To Cap or Not to Cap? Energy Crises in a Currency Union

Board of Governors of the Federal Reserve System International Finance Discussion Papers ISSN 1073-2500 (Print) ISSN 2767-4509 (Online) Number 1428 December 2025 To Cap or Not to Cap? Energy Crises in a Currency Union Momo Komatsu Please cite this paper as: Komatsu, Momo (2025). “To Cap or Not to Cap? Energy Crises in a Currency Union,” International Finance Discussion Papers 1428. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/IFDP.2025.1428. NOTE: International Finance Discussion Papers (IFDPs) 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 International Finance Discussion Papers Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

To Cap or Not to Cap? Energy Crises in a Currency Union Momo Komatsu1 Federal Reserve Board Abstract During the energy crisis in 2022 some Euro Area countries introduced price caps on energy, while others did not, leading to about 30 percentage points higher energy inflationinuncappedcountries. Thispaperinvestigatesthetrade-offspolicymakersfacewith energypricecapsinatwo-countrycurrencyunionmodelwithsharedenergysupply. The cooperative,optimaloutcomeisforneithercountrytoimposeapricecap,sincethecapis acostlymarketdistortion. However,cappingallowsacountrytoavoidacrisisatthecost ofnegativespilloversontheuncappedcountry,characterizedbyhighinflationandlower output. The quantitative model with non-homothetic preferences and substitutability of energysourcesshowsthatthecostofthepricecapexceedsthecostofsuchspillovers,explaining why some countries capped prices while others did not. Moreover, I show that the spillovers from price caps contributed to about 10 (0.5) percentage points of energy (headline) inflation in the uncapped Euro Area countries in 2022. Targeted transfers, an alternativepolicytothepricecap,isacheaperandmoreeffectivewaytoboostconsumptionofthepoorwithoutcreatingdivergencewithintheunion. Keywords: Energycrisis,energypricecap,inflation,internationalspillovers JELcodes: E31,E63,F45,Q41 1I am grateful to Andrea Ferrero for his valuable support and feedback. I am also grateful to Jenny Chan, Sergio de Ferra, Simon Lloyd, Derrick Kanngiesser, Federica Romei, Rana Sajedi, Rustam Yamilov, andFrancescoZanettifortheirvaluablecommentsandconversations. Ithanktheseminarandconference participants at the University of Oxford, Imperial PhD Conference in Economics and Finance, MMF PhD Conference (University of Surrey), and the Workshop for Young Macroeconomists (Hitotsubashi University) for their discussions. The views expressed in this manuscript are solely my own and should not be interpretedasreflectingtheviewsoftheBoardofGovernorsoftheFederalReserveSystemorofanyperson associatedwiththeFederalReserveSystem. Email: momo.komatsu@frb.gov|Website: mkomatsu.com

1 Introduction The Euro Area, and Europe as a whole, experienced a large energy crisis in 2022. Energy pricesbeganrisinginmid-2021andsoaredin2022aftertheRussianinvasionofUkraine, as shown in Figure 1. The increase in energy prices triggered energy price cap decisions from many, but not all governments. This paper investigates the effects of an energy pricecapinasubsetofcountriesinacurrencyunionduringanenergycrisis,focusingon internationalspilloversandpolicytrade-offsofthepricecap. Figure1: NaturalgaspriceinEurope Notes: ThepriceindexoftheTitleTransferFacility(TTF)gasintheNetherlands. Mostpricesettersusethis priceasthereferencepriceandgascontractsareindexedtothisprice,theTTFpriceindexisthestandard gaspricebenchmarkforEurope(Rogge,2024).2 Forlongertimeseries,seeFigure16inAppendixC.Data source: IMFData(2024). Inflation rates in uncapped countries were considerably higher than in capped countries, as shown in Figure 2. In 2022, France and Germany, among others, decided to impose an energy price cap, whereas other countries including The Netherlands and Italy did not. The energy inflation in countries without an energy price cap was about 30 percentage points higher than in the capped countries in 2022Q3, and for headline inflation the difference was about 5 percentage points. Figure 17 in Appendix C shows that the divergence of both energy and headline inflation in 2022 was at an unprecedented magnitudesincethestartoftheEuroArea. The paper uses a two-country currency union model with a shared energy supply to investigate the trade-offs of an energy price cap. In the absence of price caps, an adverse energy supply shock not only reduces energy consumption, but also acts as a cost-push shock to an economy: a decline in the exogenous supply of energy depresses output while increasing inflation. The assumption of an exogenous energy supply reflects the highdependencyofEuropeonRussiangasbefore2022(Pescatori&Steurmer,2022;Moll 2Moreover, gaspricesdrivewholesaleelectricitypricesasthehighestmarginalcostofenergyproduction,sotheyareagoodindicatorforenergy/electricitypricemovements(Pescatori&Steurmer,2022). 1

Figure2: InflationintheEuroArea2020–2022 (a)Energyinflation (b)Headlineinflation Notes: AnnualizedenergyandheadlineinflationratesintheEuroArea. Source: Eurostat. Countrieswith anenergypricecapin2022areAustria,Estonia,France,Germany,Luxembourg,Malta,Portugal,Slovakia, Slovenia,Spain. CountrieswithoutareBelgium,Cyprus,Finland,Greece,Ireland,Italy,Latvia,Lithuania, TheNetherlands. Boldlinesareweightedaveragesforeachgroup. et al., 2023). Since the shock is essentially a relative price shock, it is a standard result in NewKeynesianframeworksthatitactslikeacost-pushshock. First, I find that capping the energy price allows a country to avoid the crisis while imposing negative spillovers on the uncapped country, causing divergence within the currency union. I define the energy price cap as a policy that fixes energy price. The government then pays the difference between the actual price of energy and its retail price. This setup is somewhat stylized, because in reality most governments imposed a price cap that was less strict. However, this assumption is not overly restrictive, as the strictness of the price cap shifts the overall level of responses but does not alter the underlyingmechanisms. With the price cap, a country does not suffer from the energy supply shock: output does not suffer a fall and inflation does not soar. Moreover, households can maintain their energy consumption. On the contrary, the uncapped country experiences a larger cost-push shock compared to the case of no price caps in the entire union, and also a larger decline in energy consumption. A crucial assumption behind these spillovers is the shared energy supply between the capped and uncapped countries, as is the case in Europe for Russian gas. During the adverse energy supply shock the capped country’s energy consumption does not decrease because the price does not change. Then, the energy supply available in the uncapped country declines even more, which causes an evenhigher(energy)inflationandalowerwelfare. Second, the paper demonstrates that for policymakers facing the decision – to cap or not to cap – there is a trade-off between the cost of the price cap and the cost of the spillovers. I analyze the welfare implications of energy price caps in the two-country 2

model,inwhicheachcountryfacestwopolicydecisionsattheonsetofanenergysupply shock. The optimal, cooperative outcome occurs when both countries refrain from imposing a price cap. This outcome avoids market distortions, as energy prices reflect true scarcityduringthecrisis. However,becauseofthelackofcooperationbetweencountries within the union, each country has an incentive to deviate from the cooperative strategy: ifonecountrydoesnotimposeanenergypricecap,theothercountryhasanincentiveto imposeonetoavoidtheenergycrisisasdescribedabove. Ifonecountryimposesapricecap,shouldtheothercountryfollowsuit? Ononehand, imposing the price cap leads to inefficiencies and high costs, as artificially low prices in both countries encourage excessive energy consumption despite scarcity. On the other hand,notimposingthecapmeansbearingthenegativespilloversfromthefirstcountry’s cap, which is also costly. In the quantitative model the cost of imposing the price cap are larger than the cost of these negative spillovers. Hence, the outcome of a static decision game is a price cap in one of the two countries, even though the cooperative strategy is for neither of the countries to impose the cap. This result explains why in reality some countriescappedtheenergypricewhileothersdidnot. The key assumptions to this result are the type of household preferences and the substitutability of the energy source that is shared between the countries: they both affect themagnitudeofthenegativespilloversexperiencedbytheuncappedcountry. ThenonhomotheticpreferencesareasinBoppart(2014): householdsspendahighershareoftheir income on energy when their income is low, which means they dislike reducing their energyconsumption. Thesetypeofpreferencesarecommoninliteraturehandlingnecessity goods like energy and food (Blanco & Diz, 2021; Olivi et al., 2023). Non-homothetic preferences amplify the negative spillovers, which results in costs from spillovers exceeding the costs of implementing the price cap depending on the substitutability of the energy source. Endogenous energy production dampens the spillovers and explains why a country might prefer to bear the negative spillovers from the price cap as an uncapped country rather than imposing a price cap themselves. During the European energy crisis in 2022,theelasticityofsubstitutionbetweengas,subjectedtotheexogenoussupplyshock, and non-gas energy was crucial, since the total energy consumption per capita did not decrease (Energy Institute, 2024). In the quantitative model, I estimate this elasticity of substitutionandfindthatwiththeestimatedparametersthecostofbearingthespillovers issmallerthanthecostofimplementingapricecap. Third,IprovidecounterfactualexercisesofpricecappoliciesinEuropein2022. Iperformahistoricalshockdecompositionoftheenergyandheadlineinflationratesinwhich the energy price cap is one of the shocks. I find that the energy price cap contributed to 10 percentage points of energy inflation and 0.5 percentage points to headline inflation 3

in the uncapped countries in 2022. Moreover, the inflation rates in the capped countries would not have been much higher without the energy price cap, reinforcing the model resultthatthecooperative,optimaloutcomeinvolvesnopricecaps. Last, I introduce a version of the model with hand-to-mouth households to compare theenergypricecaptotargetedtransfersasanalternativepolicymeasure. Shieldinglowincome households from rising energy costs was a key priority for many governments (Sgaravatti et al., 2023). I show that targeted transfers are a cheaper and more effective way to boost the consumption of the poor during an energy crisis. Moreover, because targeted transfers do not distort the energy market in the union, they do not cause large spilloversnordivergencewithintheunion. Thecontributionsofthispaperarebothgeneral,oninternationalpolicycoordination, and specific to the European energy crisis in 2022: first, I show that in a decision game of two countries and two cap options, the degree of non-homotheticity and the substitutability of the shared energy source determines the magnitude of the spillovers, and hence the incentives for policymakers to implement the price cap. Second, I quantify the model by estimatingit with European data. I confirm the general result thatthe negative spillovers from the capped to uncapped country are much larger than the benefits the cappedcountryexperiencesbyimplementingthecap. Related Literature. This paper contributes to two strands of the literature. First, it buildsontothevastliteratureonmonetaryandfiscalpolicyinacurrencyunion. Beetsma et al. (2001), Beetsma and Jensen (2005), Gal´ı and Monacelli (2008), Ferrero (2009), and Hjortsoe (2016) are pioneers of this strand of literature and explore the optimal joint conduct of monetary and fiscal policy as stabilization tools under asymmetric shocks. Other authors likeAnderson (2007)and Keen andKonrad (2013)focus on strategicinteractions of regulatory policies, like taxes, trade policies, and industrial regulation. Later, papers on this topic consider long-term coordination, with the sovereign debt crises in mind (Trichet, 2013; Chang, 2015). In this paper, I introduce a new dimension of integration: a shared energy supply. I analyze the international coordination in energy price cap policies during a union-wide energy shock. The analysis focuses on the determinants of the magnitude of the cap’s spillovers, and finds that fiscal coordination between countries is favorable. However,Ishowthatcountriesdonotalwayshavetheincentivestocooperate. Second, this paper contributes to the rapidly expanding literature on energy crises. This paper is closest to Bayer et al. (2023), who also evaluate different fiscal responses to an energy shock in a currency union. They compare two types of energy price caps and thetrade-offbetweenstabilizationofthedomesticeconomyandcostlyspilloversabroad. Auclert et al. (2023) and Chan et al. (2024) study the macroeconomic effects of an energy price shock and look at the coordination of fiscal policies and optimal monetary policy, 4

respectively. Moreover, Erceg et al. (2024) and Glocker and Wegmu¨ller (2024) study the effectiveness of fiscal policies, including energy subsidies, in stabilizing inflation. This paper analyzes the trade-offs and the spillovers of the energy price cap in an internationalsetting. Iapproachthetopicwithanovelangle: Iadoptatractable,game-theoretic approachtodeterminethecooperativeenergypricecappolicyaswellastheequilibrium thatariseswhencountrieshavetheirownincentives. The rest of this paper is organized as follows: Section 2 outlines the model with nonhomothetic preferences, the price cap setup, and the model calibration. Section 3 discusses the results of the baseline model, including the magnitude of the spillovers. I also analyze the trade-offs between headline and core-inflation targeting. Then, in Section 4 I estimate the substitutability between the shared energy source (gas) and domestically producedenergy(non-gas)andshowthatthecostsofimplementingapricecaparelarger than the costs of bearing the negative spillovers. I also quantify the contribution of the energy price cap to (energy) inflation in 2022. Lastly, in Section 5 I investigate targeted transfersbyaddinghand-to-mouthhouseholdstothemodel. Section6concludes. 2 Model The model considers a currency union with two countries, Home and Foreign {H,F}, andincompletefinancialmarkets. TherelativesizeoftheHomecountryisΘ ∈ (0,1)and henceoftheForeignis1−Θ. Timeisdiscreteandindexedbyt ∈ {0,...}. Bothcountries areinhabitedbyhouseholds,firmsandafiscalauthority. Thereisonecentralbanksetting monetarypolicyfortheentirecurrencyunion. Energy supply to the union is exogenous which follows from the high dependency of Europe on imported energy (Eurostat, 2023a). The energy market clears with a single price for the whole union reflecting the well-integrated energy market in Europe (Pescatori & Steurmer, 2022). When there is an energy price cap, the government pays for the differencebetweentheactualpriceofenergyandtheretailpriceofenergy. Thissetupfor theenergymarketissimilartotheoneintroducedbyBayeretal.(2023). Households consume energy as part of their consumption basket. Households have non-homothetic preferences for energy, which ensure that they consume a higher expenditure share of energy when their income decreases. Firms in both countries produce tradable goods under monopolistic competition, using energy and labor as inputs. The lawofonepriceholdsforthosegoodsandthereishomebias. Since the Home and Foreign country are symmetric, I explain only the Home-side of the union, unless otherwise stated. Foreign variables are denoted with an ∗. Appendix A providesamoredetaileddescriptionofthemodel,includingalistofrelevantequilibrium 5

conditionsandthesteadystate. This modelincludes non-homotheticpreferences butnot thesubstitutability ofdifferent energy sources. This version of the model is useful for understanding the core intuition and mechanics before introducing the substitutability of energy sources. In Section 4 I complete the model by adding domestic energy production and allowing for substitutability of the exogenous source of energy (gas) and domestically produced energy (non-gas). 2.1 Households 2.1.1 Preferences Households derive utility from two types of goods: energy goods, Eh, and non-energy, t other goods, C . Preferences of the households are non-homothetic as introduced by Ot Boppart (2014). In this specification of preferences, the total nominal expenditure of the household, defined as exp = P Eh +P C , matters for the share of expenditure spent t Et t Ot Ot onenergyandtheothergoods. P andP arepricesforenergyandothergoodsrespec- Et Ot tively. The indirect utility function of the representative household with non-homothetic preferencesis:3 (cid:88) ∞ (cid:26) 1 (cid:20)(cid:18) exp (cid:19)ε1 (cid:21) α (cid:20)(cid:18) P (cid:19)ε2 (cid:21)(cid:27) E βt t −1 − E Et −1 (1) 0 ε P ε P 1 Ot 2 Ot t=0 whereα > 0istheshareofenergyconsumptioninthesteadystateandβ isthediscount E factor. ε governs the expenditure elasticity of demand: when ε > 0, the expenditure 1 1 elasticity of demand is strictly smaller than unity for energy and larger than unity for othergoods. So,whentotalnominalexpenditure,ortotalincomeavailableforconsumption, decreases, the demand for energy decreases less than proportionally with income andthedemandforothergoodsdecreasesmorethanproportionally. ε controlstheelas- 2 ticity of substitution between energy and other goods. In steady state, the elasticity of substitutionbetweenenergyandothergoodsisσ¯ = 1−ε − αE (ε −ε ).45 2 1−αE 2 1 3Anindirectutilityfunctionv(p,exp)expressesthehousehold’smaximalattainableutilitywhenfaced withvectorpofgoodspricesandanamountofexpenditureexp. Ingeneral,adirectutilitycounterpartof thisindirectutilityfunctiondoesnotexist. 4SeeLemma3inBoppart(2014)forthederivationandAppendixA.4forthesteadystate. 5Anothercommonlyusednon-homotheticpreferencespecificationistheStone-Gearypreferences,with which the consumer derives utility from consumption that exceeds the subsistence level. Under Stone- Gearypreferencestheexpenditureshareofenergydoesnotincreaseafterapriceincrease. Sinceinpractice the household energy expenditure share increased in Europe in 2022, I chose to use the preferences from Boppart(2014)(OECD,2023;EuropeanCommission,2024). 6

Relative demand between energy and other goods. The relative demand for energy andothergoodsobtainedusingRoy’sidentityreadsas:6 1−α ϖ P C = E t Et Eh (2) Ot α ϖ P t E t Ot where (cid:18) P (cid:19)ε1 (cid:18) P (cid:19)ε2 Ot Et ϖ = (3) t exp P t Ot istheenergyexpendituresharewedge. Thiswedgeincreaseswhenthetotalexpenditure decreases or when the price of energy increases, both relative to the price of the other goods. Consequently, the consumption demand is non-homothetic in income since the share of expenditure on energy, α ϖ , increases when the household becomes poorer. E t When ε = ε = 0, Eq. (2) simplifies to C = 1−αE PEtEh, which is the standard Cobb- 1 2 Ot αE POt t Douglas result. In this case, the expenditure elasticity for both types of goods are equal to unity. Section 3.2.4 discusses the case under preferences with Constant Elasticity of Substitution(CES). Relative demand between Home and Foreign goods. The consumption of non-energy goods is a composite index, bundling consumption of Home-produced goods C and Ht Foreign-producedgoodsC : Ft (cid:104) (cid:105)γ/(γ−1) C = (1−α )1/γ(C )(γ−1)/γ +(α )1/γ(C )(γ−1)/γ (4) Ot I Ht I Ft where α ∈ (0,1) is the share of imported goods in the consumption basket and γ is the I elasticity of substitution between Home and Foreign goods. Since the preferences between Home and Foreign-produced goods are homothetic, the intratemporal consumptiondemandbetweenHomeandForeigngoodsarestandard: C 1−α (cid:18) P (cid:19)−γ Ht I Ht = (5) C α P Ft I Ft where P and P are the price indices of Home and Foreign goods respectively. The Ht Ft aggregateexpenditureonotherconsumptionisthen: (cid:90) 1 (cid:90) 1 P (i)C (i)di+ P (i)C (i)di = P C +P C = P C (6) Ht Ht Ft Ft Ht Ht Ft Ft Ot Ot 0 0 6SeeAppendixAforadetailedderivationofthefirstorderconditions. 7

whereP istheaggregatepriceindexfornon-energygoods: Ot P = (cid:2) (1−α )P1−γ +α P1−γ(cid:3) 1− 1 γ (7) Ot I Ht I Ft 2.1.2 Intertemporalchoices Therepresentativehouseholdmakesintertemporalchoicessinceitcantradeinone-period bonds B with gross interest rate R . The household’s income sources are from labor N t t t foranominalwageW perunitandfromprofitsofdomesticfirms,D ,andenergysellers, t t DE.7 Thenominalbudgetconstraintofthehouseholdsisthefollowing: t exp = P Eh +P C = W N +D +DE +R Bh −Bh −HC −T (8) t Et t Ot Ot t t t t t−1 t−1 t t t where HC = ν˜(Bh − B ¯h)2 are the portfolio adjustment costs of the household and T t 2 t t lump-sumtaxes. Thegovernmentusesthosetaxestofinanceenergypricecaps. Whenthe householdmaximizestheirutilityfunction(1)subjecttotheconstraint,theEulerequation becomes: (cid:18)E [exp ] (cid:19)1−ε1 R (cid:20)(cid:18) 1 (cid:19)ε1 (cid:21) t t+1 = β t E (9) exp 1+P ν˜(bh − ¯ bh) t Π t t t O,t+1 where bh = B t h denotes real bond holdings and Π = POt gross inflation of the other t Pt Ot PO,t−1 goods. P istheaggregatepriceindex,explainedbelow. Householdssupplylaborinelast ¯ tically, such that N = N ∀t. In Section 3.2.4, I briefly discuss the results under elastic t laborsupply. 2.2 Firms Final good producer. The final good firms produce the final consumption good, Y , ust ingintermediategoods,Y (i),accordingto: t (cid:20)(cid:90) 1 (cid:21)ϵ/(1−ϵ) Y = Y (i)(1−ϵ)/ϵdi (10) t t 0 where Y (i) is the output of the intermediate firm i and ϵ the elasticity of substitution t between different varieties of the intermediate good. The firms produce in a competitive (cid:82)1 market and maximize profits given by P Y − P (i)Y (i)di. The first-order condition to t t 0 t t themaximizationproblemgivesthedemandfunctionoftheintermediategoodi: (cid:18) P (i) (cid:19)−ϵ Ht Y (i) = Y (11) t t P Ht 7As in Bayer et al. (2023), households earn profits determined by deviations from steady state when sellingenergy. 8

andthepriceofthefinalgoodY : t (cid:20)(cid:90) 1 (cid:21)1−ϵ P = P (i)−(1−ϵ) (12) Ht Ht 0 whereP (i)isthepriceoftheintermediategoodi. Ht Intermediate good producers. The country has a continuum of i ∈ [0,1] firms who produce the (non-energy) other goods under monopolistic competition. They use both labor N and energy Ef as production inputs in their Constant Elasticity of Substitution t t (CES)productionfunction: Y (i) = A (cid:20) (cid:0) αf (cid:1)1/θf (cid:16) Ef(i) (cid:17)(θf−1)/θf + (cid:0) 1−αf (cid:1)1/θf (N (i))(θf−1)/θf (cid:21)θf/(θf−1) (13) t t t t where αf is the share of energy used in production and θf is the elasticity of substitution between input factors energy and labor. A is the total factor productivity which follows t an AR(1) shock process. The firms face adjustment costs a` la Rotemberg (1982), so their profitmaximizationproblemis:8 ∞ (cid:20) (cid:21) (cid:88) P (i) W max E β Ht Y (i)− t N (i)−P Ef(i)−Y FC (14) PHt(i),Nt(i) 0 t=0 P Ht t P Ht t Et t t t subjectto (cid:18) P (i) (cid:19)−ϵ Ht demandcurve Y (i) = Y (15) t t P Ht ξ (cid:18) P (i) (cid:19)2 Ht priceadjustmentcosts FC (i) = −1 (16) t 2 P (i) H,t−1 whereξ governsthelevelofpriceadjustmentcosts. The first-order condition with respect to P (i) leads to the standard New Keynesian Ht Philips Curve (NKPC). See Appendix A for detailed derivations. As in Aoki (2001), the relative price of energy shows up as a shift of the NKPC, like a cost-push shock, when re-writingtheNKPCintermsofheadlineinflation. Since all intermediate goods are identical, P (i) = P and N (i) = N . Aggregate Ht Ht t t firm’sprofitsreads: D = P Y (1−FC )−W N −P Ef (17) t Ht t t t t Et t 8Sinceenergyisanexogenoussuppliedgood,Rotemberg(1982)andCalvo(1983)pricingareidentical uptofirstorder,unlikeconventionaltwo-sectormodels. 9

2.3 Monetary policy The monetary authority targets the headline inflation of the two countries, Home and Foreign, with a Taylor rule set accordingly to their respective size. So, the Taylor rule for thenominalinterestrateR is: t 1 (cid:18) ΠW(cid:19)ϕπ R = t exp(ν ) (18) t β Π ¯W t wheresuperscriptW indicatesaunion-widevariable,definedas: ΠW = (Π )Θ(Π∗)1−Θ (19) t t t Themonetaryauthorityonlytargetsinflation,andnooutputgap,reflectingtheEuropean CentralBank’spricestabilitymandate. Theinflationthatthecentralbanktargetsis: Π = (Π )αE(Π )1−αE (20) t Et Ot which corresponds to the Consumer Price Index (CPI), or headline inflation in common literatureanddatasources. ThepricelevelfollowsfromtheinflationtermΠ = P /P .9 t t t−1 2.4 Fiscal policy: The energy price cap If the Home country introduces a cap on energy prices, the fiscal policy and the government budget constraint of the country become relevant. With an energy price cap in the Homecountry,theeffectiveenergypricebecomes:  ¯ ¯ P withcapandP > P Peff = E Et E (21) Et P otherwise Et Hence,underthecap,theeffectivepriceforenergyforthehouseholdsandfirmsisequal ¯ tothesteadystatepriceofenergy,P . Consequently,whenthefiscalauthorityintroduces E the price cap, the price of energy in households’ and firms’ equilibrium conditions is givenbytheeffectivepriceofenergyPeff = P ¯ .10 Thegovernmentrunsabalancedbud- Et E get and finances the cap by a lump-sum tax, such that the government budget constraint reads: COST (Eh +Ef) = T (22) t t t t 9Themainresultsarerobusttochangingthecentralbank’stargetfromCPItocoreinflation. InSection 3.2.3Idiscusstheimplicationsofcoretargetingoninflationrates. 10The results are robust to changing the price cap target from both household and firms to just households. 10

¯ where COST = P − P denotes the cost of the cap per unit of energy for the governt Et E ment. RicardianEquivalenceholdsinthismodelbecauseconsumptiondoesnotrespond to changes in future expected taxes and government spending. Later, I introduce handto-mouth agents to the model as an extension in which case Ricardian Equivalence does nothold. 2.5 Equilibrium Theequilibriumofthiseconomyischaracterizedbyasequenceofprices{W ,W∗,P ,P ,P } t t Ht Ft Et and allocations {Eh,Eh∗,C ,C ,C∗ ,C∗ ,N ,N∗,Ef,Ef∗,bh,bh∗} such that the goods t t Ht Ft Ht Ft t t t t t t market is cleared for both Home and Foreign-produced goods, the energy market is cleared, the assets are in zero net supply between the countries, and the labor market isclearedinbothcountries. ThefulllistofequilibriumconditionsareinAppendixA.3. 2.5.1 Goodsmarketclearing The goods market clears for the Home country when the production in that country is equal to the demand for consumption goods produced in that country. Hence, the market clearing condition includes the demand for Home-produced goods in the Foreign country: Y = C +C∗ +HC +FC (23) t Ht Ht t t (cid:18) P (cid:19)−γ (cid:18) P (cid:19)−γ = (1−α ) Ht C +α∗ Ht C∗ +HC +FC (24) I P Ot I P∗ Ot t t Ot Ot where C∗ is the consumption of Home-produced goods in Foreign, and C∗ is the con- Ht Ot sumptionofothergoods(bothHomeandForeign-produced)inForeign. 2.5.2 Energymarketclearing. The energy market is fully integrated across the two countries in the union. As in Bayer etal.(2023),ImodelthesupplyofenergyE asexogenous. So,thesupplyofenergydoes t not respond to price movements and the price of energy has to adjust for the market to clear. The energy market clears when the demand for energy by households and firms frombothcountriesequalstheexogenoussupply: E = Eh +Ef +Eh∗ +Ef∗ (25) t t t t t 11

2.5.3 Currentaccountandthedynamicsofnetforeignassets Iderive thedynamicsofnet foreignassets,and hencethecurrent account,byconsolidatinghouseholds’andfirmsresourceconstraints,(8)and(17): Bh −Bh = r Bh +P Y (1−FC )−P C −HC (26) t t−1 t−1 t−1 Ht t t Ot Ot t wherer = R −1isthenetnominalinterestratesetbythemonetaryauthority. Sincethe t t right-handsideoftheequationisthecurrentaccountIcanexpresstheaboveequationas thefollowing: CA = bh −bh (27) t t t−1 P P 1 CA = r bh + Ht Y (1−FC )− Ot C − HC (28) t t−1 t−1 P t t P Ot P t t t t where bh = B t h is real bond holdings. Since the union is a closed economy, to ensure t Pt mutual consistency of current accounts CA = −CA∗ needs to hold. Moreover, the assets t t areinzeronetsupplybetweencountries. 2.6 Calibration The model is calibrated at quarterly frequency. In the extended model, I perform a Bayesian estimation of some of the model parameters. Table 1 provides an overview of thebaselinecalibrationvalues. Thecountriesareidenticalexceptfortheirrelativesizes. Table1: Baselinecalibrationofparameters Parameter Description Value Households α Shareofimportsinconsumption 0.25 I α Shareofenergyinconsumption 0.066 E γ ElasticityofsubstitutionbetweenHomeandForeigngoods 6 ϵ Elasticityofsubstitutionwithingoods 9 ε Non-homotheticityparameter 1 1 ε Non-homotheticityparameter 0.77 2 ν˜ Adjustmentcostforbonds 0.001 β Discountfactor 0.99 Firms αf Shareofenergyinproduction 0.011 θf Elasticityofsubstitutionbetweenenergyandlabor 0.2 ξ Price-adjustmentcost 15.84 Monetarypolicy ϕ Taylor-coefficientoninflation 1.5 π αCB Shareofenergyforcentralbank’sconsideration 0.066 Currencyunion Θ RelativeGDPsizeHomecountry(withcap) 0.68 12

On the household side, Eurostat (2023b) reports that in 2022, the share of internationally traded goods and services relative to GDP was 25%. Hence, the share of imports in consumption,α ,is0.25. Theshareofenergyintotalconsumptionexpenditureisonaver- I age 6.6%, so I set α as 0.066.11. The elasticity of substitution within different varieties of E Home and Foreign, ϵ, is 9, in line with standard literature. The adjustment cost for bondholdings,ν˜,is0.001,tomatchthecanonicalworkbySchmitt-Grohe´ andUribe(2003). The discountfactorβ is0.99asisstandardintheliterature. Iperformadatamatchingexercise attheendofthesubsectiontocalibratethenon-homotheticityparametersε andε . 1 2 For the firms, I set the share of energy in production, αf, to 1.1% to target the steadystate energy expenditure of the industry as share of total production value of 1%.12 The elasticity of substitution of energy and labor, θf, is 0.2, following Bayer et al. (2023) and Bachmann et al. (2024).13 I calibrate the Rotemberg (1982) price-adjustment cost parameter, ξ, such that the slope of the New Keynesian Philips Curve matches that of the Calvo (1983) price rigidities for the Calvo parameter 0.5. This value implies an expected price duration of two quarters, which is more frequent than standard, to reflect the fast change in prices in 2022. The corresponding price-adjustment cost parameter is ξ = [(ϵ−1)0.5]/[(1−0.5)(1−0.5β)] ≈ 15.84 MonetarypolicyfollowsastandardTaylor(1993)rule,withthecoefficientoninflation ϕ as 1.5. The monetary authority targets headline inflation, following the official target π oftheEuropeanCentralBank(ECB,2021).14 To obtain the relative size of the two countries, I calculate the GDP ratio of countries that introduced a cap in 2022 and that did not introduce a cap in 2022.15 Since the sum of GDPsofcountrieswithanenergypricecapin2022wasabout68%ofthetotalofcountries intheEuroArea,IsetthesizeoftheHomecountryΘ = 0.68.16 11Eurostatdata,onlinedatacode: hbs str t223. 12Icalculatethesteady-stateenergyexpenditureasshareoftotalproductionvaluewithdatafromRademaekers et al. (2020) and Eurostat data (online data code: sbs sc ovw). The sectors included are selected manufacturing sectors, wholesale and retail trade, accommodation and restaurants, and information and communication,andthecountriesincludedarethe27EuropeanUnionmembersin2020. 13Bachmannetal.(2024)showthatwhenotherproductioninputsareconstant, theown-priceelasticity maps directly to the elasticity of substitution. They estimate the own-price elasticity of energy to range from−0.15to−0.20. 14Moreover, thepressreleasesoftheECBmonetarypolicydecisionsbetweenJune2022andSeptember 2023,whentheECBkeptincreasinginterestrates,oftenmentionenergypricesasoneofthekeydriversof upwardspressuresforinflation. Thedecisionreportsmentionheadlineinflationfigurestoindicatehowfar theeconomyisoffthe2%target(EuropeanCentralBank,2024). 15EuroAreacountrieswithanenergypricecapin2022:Austria,Estonia,France,Germany,Luxembourg, Malta, Portugal, Slovakia, Slovenia, Spain. EuroAreacountrieswithoutanenergypricecapin2022: Belgium,Cyprus,Finland,Greece,Ireland,Italy,Latvia,Lithuania,TheNetherlands. CroatiajoinedtheEuro Areain2023andthereforeexcludedfromtheanalysisinthispaper. 16Theresultsarerobustagainstacalibrationwithequal-sizedcountries,soΘ=0.5. 13

Non-homotheticity parameters. For the calibration of the non-homotheticity parameters ε and ε , I conduct a data matching exercise. I take the gas inflation data for France 1 2 andtheNetherlandsfromEurostatfrom2019to2022,andfeeditintothemodelasperfect foresight energy price shocks, as shown Figure 3.17 At the peak in 2022Q3, the Netherlandsexperiencedagaspriceinflationofabout30%inquarterlyrates(over90%inannual rates). France, on the other hand, imposed a price cap on gas inflation which barely exceeded 10% in quarterly rates (about 30% in annual rates). The gas consumption data reflect the policies: in the Netherlands, the gas consumption decreased about 15 percentage points more than in France. I conduct a parameter search for ε and ε , imposing 1 2 0 ≤ ε ≤ 1 and 0 ≤ ε ≤ 1 to minimize the Mean Squared Error (MSE) between the 1 1 energy consumption from the model in which I feed in the gas inflation data, and the realized gas consumption in 2020Q3 to 2022Q4 for both countries.18 The results give ε = 1 1 andε = 0.77,whichimpliesanelasticityofsubstitutionbetweenenergyandothergoods 2 of 0.25 in steady state, which is in line with other literature on energy shocks like Bayer et al. (2023) and Chan et al. (2024). I set the elasticity of substitution between Home and Foreign goods, γ, to 6, the upper bound of standard literature (Benigno, 2009), since the dataexercisegivesthelowesterrorunderthiscalibration. Figure3: Dataexercisetocalibratethenon-homotheticityparameters Notes: TheleftpanelshowsthegasinflationofFrance(cap)andtheNetherlands(nocap). Ifeedthisdata into the model and find the parameters ϵ and ϵ that minimize the Mean Squared Error (MSE) between 1 2 thegasconsumptiondata(rightpanel,dotted)andthemodel-impliedgasconsumption(solid).Sourcesfor data: Eurostat. 17ImanipulatethedatafromEurostat(onlinedatacode: prc hicp manr)toobtainquarterlyrates. Iuse theobservationsfrom2020Q3to2021Q3tocomputethesteadystatetoexpressalldataindeviationsfrom steadystate. FranceandtheNetherlandsareoneofthemostextremecasesofinflationdivergencewithin theEuroArea. Thecountriesarerelativelyclosegeographicallyandsocio-economically,whichmakethem goodcandidatesforthisdataexercise.IncludingallcountriesintheEuroAreamakesthisexerciselessclear cut, since idiosyncrasies, like proximity to Ukraine or Russia, affect the price dynamics in different ways thanthisreducedformexercisecanhandle. IntheBayesianestimationoftheextendedmodel,Iincludeall countriesintheEuroArea. 18The data is from Eurostat (online data code: nrg c gasm). With population data (intrapolated for the quarters), Iobtainthegasconsumptionpercapita. IseasonallyadjustthedatausingX-13ARIMA-SEATS inRbeforetakingquarterlydatapointsandsteady-statedeviationsfromsteadystate. 14

Shock specification. In the numerical analyses in the following sections, I shock the modelwithanadverseenergysupplyshockof15%thatlastsforsixquarters. Inthisway, I capture the decline in the supply of Russian gas in summer 2022 and the expectations ofgovernmentsthattheshockwouldlastuntilspring2023. Moreconcretely,inJuly2022 the European Union member states agreed to a gas consumption reduction target of 15% betweenAugust2022andMarch2023,andanotherextensionuntilMarch2024,toprepare forpossiblesupplydisruptions(EuropeanCommission,2023). Moreover,mostcountries introducedenergypricecapslastingfourtoninequartersin2022. 3 Results In this section, I conduct a series of simulations with the dynamic model to investigate the effect of an adverse energy supply shock on a currency union. First, I show how an adverse energy supply shock affects the economy in absence of price caps. The shock causes an increase in the price of energy, and a cost-push shock in the economy. Second, ItakethescenariooftheEuroAreain2022,andimposeanenergypricecapinthebigger country in the union. I find that the capped country can avoid most of the crisis, while the uncapped country experiences a cost-push shock double the size. The size of such negativespilloversdependonthedegreeofnon-homotheticityofenergyandaffectpolicy decisions. Moreover, I discuss the consequences of headline and core targeting and the trade-offstheyimpose. 3.1 Energy crisis without energy caps Theshockisa15%shocktotheenergysupplyofthecurrencyunionandlastssixquarters. Since there are no energy price caps in either country and the countries are otherwise symmetric, the responses for the two countries are the same. Hence, the results in Figure 4showoneresponsepervariable. Theadverseenergysupplyshockpushesupontheenergyinflation. Therecessionary shock decreases inflation for other goods on impact. In the later periods, the other goods inflation increases since energy is one of the production inputs. Consumer Price Index (CPI) inflation, or headline inflation, is a weighted average of energy inflation and other goods inflation, and hence peaks when other goods inflation is highest. Production and consumption of other goods decrease as a consequence of the energy supply declining.19 Energy consumption by households decreases by about the same amount as the shock. Since the energy shock increases CPI inflation while depressing output, the shock acts 19Shown in the Outputpanel, since output is the production of othergoods, which is equal to the consumptionofothergoods. 15

as a cost-push shock. The monetary authority conducts contractionary policy to dampen inflationarypressures,andreturnstosteadystatetogetherwithCPIinflation. Figure4: Responsestoanadverseenergysupplyshock|Nocaps Notes: Impulse responses to a 15% decline in energy supply. Preferences are non-homothetic. Output is equaltotheoutputgap. They-axisisintermsofpercentagedeviationsfromsteadystate. Thex-axisisin quarters. Inflationandinterestratesareannualized. 3.2 Energy crisis with one cap and one no-cap country Now consider the case in which the the larger country introduces a price cap on the energy price, such that the retail energy price stays constant. Figure 5 shows the impulse responses for the economy when households have non-homothetic preferences. When the energy supply decreases by 15%, the bigger country (blue solid lines) introduces an energy price cap which costs about 2.5% of the annual GDP for the government. In the uncapped country (red dotted lines) the adverse energy supply shock is essentially doubledcomparedtothecasewithoutanypricecaps,sincethecappedcountry’sshareofthe shockspillsover. In the capped country the economy avoids most of the energy crisis. Because their energy prices do not increase, households in this country have more purchasing power than households in the uncapped country. Therefore, they consume more of the other goodsproducedintheirowncountry,butalsomoreoftheonesproducedintheuncapped country. Moreover, the other goods inflation responses show that the goods from the uncapped country have become relatively cheaper because of the large recession in that country. Hence,totalconsumptioninthecappedcountryincreases. In the country without the cap the energy price increase doubles compared to the previous case without any caps. The energy scarcity is more severe due to the capped country not reducing their energy consumption. Since the capped country does not de- 16

creasetheirenergyconsumption,energyisanevenscarcergoodintheuncappedcountry. The adverse energy supply shock causes households in this country to reduce their energy consumption by twice as much relative to the case when the other country also did notintroducetheenergypricecap. Theenergysupplyshockisessentiallydoublethesize in the uncapped country. Crucially, the other inflation fluctuations in both countries imply a terms-of-trade depreciation for the uncapped country, because their other inflation decreases more than the one in the capped country. Since the terms of trade depreciates for the uncapped country, their purchasing power decreases. Imports become more expensive and exports to the households in the capped country increases, which decreases total consumption in the uncapped country drastically.20 Despite the increased demand from the capped country for goods produced in the uncapped country, the output in the uncappedcountrydecreasesduetothelargeenergysupplyshock. Thecommonmonetarypolicyadoptsalesscontractionarystancethanwhenneitherof the countries implemented the energy price cap, in Figure 4. The central bank targets the weighted-average headline inflation in its Taylor rule, which is the black line in the CPI inflationgraphinFigure5. Sincetheinflationratesofthecappedanduncappedcountries peakatdifferenttimes,theweighted-averageinflationdoesnotincreaseasmuchasinthe casewithoutanypricecaps,whichcausesamilderresponsefromthecentralbank. Figure5: Responsestoanadverseenergysupplyshock|Capvs. nocap Notes: Impulse responses to a 15% decline in energy supply. Preferences are non-homothetic. The bigger country,ofsizeΘimposesapricecapontheenergyprice(blue,solid)andthesmallercountry,ofsize1−Θ doesnot(red,dashed). Theblacksolidlinesshowtheunion-widevariables. Outputisequaltotheoutput gap. Governmentexpenditureonthepricecap(Govt. exp. cap)isthecostofthecapasashareofannual total output of the country (GDP). The y-axis is in terms of percentage deviations from steady state. The x-axisisinquarters. Inflationandinterestratesareannualized. 20This result is similar to the terms-of-trade externality by for example Corsetti and Pesenti (2001), in which an expansionary fiscal policy causes a terms-of-trade appreciation for the country, hurting trading partners. 17

3.2.1 Energycrisiswithenergypricecaps Figure 6 shows the responses when both countries impose an energy price cap. In this case,thepriceofenergystaysconstantintheentireunion. Becauseofthedistortedprice, consumers try to maintain their energy and other goods consumption. However, since the supply of energy is exogenous, the supply side of the economy cannot increase its production accordingly. Hence, there is high pressure on other goods inflation as well as the actual price of energy. Because the government pays the difference between the actual and retail price of energy, the price cap becomes a large cost for the government and ultimately for the consumers. Combined with the high other goods inflation, such highcostscauseabigdeclineinthetotalconsumptionofthehouseholds. Figure6: Responsestoanadverseenergysupplyshock|Caps Notes: Impulseresponsestoa15%declineinenergysupplyandapricecapinbothcountries. Preferences arenon-homothetic.Outputisequaltotheoutputgap.They-axisisintermsofpercentagedeviationsfrom steadystate. Thex-axisisinquarters. Inflationandinterestratesareannualized. 3.2.2 Implicationsforpolicydecisionsandwelfare Inthissubsection,Ianalyzethewelfareimplicationsforeachcombinationofpolicystrategies(capandnocap,forbothcountries)andshowthedecisiongameisaclassicPrisoner’s Dilemma. Policy outcomes under symmetric price cap policy. Table 2 summarizes the welfare resultsinatwo-by-twomatrix. Therelevantmetricistheconsumptionequivalentwelfare gains and losses relative to the steady state of the economy.21 First, focus on the cells on 21Fortheconsumptionequivalent,Ifindχwhichsatisfies: E (cid:88) ∞ βt (cid:26) 1 (cid:20)(cid:18) exp t (cid:19)ε1 −1 (cid:21) − α E (cid:20)(cid:18) P Et (cid:19)ε2 −1 (cid:21)(cid:27) = (cid:88) ∞ βt 1 {[exp(1+χ)]ε1 −1} (29) t ε P ε P ε 1 Rt 2 Rt 1 t=0 t=0 18

the diagonal where the policies are symmetric (cap-cap and no cap-no cap), which is the symmetric benchmark. Since the currency union is a closed economy, the symmetric benchmark is equivalent to the closed economy case. In this closed economy case, the adverse energy supply shock causes a welfare loss of 0.1% without an energy price cap and1%withthecap. Withanenergypricecapintheentireunionthewelfarelossesareatenfoldbigger. As discussed, such a scenario is detrimental for household consumption since the demand distortions under the price cap increases the fiscal cost for the government to finance the cap, ultimately born by households, and the inflation of other goods. The consumption equivalentwelfarelossis1%whenbothcountriesintroduceanenergypricecap. Table2: Welfaregains/lossesafterenergysupplyshock Non-homotheticpreferences 1/3ofunion % Cap Nocap Cap ( −1.0 , −1.0 ) ( 0.5 ,−1.1) 2/3ofunion Nocap (−1.1, 0.4 ) (−0.1,−0.1) Notes:Welfaregainsandlossesaftera15%energysupplyshock.Preferencesarenon-homothetic.Thegains and losses are in terms of the consumption equivalent relative to the steady state. The circles are around thepreferredpolicychoices(CaporNocap)forthecountries. Policy outcomes under asymmetric price cap policy. When the countries do not have to cooperate, is the non-distortionary no-cap strategy still the Nash equilibrium? As the circles around the welfare values indicate in Table 2, the cooperative case is not the Nash equilibrium. Instead, as in the classic Prisoner’s Dilemma, the non-cooperative decision, imposingthepricecap,isthedominantstrategyforbothfiscalauthorities. First, given 1/3 of the union does not impose an energy price cap, does the rest, 2/3 of the union, have an incentive to deviate from the no-cap strategy? If they keep to the no-cap strategy,theunionisinthecooperativecase,inwhichbothcountriesexperienceawelfare lossof1%. However,thecountryrepresenting2/3oftheunionhasanincentivetodeviate to the cap policy, which improves the welfare in that country (0.5%) at the expense of the no-cap country (−1.1%). This result is a summary of the impulse responses in Figure 5, withlargespilloversfromthecappedtotheuncappedcountry. Second,given2/3oftheunionimposesanenergypricecap,doestherest,1/3oftheunion, also have an incentive to impose the price cap? The large, negative spillovers are very costly for the uncapped country; they cause a welfare loss of 1.1%. Hence, the country has an incentive to also impose the price cap, even though both countries imposing the So, χ is the fraction of total expenditure, i.e. total consumption, that the household would be willing to forgointheeconomyinsteadystate(right-handside)toliveintheeconomywiththeenergysupplyshock, asevaluatedbytheleft-handsideoftheequation. 19

cap causes a welfare loss of 1%. The loss is relatively big because when both countries implement an energy price cap, the cost for the cap spirals upwards: the only benefit from the price cap emerges from creating spillovers to the other country, which is not possiblewhenbothcountriesimposethecap. The above argument also applies when the small and large countries switch: for both countries, it is better to impose the energy price cap when the other does, despite the largecostofthedistortion,ratherthanbearingthenegativespillovers.22 Hence,imposing an energy price cap is the dominant strategy for both countries, leading to a Prisoner’s Dilemma: bothcountriescangainfromcooperating,butitisnotrationaltodoso. ThePrisoner’sDilemmaastheNashequilibriumdoesnotreflectthechoicesthatpolicymakersmadeinreality. InthenextsectionIintroducesubstitutabilitybetweenenergy sources,whichisanimportantfeatureinenergycrises,andshowthatthemodelachieves theNashequilibriainwhichonecountryimposesanenergypricecapandtheothercountrydoesnot. Policy outcomes under homothetic preferences. How much do these results depend on the non-homotheticity of energy? Here, I illustrate that the above results are highly dependent on the degree of non-homotheticity. The welfare table for homothetic preferences is in Table 3. The degree of non-homotheticity does not affect the values on the diagonalofthesymmetriccappolicies. Again,thenon-cooperativecasewhenbothcountries impose a cap are much worse (−1%) than in the cooperative case without any caps (−0.1%)duetothemarket-distortingpricecap. Table3: Welfaregains/lossesafterenergysupplyshock Homotheticpreferences 1/3ofunion % Cap Nocap Cap (−1.0,−1.0) ( 0.1 , −0.4 ) 2/3ofunion Nocap ( −0.3 , 0.0 ) (−0.1,−0.1) Notes: Welfare gains and losses after a 15% energy supply shock. Preferences are homothetic (Cobb- Douglas). The gains and losses are in terms of the consumption equivalent relative to the steady state. Thecirclesarearoundthepreferredpolicychoices(CaporNocap)forthecountries. However, non-homotheticity of preferences affects the magnitude of the spillovers significantly. Underthehomotheticcase,theexternalitiesofthepricecaparenotaslarge, because energy is not a necessity. Hence, when the counterpart country implements a cap, the welfare losses associated with negative spillovers are not as large: −0.4% when thelargercountryimposesacapand−0.3%whenthesmallercountryimposesacap. So, implementingthepricecapisnotworththecostwhentheothercountryalsohasthecap. Thus, with homothetic preferences, there are two Nash equilibria, in which one country 22Theexternalitiesaresmallerwhenthesmallercountryimplementstheenergypricecap. However,not smallenoughtobreaksymmetryinthepreferredstrategiesinTable2. 20

implements a price cap and the other does not. Although this reflects what occurred in reality, the assumption that energy is not a necessity is unrealistic. Instead of relying on homothetic preferences to dampen spillovers, I show in the next section that introducing substitutabilitybetweenenergysourcescanachievethesameeffect,leadingtothedesired outcome. Recall that parameters ε and ε govern the degree of non-homotheticity of energy. 1 2 Welfare outcomes under the baseline calibration with non-homothetic preferences, ε = 1 ε = 0.77, are in Table 2: one Nash equilibrium which is a price cap in both countries. 2 Welfare outcomes in the homothetic case, ε = ε = 0, are in Table 3: two Nash equilibria 1 2 for differing cap policies. The magnitude of the spillovers are crucial in determining the sizeofthenegativespilloverstotheuncappedcountryanddependonthedegreeofnonhomotheticity. The value for ε and ε for which the smaller country is indifferent about 1 2 imposingacapornot,whenthebiggercountryhasimposedacap,isε = ε = 0.72. 1 2 3.2.3 Implicationsforthecentralbank: Headlinevs. coreinflationtargeting In this subsection, I explore the different implications for the monetary authority when targeting Consumer Price Index (CPI) inflation, i.e. headline inflation, or other goods inflation, i.e. core inflation.23 I show that there is a trade-off between stabilizing different inflation rates. However, since the relative inflation rates between the capped and uncapped country are similar in either targeting regime, the magnitude of the spillovers do notchangemuch. Inthebaselineanalysis,themonetaryauthoritytargetsheadlineinflationinitsTaylor rule. In that case, Figure 5 shows that the monetary authority conducts contractionary monetarypolicytostabilizeunion-wideheadlineinflation,whichistheweightedaverage of the headline inflation rates of the two countries. However, under headline inflation targeting,therearelargefluctuationsinthecoresectorinbothcountries,butworseforthe uncappedcountry. Ontopoftheadverseenergysupplyshock,acontractionarymonetary policyworsensthecost-pushshockinthecoresectoroftheeconomies. Figure 7 present the responses of the interest rates and inflation rates with a central bank that targets core inflation in its Taylor rule. The figure shows that the central bank conducts expansionary policy in this case. Because the adverse energy supply shock is a cost-push shock to the core-goods sector, a central bank that targets core-goods inflation decreases its rates to stabilize the fluctuations in that sector. As expected, the expansionarymonetarypolicycomesatthecostofahigherheadlineinflation. All in all, a central bank with a target for other goods inflation should conduct relativelyexpansionarypolicyduringanenergycrisiswithheterogeneouscappolicies. When 23In the literature, core inflation refers to CPI inflation excluding food and energy inflation. Since my modeldoesnothavefoodinflation,IrefertoCPIinflationexcludingenergyinflationascoreinflation. 21

Figure7: Responsestoanadversesupplyshock|Coreinflationtargeting Notes: Impulseresponsestoa15%declineinenergysupplyundercoreinflationtargeting. Preferencesare non-homothetic. Thebiggercountry,ofsizeΘimposesapricecapontheenergyprice(blue,solid)andthe smallercountry,ofsize1−Θdoesnot(red,dashed). Theblacksolidlinesshowtheunion-widevariables. The y-axis is in terms of percentage deviations from steady state. The x-axis is in quarters. Inflation and interestratesareannualized. the target is for headline inflation, a more contractionary monetary policy mitigates the inflationary pressures from the high energy inflation in the uncapped country, reducing the headline inflation fluctuations. In Figure 18 in Appendix C I display the rest of the impulse responses, which show that for the magnitude of the spillovers, the targeting regime does not matter. This result arises because the relative inflation rates between the capped and uncapped country, which matters for the spillovers, is similar regardless of thetargetingregime. 3.2.4 Robustnesschecksunderalternativemodelspecifications In this section I conduct robustness checks with alternative assumptions of the model: elastic labor supply, flexible prices, flexible exchange rates, and preferences with Constant Elasticity of Substitution (CES). Most alternative model specifications confirm the robustnessofthemainresults. Elastic labor supply dampens the negative spillovers, which gives result to a welfare table that is in-between the non-homothetic and homothetic case. To allow for hours worked to adjust in the model, I implement an ad-hoc approach to elastic labor supply by incorporating a labor disutility term into the utility function, deviating from the preferences outlined by Boppart (2014).24 Figure 20 and Table 8 in Appendix C.1 present the resultsforthisalternativespecification. TheimpulseresponsesinFigure20implythatelasticlaborsupplydoesnotchangethe 24Theutilityfunctionwithlabordisutilityis: E (cid:88) ∞ βt (cid:40) 1 (cid:20)(cid:18) exp t (cid:19)ε1 −1 (cid:21) − α E (cid:20)(cid:18) P Et (cid:19)ε2 −1 (cid:21) −χ N t 1+φ (cid:41) (30) 0 ε P ε P 1+φ 1 Rt 2 Rt t=0 where χ is the disutility of labor and 1/φ the Frisch labor elasticity. The calibration is χ = φ = 1, as is standardintheliterature. 22

inflation dynamics of the two countries. However, because firms can now increase their production by employing more labor, the output responses are different to the baseline case. As discussed earlier, due to the increased demand in purchasing power of households in the capped country, they import more goods from the uncapped country. Contrarytothebaselinecase,thefirmsintheuncappedcountrynowscaleuptheirproduction by hiring more labor, which is beneficial for households in the uncapped country. Those householdsdonotdecreasetheirconsumptionasmuchasinthebaselinecasebecauseof theincreasedlabordemand. Accordingly,Table8showsthatthecostsofimplementinga cap are higher than the cost of bearing the spillovers as the uncapped country, which is a similarresulttothehomotheticcase.25 Under flexible prices the main results under the asymmetric price cap policies and the welfare table are robust. The results are in Appendix C.2. While the spillovers underasymmetricpricecappoliciesaresimilartothestickypricecase,thesymmetriccases shows some differences. Specifically, output does not decline following the energy supply shock, since price flexibility allows for efficient adjustments. Firms can adjust prices immediately,enablingtheeconomytoabsorbtheshockwithoutreducingoutput. The tables and figures in Appendix C.2, C.3, and C.4 show the results for the last two alternative assumptions, flexible exchange rates and CES preferences. Even though the impulse responses are somewhat different, the welfare table is the same as under the baseline case. This result confirms that the crucial assumption to the mechanism is the shared energy supply, not the shared monetary authority. Moreover, CES preferences dampen the overall effect of the energy supply shock and the price cap spillovers on the economy, because the expenditure share of energy remains constant and energy is not a necessitygood. 4 Endogenous energy production So far, the analysis uses the model which only has an exogenous source of energy. This setup is valuable for understanding the core intuition and mechanics, as well as investigating the dynamics of the economies and their spillovers. However, during the European energy crisis in 2022 total energy consumption per capita did not decrease. When the supply of gas fell, other energy sources substituted out for gas, such that total energy consumption stayed roughly constant (Energy Institute, 2024). Therefore, to estimate the model, I add a domestic energy production sector to both countries. There is still an exogenoussupplyofgaswhichthetwocountriesintheunionshare. 25Inthereversecase, whenthesmallercountryimposesthepricecapandthebiggercountrydoesnot, the positive effect on welfare from increased labor demand is not big enough to outweigh the costs of implementingthepricecap. 23

With the extended model, I first perform a Bayesian estimation of the parameters. Then,Irevisitthemechanismandtheincentivesaboutwhethertocapornot. Ishowthat the domestic production of energy dampens the negative spillovers of the energy price cap, such that Nash equilibria, with one capped and one uncapped country, matches the reality in 2022. Moreover, I demonstrate with a historical shock decomposition that the energypricecapcontributedto40%ofenergyinflationand20%ofCPIinflationin2022Q3 intheuncappedcountries. 4.1 Energy sector and energy market clearing in the model To make sure there is a substitute for the exogenous supply of gas, I add energy firms to bothcountriesintheunion. Unlessotherwisestated,allotherequationsinthemodelstay unchangedfromthebaselinespecification. Energyfirmsonlyuselabor,N ,astheirinputintheirproductionY : Et Et Y = A Nη (31) Et Et Et where A is the total factor technology in the energy sector. η determines the share of Et profitsfromtotalrevenue. Theproductionfunctionusesadiminishing-returntechnology, asinFerreroandSeneca(2019),tomatchtheoligopoliesintheenergysector. The representative energy producer takes the wages as given. I assume that the energy firms sell any quantity of energy at the prevailing price. This assumption reflects the findings by Zakeri et al. (2022) who find that the European electricity prices depend highlyonnaturalgasprices. Theenergyfirm’sproblemis maxP Y −W N (32) Et Et t Et NEt subjecttotheproductionfunction(31). Thefirst-orderconditionsareinAppendixA.5. Energy market clearing. Energy supply comes from the exogenous, union-wide gas supply GASW and the domestically produced energy. Hence, the market clearing condit tionsforenergyare: (cid:104) (cid:105)ζ/(ζ−1) Eh +Ef = (1−α )1/ζ (Y )(ζ−1)/ζ +α1/ζ (GAS )(ζ−1)/ζ (33) t t G Et G t (cid:104) (cid:105)ζ/(ζ−1) Eh∗ +Ef∗ = (1−α∗)1/ζ (Y∗ )(ζ−1)/ζ +α∗1/ζ (GAS∗)(ζ−1)/ζ (34) t t G Et G t GAS +GAS∗ = GASW (35) t t t 24

where α is the share of gas in energy use and ζ governs the substitutability of gas and G otherenergy. 4.2 Calibration and estimation of the parameters In the model with domestic energy production, there are a few extra parameters to consider. Moreover, since the goal is to estimate the contributions of the energy price cap, I divert from the symmetric setup and calibrate some extra parameters differently for the capped and uncapped countries when there is distinguishing data. I calibrate the share of gas in energy use, α and α∗, the steady-state productivity of the energy sector, A ¯ , G G E and the share of profits, η, with data and matching targets. For the non-homotheticity parameters, ε ,ε∗,ε and ε∗, and the elasticity of substitution between gas and non-gas 1 1 2 2 energy,ζ andζ∗,IuseBayesianestimation. 4.2.1 Calibration For the share of gas in energy use, α and α∗, I use the Harmonized Index of Consumer G G Prices (HICP) item weights from Eurostat and set them to 0.18 and 0.22 respectively for thecappedanduncappedcountries.26 Eventhoughthedataforestimationstartsadecade earlier than 2022, I group the countries already into capped and uncapped countries, referring to the energy price cap policy in 2022. To set the steady-state productivity of ¯ the energy sector, A , and the share of profits, η, I match the following targets: the share E of workers in the energy sector of 3.66% in Europe27 and the relative price of energy and other goods of 1, as in the baseline model. The values that match the targets are η = 0.19 ¯ and A = 0.17. These parameters are symmetric across the countries. Table 4 provides a E summary. 4.2.2 Estimation I estimate the non-homotheticity parameters, ε = ε , and and the elasticity of substitu- 1 2 tionbetweengasandnon-gasenergy,ζ andζ∗. Here,Ioutlinethemethodusedandsteps takenforBayesianestimationandpresenttheoutcome. I use the Bayesian estimation techniques programmed in Dynare (Adjemian et al., 2024). I include the following shocks and measurement errors in the model: total factor productivity (TFP) shocks for other goods and energy sector, demand shocks, cost-push 26Eurostat data, online data code: prc hicp inw. I take the weighted average according to Eurostat’s country weights (data code: prc hicp cow) when calculating the values for capped and uncapped countries. ThecategorizationofcappedanduncappedcountriesisinFootnote15. 27Own calculations from the World Energy Employment report in 2022 by the International Energy Agency(IEA,2022)andEurostatdata. 25

Table4: Extraparametersinmodelwithdomesticenergyproduction Parameter Description Value α Shareofgasinenergy,“Cap” 0.18 G α∗ Shareofgasinenergy,“Nocap” 0.22 G η Shareofprofitsforenergyfirms 0.19 A¯ Steady-stateproductivityenergysector 0.17 E ε Non-homotheticityparameter 0.25 1 ε Non-homotheticityparameter 0.25 2 ζ Elasticityofsubstitutionbetweengasandnon-gasenergy,“Cap” 14.88 ζ∗ Elasticityofsubstitutionbetweengasandnon-gasenergy,“Nocap” 34.89 Notes: Calibrationoftheextraparametersinthemodelwithenergyproduction. Allothervariablesarethe sameasinthebaselinecaseasinTable1. shocks in the other goods sector, shocks to gas supply, monetary policy shock, and measurement errors for energy consumption and energy inflation. Those shocks and measurement errors are separate for the two countries in the union, except for the monetary policyshockandtheenergyinflationmeasurementerror.28 First, I compute the mode of the posterior distribution with the Monte-Carlo based optimization routine. Second, the Metropolis-Hastings algorithm evaluates the marginal likelihood of the model and produces the posterior distributions of the parameters. This method closely follows the Bayesian estimation approach in Smets and Wouters (2007). MoredetailsontheestimationmethodareinAppendixB. Prior distributions. I only estimate the parameters which have no direct counterpart in the data or a sensible target to match. The non-homotheticity parameter ε = ε is 1 2 bounded by zero and one.29 Hence, I use the Beta distribution as the prior distribution. The prior mean is set to 0.77, the calibration value from the data exercise in the baseline model. For the elasticity of substitution between gas and non-gas energy, ζ and ζ∗, I use the Gamma distribution as the prior distribution. I set the prior mean to 2 with a loose standard error. Following Krause et al. (2008), all shock processes follow an AR(1) process. The prior means of all AR-coefficient parameters are 0.9 and the standard deviations are 0.01. The AR-coefficients are bounded by one and zero, so they follow a Beta distribution. ThestandarddeviationsfollowanInverse-gammadistribution. 28Iaddthemeasurementerrorforenergyinflationwithatightpriortoavoidstochasticsingularity. 29Iestimatewithε = ε∗ = ε = ε∗. First, Iassumethatthe“Cap”and“Nocap”donotdifferintheir 1 1 2 2 non-homotheticitytoenergy. Sincethedataseriesisnottoolongandthe“Cap”and“Nocap”-blocksonly arosein2022,Iassume,asinthebaselinecalibration,thatthecountriesaresymmetric. Theonlyexception Imakeistheelasticityofsubstitutionbetweengasandnon-gas,asexplainedinthisparagraph. Second,I setε =ε tokeeptractability. Whenε =ε theelasticityofsubstitutionbetweenenergyandothergoods 1 2 1 2 is1−ε . 2 26

Data. I use the following data series from 2008Q1 to 2019Q4 in the Bayesian estimation:30 Energy inflation, gas inflation, CPI inflation, energy consumption, gas consumption, output, and the nominal interest rate. Since the union has an integrated energy market, and therefore also gas market, there is one energy and gas inflation rate each for the entire union. Moreover, since the model implies a shared supply of gas, the gas consumptionisthesameaswell. AlldataarefromEurostatData. Iseasonallyadjustthedata anddetrendthemtogetthecyclicalcomponent. MoredetailsareinAppendixB. Estimation results. Table 5 presents the results of the Bayesian estimation. The nonhomotheticity parameters, ε , and therefore also ε∗, ε , and ε∗, are 0.27.31 Moreover, the 1 1 2 2 substitutabilityofgasandnon-gasenergy,ζ andζ∗,are15.21and35.31respectively. Interestingly,thecountry-blocthatin2022implementsanenergypricecaphaveamuchlower elasticity of substitution between gas and non-gas energy. This policy decision seems to make sense given the relatively low ability to substitute away from gas. The parameters are well-identified because I use both gas and energy inflation rates and gas and energy consumptionfortheestimation.32 Theposteriordistributionsplotsandsomemoredetails abouttheestimationresultsareinAppendixB. Table5: Priorsandposteriors Parameter Priordist. Priormean Priorstd. Post. mean Post. std. 90%HPDinterval ε Gamma 0.8 0.1 0.27 0.07 [0.165,0.380] 1 ζ Beta 2 1 15.21 1.86 [12.190,18.265] ζ∗ Beta 2 1 35.31 3.31 [29.981,40.802] Notes: The prior distribution, mean, standard deviation, posterior mean and standard deviation, and the HighestPosteriorDensity(HPD)intervaloftheBayesianestimation. 4.3 Results Inthissubsection,Ifirstshowthesimulationresultsoftheextendedmodelwithparameter values from the calibration and the estimation, as summarized in Table 4. I show that domestic energy production dampens the effect of the gas supply shock on the economy. 30IdeliberatelyomittheCOVID-19pandemicyeartokeeptheobservablesstable. Fortheestimationof theshockslater,Icannotavoidthepandemicyear. Thesamplestartsin2008Q1duetodataavailability. 31The estimated non-homotheticity values, 0.27, are substantially lower than the values from the data exerciseinthebaselinemodel,0.77. Acouplereasonstoexplainthisdifference: inthebaselinemodel,the parametercapturesthenon-homotheticityofgas,whereastheextendedmodelcoversallenergy.Moreover, the sample period of the data exercise was very short, 2020Q3 – 2022Q4, and not overlapping with the sampleperiodoftheestimationexercise. Despitethedifference,theresultsoftheextendedmodeldoesnot changequalitativelywhenIsetthenon-homotheticityparameterto0.77insteadof0.27. 32Sincegasinflation/consumptionisafractionofenergyinflation/consumption,thedataimpliesinflation/consumptionofnon-gasenergy. 27

Consequently, the negative spillovers are also smaller, which leads the costs of implementing the price cap exceeding the costs of bearing the spillovers. Then, I conduct a historical shock decomposition to quantify the contribution of the energy price cap in 2022totheenergyandCPIinflationlevelsinboththecappedanduncappedcountries. 4.3.1 Simulationresults Figure 8 shows the impulse responses to an adverse energy supply shock when one country implements an energy price cap, with the model that allows for domestic energy production. The energy production in the uncapped country dampens the negative spillovers from the capped to the uncapped country substantially. For example, energy consumption for the households only decreases by about 10% compared to about 20% in the case without energy production in Figure 5. The response of CPI inflation, about 3% onimpact,isalsomuchlowerthanthe20%inthepreviouscase. Figure8: Responsestoanadversegassupplyshock|Energyproduction Notes:Impulseresponsestoa15%declineingassupply,inamodelwithenergyproduction.Preferencesare non-homothetic. Thebiggercountry,ofsizeΘimposesapricecapontheenergyprice(blue,solid)andthe smallercountry,ofsize1−Θdoesnot(red,dashed). Theblacksolidlinesshowtheunion-widevariables. The y-axis is in terms of percentage deviations from steady state. The x-axis is in quarters. Inflation and interestratesareannualized. The welfare outcomes for the combinations of price cap strategies are in Table 6. Because the energy sector dampens the effect of the exogenous gas supply shock, the loss from the gas supply shock is 0.02% instead of 0.1% in the baseline case, when there are no price cap policies in place. Moreover, when both countries impose a price cap, in the baseline case the losses rose to 1%. The domestic energy production dampens this effect to a loss of 0.3%, implying that the actual price of energy, and therefore the cost for the government to implement the cap, does not rise as high as in the baseline case. Importantly, Table 6 shows that imposing a price cap is not the dominant strategy as it was in 28

the baseline case in Table 2. Under the extended model there are two Nash equilibra in which one country imposes the price cap and the other country does not, explaining the reality in 2022. Because the energy sector dampens the negative spillovers of the energy pricecap,imposingthecapwhentheopponentcountryalsohasoneisnotworththecost. Table 6 also displays the union-wide welfare losses, outside of the parentheses in case of differing cap policies. The union-wide welfare loss is biggest when both countries impose the energy price cap, 0.5%, because the cost of imposing the cap is high for the government, and there is no other country to spillover to. The cooperative outcome whentherearenopricecapsintheentireunionhasthesmallestunion-widewelfareloss, −0.03%, compared to the weighted averages −0.03% (with higher precision) and −0.07%, whenoneofthecountriesimposethepricecap. So,theNashequilibraarenottheoptimal outcomeforunion-widewelfare,eveniftheybenefitthebiggercountry. Table6: Welfaregains/lossesaftergassupplyshock Modelwithdomesticenergyproduction 1/3ofunion % Cap Nocap Cap (−0.5,−0.5) ( 0.1 , −0.3 );−0.03 2/3ofunion Nocap ( −0.3 , 0.1 );−0.2 (−0.03,−0.03) Notes: Welfaregainsandlossesaftera15%gassupplyshock, inamodelwithenergyproduction. Preferencesarenon-homothetic. Thegainsandlossesareintermsoftheconsumptionequivalentrelativetothe steady state. The circles are around the preferred policy choices (Cap or No cap) for the countries. The valuesoutsidetheparenthesesareweightedaverages,i.e. union-widewelfare. To understand the forces behind the welfare gains and losses, I decompose the loss value −0.3 of the uncapped 1/3 of the union in Table 7. The welfare loss of the uncapped country when the countries sharing a gas supply are in a currency union (iii), is, with higher precision, −0.26. By computing the welfare losses in the cases of (i) two autarkies sharing a gas supply and (ii) two trading countries not in a currency union sharing a gas supply, the Table decomposes the total welfare loss of the uncapped country into three components: loss coming from energy price distortions, loss coming from the terms-oftradeeffect,andthelosscomingfrombeinginacurrencyunion. Table7showsthatthewelfarelosscomingfromtheterms-of-tradeeffectisthelargest. Energy price distortions, though they affect the inflation rates, do not seem to have a big contribution to the welfare losses. Lastly, being in a currency union does not seem to affect the welfare losses too much, as expected from the robustness checks in Appendix C.3. 29

Table7: Decompositionofthewelfarelossforuncappedcountry (i)Autarkies (ii)Tradepartners (iii)Inunion % (α = 0,indep. CBs) (α = 0.25,indep. CBs) (α = 0.25,oneCB) I I I Energyprice −0.03 −0.03 −0.03 Notes: Termsoftrade - −0.23 −0.23 Currencyunion - - +0 Totalloss −0.03 −0.26 −0.26 Decompositionofthewelfarelossoftheuncappedcountryaftera15%gassupplyshock,inamodelwith energyproduction. Preferencesarenon-homothetic. Thelossesareintermsoftheconsumptionequivalent relativetothesteadystate. Thefirstcolumnindicatesthewelfarelossinthecaseinwhichthecountriesare autarkies, i.e. donottrade(α = 0)andhaveindependentcentralbanks. Thesecondcolumnrelaxesthe I no-trading assumption (α = 0.25) but still assumes independent central banks. The third column is the I fullmodel,withcountriestradingandinacurrencyunion. 4.3.2 Historicalshockdecomposition UsingthecalibratedandestimatedvaluesinTable4,Iperformahistoricalshockdecomposition for the period 2008Q1–2022Q4. Again, I use the Bayesian estimation techniques in Dynare (Adjemian et al., 2024). I use the same shock processes and data series as described for the estimation of the parameters. I add the energy price cap as an additional shock. Asbefore,allshocksfollowanAR(1)processandIestimatethecoefficientsforthe shock in the same way as before. After the estimation of the shock processes, I perform a historical shock decomposition. More details on the data and estimation method are in AppendixB. Thehistoricalshockdecompositiondecomposesthefluctuationsinthedataseriesinto the contributions from the shocks. The results are in Figure 9. I group all shocks but the energypricecapinone(bluebars)andkeepthecontributionsfromthecapseparate(red bars). The top-right graph in Figure 9 shows that the energy price cap contributed to about 10 percentage points of energy inflation in the uncapped countries in 2022. In the last quarter of 2022, the price cap was responsible for virtually all of energy inflation in theuncappedcountries. Eventhoughthespilloversthatthepricecapcreatedwerelarge, thetop-leftgraphshowsthatinthecountrieswiththecaptheenergyinflationwouldnot have been much higher without it. If there were no energy price caps, the burden of the gassupplyshockswouldhavebeensharedequallyintheunion. Thepartialsubstitution to non-gas energy mitigates the upward pressure on energy inflation across the entire union. Similarly, the bottom graphs show that there were negative spillovers of the price cap to the uncapped countries, the upward pressure on CPI inflation: the price cap contributed to about 0.5 percentage points of CPI inflation in the uncapped countries, depending on the quarter. Moreover, the contribution increasing CPI inflation in the uncapped countries was a lot larger than the cap’s contribution lowering inflation in the 30

Figure9: Historicalshockdecomposition|Contributionsfromtheenergypricecap Notes: Historical shock decomposition of the annual inflation rates in deviations from the sample mean. “Othershocks”consistoftotalfactorproductivity(TFP)shocksforothergoodsandenergysector,demand shocks,cost-pushshocksintheothergoodssector,shockstogassupply,monetarypolicyshock,andmeasurementerrorsfor energyconsumptionandenergyinflation. Thoseshocksandmeasurement errorsare separate for the two countries in the union, except for the monetary policy shock. Mean energy inflation is 2.15% and 2.72% for capped and no-cap countries respectively. Mean CPI inflation is 1.42% and 1.46% forcappedandno-capcountriesrespectively. Theredbarsindicatethecontributionsfromtheenergyprice cap,whereasthebluebarsaggregateallothershocks. MoredetailsareinAppendixB. cappedcountries. 5 TANK results In this section I compare the energy price cap with targeted transfers. Most national governmentsconductedtransferstovulnerablegroupsin2022,sincetheenergycrisisaffected them the most (Sgaravatti et al., 2023). To create heterogeneity within households, I add poor hand-to-mouth to the model with domestic energy production. A targeted transfer is a transfer to just those hand-to-mouth households. Since labor income is the only source for hand-to-mouth households and hence an important model feature, this version of the model includes elastic labor supply as discussed in Section 3.2.4. Therefore,themodelbecomesaTwo-AgentNewKeynesian(TANK)model. The TANK version of the model does not alter in the transmission mechanism and thespilloversofthepricecapandtheaggregatewelfareresultsdiscussedintheprevious sectionshold. AninterestingaspectoftheTANKmodelisitspotentialforevaluatingthe effects of targeted policies, such as the transfer program, on different household groups: I compare a country-wide energy price cap (to all households and firms) to a targeted transfertothehand-to-mouthhouseholds. Sincelow-income,hand-to-mouthhouseholds 31

spendalargershareoftheirincomeonenergy,anadverseenergysupplyshockisparticularly burdensome for them (Bayer et al., 2023). I find that with much lower cost for the government, the targeted transfers achieve more favorable results in terms of boosting consumption for the poor. Moreover, since the transfer does not distort the energy market,thereisbarelyanydivergencewithintheunionevenifonlyonecountryimplements thetransfers. 5.1 Adding hand-to-mouth households to the model In the two-agent version of the model, there are financially constrained households who representshareλ ∈ [0,1]ofthepopulation,andunconstrainedhouseholdswhoareshare 1−λ. Financiallyconstrainedhouseholdshavenoaccesstotheone-periodbonds. Moreover,theyearnnoprofitsfromfirmsnortheenergysellers. Thebudgetconstraintsofthe constrainedandunconstrainedhouseholdsarerespectively: expc = P eh,c +P cc = W nc +P τc +T −Tc (36) t Et t Rt Rt t t t t t 1−δ 1 B B expu = P eh,u +P cu = W nu + D + DE +R t−1 − t −HC +P τu −Tu t Et t Rt Rt t t 1−λ t 1−λ t t−1 1−λ 1−λ t Rt t t (37) where superscript c refers to variables belonging to constrained households and u to unconstrainedones. τ areredistributivetransfersfromthegovernmentexplainedbelow. T t is a steady-state transfer from the constrained to unconstrained, to make sure their consumption is equal in steady state. The preferences are the same for both households and includethedisutilityforlaborsupplyasin3.2.4. Iaggregateenergyandothergoodsconsumptionandlaboras: λeh,c +(1−λ)eh,u = Eh (38) t t t λcc +(1−λ)cu = C (39) Rt Rt Rt λnc +(1−λ)nu = N (40) t t t Labor supply of constrained and unconstrained households are therefore identical to the firms. Following Debortoli and Gal´ı (2018) and Komatsu (2023), the fiscal authority redistributesthetaxedprofitsfromfirmsD astransferstotheconstrainedhouseholds,τc,and t t 32

unconstrainedhouseholds,τu,accordingtotherules: t τc = (1−τ )δD (41) t 0 t (cid:18) (cid:19) τ λ τu = 1+ 0 δD (42) t 1−λ t where δ is the tax rate on firms’ profits, where τ indicates how much of the profits go to 0 (un)constrained households, using λτc + (1 − λ)τu = δD . So, when τ is equal to unity, t t t 0 allprofitsgobacktotheunconstrainedhouseholds. Calibration. The Household Finance and Consumption Survey (HFCS, 2022) collects household-level data in the Eurozone and estimate that credit-constrained households make up around 5-10% of the population. Hence, in the TANK version, the share of hand-to-mouth households, λ, is 0.1.33 For the redistribution of taxed firms’ profits, I set the tax rate on firm’s profits at δ = 0.215, which was the average corporate tax rate in 2022 of European OECD countries (Bray, 2023). The redistribution rule, τ, is equal to unity, such that all profits go to unconstrained households. All other calibration values areidenticaltothebaselinemodelandthemodelwithdomesticenergyproduction. Consumption response decomposition. In the next subsection I investigate the consumptionresponsesofconstrainedandunconstrainedhouseholdsindetail. Hence,Iperform an impulse response decomposition by rearranging the log-linearized equations. Hattedvariablesindicatelog-lineardeviationsfromsteadystate. For constrained households, take total consumption as a sum of energy consumption andothergoodsconsumption: e¯c c¯c cˆc = eˆc + Rcˆc (43) t c¯c t c¯c Rt Using the choice between energy and other goods, Eq. (2), the definition of the energy expenditure wedge, Eq. (3), and their budget constraint, Eq. (36), I decompose the consumptionoftheconstrainedhouseholds: cˆc = Aceˆc +Bpˆrel,ER+ Cwˆ − Dtˆ (44) t t t t t (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) energyconsumption realwage taxes (cid:104) (cid:16) (cid:17)(cid:105) where Ac = 1 (e¯c +c¯c ), B = c¯c R 1+ 1 1 ε W ¯ N ¯ α −ε , C = c¯c R 1 ε W ¯ N ¯ , c¯c R c¯c 1−αE exp 1 E 2 c¯c (1−αE)exp 1 andD = c¯c R 1 ε λ. c¯c (1−αE)exp 1 335-10% is the share of so-called “poor” hand-to-mouth households. When including the share of “wealthly” hand-to-mouth households, who own illiquid assets, the share of hand-to-mouth households risestoabout30%. 33

Analogously for unconstrained households, decompose total consumption using the choice between energy and other goods, Eq. (2), the definition of the energy expenditure wedge,Eq. (3): cˆu = Aueˆu +Epˆrel,ER+ Fexˆp (45) t t t t (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) energyconsumption consumptionsmoothing (cid:16) (cid:17) where Au = 1 (e¯u +c¯u), E = c¯u R 1− 1 ε , and F = c¯u R 1 ε . I call the last term c¯u R c¯u 1−αE 2 c¯u 1−αE 1 “consumption smoothing”, since the Euler equation (9) determines the total nominal expendituresoftheunconstrainedhousehold,exˆp . t 5.2 Results: Price cap vs. targeted transfers The TANK impulse responses after an adverse gas supply shock with one capped and one uncapped country are quantitatively and qualitatively similar to the representative agent model in Figure 8.34 So, the analysis of the macroeconomic responses and welfare intheprevioussectionstillappliestotheTANKmodel. Toinvestigatetheconsumptionresponsesforconstrainedandunconstrainedindetail, I decompose the consumption responses for the constrained and unconstrained as in Eq. (44) and (45). The results are in Figure 10. For the uncapped country, I decompose the aggregate consumption response into contributions from constrained and unconstrained households. In the capped country, the consumption of the unconstrained increases, whereas the consumption of the constrained decreases. The unconstrained households increase their consumptionbothbyincreasingtheirenergyconsumptionandfromconsumptionsmoothing. Recallthemechanismthroughwhichhouseholdsbenefitfromtheenergypricecapin thebaselinemodel: householdsincreasetheirconsumptionbecausetheyconsumecheap goods from the uncapped country, i.e. the capped country consumes more than it produces. This mechanism is intertemporal, since the capped country temporarily runs a current account deficit and borrows from abroad while the energy shock takes place. In the two-agent version, only unconstrained households make intertemporal decisions. Hence, unconstrained households can increase their consumption, whereas constrained householdscannot. Therightmostgraphdisplaysthelargespilloversfromthecappedtouncappedcountry, similar to previous versions of the model. Because the price cap distorts the energy market in the union, it creates spillovers to the uncapped country. Next, I analyze whether targeted transfers are more effective in helping poorer, constrained households, andwhethertheycreatelessdistortionsandspillovers. 34TheresponsesfortheTANKmodelareinFigure19inAppendixC. 34

Figure10: Consumptionresponsedecomposition|Capandnocap Notes: Impulse responses to a 15% decline in energy supply (black lines) and the decomposition of the responses(coloredbars).Preferencesarenon-homothetic.Thebiggercountry,ofsizeΘimposesapricecap ontheenergyprice(lefttwopanels)andthesmallercountry,ofsize1−Θdoesnot(rightmostpanel). The y-axisisintermsofpercentagedeviationsfromsteadystate. Thex-axisisinquarters. Inflationandinterest ratesareannualized. Targetedtransfers. Foreasycomparisonwiththepricecap,Isetthetargetedtransfersto the same per person government expenditure, but only for the constrained households. Since they are 10% of the population, the specified targeted transfer only costs 10% of the cost of the price cap. The consumption responses are in Figure 11. The left graph shows that the targeted transfers are effective in increasing the constrained household’s consumption,whilenotloweringtheconsumptionoftheunconstrainedtoomuch. Moreover, the spillovers to the country without transfers substantially smaller than under the cap. Figure11: Consumptionresponsedecomposition|Transfersandnotransfers Notes: Impulse responses to a 15% decline in energy supply (black lines) and the decomposition of the responses(coloredbars). Preferencesarenon-homothetic. Thebiggercountry,ofsizeΘconductstargeted transfers to the constrained households (left two panels) and the smaller country, of size 1−Θ does not (rightmostpanel). Thedashedlineindicatesthebaselinescenariowiththeenergypricecap. They-axisis in terms of percentage deviations from steady state. The x-axis is in quarters. Inflation and interest rates areannualized. The responses of some other macroeconomic variables for both the country with and without transfers are in Figure 12. Because transfers do not distort the integrated energy market in the currency union, they do not create much divergence within the union. The inflation response is significantly milder than in the case with price caps. So, together with the absence of divergence, the transfers are more preferable for the common central bank when stabilizing the inflation rates across the union. Moreover, the spillover in 35

termsofconsumptionarealsosubstantiallylowerthanwiththepricecap. Figure12: Responsestoanadverseenergyshock|Transfersvs. notransfers Notes: Impulse responses to a 15% decline in energy supply. Preferences are non-homothetic. The bigger country,ofsizeΘconductstargetedtransferstotheconstrained(c)households(blue,solid)andthesmaller country,ofsize1−Θdoesnot(red,dashed). ustandsfortheunconstrainedhouseholds. Outputisequalto theoutputgap. They-axisisintermsofpercentagedeviationsfromsteadystate. Thex-axisisinquarters. Inflationandinterestratesareannualized. 5.3 Discussion: A debt-financed energy price cap in a non-Ricardian monetary union Contrarytothebaselinemodel,RicardianEquivalencedoesnotholdintheTANKversion of the model. If a country finances its cost of implementing the price cap through debt, beingpartofamonetaryunionpreventsthatcountrytofullybenefitfromthecap. Considering the case of a two-country monetary union with one capped and one uncappedcountry,thecappedcountrycanlimitthesurgeininflationrates,whereastheuncappedcountrycannot. Sincethecountriesareinamonetaryunion,theinterestratesset by the central bank are too high for the capped country while too low for the uncapped country. Assuming the capped country financed its cap implementation costs through debt,thecountryfacesdebt-servicingcostshigherthandesiredduetonothavingitsown monetarypolicy. Iintendtoexplorethisideaingreaterdepthinfuturework. 36

6 Conclusion This paper investigates the trade-offs of imposing an energy price cap during an energy crisis, based on the Euro Area energy crisis in 2022. I introduce a shared energy supply as an additional dimension of integration to a New Keynesian currency union model with two countries and rationalize the decisions that policymakers made in 2022. An adverse energy supply shock causes high energy inflation and a cost-push shock in the non-energy, core sector. I show that the cooperative policy is to refrain from introducing price caps. However, for an individual country it is welfare improving to impose a price cap, if the other country does not: the capped country avoids the crisis while the uncappedcountryexperiencesasupplyshockoftwicethemagnitude. The magnitude of those spillovers determine the preferred policy decisions – To cap ornottocap? Ontheonehand,apricecapensuresthathouseholdscanmaintainenergy consumption levels. On the other hand, a cap is a cost to the government, and therefore ultimatelythehouseholds. Whenonecountryimposesapricecap,theuncappedcountry incurs negative spillovers. So, there is a trade-off between paying for the cap and the paying for the negative spillovers. The magnitude of the spillovers depend on the nonhomotheticity of preferences and the substitutability of energy sources. The quantitative modelwithbothingredientsshowthatthecostoffundingthepricecapexceedsthecosts of bearing negative spillovers. This result explains why some Euro Area countries did notintroduceapricecapin2022,whileothersdid. Moreover, I perform some counterfactual exercises. First, I use a historical shock decompositiontoexaminewhatenergyandheadlineinflationwouldhavebeenintheEuro Area in 2022 if none of the countries imposed an energy price cap. I find that the energy pricecapcontributedtoabout10percentagepointstoenergyinflationand0.5percentage pointstoheadlineinflationin2022. Second,Icomparetheenergypricecapwithtargeted transfersandshowthattargetedtransferstothosehouseholdsischeaperandmoreeffectiveinboostingconsumptionofthepoor. Inaddition,becausethetransfersdonotdistort theenergyprice,thereisnodivergencewithintheunion. References Adjemian, S., Juillard, M., Karame´, F., Mutschler, W., Pfeifer, J., Ratto, M., Rion, N., & Villemot, S. (2024). Dynare: Reference manual, version 6 (Dynare Working Papers No.80).CEPREMAP.(Cit.onpp.25,30,53). Anderson, T. M. (2007). Fiscal policy coordination and international trade. Economica, 74(294), 235–257. https://doi.org/https://doi.org/10.1111/j.1468-0335.2006. 00536.x(cit.onp.4). 37

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where 0 ≤ ε ≤ ε < 1 and α > 0. The utility assumes an inelastic labor supply and 1 2 E a per-period utility of the form v = 1 (cid:104)(cid:16) expt (cid:17)ε1 −1 (cid:105) − αE (cid:104)(cid:16) PEt (cid:17)ε2 −1 (cid:105) . exp is the total ε1 POt ε2 POt t nominal expenditure of the household on Energy and non-energy (Rest) goods, defined asexp = P Eh +P C . t Et t Ot Ot Choice between energy and non-energy other goods. Marshallian demand functions obtainedwithRoy’sidentity: α (cid:16) PEt (cid:17)ε2−1 Eh = − ∂v/∂P Et = E POt (A.2) t ∂v/∂exp t (cid:16) expt (cid:17)ε1−1 POt (cid:16) expt (cid:17)ε1 −α (cid:16) PEt (cid:17)ε2 C = − ∂v/∂P Ot = POt E POt (A.3) Ot ∂v/∂exp t (cid:16) expt (cid:17)ε1−1 POt RearrangetoexpressC intermsofEh togetrelativedemand:36 Ot t (cid:16) expt (cid:17)ε1 −α (cid:16) PEt (cid:17)ε2 1−α (cid:16) POt (cid:17)ε1 (cid:16) PEt (cid:17)ε2 C = POt E POt = E expt POt = 1−α E ϖ t (A.8) Ot (cid:16) e P x O p t t (cid:17)ε1−1 e P x O p t t e P x O p t t 1−α ϖ P = E t Et Eh (A.9) α ϖ P t E t Ot where (cid:18) P (cid:19)ε1 (cid:18) P (cid:19)ε2 Ot Et ϖ = (A.10) t exp P t Ot is the energy expenditure share wedge. When ε = ε = 0 (Cobb-Douglas case), then 1 2 36AnotherwaytorearrangetheMarshalliandemands: exp exp (cid:18) P (cid:19)ε1 (cid:18) P (cid:19)ε2 Eh =α tϖ =α t Ot Et (A.4) t E P t E P exp P Et Et t Ot exp exp (cid:20) (cid:18) P (cid:19)ε1 (cid:18) P (cid:19)ε2 (cid:21) C = t (1−α ϖ )= t 1−α Ot Et (A.5) Ot P E t P E exp P Ot Ot t Ot • With ε > 0, the expenditure elasticity of demand is positive, but strictly smaller than unity for 1 energyandlargerthanunityforRest. Withε =0,theyarebothequaltounity. 1 • Theexpenditureelasticityofdemandforenergyis1−ε . 1 Theexpendituresharesofthetwotypesofgoodsare: P Eh (cid:18) P (cid:19)ε1 (cid:18) P (cid:19)ε2 η = Et t =α ϖ =α Ot Et (A.6) Et exp E t E exp P t t Ot P C (cid:18) P (cid:19)ε1 (cid:18) P (cid:19)ε2 η = Ot Ot =1−α ϖ =1−α Ot Et (A.7) Ot exp E t E exp P t t Ot 42

C = 1−αE PEtEh. Definerelativetotalexpenditureas: Ot αE POt t exp P exprel ≡ t = Et Eh +C (A.11) t P P t Ot Ot Ot Choice between Home and Foreign goods. The non-energy goods are bundled in a compositeindex: (cid:104) (cid:105)γ/(γ−1) C = (1−α )1/γ(C )(γ−1)/γ +(α )1/γ(C )(γ−1)/γ (A.12) Ot I Ht I Ft where α ∈ (0,1) is the share of imported goods in the consumption basket and γ is the I elasticity of substitution between Home and Foreign goods. C ,C are consumption Ht Ft indicesofH-producedandF-producedgoodsrespectively: (cid:20)(cid:90) 1 (cid:21)ε/(ε−1) (cid:20)(cid:90) 1 (cid:21)ε/(ε−1) C ≡ C (i)(ε−1)/εdi C ≡ C (i)(ε−1)/εdi (A.13) Ht Ht Ft Ft 0 0 where ϵ is the elasticity of substitution between different varieties within Home and Foreign goods. The intratemporal consumption choice between different varieties of HproducedandF-producednon-energygoodsis:37 (cid:18) P (i) (cid:19)−ε (cid:18) P (i) (cid:19)−ε Ht Ft C (i) = C C (i) = C (A.16) Ht Ht Ft Ft P P Ht Ft where P ,P are indices of prices of of H-produced and F-produced goods respec- Ht Ht tively: (cid:18)(cid:90) 1 (cid:19)1/(1−ε) (cid:18)(cid:90) 1 (cid:19)1/(1−ε) P ≡ P (i)1−εdi P ≡ P (i)1−εdi (A.17) Ht Ht Ft Ft 0 0 From Eq. (A.16) and (A.17), aggregate expenditure on H-produced and F-produced goodsrespectively: (cid:90) 1 (cid:90) 1 P (i)C (i)di = P C P (i)C (i)di = P C (A.18) Ht Ht Ht Ht Ft Ft Ft Ft 0 0 Intratemporal concumption choice between H-produced and F-produced goods bun- 37Solutionstothefollowingproblems: (cid:90) 1 (cid:20)(cid:90) 1 (cid:21) ε− ε 1 ε−1 min P Ht (i)C Ht (i)di s.t. C Ht (i) ε di ≥C Ht (A.14) CHt(i) 0 0 (cid:90) 1 (cid:20)(cid:90) 1 (cid:21) ε− ε 1 ε−1 min P Ft (i)C Ht (i)di s.t. C Ft (i) ε di ≥C Ft (A.15) CFt(i) 0 0 43

dle:38 (cid:18) P (cid:19)−γ (cid:18) P (cid:19)−γ Ht Ft C = (1−α ) C C = α C (A.19) Ht I Ot Ft I Ot P P Ot Ot whereP istheaggregatepriceindexfornon-energygoods: Ot P = (cid:2) (1−α )P1−γ +α P1−γ(cid:3) 1− 1 γ (A.20) Ot I Ht I Ft Combining the intratemporal consumption choice between Home and Foreign goods, I get: C 1−α (cid:18) P (cid:19)−γ Ht I Ht = (A.21) C α P Ft I Ft FromEq. (A.19)and(A.20),aggregateexpenditureonnon-energyconsumptionis: (cid:90) 1 (cid:90) 1 P (i)C (i)di+ P (i)C (i)di = P C +P C = P C (A.22) Ht Ht Ft Ft Ht Ht Ft Ft Ot Ot 0 0 A.1.2 Intertemporalconsumptionchoicesandlaborsupply Nominalbudgetconstraint: exp = P Eh +P C = W N +D +DE +R B −B −HC −T (A.23) t Et t Ot Ot t t t t t−1 t−1 t t t where D is the nominal profit paid by the domestic firms to the representative domestic t householdandDE theprofitsfromtheenergysellersgivenby: t P (cid:16) (cid:17) DE = Et Eh +Ef (A.24) t P ¯ t t E HC aretheportfolioadjustmentcostsofthehousehold: t ν˜ HC = (B −B ¯ )2 (A.25) t t 2 where B is nominal bond holdings of the household. T are lump-sum taxes for the t t governmenttofinancetheenergypricecap. 38Solutiontothefollowingproblem: (cid:20) (cid:21) γ min P Ht C Ht +P Ft C Ft s.t. (1−α)γ 1 C H γ− γ t 1 +αγ 1 C F γ− t γ 1 γ−1 ≥C Ot CHt,CHt 44

(cid:88) ∞ (cid:26) 1 (cid:20)(cid:18) exp (cid:19)ε1 (cid:21) α (cid:20)(cid:18) P (cid:19)ε2 (cid:21) L = E βt t −1 − E Et −1 0 ε P ε P 1 Ot 2 Ot t=0 (cid:2) (cid:3)(cid:9) + λ W N +D +DE +R B −B −HC −exp (A.26) t t t t t t−1 t−1 t t t ∂L : expε1−1P−ε1 −λ = 0 (A.27) ∂exp t Ot t t ∂L : λ = βR E [λ ] (A.28) t t t t+1 ∂B t Eulerequation: (cid:18)E [exp ] (cid:19)1−ε1 R (cid:20)(cid:18) 1 (cid:19)ε1 (cid:21) t t+1 = β t E (A.29) exp 1+P ν˜(b − ¯ b) t Π t t t O,t+1 where b = Bt is real bond holdings and Π = POt is gross inflation. Inelastic labor t Pt Ot PO,t−1 ¯ meansN = N. t A.2 Firms There is monopolistic competition among intermediate firms producing the other consumptiongoods. Theyfaceadjustmentcostsa` laRotemberg(1982). Costminimization min W N (i)+P Ef(i) (A.30) t t Et t Nt(i),E t f(i) (cid:18) P (i) (cid:19)−ϵ Ht s.t. demandcurve Y (i) = Y (A.31) t t P Ht productionfunction Y (i) = A (cid:20) (cid:0) αf (cid:1)1/θf (cid:16) Ef(i) (cid:17)(θf−1)/θf + (cid:0) 1−αf (cid:1)1/θf (N (i))(θf−1)/θf (cid:21)θf/(θf−1) t t t t (A.32) 45

Firstorderconditionw.r.t. N (i)andEf(i):39 t t W = (cid:0) 1−αf (cid:1)1/θf µnom (cid:18) Y t (i) (cid:19)1/θf A(θf−1)/θf (A.33) t t N (i) t t P = (cid:0) αf (cid:1)1/θf µnom (cid:18) Y t (i) (cid:19)1/θf A(θf−1)/θf (A.34) Et t Ef(i) t t Combining these equations, we get that the relative price of the production inputs determinethetrade-offbetweenthem: Ef(i) αf (cid:18) P (cid:19)−θf t = Et (A.35) N (i) 1−αf W t t Thetotalfactorproductivity: ln(A ) ≡ a = ρ a +εa (A.36) t t a t−1 t Pricesetting ∞ (cid:20) (cid:21) (cid:88) P (i) W max E Λ Ht Y (i)− t N (i)−P Ef(i)−Y FC (A.37) PHt(i),Nt(i) 0 t=0 t+1 P Ht t P Ht t Et t t t (cid:18) P (i) (cid:19)−ϵ Ht s.t. demandcurve Y (i) = Y (A.38) t t P Ht ξ (cid:18) P (i) (cid:19)2 Ht priceadjustmentcosts FC (i) = −1 (A.39) t 2 P (i) H,t−1 whereΛ = β (cid:16) Ct+1 (cid:17)−σ isthestochasticdiscountfactorand (cid:82)1 P (i) = P theaverage t+1 Ct 0 Ht Ht priceofH goods. Firstorderconditionw.r.t. P (i)is:40 Ht (cid:18) P (i) (cid:19)−ϵ 1 (cid:18) P (i) (cid:19) 1 Ht Ht (1−ϵ) Y (i)−ξ −1 Y (i) t t P P P (i) P (i) Ht Ht H,t−1 H,t−1 (cid:18) 1 (cid:19)−ϵ (cid:20) (cid:18) P (i) (cid:19) (cid:18) P (i) (cid:19)(cid:21) +µ ϵP (i)−ϵ−1 Y (i)+E Λ ξ H,t+1 −1 Y (i) H,t+1 = 0 t Ht P t t t+1 P (i) t+1 P (i)2 Ht Ht Ht (A.40) Imposing the symmetric equilibrium conditions P (i) = P and Y (i) = Y , derive the Ht Ht t t NewKeynesianPhilipsCurve(NKPC): (cid:20) (cid:21) Y (1−ϵ)−ξ(Π −1)Π +µ ϵ+βE ξ(Π −1)Π t+1 = 0 (A.41) Ht Ht t t H,t+1 H,t+1 Y t 39TheLagrangemultiplieronthedemandcurveµnomisthenominalmarginalcost. t 40TheLagrangemultiplieronthedemandcurveµ istherealmarginalcost. t 46

(cid:16) (cid:17) whereΠ = PHt(i) isinflationoftheHome-producedgood. Ht PHt Aggregatepriceadjustmentcosts: (cid:90) 1 FC = FC (i)di (A.42) t t 0 Aggregatenominalprofits: D = P Y (1−FC )−W N −P Ef (A.43) t Ht t t t t Et t A.3 Summary of model equations Relativeprices P = (cid:2) (1−α )P1−γ +α P1−γ(cid:3) 1− 1 γ (A.44) Ot I Ht I Ft P∗ = (cid:2) α∗P1−γ +(1−α∗)P1−γ(cid:3) 1− 1 γ (A.45) Ot I Ht I Ft P = α logP +(1−α )logP (A.46) t E Et E Ot P∗ = α logP∗ +(1−α )logP∗ (A.47) t E Et E Ot P Ft S = (A.48) t P Ht P Prel,EO = Et (A.49) t P Ot P∗ Prel,EO∗ = Et (A.50) t P∗ Ot P = P∗ (A.51) Et Et 47

Households. C = Eh +C (A.52) t t Ot C∗ = Eh∗ +C∗ (A.53) t t Ot (cid:104) (cid:105)γ/(γ−1) C = (1−α )1/γ(C )(γ−1)/γ +(α )1/γ(C )(γ−1)/γ (A.54) Ot I Ht I Ft (cid:104) (cid:105)γ/(γ−1) C∗ = (1−α∗)1/γ(C∗ )(γ−1)/γ +(α∗)1/γ(C∗ )(γ−1)/γ (A.55) Ot I Ft I Ht C 1−α (cid:18) P (cid:19)−γ 1−α Ht = I Ht = I Sγ (A.56) C α P α t Ft I Ft I C∗ 1−α∗ (cid:18) P (cid:19)−γ 1−α∗ Ht = I Ht = ISγ (A.57) C∗ α∗ P α∗ t Ft I Ft I [1−α ϖ ] C = E t Prel,EOEh (A.58) Ot α ϖ t t E t [1−α ϖ∗] C∗ = E t Prel,EO∗Eh∗ (A.59) Ot α ϖ∗ t t E t ϖ = (cid:18) P Ot (cid:19)ε1 (cid:18) P Et (cid:19)ε2 = (cid:0) exprel (cid:1)−ε1 (cid:16) Prel,EO (cid:17)ε2 (A.60) t exp P t t t Ot ϖ∗ = (cid:18) P O ∗ t (cid:19)ε1 (cid:18) P E ∗ t (cid:19)ε2 = (cid:0) exprel∗ (cid:1)−ε1 (cid:16) Prel,EO∗ (cid:17)ε2 (A.61) t exp∗ P∗ t t t Ot exprel = Prel,EOEh +C (A.62) t t t Ot exprel∗ = Prel,EO∗Eh∗ +C∗ (A.63) t t t Ot ¯ N = N (A.64) t N∗ = N ¯ (A.65) t (cid:32) E (cid:2) exprel (cid:3)(cid:33)1−ε1 R t t+1 = β t E (cid:2) Π−1 (cid:3) (A.66) exprel 1+P ν˜(b − ¯ b) t O,t+1 t Ot t (cid:32) E (cid:2) exprel∗ (cid:3)(cid:33)1−ε1 R (cid:104) (cid:105) t t+1 = β t E (cid:0) Π∗ (cid:1)−1 (A.67) exprel∗ 1+P∗ ν˜(b∗ − ¯ b) t O,t+1 t Ot t 48

Firms. Y = A (cid:20) (cid:0) αf (cid:1)1/θf (cid:16) Ef (cid:17)(θf−1)/θf + (cid:0) 1−αf (cid:1)1/θf (N )(θf−1)/θf (cid:21)θf/(θf−1) (A.68) t t t t Y∗ = A∗ (cid:20) (cid:0) αf (cid:1)1/θf (cid:16) Ef∗ (cid:17)(θf−1)/θf + (cid:0) 1−αf (cid:1)1/θf (N∗)(θf−1)/θf (cid:21)θf/(θf−1) (A.69) t t t t ln(A ) ≡ aˆ = ρ aˆ +εa (A.70) t t a t−1 t ln(A∗) ≡ aˆ∗ = ρ aˆ∗ +εa∗ (A.71) t t a t−1 t W t = (cid:0) 1−αf (cid:1)1/θf µ (cid:18) Y t (cid:19)1/θf A(θf−1)/θf (A.72) P t N t t t W t ∗ = (cid:0) 1−αf (cid:1)1/θf µ∗ (cid:18) Y t ∗(cid:19)1/θf (A∗)(θf−1)/θf (A.73) P∗ t N∗ t t t P Et = (cid:0) αf (cid:1)1/θf µ (cid:18) Y t (cid:19)1/θf A(θf−1)/θf (A.74) P t Ef t t t P Et = (cid:0) αf (cid:1)1/θf µ∗ (cid:18) Y t ∗ (cid:19)1/θf (A∗)(θf−1)/θf (A.75) P∗ t Ef∗ t t t (cid:20) (cid:21) ϵ Y (Π −1)Π = (µ −µ¯)+βE (Π −1)Π t+1 (A.76) Ht Ht t t H,t+1 H,t+1 ξ Y t ϵ (cid:20) Y∗ (cid:21) (Π −1)Π = (µ∗ −µ¯)+βE (Π −1)Π t+1 (A.77) Ft Ft ξ t t F,t+1 F,t+1 Y∗ t Goodsmarketclearing. Y = (1−α ) (cid:0) Prel (cid:1)−θ C +α (cid:0) Prel∗ (cid:1)−θ C∗ +AC +FC +T (A.78) t I Ht Ht I Ht Ht t t t Y∗ = α (cid:0) Prel (cid:1)−θ C +(1−α ) (cid:0) Prel∗ (cid:1)−θ C∗ +AC∗ +FC∗ (A.79) t I Ft Ft I Ft Ft t t Energymarketclearing. E = Eh +Eh∗ +Ef +Ef∗ (A.80) t t t t t E = Eρe E ¯1−ρeexp(εe) (A.81) t t−1 t Bondsmarketclearing. CA = r bh +P Y (1−FC )−P C −HC (A.82) t t−1 t−1 Ht t t Ot Ot t CA∗ = r bh∗ +P Y∗(1−FC∗)−P∗ C∗ −HC∗ (A.83) t t−1 t−1 Ft t t Ot Ot t (CA )Θ = −(CA∗)1−Θ (A.84) t t (cid:0) bh (cid:1)Θ = − (cid:0) bh∗ (cid:1)1−Θ (A.85) t t 49

Monetarypolicy. 1 (cid:18) ΠW(cid:19)ϕπ (cid:18) YW(cid:19)ϕy R = t t exp(ν ) (A.86) t β Π ¯W Y ¯W t t ν = ρνν +εν (A.87) t t−1 t Fiscalpolicy. Peff = P −CAP (A.88) Et Et t ¯ CAP = P −P (A.89) t Et E CAP (Eh +Ef) = T (A.90) t t t t (cid:16) (cid:17) CAP Eh +Ef t t t CAPexp = (A.91) t Y t A.4 Steady state This section characterizes the steady state of the Home economy. The Foreign economy isidentical. Insteadystate,thepricesareconstant. Hence,theinflationratesareallequal tounity. ¯ Π = 1 (A.92) ¯ Π = 1 (A.93) E ¯ Π = 1 (A.94) O ¯ I take P = 1 as the numeraire. With the below calculations, I get the exogenous level of O ¯ ¯ energyE whichsetsthesteady-statepriceofenergyalsoequaltounity,soP = 1. E Demandside. TakingtheEulerequationinsteadystate,Icanexpressthesteadystate nominalinterestrateasafunctionofthediscountfactor: 1 ¯ R = (A.95) β Moreover,Iassumethattheenergyexpenditurewedgeϖ isunityinsteadystate,sothat t theexpendituresharesofenergyandtheotherconsumptiongoodsarethesameasinthe benchmarkCobb-Douglascase: ϖ¯ = 1 (A.96) Then,sincepricesareequaltounityinsteadystate,Iobtainthatsteady-statetotalexpen- 50

ditureofthehouseholdfromthefoodexpenditurewedgeequation: ex¯p = ϖ¯1/ε1 (A.97) From the Marhsallian demands from Footnote 36, derive the steady-state values for energyandothergoodsconsumption: E ¯h = α ex¯pϖ¯ (A.98) E ¯ C = (1−α ϖ¯)ex¯p (A.99) O E Then,fromthegoodsmarketclearingcondition,getthesteady-stateoutputvalue: Y = (1−α )C ¯ +α∗C ¯∗ (A.100) I O I O Supply side. From the price-setting equation of the firms, get the steady-state real marginalcost: ϵ−1 µ¯ = (A.101) ϵ Sinceexp(a¯)scalestheeconomy,Isetthetotalfactorproductivitya¯ suchthatexp(a) = 1: a¯ = 0 (A.102) From the energy demand equation of the firms, get the steady-state value for the firms’ energyuse: E ¯f = αf(µ¯)θfY ¯ (A.103) Then,fromtheproductionfunction,obtainthesteady-statevalueforlabor: (cid:34) Y − (cid:0) αf (cid:1)1/θf (cid:0) E ¯f (cid:1)(θf−1)/θf(cid:35)(θf)/(θf−1) ¯ N = (A.104) (1−αf)1/θf Usingthesteady-statevaluesforlabor,outputandmarginalcost,gettherealwage: (cid:18) ¯ (cid:19)1/θf W ¯ real = (cid:0) 1−αf (cid:1)1/θf µ¯ Y (A.105) ¯ N Theprofitsinsteadystateare: D ¯ = Y −W ¯ realN ¯ −E ¯f (A.106) Check supply and demand side are consistent. From the budget constraint of the 51

household,checkthatthefollowingequationholds: ex¯p = W ¯ realN ¯ +D ¯ +E ¯h +E ¯f (A.107) For the two-agent version, check that the aggregate budget constraint (constrained and unconstrainedhouseholdholdscombined)holds: ex¯p = W ¯ N ¯ +(1−δ)D ¯ +E ¯h +E ¯f +λτc +(1−λ)τu (A.108) t t A.5 Domestic energy production sector Theoligopolisticenergyfirm’sproblemis maxP Y −W N (A.109) Et Et t Et NEt s.t. productionfucntionY = A Nη (A.110) Et Et Et Thefirst-orderconditiongivesrisetothelabordemand: (cid:18) (cid:19) 1 P 1−η Et N = ηA , (A.111) Et Et W t whichdeterminestheenergyproduction: (cid:18) (cid:19) η 1 P 1−η Y = A1−η η Et (A.112) Et Et W t andtheprofitsoftheenergyfirm: D = (1−η)P Y (A.113) Et Et Et B Bayesian estimation: Details on data used and results In this section, I describe and present the data used for Bayesian estimation. For the first estimation, to estimate the parameters, I use data from before the COVID-19 pandemic, so 2008Q1 – 2019Q4. For the second estimation, to perform a historic shock decomposition of the shock, I use data up to 2022Q4. I seasonally adjust the all data series withX-13ARIMA-SEATS.Whenthedataismonthly,Itransformthedatatogetquarterly equivalents. To get the aggregates for “Cap” and “No cap” countries, I take weighted averages with country weights from Eurostat Data. Finally, I detrend the data with the one-sidedHodrick-Prescottfilteranddemeantheseriestomatchthemodelvariables. 52

Pre-pandemic data for estimating parameters. The data used for the estimation parameters are in Figure 13.41 Since the price cap policy only took place in 2022, the energy and gas inflation in the union is the same across countries. Using gas consumption as a common variable avoids stochastic singularity. Since I assume that the countries in the union share one supply of gas, when the gas price is the same across countries the gas consumption also needs to be the same. Nominal interest is the rate that the European Central Bank sets. Energy consumption is yearly data. Hence, I allow for measurement errorsinthemodeltocapturequarterlyfluctuations. Figure13: Datausedforestimationofparameters Notes: PlotoftheEuroAreadatausedintheBayesianestimationoftheparameters. Forthelowerpanels I separate the Euro Area countries into “Cap” and “No cap” countries, according to whether countries imposedanenergycapin2022ornot. SeeFigure2fordetails. Data for historical shock decomposition. The data used for the historical shock decompositionareinFigure14. Thedatasourcesareidenticaltothoseforthepre-pandemic data. However, since I detrend the data over a slightly longer sample, the values are somewhat different. Moreover, I only let the energy and gas inflation diverge in 2022. Before 2022, I take the weighted average of the two blocs, since there are no price caps in place. Figure14showsthattheenergyandgasinflationratesmovedverycloselybetween “Cap”and“Nocap”countriesbefore2022. Estimation method. I use the Bayesian estimation approach built in Dynare Adjemian et al. (2024). For the estimation of the parameters, I use a slice optimizer to find the 41Data sources: Energy, gas, and CPI inflation (Eurostat, prc hicp manr), gas consumption (Eurostat, nrg cb gasm), nominal interest rate (Eurostat, irt st q), total output (Eurostat, namq 10 pc), energy consumption(OurWorldinData,Percapitaprimaryenergyconsumptionbysource). Whendataarenotper capitaandtheyneedtobe,Iuseintrapolatedpopulationdata(Eurostat,demo pjan)totransformthemto percapitavariables. 53

Figure14: Datausedforhistoricalshockdecomposition Notes: Plotofthedatausedinthehistoricalshockdecomposition. IseparatetheEuroAreacountriesinto “Cap”and“Nocap”countries,accordingtowhethercountriesimposedanenergycapin2022ornot. See Figure 2 for details. Grey-shaded area are 2020Q1 and Q2, the quarters most affected by the COVID-19 pandemic. Figure15: Priorandposteriordistributions Notes: Plot of the prior distribution (gray) and the posterior distribution (black). The vertical blue line indicatestheposteriormode. They-axisdisplaysthedensityofthedistributions. mode of the posterior distribution.42 Then, the Metropolis-Hastings algorithm evaluates the marginal likelihood of the model and produces the posterior distributions. I use one million replications for each chain of the algorithm and four parallel chains. I check that theMonteCarloMarkovChainconvergesandthattheposteriorchainforeachparameter isstable. TheposteriorplotsareinFigure15. Forthehistoricalshockdecomposition,Iuse the shock decomposition-command in Dynare, which uses the Kalman smoother to decomposethehistoricalfluctuationsofthevariablesintocontributionsfromeachshock. 42Option5ofthemode compute-optionintheestimation-commandinDynare. 54

C Additional figures Figure16: NaturalgaspriceinEurope Notes: The price index of the Title Transfer Facility (TTF) gas in the Netherlands. Data source: IMF Data (2024). Figure17: Decompositionofthevarianceofheadlineinflation Notes: The headline inflation in country i in quarter t is Π = (1−αE)ΠO +αEΠE where is αENG the it it it it it shareofenergyintheconsumptionbasket,ΠE andΠO energyandothergoodsinflation. Thevariancedeit it compositionisacrosscountriesforeachquarter,soVar (Π ) = Var (cid:2) (1−αE)ΠO+αEΠE(cid:3) . Datasource: t it t it it it it Eurostat. 55

Figure 18: Responses to an adverse energy supply shock | Cap vs. no cap under coreinflationtargeting Notes: Impulse responses to a 15% decline in energy supply, in a model in which the central bank targets other inflation (core inflation). Preferences are non-homothetic. The bigger country, of size Θ imposes a price cap on the energy price (blue, solid) and the smaller country, of size 1−Θ does not (red, dashed). Theblacksolidlinesshowtheunion-widevariables. They-axisisintermsofpercentagedeviationsfrom steadystate. Thex-axisisinquarters. Inflationandinterestratesareannualized. Figure19: Responsestoanadverseenergysupplyshock|Capvs. nocapwithTANK Notes: Impulse responses to a 15% decline in energy supply, in the two-agent version of the model. Preferences are non-homothetic. The bigger country, of size Θ imposes a price cap on the energy price (blue, solid)andthesmallercountry,ofsize1−Θdoesnot(red,dashed). Theblacksolidlinesshowtheunionwidevariables. They-axisisintermsofpercentagedeviationsfromsteadystate. Thex-axisisinquarters. Inflationandinterestratesareannualized. 56

C.1 Results under elastic labor supply Figure 20: Responses to an adverse energy supply shock | Cap vs. no cap with elastic laborsupply Notes: Impulse responses to a 15% decline in energy supply, in a model with elastic labor supply. Preferencesarenon-homothetic.Thebiggercountry,ofsizeΘimposesapricecapontheenergyprice(blue,solid) and the smaller country, of size 1−Θ does not (red, dashed). The black solid lines show the union-wide variables. Outputisequaltotheoutputgap. Governmentexpenditureonthepricecap(Govt. exp. cap)is thecostofthecapasashareofannualtotaloutputofthecountry(GDP).They-axisisintermsofpercentagedeviationsfromsteadystate. Thex-axisisinquarters. Inflationandinterestratesareannualized. Table8: Welfaregains/lossesafterenergysupplyshock (a)Baseline (b)Elasticlaborsupply 1/3ofunion 1/3ofunion % Cap Nocap % Cap Nocap Cap ( −1.04 , −1.04 ) ( 0.49 ,−1.08) Cap ( −0.86 ,−0.86) ( 0.40 , −0.72 ) 2/3 2/3 Nocap (−1.08, 0.42 ) (−0.05,−0.05) Nocap (−0.97, 0.53 ) (−0.03,−0.03) Notes: Welfaregainsandlossesaftera15%energysupplyshock,inamodelwithelasticlaborsupply. The gains and losses are in terms of the consumption equivalent relative to the steady state. The circles are aroundthepreferredpolicychoices(CaporNocap)forthecountries. 57

C.2 Results under flexible prices Figure 21: Responses to an adverse energy supply shock | No price caps with flexible prices Notes: SeeFigure20. Impulseresponsestoa15%declineinenergysupply,inamodelwithflexibleprices. Figure22: Responsestoanadverseenergysupplyshock|Pricecapswithflexibleprices Notes: SeeFigure20. Impulseresponsestoa15%declineinenergysupply,inamodelwithflexibleprices. 58

Figure 23: Responses to an adverse energy supply shock | Cap vs. no cap with flexible prices Notes: SeeFigure20. Impulseresponsestoa15%declineinenergysupply,inamodelwithflexibleprices. Table9: Welfaregains/lossesafterenergysupplyshock (a)Baseline (b)Flexibleprices 1/3ofunion 1/3ofunion % Cap Nocap % Cap Nocap Cap ( −1.04 , −1.04 ) ( 0.49 ,−1.08) Cap ( −1.04 , −1.04 ) ( 0.49 ,−1.08) 2/3 2/3 Nocap (−1.08, 0.42 ) (−0.05,−0.05) Nocap (−1.08, 0.42 ) (−0.05,−0.05) Notes:SeeTable8.Welfaregainsandlossesaftera15%energysupplyshock,inamodelwithflexibleprices. 59

C.3 Results under flexible exchange rates Figure 24: Responses to an adverse energy supply shock | Cap vs. no cap in world with flexiblenominalexchangerates Notes:SeeFigure20. Impulseresponsestoa15%declineinenergysupply,inamodelwithflexiblenominal exchangerates. Table10: Welfaregains/lossesafterenergysupplyshock (a)Baseline (b)Flexiblenominalexchangerate 1/3ofunion 1/3ofunion % Cap Nocap % Cap Nocap Cap ( −1.04 , −1.04 ) ( 0.49 ,−1.08) Cap ( −1.04 , −1.04 ) ( 0.49 ,−1.08) 2/3 2/3 Nocap (−1.08, 0.42 ) (−0.05,−0.05) Nocap (−1.07, 0.42 ) (−0.05,−0.05) Notes: See Table 8. Welfare gains and losses after a 15% energy supply shock, in a model with flexible nominalexchangerates. 60

C.4 Results under CES preferences Figure25: Responsestoanadverseenergysupplyshock|NocapswithCESpreferences Notes: Impulse responses to a 15% decline in energy supply. The graph compares three types of preferences: Cobb-Douglas(solid,thin),ConstantElasticityofSubstitution(CES)(dashed),andnon-homothetic preferences(solid,bold). Outputisequaltotheoutputgap. They-axisisintermsofpercentagedeviations fromsteadystate. Thex-axisisinquarters. Inflationandinterestratesareannualized. Figure 26: Responses to an adverse energy supply shock | Cap vs. no cap with CES preferences Notes:SeeFigure20.Impulseresponsestoa15%declineinenergysupply,inamodelwithCESpreferences. 61