agreement between national fiscal authorities
or from a supranational (or federal) fiscal authority.
Four different types of fiscal policy cooperation are possible: a combined tax and transfer
union; a tax union with no transfer union; a transfer union with no tax union; and no tax nor
transfer union.11 The objective functions for the various types of fiscal union are written below.
Cit1−σ
Nit1+ϕ
max
max Et−1
−χ
di
Nit
∀τi
1−σ
1+ϕ
0
1−σ
Z 1
Nit1+ϕ
Cit
max
−χ
Et−1
di
∀τi
1−σ
1+ϕ
0
Z
1
(21a)
(21b)
Objective functions (21a) and (21b) refer to a tax union outside of and within a currency
union, respectively. Here, the fiscal authorities in each country jointly maximize the welfare of
all countries in the union by choosing the same steady state income tax rate.
1−σ
Nit1+ϕ
Cit
−χ
di
max
max max Et−1
τi
Nit
∀Tit
1−σ
1+ϕ
0
1−σ
Z 1
Cit
Nit1+ϕ
max
max Et−1
−χ
di
τi
∀Tit
1−σ
1+ϕ
0
Z
1
(21c)
(21d)
Objective functions (21c) and (21d) refer to a transfer union outside of and within a currency
union, respectively. Here, a supranational (or federal) fiscal body optimally chooses crosscountry transfers in order to maximize union-wide welfare.
Nit1+ϕ
Cit1−σ
−χ
di
max
max Et−1
Nit
∀τi ,Tit 0
1−σ
1+ϕ
1−σ
Z 1
Cit
Nit1+ϕ
max
Et−1
−χ
di
∀τi ,Tit 0
1−σ
1+ϕ
Z
11
1
(21e)
(21f)
Because our focus is on the optimal design of a fiscal union, we ignore the implications of monetary policy
cooperation. With one common central bank, monetary cooperation within a currency union is not possible.
In another paper (Dmitriev and Hoddenbagh (2013)), we show that in a continuum of small open economies
monetary cooperation yields no welfare gains, as non-cooperative and cooperative equilibria exactly coincide.
16
Finally, (21e) and (21f) refer to a tax and transfer union outside of and within a currency
union, respectively. Here, countries not only agree on income tax rates, but also agree to send
contingent cash transfers across countries.
Proposition 5 Tax Unions Policymakers in a tax union will internalize the impact of their
income tax rate on all union members. As a result, a tax union will remove the incentive
for policymakers to manipulate their terms of trade. The optimal tax rate in a tax union
is τi = 1 − µε , which will remove the markup on domestic production in each country, µε ,
while preventing the imposition of a terms of trade markup on exports, µγ , from all equilibrium
allocations.
Proof See Appendix E. A tax union forces domestic fiscal authorities to internalize the impact of their terms of trade
externality on union wide welfare. As a result, policymakers will not impose a terms of trade
markup on the export of their country’s unique good in a tax union. This improves welfare for
the entire union as well as for each individual country, particularly for low values of elasticity
when countries have a high degree of monopoly power. The distortive impact of the terms of
trade externality increases as the degree of substitutability decreases, which will become clear
in Section 6 when we calculate the welfare gains from a tax union. In other words, the benefits
of a tax union are increasing in the degree of country-level monopoly power.
Members of a transfer union agree to send contingent cash transfers across countries in order
to insure against idiosyncratic consumption risk. In complete markets the presence of crosscountry transfers will alter the goods market clearing constraint, so that (12) is replaced by the
following two conditions:
γ−1 Cit = Cwt Et−1 Yit γ
+ Tit ,
1
γ
(22)
In financial autarky the presence of cross-country transfers will alter the goods market clearing
constraint, so that (13) is replaced by the following two conditions:
γ−1
1
γ
Cit = Cwt
Yit γ + Tit .
(23)
Note that the sum of all cross-country transfers must add up to zero in both complete markets
R1
and in financial autarky: 0 Tit di = 0.
Proposition 6 Transfer Unions Policymakers in a transfer union agree to send contingent
cash transfers across countries in order to insure against idiosyncratic consumption risk. The
equilibrium allocation within a transfer union will be identical with the equilibrium allocation
under complete markets. As a result, transfer unions are redundant when international asset
17
markets are complete or when substitutability is one, but yield large welfare gains when markets
are incomplete.
Proof See Appendix F. As Proposition 6 states, a transfer union guarantees complete cross-country consumption
insurance and thus replicates the effect of complete markets.12 The welfare benefits of a transfer
union are increasing in the Armington elasticity: as goods become closer substitutes, the natural
risk-sharing role played by the terms of trade begins to disappear. This will be seen more clearly
in Section 6 when we calculate the welfare gains from a transfer union.
It is also possible to have a tax and a transfer union. If countries agree to both, they will
enjoy complete risk-sharing and eliminate the distortive impact of the terms of trade externality.
Proposition 5 and 6 show that a tax union, a transfer union or a combination of the two will
move countries toward the Pareto efficient allocation.
Proposition 7 Pareto Optimum The Pareto efficient allocation is achieved through a combination of: (1) independent monetary policy outside of a currency union or contingent fiscal
policy within a currency union; (2) internationally complete asset markets or a transfer union;
and (3) a tax union.
Proof (1) eliminates the distortionary impact of wage rigidity, (2) provides cross-country
risk-sharing, and (3) prevents terms of trade manipulation. Any combination of (1), (2) and
(3), for example a tax and transfer union whose members control their own monetary policy
outside of a currency union, will yield the Pareto efficient allocation. Although each of these ingredients is necessary to achieve the Pareto efficient allocation,
the relative importance of these ingredients is highly sensitive to the elasticity of substitution
between products from different countries. In Section 6, we show that the optimal design of a
fiscal union, and the emphasis given to a tax versus a transfer union, will depend on the value
of the elasticity parameter.
5 Labor Mobility
If deeper fiscal integration is not possible, what should governments do? One possibility, heralding back to James Meade (1957), is to pursue policies that increase labor mobility. Discussing
the creation of a common currency area in Western Europe, Meade argued that without the free
movement of goods, capital and labor, the idea was doomed to failure. We demonstrate the benefits of labor mobility in a rigorous microfounded setup using using our analytical closed-form
model.
12
When we introduce home bias in consumption to the model in Section 7, we see that a transfer union actually
improves upon the complete markets allocation. This matches the results of Farhi and Werning (2013), who
show that generically private risk sharing is inefficient in the presence of consumption home bias because of
an aggregate demand externality.
18
We assume perfect labor mobility, such that workers within a household can reallocate immediately without any costs after learning the realization of technology shocks. Workers send
transfers to their family members. Disutility from labor results from the total number of hours
worked in the household. Wages are set one period in advance by firms, and firms have rational
expectations about worker’s reallocation. Nominal wages will equalize across countries in the
currency union since consumption baskets are identical across countries. However, producer
price indices (PPI) will be higher in the countries with smaller production and productivity.
As a result, the real wage defined using the consumer price index (CPI) will be identical across
the currency union, while real wages defined using the PPI will differ across countries.
When labor is mobile, non-cooperative policymakers in a currency union in financial autarky
maximize the familiar objective function, but face a new set of constraints (found in Appendix
G). As each economy in the currency union is hit with idiosyncratic shocks, labor will move
from low demand recession countries to high demand boom countries. The movement of workers across borders equalizes real wages across countries and thereby acts as a natural shock
absorber, enabling efficient adjustment of the economy without any policy actions taken by
monetary or fiscal authorities.
Proposition 8 Labor Mobility Labor mobility will eliminate two distortions: wage rigidity
and the lack of risk-sharing in financial autarky. The resulting equilibrium allocations yield
higher welfare than the flexible wage allocations in complete markets and financial autarky.
When labor can move freely across borders, the solution under commitment for non-cooperative
policymakers outside of or within a currency union, in complete markets or financial autarky,
is:
Ci =
Ni =
1
χµγ
1
χµγ
1
σ+ϕ
Z
1
Ziγ−1 di
1+ϕ
(σ+ϕ)(γ−1)
(24a)
0
1
σ+ϕ
Z
1
Ziγ−1 di
1+ϕ−γ(σ+ϕ)
(γ−1)(σ+ϕ)
Ziγ−1 .
(24b)
0
The optimal tax rate for non-cooperative fiscal authorities is τi = 1 −
µε
.
µγ
Proof See Appendix G. Labor mobility achieves three goals from a policy perspective. First, households whose members can move across countries without any restrictions diversify their source of income, which
provides cross-country risk-sharing even in the absence of internationally complete asset markets. Second, shifting labor hours across countries generates labor risk-sharing, so that hours
worked do not fluctuate with country-specific shocks. Third, the stabilizing influence of labor
mobility on both consumption and labor eliminates the distortionary impact of wage rigidity.
So the fixed wage allocation under labor mobility will mimic the flexible wage allocation. Al-
19
though the distortive effect of the terms of trade externality remains so there is still a need for
a tax union, the benefits of labor mobility are potentially massive and ease the burden on fiscal
policy greatly. We quantify these gains in Sections 6 and 7.
Legally, labor mobility is enshrined as one of the four pillars of economic integration within the
European Union. EU citizens are free to migrate to any other EU country to seek employment
(Kahanec 2012, Zimmermann 2005). Workers are also free to move across state borders in the
U.S., as well as provincial borders in Canada and state borders in Australia. Despite similar
legal guarantees for labor mobility, the data show that labor mobility is much higher within
the U.S., Canada and Australia than within the EU.
Figure 2 plots the extent of labor mobility across countries in the EU as well as across U.S.
states, Canadian provinces, and Australian states and territories. Over and above regulatory
and legal barriers to mobility, language seems to rule: linguistic and cultural differences across
countries make emigration much more difficult. For example, notice the high degree of labor
mobility within unilingual currency unions (Australia and the U.S.), the slightly lower degree
of labor mobility in a bilingual currency union (Canada), and the much lower degree of labor
mobility in a multilingual currency union (euro area). One can see the importance of language
most clearly by focusing on the much higher degree of mobility within EU countries (0.95%)
where languages are uniform than across EU countries (0.29%) where they differ, as well as the
the high mobility across Canada as a whole (0.98%) versus the low degree of mobility between
French-speaking Quebec and the English-speaking provinces (0.39%).
These data reinforce the notion that labor mobility is vital to the sound functioning of a
currency union. High mobility in the US, Australia and Canada dampens the distortionary
impact of internal wage rigidity, lowers unemployment and improves risk-sharing. On the other
hand, low mobility in the EU leads to high unemployment and overvalued wages in areas that
are hit by large negative shocks. But the main takeaway is surely this: achieving a fiscal union
within the euro area may prove politically difficult, but the lack of labor mobility within the euro
area suggests an even stronger need for a fiscal union.
6 Welfare Analysis in the Closed-Form Model
We now analyze the welfare gains resulting from the introduction of a tax union, a transfer
union as well as labor mobility. The advantage of the closed-form solution is most apparent
here: rather than approximating a quadratic welfare function around a particular steady state,
we can calculate welfare explicitly at any steady state, whether distorted or otherwise. This is
particularly important when we focus on the welfare gains from a tax union, which eliminates
the terms of trade markup from the steady state allocation. In a log-linear model, comparing
welfare between the tax union and no tax union cases is infeasible because of the different
steady states.
20
Our calibration for the closed-form model at quarterly frequency follows standard benchmarks
from the literature and is reported in Table 1. In our welfare analysis, we allow the Armington
elasticity to vary while fixing the other parameters of the model.
Expected Utilities
Below we calculate the log of expected utility for five allocations: flexible exchange rates with
complete risk-sharing (25a); flexible exchange rates in financial autarky (25c); currency union in
complete markets (25b); currency union in financial autarky (25d); and labor mobility (25e).13
We ignore the constant terms and focus only on the exponents of Z. Details on how we compute
welfare analytically for each allocation are found in Appendix H. We assume technology is lognormally distributed in all countries, log(Zi ) ∼ N (0, σZ2 ), and is independent across time and
across countries.
log E {Uflex complete } =
log E {Ufixed complete } =
log E {Uflex autarky } =
log E {Ufixed autarky } =
log E {Ulabor mobility } =
(γ − 1)(1 − σ)(1 + ϕ)2 2
σZ
(1 + γϕ)(σ + ϕ)
(γ − 1)(1 − σ)(1 + ϕ)(1 + ϕ − γϕ) 2
σZ
(σ + ϕ)
(γ − 1)(1 − σ)(1 + ϕ)2 [1 + ϕ + (γ − 1)(1 − σ)(σ + ϕ)] 2
σZ
(σ + ϕ)[1 − σ + γ(σ + ϕ)]2
(γ − 1)(1 − σ)(1 + ϕ) [1 − (γ − 1)(σ + ϕ)] 2
σZ
σ+ϕ
(γ − 1)(1 + ϕ) 2
σZ
σ+ϕ
(25a)
(25b)
(25c)
(25d)
(25e)
Using these expected utilities, it is straightforward to show that: (1) improved risk-sharing
always has positive welfare consequences, (2) flexible exchange rates always have positive welfare consequences, and (3) the gains from improved risk-sharing are always higher within a
currency union than outside of one.14 We also see that under unitary elasticity the expected
utility for all policy coalitions is identical. Recall that unitary elasticity leads to complete risksharing and eliminates wage rigidity via movements in the terms of trade. This explains in part
why the literature has found very small gains from international policy cooperation: the common assumption of unitary elasticity precludes gains from cooperation because terms of trade
movements provide cross-country risk-sharing and negate the influence of nominal rigidities.15
13
Allocations that eliminate wage rigidity (via flexible exchange rates or contingent fiscal policy) are denoted
by f lex, while those that do not are denoted by f ixed (i.e. currency unions). We assume that there is no
contingent fiscal policy in the currency union allocations, so that wages are rigid. Similarly, allocations with
complete international risk-sharing (via complete markets or a transfer union) are denoted by complete,
while autarky allocations with no risk-sharing are denoted by autarky.
14
See Appendix H equations (H.2a) – (H.2d) for calculation of explicit welfare differences.
15
Papers which employ unitary elasticity include Corsetti and Pesenti (2001, 2005), Obstfeld and Rogoff (2000,
2002), and Farhi and Werning (2012).
21
Tax Union
First, we compare the welfare of a country in a tax union with the welfare of a country outside of
a tax union, assuming the two countries are identical in all other respects.16 The log difference
in expected utility between a country inside a tax union and a country outside of a tax union
is:
log E {Utax union } − log E {Uno tax union } =
1−σ
σ+ϕ
log µγ =
1−σ
σ+ϕ
log
γ
γ−1
.
(26)
Equation (26) shows that the welfare gains from a tax union are decreasing in the elasticity
γ, the degree of substitutability between products across countries. As goods become closer
substitutes, country level monopoly power falls and the distortionary impact of the terms of
trade externality decreases due to the declining markup on exports. In the limit, as γ → ∞
and goods become perfect substitutes, the terms of trade markup will go to zero, and a tax
union will be unnecessary. On the other hand, as the Armington elasticity decreases, countries
gain a higher degree of monopoly power over their export good and thus increase their terms
of trade markup µγ . In the limit, as γ → 1, the terms of trade markup approaches infinity and
the benefits of a tax union dwarf the gains from other policy measures.
Figure 3 illustrates the importance of a tax union for low values of the Armington elasticity
and plots the loss in consumption from the terms of trade distortion relative to the Armington
elasticity. The figure makes it clear that as γ → 1, a tax union becomes imperative. In
the absence of a tax union, optimizing non-cooperative fiscal authorities charge extremely
large markups on their export good, leading to a dismal equilibrium for all countries. Noncooperative fiscal policy inflicts extremely large welfare losses on other countries under low
Armington elasticity. It is important reemphasize that the benefits of a tax union are steady
state benefits because the terms of trade markup is itself a steady state markup. Business cycle
fluctuations and shocks have no bearing on the welfare gains from a tax union. As the gains
from a tax union are a steady state phenomenon they cannot be analyzed in a log-linear model.
Transfer Union, Flexible Wages and Labor Mobility
We now turn to the welfare gains achieved through improved risk-sharing (via a transfer union
or deeper financial integration), the elimination of wage rigidity (via optimal monetary policy
outside of a currency union or contingent fiscal policy within a currency union) and labor
mobility. In what follows, we assume the presence of a tax union, which removes the constant
terms of trade markup.
Figure 4 plots the consumption in each allocation as a percentage of the Pareto optimum.
The negative impact of the wage rigidity distortion dominates the negative impact of financial
16
They differ only in the fact that one country faces a terms of trade markup in it’s steady state allocation (the
country outside of a tax union) and the other does not (the country within a tax union).
22
autarky, as both fixed wage allocations perform quite poorly relative to the flexible wage allocations, particularly as the degree of substitutability increases. The relative similarity of all
flexible wage allocations (Flex Complete, Flex Autarky, and Labor Mobility) is quite striking.
Even in financial autarky, flexible wages enable households to stabilize consumption with small
movements in their labor hours. The benefit of consumption risk-sharing is thus very small
when wages are flexible. The gains from flexible wages approach 2% of permanent consumption under complete risk-sharing and 4% of permanent consumption under financial autarky for
γ = 10. On the other hand, the welfare gains from perfect risk-sharing via a transfer union or
complete markets equal 2% of permanent consumption within a currency union when γ = 10.
Are countries better off in a currency union?
One of the arguments in support of a currency union, advanced by Mundell (1961, 1973) among
others, is that the formation of such a union leads to deeper financial integration and improves
cross-country risk-sharing. Using this logic, we conduct a thought experiment on the potential
benefits of a currency union. We compare the welfare of a country outside a currency union in
financial autarky with a member of a currency union in complete markets.
Is a country better off with a flexible exchange rate and no risk-sharing, or in a currency
union with perfect risk-sharing?17 For plausible calibrations, welfare is higher in the former.
When households are completely risk neutral (σ = 0), they always prefer to be outside of a
currency union in financial autarky. As households become more risk averse, they may prefer a
country that is a member of a currency union with full risk-sharing.
But even in the limit, as
σ → ∞, households prefer a currency union only if γ ∈ 1, 1+2ϕ
. For standard calibrations of
ϕ
ϕ, this means households will prefer a currency union when γ is between 1 and 2. So there is
a very small range of γ for which extremely risk averse households are better off in a currency
union in complete markets than outside of one in financial autarky. Even under extreme risk
aversion, deeper financial integration is not worth the loss of independent monetary policy.
Why is welfare higher under flexible exchange rates and financial autarky? Assume country
i is a member of a currency union and is hit with a negative productivity shock. Wage rigidity
will force its producers to charge a higher price to stay afloat. With a flexible exchange rate,
the higher domestic price would be offset by a depreciated currency, but in a currency union
this effect is absent. Given the higher price, consumers in country i and in other countries
will switch to cheaper substitutes. If the elasticity of substitution is very high, demand for
country i0 s good will collapse, and country i will produce almost nothing. If markets within the
currency union are complete or a transfer union is in place, consumption must be equal across
countries. However, only a few countries will produce any output, and households in those few
countries will have to work long hours to supply goods for the whole currency union. As a
17
The answer depends on the degree of risk aversion as defined by σ, the inverse Frisch elasticity of labor supply
ϕ, as well as the Armington elasticity γ. Details are found in Appendix I. Here we focus on the intuition.
23
result, average consumption and welfare will fall. This effect is exacerbated as goods become
closer substitutes. In the limit, when goods are perfect substitutes (γ = ∞) and shocks are
asymmetric, only one country in the currency union will produce any output, and consumption
and welfare will equal zero for all countries in the union.
On the other hand, when substitutability is close to one, the welfare losses from wage rigidity
and the gains from risk-sharing go to zero. Terms of trade movements will provide risk-sharing
and insulate economies from the negative impact of asymmetric productivity shocks and nominal rigidities. In this case, a country will be indifferent between remaining outside a currency
union in financial autarky and joining a currency union with full risk-sharing.
In reality of course, membership in a currency union does not guarantee perfect risk-sharing
through access to complete markets, nor does lack of membership in a currency union prevent
countries from accessing international financial markets. Whether countries enjoy some degree
of cross-border risk-sharing seems to be largely unrelated to their membership in a currency
union, although it is true that the introduction of the euro led to an increase in cross-border
lending within the euro area, as well as an initial convergence of borrowing rates within the
union. However, from a theoretical standpoint it is hard to argue that the potential risk-sharing
benefits of a currency union outweigh the loss of independent monetary policy.
7 Welfare Analysis in the Extended Model
In this section we relax some earlier simplifying assumptions and conduct welfare analysis in
a model with home bias, Calvo wage rigidities and incomplete markets. The model presented
here, which we refer to as the “Extended Model,” is identical in all other respects to the
closed-form model described in Section 2 and is laid out in detail in Appendix J.
Calibration
As in the closed-form model, we calibrate the extended model at quarterly frequency. All
parameter values are found in Table 1. The elasticity of substitution between home and foreign
products is defined by η, while the elasticity of substitution between the goods of different
countries remains γ. The relative weight of these goods in the consumption basket is defined
by the degree of home bias, 1−α. When α = 0, households only consume domestic goods, while
when α = 1, the economy is fully open and households will consume a basket made up entirely
of imports from all other countries in the world. The strength of wage rigidity is defined by
θW , the fraction of households who are able to reset wages in each period. We consider three
primary settings for Calvo wage rigidity: flexible wages (θW = 0), low wage rigidity (θW = 0.75)
which implies that the average household resets wages once every four quarters, and high wage
rigidity θW = 0.87 which implies that the average household resets wages every two years.18
18
We take θW = 0.75 as a conservative estimate of wage rigidity from Basu, Barattieri and Gottschalk (2014),
who find strong empirical evidence for θW in the range of 0.75 and 0.8 using U.S. micro data. Even
24
Tax Union
First, we analyze the welfare gains from a tax union by comparing the difference in steady state
consumption between a set of countries outside of a tax union with those inside a tax union.
We set η = 4 to match the average elasticity between home and foreign goods for the euro
area countries from Table 4. Figure 5 plots the loss in permanent consumption from the terms
of trade externality as a function of openness and the Armington elasticity. The distortionary
impact of the terms of trade markup is increasing in both openness and the Armington elasticity.
One can see this by
examining the optimal non-cooperative income tax rate in the extended
γ−1+(1−α)η
model, τi = 1 − µε γ−(1−α)(1−η)
, which we derive in Appendix K. This tax rate is increasing
in openness and both elasticities, η and γ.
As the degree of home bias increases, self-interested fiscal authorities find it less desirable
to impose a large markup on their export good. Why? Because the same good makes up an
increasingly significant portion of the home consumption basket. A tradeoff is thus introduced
between exploiting the terms of trade and driving up the price of the home good for domestic
consumers. This is a result of the law of one price: if the fiscal authority taxes workers in order
to reduce supply and increase the price of its unique final good, households in all countries will
suffer but home households will suffer more because the home good makes up (1-α) fraction of
the consumption basket. In contrast, when economies are completely open, home households
consume measure zero of their domestic good, and the non-cooperative fiscal authority is no
longer concerned about reducing home welfare through the terms of trade markup. As we
saw earlier in the closed-form version of the model, the optimal income tax rate converges to
τi = 1 −
γ
γ−1
when economies are completely open. Within a tax union, the optimal tax rate
remains τi = 1 − µε .
Why does the welfare loss from the terms of trade externality increase in the degree of openness? When an economy is more open, imports make up a larger portion of the consumption
basket. These imports are subject to a terms of trade markup. As a result, the more a country
needs to import, the greater the negative welfare impact of the terms of trade externality. In
addition, as goods become less substitutable (as the Armington elasticity decreases), fiscal authorities have an incentive to impose larger terms of trade markups on their exports through
income tax setting. Country-level monopoly power is thus increasing in both openness and the
Armington elasticity.
θW = 0.87 is a relatively conservative parameterization given recent estimates by Schmitt-Grohe and Uribe
(2012). They find very strong downward wage rigidity in a number of countries in Europe from 2008-2011,
including Greece, Portugal and Spain. Although wages are more flexible in the upward direction, our focus
here is on the negative effect of downward wage rigidity and the large welfare losses that accrue in a currency
union under this scenario. Cacciatore, Ghironi and Fiori (2015) show that labor market reforms which reduce
downward wage rigidity in a monetary union provide large welfare gains.
25
Transfer Union
Next, we calculate the welfare losses from business cycle fluctuations in financial autarky and
incomplete markets. We follow Lucas (2003) and estimate the utility from a deterministic
consumption path and a risky consumption path with the same mean. We then calculate
the amount of consumption necessary to make a risk averse household indifferent between the
deterministic and risky consumption streams. In what follows we assume that there is no steady
state terms of trade markup.
Figure 6 plots the loss in permanent consumption from business cycle fluctuations in financial
autarky for high wage rigidity (θW = 0.87). Figure 6 shows that home bias lowers welfare for
every value of the Armington elasticity. In other words, home bias exacerbates the negative
welfare impact of wage rigidity in the absence of risk-sharing. Our closed-form model thus
provides a lower bound estimate of the welfare benefits of a transfer union under financial
autarky. The losses in permanent consumption in financial autarky in a currency union are
as high as 16% when economies are completely open (α = 1) and 17% under full home bias
(α → 0). The Armington elasticity is far more important than consumption home bias in
determining welfare losses: the losses in permanent consumption can be as low as 0% for low
elasticity (γ = 1) and as high as 17% for high elasticity (γ = 10).
Why does home bias increase the welfare losses from business cycle fluctuations in financial
autarky? Consider the following thought experiment. Assume that wages are completely rigid,
that α = 0.01 so that 1% of consumption always goes to imports and the remainder goes towards
home goods, and that home and foreign consumption baskets are perfect complements. Under
financial autarky in a currency union, the cash value of imports must equal the cash value
of exports. Therefore, total consumption is equal to the value of exports multiplied by 100.
Also, when the home and foreign consumption baskets are perfect complements, fluctuations
in total household consumption are equal to fluctuations in export revenues. This effect is
only strengthened when home and foreign goods are imperfect complements or substitutes.
A negative shock that raises the price of the home good will lead domestic households to
substitute home goods for foreign goods in their consumption basket, increasing the share of
imports in total consumption. But in financial autarky the value of imports must equal the
value of exports, and exports are now uncompetitive on the world market due to the rise in
price caused by the negative shock. As a result, total home consumption must fall.
Figure 7 plots the loss in permanent consumption from business cycle fluctuations in incomplete markets. The ability to trade bonds greatly improves welfare for countries in a currency
union who are exposed to asymmetric shocks. Different from financial autarky, an increase in
home bias improves the ability of countries to stabilize business cycles when markets are incomplete. If a country is completely open, bonds allow households to stabilize consumption but
not hours worked, because exports become uncompetitive following a negative technology shock
26
and wages are rigid. When home bias increases, stabilization of both consumption and hours
worked is possible, because firms supply goods mainly to the home market. In contrast, under
financial autarky more home bias is bad for welfare because consumption is never stabilized.
As a robustness check, we plot the loss in permanent consumption from business cycle fluctuations for varying levels of Calvo wage rigidity in Figure 8. We set consumption home bias
equal to the euro area average of 65% (i.e. (1 − α) = 0.65). Again, we see that it is not only
wage rigidity that leads to large welfare losses, but the combination of wage rigidity and a
high Armington elasticity. Simply put, when a country in a currency union produces exports
that are easily substitutable, the welfare consequences are dramatic. When wages are quite
rigid, as the evidence in Schmitt-Grohe and Uribe (2011) suggests for some European countries, the losses in permanent consumption are small for low values of elasticity, but approach
80% for high values of elasticity. Even for conservative estimates of wage rigidity, the losses in
permanent consumption are quite large when the Armington elasticity is high.
Overall our closed-form results are robust to the inclusion of consumption home bias, incomplete markets and Calvo rigidities. The Armington elasticity remains an essential parameter in
the optimal design of a fiscal union. A tax union is more important when exports are imperfect
substitutes, while a transfer union is more important when exports are close substitutes. We
find that consumption home bias increases the need for a transfer union in financial autarky, as
the losses from incomplete risk-sharing rise with 1 − α. On the other hand, home bias reduces
the need for a tax union, as monopoly power at the export level declines when domestic goods
make up a larger percentage of the domestic consumption basket. In the limit, when countries
only consume domestic goods, there are no terms of trade externalities. In addition, home
bias strengthens the distortionary impact of wage rigidity, raising the importance of contingent
fiscal policy or labor mobility within a currency union. Our welfare analysis also demonstrates
that under the commonly assumed Cole-Obstfeld calibration, which sets σ = η = γ = 1, the
welfare losses from business cycle fluctuations, and thus the gains from a transfer union, are
extremely small. We nest the Cole-Obstfeld calibration as a special case here, and show that
the the distortionary impact of imperfect risk-sharing and wage rigidity is much larger as the
Armington elasticity moves away from unity and goods become more substitutable.
Welfare Losses for Country-Specific Elasticity Estimates
In Table 2 we examine the steady state losses arising from terms of trade externalities for
particular countries in Europe using the elasticity estimates calculated by Corbo and Osbat
(2013) and Imbs and Méjean (2010). These are two of the only papers to estimate the aggregate
elasticity of substitution between home and foreign products (η) and the aggregate elasticity
of substitution between the products of different countries (γ). Most trade papers focus only
on sector-specific estimates. We find that more open countries with low elasticities (e.g. the
Czech Republic and the Netherlands) gain the most from the presence of a tax union.
27
Next, we compute the welfare losses for a number of European countries under financial
autarky, incomplete markets, complete markets and the optimal transfer union resulting from
1% technology shocks. We consider three different values for Calvo wage rigidity in our computations: high wage rigidity (θW = 0.87), low wage rigidity (θW = 0.75) and flexible wages
(θW = 0). Table 3 reports the loss in permanent consumption for the country-specific elasticity estimates of Corbo and Osbat (2013), while Table 4 reports the results for the countryspecific elasticity estimates of Imbs and Méjean (2010). The results are particularly striking
for countries in financial autarky, where the losses are as high as 7.22% (Greece). Access to a
non-contingent bond for households significantly improves welfare, as permanent consumption
losses drop to a range of 1.19% (Slovakia) to 2.88% (Austria) under high wage rigidity. Moving
from incomplete markets to complete markets yields a smaller welfare gain of between 0.5 to 2
percent of permanent consumption for most countries in the sample. A transfer union is the
most efficient allocation, as losses range from 0% (Italy) to 0.56% (Czech Republic).
In Table 5 we compute the welfare losses in incomplete markets resulting from productivity
shocks calibrated to match the volatility and autocorrelation of output in the data. The autocorrelation of HP-filtered output (ρY ) in the data is approximately 0.9998. We conservatively
set ρZ = 0.99 for our simulations which implies ρY = 0.99 in the model; if we increase ρZ , the
welfare losses also increase. The volatility of productivity shocks (σZ,1 , σZ,2 ) are calibrated to
match the volatilities of HP-filtered output (σY,1 , σY,2 ). We conduct welfare analysis for two
scenarios. In Scenario 1 we take the volatility of output (σY,1 ) from HP-filtered GDP data
for each country with no adjustments and calibrate productivity shocks (σZ,1 ) to match the
output volatilities from the data. In Scenario 2 we compute the volatility of output (σY,2 ) from
HP-filtered GDP data for each country after subtracting euro area GDP in order to construct a
valid measure of asymmetric productivity shocks. We then calibrate the volatility of asymmetric productivity shocks (σZ,2 ) to match the volatility of asymmetric output in the data (σY,2 ).
The volatility of output resulting from asymmetric shocks (σY,2 ) is roughly half of the overall
volatility of output for each country (σY,1 ). Scenario 1 may be thought of as a set of upper
bound estimates and Scenario 2 as a set of lower bound estimates of empirically plausible welfare losses resulting from the absence of perfect cross-country risk sharing in a monetary union,
where a common central bank cannot respond to asymmetric shocks across countries. The
countries with the largest to gain from a transfer union are the smaller “periphery” countries,
including Greece, Hungary, the Czech Republic, Portugal and Slovakia.
Our results on the welfare gains of a transfer union over and above the complete markets
allocation match the findings of Farhi and Werning (2013), who demonstrate that the privately
optimal allocation under complete markets is inefficient and can be improved upon by government intervention in the form of fiscal transfers between countries. Here we go one step further
and explicitly quantify the welfare gains from a transfer union. These results confirm that the
28
potential gains from international fiscal cooperation are economically significant for a number
of countries in the euro area, but particularly for countries that lose access to international
financial markets and have high trade elasticities. Greece is a prime example of both.
8 Conclusion
We provide a unique perspective on the welfare gains that would result from the construction
of a fiscal union in the euro area. We derive a novel, global, closed-form solution for an open
economy model in which currency union members face limited cross-country risk-sharing, selfinterested fiscal authorities and wage rigidity. Our novel closed-form solution allows us to
examine the optimal structure of a fiscal union and analytically calculate the exact welfare
gains that would result from a fiscal union for a wide range of scenarios. We confirm our
analytical findings in a larger model that incorporates consumption home bias and Calvo wage
rigidities.
In our analytical framework we prove that the optimal design of a fiscal union depends
crucially on the Armington elasticity, which defines the degree of substitutability between the
products of different countries. When substitutability is low (around one), risk-sharing occurs
naturally via terms of trade movements so that a transfer union is unnecessary. Terms of trade
externalities are large however, and optimal policy will implement a tax union to prevent terms
of trade manipulation. The welfare gains from a tax union can be as high as one percent of
permanent consumption for standard calibrations. When substitutability is high (above one),
risk-sharing no longer occurs naturally via terms of trade movements. If financial markets do
not provide cross-country risk-sharing, there is a role for a transfer union to provide insurance
against idiosyncratic shocks. The relative importance of a transfer union increases as goods
become more substitutable. The welfare gains from a transfer union are negligible outside of a
currency union, but are significant for countries inside the euro area.
The euro area, a currency union with self-interested fiscal authorities, low cross-border risksharing and low labor mobility, would benefit immensely from a fiscal union between member
states. Although it may be difficult to achieve, our findings suggest that the preservation of
the currency union depends on it.
29
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33
Table 1:
Parameter Value
σ
2
ϕ
3
χ
1
ε
6
θW
Varies
β
0.99
γ
Varies
α
Varies
η
1
ρZ
0.95
σZ
0.01
Calibration of the Closed-Form and Extended Models
Description
Risk aversion parameter
Inverse labor supply elasticity (Gali and Monacelli (2005, 2008))
Following Gali and Monacelli (2005, 2008)
Elasticity between different types of labor
Calvo parameter for wage rigidity
Household discount factor
Armington elasticity
Openness in extended model
Elasticity between home and foreign goods in extended model
Persistence of technology shock in extended model
Standard deviation of technology
Table 2: Losses in Permanent Consumption from Terms of Trade Externalities
Austria
Czech
Denmark
Finland
France
Germany
Greece
Hungary
Italy
Netherlands
Portugal
Slovakia
Spain
Sweden
UK
European Avg
α
0.55
0.70
0.40
0.32
0.31
0.34
0.35
0.49
0.22
0.62
0.41
0.30
0.30
0.42
0.37
0.35
Corbo & Osbat (2013)
η
γ
Loss in percent
4.5 3.8
0.114
3.4 3.8
0.273
3.3 3.4
0.075
3.5 3.4
0.041
3.7 3.8
0.031
3.7 4.3
0.033
2.9 4.1
0.045
3.3 4.2
0.090
3.2 3.2
0.021
3.5 3.5
0.218
3.3 3.9
0.065
3.7 3.9
0.028
3.4 3.2
0.040
4.2 4.5
0.046
2.9 3.0
0.084
3.5 3.7
0.046
Imbs and Méjean (2010)
η
γ
Loss in percent
1.9 3.9
0.180
3.5
3.1
3.5
4.8
2.4
3.9
3.4
3.5
3.8
4.2
3.1
3.8
0.042
0.041
0.041
0.029
0.186
0.014
3.6
3.2
3.5
3.2
3.2
3.3
4.9
2.7
4.4
4.0
3.6
3.8
0.044
0.051
0.025
0.067
0.060
0.046
Openness (α) is taken from Balta and Delgado (2013). The elasticity of substitution between home and foreign
products (η) and the elasticity of substitution between the products of different countries (γ) for European
countries is taken from Imbs and Méjean (2010).
34
Table 3: Losses in Permanent Consumption for Country-Specific Elasticity Estimates of Corbo & Osbat (2013)
Parameters
35
Austria
Czech
Denmark
Finland
France
Germany
Greece
Hungary
Italy
Netherlands
Portugal
Slovakia
Spain
Sweden
UK
Europ. Avg
α
0.55
0.70
0.40
0.32
0.31
0.34
0.35
0.49
0.22
0.62
0.41
0.30
0.30
0.42
0.37
0.35
η
4.5
3.4
3.3
3.5
3.7
3.7
2.9
3.3
3.2
3.5
3.3
3.7
3.4
4.2
2.9
3.5
γ
3.8
3.8
3.4
3.4
3.8
4.3
4.1
4.2
3.2
3.5
3.9
3.9
3.2
4.5
3.0
3.7
High Wage Rigidity (θW = 0.87)
Fin.
Incomp. Complete Transfer
Autarky Markets Markets
Union
6.05
2.88
2.09
0.37
5.13
2.86
2.21
0.56
5.48
1.88
1.08
0.06
5.83
1.64
0.81
0.02
6.38
1.82
0.95
0.03
6.75
2.18
1.27
0.07
6.06
1.90
1.06
0.04
6.06
2.62
1.80
0.23
5.67
1.01
0.30
0.00
5.07
2.55
1.85
0.34
5.97
2.18
1.34
0.10
6.50
1.81
0.93
0.03
5.61
1.44
0.65
0.01
6.97
2.77
1.87
0.21
4.84
1.45
0.71
0.02
6.05
1.90
1.05
0.05
Low Wage Rigidity (θW = 0.75)
Fin.
Incomp. Complete Transfer
Autarky Markets Markets
Union
2.76
1.41
1.08
0.33
2.40
1.41
1.13
0.47
2.54
0.94
0.58
0.06
2.68
0.82
0.44
0.02
2.88
0.91
0.51
0.03
3.02
1.08
0.68
0.07
2.76
0.95
0.57
0.04
2.76
1.29
0.94
0.21
2.61
0.51
0.17
0.00
2.38
1.26
0.96
0.30
2.73
1.08
0.71
0.09
2.92
0.90
0.50
0.02
2.59
0.72
0.35
0.01
3.09
1.36
0.97
0.20
2.29
0.73
0.39
0.02
2.76
0.95
0.56
0.04
Flex Wages (θW = 0)
Fin.
Incomp.
Autarky
Markets
0.032
0.011
0.029
0.010
0.033
0.014
0.034
0.016
0.035
0.016
0.035
0.015
0.035
0.015
0.033
0.013
0.036
0.019
0.029
0.010
0.034
0.014
0.035
0.016
0.035
0.017
0.034
0.013
0.033
0.015
0.034
0.015
We compute the welfare losses in percent from a one percent technology shock (σZ = 0.01, ρZ = 0.95). We follow Lucas (2003) and estimate the utility from
a deterministic consumption path and a risky consumption path with the same mean. We then calculate the amount of consumption necessary to make a risk
averse household indifferent between the deterministic and risky consumption streams. The result is the loss in permanent consumption in percentage points for
four scenarios: autarky, incomplete markets, complete markets, and a transfer union. There is no steady state terms of trade markup here. Openness (α) is taken
from Balta and Delgado (2009). The elasticity of substitution between home and foreign products (η) and the elasticity of substitution between the products of
different countries (γ) for European countries is taken from Table 4 and 5 of Corbo and Osbat (2013).
Table 4: Losses in Permanent Consumption for Country-Specific Elasticity Estimates of Imbs and Méjean (2010)
Parameters
36
Austria
Finland
France
Germany
Greece
Hungary
Italy
Portugal
Slovakia
Spain
Sweden
UK
European Avg
α
0.55
0.32
0.31
0.34
0.35
0.49
0.22
0.41
0.30
0.30
0.42
0.37
0.35
η
1.9
3.5
3.1
3.5
4.8
2.4
3.9
3.6
3.2
3.5
3.2
3.2
3.3
γ
3.9
3.4
3.5
3.8
4.2
3.1
3.8
4.9
2.7
4.4
4.0
3.6
3.8
High Wage Rigidity (θW = 0.87)
Fin.
Incomp. Complete Transfer
Autarky Markets Markets
Union
4.94
2.21
1.50
0.18
5.81
1.63
0.81
0.02
5.70
1.53
0.72
0.01
6.19
1.91
1.05
0.04
7.22
2.48
1.53
0.12
4.32
1.66
0.98
0.07
6.71
1.36
0.52
0.00
6.99
2.73
1.82
0.19
4.99
1.19
0.47
0.00
6.80
1.95
1.04
0.03
6.01
2.25
1.41
0.11
5.65
1.82
1.00
0.04
6.05
1.90
1.05
0.05
Low Wage Rigidity (θW = 0.75)
Fin.
Incomp. Complete Transfer
Autarky Markets Markets
Union
2.33
1.10
0.79
0.16
2.67
0.81
0.44
0.02
2.63
0.76
0.39
0.01
2.81
0.95
0.56
0.04
3.18
1.22
0.80
0.11
2.08
0.84
0.53
0.06
3.00
0.68
0.29
0.00
3.10
1.33
0.95
0.18
2.35
0.60
0.26
0.00
3.03
0.97
0.56
0.03
2.74
1.11
0.74
0.11
2.61
0.91
0.54
0.04
2.76
0.95
0.56
0.04
Flex Wages (θW = 0)
Fin.
Incomp.
Autarky
Markets
0.031
0.012
0.034
0.016
0.035
0.016
0.035
0.015
0.035
0.015
0.030
0.013
0.036
0.019
0.035
0.014
0.034
0.017
0.036
0.016
0.034
0.014
0.034
0.015
0.034
0.015
We compute the welfare losses in percent from a one percent technology shock (σZ = 0.01, ρZ = 0.95). We follow Lucas (2003) and estimate the utility from
a deterministic consumption path and a risky consumption path with the same mean. We then calculate the amount of consumption necessary to make a risk
averse household indifferent between the deterministic and risky consumption streams. The result is the loss in permanent consumption in percentage points for
four scenarios: autarky, incomplete markets, complete markets, and a transfer union. There is no steady state terms of trade markup here. Openness (α) is taken
from Balta and Delgado (2013). The elasticity of substitution between home and foreign products (η) and the elasticity of substitution between the products of
different countries (γ) for European countries is taken from Imbs and Méjean (2010).
Table 5: Losses in Permanent Consumption in Incomplete Markets from Productivity Shocks Calibrated to the Data
37
Austria
Czech
Denmark
Finland
France
Germany
Greece
Hungary
Italy
Netherlands
Portugal
Slovakia
Spain
Sweden
UK
European Avg
α
0.55
0.70
0.40
0.32
0.31
0.34
0.35
0.49
0.22
0.62
0.41
0.30
0.30
0.42
0.37
0.35
Data
σY,1
0.013
0.019
0.015
0.021
0.010
0.017
0.024
0.017
0.014
0.013
0.013
0.023
0.012
0.019
0.014
0.013
σY,2
0.005
0.010
0.007
0.010
0.005
0.006
0.024
0.010
0.004
0.005
0.011
0.014
0.007
0.009
0.008
0.000
η
4.5
3.4
3.3
3.5
3.7
3.7
2.9
3.3
3.2
3.5
3.3
3.7
3.4
4.2
2.9
3.5
Corbo &
γ
σZ,1
3.8 0.010
3.8 0.015
3.4 0.013
3.4 0.018
3.8 0.008
4.3 0.014
4.1 0.020
4.2 0.013
3.2 0.012
3.5 0.010
3.9 0.010
3.9 0.019
3.2 0.010
4.5 0.014
3.0 0.011
3.7 0.010
Osbat
Losses
1.087
2.065
1.318
2.500
0.576
1.790
3.393
1.767
0.954
0.987
1.006
3.020
0.848
2.168
0.936
0.978
(2013)
σZ,2
0.004
0.007
0.006
0.008
0.004
0.005
0.020
0.008
0.003
0.004
0.009
0.012
0.006
0.007
0.006
0.000
Losses
0.147
0.514
0.291
0.562
0.142
0.207
3.340
0.629
0.079
0.141
0.750
1.200
0.291
0.505
0.299
0.000
η
1.9
Imbs & Méjean (2010)
γ
σZ,1 Losses σZ,2 Losses
3.9 0.011 0.982 0.004 0.132
3.5
3.1
3.5
4.8
2.4
3.9
3.4
3.5
3.8
4.2
3.1
3.8
0.018
0.008
0.014
0.019
0.014
0.011
2.502
0.542
1.722
3.681
1.458
1.050
0.008
0.004
0.005
0.019
0.008
0.003
0.561
0.133
0.198
3.633
0.521
0.087
3.6
3.2
3.5
3.2
3.2
3.3
4.9
2.7
4.4
4.0
3.6
3.8
0.010
0.019
0.010
0.015
0.011
0.011
1.081
2.600
0.941
2.016
1.030
0.982
0.009
0.012
0.006
0.007
0.006
0.000
0.811
1.033
0.323
0.470
0.326
0.000
Here we compute the welfare losses in incomplete markets resulting from productivity shocks calibrated to match the volatility and autocorrelation of output in
the data. The autocorrelation of HP-filtered output (ρZ ) in the data is approximately 0.9998, so we set ρZ = 0.99 for the simulations above. The volatility of
productivity shocks (σZ,1 , σZ,2 ) are calibrated to match the volatilities of HP-filtered output (σY,1 , σY,2 ). In Scenario 1 we take the volatility of output (σY,1 ) from
HP-filtered GDP data for each country with no adjustments and calibrate productivity shocks (σZ,1 ) to match the output volatilities from the data. In Scenario
2 we compute the volatility of output (σY,2 ) from HP-filtered GDP data for each country after subtracting euro area GDP in order to construct a valid measure
of asymmetric productivity shocks. We then calibrate the volatility of asymmetric productivity shocks (σZ,2 ) to match the volatility of asymmetric output in the
data (σY,2 ). We follow Lucas (2003) and estimate the utility from a deterministic consumption path and a risky consumption path with the same mean. We then
calculate the amount of consumption necessary to make a risk averse household indifferent between the deterministic and risky consumption streams: the result is
the losses in permanent consumption in percentage points. There is no steady state terms of trade markup here. Openness (α) is taken from Balta and Delgado
(2009). The elasticity of substitution between home and foreign products (η) and the elasticity of substitution between the products of different countries (γ) for
European countries is taken from Corbo and Osbat (2013) and from Imbs and Méjean (2010).
Figure 1: Model Timeline
-1
0
1
Policymaker
declares
fiscal and
monetary
policy
Household
makes
state-contingent
plan
Period 1
shocks are
realized
t
2, 3, ..., t-1
Period t shocks
are realized
Figure 2: Annual Cross-Border Mobility, % of Population (2010)
EU27: across countries
Canada: between Quebec and 9
other provinces/territories
EU15: across regions within
countries
Canada: across 10
provinces/territories
US: across 4 main regions
Australia: across 8 states/territories
US: across 50 states
0
1
2
3
Source: OECD. This figure plots the percentage of the population that moved across national
borders (for the EU27), or across regions within the same country (all others).
38
Figure 3: Welfare Losses From Terms of Trade Externalities in the Closed-Form Model
Consumption as % of Pareto Optimum
100
98
96
94
Flex Complete
Fixed Complete
Flex Autarky
Fixed Autarky
Labor Mobility
92
90
1
2
3
4
5
6
7
Armington Elasticity (γ)
8
9
10
Consumption as % of Pareto Optimum
Figure 4: Welfare Losses from Business Cycle Fluctuations in the Closed-Form Model
100
99
98
97
Flex Complete
Fixed Complete
Flex Autarky
Fixed Autarky
Labor Mobility
96
95
2
4
6
8
10
12
14
Armington Elasticity (γ)
39
16
18
20
Losses In Permanent Consumption
Figure 5: Welfare Losses from Terms of Trade Externalities
10
8
6
4
2
0
1
0.7
2
0.6
0.5
3
γ
0.4
α
4 0.3
Losses In Permanent Consumption
Figure 6: Welfare Losses from Business Cycle Fluctuations in Financial Autarky
20
15
10
5
0
10
8
1
6
4
2
Armington Elasticity (γ)
0 0
40
0.2
0.4
0.6
0.8
Openness (α)
Losses In Permanent Consumption
Figure 7: Welfare Losses from Business Cycle Fluctuations in Incomplete Markets
8
6
4
2
0
10
8
1
6
4
2
0 0
Armington Elasticity (γ)
0.4
0.2
0.6
0.8
Openness (α)
Losses In Permanent Consumption
Figure 8: Welfare Losses from Business Cycle Fluctuations in Financial Autarky for Different
Levels of Wage Rigidity
10
8
6
4
2
0
10
8
0.95
6
0.9
4
0.85
2
Armington Elasticity (γ)
0.8
00.75
Calvo Wage Rigidity (θW )
41
For Online Publication: Technical Appendix
A Global Social Planner
We begin by describing the maximization problem faced by a benevolent global social planner
who has complete control over the monetary and fiscal policies of each country. Since the
economies in our model are identical ex-ante, the global social planner will maximize a weighted
utility function over all i countries,
Z 1 1−σ
Ni1+ϕ
Ci
−χ
di,
(A.1)
1−σ
(1 + ϕ)
0
subject to the consumption basket and the aggregate resource constraint:
Z
Ci =
1
γ−1
γ
γ
γ−1
cij dj
,
Z 1
Yi =Ni Zi =
cji dj.
(A.2)
0
(A.3)
0
Proposition 9 The global social planner will maximize (A.1), subject to (A.2) and (A.3). The
solution to the global social planner problem is:
1
σ+ϕ
1+ϕ
1
Ci =
Zwσ+ϕ ,
χ
1
σ+ϕ
(1−γσ)(1+ϕ)
γ−1
1
Ni =
Zw(1+γϕ)(σ+ϕ) Zi1+γϕ ,
χ
1+γϕ
Z 1 (γ−1)(1+ϕ) (γ−1)(1+ϕ)
1+γϕ
Zi
Zw =
di
.
(A.4a)
(A.4b)
(A.4c)
0
Proof See Appendix C in Dmitriev and Hoddenbagh (2013). The global social planner solution characterizes the Pareto efficient allocation. From (A.4a),
we see that domestic consumption depends on average world technology Zw , which is a constant
because technology shocks are identically and independently distributed. Consumption is thus
stabilized at the country level, insuring risk averse households from consumption risk. On the
other hand, (A.4b) shows that labor will fluctuate with technology shocks, increasing in booms
and decreasing in recessions. There are no distortions in the efficient allocation: wage ridigity,
incomplete risk-sharing, and the terms of trade externality are absent.
The efficient allocation provides a natural benchmark to evaluate different policy regimes.
In Sections 3 and 4 we look closely at optimal monetary and fiscal policy in non-cooperative
and cooperative settings and see what conditions are necessary to replicate the Pareto efficient
allocation outside of and within a currency union. In Section 5 we study the effect of labor
mobility and see if it can replicate the Pareto efficient allocation.
B Proof of Proposition 1 and 2: Flexible Exchange Rate Allocations
Non-cooperative central banks will maximize their objective function (14) subject to (1), (8), (9)
and (12) in complete markets and (1), (8), (9), and (13) in financial autarky. The Lagrangian
42
for the non-cooperative and cooperative cases is:
1−σ 1+ϕ Et−1 Nit1+ϕ
Et−1 Cit1−σ
χµε
−χ
+ λi Et−1 Cit
−
Et−1 Nit
L=
1−σ
1+ϕ
1 − τi
γ−1 γ−1 γ−1
γ−1
1
1
γ
γ
Using Cit = Cwt Et−1 Nit γ Zit γ
for complete markets, or Cit = Cwt
Nit γ Zit γ for financial
autarky, we can take the first order condition with respect to Nit .19 The FOC will be identical
in both cases.
 γ−1 1 
γ
γ − 1  Yit γ Cwt
µ
1 1+ϕ
∂L
ε
−σ
−χ 1+λ
= Cit (1 + λi (1 − σ))
(1 + ϕ)
N
=0
∂Nit
γ
Nit
1 − τi
Nit it
In equilibrium, this equals:
1=χ
|
ε (1+ϕ)
1 + λi µ1−τ
i
λi (1−σ)(γ−1)
1+
γ
{z
!


Nit1+ϕ
γ−1
γ
Cit−σ Yit
1
γ
.
(B.1)
Cwt
}
Constant
This equation holds in both complete markets and financial autarky, and differs from the flexible
price equilibrium only by the constant term. However, subject to labor market clearing, this
constant will coincide with the flexible price equilibrium. The flexible wage equilibrium in
complete markets and financial autarky is found by taking expectations out of the labor market
clearing condition (9) and substituting in goods market clearing (12):
1=
χµε
1 − τi
1+ϕγ
γ
Yit
1
.
(B.2)
γ
Zit1+ϕ
Cit−σ Cwt
For complete markets, we can express output as a function of technology and a constant term
γ(1+ϕ)
1+γϕ
by substituting (12) into (B.2): Yit = Ai Zit
. (We can do the same for exercise for autarky
by substituting (13) into (B.2), but leave that to the reader). Using this expression for output,
consumption in complete markets in country i can be expressed as
(γ−1)(1+ϕ) γ−1
1
γ
γ
Cit = Ai Cwt Et−1 Zit 1+γϕ
.
(B.3)
Now substitute (B.3) back into the flexible price equilibrium (B.2)
σ (γ−1)σ
(γ−1)(1+ϕ) σ 1+ϕγ
−1
χµε
γ
γ
1=
Cwt Ai
Et−1 Zit 1+γϕ
Ai γ Cwtγ ,
1 − τi
19
(B.4)
Remember that we are optimizing given the fact that state st is realized. Expectations in our context thus
refer to a summation over
P all possible states multiplied by the probability of each state occuring. For
1−σ
example, Et−1 {Cit
} = st Ci1−σ (st )Pr(st ).
43
and rearrange and solve for Ai :
"
Ai =
1 − τi
χµε
γ
(γ−1)(1+ϕ)
1+γϕ
1−σ
Et−1 Zit
Cwt
1
−σγ # 1−σ+γ(ϕ+σ)
.
(B.5)
Now, substitute the solution (B.5) into (B.3):
"
Cit =
Using the fact that Cwt =
χµε
1 − τi
R1
0
1−γ
1
(γ−1)(1+ϕ) 1+γϕ # 1−σ+γ(ϕ+σ)
ϕ+1
Cwt
Et−1 Zit 1+γϕ
.
Cit di = Cit in equilibrium, and setting Zw =
R1
0
(B.6)
(γ−1)(1+ϕ)
1+γϕ
Zit
1+γϕ
(γ−1)(1+ϕ)
di
,
integrate (B.6) over all i and solve for consumption for country i in complete markets:
Cit =
1 − τi
χµε
1
σ+ϕ
1+ϕ
Zwσ+ϕ .
(B.7)
Solving for labor and output using (B.7) is a straightforward exercise. The solution to the
central bank’s problem in complete markets and financial autarky for cooperative and noncooperative equilibria coincides exactly with the flexible wage allocation.
Non-cooperative fiscal authorities will set a labor tax rate of τi = 1 − µµγε , introducing a terms
of trade markup to exploit their country-level monopoly power. C Proof of Proposition 3 and 4: Currency Union Allocations
Non-cooperative policymakers in a currency union in complete markets will maximize (17) by
choosing a non state contingent income tax rate, subject to (1), (8), (9), (12), and (18). From
(18) we can compute labor using Yit = Zit Nit :
Nit = AZitγ−1 .
(C.1)
Given the above, consumption will be
Z
Cit = Cwt = A
Zitγ−1 di
γ
γ−1
.
(C.2)
Using labor market clearing (9), and substituting in Yit , Cit , Nit expressed as functions of A and
Zit from above, we find:
R 1 (γ−1)(1+ϕ)
A1+ϕ 0 Zit
di
χµε
1=
(C.3)
γ(1−σ)
R 1 (γ−1)
1 − τi
γ−1
A1−σ 0 Zit di
Now we can solve for A:
A=
χµε
1 − τi
−1 Z
σ+ϕ
1
(γ−1)(1+ϕ)
Zit
0
−1 Z
σ+ϕ
di
0
44
1
Zitγ−1 di
γ (1−σ)
γ−1
σ+ϕ
.
(C.4)
Given this solution for the constant A, one can solve for Cit and Nit by substituting A into the
expressions above, resulting in (19a) for Cit and (19b) for Nit . The same exercise in financial
autarky will yield (20a) for Cit and (20b) for Nit . D Contingent Tax Policy
Up to this point we have assumed that tax policy is non-contingent, so that fiscal authorities
can only set a constant income tax rate. If we allow fiscal policymakers to adjust tax rates over
the business cycle, the objective function under flexible exchange rates is
1−σ
Nit1+ϕ
Cit
−χ
,
(D.1)
max max Et−1
τit
Nit
1−σ
1+ϕ
and within a currency union is
max Et−1
τit
N 1+ϕ
Cit1−σ
− χ it
1−σ
1+ϕ
.
(D.2)
As we showed in Proposition 1 and 2, when exchange rates are flexible national central banks
will mimic the flexible wage allocation and a constant labor tax rate will be optimal for both
contingent and non-contingent fiscal policymakers. The role of fiscal policy under flexible
exchange rates is simply to ameliorate the monopolistic markup on differentiated labor inputs
and impose a terms of trade markup in the non-cooperative case. In other words, contingent
fiscal policy is redundant when exchange rates are flexible because national central banks adjust
monetary policy over the business cycle to counteract wage rigidity.
In contrast, the role of fiscal policy in a currency union is twofold: to impose a terms of trade
markup, but also to eliminate wage rigidity at the national level resulting from asymmetric
shocks. Optimal contingent fiscal policy within a currency union will not set a constant tax
rate. Within a currency union the union-wide central bank has only one policy instrument
at its disposal and cannot offset the effect of asymmetric shocks across countries. Contingent
national fiscal policy can fill the void, setting domestic tax rates in each period to remove
domestic wage rigidity and mimic the flexible wage equilibrium. One already begins to see
that fiscal policy is more important within a currency union than outside of one. Given that
contingent fiscal policy is only necessary in a currency union, we ignore flexible exchange rate
allocations in Proposition 10.
Proposition 10 Contingent Fiscal Policy Contingent non-cooperative policymakers within
a currency union will maximize (D.2), subject to (9), (12), (1), (8), and (18) in complete
markets and subject to (9), (1), (8), (13) and (18) in autarky. The optimal allocation will
exactly coincide with (15a) in complete markets and (16a) in autarky, replicating the flexible
wage allocation with a terms of trade markup.
Proof
We now assume that fiscal policymakers can choose their income tax rate in each period. In
both complete markets and financial autarky, the fiscal authority in country i will maximize
utility in each period, choosing the optimal contingent labor tax τit , given the realization of Zit
in that period. To solve for the optimal allocation, follow the exact same steps as in Appendix
C, but use τit as the policy instrument rather than Nit . The optimal allocations will remove
the wage rigidity distortion and mimic the flexible exchange rate allocations, given by (15a) in
complete markets and (16a) in financial autarky. 45
The optimal contingent labor tax within a currency union will be equal to τit = 1 − µit . The
realized markup µit is defined by
M P Lit = µit · M RSit ,
where M P Lit = Zit is the marginal product of labor and M RSit is the marginal rate of
substitution. A positive productivity shock increases M P Lit , but since wages remain fixed,
there will be a rise in demand for labor. The rise in demand for labor will induce households
to work more hours and cause an even larger increase in M RSit . As a result, the markup will
be countercylical and the optimal contingent labor tax will be procyclical: taxes will increase
when productivity shocks are positive and decline when productivity shocks are negative.
Note that international policy cooperation is not necessary to eliminate the distortionary
impact of wage rigidity: non-cooperative contingent fiscal policy is all that is required. For the
remainder of the paper, we will assume that fiscal policy is non-contingent. However, keep in
mind that contingent fiscal policy within a currency union can play the same role as monetary
policy outside of a currency union, eliminating nominal rigidities and mimicking the flexiblewage allocation. While the ability of contingent fiscal policy to eliminate nominal rigidities has
been shown in other closed and open economy studies, we are the first to emphasize that the
distortionary impact of wage rigidity and hence the importance of contingent fiscal policy is
increasing in the Armington elasticity γ.
E Proof of Proposition 5: Tax Union
Under flexible exchange rates, policymakers in a tax union will maximize (21a) if they are not
in a transfer union or (21e) if they are in a transfer union. In a currency union, policymakers
in a tax union will maximize (21b) if they are not in a transfer union or (21f) if they are in a
transfer union. Policymakers face the constraints outlined in Propositions 1 (flexible exchange
rates and complete markets), 2 (flexible exchange rates and financial autarky), 3 (currency
union and complete markets) or 4 (currency union and financial autarky), respectively.
In the non-cooperative case, policymakers do not internalize the impact of their tax rate on
other countries. As a result, non-cooperative policymakers set τi to eliminate the markup on
domestic intermediates, µε , but also introduce a terms of trade markup, µγ , to take advantage
of the monopoly power they exercise over their unique export good. The solution to the noncooperative problem is τi = 1 − µµγε .
In a tax union policymakers do internalize the impact of their tax rate on other countries. The
solution to the cooperative problem defines the optimal tax rate within a tax union: τi = 1−µε .
The optimal tax rate in a tax union eliminates the markup on domestic intermediate goods
without imposing a terms of trade markup.
F Proof of Proposition 6: Transfer Union
In a transfer union, policymakers agree to send contingent cash payments across countries.
Under flexible exchange rates, policymakers in a transfer union will maximize (21c) if they are
not in a tax union or (21e) if they are in a tax union. In a currency union, policymakers in
a transfer union will maximize (21d) if they are not in a tax union or (21f) if they are in a
tax union. Policymakers face the constraints outlined in Propositions 1 (flexible exchange rates
and complete markets), 2 (flexible exchange rates and financial autarky), 3 (currency union
and complete markets) or 4 (currency union and financial autarky), respectively. Because
the transfers are state contingent, countries jointly agree on a state contingent plan to insure
households against idiosyncratic consumption risk resulting from asymmetric shocks. Outside
46
of a tax union, the solution to the transfer union optimization problem under flexible exchange
rates will replicate the complete markets allocation detailed in Proposition 1. Outside of a tax
union, the solution to the optimization problem in a currency union will replicate the complete
markets allocation detailed in Proposition 3. Within a tax union, the solution to the transfer
union optimization problem under flexible exchange rates will replicate the complete markets
allocation detailed in Proposition 1 without a terms of trade markup. Within a tax union, the
solution to the optimization problem in a currency union will replicate the complete markets
allocation detailed in Proposition 3 without a terms of trade markup.
G Proof of Proposition 8: Labor Mobility
In the presence of labor mobility, non-cooperative policymakers in a currency union in financial
autarky will face the following problem
max
τi
N 1+ϕ
C 1−σ
−χ
1−σ
1+ϕ
(G.1)
s.t.
W N (h) = CP
γ
Z 1 γ−1 γ−1
γ
cj dj
C=
(G.2a)
(G.2b)
0
W
= Zj
pj
1
Z 1
1−γ
1−γ
P =
pj dj
(G.2c)
(G.2d)
0
cj = Zj Nj
Z 1
Nj (h)dj
N (h) =
Z
0
1
Nj =
Nj (h)
ε−1
ε
(G.2e)
(G.2f)
ε
ε−1
(G.2g)
dj
0
where W is equalized across countries because of labor mobility. As each economy in the
currency union is hit with idiosyncratic shocks, labor will shift from low demand bust countries
to high demand boom countries.
FOCs:
p −γ
j
C
(G.3)
cj =
P
W
χµε
=
CσN ϕ
(G.4)
P
1 − τi
To solve for the optimal allocation, begin with (G.2d)
Z
P =
0
1
p1−γ
dj
j
1
1−γ
Z
=
0
1
W
Zj
1
! 1−γ
1−γ
47
dj
Z
=W
0
1
Zjγ−1 dj
1
1−γ
and solve for the real wage
Z
W
=
P
1
Zjγ−1 dj
1
γ−1
.
(G.5)
0
Now substitute this expression for the real wage into (G.2a):
W
C = N (h)
= N (h)
P
1
Z
Zjγ−1 dj
1
γ−1
1
Z
Z
Nj (h)dj
=
0
0
1
Zjγ−1 dj
1
γ−1
.
0
Take (G.3) and substitute in the expression for the real wage from (G.2c):
!γ
p −γ
Z
j
j
cj =
C= W
C,
P
P
(G.6)
and then substitute cj = Zj Nj in G.6
Nj =
W
P
−γ
Zjγ−1 C.
(G.7)
Integrating Nj over j will yield
Z
1
Nj dj = N = C
0
W
P
−γ Z
1
Z
Zjγ−1 dj
=C
1
Zjγ−1 dj
1
− γ−1
.
(G.8)
0
0
Using (G.4), we can substitute in our expression for N from (G.8) and our expression for W/P
from (G.5):
W
=
P
Z
1
Zjγ−1 dj
1
γ−1
=
0
χµε
1 − τi
C σ N ϕ.
Solving for C, we find:
C=
1 − τi
χµε
1
σ+ϕ
Z
1
Zjγ−1 dj
1+ϕ
(σ+ϕ)(γ−1)
.
(G.9)
0
To solve for Nj simply substitute (G.9) into (G.7):
Nj =
1 − τi
χµε
1
σ+ϕ
Z
1
Zjγ−1 dj
1+ϕ−γ(σ+ϕ)
(γ−1)(σ+ϕ)
Zjγ−1 .
(G.10)
0
As we’ve mentioned a number of times, the optimal tax rate outside of a tax union in the
decentralized Nash equilibrium will be τi = 1 − µµγε .
When labor can move freely across borders, the equilibrium allocation outside of a tax union
48
will be:
Ci =
Ni =
1
χµγ
1
χµγ
1
Z
σ+ϕ
1
Ziγ−1 di
1+ϕ
(σ+ϕ)(γ−1)
0
1
σ+ϕ
Z
1
Zjγ−1 dj
1+ϕ−γ(σ+ϕ)
(γ−1)(σ+ϕ)
Zjγ−1 .
0
This allocation holds under flexible exchange rates and within a currency union, in both complete markets and financial autarky. H Welfare Derivations
Below, we outline the steps necessary to derive the expected utility functions contained in
Section 6 of the paper. Here we only conduct the exercise for flexible exchange rates in complete
markets, but following the steps presented here will also yield the expected utility functions for
the other allocations.
1+γϕ
1 Z 1 (γ−1)(1+ϕ) (γ−1)(σ+ϕ)
1 − τi σ+ϕ
Zi 1+γϕ di
Cf lex,complete =
χµε
0
n
o
1
1 − τi
E {Uf lex,complete } =
−
E Cf1−σ
lex,complete
1 − σ µε (1 + ϕ)


1−σ
Z 1 (γ−1)(1+ϕ) (1+γϕ)(1−σ)
σ+ϕ

(γ−1)(σ+ϕ) 
1
1 − τi
1 − τi
Z 1+γϕ di
=
E
−

 0 i
1 − σ µε (1 + ϕ)
χµε
For normative analysis, we assume that technology is log-normally distributed and is independent across time and across countries: log Zit ∼ N (0, σZ2 ). The expectation above can then be
rewritten as:


Z 1 (γ−1)(1+ϕ) (1+γϕ)(1−σ)
(γ−1)(σ+ϕ) 
(γ−1)(1+ϕ) 2 (1+γϕ)(1−σ) 2
E
Zi 1+γϕ di
= e[ 1+γϕ ] (γ−1)(σ+ϕ) σZ
 0

=e
(γ−1)(1+ϕ)2 (1−σ) 2
σZ
(1+γϕ)(σ+ϕ)
.
Now, we insert this expression back into the original equation and get:
1 − τi
1
−
E {Uf lex,complete } =
1 − σ µε (1 + ϕ)
1 − τi
χµε
1−σ
σ+ϕ
e
(γ−1)(1+ϕ)2 (1−σ) 2
σZ
(1+γϕ)(σ+ϕ)
.
Taking logarithms, we can rewrite the log of expected utility as:
1
1 − τi
1−σ
1 − τi
(γ − 1)(1 + ϕ)2 (1 − σ) 2
−
+
log
+
σZ .
log E {Uf lex,complete } = log
1 − σ µε (1 + ϕ)
σ+ϕ
χµε
(1 + γϕ)(σ + ϕ)
(H.1)
Calculating the expected utility for the other coalitions simply requires that one follow the
steps outlined here. Notice that when we calculate welfare differences between allocations, the
first and second terms on the right hand side of equation (H.1) will cancel out, leaving only the
49
difference between the remaining term on the right hand side.
Using the expected utilities from (25a) – (25d), and the fact that any constant terms will
cancel out when subtracted from each other, we calculate the welfare differences for four scenarios: (1) complete markets vs. autarky for flexible wages; (2) complete markets vs. autarky
for fixed wages; (3) flexible vs. fixed wages for complete markets; and (4) flexible vs. fixed
wages for autarky. When comparing welfare across different allocations, it is important to keep
in mind that as risk-aversion decreases, (i.e. as σ → 1), the welfare differences expressed in
logarithms also decrease but the absolute values of utility increase. In other words, when risk
aversion is low, the welfare differences shown in (H.2a) – (H.2d) will shrink, but this does not
mean that the welfare differences are decreasing in absolute value.
σ(γ − 1)2 (1 − σ)(1 + ϕ)2
σ 2 (H.2a)
(σ + ϕ)(1 + γϕ)[1 − σ + γ(σ + ϕ)] Z
σ(γ − 1)2 (1 − σ)(1 + ϕ) 2
log E {Uf ixed,complete } − log E {Uf ixed,autarky } =
σZ
(H.2b)
σ+ϕ
γϕ2 (γ − 1)2 (1 − σ)(1 + ϕ) 2
log E {Uf lex,complete } − log E {Uf ixed,complete } =
σZ
(H.2c)
(1 + γϕ)(σ + ϕ)
(γ − 1)2 (1 − σ)(1 + ϕ)[γ(σ + ϕ) − σ] 2
log E {Uf lex,autarky } − log E {Uf ixed,autarky } =
σZ
1 + γ(σ + ϕ) − σ
(H.2d)
log E {Uf lex,complete } − log E {Uf lex,autarky } =
I Are countries better off in a currency union?
A country with a flexible exchange rate and no risk-sharing is better off than a country in a
currency union with perfect risk-sharing whenever
(1 + ϕ) [1 + ϕ + (γ − 1)(1 − σ)(σ + ϕ)] − (1 + ϕ − γϕ) [1 − σ + γ(σ + ϕ)]2 ≥ 0,
(I.1)
which can be rewritten in cubic form as
(
h
i
γ − 1 ϕ(σ + ϕ)2 (γ − 1)2 + (γ − 1) 2ϕ(σ + ϕ)(1 + ϕ) − (σ + ϕ)2
)
+ ϕ(1 + ϕ)2 + (1 − σ)(1 + ϕ)(σ + ϕ) − 2(σ + ϕ)(1 + ϕ)
The roots to this cubic equation are:
 2 2
3√
σ +2σ ϕ−ϕ2 +2σϕ2 −(σ+ϕ) 2 σ+ϕ+4σϕ+4σϕ2



2(σ 2 ϕ+2σϕ2 +ϕ2 )
1
γ=

3√

 σ2 +2σ2 ϕ−ϕ2 +2σϕ2 +(σ+ϕ) 2 σ+ϕ+4σϕ+4σϕ2
≥ 0.
(I.2)
(I.3)
2(σ 2 ϕ+2σϕ2 +ϕ2 )
where the first root is less than one, and the third root is greater than one. When γ is less than
one or greater than the third root expressed in (I.3), a country will be better off outside of a
currency union in financial autarky than as a member of a currency union in complete markets.
In the limiting case as households become completely risk neutral (σ → 0), the roots of (I.2)
50
will be

 −1
0
γ=

1
(I.4)
while in the opposite limiting case, as households become extremely risk averse (σ → ∞), the
roots of (I.2) will be

 0
1
(I.5)
γ=
 1+2ϕ
ϕ
Internationally traded goods are perfect complements as γ approaches zero and perfect substitutes as γ approaches infinity, so γ must be non-negative. We can safely ignore any negative
roots and focus only on positive roots. There is a very small window for which countries are
better off as members of a currency union than as non-members. In a standard calibration
with σ = 2 and ϕ = 3, countries are better off as members of a currency union only when
1 ≤ γ ≤ 1.12. For all other values of γ outside this narrow range, households prefer to be
outside of a currency union in financial autarky.
J Extended Model with Home Bias and Calvo Wage Rigidity
In each country i, the consumption basket consists of home (CitH ) and foreign (CitF ) goods,
i η
h
η−1
η−1 η−1
1
1
Cit = (1 − α) η (CitH ) η + α η (CitF ) η
(J.1)
where CitF and CitH are defined as
CitF
Z
=
F
(Cijt
)
γ−1
γ
γ
γ−1
dj
and
CitH
Z
=
ε
ε−1
ε−1
(CitH (h)) ε dh
.
(J.2)
F
Cijt
denotes consumption by households in country i of the variety produced by country j,
while CitH (h) denotes consumption by households in country i of the domestic variety produced
by intermediate firm h. The elasticity of substitution between home and foreign products is
defined by η, while the elasticity of substitution between the goods of different countries remains
γ. The relative weight of these goods in the consumption basket is defined by the degree of
home bias, 1 − α. When α = 0, home bias is complete and households only consume domestic
goods. In the opposite extreme, when α = 1, the economy is fully open and households will
consume a basket made up entirely of imports from all other countries in the world.
Similarly the price index will consist of goods prices of both home and foreign products:
h
i 1
H 1−η
F 1−η 1−η
Pit = (1 − α)(Pit )
+ α(Pit )
.
51
(J.3)
Relative demand for home and foreign products is given by
CitH
= (1 − α)
CitF
=α
PitH
Pit
−η
PitF
Pit
−η
Cit ,
(J.4)
Cit ,
(J.5)
where CitF and CitH are defined as
CitF
CitH
Z
=
Z
=
F
(Cijt
)
γ−1
γ
γ
γ−1
dj
,
(J.6)
.
(J.7)
−1
−1
(CitH (h)) dh
F
denotes consumption by households in country i of the variety produced by country j.
Cijt
H
Cit (h) denotes consumption by households in country i of the domestic variety produced by
intermediate firm h.
Production in each country i and the demand for country i0 s goods are given by:
Yit = Zit Nit
(J.8)
Yit = CitH + PitH /PitF
−γ
CtH∗
(J.9)
where CtH∗ represents foreign consumption of the home good. The formula for consumption depends on whether we have complete markets, incomplete markets, financial autarky or financial
autarky with a transfer union. Correspondingly:
nP
o
PH,it
F σ1
t
β Yit P F
E0
Pt
it
Cit =
(J.10a)
σ−1 P t Pit σ
Pt
E0
β PF
it
Yit PH,it
− Bit + Bit−1 (1 + iit−1 )
Pt
Yit PH,it
Cit =
Pt
Yit PH,it Tit
Cit =
+
Pt
Pit
Calvo wage setting can be expressed as
Cit =
1−ε
Wt1−ε = (1 − θW )W̃t1−ε + θW Wt−1
(J.10b)
(J.10c)
(J.10d)
(J.11)
where Wt is the actual wage, W̃ is the optimal reset wage and θW is the fraction of households
52
who are able to reset wages in each period. The equations describing Calvo wage setting are:
VW,t = Nt1+ϕ + βθW Et VW,t+1
Nt
ṼW,t = Ct−σ
+ βθW ṼW,t+1
Pt
VW,t
ε
W̃ = χ
ε − 1 ṼW,t
1−ε
Wt1−ε = (1 − θW )W̃t1−ε + θW Wt−1
(J.12)
(J.13)
(J.14)
(J.15)
There is a nominal government bond that pays in units of the import basket CF for incomplete
markets. Households will maximize utility from (2) subject to the following budget constraint:
Wit (h)
Bit−1 (h)
Bit (h)
Cit (h) +
= (1 − τi )
Nit (h) + Dit (h) + Tit (h) + Γit (h) + (1 + it−1 )
.
Pit
Pit (h)
Pit
(J.16)
The domestic interest rate it equals the world interest rate plus a country specific interest rate
premium p() that is strictly increasing in the amount of debt Bt :
it = i∗ + p(Bt ).
(J.17)
Financial autarky is the case for which p goes to infinity. The interest rate premium is necessary
to ensure stationarity.
To compute welfare under financial autarky, incomplete markets, complete markets and a
transfer union, we solve a second-order approximation of the model for each country assuming
that the home country is a small open economy and that the rest of the union is in the steady
state. In this case the calibration of parameters for other members of the union has no effect.
The correct specification yields the ergodic mean for the rest of the union. We compute the
ergodic mean (assuming that the rest of the union has an identical calibration to the small open
economy, but faces asymmetric shocks) and find that it has no effect on allocation or welfare
analysis. Indeed, since shocks are asymmetric and positive shocks cancel out negative shocks
across the union, it does not matter whether the rest of the union is at rest or experiences
asymmetric shocks. This is also true if we consider two economies rather than a continuum of
small open economies. The computation of the ergodic mean remains unaffected as positive
shocks for the rest of the union would cancel out with negative shocks for the home country,
and the welfare results should be robust in that respect.
To compute welfare for the optimal transfer union we numerically search over the optimal
transfers as a function of technology shocks and then estimate the ergodic mean for welfare
using second order perturbation methods.
K Non-cooperative Tax Policy in the Extended Model
In this section we solve for the optimal steady state income tax rate for non-cooperative fiscal
authorities in the extended model with home bias. Because the optimal income tax rate is a
steady state object, we will drop all time subscripts in this section. Using the demand equation
from (J.4) and the fact that C = Y P H /P , we can write
C H = (1 − α)C 1−η Y η .
53
(K.1)
We can also rearrange the price index (J.3) to get
η−1
PH
P
PH
+α−1=α
PF
η−1
.
(K.2)
Substitute C/Y = P H /P into (K.2) and rearrange so that
1
PH
(C/Y )η−1 + α − 1 η−1
.
=
PF
α
(K.3)
Now we plug this expression into the demand equation for home products (J.9):
Y = (1 − α)C
1−η
(C/Y )η−1 + α − 1
Y +
α
η
−γ
η−1
C H∗
(K.4)
Using the implicit function theorem, we define C = f (Y )=(K.4), so that the non-cooperative
policymaker’s objective function becomes
max
Y
f (Y )1−σ
Y 1+ϕ
−χ
.
1−σ
1+ϕ
(K.5)
The policymaker’s first order conditions with respect to Y are
f 0 (Y )f (Y )−σ − χY ϕ = 0
(K.6)
where we’ve used Y = N because we are in steady state, and C = f (Y ). From the implicit
function theorem, we know that f 0 (Y ) = −gY /gC , where
g(C, Y ) = −Y + (1 − α)C
1−η
C η−1 Y 1−η + α − 1
Y +
α
η
−γ
η−1
C H∗ .
(K.7)
Solving for gC gives
−γ
η−1 1−η
η−1
−1
γ
C
Y
+
α
−
1
−η η
C H∗ C η−2 Y 1−η .
gC = (1 − η)(1 − α)C Y −
α
α
(K.8)
In steady state, we know that αC = αY = C H∗ , so that gC = (1 − η)(1 − α) − γ. Similarly, we
solve for gY :
gY = −1 + (1 − α)ηC
1−η
Y
η−1
−γ −1
γ C η−1 Y 1−η + α − 1 η−1
+
C H∗ C η−1 Y −η ,
α
α
(K.9)
which becomes gY = −1 + (1 − α)η + γ in steady state. Using these two simplified expressions
for gC and gY , we can rewrite the FOC from (K.6):
f 0 (Y )f (Y )−σ − χY ϕ = −
gY −σ
1 − (1 − α)η − γ −σ
C − χY ϕ =
Y − χY ϕ = 0.
gC
(1 − η)(1 − α) − γ
54
(K.10)
Using the implicit function theorem and solving for steady state Y yields:
Y =
1 − τi
χµε
1
σ+ϕ
1
− σ+ϕ
1 γ − 1 + (1 − α)η
=
χ γ − (1 − α)(1 − η)
(K.11)
γ−1+(1−α)η
where the optimal tax rate for non-cooperative fiscal authorities is τi = 1 − µε γ−(1−α)(1−η)
.
As in the closed-form model, the optimal tax rate in a tax union will be τi = 1 − µε .
55