Optimal Unemployment Insurance and International Risk Sharing

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Optimal Unemployment Insurance and International Risk Sharing St´ephane Moyen, Nikolai St¨ahler, and Fabian Winkler 2016-054 Please cite this paper as: Moyen, St´ephane, Nikolai St¨ahler, and Fabian Winkler (2016). “Optimal Unemployment Insurance and International Risk Sharing,” Finance and Economics Discussion Series 2016-054. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2016.054. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Optimal Unemployment Insurance (cid:73) and International Risk Sharing StéphaneMoyena,NikolaiStählera,FabianWinklerb aDeutscheBundesbank,ResearchDepartment,Wilhelm-Epstein-Str. 14,60431Frankfurt,Germany bFederalReserveBoard,20thStandConstitutionAveNW,WashingtonDC20551,U.S.A. Abstract We discuss how cross-country unemployment insurance can be used to improve international risk sharing. We use a two-country business cycle model with incomplete financial markets and frictional labor markets where the unemployment insurance scheme operates across both countries. Cross-country insurance through the unemployment insurance system can be achieved without affecting unemployment outcomes. The Ramsey-optimal policy however prescribes a more countercyclical replacement rate when international risk sharing concerns enter the unemployment insurance trade-off. We calibrate our model to Eurozone data and find that optimal stabilizing transfers through the unemployment insurance system are sizable and mainly stabilize consumption in the periphery countries, while optimal replacement rates are countercylicaloverall. Moreover,wefindthatdebt-financednationalpoliciesareapoorsubstituteforfiscaltransfers. Keywords: UnemploymentInsurance,InternationalBusinessCycles,FiscalUnion, InternationalRiskSharing JEL:,E32,E62,H21,J64 1. Introduction Europe has seen business cycle movements differ greatly across countries. This development, together with the resulting strains on public budgets, has renewed calls to (cid:73) The views herein are those of the authors and do not represent the views of the Deutsche Bundesbank, the Eurosystem or its staff, the Board of Governors of the Federal Reserve System or the Federal ReserveSystem. Anyerrorsareoursalone. WewouldliketothankKlausAdam,JulienAlbertini,Adrien Auclert,MichaelBurda,FrancescoCaselli,WouterdenHaan,SusanneEk,JavierFernandezBlanco,Etienne Gagnon, Sebastian Grisse, Mathias Hoffmann, Josef Hollmayr, Michael Krause, Jochen Mankart, Christian Merkl, Pascal Michaillat, Kurt Mitman, Martin Scheffel, Thepthida Sopraseuth and seminar participantsatBundesbank,FederalReserveBoard,LSE,theEEAannualmeetinginMannheim,the18th “TheoriesandMethodsinMacroeconomics”conferenceinLausanne,IAB-Bundesbankjointconference 2014,Oenb-Bundesbankannualworkshop2015,BanquedeFrance-Bundesbankannualworkshop2014, Humboldt University, University of Cologne, GATE University of Lyon and UAB Barcelona for their helpfulcomments.

introducesomeformofpubliccross-countryrisksharing,sometimesunderthenameof a“fiscalunion”. Indeed,awidelyheldviewisthatacommoncurrencyexacerbatesthe need for international risk sharing mechanisms, and that fiscal transfers become desirable when the private sector lackssuch mechanisms (Mundell (1961), McKinnon(1963) andKenen(1969)).1 At the same time, high unemployment levels in many developed countries have led to renewed interest in the design of unemployment insurance. In the Eurozone in particular, policy makers have argued that the unemployment insurance system is a goodandpoliticallyviablechanneltoshareriskacrosscountries. TheEUCommissioner for Employment, László Andor, states that “based on the expert work available to date, I consider that the best form of [...] a countercyclical fiscal capacity at the EMU level would be a scheme where the participating countries share part of the costs of short-term unemployment insurance”(Andor,2014). 2 Ourquestionis: Ifagroupofcountriesweretointroduceacommonunemployment insurance system, what should it look like? We answer it using a two-country business cycle model with search frictions in labor markets. Financial markets are incomplete and labor is immobile across countries, so that country-specific risk and idiosyncratic unemployment risk can only be partially insured privately. The government in each countrymaintainsamandatoryunemploymentinsurancesystem. Inaddition,asupranational unemployment insurance agency is able to administer an additional component of the unemployment insurance system. This component can differ across countriesasafunctionofcountry-specificshocks. Starting with a simplified version of our model, we derive two theoretical insights. First, a supranational unemployment insurance system can be used to insure against country-level risk without affecting unemployment levels. The intuition is as follows. Unemployment insurance affects unemployment levels by changing the relative value of employment over unemployment, which determines incentives to search and wage bargaining outcomes. When a country is to receive a fiscal transfer, this relative value canbekeptconstantbysimultaneouslyincreasingthelevelofbenefitsandloweringthe rateofcontributionstotheunemploymentinsurancesystem. Theoppositecanbedone inthecountrywhichistosendthetransfer. Second, the presence of an international risk sharing motive introduces a countercyclical element to the optimal unemployment insurance policy. Here, the intuition is 1SeeFurceriandZdzienicka(2015)andKalemli-Ozcanetal.(2014)forrecentevidenceonthelackof risksharingmechanismsintheEuroarea.SørensenandYosha(1998)alsoprovidedcomparableestimates inthepast. 2AharmonizedunemploymentinsurancesystemwithintheEurozoneasatoolforinternationalrisk sharing has also been suggested by the President of the European Council (van Rompuy, 2012), the InternationalMonetaryFund(Blanchardetal.,2014),theGermanInstituteforEconomicResearch(Bernoth andEngler,2013),theFrenchAdvisoryCouncil(Artusetal.,2013),Dollsetal.(2014),theBancad’Italia (Brandolinietal.,2015)andBénassy-Quéréetal.(2016). Brenke(2013)alsodiscussessomeofthedrawbacks. 2

as follows. The classic unemployment insurance trade-off for a social planner is between efficiency of employment and insurance. Too much insurance reduces search effort and/or job creation, while too little insurance harms risk-averse workers who cannot insure privately against unemployment risk. When international risk sharing is present, the planner is shielding local consumption from fluctuations in local output. After a negative productivity shock in one country, the planner can then afford to providemoregenerousinsuranceandshiftemploymenttowardscountrieswhereitismore productive. Therefore,insurancebecomesmorecountercyclical. Wethenmoveontoaquantitativelyrichermodel,wherewecalibratethetwocountries to the core and the periphery of the Euro area. We compute the Ramsey-optimal policy and compare it with a policy of constant replacement rates and no international transfers,assuchpoliciesarecurrentlyinplaceinmostcountries. Incomputingoptimal policy, we rule out permanent or perpetual transfers: A country can receive a transfer payment in response to a shock (and does not have to pay it back in the future), but transfers have to average out over time and in expectation, so that no country can expect to be a permanent net contributor or net recipient to the scheme. In our baseline simulation, the optimal unemployment insurance policy is countercyclical even without transfers, and it becomes more generous when a country receives a transfer. The transfers themselves are sizable: The periphery receives 0.70 percent of GDP for every percentage point drop in its own GDP, but has to transfer an almost equal amount to the core for a percentage point reduction in core GDP. However, there is overall very little country-specific business cycle risk in the model, owing to it being calibrated to the high correlation of GDP among Eurozone countries. As a result, transfers are unable to reduce consumption risk everywhere, but rather they distribute it more evenly byshiftingsomeriskfromthehighlyvolatileperipherytothemorestablecore. Acommoncritiquetoasupranationalunemploymentinsuranceisthatitsobjectives could be achieved just as well by national policies financed with government debt. We show that this argument does not hold: When transfers are replaced by debt as the instrumentoftheRamseyplanner,thestabilizationgainsallbutdisappear. Thereasonfor this is that budget deficits have to be reversed in the future, but international transfers onlyhavetoaverageoutovertime. Thismakesthemmuchmoreeffectiveatstabilizing theeconomy. Fiscalrisksharingworksmuchlikeafairlypricedinsurancepolicy: Even though,onaverage,thepremiaandexpectedpaymentsnetouttozero,oneisstillbetter better off buying the insurance than taking out a bank loan in the event of “damage”, becausetheloancreatesalargeliabilitywhentimesarebad. Ourresultshaveonepracticallimitation,whichisthatweabstractfromthepolitical moral hazard induced by risk sharing (Persson and Tabellini, 1996). It is plausible that a fiscal transfer mechanism reduces incentives for national governments to carry out structuralreforms,andthisisprobablythemainpoliticalreasonforitsoppositioninthe Eurozone. In this paper, we acknowledge this concern insofar as we rule out policies with permanent transfers from one country to another. However, our main focus is on thepotentialeconomicbenefitsratherthanthepoliticaleconomy. Theremainderofthepaperisorganizedasfollows. Webrieflyreviewtherelatedlit- 3

eratureinSection2. InSection3,welayoutasimplifiedversionofourmodelwithonly two periods. This allows us to show our theoretical insights analytically and provide intuition. In Section 4, we lay out the full dynamic model that we use for quantitative analysis and calibrate it to the Euro area. Section 5 contains the numerical results from our calibrated model. Section 6 discusses why government debt is not an effectivereplacementfortherisksharingachievedbyourunemploymentinsurancescheme. Section7concludes. 2. Relatedliterature Our analysis relates to the literature on international risk sharing and fiscal unions on one hand, and the literature on the design of optimal unemployment insurance on theotherhand. It is well known that the search externality in frictional labor markets can be corrected using unemployment insurance. Because of costly search, employment – and the corresponding fluctuations – may be too low or too high, depending mainly on the relation of the workers’ bargaining power to the matching elasticity. In the steady state, this can be resolved by changing the outside option of workers through unemployment benefits (Hosios, 1990). When workers are risk-averse, the correction of the search externality needs to be weighed against the provision of insurance (Baily, 1978). Fredriksson and Holmlund (2006) survey the literature on optimal unemployment insurance in static and steady-state situations. More recently, interest has emerged in unemployment insurance policies that depend on the state of the business cycle. Here, a central point of debate is whether benefits should become more generous in a recession in order to increase insurance (countercyclical policy), or less generous in order to mitigate thefall in employment (procyclicalpolicy). Earliercontributions such asKiley (2003)andSanchez(2008)suggestthatthereisroomforcountercyclicalunemployment benefits. Landais et al. (2015) analyze a model with sticky wages and job rationing and find that a countercyclical policy is optimal as the effect of insurance on equilibrium unemployment is smaller in recessions. Albertini and Poirier (2015) and Kekre (2016) find that more generous unemployment insurance can mitigate the aggregate demand deficiencywhenthezerolowerboundis binding. Ontheotherside,MitmanandRabinovich(2015)numericallycomputeoptimaldynamicpoliciesandshowthatthecyclical stanceoftheunemploymentinsuranceisprocyclicalinasettingwhenworkers’outside option leads to inefficiently high wages. Moyen and Stähler (2014) analyze the optimal cyclicality of benefits holding their average level fixed. They show that there are situations in which unemployment insurance should be countercyclical even when wages are directly affected and the bargaining power of workers is too high relative to the Hosios condition.3 Jung and Kuester (2015) analyze first-best policy with sufficiently 3MoyenandStähler(2014)comparetheoptimalbenefitdurationpolicyinEuropeandtheUS.Intheir European calibration, the bargaining power of workers is larger than the matching efficiency, implying 4

many fiscal instruments. They find that benefits should rise in recessions if hiring subsidies and layoff taxes also rise at the same time. These two instruments increase the incentives to hire and decrease those to fire workers, which may compensate partly for increased wage costs. However, if hiring subsidies and layoff taxes are not adjusted, theyalsofindprocyclicalbenefitstobeoptimal. Importantly,theliteratureexclusivelyanalyzesclosedeconomies. Ourpaperinstead analyzes optimal policy when unemployment insurance can operate across multiple countriesandfacestheadditionalobjectiveofsharingcross-countryrisk. Turning to the literature on fiscal unions, Leduc et al. (2009) have shown that, when assetmarketsareincomplete,country-specificproductivitydisturbancescanhavelarge uninsurable effects on wealth and consumption paths. In a prominent recent paper, FarhiandWerning(2012)findthatsuchuninsurableeffectsmaybeespeciallylargeina currencyunionwithnominalrigidities. Theysuggestformingatransferuniontoinsure against this risk. Many economists follow their view that, in federal unions, a (fiscal) transfer mechanism to at least compensate for the uninsurable effects due to nominal rigiditiesmaybedesirable. However,thereisstillsomedebateonhowtoideallyestablishsuchatransfermechanismorafiscalunion(seeBargainetal.,2013andBordoetal., 2011foradiscussion). Evers(2012)providesaquantitativeassessmentoffederaltransfer rules and finds that targeting regional differences in labor income generates highest welfaregains,whichprimarilystemfromreducingtheallocativeinefficienciesoffactor inputs caused by nominal rigidities. Dmitriev and Hoddenbagh (2013) find that a tax union,inwhichthesteadystateincometaxareharmonized,hastobepreferredtocross countries fiscal transfers only if the Armington elasticity is low. Evers (2015) shows thatafiscalrevenueequalizationsystem,thatsharesnominaltaxrevenues,destabilizes business cycles and worsens welfare while a fully centralized fiscal authority does the opposite. While these papers have mostly focused on symmetric countries, we show, in a model calibrated to Eurozone data, that a transfer mechanism is desirable even withoutnominalrigidities. 3. Simplifiedmodel The intuition for our results can best be seen in a two-period model that allows us to analytically prove our results and provide a graphical representation. The model is highlysimplified: Inparticular,therealexchangerateisconstantandthetwocountries are in financial autarky. We will relax these assumptions in the quantitative part in the nextsection. theoptimalbenefittobenegativeinlightoftheHosioscondition. However,itisrestrictedtobepositive. Additionally, rule-of-thumb households make average marginal utility of consumption fluctuate relativelymuch. Itcanbeshownthatsteady-statebenefitsaboveoptimumandrelativelyvolatilemarginal utility of consumption make optimal benefit policy countercyclical even when the bargaining power of workersisalreadyhigh. 5

3.1. Modelsetup There are two countries, which we call Home and Foreign. Home is inhabited by a mass ω ∈ (0,1) of workers, while Foreign is inhabited by a mass 1−ω of workers. In eachcountry,firmstransformlaborintoahomogenousconsumptiongood(inthequantitative model of Section 4 we will introduce imperfect substitutability between Home andForeigngoods). Firmsareownedbyworkers,butmakezeroprofitsinequilibrium. Whileconsumptiongoods canbetradedacross countries incompetitivemarkets,labor is immobile across countries and labor markets are subject to search frictions within eachcountry. In the first period, all workers start out as unemployed. In the second period, firms post vacancies and are matched with workers. Production and consumption take place onlyinthesecondperiod. ExpectedutilityofaworkeratHomeinthefirstperiodis: U = E[nu(c )+(1−n)u(c )] (1) e u wherec ishisconsumptionlevelifheturnsouttobeemployed,andc hisconsumption e u level if he turns out to be unemployed. n is the employment level per capita in the secondperiod.4 Weassumelogarithmicutility: u(c) = log(c). In the second period, the number of vacancies is v and the number of matches is givenby n = κ θ1−µ. (2) m whereθisthetightnessinthelabormarket. Sincetheinitialstockofemploymentiszero, employment in the second period equals the number of matches and market tightness equalsthenumberofvacancies. Afirmthatpostsavacancyincursacostκ . Theprobabilitythatthevacancyisfilled v is q = κ θ −µ. In that case, the match produces a units of output and the worker gets m paid a wage w. This wage is determined using Nash bargaining, where the bargaining power of workers is denoted ξ (the bargaining solution is described further below). A zero-profitconditionforvacancycreationprescribes κ = q(a−w) (3) v Wedenoteby y aggregateoutputintheHomecountrynetofvacancycosts: y = an−κ v (4) v Theproductivityaisarandomvariablewhichisonlyrevealedinthesecondperiod. The ForeigncountryhasasimilarstructuretotheHomecountry,butwithpossiblydifferent ∗ parameters. We denote Foreign variables with an asterisk, e.g. y for foreign output. Homeandforeignproductivity (a,a ∗) aretheonlysourcesofaggregateuncertainty. 4Throughoutthepaper,quantitieswillbeexpressedinpercapitatermsunlessotherwiseindicated. 6

Employedworkersreceivewageswwhicharetaxedattherateτ,whileunemployed workers receive unemployment benefits b. Payroll taxes τ and benefits b are administered by an unemployment insurance agency. We assume that the two countries are part of an insurance union, such that the agency operates across both countries. It has torunabalancedbudgetwiththeconstraint: ω[(1−n)b−nτw]+(1−ω)[(1−n ∗)b ∗ −n ∗ τ ∗ w ∗] = 0. (5) 3.2. Socialplannerproblem We first look at the social planner problem. A utilitarian social planner maximizes a weighted average of worker utilities subject only to the resource constraint and the searchfrictionbysolvingthefollowingproblem: ω˜ E[nu(c )+(1−n)u(c )] e u max (cid:32) n,θ,ce,cu, (cid:33) +(1−ω˜)E[n ∗ u(c ∗ u )+(1−n ∗)u(c ∗ u )] n ∗ ,θ ∗ ,ce∗ ,cu∗ s.t. n = κ θ1−µ m n ∗ = κ ∗ (θ ∗)1−µ∗ m ω(nc +(1−n)c ) ω(an−κ θ) e u = v +(1−ω)(n ∗ c ∗ +(1−n ∗)c ∗) +(1−ω)(a ∗ n ∗ −κ ∗ θ ∗) e u v Here, ω˜ is the relative weight the planner puts on workers in the Home country, which might be more or less than the size of its population ω. Within a country, all workers are ex-ante homogenous and so weighting of individual workers is inconsequential. Thefirstorderconditionsoftheplannerproblemarestandard: κ = κ θ −µ(1−µ)a v m κ ∗ = κ ∗ θ ∗−µ∗ (1−µ ∗)a ∗ v m c = c u e c ∗ = c ∗ u e ω 1−ω c = c ∗ ω˜ e 1−ω˜ e ThefirsttwoconditionsaretheHosiosconditionsineachcountry,whichdeterminethe numberofvacanciesthatmaximizeaggregateoutputnetofvacancycosts. Theremainingconditionsprescribefullrisksharingwithinandacrosscountries. Theconsumption levels of employed and unemployed workers within each country should be identical, andeachcountryshouldconsumeaconstantfractionofunionoutput. 7

3.3. Optimalpolicywithprivateinsurance We now come back to the competitive equilibrium. Throughout this chapter, we assume some form of international market incompleteness, since our focus is on how unemploymentinsurancecanbeusedtoovercomealackofrisksharing. Inthissection, weallowworkerstoonlyinsuredomesticallyagainstidiosyncraticunemploymentrisk. Sinceallworkersareex-anteidentical,itisoptimalforaHomeworkertofullydiversify his risk by selling his entire future income in exchange for a diversified portfolio of the income of all other Home workers’ income. In this case, the consumption levels of all Homeworkersareequalized: c = c = c. (6) e u The wage w is assumed to be set by Nash bargaining. When workers have bargaining power ξ,theNash-bargainedwageissimply: ξa w(a,ρ) = (7) ξ +(1−ξ)(1−ρ) where ρ isthenetreplacementrate,definedas b ρ = . (8) (1−τ)w A higher replacement rate improves workers’ outside option and drives up wages. It therebylowerstheincentivesforjobcreationandreducesemployment. We want to know what the optimal unemployment insurance scheme looks like in this situation. Our first result is that a transfer of resources from one country to another can be implemented through the unemployment insurance system without affecting unemployment levels. We first note that the budget constraint of the unemployment insurance agency can berewrittenas 0 = ω(c−y)+(1−ω)(c ∗ −y ∗). (9) ∗ We can therefore choose replacement rates ρ,ρ and a transfer from the Foreign to the Homecountry T = c−yasapolicyandbackoutthenecessarybenefitsb,b ∗ ,τ,τ ∗ from thebudgetconstraint. Weobtain: nw+T b = ρ (10) n+(1−n)ρ 1 nw+T τ = 1− . (11) wn+(1−n)ρ The unemployment level n and the wage w depend only on policy through the replacement rate ρ. Therefore, from the formula above we can see that a positive transfer from Foreign to Home (T > 0) can be implemented by increasing unemployment 8

benefits b and at the same time lowering payroll taxes τ. This way, all workers get to consume more, but the net replacement rate ρ stays constant and the relative bargaining position of workers is unchanged. Since we generally have n > 1−n, most of this transfer is achieved by reducing the contributions by the employed. Increased benefits oftheemployedareonlyaminorpartofsuchaneutraltransfer. While it is possible to make transfers without affecting replacement rates, is this also optimal? With privately insured unemployment risk, the insurance agency has to mitigatethreeinefficiencies: searchexternalitiesintheHomeandForeigncountriesand lack of international risk sharing. It also has three policy instruments: the Home and Foreignreplacementratesandacross-countrytransfer. Indeed,thereexistsapolicythat eliminatesallthreeinefficiencies. ThereplacementratessatisfyingtheHosioscondition are µ−ξ µ ∗ −ξ ∗ ρ = , ρ ∗ = . (12) µ(1−ξ) µ∗(1−ξ∗) ∗ These rates do not depend on the realizations of a and a and are therefore constant. They also do not depend on the planner weight ω˜. The transfer T however does depend on this weight. In principle, one can choose any value for ω˜. We determine it by imposingthattransfersarezeroinexpectation: E[T] = 0. (13) This condition implicitly defines a value for ω˜. This choice of the weight implies that neither country expects to be systematically subsidizing the other country through the unemploymentinsurancesystem. Thevalueoftheweightis E[ωy] ω˜ = (14) E[ωy+(1−ω)y∗] andtheresultingtransferpolicyis E[y]E[y ∗] (cid:18) y y (cid:19) T = E(cid:2) ω y+y∗ (cid:3) E[y∗] − E[y] . (15) 1−ω The Home country receives a transfer when its output is below average, but has to pay atransferwhenoutputintheForeigncountryisbelowaverage. 3.4. Optimalpolicywithoutprivateinsurance So far we have abstracted from the most important objective of unemployment insurance, namely to insure against unemployment. We now remove the possibility to privatelyinsureunemploymentrisk. Inthiscase,theoptimalpolicybecomesgenuinely second-best and trade-offs emerge between all three policy objectives: maximizing net output,providinginsurancebetweenemployedandunemployed,andprovidinginsuranceacrosscountries. 9

WeeliminateallassettradeinPeriod1(inthequantitativemodelofSection4wewill allow for trade of a non-contingent bond across the border). The consumption levels in Period2aresimply: c = (1−τ)w (16) e c = b = ρc (17) u u The Nash-bargained wage now takes into account the curvature in the worker’s utility function. Whenworkershavebargainingpower ξ,thebargainedwageis: ξa w(a,ρ) = . (18) ξ −(1−ξ)logρ In this situation, the social planner allocation is no longer feasible. Providing full insurance against idiosyncratic unemployment risk calls for ρ = 1, but then the worker capturesthewholesurplus(w = a)andjobcreationcollapsestozero. Wethereforehave tosolveforthe(second-best)Ramsey-optimalpolicyasfollows: ω˜ E[nu(c )+(1−n)u(ρc )] e e max (cid:32) n,θ,ce,ρ, (cid:33) +(1−ω˜) E[n ∗ u(c ∗ e )+(1−n ∗)u(ρ ∗ c ∗ e )] n ∗ ,θ ∗ ,ce∗ ,ρ ∗ s.t. n = κ θ1−µ m n ∗ = κ ∗ (θ ∗)1−µ∗ m κ = κ θ −µ(a−w(a,ρ)) v m κ ∗ = κ ∗ (θ ∗) −µ∗ (a ∗ −w ∗(a ∗ ,ρ ∗)) v m ω(nc +(1−n)ρc ) ω(an−κ θ) e e = v +(1−ω)(n ∗ c ∗ +(1−n ∗)ρ ∗ c ∗) +(1−ω)(a ∗ n ∗ −κ ∗ θ ∗) e e v Here, we have substituted out most equilibrium conditions. In particular, choosing an unemployment insurance policy (b,b ∗ ,τ,τ ∗) subject to the insurance agency’s budget constraint is equivalent to choosing replacement rates and consumption levels (ρ,ρ ∗ ,c ,c ∗) subjecttotheaggregateresourceconstraint. Thebenefitandtransferlevels e e can be inferred from (10)-(11) as before. Again, we choose the social planner weight ω˜ suchthatanytransfersmadeacrosscountriesnetoutinexpectation: E[T] = 0. Theproblemhaseightchoicevariablesandfiveconstraints,leavingthreedegreesof ∗ freedom. These correspond to the three policy instruments ρ, ρ and the cross-country transfer T. Thefirstorderconditiondeterminingtheoptimaltransferisderivedas: E[y]E[y ∗] (cid:18) y ∗ y (cid:19) T = E(cid:2) ω y+y∗ (cid:3) E[y∗] − E[y] . (19) 1−ω 10

This is the exact same expression as in (15). Each country consumes a constant share of union output. The Home country receives a transfer when its output is below average, but has to pay a transfer when output in the Foreign country is below average. The additionalquestionhereishowtransfers(andallotherresources)aredistributedamong theemployedandtheunemployed. The answer to this question is contained in the first order condition with respect to thereplacementrate. FortheHomecountry,itreadsasfollows: (cid:18) (cid:19) (1−n)(1−ρ) 1−ρ 1 y −(cid:101)n logρ+ = −(cid:101) y (20) n+(1−n)ρ ρ n+(1−n)ρ ρ ny+T (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) =:I(ρ) =:H(ρ) where (cid:101)n = dnρ is the elasticity of Home employment with respect to the Home re- ρ dρn placement rate, and (cid:101) y = dyρ is the elasticity of net Home output with respect to the ρ dρy Home replacement rate. A symmetric condition is obtained for the Foreign country. The left-hand side I(ρ) is the marginal benefit of insurance when raising the replacement rate, at a fixed quantity of output available to the country. By raising ρ, the unemployed’s marginal utility increases relative to average marginal utility. This is the first term on the left-hand side of Equation (20). At the same time, a higher ρ reduces employment (through higher wages and lower job creation), which shifts the composition of the workforce towards the unemployed. This means that one marginal worker suffers a utility loss, which is the “logρ” term in the left-hand side of Equation (20). It also implies a composition effect on the insurance budget, captured by the remaining term on the left-hand side. The right-hand side H(ρ) is the marginal cost of raising the replacement rate in terms of net output lost (output minus vacancy costs. This cost is where the level of transfers T enter. A high transfer implies that the planner is relying less on domestic production to support domestic consumption, lowering the marginal costofraisingdomesticoutput. The determination of the optimal replacement rate is graphically depicted in Figure 1,whichplotsthefunctions H(ρ) and I(ρ).5 We can see that the insurance term I(ρ) is positive and only equals zero at ρ = 1. Intuitively, holding output constant, it is always desirable to increase the replacement rateuntilfullinsuranceisachieved. The efficiency term H(ρ) is first negative and then turns positive, approaching infinity as ρ → 1. Intuitively, when ρ is too high, there is too little job creation and the amount of resources available for consumption can be increased by lowering replacement rates, thereby lowering bargained wages and increasing job creation. In this case y (cid:101) < 0, and therefore H(ρ) is positive. Conversely, when ρ is too low, there is too ρ much vacancy posting and the amount of resources available for consumption can be increasedbyraisingreplacementrates. Inthiscase, H(ρ) isnegative. 5Proposition1intheappendixprovesthattheshapeofthe I andHcurvesareindeedasdepicted. 11

Figure1: Optimalreplacementrate. H(ρ) I(ρ) ρ 0 1 The optimal replacement rate lies at the intersection between the two curves. Employment is always lower than in the social planner solution where H(ρ) = 0. The marginal benefit of insurance is always positive and so the optimal ρ is higher than whattheHosiosconditionwouldcallfor. ∗ What happens to the replacement rate when shocks to a or a hit the economy? We first keep the ratio y/(y+T) constant. This is the case in particular when T = 0, as in the limit ω → 1 of a closed-economy. The left panel of Figure 2 depicts the effects of a recessioncausedbyareductioninproductivity a. A reduction in a increases the insurance term I(ρ) and scales up the efficiency term H(ρ). The intuition is as follows. A reduction in productivity reduces employment. Holding total resources constant, this translates into an increase in unemployment risk for each worker, raising the social benefit of insurance. I(ρ) shifts up for any value of ρ. At the same time, lower productivity directly reduces net output for any level of employment. Therefore, the average marginal utility of increasing output towards its efficient level increases and H(ρ) is scaled up for any value of ρ. The figure shows that the effects of these two forces on the replacement rate work in opposite directions, so thatthechangeinρisambiguous. Thisambiguityreflectsthedebateintheexistingliteratureaboutthecyclicalityofoptimalunemploymentinsuranceinclosedeconomies—it is not clear whether, in a recession, it is more important to keep output from falling or toprotectworkersfromunemploymentrisk. In our supranational insurance scheme, however, a drop in productivity a also triggers an incoming transfer T. The risk sharing condition (19) prescribes that T is a decreasingfunctionof y,inordertokeepaverageconsumptionoftheHomecountryproportional to union output. The increase in T then affects the optimal replacement rate, anditisthisrisksharingaspectthatisnoveltotheliteratureonoptimalunemployment insurance. Specifically, the presence of international risk sharing makes the replacement rate more 12

Figure2: Changeoftheoptimalreplacementrateinarecession. (a)Reductionina. H(ρ) I(ρ) ρ 0 1 (b)IncreaseinT. H(ρ) I(ρ) ρ 0 1 13

countercyclical. Thiscanbeseenfromtheoptimalitycondition(20). Whenafalls,Home’s output will relatively low compared to union output, and T will increase. The right panelofFigure2showsthattheoptimalreplacementraterisesasaresponse,introducingacountercyclicalelementtotheoptimalpolicy. Whentransfersincrease,theplanner is relying less on domestic production to support domestic consumption, lowering the marginal cost of raising domestic output. The efficiency term H(ρ) shrinks and the trade-off between efficiency and insurance shifts towards the latter. The replacement rate ρ risesasaresult.6 ∗ Next,theHomereplacementrateisincreasinginForeignproductivitya . Foreignproductivity only affects the Home replacement rate because of its effect on Foreign output y and therefore transfers to the Home country. When the Foreign country experiences a drop in productivity, it receives a transfer and T falls. The planner is now relying more on Home output relative to Foreign output, and the trade-off between efficiency and insurance shifts towards the former. The replacement rate ρ falls in order to increase productionatHome. ∗ Finally, we can show that when a and a are independent, the optimal Home replacement rate is countercyclical in the limit as ω → 0. When the Home country is small and its shocksareuncorrelatedwiththerestoftheunion,itsriskcanbecompletelydiversified atthesupranationallevel. Theappendixshowsthatinthiscase,shiftsofthe H function due to transfers (right panel of Figure 2) dominate those due to movements in productivity (left panel). The Home replacement rate unambiguously rises when productivity falls. It is worth mentioning that even in the case ω → 0, the optimal replacement rate is not one. This might seem counterintuitive as it would be costless for the union to perfectly insure all workers in an infinitesimally small country. But this would imply that the country receives positive transfers in all states of the world, which we have ruled out with condition (13). The planner weight ω˜ shrinks together with ω to ensure that this condition holds. Every country, regardless how small, has to be self-financed inexpectation,evenasitsidiosyncraticbusinesscycleriskcanbebetterinsured. 4. Modelforquantitativeanalysis Thesimplemodeloftheprevioussectionillustratestherelevanttrade-offsinvolved in supranational unemployment insurance. In this section, extend the model along a number of dimensions and calibrate it to the core and periphery of the Eurozone. We numerically solve for the Ramsey-optimal policies with and without the possibility of transfers. Our simulation results of the optimal policies confirm the insights from the simplemodel. We extend the simple model in the following dimensions. First, the model is dynamic with an infinite horizon, where workers transition back and forth between employment and unemployment. Second, we allow for variable search effort, so that unemploymentbenefitsaffectthelabormarketnotonlythroughwagebargainingbutalso 6Theappendixprovidesaformalproof. 14

through their effect on search effort. Third, we add imperfect substitutability between Home and Foreign goods, so that movements in the terms of trade can partly insure againstcountry-specificshockstoproduction. Fourth,weaddpriceandwagerigidities to help the quantitative fit of the model, in particular the volatility of unemployment. Finally, we allow for a limited degree of trade of intertemporal non-contingent bonds, so that even in the absence of supranational risk sharing mechanisms agents can somewhatsmoothbusinesscycleshocksintertemporally. 4.1. Modelsetup Time is discrete at t = 0,1,2,.... As before, a unit mass of workers and firms populates the economy, where ω ∈ (0,1) workers live in the Home country and (1−ω) workersliveintheForeigncountry. WewillonlydescribethemodelsetupintheHome country. The structure of the Foreign economy is identical up to potentially different parametervalues. IfweneedtoshowvariablesandparametersofForeign,theywillbe indicatedbyanasterisk. 4.1.1. Matching At the beginning of period t, a fraction u of workers at Home are unemployed. We t assumethatlaborisimmobileacrosscountries,sothatworkerscanonlysearchforjobs domestically. The number of total new hires is determined by the number of searching workers u , the search effort e , and the number of vacancies v . Workers and vacancies t t t arematchedaccordingtoastandardCobb-Douglasmatchingfunction m = κ (e u )µ v 1−µ (21) t m t t t where κ is a matching efficiency parameter and µ is the elasticity of matches with rem specttounemployment. Defininglabormarkettightnessasθ = v /e u ,theprobability t t t t thatavacancygetsfilled,andtheprobabilitythataworkerputtinginoneunitofsearch −µ effort finds a job, are given by q = κ θ and f = q θ . Separation occurs at the t m t t t t exogenous rate s. Unemployed workers who separate have to wait one period before they can start searching again. Accordingly, the laws of motion for employment and unemploymentaregivenby: n = (1−s)n +q v (22) t t−1 t t u = 1−n . (23) t t−1 4.1.2. Workers A worker can be employed or unemployed. He maximizes expected lifetime utility, definedrecursivelyasfollows: W = u(c )+β E {(1−s)W +sU } (24) t et t t+1 t+1 U = u(c )+β E {(1− f e )U + f e W −k(e )}. (25) t ut t t+1 t+1 t+1 t+1 t+1 t+1 t+1 15

whereW istheutilityofanemployedworkerwithconsumptionc andU istheutility t et t of an unemployed worker with consumption c . Unemployed workers have to exert ut search effort e at the beginning of the next period to find a job. We assume the folt+1 lowingfunctionalforms: c1−γ u(c) = , γ ≥ 0 1−γ e1+φ k(e) = κ , φ > 0 e 1+φ Theconsumptionflowc ,j ∈ {e,u}denotesexpenditureonaconsumptionbasket. This jt basketconsistsofgoodsproducedintheHomeandForeigncountriesandisgivenby: (cid:16) (cid:0) (cid:1)σ (cid:0) (cid:1)σ (cid:17)1/σ c = ψ c +(1−ψ) c (26) jt jt,H jt,F Here, c is the amount of goods consumed and produced at Home, while c is jt,H jt,F the amount of goods consumed at Home but produced in Foreign. The parameter σ ∈ (−∞ ,1) governstheelasticityofsubstitutionbetweenforeignanddomesticgoods, whichequals1/(1−σ),andtheparameterψrepresentstherelativevaluationofHome goods.7 We abstract from international trade costs so the law of one price holds for both goods. We take the Home good as the numéraire and let p be the relative price of t Foreign goods. Thus, p also represents the terms of trade of the Foreign country. Next, t (cid:0) (cid:1) we define the consumer price index (CPI) at Home by P = c + p c /c . Utility t jt,H t jt,F jt maximizationimpliesthat8 1 (cid:18) (cid:19) c jt,H ψ 1−σ = p (27) c t 1−ψ jt,F P t = (cid:16) ψ1− 1 σ +(1−ψ)1− 1 σ p − t 1− σ σ (cid:17)−1− σ σ (28) Next,wespecifyworkers’budgetconstraintsandthefinancialassetstowhichtheyhave access. We want to capture an incomplete market setting in which workers can neither obtain perfect insurance of their idiosyncratic unemployment risk nor of countryspecific risk. We opt for a setting in which workers do not have access to individual 7Inthecaseofunitaryelasticityofsubstitution(σ =0),theconsumptionbasketisoftheCobb-Douglas formc = (cid:0) c (cid:1)ψ(cid:0) c (cid:1)1−ψ ,sothattheexpenditureshareonHomegoodsisexactlyψ.Asituationwhere jt jt,H jt,F φ > ωthencorrespondstohomebiasinconsumption. 8Having normalized the price of the Home good to one, note that P is Home CPI relative to Home t PPI.Itcorrespondinglyholdsthat P∗ isForeignCPIexpressedrelativetogoodsproducedinHome. We t will,hence,expressallForeignvariablesrelativetothepriceofgoodsproducedinHome. 16

savings at all, so that they cannot insure their idiosyncratic unemployment risk. However, both unemployed and employed workers will own a fixed amount of shares in firms,andfirmsareabletoaccessanon-contingentinternationalbond. Aswillbecome clear below, this allows us to introduce limited intertemporal trade while escaping the needtokeeptrackofthedistributionofassetswhensolvingthemodel. Employedworkersreceiveawagepaymentw (inunitsofHomegoods),ofwhichan t amount τ of payroll taxes is deducted. The unemployed receive unemployment insurt Π ance benefits b . All workers receive an equal share of profits from the firms in their t t countryofresidence. Sincefirmsdiscountprofitsatahouseholdrate,holdingsharesin firms effectively gives agents a form of savings through firms’ intertemporal decisions, buttheycannotsaveindividuallyandmustconsumetheirperperiod-income,whichin realtermsreadsasfollows: c = (w −τ +Π )/P (29) et t t t t c = (b +Π )/P. (30) ut t t t The only choice variable for workers is the search effort e when unemployed. Maxt imizing the utility of the unemployed with respect to effort leads to the following optimalitycondition: k (cid:48)(e ) = f (W −U ). (31) t t t t 4.1.3. Firms We assume that production is divided into a final and an intermediate goods sector, with the latter being subject to search frictions. Each country produces a distinct final goodfromdomesticintermediates. Intermediateproducersoperateundermonopolistic competitionandareabletosetprices. More precisely, at Home there is a representative final good producer which purchases a variety of differentiated intermediate goods, bundles these into a final good and sells the latter as a price taker. The price of the Home country’s final good is the same in both countries and equal to one (the price for the good produced in Foreign is equalto p ). Themaximizationproblemoftherepresentativeretailfirmreads t ˆ 1 ω max Y − p˜ (j)y (j)dj, (32) t t t {y˜t (j):j∈[0,ω]} ω 0 where p˜ (j) isthepriceofaspecificvariety j,and t ˆ (cid:18) 1 ω (cid:19)(cid:101)/((cid:101)−1) Y = y (j)((cid:101)−1)/(cid:101)dj , (cid:101) > 1, (33) t t ω 0 is the final goods producer’s production function. y (j) is the final goods producer’s t demand for the differentiated input j ∈ [0,ω]. The first-order condition for each input 17

reads: y (j) = (p˜ (j)) −(cid:101) Y. (34) t t t Combining the latter with (32) and the zero profit condition, we obtain the producer priceindexinthehomecountryandnormalizeittoone: ˆ 1 ω 1 = p˜ (j)1−(cid:101)dj. (35) t ω 0 Now,eachintermediateproduceroperatesthefollowingtechnology: y (j) = a n (j) (36) t t t whichislinearinlabor,anda isaproductivityshockcommontoallfirms. Employment t is subject to search frictions. Firm j needs to post a number of vacancies v (j), each of t which leads to successful matching with a worker with probability q . The vacancy t fillingrateistakenasgivenbythefirm. Successfulmatchesstartproductioninthenext period. Furthermore, firm j needs to pay its workers a wage w (j), and it needs to pay a t cost for each vacancy, which takes the form of a constant quantity κ of domestically v produced goods. Following Arseneau and Chugh (2007), the firm faces a quadratic cost of adjusting real wages.9For each of its workers, the real cost of changing wages betweenperiod t−1and t is κ (cid:18) w (j) (cid:19)2 κ w t −1 = w (π (j)−1)2 , 2 w (j) 2 wt t−1 where π (j) = w (j)/w (j) is the gross real wage growth rate. The wage itself is wt t t−1 determined through Nash bargaining, as described below. The firm can further save in non-contingent international bonds d (j) which pay a gross interest R , subject to portt t folio adjustment cost κ /2d2(j), as is standard in the international economics literature d t (e.g. Benigno, 2009). Given that firms operate in monopolistic competition and the fact thatfirmsalsosettheprice p˜ (j) fortheirgoodsvariety j,weassumethattheyalsoface t Rotemberg price adjustment costs that are similar to wage adjustment costs described above,withκ beingthecostparameter. Finally,thefirmhastopayafixedcost F every p period. Insum,firm j’sprofitisgivenby κ d (j) Π (j) = p˜ (j)y (j)−w (j)n (j)−κ v (j)−d (j)− d d2(j)+R t−1 t t t t t v t t t t−1 2 π pt 9Aspointedoutbytheseauthors,weadoptedthisassumptionascostofnominalwageadjustmentdo nothelpincreasingsignificantlythevolatilityofunemployment. Arequirementthatwillbecomeclearin thecalibrationsection. 18

− κ w (π (j)−1)2 n (j)− κ p (cid:18) p˜ t (j) π −1 (cid:19)2 y (j)−F, (37) 2 wt t 2 p˜ (j) pt t t−1 p where π isaggregatePPIinflationatHome. t Thefirmmaximizesthediscountedsumofprofits ∞ E ∑ Q Π (j) 0,t t t=0 where Q is the discount factor between times s and t. Since the firm is owned in part s,t by employed and unemployed workers, it is not obvious what discount factor the firm shoulduse. AsinJungandKuester(2015)profitsaresharedequallyacrosshouseholds, implying that the firm discount factor is a weighted average of the worker discount factors: n u (cid:48)(c )+(1−n )u (cid:48)(c ) P Q = βt−s t et t ut s . (38) s,t n u(cid:48)(c )+(1−n )u(cid:48)(c ) P s es s us t Maximizing profits with respect to employment while taking into account the employment law-of-motion as well as the demand for each intermediate goods’ variety leads toanexpressionforthevalueofafilledjob J : t (cid:18) (cid:19)2 κ w J = mc a −w − w t −1 +(1−s)E Q [J ], (39) t t t t t t,t+1 t+1 2 w t−1 where mc are marginal costs (formally, the Lagrangian parameter on equation (34)). t The optimality condition of the firm with respect to vacancy creation takes the familiar form: κ = q J . (40) v t t Finally,theoptimalityconditionwithrespectto p˜ (j) canbewrittenasfollows: t y 1−κ (π −1)π +E Q κ (π −1)π t+1 = (1−mc )(cid:101). (41) p pt pt t t,t+1 p pt+1 pt+1 t y t Note that in the last three equations we made use of the fact that, in equilibrium, all firms will chose the same price and allocation; thus, we dropped the index j due to symmetryandimposed p˜ (j) = 1fromEquation(35). t 4.1.4. Wagedetermination The wage paid to workers is determined by Nash bargaining in which workers and firmssharethesurplusfrommatchingaccordingto max(W −U )ξ J1−ξ t t t wt 19

where ξ is the bargaining power of workers. The wage is determined implicitly by the first-orderconditiontotheaboveproblem: (cid:18) π π (cid:19) ξ u (cid:48)(c ) 1+κ (πw −1) wt +(1−s)κ E Q (π −1) wt+1 [W −U ] = et J . w t w w t t,t+1 wt+1 w t t 1−ξ P t t t t (42) 4.1.5. Government Unlike in the simple model of the previous section, we explicitly spell out national governmentsaswellasasupranationalunemploymentinsuranceagency,eachofwhich independentlymanagesitsfinances. The government in the Home country gains revenue exclusively from payroll taxes τ . Thesetaxesareusedtofundbenefitsforunemployedworkersb aswellasgoverngt gt ment expenditure g . Government expenditure is spent entirely on domestically prot duced goods.10 The government has to balance its budget every period. Its budget constraintwrites g +u b = τ n . (43) t t gt gt t Thesupranationalunemploymentinsuranceagencycanlikewiseadministeracomponent of unemployment insurance. This agency also has to balance its budget every period. It collects payroll taxes τ and disburses unemployment benefits b in the xt xt ∗ ∗ Homecountry,andcollectpayrolltaxesτ anddisbursesunemploymentbenefitsb in xt xt theForeigncountry. Theagency’sbudgetconstraintwrites ω(1−n )b +(1−ω)(1−n ∗)p b ∗ = ωn τ +(1−ω)n ∗ p τ ∗ . (44) t xt t t t t xt t t xt Total taxes on employed workers, total benefits received by unemployed workers, and thenetreplacementratearethengivenby: τ = τ +τ (45) t gt xt b = b +b (46) t gt xt b ρ = t (47) t w −τ t t In our benchmark calibration, the supranational agency is inactive (b = b ∗ = τ = xt xt xt τ ∗ = 0)andnationalgovernmentstargetconstantreplacementratesρ = ρ¯andρ ∗ = ρ¯ ∗ . xt t t Since this situation is close to the current system in place in the Eurozone, we call it the “statusquo”. 10Oursetupimplicitlyassumesthatanyutilityworkersreceivefromgovernmentexpenditureisseparablefrommarketconsumption,sothatwecanignoreitintheutilityfunction. 20

MonetarypolicyissetaccordingtoaTaylor-typerule (cid:20) (cid:18) (cid:18) (cid:19)(cid:19) Y R = ρ R +(1−ρ ) R¯ +(cid:118) φ π +φ log t t R t−1 R t π t y Y t−1 (cid:32) (cid:32) (cid:33)(cid:33)(cid:35) ∗ p Y +(1−(cid:118) ) φ π ∗ +φ log t t , (48) t π t y p Y∗ t−1 t−1 whereπ = π P/P isHomeCPIinflationandπ ∗ = π P ∗ /P ∗ isForeignCPIinflat pt t t−1 t pt t t−1 tion. The monetary authority reacts to a weighted average of inflation deviations from target and output growth in Home and Foreign with strength φ and φ , respectively, π y wheretheweightisgivenbytheirrealGDP (cid:118) = ωYt . Theparameterρ isan t ωYt +(1−ω)ptY t ∗ R interestratesmoothingparameter. 4.1.6. Marketclearingandshocks The market clearing conditions for consumption goods produced in each country taketheform: ω(Y −κvv −g ) = ω(n c +(1−n )c )+(1−ω) (cid:0) n ∗ c ∗ +(1−n ∗)c ∗ (cid:1) t t t t et,H t ut,H t et,H t ut,H (49) (1−ω)(Y ∗ −κ ∗vv ∗ −g ∗) = ω(n c +(1−n )c )+(1−ω) (cid:0) n ∗ c ∗ +(1−n ∗)c ∗ (cid:1) t t t t et,F t ut,F t et,F t ut,F (50) Theinternationalbondisinzeronetsupplysothatbondmarketequilibriumprescribes 0 = ωd +(1−ω)p d ∗ . (51) t t t Finally, the exogenous shocks in our model are persistent shocks to productivity and governmentspending. Theprocessesforgovernmentspendingandproductivityinthe Homecountryareasfollows: loga = ρ loga +(1−ρ )loga¯+ε (52) t a t−1 a at (cid:0) (cid:1) logg = ρ logg + 1−ρ logg¯+ε (53) t g t−1 g gt where all shocks are i.i.d. normally distributed with zero mean. Technology and governmentspendingshocksareindependentofeachother,butwedoallowforcorrelation oftheHomeandForeigntechnologyandgovernmentspendingshocks,respectively. We require the autoregressive coefficients to be less than one to rule out permanent shocks. Thischoiceisnotinnocuousinitspolicyimplications. Thefirstbestallocationin ourmodelwouldcompletelyshielddomesticconsumptionfromdomesticemployment and instead tie it to union output. In the presence of permanent shocks that differentially affect the long-run level of GDP in each country, this would effectively prescribe permanent transfers from the country with higher per capita income to the one with 21

lower per capita income, and the Ramsey-optimal policy would then implement this prescription by permanent fiscal transfers. We do not see much practical relevance in suchanextremeformofrisksharingandthereforefocusexclusivelyoncyclicalshocks. Any cross-country transfers under the Ramsey planner will always fall back to zero in expectation. 4.2. Optimalpolicy When we characterize optimal unemployment insurance policies, we define “optimal”tobemaximizingautilitarianwelfarefunction: E[ω˜ (n W +(1−n )U )+(1−ω˜)(n ∗W∗ +(1−n ∗)U∗)] (54) t t t t t t t t ∗ We solve for Ramsey-optimal policies involving the replacement rates ρ ,ρ as well as a t t transferpolicyofthesupranationalagency,definedas T = (1−n )b −n τ. (55) t t t t t Unless stated otherwise, we implement transfer policies by setting b = T, b ∗ = xt t xt ω/(1−ω)T /p ,andτ = τ ∗ = 0. Thisiswithoutlossofgenerality: Foranyotherset t t xt xt (cid:16) (cid:17) of policies b ,b ,τ ,τ ,b ∗ ,b ∗ ,τ ∗ ,τ ∗ , one can define T as above and set b (cid:48) = T, xt gt xt gt xt gt xt gt t xt t b (cid:48) = b +b − T, τ (cid:48) = 0, τ (cid:48) = τ +τ etc. and achieve exactly the same allocagt gt xt t xt gt gt xt tion, as the budget constraints of the public sector can be consolidated from a planner perspective. As in the simple model, we choose the welfare weight on the Home country ω˜ to ruleoutpermanenttransfersfromonecountrytoanother: E[T ] = 0. (56) t We also compute what we call a “no-transfer” Ramsey problem. There, we keep the planner weight ω˜ at the value imposed by condition (56), but now impose T = 0, t so that the Ramsey planner is left with the Home and Foreign replacement rates as instruments. Comparing the “full” and the “no-transfer” Ramsey solutions will allow ustoisolatetheinteractioneffectoftransfersandunemploymentinsurancepolicies. 4.3. Calibration We calibrate the model to the Eurozone, where we identify the Home country with six Eurozone core countries (Austria, Belgium, Germany, Finland, France, Luxemburg, Netherlands) and the Foreign country with six periphery countries (Spain, Greece, Ireland, Italy, Portugal). In what follows, we use the term “country” in the model sense and use the words “Home”/”Foreign” and “Core”/”Periphery” interchangeably. Our calibrationissummarizedinTable1. ThenumberofworkersintheHomecountryissetto60percent,whichcorresponds to the relative size of the labor force in the Core. We set the discount factor β in both 22

Table1: Calibratedparametervalues. Parameter Symbol Core Periphery Numberofworkers ω 0.601 0.399 Discountfactor β 0.99 Riskaversion γ 1.5 Elasticityparam. onHome/Foreigngoods σ 0.744 UtilityweightonHomegoods ψ 0.574 0.479 Inverseelasticityofsearcheffort φ 1.447 0.402 Effortcostscalingparameter κ 0.645 1.47 e Matchingelasticity η 0.5 Workerbargainingpower ξ 0.9 Matchingefficiency κ 0.4583 m Separationrate s 0.0275 0.0418 Vacancycosts κ 0.0369 0.0686 v Firmfixedcosts F 0.214 0.171 Priceadjustmentcost κ 6.60 p Elasticityofsubst. forintermediates (cid:101) 4.3 Wageadjustmentcost κ 653 87.4 w Portfolioadjustmentcost κ 0.01 d Netreplacementrate ρ¯ 0.725 0.523 Coefficientoninflation φ 1.5 π Coefficientonoutputgrowth φ 0.5 y Interestratesmoothing ρ 0.85 R Steady-stateTFPlevel a¯ 1 0.835 AutocorrelationofTFP ρ 0.95 A Std. dev. ofTFPshock σ(ε ) 0.00348 0.00985 a Corr. ofTFPshockH/F ρ(ε ,ε∗) 0.829 a a Steady-statelevelofgovt. spending g¯ 0.257 0.180 Autocorrelationofspending ρ 0.775 0.855 g (cid:0) (cid:1) Std. dev. ofspendingshock σ ε 0.00380 0.00443 g (cid:16) (cid:17) Corr. ofspendingshockH/F ρ ε ,ε∗ 0.293 g g 23

Table2: Targetedmoments. Moment Core Periphery Source Laborforceshare 0.601 0.399 OECD RealGDPshare 0.653 0.347 OECD GovernmentshareinGDP 28.1% 24.6% OECD S.d. ofrealGDP(filtered) 0.88% 1.64% OECD Meanunemploymentrate 8.38% 12.23% OECD Sd. ofunemployment(filtered) 0.380% 0.646% OECD Corr. ofunemployment(filtered) 0.510 OECD Netreplacementrate 0.725 0.523 Christoffeletal.(2009) SSjobfindingrate 0.3 BaltaandDelgado(2009) SSvacancyfillingrate 0.7 Christoffeletal.(2009) Consumptionhomebias 0.85 CorboandOsbat(2013) OECDdataistakenintherange1984Q1–2014Q4. GDPinthistableisdefinedasthesumoffinalprivate andgovernmentexpenditure. FilteredstandarddeviationsarecalculatedfromHodrick-Prescottfiltered datawithsmoothingparameter1600,whereGDPisinlogarithm. countries to the standard value of 0.99 which yields an annual real interest rate of 4 per cent. The curvature of consumption γ in both countries is set to 1.5 as reported in Smets and Wouters (2003). The parameter σ in both countries is set to 0.744, implying an elasticity of substitution between Home and Foreign goods of 3.9 matching the European average of estimates reported in Corbo and Osbat (2013). Given that value, ∗ we calculate values for the home good preferences ψ,ψ to match a GDP-weighted average of domestic expenditure shares of 85 percent as estimated in Balta and Delgado (2009),andensuringthattherelativepriceofforeigngoodsinsteadystateequals p = 1. ∗ The inverse elasticities of search effort φ,φ are chosen to match the estimate in Meyer (1990) of an elasticity of unemployment duration with respect to the level of benefits of ∗ 0.9. The effort scaling parameters κ ,κ are set to normalize steady state effort in each e e countrytounity. Wesetthematchingelasticity µ totheconventionalvalue0.5accordingtoestimates by Burda and Wyplosz (1994). The bargaining power of workers ξ is set at 0.9. As observed by Hagedorn and Manovskii (2008), workers need to capture a high share of the match surplus for a search model to reproduce the volatility of unemployment in thedata. Thisholdstrueeveninthepresenceofthewagerigiditiesweimposehere.11 ∗ ∗ ∗ The matching efficiencies κ ,κ ; separation rates s,s ; and vacancy costs κ ,κ are m m v v set in each country to jointly match an average quarterly vacancy-filling probability of 11Many ways have been proposed to address the fact that the standard search and matching model fails, for a standard calibration, to account for the cyclical properties of unemployment and vacancies, the so called ”Shimer puzzle” (Shimer, 2005). Contributing to this debate is beyond the scope of our paper. 24

70percent(Christoffeletal.,2009),aquarterlyjobfindingrateof30%(Elsbyetal.,2013) and unemployment rates of 8.4 percent in the Core and 12.1 percent in the Periphery. ∗ The fixed cost F,F is set in each country such that profits π are zero in the steady t state.12 The replacement rates ρ,ρ ∗ of the unemployment insurance system are set to matchaveragevaluesreportedinChristoffeletal.(2009). The cost coefficient of price adjustment κ is set to 6.06 and the demand elasticity p for varieties is set to (cid:101) = 4.3 in both countries. This leads to a steady-state markup of intermediate good producers of 30 percent and a Phillips curve that has the same slope coefficient on marginal cost as a corresponding Phillips curve with price rigidities á la Calvo with an average price duration of two quarters. This value is lower than what is common in the literature; stronger price rigidities would imply that unemployment rises after positive technology shocks because of the associated fall in markups. This is a common problem in New-Keynesian models with search frictions (see for example Gertler et al., 2008). We do, however, allow for substantial wage rigidity and set the ∗ wage adjustment costs κ ,κ to match the volatility of (HP-filtered) unemployment w w ratesineachcountry. Monetary policy is described by a simple Taylor rule with a coefficient of inflation of φ = 1.5, a coefficient on output growth of φ = 0.5, and an interest rate smoothing π y coefficient of ρ = 0.85. The portfolio adjustment costs on the internationally traded R non-contingentbondissetto κ = 0.01followingBenigno(2009). d The steady-state level of Home productivity a¯ is normalized to one and the Foreign ∗ level a¯ is set to reproduce the ratio of Periphery to Core GDP. We set the technology shockpersistencetoρ = 0.95inbothcountries. Thestandarddeviationoftheshocksin a Home and Foreign are chosen to match the HP-filtered standard deviation of real GDP in the Core and the Periphery. The correlation of the two shocks is chosen to match the correlation between filtered unemployment rates in the Core and Periphery of 51 percent. The resulting correlation is approximately 0.8, which implies that the synchronization of the business cycle across the two countries is quite high. The steady-state levels of government spending g¯ are set to match the share of government spending inGDPgovernmentspendingprocessisparametrizedtomatchdetrendedgovernment expendituredata. The Eurozone-specific moments we target are summarized in Table 2. The Periphery contributes about 40 percent of the EZ-12 labor force, but only about 35 percent of real GDP. It also has a higher volatility of GDP and unemployment. Nevertheless, our calibration features a higher degree of wage rigidity in the core than in the periphery. This is because the relative standard deviation of unemployment to output is actually higher in the core, so that for a shock of a given size unemployment has to rise more 12DoingsoavoidstheproblemofendingupwithanegativereplacementrateintheRamseysolution. Since we are setting a high bargaining power for workers, the Ramsey planner tends to undo this bargaining power with a lower replacement rate. Without the fixed cost F, it is possible to set a negative replacementrate(i.e. ataxonunemploymentinsteadofaninsurance)becauseunemployedworkerscan stillconsumeapositiveamountofprofits. 25

in the core, which is achieved by setting a higher degree of wage rigidity. Finally, the peripheryhasahigheraverageunemploymentrate,butalessgenerousunemployment insurancesystemasmeasuredbynetreplacementrates,whichinourcalibrationtranslatesintoahighercalibratedvaluefortheseparationrate. 5. Results We now present the results from our numerical simulations. We first confirm that our calibration of the status quo produces second moments that are close to the data. WethencomputeRamsey-optimalunemploymentinsurancepolicies,andalsocalculate optimalpolicywithouttransfers,i.e. imposingthatT = 0holdsatalltimes. Evenwitht out transfers, we find that optimal replacement rates are countercyclical in the model. With regards to transfers, we confirm our predictions from the simple model: Replacement rates rise when a country receives a transfer, so the generosity of unemployment insurance becomes more countercyclical in the presence of transfers. Nevertheless, the impact of transfers on output and unemployment rates is relatively minor, indicating that transfers can be implemented through the unemployment insurance system without causing large distortions in labor markets. We then explore how our results differ underalternativescenarios. 5.1. Statusquo Table3summarizeskeysecondmomentsofourcalibratedmodel. The standard deviations of real GDP and unemployment in each country are targeted by the calibration. The standard deviation of consumption is higher than in the data. This is an outcome of the fact that our model does not include capital and has a sizableshareofoutputabsorbedbyarelativelystablegovernmentexpenditureprocess. Weseethisoutcomeasanacceptablecostofsimplification,butthegainsfromstabilizationmightthereforebesomewhatoverstated. Thestandarddeviationofwages,bycontrast,isreasonable. ThebottomrowalsoreportsstatisticsontheHometradebalanceas a percentage of domestic GDP. The standard deviation is of a realistic magnitude. This implies that even in the absence of cross-country fiscal transfers, households are able to smooth international consumption risk to some extent through savings in the bond market. The contemporaneous correlations with GDP are close to the data, with the exception of real wages. Our HP-filtered wage series display very little correlation with filtered output. As for the cross-correlations across countries, the cross-correlation of unemployment is targeted by the calibration, and that of output and wages is within reasonable range. But the correlation of real consumption across countries is very close to one. Our model therefore suffers from the Backus et al. (1992) consumption correlation puzzle, overstating the amount of international risk sharing present in the data. Addressingthepuzzlewilllikelystrengthenourresults. 26

Table3: Secondordermomentsinbenchmarkcalibration. Variable s.d. s.d. rel. to corr. with cross-corr. dom. GDP dom. GDP RealGDP Home 0.88*[0.88] 1.00[1.00] 1.00[1.00] 0.86[0.91] Foreign 1.64*[1.64] 1.00[1.00] 1.00[1.00] Consumption Home 2.00[1.16] 2.27[1.32] 0.98[0.93] 0.96[0.58] Foreign 2.56[1.91] 1.56[1.16] 0.98[0.98] Realwage Home 0.28[0.39] 0.32[0.44] 0.80[-0.07] 0.81[0.70] Foreign 0.65[0.70] 0.40[0.43] 0.62[0.15] Unemployment Home 0.38*[0.38] 0.43[0.43] -0.71[-0.68] 0.51*[0.51] Foreign 0.67*[0.67] 0.41[0.41] -0.71[-0.60] Tradebal.,%GDP Home 0.26 0.34 -0.51 Secondmomentsasobtainedfromsimulatingalinearapproximationofthemodelatbenchmarkcalibration. RealGDP,consumptionandrealwageareinlogarithms. AllseriesareHP-filteredwithsmoothing parameter 1600. Standard deviations are reported in percentage points. Moments targeted by the calibrationaremarkedwithanasterisk. CorrespondingmomentinOECDdatainbracketswhereavailable (maximaldaterange1984Q1–2014Q4), whereGDPistakenasthesumofprivateandgovernmentfinal consumptionexpenditure,andrealwagesaretakenasmanufacturingwagesdividedbyCPI. 5.2. Ramseypolicy We now compute Ramsey-optimal policies with and without transfers and confirm our predictions from the simple model about the cyclicality of policy instruments. The resultsaresummarizedinTable4. The left half of the table documents the standard deviations of optimal replacement rates and transfers and their correlation with output. The standard deviations of replacement rates are relatively large, around three percentage points. Remarkably, all replacement rates are countercyclical. Transfers to the Foreign country as a percentage of Foreign GDP are also large and countercyclical. In the Foreign country, replacement rates are more volatile and more negatively correlated when the planner has access to transfers than without transfers (T = 0 imposed). However, in the Home country the t oppositepatterncanbeobserved. However, looking at correlation coefficients does not say much about the behavior ofpolicybecauseitdoesnotsaybyhowmuchreplacementrateschangeinresponseto economicconditions. Moreover,GDPacrossthetwocountriesishighlycorrelated,and so correlation coefficients mask differences in policy reactions to shocks originating at HomeandinForeign. For these reasons, our preferred measures of the cyclicality of optimal policy are the coefficients from bivariate regressions of replacement rates and transfers to Home and Foreign GDP. These are shown in the right half of Table 4, and they clearly con- 27

Table4: CyclicalityoftheRamsey-optimalunemploymentinsurance. summarystatistics: bivariateregression: s.d. corr. with coefficient coefficient dom. GDP HomeGDP ForeignGDP Homereplacementrate 2.38 -0.33 -1.37 0.43 (withouttransfers) 2.45 -0.49 -1.08 0.14 Foreignreplacementrate 3.71 -0.61 0.83 -1.16 (withouttransfers) 3.10 -0.52 0.15 -0.58 Foreigntransfer,%GDP 1.18 -0.35 0.69 -0.67 Moments as obtained from simulating a linear approximation of the model at benchmark calibration. Standarddeviationsarereportedinpercentagepoints. RegressioncoefficientsareobtainedfromregressingeachrowvariablesimultaneouslyonthelogofHomeandForeignrealGDP. firm our predictions from the simple model. On average, when Home GDP rises by one percent, replacement rates fall by 1.37 percentage points in the Home country with transfers. Without transfers, they only fall by 1.08 percentage points. Foreign replacement rates rise by 0.83 percentage points in the presence of transfers, compared to only 0.15percentagepointswithouttransfers. TheForeigncountryalsoreceives,onaverage, atransferof0.69percentofitsGDPwhenGDPatHomerisesbyonepercent. Thesame patternsappearinresponsetochangesinForeignGDP:Replacementratesarecountercyclical with respect to domestic GDP and procyclical with respect to GDP abroad, and thiscyclicalitybecomesstrongerinthepresenceoftransfers. To further illustrate these results, we compare impulse response functions to a negativeproductivityshockintheperipheryinFigure3(thepatternsarequalitativelysimilar for a shock in the core). Specifically, we contrast the optimal Ramsey planner response with the no-transfer Ramsey solution and the status quo policy (no transfers, constantreplacementrates). The upper panel of the figure depicts the four direct policy instruments: Home and ∗ ∗ Foreignbenefits b ,b andHomeandForeignpayrolltaxes τ,τ . Thesecanbemapped t t t t intothereplacementratesandtransferpolicies,butwewanttohighlighthowvariations in these policy variables are implemented by the planner. After a negative productivity shock in the periphery, the Ramsey planner (solid red line) implements a transfer from the core to the periphery. It does so by increasing unemployment benefits in the peripheryandatthesametimecuttingpayrolltaxes,andeffectingtheoppositepattern inthecore. Benefitsdropbelowtheirinitiallevelaftersixquartersinordertomakethe increase in unemployment dissipate more quickly, while taxes stay low for a long time. A large transfer finds its way to the economy mainly by a reduction in payroll taxes which affect most of the population, rather than benefits which only affect only the relatively few unemployed workers. When the planner does not have access to transfers (blue dashed line), the behavior of the instruments is markedly different. In the core, 28

Figure3: Impulseresponses,negativeForeignproductivityshock. (a)Policyinstruments. Benefits (H) Benefits (F) Taxes (H) Taxes (F) 0.5 0.5 10 10 0 0 Ramsey Ramsey (no transfers) 5 status quo 5 -0.5 -0.5 0 0 -1 -1 0 8 16 0 8 16 0 8 16 0 8 16 (b)Aggregateoutcomes. GDP (H) Unemployment (H) Consumption (H) Trade balance (H) 0 0 0.2 -0.1 0.15 -0.2 0.1 -0.2 0.1 -0.4 0.05 -0.3 0.05 -0.6 -0.4 0 -0.8 0 -0.5 -0.05 0 8 16 0 8 16 0 8 16 0 8 16 GDP (F) Unemployment (F) Consumption (F) Transfer (F) 0.3 0 0 0.2 0.25 -0.1 0.15 -0.2 0.2 -0.2 0.1 -0.4 0.15 -0.3 0.05 -0.6 0.1 -0.4 0 -0.8 0.05 -0.5 -0.05 0 0 8 16 0 8 16 0 8 16 0 8 16 ImpulseresponsefunctionstoaonestandarddeviationnegativeproductivityshockinForeign. Benefits,taxes,GDPandconsumptionareinpercentdeviationfromthesteadystate. Unemploymentis inpercentagepointdeviationfromthesteadystate. Tradebalance(H)isinpercentofHomerealGDP andtransfer(F)isinpercentofForeignrealGDP. 29

Table5: Gainsfromstabilization. (1) (2) (3) (4) s.d. % statusquo Ramsey(notransfers) Ramsey Debt Consumption Home 4.17 2.91 3.59 2.99 Foreign 6.59 5.21 3.85 5.05 Unemployment Home 0.72 0.26 0.27 0.26 Foreign 1.03 0.54 0.52 0.53 Standarddeviationsfromsimulatedmodeldata,unfiltered. Consumptionisinlogarithms. benefits and taxes stay mostly flat after the shock. The level of benefits still rises in the periphery,butthistimetheincreaseinbenefitshastobefinancedbyhighertaxes. Inthe status quo calibration with constant replacement rates, taxes also increase even though the level of benefits barely moves. This is because unemployment increases markedly and persistently, and more recipients of benefits and fewer payroll contributors necessitate a higher level of contributions to balance the national unemployment insurance budget. The lower panel of the figure depicts the aggregate outcomes under the different policies. The responses of GDP and unemployment in the core are barely affected, but therearemarkeddifferencesintheperipherywheretheshockhits. TheRamseyplanner manages to stabilize GDP and unemployment with respect to the status quo policy. However, the response of these variables is nearly identical whether the planner can make use of cross-country transfers or not. This result obtains despite the fact that the transfer is sizable—0.28 percent of Foreign GDP at its peak—and underscores our predictionthattransferscanbechanneledthroughtheunemploymentinsurancesystem withoutdistortinglabormarketoutcomes. Of course, transfers have a stabilizing function on consumption. Compared to the status quo, the planner without access to transfers already manages to stabilize consumption in both countries by reducing the size of unemployment fluctuations. Still, consumption drops by more in the periphery where the shock originates. Transfers even out the burden of reducing consumption, raising it in the periphery and reducing it in the core. Finally, the figure also depicts the response of the core trade balance. It is positive under all three policies, reflecting the fact that the periphery runs a current accountdeficittoprivatelysmoothoutthedropinconsumption. Itdoessobytakingup debt. However,non-contingentdebtisnotthesameasfullrisksharingandtheplanner can improve on this allocation. The transfer of course results in an even larger trade balance. How important are the stabilization gains? We measure stabilization in terms of the reductioninthevariancesofconsumptionandunemployment,documentedinTable5. Column (1) shows the standard deviations of unfiltered (log) consumption and un- 30

employment in both countries at the status quo policy. Column (2) shows the same statistics at the Ramsey policy without transfers. The Ramsey planner can achieve a reduction of at least 20 percent in the volatility of consumption and at least 45 percent in the volatility of unemployment by adjusting national replacement rates alone. In Column(3),theplannercanadditionallypooltheunemploymentinsurancesystemsatthe supranational level. In that case, it achieves additional stabilization in the periphery: unemploymentvolatilitydropsthreepercent,andconsumptionvolatilitydrops26percent. In the core however, the volatility of unemployment and consumption increase by 4 percent and 24 percent, respectively, although they remain lower than under the statusquo. Why is the planner not able to stabilize consumption in both countries? The answeristhatbusinesscyclesinthemodelarehighlycorrelated,owingtothecalibration. There is relatively little country-specific risk that can be reduced through diversification. Instead, the planner is implementing transfers to shift risk from the periphery to the relatively more stable core. As a result, the standard deviations of consumption in the core and the periphery move closer to each other. The policy distributes risk more evenlyacrossthetwocountries,butitisnotaPareto-improvementovertheno-transfer policy: The core would be better off without transfers. We conjecture that a Pareto improvement could be achieved by a perpetual steady-state payment from the periphery tothecoreasacompensationfortheaddedrisk. Butthisisruledoutbyourrequirement that transfers be zero in expectation. We think that this point has not been appreciated in the existing literature: A fiscal risk sharing mechanism in the Eurozone is likely to bedetrimentaltothemorestableeconomiesofthecoreevenwhenpermanenttransfers areruledoutbydesign,simplybecauseriskisperpetuallyshiftedfrommorevolatileto morestablecountries. 13 5.3. Alternativescenarios How sensitive are our results to the calibration? Our main qualitative result— transfers make optimal unemployment insurance more countercyclical—hold up for any reasonable parametrization of the model, but the quantitative results are sensitive toseveralparameters. Table6computesthecyclicalitiesofoptimalpolicyunderavarietyofalternativeassumptionsontheparameters. Column (1) repeats the baseline calibration regression coefficients from Table 4. In Column (2), the Home population share is increased to ω = 0.8. This scenario implies that the size of the Foreign country is roughly that of Spain within the Eurozone. Compared to the baseline, the Home replacement rate becomes less countercyclical, Foreign replacement rates more countercyclical, and Foreign transfers become larger. These resultsarefullyinlinewiththepredictionsfromoursimplifiedmodel: Asmallercountry 13Ourcalibrationtoonlytworegionsmightunderstatetheamountofdiversifiablebusinesscyclerisk. Atthesinglecountrylevel,thecomponentofriskthatisdiversifiableacrosstheunioniscertainlylarger, so that it might be possible to reduce consumption volatility in every country with appropriate risksharingpolicies. 31

Table6: Sensitivityofoptimalpolicytoalternativescenarios. (1) (2) (3) (4) (5) (6) bivariateregressioncoefficients Baseline ω = 0.8 σ A∗ = 0 u¯∗ = u¯ σ = 0.01 ξ = µ Replacement HomeGDP -1.37 -1.06 -2.90 -1.55 0.59 -1.34 rate,Home ForeignGDP 0.43 0.25 7.42 0.53 -1.69 0.29 Replacement HomeGDP 0.83 1.16 2.53 0.45 0.05 0.73 rate,Foreign ForeignGDP -1.16 -1.44 -9.96 -1.08 -1.12 -1.55 Transfer,Foreign, HomeGDP 0.69 1.11 1.17 0.71 -0.03 0.65 %GDP ForeignGDP -0.67 -1.03 -2.69 -0.69 -0.01 -0.66 RegressioncoefficientsareobtainedfromregressingeachrowvariablesimultaneouslyonthelogofHome andForeignrealGDP. can be better insured by the union, and the countercylical effect of risk sharing on replacementratesbecomesstronger. In Column (3), the variance of Foreign productivity shocks is set to σ A∗ = 0, so that almost all output fluctuations come from shocks originating in the Home country. This scenario isolates the policy reaction to country-specific shocks, which arere not clearly visible in the baseline because of the high correlation of shocks. The cyclicalities of replacementratesandtransfersincreaseeverywhere. ∗ In Column (4), the steady-state unemployment rate in the Foreign country is u¯ reducedtothevaluetothatintheHomecountryu¯. Thisscenariocanbethoughtofasthe implementation of labor market reforms in the Eurozone periphery that reduce structuralunemployment. Asaresult,replacementratesbecomemorecountercyclicalinthe Home country and less countercyclical in the Foreign country, while the magnitude of ∗ transfers is roughly unchanged. Intuitively, when u¯ falls, Foreign GDP becomes less volatile and its share in total GDP rises, meaning that the core will insure less risk of the periphery. Replacement rates become less countercyclical in the less well-insured Foreigncountry,whiletheoppositehappensintheHomecountry.14 In Column (5), the elasticity of substitution between Home and Foreign goods is set close to unity. This scenario puts the economy close to the case studied in Cole and Obstfeld (1991) where movements in the terms of trade achieve perfect risk sharing even under financial autarky. In line with their finding, the size of optimal transfers is nowclosetozero. Thereplacementratesbecomemoreprocyclicalasaresult. Most other parameters of the model have little impact on the quantitative results. For example, the choice of the bargaining weight ξ is crucial to match the volatility of unemployment in the model, but does not affect optimal policy. In Column (6), we set ξ = µ soastobringthemodelclosertotheHosios(1990)condition. Thechangestothe 14AsimilarresultobtainswhenthedegreeofwagerigidityintheForeigncountryκ∗ isreduced. w 32

Figure4: Sensitivityofoptimalpolicytopricerigidities. Replacement rate (H) Replacement rate (F) Transfer/GDP (F) 1 1.5 1 0.5 0 0.5 GDP (H) GDP (F) 0 0 GDP (H), no trans. GDP (F), no trans. -1 -0.5 -1 -0.5 -2 -1.5 10 20 30 40 50 10 20κ 30 40 50 10 20 30 40 50 p Regression coefficients obtained from regressing each row variable simultaneously on the log of Home andForeignrealGDP,asafunctionoftheparameterκ (commonacrossbothcountries),fortheRamseyp optimalpolicywithandwithouttransfers. Theblackdottedlinerepresentsthebaselinecalibrationvalue κ =6.601. p coefficientsinthetablearerelativelysmallandthereisnoclearpatterninthechangeof cyclicality. Similarresultsobtainwhenvaryingotherparameters. Likewise, the degree of nominal price rigidities has little impact on the optimal policy. Figure4plotsthebivariateregressioncoefficientsontheoptimalpolicyinstruments as a function of the Rotemberg price rigidity parameter κ . The dotted line marks the p baseline value of κ = 6.601. This is at the lower end of values used in the literature, p as discussed above. But a lower or higher value for κ does little change to the cyclip cality of the Home replacement rate. It has some effect on the response of the Foreign replacement rate to changes in Home GDP (red lines in the middle panel of the figure): Larger price rigidities induce the Foreign replacement rate rise less, on average, when Home GDP rises. Still, the qualitative conclusion of more countercyclical replacement rates in the presence of transfers is unaffected by the size of price rigidities. 15 What’s more surprising is that even the transfer policy is little changed as κ varies. The ineffip cienciescausedbythepresence ofacurrencyunionandnominalrigiditiesemphasized byFarhiandWerning(2012)seemtobequantitativelyunimportantcomparedwiththe otherfinancialmarketimperfectionsinthemodel. 6. Thedifferencebetweendebtandinsurance Inwritingthispaper,wefrequentlyencounteredtheargumentthatafiscalrisksharing mechanism without permanent transfers would be no different from using government debt to run countercyclical deficits. Here, we show that this argument does not 15ThisfindingisinlinewithKekre(2016)whofindsthatoptimalunemploymentinsuranceisaffected bynominalrigiditiesandtheensuingdemandexternalitiesonlywhenthezerolowerboundoninterest ratesisbinding. 33

hold. Our fiscal risk sharing mechanism works like a fairly priced insurance policy: Even though on average, the premia and expected payments net out to zero, one is still better better off buying the insurance than taking out a bank loan in the event of damage. We modify our quantitative model to introduce national government debt. All debt is denominated in nominal terms in the common currency of the two countries. The budgetconstraint(43)oftheHomegovernmentismodifiedasfollows: g +u b = τ n +d − κ d d2 −R d gt−1 . (57) t t gt gt t gt gt t−1 2 π pt Note that national governments face the same quadratic portfolio adjustment costs astheprivatesector. Ifthiswerenotthecase,theprimaryuseofgovernmentwouldbe to replace frictional private borrowing with frictionless public borrowing. The Foreign ∗ government similarly can similarly issue debt d . The clearing condition (51) in the gt bondmarketismodifiedtoread: (cid:16) (cid:17) 0 = ω (cid:0) d −d (cid:1) +(1−ω)p d ∗ −d ∗ . (58) t gt t t gt We then compute the Ramsey-optimal policies where the planner does not have access to cross-country transfers T but can instead optimally choose the national govt ∗ ernment debt levels d and d . As before, the planner can also choose the national gt gt ∗ replacement rates ρ ,ρ . From a welfare perspective, the value of the Ramsey problem t t with debt is bounded from above by that with transfers, and from below by the value ofthatwithnationallybalancedbudgets. It turns out that the planner is not able to distribute risk very effectively with debt. This can be seen immediately from Column (4) in Table 5: The values for the volatility ofconsumptioninparticularareveryclosetothosewithouttransfers(Column2). InFigure5,weplotimpulseresponsesafteranegativecountry-specificproductivity shockintheperiphery,comparingthefullRamseysolution,thesolutionwithdebt,and thesolutionwithnationallybalancedbudgets(labeled“notransfers”). It is immediately clear that the solution with debt is very close to the solution with balancedbudgets. ThedropinconsumptionisminimallyshiftedfromForeigntoHome, whichisreflectedinaslightlyhighertradebalance. ItisnotadvantageousfortheplannertomoveresourcesfromHometoForeignwithdebt,andthereasonisofcoursethat debt has to be repaid. The rightmost graph in Figure 5, labeled “Transfer (F)”, depicts the budget deficit of the Foreign government in the solution with debt. This is the closestcorrespondencetothesupranationaltransfersinthefullRamseysolution,whichare also plotted. The planner does run a budget deficit in the periphery after the shock, but the optimal size of the deficit is only about 0.08 percent of GDP at the peak, since it will have to be repaid eventually (in the graph, starting four years after the shock). By contrast, when the planner has access to transfers the stabilization gains are much larger, despite the fact that transfers are temporary and equal zero in expectation. With 34

Figure5: Debtisnotthesameasinsurance. Consumption (H) Consumption (F) Trade balance (H) Transfer (F) 0.3 0 0 0.25 -0.2 -0.2 0.1 0.2 -0.4 -0.4 0.15 0.05 -0.6 -0.6 0.1 Ramsey (no transfers) 0.05 -0.8 (debt) -0.8 0 0 0 8 16 24 32 40 0 8 16 24 32 40 0 8 16 24 32 40 0 8 16 24 32 40 Impulseresponsestoaone-standarddeviationnegativeForeignproductivityshock. debt, any “transfer” has to be repaid with certainty at some point in the future; with fiscalrisksharing,transfersonlyhavetoaverageoutovertime. 7. Conclusions In this paper, we have used an international business cycle model with frictional labor markets and incomplete financial markets to discuss optimal unemployment insurance policy operating across multiple countries. Theoretically, the possibility to set different policies across countries augments the classic policy trade-off between efficientemploymentandinsuranceofunemploymentriskwithaconcernforinternational risk-sharing. We have shown that cross-country insurance through the unemployment insurance system can in principle be achieved without affecting unemployment levels; and that the desirability of international risk sharing introduces a countercyclical elementtotheoptimalunemploymentinsurancepolicy. CalibratedtoEurozonecoreandperiphery,ourtwo-countrymodelimpliedthatoptimal replacement rates are countercyclical even from a national perspective without transfers. Adding transfers markedly increased this countercyclicality, and the optimal transfers were found to be sizeable (about 0.7 percent of GDP following a one percentage point decrease in GDP). However, we also found that, due to the high correlation of business cycles across the Eurozone, there is very limited scope to diversify risk. Instead, the optimal planner policy mainly implied a shift of consumption risk from the peripherytothecoreinordertodistributeitmoreevenly. There is one important direction in which our findings could be extended in further research. The optimal policy we compute here is one in which the planner has perfect knowledge of the structure of the economy. One of the most difficult issues in implementing a policy such as the one in this paper is that the structural rate of unemployment can only be reliably estimated in hindsight, if at all. It would be useful to see whether simple policy rules that are more easily implementable under imperfect informationcanreasonablyapproximatetheoptimalpolicy. 35

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Appendix A. PropositionsforSection1 TheoptimalreplacementrateintheabsenceofprivaterisksharingsatisfiesEquation (19)inthemaintext: (cid:18) (cid:19) (1−n)(1−ρ) 1−ρ x −(cid:101)n logρ+ = −(cid:101) y n+(1−n)ρ ρ n+(1−n)ρ ρ n (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) =:I(ρ) =:H(ρ) where x = y/(y+T) is the income to consumption ratio of the Home country. Here, we prove the properties of the optimal policy as discussed in the text. Throughout, we makethefollowingassumption: Assumption. I isstrictlyconcave, H and Hy arestrictlyconvexin ρ given x. We numerically verified this assumption for a wide range of parameters. The limit behavior of the functions at the corners is easy to prove and together with our assumptiondeterminestheshapeofthecurvesinthemaintext. (cid:16) (cid:17)(1−µ)/µ Proposition 1. I(0) = 1−n¯ where n¯ = κ κma and I(1) = 0. Also, we have n¯ m κv (cid:16) (cid:16) (cid:17)(cid:17) H(0) = 0, H exp −1−µ ξ = 0,lim = +∞ ,and H (cid:48)(ρ) stronglyconvexgiven x. µ 1−ξ ρ→1 Proof. We start with the insurance term I(ρ). At the limit when ρ → 0, we have w → 0 (cid:16) (cid:17)(1−µ)/µ and n = κ κm (a−w) → n¯. Therefore: m κv (1−n)(1−ρ) ρ→0 1−n¯ −→ . n+(1−n)ρ n¯ Theremainingtermof I(ρ) mustthereforegotozero. Indeed, dn ρ a 1−µw21−ξ (cid:101)n = =− ρ dρ n a−w µ a2 ξ 1 1−µ ξ = logρ µ ξ −(1−ξ)logρ andtherefore (cid:18) (cid:19) 1−ρ (cid:101)n logρ+ ρ n+(1−n)ρ (cid:18) (cid:19) 1−µ ξ 1−ρ 1 ρ→0 = 1+ −→ 0. µ ξ −(1−ξ)logρ n+(1−n)ρlogρ 40

Forthecase ρ → 1,thefirsttermclearlydisappears: (1−n)(1−ρ) ρ→1 −→ 0 n+(1−n)ρ andforthesecondterm,wehave: (cid:18) (cid:19) (cid:18) (cid:19) 1−µ ξ 1−ρ 1 ρ→0 1−µ 1−ρ 1+ −→ 1+ lim = 0. µ ξ −(1−ξ)logρ n+(1−n)ρlogρ µ ρ→1 logρ Turn now to the efficiency term H(ρ). As ρ → 0, n → n¯ > 0 and and w → 0. Therefore (cid:18) (cid:19) y x xw 1−ξ 1−µ 1 ρ→0 −(cid:101) = − + −→ 0. ρ n n a ξ µ logρ Andas ρ → 1, w → 1and n → 0+,sothat (cid:18) (cid:19) − xw 1−ξ + 1−µ 1 − ρ→ → 1 +∞ . n a ξ µ logρ (cid:16) (cid:17) µ 1−ξ Proposition 2. The optimal replacement rate is unique and strictly between exp 1−µ ξ andone. Proof. Since f (ρ) = H(ρ)− I(ρ) is continuous on [0,1] and a strictly concave by our assumption, it crosses zero at most twice. But f (0) > 0 and lim ρ→ −∞ , so there is a uniqueinteriorsolution ρ ∗ to f (ρ) = 0. Since I(0) > I(1) = 0and I isstrictlyconcave, I(ρ) > 0∀ρ ∈ (0,1) and the optimum has H(ρ ∗) > 0. Since H is a strictly convex function, H(0) = 0 and lim H(ρ) = +∞ and , H(ρ ) = 0 for exactly one ρ ∈ (0,1) ρ→1 0 0 (cid:16) (cid:16) (cid:17)(cid:17) and ρ ∗ > ρ . Finally, H exp µ 1−ξ = 0. 0 1−µ ξ Proposition3. Theoptimalreplacementrateisstrictlydecreasingin x. Proof. Takingthetotalderivativeoftheoptimalityconditionwithrespectto x,wehave ∂I ∂H ∂I dρ ∂H dρ 0 = − + − ∂x ∂x ∂ρdx ∂ρ dx dρ ∂I − ∂H ⇔ = −∂x ∂x . dx ∂I − ∂H ∂ρ ∂ρ Clearly, dI/dx = 0 and at the optimal ρ, we have dH/dx = H(ρ)/x > 0. Furthermore, we know that I(0) > H(0) and I(ρ) = H(ρ) for exactly one value of ρ, so it must be thecasethat dH/dρ > dI/dρ attheoptimal ρ. Therefore dρ/dx < 0. 41

Thisresultshowsinparticularthatthereplacementrateisincreasinginforeignpro- ∗ ∗ ∗ ductivity a ,sinceanincreasein a raises y andthereforedecreases x. Finally,wearegoingtoestablishcountercyclicalityofρinthelimitcasewhenHome ∗ countrybecomesverysmall,andwhen a and a areindependent. Proposition 4. In the limit as ω → 0, the optimal replacement rate is unique, strictly below one,andstrictlydecreasingin a. Proof. As ρ → 0,therisksharingcondition(19)becomes E[y ∗] y x = . E[y] y∗ Theoptimalchoiceof ρ when x ischosenoptimallycannowbedescribedas I(ρ) = H˜ (ρ) 1 E[y ∗] y where H˜ (ρ) = −(cid:101) y . ρ n E[y] y∗ Byourassumption, H˜ (ρ) isastrictlyconvexfunction. Thevalueof H˜ atzerois H˜ (0) = H(0) E[y ∗]lim ρ→0 wn = H(0)·0 = 0. E[y] y∗ Forthelimitatone,wenote H˜ (ρ) E[y] y ∗ = − w2 (cid:18) 1−ξ + 1−µ 1 (cid:19) − ρ→ → 1 +∞ E[y∗] a ξ µ logρ since w → a as ρ → 1. Therefore, the optimal ρ when y/c is chosen optimally has the same properties that we used before holding x constant. In particular, the optimal replacementrateisuniqueandstrictlybelowone. Also,wehavedH˜/dρ > dI/dρatthe optimal ρ asinProposition(3). Takingthetotalderivativeagain,wehave dρ ∂I − ∂H˜ = −∂a ∂a da ∂I − ∂H˜ ∂ρ ∂ρ wherethedenominatorofthefractionisnegative,sodρ/dahasthesamesignas ∂I − ∂H˜ . ∂a ∂a Thederivativesof I and H˜ withrespecttoproductivity a are: ∂I ∂I ∂n = ∂a ∂n ∂a ∂n (cid:18) 1−ρ (cid:19)2(cid:18) w1−µ 1 1 (cid:19) = − < 0 ∂a n+(1−n)ρ a µ logρ 1−ρ 42

and ∂H˜ H˜ = > 0. ∂a a Therefore dρ/da < 0. The last two propositions combined establish the final result of Section 3. A decline ∗ in a raises ρ, and at the same time output y falls. An increase in a also raises ρ, and the lowerreplacementratecausesytofall. Therefore,thereplacementrateiscountercyclical conditionaloneitherproductivity. Independenceoftheproductivitiesthenimpliesthat itisalsocountercyclicaloverall. 43