Supply Chain Constraints and Inflation

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Supply Chain Constraints and Inflation Diego Comin, Robert Johnson, Callum Jones 2023-075 Please cite this paper as: Comin, Diego, Robert Johnson, and Callum Jones (2023). “Supply Chain Constraints and Inflation,” Finance and Economics Discussion Series 2023-075. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2023.075. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Supply Chain Constraints and Inflation∗ Diego Comin† Robert C. Johnson‡ Callum Jones§ October, 2023 Abstract Wedevelopamultisector,openeconomy,NewKeynesianframeworktoevaluatehowpotentiallybindingcapacityconstraints,andshockstothem,shapeinflation. Weshowthatbinding constraints for domestic and foreign producers shift domestic and import price Phillips Curves up, similar to reduced-form markup shocks. Further, data on prices and quantities together identify whether constraints bind due to increased demand or reductions in capacity. Applying the model to interpret recent US data, we find that binding constraints explain half of the increase in inflation during 2021-2022. In particular, tight capacity served to amplify theimpactofloosemonetarypolicyin2021,fuelingtheinflationtakeoff. ∗WethankBrentNeiman,SebastianGraves,RobertKollmann,WernerRoeger,NarayanaKocherlakota,andDavid Lopez-Salidoforhelpfuldiscussions,andseminarparticipantsatDukeUniversity,BostonUniversity,ErasmusUniversity,UniversidadCarlosIIIdeMadrid,theEinaudiInstitute,theFederalReserveBankofDallas,theNBERconferenceon“TheRiseofGlobalSupplyChains”(December2021),theInternationalResearchForumonMonetaryPolicy Conference(May2022),theCEPR/EC/EERconferenceon“TheCOVID-shockandtheNewMacroeconomicLandscape” (October 2022), the BOJ-CEPR 7th International Macroeconomics and Finance Conference (March 2023), theSocietyforEconomicDynamicsAnnualMeeting(June2023),theNBERSummerInstituteMonetaryEconomics meeting(July2023), andtheCEPRSalentoMacroMeetings(July2023)forcomments. WeespeciallythankDiego Anzoategui,whoassistedusduringintermediatestagesofthisresearch. Thismaterialisbaseduponworksupported by the U.S. Department of Homeland Security under Grant Award Number 18STCBT00001-03-00. The views and conclusionscontainedinthisdocumentarethoseoftheauthorsandshouldnotbeinterpretedasnecessarilyrepresentingtheofficialpolicies, eitherexpressedorimplied, oftheU.S.DepartmentofHomelandSecurity. Thismaterialis baseduponworksupportedbytheNationalScienceFoundationunderGrantNo. SES-2315629. Anyopinions,findings, andconclusionsorrecommendationsexpressedinthismaterialarethoseoftheauthorsanddonotnecessarily reflect the views of the National Science Foundation. Finally, the views expressed are those of the authors and not necessarilythoseoftheFederalReserveBoardortheFederalReserveSystem. †DartmouthCollege,NBER,andCEPR.Email: diego.comin@dartmouth.edu. ‡UniversityofNotreDame&NBER.Email: rjohns24@nd.edu. §FederalReserveBoard. Email: callum.j.jones@frb.gov.

In the later half of 2021 and into 2022, the United States experienced a burst of inflation as it emerged from the COVID-19 pandemic, led by a large increase in goods price inflation. Popular narratives suggest that strong consumer demand bumped up against constraints on the supply of goods, fueling inflation.1 Further, in their public statements, policymakers frequently blamed disruptions in both domestic and foreign segments of supply chains for restraining the supply of goods.2 Despite the plausibility of this narrative, it has been difficult to evaluate the quantitative importanceofsupplychainconstraintsforinflation,notleastbecausewelackmodelsthatcapture theirimpact. In this paper, we investigate how potentially binding capacity constraints for domestic and foreign producers shape inflation in a multisector, open economy, New Keynesian (NK) model with imported inputs and input-output linkages across sectors. Solving for the model’s non-linear equilibriumdynamicsviapiecewiselinearapproximations,wedevelopaBayesianmaximumlikelihood procedure to estimate key parameters and infer when constraints bind. We then apply the model to quantify how constraints in the supply chain, and potential shocks to them, have influenced recent data outcomes. We find that binding constraints account for about half (two percentagepoints)oftheincreaseininflationduring2021-2022. Interestingly,nosinglesetofshockscan explaintheinflationtakeoff. Rather,shocksthattightenedcapacitysetthestagefordemandshocks –mostimportantly,monetarypolicyshocks–totriggerbindingconstraintsandaccelerateinflation in 2021. Relaxation of the constraints, in part due to monetary tightening, then also explains the rapiddeclineingoodspriceinflationinthelatterhalfof2022. The framework we develop features occasionally binding constraints in two different places. The first is a constraint that applies at the level of individual foreign firms, whereby foreign producers are able to supply output at constant marginal costs up to a predetermined level, at which point production is quantity-constrained. Motivated by evidence on disruptions in markets for imported inputs, we devote particular attention to binding constraints on foreign input supply. The second constraint is a similar limit on production capacity for domestic firms, which impacts both downstreamfirmsandconsumers. Thesedualconstraintsallowustoseparatelycapturetheroleof domesticversusforeignsupplychaindisruptionsoninflation. Further, this framework features a distinction between supply-side versus demand-side explanations for binding constraints, with potentially important implications for policy. On the supply side, we assume the levels of the capacity constraints are exogenous and subject to stochastic 1SeeTheEconomist(2021),ReesandRungcharoenkitkul(2021),anddeSoyres,SantacreuandYoung(2023). 2In International Monetary Fund (2021), Gita Gopinath writes: “Pandemic outbreaks in critical links of global supplychainshaveresultedinlonger-than-expectedsupplydisruptions, furtherfeedinginflationinmanycountries.” Smialek and Nelson (2021) characterize the views of the US Federal Reserve chair as follows: “[Jerome Powell] notedthatwhiledemandwasstrongintheUnitedStates,factoryshutdownsandshippingproblemswereholdingback supply, weighing on the economy and pushing inflation above the Fed’s goal.” See Lane (2022) for a discussion of viewsattheEuropeanCentralBank,andGoodman(2021)foranarrativeofsupplychainbreakdowns. 1

shocks.3 This formulation captures the type of time-varying input shortages that occurred during the COVID period, both in the United States and abroad.4 On the demand side, an increase in demand may also exhaust excess capacity and induce capacity constraints to bind in the model. This alternative mechanism is salient, because the abrupt recovery of demand in 2021 seemed to stressexistingsupplychaincapacity. Separatingthesetwomechanisms–thatbindingconstraintsmaybetheresultofstrongdemand, or disruptions to capacity – represents a key quantitative challenge. Breaking the challenge into twopieces,wemustascertainwhetherconstraintsbind,whilealsoidentifyingwhytheybind. To shed light on how binding constraints may be detected, we note that binding constraints impactpricingdecisions. Inthemodel,constraintsareinternalizedbyeachfirmasitsetsitsprice, suchthatthefirm’soptimalmarkupdiffersdependingonwhethertheconstraintisbinding. Assuming that both exports and imports are invoiced in US Dollars, and prices are subject to adjustment frictions, then domestic and import price inflation satisfy Phillips Curve type relationships. When the domestic constraint binds, we show that there is an additional term in sector-level, domestic pricePhillipsCurvesthatresemblesamarkup(equivalently,cost-push)shock. Similarly,thereisa quasi-markupshockintheimportpricePhillipsCurvewhentheimportconstraintbinds. Thus,our framework provides a structural interpretation for reduced-form markup/cost-push shocks, based onbindingconstraints. This“markupshock”interpretationoftheroleofbindingconstrainsdovetailswellwithrelated work by Bernanke and Blanchard (2023), which uses an empirical model to argue that product market shocks (which raise prices given wages) explain a large share of recent US inflation. Importantly, our work investigates the structural origins of these empirically plausible shocks.5 The markup shock interpretation also highlights the contrast between binding constraints and other competing mechanisms that work through marginal costs, such as factor reallocation frictions or labor shortages. Finally, the markup shock interpretation is also prima facie consistent with the fact that US profit margins increased as inflation took off in 2021, so binding constraints can also 3Pre-determinedproductioncapacityisshapedbypastdecisionsaboutorganization,installedcapital,investments inworkercapabilities,andthefirm’sstockofbuyer/supplierrelationships. Thoughwetreatcapacityasanexogenous stochastic variable, one could extend the model to allow for endogenous capacity investment decisions. We eschew thisextensionfornow,becauseitdistractsfromourmainfocusonaccountingforrecentinflationdynamics. 4Onesourceoftheseshockswouldbepandemic-relatedfactoryshutdowns,asoccurredintheUS,China,Vietnam andelsewhere. Theyalsocaptureshortagesofinputsdueotherdisruptionstoglobalsupplyrelationships(e.g., cancellationofsupplycontractsearlyinthepandemicledtoshortagesofforeign-suppliedsemiconductorsthatcurtailed US auto production). Other historical shocks are also plausibly thought of as shocks to capacity – for example, the 2011To¯hokuearthquake/tsunamitookproductioncapacityofflineinJapan.Wenoteherethatdisruptionsintheglobal shippingindustry(e.g.,portcongestion)andbottlenecksindistributionnetworks(e.g.,truckingshortages)alsomade itdifficulttodelivergoodstobuyersduringthepandemicrecovery. Wefocusonconstraintsonthesupplyofgoods, ratherthandistributionofthem,inourmodel. 5Inablogpost,DelNegroetal.(2022)alsoarguethatmarkupshocksareimportant,basedonanalyzingUSdata throughthelensofaclosedeconomymodelwithoutcapacityconstraints(theNYFedmodel). 2

helprationalizeconcernsabout“greedflation”intheU.S. Turning to the second challenge, we demonstrate that data on quantities and prices together serve to identify the reasons why constraints bind – i.e., to disentangle whether demand shocks or supply-side constraint shocks lead constraints to bind. While either a positive demand shock or negative constraint shock may trigger binding constraints and thus lead inflation to rise, these shocks have distinct implications for quantities. A demandshock pushes both inflationand output quantity up, while a negative constraint shock raises inflation whilst lowering output. Put differently, adverse constraint shocks lead to negative comovement between inflation and quantities (of output or imports) in the model. In contrast, there is positive comovement in these variables following a goods-biased demand shock. Implicitly, we use these quantitative patterns to identify shocks when applying the model to filter data. In particular, constraints help the model explain whybothUSgoodsoutputandimportsofintermediateinputsdidnotriseinresponsetostrongUS demand. To lay out the structure of the paper, we start by collecting stylized facts in Section 1, which both motivate elements of the framework and serve as inputs into quantification. Some are well known: headline consumer price inflation rose a lot, more for goods than services. And consumer expenditure shifted from services to goods, driving real goods expenditures above trend. On the import side, prices for imported industrial materials (inputs) rose rapidly in 2021, while prices for imported consumer goods were essentially flat. As for quantities, production of goods has recoveredfromitstemporarypandemicdownturn,butithasnotincreasedinresponsetothesurge in consumer demand for goods.6 Stagnant domestic production in the face of surging demand (and the corresponding lack of imported inputs) hints at potentially binding constraints, whether domesticorforeigninnature. In Section 2, we develop a model to organize our interpretation of these facts, in which we study the impact of constraints for domestic goods producers and foreign goods input suppliers. Using impulse responses from the model in Section 3, we describe how prices and quantities respond to demand shocks and shocks to constraints. In Section 4, we then apply the model to filter shocks from US national accounts data. To capture the rich data dynamics, we allow for a number of different shocks, including shocks to aggregate demand (time preference), demand for goods (preferences for goods versus services), monetary policy, capacity levels at home and abroad, sector-specific productivity, and foreign production costs. In an extended version of the model, we also allow for labor supply shocks (disutility of labor) and stochastic constraints on laborsupply. Asakeyintermediatestep,wedevelopaBayesianMaximumLikelihoodestimationprocedure 6Correspondingly,importsoffinalgoodshaverisenby40%,whileimportsofindustrialmaterialsandotherintermediateshavebarelyrecoveredtopre-pandemiclevels. 3

to infer when constraints are binding and estimate structural parameters.7 Our model presents severalchallengesforestimation. Onechallengeisthatitfeaturescapacityshocks,andcapacityis alatentvariablethathasnofirstorderimpactonotherpotentiallyobservableequilibriumvariables when constraints are slack. As a result, prior estimation routines (e.g., Guerrieri and Iacoviello (2017)) that use inversion filters to construct the likelihood function are not applicable in our context. Instead, our estimation procedure builds on prior work by Kulish, Morley and Robinson (2017), Kulish and Pagan (2017), and Jones, Kulish and Rees (2022), which treats the duration of binding constraints as a parameter to be estimated. In this, a second challenge is that the duration of binding constraints is an equilibrium outcome in our model, unlike prior applications of the duration-based estimation approach. Therefore, we adapt the maximum likelihood procedure to imposeconstraintsonadmissibledurationparameterdraws.8 Overall,ourestimatedmodelfitsthe data well; most importantly, it captures the evolution of inflation for goods, services, and imports during the post-2020 period, making it a useful laboratory for analysis. Further, smoothed values for multipliers on the constraints imply that constraints bind during most of 2021-2022, and how tighttheyarefluctuatesovertime. Withthemodelandestimatesinhand,weevaluatetheroleofbindingconstraintsinexplaining the evolution of inflation through a sequence of counterfactual exercises. The first counterfactual allows all shocks to be active, but exogenously relaxes the capacity constraints in all periods. Comparing this counterfactual to the data, we find that binding constraints explain about half of the increase in inflation in 2021-2022, about two percentage points of the four percentage point increase in overall inflation. Further, easing of constraints in the latter half of 2022 helps explain recentdeclinesingoodsandimportpriceinflation. Toevaluatetheroleofindividualshocks,werunaseriesofcounterfactualsinwhichweintroduce shocks one at a time and in combination. We find that tight capacity, in part due to negative capacity shocks, set the stage for expansionary monetary policy – looser policy than suggested by an extended Taylor rule – to generate excess inflation in 2021. By implication, neither aggregate norgoods-biasedconsumerdemandshocksplayanimportantrolein2021,thoughtheydoaccount for inflation dynamics in 2020.9 As monetary policy was tightened in 2022, demand shocks then 7Thestructuralparametersweestimatearesubstitutionelasticitiesbetweenhomeandforeigninputs,coefficients inthemonetarypolicyrule,themeanlevelofcapacity,andthestochasticprocessesforshocks. 8In Kulish, Morley and Robinson (2017) and Jones, Kulish and Rees (2022), the binding constraint is the zero lowerboundoninterestrates,sothedurationtobeestimatedreflectsbeliefsabouthowlongthecentralbankwillhold the interest rate at zero. Because this is a free policy variable, these papers treat durations as unconstrained in the estimation. Inourapplication, theanticipateddurationofbindingcapacityconstraintsisdeterminedbytherealized shock today and the state of the economy. Thus, we adapt the estimation procedure to this new environment; see Sections4.1andA.3.2forfulldetails. 9Thoughwedonotdirectlyaccountforfiscalpolicy,wenotethattaxandtransferpolicychangesenactedduring thepandemiclargelyworkedbysupportingconsumption. Thus,theconsumptiondemandshocksthatweinferfrom datapartlycapturetheimpactofthesefiscalpolicies. 4

playalargerroleinaccountingforsustainedinflation. Probing the robustness of these results, we show that these results are not spuriously driven by fluctuations in energy prices, by re-estimating and simulating the model using inflation data that excludes energy. We also investigate how our mechanism compares to a leading alternative – labormarketshocks–inaccountingforinflation. Specifically,weenrichthelabormarkettoallow forwagerigidity,laborsupplyshocks,and(novel)potentiallybindingconstraintsonlaborsupply. While these additional features help us account for labor market dynamics (labor quantities and real wages) and the absence of disinflation in 2020, binding capacity constraints continue to play animportantroleinexplaininginflationdynamicsin2021-2022. In addition to work cited above, our paper is related to two distinct strands of work. First, our approach to modeling capacity constraints is related to models developed in Álvarez-Lois (2006) and Boehm and Pandalai-Nayar (2022), which feature heterogeneous firms that differ in terms of their exogenous capacity constraints on output.10 As Boehm and Pandalai-Nayar emphasize, aggregating across these heterogeneous firms yields convex industry supply curves, in which industry price indexes increase with industry output, since it is related to the share of firms whose constraints are binding. In contrast to these papers, we employ a homogeneous firms framework, which has pedagogical advantages for comparability to textbook models. Further, we allow for binding aggregate constraints, which give rise to kinked convex supply (Phillips) curves with verticalsegmentswherecapacityisexhausted. Second, our themes are related to recent work on how global value chains may have played a role in transmitting shocks during the pandemic crisis, including Bonadio et al. (2021), Lafrogne- Joussier, Martin and Mejean (2023b), Gourinchas et al. (2021), Alessandria et al. (2023), and Lafrogne-Joussier, Martin and Mejean (2023a).11 Celasun et al. (2022) provide a comprehensive analysis of the global scope of disruptions and bottlenecks in supply chains during the pandemic, andattributelargeoutputlossestothem. Severalcontributionsspecificallystudytheroleofsupplychaindistributionsinexplainingprice changes during the pandemic period. Amiti, Heise and Wang (2021), Young et al. (2021), and Santacreu and LaBelle (2022) demonstrate industry-level exposure to input price changes and/or supply chain disruptions are related to differences in output price changes across industries in the United States. Relatedly, Benigno et al. (2022) develop an index of global supply chain pressures from survey data and transportation indicators, and they find it has predictive power for inflation during the pandemic using a local projections empirical framework. di Giovanni et al. (2022) examine the role of disruptions to input markets and trade linkages on inflation during 10Fagnart,LicandroandPortier(1999)studiestheendogenousdeterminationofcapacity,inamodelthatprovides amicrofoundationforcapacityconstraintsonoutput. Aswediscussbelow,output-basedconstraintsarerelated,but somewhatdifferentthan,priorworkoncapitalutilization. 11SeealsoadiscussionoftheimpactofChineseshutdownsonUSsourcingfromChinabyHeise(2020). 5

the pandemic, using a sufficient statistics approach in a two period, multi-country, multi-sector input-output framework.12 Amiti et al. (2023) study how the combination of domestic labor market shocks and import supply chain disruptions contribute to inflation. Additional contributions focus on the impacts of fiscal policy on inflation, including di Giovanni et al. (2023), de Soyres, SantacreuandYoung(2023),andBianchi,FacciniandMelosi(forthcoming). Relative to this literature, our paper is the first (to our knowledge) to analyze occasionally bindingcapacityconstraintsinthesupplychain,withinacompleteDSGEmodel. Inthis,ourpaper extendsthenewliteratureonmonetarypolicyineconomieswithproductionnetworks[Ozdagliand Weber (2021); La’o and Tahbaz-Salehi (2022)] to accommodate supply chain constraints. Thus, webelieveitopensthedoortofurtherstudyoftheimplicationsofsupplychainbottlenecksforthe conductofpolicy. 1 Collecting Facts Webeginbycollectingseveralkeyfactsaboutrecentinflation,consumerexpenditure,production, andimportsthatmotivatevariouselementsoftheframeworkweconstruct. The first facts about consumer price inflation are well known: consumer price inflation rose substantially in 2021, led by inflation for goods. In Figure 1, we plot year-on-year growth in the pricedeflatorforUSpersonalconsumptionexpenditure(PCE),aswellasseparateseriesforgoods andservices. Theriseinheadlineinflation–fromroughly2percentin2021to6percentasofearly 2022 – is obviously startling. Importantly, this rise in inflation was led by goods price inflation, whichrosefromnearzeroto10percentin2021andthenplummetedinthesecondhalfof2022. A second set of facts concerns import price inflation: prices for imported inputs rose dramatically in 2021, while price changes for imported consumer goods were modest. Plotting import price inflation by end use in Figure 2, we see that inflation for imported industrial materials rose substantially in 2021, peaking at 50% year on year.13 While the price of oil and derivative fuels doubled during this period, the price of industrial materials excluding fuels also rose over 30% 12ThesufficientstatisticsapproachhaspreviouslybeenusedtostudymacroeconomicshockpropogationinBaqaee and Farhi (2019) in general, and the impacts of changes in trade on inflation by Comin and Johnson (2020). While well-suitedtoanalyzeshocksofforeignorigin,thesufficientstatisticsapproachislessusefulforstudyinghowtrade mitigates/exacerbates the impacts of shocks that are domestic in origin, because domestic shocks have both direct effectsandindirecteffectsthroughtheimportshare.Further,asCominandJohnson(2020)discuss,conclusionsdrawn aboutinflationfromthesufficientstatisticsapproacharesensitivetoassumptionsaboutthetimingandpersistenceof changesindomesticsourcingshares. 13This data is from the International Price Program of the Bureau of Labor Statistics. The source data consist primarily of free on board (FOB) prices (i.e., prices received by foreign producers at foreign dock). During 2021- 2022, transport costs also increased dramatically, which then would be added to these FOB prices to arrive at CIF prices (inclusive of cost, insurance, and freight) paid by the importer. We abstract from these additional transport margins,inordertofocusonchangesinsupplyprices. 6

Figure1: ConsumerPriceInflation )raey-no-raey( stnioP egatnecreP 01 5 0 5- Total Goods Services 2017 2018 2019 2020 2021 2022 2023 Note: ConsumerpricesaremeasuredusingthePersonalConsumptionExpenditure(PCE)priceindex,from theUSBureauofEconomicAnalysis(seriesidentifiersDPCERGM,DGDSRGM,andDSERRGM. in 2021. In contrast, inflation for imported consumer goods was subdued. This large difference between import price inflation for inputs versus consumer goods motivates our ensuing focus on disruptions impacting markets for imported inputs, rather than consumer goods.14 In 2022, importedinputpriceinflationdissipatesrapidly,evenexcludingvolatilefuelsprices. Tying the first and second set of facts together, goods production relies heavily on imported materials,relativeproductionofservices. Thus,thelargeincreaseinimportedmaterialspricesmay playaroleinexplainingthesurgeofinflationinthegoodssectordiscussedabove. Thisobservation is consistent with Amiti, Heise and Wang (2021), which documents that sectors that were more exposed to recent imported input price changes experienced higher producer price inflation in the U.S.15 Our model framework will include this potential mechanism, alongside other competing driversofinflation. The third set of facts relate to consumer expenditures. While consumer expenditure collapsed during the lockdown phase of the pandemic, it returned to trend by the end of 2021. At the same time, the sector composition of consumer expenditures changed dramatically, as consumers reallocated away from services toward goods. This is illustrated in terms of nominal expenditure 14Wehaveomittedseveralcategoriesofimportsfromthefigureforclarity,includingcapitalgoodsimports(IR2), imports of automotive vehicles, parts, and engines (IR3), and foods, feeds, and beverages (IR0). To verbally summarize, inflationforcapitalgoodsimportswasgenerallylow, similartoimportedconsumergoods. Inflationforthe automotive sector was also very low, and inflation for foods tracked total import price inflation closely. Thus, the behaviorofimportedmaterialspricesstandsout. 15Inarelatedvein,SantacreuandLaBelle(2022)findthatsectorsmoreexposedtoglobalsupplychaindisruptions (asmeasuredanindexofbacklogsanddeliverytimes)alsoexperiencedhigherproducerpriceinflation. 7

Figure2: ImportPriceInflationbyEndUse )raey-no-raey( stnioP egatnecreP 06 04 02 0 02- Import Price Inflation Industrial Materials & Supplies Industrial Materials & Supplies (excluding fuels) Consumer Goods (excluding autos) 2017 2018 2019 2020 2021 2022 2023 Note: Import price indexes are obtained from the US Bureau of Labor Statistics (series identifiers: IR for totalimports,EIUIR1forindustrialmaterials,EUIIR1EXFUELforindustrialmaterialsexcludingfuels,and EIUIR4forconsumergoods). shares in Figure 3a, and in terms of real quantities consumed for goods and services in Figure 3b. Further, note that the change in composition has proven remarkably persistent: real consumption of goods (correspondingly, the goods share in expenditure) remains high relative to pre-pandemic levelsthrough2023. The final set of facts point to potential supply-side constraints. In Figure 4a, we plot real US grossoutputbybroadsector. Thekeyfactisthatrealproductionofgoods(alreadystagnantbefore the pandemic) only just recovered and then trended slightly down in 2021-2022, which contrasts sharply with services output. Stagnant goods production in the face of high domestic demand for goods immediately suggests that US producers may have faced binding constraints. Correspondingly, consumer demand for goods was filled by imports: in Figure 4b, imported quantities for consumer goods (excluding autos) surge. In contrast, imports of industrial materials are flat, recoveringonlytoits2017levelsbytheendof2021andplateauingthere. Deficient US goods production and stagnant imports of industrial materials are naturally connected, though the direction of causality is not immediately clear. Limited supplies of imported materials may have constrained domestic production, or distinct binding constraints of domestic originmayhavecurtailedproductionandindirectlydepresseddemandforimportedinputs. Moreover,boththesemechanismsmightbeactivesimultaneously. Below,wediscusshowquantityand pricedatatogetherhelpdistinguishbetweenbindingdomesticversusforeignsupplyconstraintsin ourmodel. Withthisbackgroundinmind,weturntodetailsofthemodel. 8

Figure3: ConsumptionbySector (a)SectorSharesinExpenditure erahS sdooG ECP 63. 53. 43. 33. 23. 13. 96. 86. 76. 66. 56. 46. erahS secivreS ECP (b)RealQuantitiesConsumedbySector Goods Services 2017 2018 2019 2020 2021 2022 2023 xednI ytitnauQ laeR 061 041 021 001 08 Goods Services 2017 2018 2019 2020 2021 2022 2023 Note: PersonalConsumptionExpendituresharesandrealquantityindexesbysectorareobtainedfromthe USBureauofEconomicAnalysis(seriesidentifiers: DPCERC,PCES,DGDSRA3,andDSERRA3). Figure4: ProductionandImportQuantities (a)RealGrossOutputbySector )1=1Q7102( xednI ytitnauQ laeR 2.1 1.1 1 9. (b)ImportQuantitiesbyEndUse Goods Services 2017 2018 2019 2020 2021 2022 2023 )1=1Q7102( xednI ytitnauQ laeR 6.1 4.1 2.1 1 8. Consumer Goods (ex. autos) Industrial Materials & Supplies 2017 2018 2019 2020 2021 2022 2023 Note: RealgrossoutputisconstructedusingdatafromtheUSBureauofEconomicAnalysis(GDPby Industry,Table17). RealquantityindexesforimportsareobtainedfromtheUSBureauofEconomic Analysis(seriesidentifiers: IB0000043andB652RA3). 9

2 Model This section presents a small open economy model with many sectors, s ∈ {1,...,S}, which are are connected through input-output linkages. Within each sector, there is a continuum of monopolistically competitive firms, who set prices subject to Rotemberg-type adjustment costs. As in Gopinathetal.(2020),weassumethatbothexportsandimportsfortheHomecountryaredenominated in Home currency (i.e., US Dollars). Motivated by the data, we also allow import prices to differforfinalgoodsandinputs. The principal new features of the model are the output capacity constraints, for foreign and domestic firms. In writing down the model here, we allow these constraints to be potentially binding in any domestic sector, and we distinguish constraints that apply to foreign final versus input producing firms. Looking forward, we then restrict attention to particular constraints in quantitativeanalysisofthemodelforreasonsofbothtractabilityandempiricalrelevance. Wealso assume that the constraints are exogenously determined and (potentially) time varying, subject to stochastic shocks. This sets up a framework in which constraints may bind either due to negative shockstocapacity,orbecauseothershocksleadfirmstoexhausttheirexcesscapacity. 2.1 Consumers There is a representative Home consumer, with preferences over labor supply L and consumption t ofsectorcompositegoods{C (s)} representedby: t s∈S (cid:34) (cid:35) 1−ρ 1+ψ ∞ C L U({C t ,L t } t ∞ =0 )=E 0 ∑β tΘ t 1 t −ρ − 1 t +ψ (1) t=0 (cid:18) (cid:19)ϑ/(ϑ−1) with C = ∑ζ (s)1/ϑC (s)(ϑ−1)/ϑ t t t s (cid:18) (cid:19)ε(s)/(ε(s)−1) and C (s)= ∑γ(s)1/ε(s)C (s)(ε(s)−1)/ε(s)+(1−γ(s))1/ε(s)C (s)(ε(s)−1)/ε(s) , t Ht Ft s whereC (s) is consumption of a sector-composite good, which is comprised of domestic (C (s)) t Ht and foreign (C (s)) sub-composite goods. The parameter β < 1 is the usual time discount rate, Ft ρ ≥ 0 controls intertemporal substitution, ψ > 0 governs the elasticity of labor supply, ϑ ≥ 0 is the elasticity of substitution across sectors, and ε(s) ≥ 0 is the elasticity of substitution between homeandforeignconsumptioncomposites. The parameter ζ (s) is a time-varying parameter that controls tastes for goods from sector s, t and we require that ∑ s ζ t (s) = 1 throughout, so ζ t (s) should be interpreted as a relative sectoral demandshock. TheparameterΘ isanaggregatepreference(discountrate)shockatdatet. Though t 10

our setting does not directly consider fiscal policy shocks, fiscal shocks would be subsumed in thesediscountrateshockswhenRicardianequivalencefails,asinrecentmodelsbyGabaix(2020) and Angeletos, Lian and Wolf (2023). Therefore, our framework parsimoniously captures the combinedeffectoffiscalpolicyandotherdriversofdiscountratesinthissingleexogenousvariable. Financialmarketsarecomplete,andtheagent’sbudgetconstraintisgivenby: PC +E [S B ]≤B +WL , (2) t t t t,t+1 t+1 t t t whereP t C t =∑ s P t (s)C t (s),withP t beingthepriceforoneunitofthecompositeconsumptiongood and and P(s) being the price of the sector composite good. B denotes the portfolio of Arrowt t Debreu securities that pay off in domestic currency, and S is the Home consumer’s stochast,t+1 tic discount factor (defined below). Further, sectoral consumption expenditure is P(s)C (s) = t t P (s)C (s)+P (s)C (s), where P (s) and P (s) are the prices of the home and foreign con- Ht Ht Ft Ft Ht Ft sumptioncomposites. Given prices {P,P(s),P (s),P (s),S ,W} and initial asset holdings B , the consumer t t Ht Ft t,t+1 t 0 choosesconsumption,laborsupply,andassetholdingstomaximizeEquation1subjecttoEquation 2andthestandardtransversalitycondition. Optimalconsumptionandlaborchoicessatisfy: (cid:18) (cid:19) W −ρ t ψ C =L (3) t t P t (cid:18) P(s) (cid:19)−ϑ t C (s)=ζ (s) C (4) t t t P t (cid:18) P (s) (cid:19)−ε(s) Ht C (s)=γ(s) C (s) (5) Ht t P(s) t (cid:18) P (s) (cid:19)−ε(s) Ft C (s)=(1−γ(s)) C (s) (6) Ft t P(s) t (cid:20) (cid:21) P t 1=E S (1+i ) (7) t t,t+1 t P t+1 where S t,t+1 =β Θ Θ t+ t 1 (cid:16) C C t+ t 1 (cid:17)−ρ is the stochastic discount factor, P t = (cid:0) ∑ s ζ t (s)P t (s)1−ϑ (cid:1)1/(1−ϑ) is (cid:16) (cid:17)1/(1−ε(s)) the aggregate price index, P(s)= γ(s)(P (s))1−ε(s) +(1−γ(s))(P (s))1−ε(s) is the t Ht Ft sector-compositepriceindex,andi isaoneperiod,riskfreenominalinterestrate. t 2.2 Domestic Producers ThereisacontinuumoffirmswithineachsectorinHome,eachofwhichproducesadifferentiated good (indexed by ω). In addition, there exist competitive intermediary firms that aggregate these 11

varieties into composite goods, which are then consumed, used as inputs, and exported. We start bydescribingtheseintermediaries,andthenturntoindividualfirms. 2.2.1 CompositeDomesticGood Eachcompetitiveintermediaryfirmpurchasesoutputfromdomesticproducerstoformadomestic (cid:16) (cid:17)ε/(ε−1) composite. Theproductionfunctionfortheintermediaryis:Y(s)= (cid:82)1Y(s,ω)(ε−1)/εdω , t 0 t whereY(s,ω)istheamountofoutputpurchasedfromfirmω insectors,andε >1istheelasticity t of substitution. Given prices P(s,ω) for individual domestic varieties, cost minimization implies t (cid:16) (cid:17)−ε (cid:104) (cid:105)1/(1−ε) demandsY(s,ω)= Pt(s,ω) Y(s),whereP (s)= (cid:82)1P(s,ω)1−εdω isthepriceofthe t P (s) t Ht 0 t Ht sectorcompositegood. 2.2.2 DomesticFirms Eachdomesticproducerinsectorsisabletosupplyoutputuptoapre-determinedcapacityofY¯ (s), t whichwerefertoasafirm-levelcapacityconstraint. Weassumethiscapacitylevelisexogenously determinedandequalacrossfirmswithineachsector. Theproductionfunctionfordomesticvarietyω insectorsis: Y(s,ω)=Z (s,ω)A(s)(L (s,ω))1−α(s) (M (s,ω))α(s) (8) t t t t (cid:32) (cid:33)κ/(κ−1) (cid:16) (cid:17)1/κ M (s,ω)= ∑ α(s (cid:48) ,s)/α(s) M (s (cid:48) ,s,ω)(κ−1)/κ (9) t t s(cid:48) (cid:48) η(s) (cid:34) (cid:48) (cid:48) (cid:35) (cid:48) 1 η(s)−1 1 η(s)−1 η(s)−1 (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) M t (s,s,ω)= ξ(s,s)η(s)M Ht (s,s,ω) η(s) +(1−ξ(s,s))η(s)M Ft (s,s,ω) η(s) , (10) where L (s,ω) is the quantity of labor used by the firm, M (s,ω) is the firm’s use of a composite t t input, Z (ω) is productivity, and A(s)=α(s)−α(s)(1−α(s))−(1−α(s)) is a normalization constant. t (cid:48) The composite input combines inputs purchased from upstream sectors M (s,s,ω), with elastict ity of substitution κ ≥ 0. And those upstream inputs are themselves a CES composite of Home (cid:48) (cid:48) (M (s,s,ω)) and Foreign (M (s,s,ω)) composite inputs. The parameters η(s) ≥ 0 are elas- Ht Ft ticities of substitution across country sources for inputs (conventionally termed the Armington (cid:48) elasticity),whileξ(s,s)∈(0,1)controlsrelativedemandforhomeinputsconditionalonprices. Producers set prices in domestic currency under monopolistic competition, and they select the inputmixtosatisfytheimplieddemand. Thesetwoproblemscanbeanalyzedseparately. Thefirm (cid:110) (cid:111) chooses L (s,ω),M (s,ω),M (s (cid:48) ,s,ω),M (s,ω),M (s(cid:48),s,ω) to minimize the cost of product t t Ht Ft ing Y t (s,ω), which is W t L t (s,ω)+P Mt (s)M t (s,ω), with P Mt (s)M t (s,ω) = ∑ s(cid:48) P t (s(cid:48),s)M t (s (cid:48) ,s,ω) 12

and P(s(cid:48),s)M (s (cid:48) ,s,ω) = P(s(cid:48))M (s(cid:48),s,ω)+P (s(cid:48))M (s(cid:48),s,ω), where P (s(cid:48)) is the (domestic t t t Ht Ft Ft Ft currency) price of the foreign composite input from sector s(cid:48) . The first order conditions to this problemcanbewrittenasfollows: WL (s,ω)=α(s)MC (s,ω)Y(ω) (11) t t t t P (s)M (s,ω)=(1−α(s))MC (s,ω)Y(s,ω) (12) Mt t t t α(s(cid:48),s) (cid:18) P(s(cid:48),s)/P (cid:19)−κ M (s(cid:48),s,ω)= t t M (s,ω) (13) t t α(s) P (s)/P Mt t (cid:18) P (s(cid:48))/P (cid:19)−η(s(cid:48)) M (s(cid:48),s,ω)=ξ(s(cid:48),s) Ht t M (s(cid:48),s,ω) (14) Ht P(s(cid:48),s)/P t t t (cid:18) P (s(cid:48))/P (cid:19)−η(s(cid:48)) M (s(cid:48),s,ω)=(1−ξ(s(cid:48),s)) Ft t M (s(cid:48),s,ω), (15) Ft P(s(cid:48),s)/P t t t whereP Mt (s)= (cid:16) ∑ s(cid:48) (cid:16) α α (s ( (cid:48) s , ) s) (cid:17) P t (s(cid:48),s)1−κ (cid:17)1/(1−κ) isthepriceofthecompositeinput,P t (s(cid:48),s)isthe priceofM (s,ω),andthefirm’smarginalcostisMC (s,ω)=(Z (s,ω)) −1W 1−α(s) (P (s))α(s) . t t t t Mt Given this solution for marginal costs, the domestic firm chooses a sequence of prices to maximize profits, with knowledge of the demand curve for its output, and subject to quadratic adjustmentcostforprices[Rotemberg(1982a,b)]. Thispricingproblemcanbewrittenas: (cid:34) (cid:35) ∞ S φ(s) (cid:18) P(s,ω) (cid:19)2 max E ∑ 0,t P(s,ω)Y(s,ω)−MC (s,ω)Y(s,ω)− t −1 P (s)Y(s) 0 t t t t Ht t {Pt(s,ω)} t=0 P t 2 P t−1 (s,ω) s.t. Y(s,ω)≤Y¯ (s), t t where the discount rate for profits reflects the domestic agent’s stochastic discounting.16 The final terminthefirstlinecapturestheadjustmentcosts,whereφ(s)governsthedegreeofpricerigidity. Note also that the firm accounts for the potentially binding constraint in its pricing decisions. DenotingtheLagrangemultiplierattachedtothecapacityconstraint µ (s,ω),optimalpricessatisfy: t (cid:18) (cid:19) (cid:18) (cid:19) MC (s,ω)+µ (s,ω) P(s,ω) P (s)Y(s) t t t Ht t 0=1−ε 1− −φ(s) −1 P(s,ω) P (s,ω) P (s,ω)Y(s,ω) t t−1 t−1 t (cid:34) (cid:35) Θ (cid:18) C (cid:19)−ρ P (cid:18) P (s,ω) (cid:19) P (s)Y (s)P (s,ω) t+1 t+1 t t+1 Ht+1 t+1 t+1 +E β φ(s) −1 . (16) t Θ C P P(s,ω) P(s,ω)Y(s,ω) P(s,ω) t t t+1 t t t t 16Ofcourse,withcompletemarkets,itisimmaterialwhetherdomesticorforeignagentsownthefirm. 13

Thecorrespondingcomplementaryslacknessconditionis: µ (s,ω)[Y(s,ω)−Y¯ (s)]=0. (17) t t t And we require µ (s,ω)≥0 and the constraint to hold in equilibrium (Y(s,ω)≤Y¯ (s)) as usual. t t t When the constraint binds, then µ (s,ω) > 0. In Equation 16, we see this is equivalent to an t increase in the marginal cost of the firm, which drives up the optimal price. When the capacity constraint is slack, such that µ (s,ω) = 0, and is expected to remain slack, then Equation 16 t collapsestoastandardintertemporalpricingequation. 2.3 Foreign Producers Turning to foreign producers, we again start with aggregation of varieties by competitive intermediaries, and then we present the pricing problem for foreign firms. Here we distinguish between producers of foreign consumption goods versus inputs, which allows us to to analyze data on importpricesbyenduse. 2.3.1 CompositeForeignGoods For each end use u ∈ {C,M}, where C and M denote consumption and intermediate use respectively,thereisaunitcontinuumofforeignfirmsthatproduceforeigninputs,indexedbyϖ. Acompetitiveintermediaryfirmaggregatesoutputproducedbyeachforeignfirm,andbundlesitintothe (cid:16) (cid:17)ε/(ε−1) foreign composite according to the production function: Y∗(s)= (cid:82)1Y∗(s,ϖ)(ε−1)/εdϖ . ut 0 ut (cid:16) (cid:17)−ε DemandforeachvarietythentakestakesthestandardCESform: Y∗(s,ϖ)= P uFt (s,ϖ) Y∗(s), ut P (s) ut uFt (cid:16) (cid:17)1/(1−ε) where P (s,ϖ) is the price of variety ϖ and P (s)= (cid:82)1P (s,ϖ)1−εdϖ is the price uFt uFt 0 uFt oftheforeigncomposite,bothdenominatedinHomecurrency. 2.3.2 ForeignFirms Each foreign firm (in sector s, producing for end use u) is able to supply output up to a predetermined capacity of Y¯∗(s), and this capacity is exogenous and equal across firms. Foreign ut marginal costs are given by MC∗(s,ϖ), and we assume this cost is exogenous (as in a small open economy),denominatedinforeigncurrency,andequalacrossenduses. EachfirmchoosesasequenceforthepriceofitsvarietyinHomecurrency{P (s,ϖ)},subject uFt 14

topriceadjustmentfrictions,tosolve: (cid:34) (cid:35) ∞ S∗ φ(s) (cid:18) P (s,ϖ) (cid:19)2 max E ∑ 0,t P (s,ϖ)Y∗(s,ϖ)−E MC∗(s)Y∗(s,ϖ)− uFt −1 P (s)Y∗(s) {PFt(s,ϖ)} 0 t=0 P t ∗E t uFt ut t t ut 2 P uFt−1 (s,ϖ) uFt ut s.t. Y∗(s,ϖ)≤Y¯∗(s), ut ut with knowledge of the demand curve for its output specified above. Here S∗ =βt (cid:16) C t ∗ (cid:17)−ρ is the 0,t C∗ 0 foreign stochastic discount factor (withC∗denoting foreign consumption), P∗ is the foreign price t t level (in foreign currency), and E is a the nominal exchange rate (units of home currency to buy t oneunitofforeigncurrency). Denoting the Lagrange multiplier attached to the capacity constraint µ∗(s,ϖ), then the first ut orderconditionis: (cid:18) E (MC∗(s,ϖ)+µ∗(s,ϖ)) (cid:19) (cid:18) P (s,ϖ) (cid:19) P (s)Y∗(s) 1−ε 1− t t ut −φ(s) uFt −1 uFt ut P (s,ϖ) P (s,ϖ) P (s,ϖ)Y∗(s,ϖ) uFt uFt−1 uFt−1 ut (cid:34) (cid:35) +E β (cid:18)C t ∗ +1 (cid:19)−ρ(cid:18) E t P t ∗ (cid:19) φ(s) (cid:18) P uFt+1 (s,ϖ) −1 (cid:19) P uFt+1 (s)Y u ∗ t+1 (s) P uFt+1 (s,ϖ) =0. (18) t C∗ E P∗ P (s,ϖ) P (s,ϖ)Y∗(s,ϖ) P (s,ϖ) t t+1 t+1 uFt uFt ut uFt Thecomplementaryslacknessconditionis: µ ∗(s,ϖ)[Y∗(ϖ)−Y¯∗]=0. (19) ut ut ut Inequilibrium, µ∗(ϖ)≥0andY∗(ϖ)≤Y¯∗. ut ut ut 2.4 Closing the Model WeassumethatdemandforexportsofthehomecompositegoodtakestheCESform: (cid:18) P (s) (cid:19)−σ(s) X (s)= Ht X∗(s), (20) t PQ t t t whereQ ≡ EtP t ∗ istherealexchangerateandX∗(s)isanexogenousforeignsector-demandfactor. t Pt t Themarketclearingconditionforthehomecompositegoodis: (cid:34) (cid:35) S (cid:90) 1 (cid:90) 1 φ(s) (cid:18) P(s,ω) (cid:19)2 Y(s)=C (s)+ ∑ M (s,s(cid:48),ω)dω+X (s)+ t −1 Y(s) dω, (21) t Ht Ht t t 2 P (s,ω) s(cid:48)=1 0 0 t−1 where the composite good is sold to consumers and domestic producers, exported, and used to coverpriceadjustmentcosts. Fortheforeigncompositegoods,weimposesimilarmarketclearing 15

conditions: (cid:34) (cid:35) (cid:90) 1 φ(s) (cid:18) P (s,ϖ) (cid:19)2 Y∗(s)=C (s)+ CFt −1 Y∗(s) dϖ (22) Ct Ft 2 P (s,ϖ) Ct 0 CFt−1 (cid:34) (cid:35) (cid:90) 1 φ(s) (cid:18) P (s,ϖ) (cid:19)2 Y∗ (s)=∑M (s,s(cid:48))+ MFt −1 Y∗ (s) dϖ. (23) Mt Ft 2 P (s,ϖ) Mt s(cid:48) 0 MFt−1 Labormarketclearingisgivenby: S (cid:90) 1 L = ∑L (s) with L (s)= L (s,ω)dω. (24) t t t t s=1 0 TradeinArrow-DebreusecuritiesimpliesthatHomeandForeignconsumerssharerisk,suchthat: (cid:18) C (cid:19)−ρ t Θ Q =Ξ, (25) t C∗ t t whereΞisaconstant. Turning to monetary policy, we specify an extended inflation-targeting rule for interest rates. Since we allow for sector-specific preference shocks, we now distinguish measured price inflation fromchangesinthewelfare-theoreticpriceindex. Wedefineanauxiliarypriceindexundertheas- (cid:16) (cid:17)1/(1−ϑ) sumption that preferences are constant over time: P¯ t = ∑ s ζ 0 (s)(P t (s))1−ϑ , where ζ 0 (s) are steady-state CES weights. Then Π¯ = P¯/P¯ is the ratio of measured prices across periods, t t t−1 (cid:16) (cid:17) andtheapproximateinflationrateisgivenbyπ¯ t =∑ s P 0 ( P s) C C 0 (s) [lnP t (s)−lnP t−1 (s)].17 Wewrite 0 0 themonetarypolicyruleintermsofmeasuredinflation: 1+i =(1+i )ρiΠ¯ω(1−ρi ) (Y/Y ) (1−ρi )ρyΨ (26) t t−1 t t 0 t where Y t = ∑ s P 0 (s)Y t (s) is aggregate real gross output and Ψ t is a monetary policy shock. The parameters ω and ρ determine how aggressively the central bank responds to inflation and the y output gap (defined as the deviation of output from steady state), while the parameter ρ controls i thedegreeofinterestrateinertia. 17Thefollowingrelationshipholdsbetweentheratiosofmeasuredandwelfare-basedpriceindexesacrossperiods: Π¯ t = P¯ t− P¯ 1 t/ / P P t t−1 Π t , where P P ¯ t t = (cid:18) ∑ s ζ 0 (s) (cid:16) Pt P ( t s) (cid:17)1−ϑ (cid:19)1/(1−ϑ) andtheratioofaggregatepricesacrossperiodsisΠ t ≡ Pt . Weincludetheseamongauxiliarypricedefinitionsinthemodelequilibrium. Pt−1 16

2.5 Equilibrium with Symmetric Firms We focus on an equilibrium with symmetric producers within each sector and country. The model (cid:8) (cid:9) parameters are ρ,ψ,χ,ϑ,β,κ,ε,Ξ,i ,ω,ρ,ρ , {ε(s),α(s),η(s),φ(s),σ(s),φ(s),γ(s)} , and 0 i y s {α(s(cid:48),s),ξ(s(cid:48),s)} . Further, values for domestic variables {Θ ,{ζ (s),Z (s)} ,Ψ}, foreign varis,s(cid:48) t t t s t ables (cid:8) C∗,{X∗(s),MC∗(s)/P∗} (cid:9) and constraints (cid:8) Y¯ (s),Y¯∗(s),Y¯∗ (s) (cid:9) are exogenously given. t t t t s t Ct Mt s Wewriteallpricesrelativetothedomesticpricelevel,andwedefineΠ ≡ Pt . t P t−1 Givenparametersandexogenousvariables,anequilibriumisasequenceofaggregatequantities {C ,L },sector-levelquantities (cid:8) C (s),C (s),C (s),L (s),Y(s),M (s),X (s),Y∗(s),Y∗ (s) (cid:9) ,int t t Ht Ft t t t t Ct Mt s putuse{{M (s(cid:48),s),M (s(cid:48),s),M (s(cid:48),s)} } ,aggregateprices (cid:8) W/P,i ,Q ,Π ,Π¯ ,P¯/P (cid:9) ,sectort Ht Ft s(cid:48) s t t t t t t t t levelprices{Π (s),Π (s),Π (s),P(s)/P,MC (s)/P,P (s)/P,P (s)/P,P (s)/P,P (s)/P} , t CFt MFt t t t t Mt t Ht t CFt t MFt t s input prices {{P(s(cid:48),s)/P} } , and (normalized) multipliers (cid:8) µ (s)/P,µ∗ (s)/P∗,µ∗ (s)/P∗(cid:9) t t s(cid:48) s t t Ct t Mt t s that satisfy the equilibrium conditions collected in Table 1. This system is 8+21S+4S2 equationsinthesamenumberofunknowns. Table1: EquilibriumConditions LaborSupply C −ρWt =χLψ t Pt t (cid:16) (cid:17)−ϑ C(s)=ζ (s) Pt(s) C Consumption t t Pt t (cid:16) (cid:17)−ε(s) Allocation C Ht (s)=γ(s) P P H t t ( ( s s ) ) / / P P t t C t (s) (cid:16) (cid:17)−ε(s) C (s)=(1−γ(s)) PCFt(s)/Pt C(s) Ft (cid:20) Pt(s)/Pt (cid:21) t (cid:16) (cid:17)−ρ(cid:16) (cid:17) EulerEquation 1=E βΘt+1 Ct+1 1+it t Θt Ct Πt+1 (cid:18) (cid:16) (cid:17)1−ϑ (cid:19)1/(1−ϑ) ConsumerPrices 1= ∑ s ζ t (s) Pt P ( t s) (cid:18) (cid:16) (cid:17)1−ε(s) (cid:16) (cid:17)1−ε(s) (cid:19)1/(1−ε(s)) Pt(s) = γ(s) P Ht (s) +(1−γ(s)) P CFt (s) Pt Pt Pt LaborDemand WtL (s)=(1−α(s)) MCt(s)Y(s) Pt t Pt t PMt(s)M(s)=α(s) MCt(s)Y(s) M P ( t s(cid:48),s) t = α(s(cid:48),s) (cid:16) Pt( P s t (cid:48),s)/P t t (cid:17)−κ M(s) InputDemand M t (s(cid:48),s)= α ξ ( ( s) s(cid:48),s) P (cid:16)Mt P (s H ) t / ( P s t (cid:48))/Pt (cid:17)−η t (s(cid:48)) M(s(cid:48),s) Ht Pt(s(cid:48),s)/Pt t M (s(cid:48),s)=(1−ξ(s(cid:48),s)) (cid:16) PMFt(s(cid:48))/Pt (cid:17)−η(s(cid:48)) M(s(cid:48),s) Ft Pt(s(cid:48),s)/Pt t (cid:16) (cid:17)1−α(s)(cid:16) (cid:17)α(s) MarginalCost MCt(s) = 1 Wt PMt(s) Pt Zt(s) Pt Pt InputPrices PM P t t (s) = (cid:18) ∑ s(cid:48) (cid:16) α α (s ( (cid:48) s , ) s) (cid:17)(cid:16) Pt( P s t (cid:48),s) (cid:17)1−κ (cid:19)1/(1−κ) Pt(s(cid:48),s) = (cid:20) ξ(s(cid:48),s) (cid:16) PHt(s(cid:48)) (cid:17)1−η(s(cid:48)) +(1−ξ(s(cid:48),s)) (cid:16) PMFt(s(cid:48)) (cid:17)1−η(s(cid:48)) (cid:21)1/(1−η(s(cid:48))) Pt Pt Pt 17

Table1: EquilibriumConditions (cid:18) (cid:19) MC(s)/P +µ (s)/P t t t t 0=1−ε 1− −φ(s)(Π (s)−1)Π (s) DomesticPricing P (s)/P Ht Ht Ht t +E (cid:34) β Θ t+1 C t − + ρ 1 1 φ(s)(Π (s)−1)Π (s)2 Y t+1 (s) (cid:35) t Θ t C t −ρ Π t+1 Ht+1 Ht+1 Y t (s) (cid:18) Q MC∗(s)+µ∗ (s) (cid:19) 0=1−ε 1− t t Ct −φ(s)(Π (s)−1)Π (s) ConsumptionImport P (s)/P P∗ CFt CFt CFt t t (cid:34) (cid:35) Pricing (cid:18)C∗ (cid:19)−ρ Q 1 Y∗ (s) +E β t+1 t φ(s)(Π (s)−1)Π (s)2 Ct+1 t C∗ Q Π CFt+1 CFt+1 Y∗(s) t t+1 t+1 Ct (cid:18) Q MC∗(s)+µ∗ (s) (cid:19) 0=1−ε 1− t t Mt −φ(s)(Π (s)−1)Π (s) InputImportPricing P (s)/P P∗ MFt MFt MFt t t (cid:34) (cid:35) (cid:18)C∗ (cid:19)−ρ Q 1 Y∗ (s) +E β t+1 t φ(s)(Π (s)−1)Π (s)2 Mt+1 t C∗ Q Π MFt+1 MFt+1 Y∗ (s) t t+1 t+1 Mt min{µ (s),Y¯(s)−Y(s)}=0 t t t Comp. Slacknessand min{µ∗ (s),Y¯∗(s)−Y∗(s)}=0 FirmConstraints Ct Ct Ct min{µ∗ (s),Y¯∗ (s)−Y∗ (s)}=0 Mt Mt Mt (cid:16) (cid:17)2 Y t (s)=C Ht (s)+∑ s(cid:48) M Ht (s,s(cid:48))+X t (s)+φ( 2 s) Pt P − t( 1 s ( ) s) −1 Y t (s) (cid:16) (cid:17)−σ(s) X(s)= PHt(s) X∗(s) t PtQt t MarketClearing Y∗(s)=C (s)+φ(s) (Π (s)−1)2Y∗(s) Ct Ft 2 CFt Ct Y M ∗ t (s)=∑ s(cid:48) M Ft (s,s(cid:48))+φ( 2 s) (Π MFt (s)−1)2Y M ∗ t (s) (cid:16) (cid:17)−ρ Θ Ct Q =Ξ t C∗ t t ∑ s L t (s)=L t 1+i =(1+i )ρiΠ¯ω(1−ρi ) (Y/Y ) (1−ρi )ρyΨ with t t−1 t t 0 t MonetaryPolicyRule Y t =∑ s P 0 (s)Y t (s) (cid:16) (cid:17) Π (s)= PHt(s)/Pt Π Ht (cid:16) PHt−1(s)/Pt−1 (cid:17) t Π (s)= PCFt(s)/Pt Π AuxiliaryInflation CFt (cid:16) PCFt−1 (s)/Pt−1 (cid:17) t Π (s)= PMFt(s)/Pt Π andPriceDefinitions MFt PMFt−1(s)/Pt−1 t Π¯ = P¯ t/Pt Π t P¯ t−1/Pt−1 t (cid:18) (cid:16) (cid:17)1−ϑ (cid:19)1/(1−ϑ) P¯ t = ∑ ζ (s) Pt(s) Pt s 0 Pt 2.6 Discussion Webrieflydiscusssometechnicalitiesassociatedwithsolvingthemodel. WethendescribePhillips Curvesinthemodel,whichcontainanimportantinsightforinterpretingsimulationresults. 18

2.6.1 SolvingtheModel Because the model features occasionally binding constraints, we need to adopt an appropriate solution technique that captures the non-linearities induced by them. Among alternatives, we adopt the piecewise linear solution technique developed by Guerrieri and Iacoviello (2015). The perturbation-based solution algorithm combines first order approximations to the model equilibriumforboththeunconstrainedandconstrainedequilibria,wherethepointofapproximationisthe unconstrained equilibrium in all cases.18 The log-linear approximation for the model used in our quantitativeanalysis,anddetailsregardingthesolutionprocedure,arepresentedinAppendixA. Collectinglogdeviationsfromsteadystateforendogenous(bothcontrolandstate)variablesin thevectorX ,thegeneralsolutionforthemodelcanbewrittenas: t X =J(X ,ε ;θ)+Q(X ,ε ;θ)X +G(X ,ε ;θ)ε , (27) t t−1 t t−1 t t−1 t−1 t t where ε is the vector of exogenous shocks in period t, θ is a collection of structural parameters, t andJ(·),Q(·),andG(·)aretime-varyingmatrices(dependentonthestateandcurrentshocks)that describe the optimal policy function. Given parameters and the initial steady-state shares needed to parameterize the approximate model, as well as lagged values X and a realization for ε , we t−1 t solveforthepolicyfunctionsusingtheOccBintoolboxinDynare. 2.6.2 DomesticandImportPricePhillipsCurves It is instructive to examine log-linear approximations for the dynamic pricing equations for domestic and imported goods. Noting that µ (s)/P and µ∗(s)/P∗ for u ∈ {C,M} take on zero t t ut t values in the unconstrained equilibrium, we define auxiliary variables µ˜ (s) ≡ µ (s)/P +1 and t t t µ˜∗(s) ≡ µ∗(s)/P∗+1, and then we log-linearize the equilibrium with respect to these auxiliary ut ut t variables. Theresultingapproximatepricingequationsare: (cid:18) (cid:19) (cid:18) (cid:19) ε−1 ε P π (s)= (rmc (s)−rp (s))+ 0 µˆ˜ (s)+βE [π (s)] (28) Ht φ(s) (cid:100)t (cid:98)Ht φ(s)P (s) t t Ht+1 H0 (cid:18) (cid:19) (cid:18) (cid:19) π (s)= ε−1 (cid:0) rmc ∗ (s)+qˆ −rp (s) (cid:1) + ε P 0 µˆ˜∗(s)+βE [π (s)], (29) uFt φ(s) (cid:100)t t (cid:98)uFt φ(s)P (s) ut t uFt+1 uF0 where hat-notation denotes deviations from steady state, π (s) ≡ lnP(s)−lnP (s), π (s) ≡ t t t−1 Ft lnP (s)−lnP (s),rmc (s)=ln(MC (s)/P),rmc∗(s)=ln(MC∗(s)/P∗),rp (s)=ln(P (s)/P), Ft Ft−1 t t t t t t Ht Ht t 18The solution procedures requires that the model satisfies two important conditions. First, it is assumed that the modelreturnstotheunconstrainedequilibriuminfinitetimeafteraonce-offshock,ifagentsexpectfutureshocksto bezero. Second,theunconstrainedequilibriummustbestable,intheusualBlanchard-Kahnsense. Bothrequirements aresatisfiedforourbaselinemodelandparametervalues. 19

rp (s) = ln(P (s)/P), and q = lnQ . Equations 28-29 are sector-level domestic and import uFt uFt t t t pricePhillipscurves. Binding Constraints as Markup Shocks An important conceptual point is that binding constraints – when µ (s) or µ∗(s) are strictly positive – appear as “markup shocks” in reduced form. t ut Thatis,bindingconstraintsleadinflationtobehigherthancanbeaccountedforgivenparameters, real marginal costs, and expected inflation. Thus, one can identify whether constraints bind in our modelusingthesameapproachesthatwouldtypicallybeusedtoidentifyexogenous,reduced-form markupshocksinstandardNewKeynesianmodels. WhereasexogenousmarkupsshocksinNewKeynesianmodelstypicallyaremicro-foundedby assuming that there are shocks to the elasticity of demand, the endogenous “markups shocks” in ourmodelhaveadifferentstructuralinterpretation. Markupshocksariseinourmodelnotbecause the competitive environment per se has changed – i.e., market structure and demand elasticities are time invariant – rather firm conduct changes when constraints bind. Firms cease to make price changes to target their ideal (flexible price, CES) markups; they instead “price to demand,” based on willingness to pay for their constrained output.19 Further, markups may rise and fall sharply (reflecting non-linearities) as constraints are triggered and relaxed, such that the evolution of markups in times of binding constraints will be different than in normal times when constraints arealwaysslack. This “markup shock” interpretation of binding constraints also serves to highlight how the mechanism we emphasize is distinct from alternative explanations for the inflation surge. First, muchattentionhasfocusedontheroleoflaborshortages. Attheaggregatelevel,thesemayreflect changesinworkerpreferencesforsupplyinglabor,orotherconstraintsonlaborsupply. Atthesector level, worker shortages may be explained by impediments to reallocating workers in response to differential changes in demand across sectors.20 In either case, demand for workers outstripping supply ought to manifest as higher wages, which would then drive marginal costs higher. Thus, one would expect to see that changes in real marginal costs explain inflation outcomes, not markups (one might even expect markups to be compressed where labor shortages are tightest). Totheextentthatconstraintsmasqueradingasmarkupshocksexplaininflation,thisthenlimitsthe scope for these alternative labor market mechanisms. All that said, we will discuss exactly how 19The conclusion that capacity constraints influence firm conduct, holding market structure fixed, is not unique to our particular monopolistic competition model. For example, in oligopoly models with symmetric firms, it is well-knownthatBertrandcompetitionleadstocompetitive(marginalcost)pricingwhenfirmsareunconstrained. In contrast,whenfirmsarecapacityconstrainedsuchthattheycannotcollectivelymeettotalmarketdemandwhenprices equalmarginalcost,thentheBertrandequilibriumfeaturespricessetabovemarginalcost. 20We have assumed that factors are perfectly mobile across sectors in our model, with a common economy-wide wage.ThiscontrastswithFerrante,GravesandIacoviello(forthcoming),whoanalyzehowasymmetricdemandshocks leadtoinflationwhenthereareworkerreallocationfrictions. 20

incorporatinglabormarketshocksandconstraintsaffectsinflationinSection5.2. In a related vein, the approach we adopt for modeling capacity differs from prior literature, which has emphasized variable capital utilization rather than output-based capacity constraints [Greenwood, Hercowitz and Huffman (1988); Gilchrist and Williams (2000)]. In this literature, it is typically assumed that higher rates of capital utilization lead capital to depreciate faster. As a result,higherutilizationraisestheeffectivemarginalcostforthefirm(includingwages,usercosts of capital, and increased capital depreciation), so utilization affects inflation through marginal costs. Further, with the functional form assumptions in Greenwood, Hercowitz and Huffman, the standard log-linear Phillips Curve relationship between marginal costs and inflation (equivalently, utilization and inflation) holds. Thus, this alternative approach to capacity utilization will struggle toexplainthehighlynon-linearresponseofinflationobservedinrecentdata,aswellastheroleof reduced-formmarkupshocksinexplainingit. Profits Our model implies that price-cost margins (realized markups) are high when firms face binding constraints. To examine the plausibility of this channel, we turn to data on profits per unit ofoutput,whichservesasanobservableproxyforprice-costmargins. Toformalizethislink,note that the absolute markup is equal to profits per unit of output in the steady state: P(s)−MC (s)= t t Ξt(s) , where Ξ (s)≡P(s)Y(s)−MC (s)Y(s) is the profit of the representative producer in sector Yt(s) t t t t t s. Thus,trackingprofitsperunitovertimeshedslightonhowmarkupsarechanging. In Figure 5, we plot indexes of US corporate profits per unit of gross output for both the manufacturing sector and the aggregate private sector.21 The takeaway is that profits per unit escalatedsharplyformanufacturingfirmsduringthepandemicrecovery,coincidingwiththetakeoff in goods price inflation and widespread complaints about binding (supply chain) constraints that limited production. Further, total profits (profits per unit times quantity sold) were at historically high levels in 2021. This pattern of high profitability alongside high inflation is a natural outcome of binding (domestic) constraints in our model. It is also remarkably consistent with concerns about “greedflation” in the U.S., wherein corporations have been criticized for fueling inflation by gouging consumers [DePillis (2022)].22 More recently, profit margins appear to be falling as 21This corporate profit measure omits profits attributable to non-corporate entities; We focus on corporate profits becausedataisavailableformanufacturingonaquarterlyfrequencyinthenationalaccounts. 22Greedflationisoftenattributedtothesecularriseofmarketpower,whichhaspotentiallyincreasedtheabilityof corporationstopasscostshocksthroughtoconsumers. Ourmechanismisnotaboutpass-throughofcostshocks;itis abouthowfirmssetpricesconditionaloncostswhenconstraintsbind.Toexplainthedifferenceinemphasis,itisuseful to refer to the automobile industry. High auto prices during the pandemic did not reflect high rates of pass-through fromdealercosts(productioncosts)toretailprices. Rather,theconstrainedsupplyofnewautomobilesledtohigher dealermarkupsandrobustprofitability[Smialek(2022)]. Tointerpretthis, thinkaboutdealersashavingaLeontief productionfunction,combiningcarswithdealerservices,suchthatconstrainedaccesstoinputs(newcars)effectively constrained dealer output. Unlike this auto example, our model allows primary factors to substitute for inputs, and allowsdomesticinputstosubstituteforforeigninputs,sooutputconstraintsaredistinctfrominputconstraints. 21

Figure5: CorporateProfitsperunitofGrossOutput )1=1Q7102( xednI 5.2 2 5.1 1 Manufacturing All Sectors 2017 2018 2019 2020 2021 2022 2023 Note: Corporate profits (with inventory valuation adjustments) and gross output are obtained from the US Bureau of Economic Analysis (series identifiers N400RC and A390RC). The figure contains the ratio of corporateprofitstogrossoutputineachquarter,expressedasanindexnumber(valuesaremeasuredrelative totheratioin2017Q1). inflationhasdeclinedin2023[Kerr(2023)]. 3 Impulse Response Analysis To illustrate how the model works, we turn to impulse response functions. We first discuss the quantitative setup, which puts restrictions on the general model presented above. We then analyze responses to particular demand and capacity constraints that play important roles in accounting for variation in the data below. Further, looking forward to the full quantitative analysis, we pay particularattentiontoillustratinghowdatamaybeusefulinidentifyingshocks. 3.1 Quantitative Setup While the model (as written above) allows for many sectors and many potentially binding constraints, we now hone in on a two sector structure with two potentially binding constraints, motivatedbythestylizedfactspresentedabove. Asfornumberingsectors,lets=1bethegoodssector and s=2 be the services sector. We then focus on equilibria with potentially binding constraints for the goods sector, since the anomalous behavior of goods price inflation in recent data requires 22

explanation.23 Further, motivated by data that shows a surge of consumption goods imports during2020-2021,weassumethattheconstraintsforforeignconsumergoodsfirmsareslackaswell. Thatis,weallowforpotentiallybindingconstraintsfordomesticfirmsandforeigninputproducing firms–i.e.,constraintsinthesupplychainforgoods. We set parameters in the model based on external calibration and an estimation procedure that wedescribeinSection4.2,withthefullsetofparametersprovidedinAppendixA.Further,exogenous variables follow independent first-order autoregressive processes, with parameters estimated below. 3.2 Analyzing a Demand Shock To start, we consider a consumer discount rate (Θ ) shock, which raises the desire to consume in t the current period. For concreteness, let Θˆ = λ Θˆ +ε , with var(ε ) = σ2. To scale the t Θ t−1 Θt Θt θ shock, we assume that the initial innovation to Θ is 0.11σ for the simulation when the domestic t θ constraint binds and 0.065σ for the simulation when the foreign constraint binds, where σ and θ θ λ are estimated below. These are relatively small shocks, so our focus is on qualitative results Θ in this section, rather than magnitudes. We contrast the impacts of this shock in an unconstrained equilibrium to those in two alternative equilibria, in which only the domestic constraint binds, or only the foreign constraint binds on impact. In each case, we set the level of the constraint so that the constraint binds for only one period (i.e., on impact) in response to the shock, and is slack thereafter. Whileaspecialcase,thisservestohighlightthekeyimpactsofhittingconstraints. 3.2.1 UnconstrainedBenchmark Figure6collectsimpulseresponsesforkeyvariablesfromtheunconstrainedmodel(i.e.,assuming constraints do not bind after the shock). In Figure 6a, we see that the demand shock raises overall consumer price inflation, and inflation for services is about double that for goods after the shock. Tobreakthisdown,sector-levelinflationisaweightedaverageofdomesticpriceinflation(π (s)) Ht andimportpriceinflationforconsumptiongoods(π (s)),soweplotπ (s)andπ (s)inFigure CFt Ht CFt 6b.24 First, we note that import price inflation for consumer goods is negative on impact in Figure 6b, so this also lowers overall consumption price inflation for goods relative to services, since the 23Taking the constraints for sector 2 to infinity – i.e.,Y¯(2)→∞,Y¯∗(2)→∞, andY¯∗ (2)→∞ – is sufficient to t Ct Mt ensure constraints are only potentially binding for goods. One ought not over-interpret this assumption. This is a sufficient,butnotnecessarycondition. Givenanysequenceofshocks,onecanbackouttheleveloffiniteconstraints neededtopreventstheseconstraintsfrombinding. (cid:16) (cid:17) (cid:16) (cid:17) 24Sector-level consumer price inflation is given by: π(s)= PH0(s)CH0(s) π (s)+ PF0(s)CF0(s) π (s), where t P0(s)C0(s) Ht P0(s)C0(s) CFt π(s)=pˆ (s)−pˆ (s)isinflationinsectors. t t t−1 23

goodssectorhasahigherimportshare. Glancingforward,notethatimportpriceinflationforgoods inputsisalsonegativeonimpactinFigure6d,matchingthedynamicsforconsumerimportprices. The reason is that on impact the exchange rate appreciates in response to the demand shock, and this appreciation lowers import price inflation symmetrically across sectors and end uses (π (s) uFt coincideacrossuands). Second, domestic price inflation is also lower for goods than services. The reason is that real wagesrise,andservicesuselabormoreintensivelythandogoods. Toillustratethis,weiteratethe PhillipsCurvesinEquation28forwardtoyieldthedecomposition: (cid:18) ε−1 (cid:19) ∞ (cid:18) ε−1 (cid:19) ∞ π (s)=(1−α(s)) ∑βrE [wˆ −pˆ (s)]+α(s) ∑βrE [pˆ (s)−pˆ (s)] Ht t t+r Ht+r t Mt+r Ht+r φ(s) φ(s) r=0 r=0 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) realwageterm realinputpriceterm + (cid:18) ε P 0 (cid:19) ∑ ∞ βrE (cid:2) µˆ˜ (s) (cid:3) . (30) t t+r φ(s)P (s) H0 r=0 (cid:124) (cid:123)(cid:122) (cid:125) bindingconstraintterm where we have substituted for rmc (s)−rp (s) and defined “real” values here in term of (cid:100)t+r (cid:98)Ht+r domestic output prices. The first term captures the role of real wages, where the labor share of gross output (1−α(s)) is higher for services than goods. The second term then accounts for real inputpricesincosts. Thethirdtermcapturestheimpactofbindingconstraintsonmarkups,which is identically zero in this simulation with slack constraints. We plot this decomposition by sector in Figure 6c. The real wage term for services clearly drives inflation for services beyond that for goods,reflectingthehigherlaborcontentofservices. Thesecondpointtonoteinthefigureisthat inputcostsactuallyrestraininflationinbothsectors,thoughtheseeffectsaresmallinmagnitude. Lookingatquantities,weplotrealgrossoutputbysectorinFigure6eandrealgoodsimportsin Figure6f. Thequantityofbothdomesticgoodsandservicesproducedrisesinresponsetoincreased demand in the unconstrained equilibrium. Further, both the quantity of imported consumer goods andinputsrise,withtheincreaseinimportedinputsoutstrippingconsumptiongoods. In all, while the demand shock raises overall inflation, it yields a mix of results that are inconsistent with recent data. Whereas goods price inflation exceeds services inflation in data, the opposite is true in the unconstrained model. Further, import price inflation is negative on impact, incontrasttodata. Finally,bothrealgoodsoutputandimportedinputsriseinthesimulation,while they are largely flat in the data [see Figure 4]. With these puzzles in hand, we turn to versions of themodelwithbindingconstraints. 24

Figure6: DemandShock: UnconstrainedEquilibrium (a)ConsumerPriceInflation stnioP egatnecreP 520. 20. 510. 10. 500. 0 (b)ConsumerInflationComponents Total Goods Services 0 4 8 12 16 20 period stnioP egatnecreP 30. 20. 10. 0 10.- (c)DomesticPriceInflation Domestic Goods Domestic Services Goods Imports Services Imports 0 5 10 15 20 period stnioP egatnecreP 520. 20. 510. 10. 500. 0 Goods Wage Term Goods Input Price Term Goods Constraint Term Services Wage Term Services Input Price Term 0 5 10 15 20 period (d)ImportedInputPriceInflation stnioP egatnecreP 500. 0 500.- 10.- 510.- (e)GrossOutputQuantity 0 5 10 15 20 period etatS ydaetS morf noitaiveD goL 200. 5100. 100. 5000. 0 (f)ImportQuantity Goods Output Services Output 0 5 10 15 20 period etatS ydaetS morf noitaiveD goL 300. 200. 100. 0 Consumption Goods Input Goods 0 5 10 15 20 period 3.2.2 BindingConstraints We now turn to discuss the impacts of the demand shock when constraints bind. We illustrate the impact of binding domestic constraints in Figure 7, and the impact of binding foreign constraints inFigure8. ForcomparisontoFigure6,recallthatwesetthevaluesoftheconstraintssothatthey bindonlyintheinitialpost-shockperiod,andareslackthereafter. In7a,weseethatbindingdomesticconstraintsleadtoabouttwiceasmuchinflationonimpact. Importantly, goods price inflation now rises more than inflation in services, increasing about ten times as much as in the unconstrained equilibrium. This reflects high inflation in domestic goods prices, in Figure 7b.25 Unpacking domestic prices using Equation 30 in Figure 7c, domestic price inflation surges due to the markup shocks induced by binding constraint, where µˆ˜ (1) > 0 on 1 impact. InFigures7dand7b,weagainseethatimportpriceinflationfallsonimpactinresponseto the demand shock, reflecting an exchange rate appreciation and slack imported input constraints. Turningtoquantities,goodsoutputrisesonimpact(asthereissurpluscapacityinsteadystate),but its rise is capped at about half the unconstrained response, so goods output rises by significantly lessthanservicesoutput. Further,reflectinglowerdomesticgoodsoutput,thequantityofimported goods inputs is dampened as well, while imports of consumption goods imports increase by more 25Likeintheunconstrainedequilibrium,low(positive)importpriceinflationforconsumergoodsactuallyattenuates overallgoodspriceinflation;equivalently,domesticgoodspricesrisemorethantheconsumerpriceindexforgoods. 25

Figure7: DemandShock: DomesticConstraintBinds (a)ConsumerPriceInflation stnioP egatnecreP 1. 80. 60. 40. 20. 0 (b)ConsumerInflationComponents Total Goods Services 0 4 8 12 16 20 period stnioP egatnecreP 1. 80. 60. 40. 20. 0 (c)DomesticPriceInflation Domestic Goods Domestic Services Goods Imports Services Imports 0 5 10 15 20 period stnioP egatnecreP 1. 50. 0 50.- Goods Wage Term Goods Input Price Term Goods Constraint Term Services Wage Term Services Input Price Term 0 5 10 15 20 period (d)ImportedInputPriceInflation stnioP egatnecreP 10. 500. 0 500.- 10.- (e)GrossOutputQuantity 0 5 10 15 20 period etatS ydaetS morf noitaiveD goL 200. 5100. 100. 5000. 0 (f)ImportQuantity Goods Output Services Output 0 5 10 15 20 period etatS ydaetS morf noitaiveD goL 5200. 200. 5100. 100. 5000. 0 Consumption Goods Input Goods 0 5 10 15 20 period thanintheunconstrainedequilibrium. Turning to the case where only foreign constraints are binding in Figure 8, we note first that binding foreign constraints raise both consumer goods and domestic goods price inflation in Figures8aand8b. Lookingat8c,highergoodspriceinflationreflectsthefactthatpricesforinputsin the goods sector increase on impact, and these are passed into domestic goods prices. In turn, the priceofinputsforgoodsproducersrisesbecauseimportedgoodsinputspriceinflationnowspikes onimpactinFigure8d,duetothebindingforeignconstraints. Thebehaviorofimportedinputprice inflationdiffersfromboththeunconstrainedequilibriumandtheequilibriumwithbindingdomestic constraints. It also differs from import price inflation for consumer goods in Figure 8b, where the effects of binding constraints on imported input prices dominates the impacts of the exchange rate appreciation, which leads imported consumer goods inflation to fall on impact. Turning to quantities, note that domestic goods output rises even though the foreign constraint binds, nearly the same amount as in the unconstrained equilibrium. Although foreign input constraints limit input availability for domestic producers, they also trigger substitution toward domestic goods producers, with offsetting effects for total production. Finally, imports of goods inputs are obviously constrainedinFigure8f,relativetothepriortwocases. 26

Figure8: DemandShock: ForeignConstraintBinds (a)ConsumerPriceInflation stnioP egatnecreP 30. 20. 10. 0 (b)ConsumerInflationComponents Total Goods Services 0 4 8 12 16 20 period stnioP egatnecreP 30. 20. 10. 0 10.- (c)DomesticPriceInflation Domestic Goods Domestic Services Goods Imports Services Imports 0 5 10 15 20 period stnioP egatnecreP 520. 20. 510. 10. 500. 0 Goods Wage Term Goods Input Price Term Goods Constraint Term Services Wage Term Services Input Price Term 0 5 10 15 20 period (d)ImportedInputPriceInflation stnioP egatnecreP 6. 4. 2. 0 2.- (e)GrossOutputQuantity 0 5 10 15 20 period etatS ydaetS morf noitaiveD goL 200. 5100. 100. 5000. 0 (f)ImportQuantity Goods Output Services Output 0 5 10 15 20 period etatS ydaetS morf noitaiveD goL 200. 5100. 100. 5000. 0 5000.- Consumption Goods Input Goods 0 5 10 15 20 period 3.3 Shocks to Constraints Wenowbrieflysummarizetheimpactsofshockstotheforeignanddomesticfirm-levelconstraints. Toanchorthemagnitudes,deviationsindomesticandforeigncapacityfromsteadystatearegiven by: yˆ¯ t (1)=λ y¯ yˆ¯ t−1 (1)+ε y¯t , yˆ¯∗ Mt (1)=λ y¯ ∗yˆ¯∗ Mt−1 (1)+ε y¯∗t , with var(ε y¯t )=σ y¯ 2 and var(ε y¯∗t )=σ y¯ 2 ∗ , andautocorrelationandshockvarianceparameterssetbasedonestimatesbelow. In Figure 9, we plot responses to a shock to the foreign capacity constraint, equal to −0.15σ y¯∗ in magnitude. As above, we set the initial steady state capacity level so that the constraint binds for only one period after the shock, and we assume that the domestic constraint is slack in all periods. Following the negative foreign capacity shocks, there is a sharp rise in imported input price inflation in Figure 9d. This feeds through to domestic goods prices, and in turn to overall goods price inflation, which rises more than services price inflation in this scenario (Figures 9a and9b). Nonetheless,theoverallchangeininflationismodest,reflectingtherelativelysmallshare of imported inputs in domestic input use, as well possibilities to substitute domestic for foreign inputs. Reflecting this substitution, domestic goods output actually rises slightly. Despite the fact thattherealquantityofimportedinputsfallsonimpact,duetoreducedforeigncapacity. Thus,this constraintshockleadstorisingimportpriceinflationtogetherwithfallingquantitiesofimports. In Figure 9d, we plot responses to a −0.15σ shock to the domestic capacity constraint, under y¯ the assumption that the foreign constraint is slack. There is again a rise in goods price inflation in 27

Figure9: ForeignFirmCapacityShock (a)ConsumerPriceInflation stnioP egatnecreP 5100. 100. 5000. 0 (b)ConsumerInflationComponents Total Goods Services 0 4 8 12 16 20 period stnioP egatnecreP 5100. 100. 5000. 0 (c)DomesticPriceInflation Domestic Goods Domestic Services Goods Imports Services Imports 0 5 10 15 20 period stnioP egatnecreP 5100. 100. 5000. 0 5000.- Goods Wage Term Goods Input Price Term Goods Constraint Term Services Wage Term Services Input Price Term 0 5 10 15 20 period (d)ImportedInputPriceInflation stnioP egatnecreP 60. 40. 20. 0 20.- (e)GrossOutputQuantity 0 5 10 15 20 period etatS ydaetS morf noitaiveD goL 60-e00.5 0 60-e00.5- 10000.- 510000.- (f)ImportQuantity Goods Output Services Output 0 5 10 15 20 period etatS ydaetS morf noitaiveD goL 0 50000.- 1000.- 51000.- 2000.- Consumption Goods Input Goods 0 5 10 15 20 period Figure 10a, driven by an increase in domestic goods price inflation (Figure 10b). The constraint shockleadsthemultiplierontheconstrainttobegreaterthanzero,whoseeffectsondomesticprice inflationarecapturedinthegoodsconstraintterminFigure10c. Duetothefallindomesticgoods capacity, actual realized goods output falls in this case (Figure 10e), and imports of intermediate goods fall on impact as well (Figure 10f). In contrast, imports of consumption goods increase, reflectingsubstitutionfromdomestictoimportsources. Thenegativecomovementbetweeninflation andrealoutput/importsisadistinctivefeatureofthisshock. 3.4 Summing Up Stepping back, we collect a few summary results that shed light on how the model will identify whichconstraintsarebindingandthestructureofunderlyingshocks. First,thesector-composition ofinflationdependsontheconfigurationofshocksandconstraints. Withslackconstraints,inflation forservicesoutstripsthatforgoods. Bindingforeignconstraintsraisegoodspriceinflationtoequal services price inflation, and binding domestic constraints lead goods price inflation to outstrip servicesinflation,whichisaprominentfeatureofrecentdata. Second,importedgoodsinputprice inflation is high only when the foreign firm constraint binds, either following a demand shock or a shock to the constraint itself. Put differently, while a binding domestic constraint may explain excess inflation for goods, it cannot also generate a sharp increase in import price inflation on its 28

Figure10: DomesticFirmCapacityShock (a)ConsumerPriceInflation stnioP egatnecreP 10. 500. 0 500.- (b)ConsumerInflationComponents Total Goods Services 0 4 8 12 16 20 period stnioP egatnecreP 10. 500. 0 500.- (c)DomesticPriceInflation Domestic Goods Domestic Services Goods Imports Services Imports 0 5 10 15 20 period stnioP egatnecreP 510. 10. 500. 0 500.- Goods Wage Term Goods Input Price Term Goods Constraint Term Services Wage Term Services Input Price Term 0 5 10 15 20 period (d)ImportedInputPriceInflation stnioP egatnecreP 5000. 4000. 3000. 2000. 1000. 0 (e)GrossOutputQuantity 0 5 10 15 20 period etatS ydaetS morf noitaiveD goL 0 20000.- 40000.- 60000.- 80000.- 1000.- (f)ImportQuantity Goods Output Services Output 0 5 10 15 20 period etatS ydaetS morf noitaiveD goL 1000. 50000. 0 50000.- Consumption Goods Input Goods 0 5 10 15 20 period own. Third, co-movement in prices and quantities differs depending on whether constraints are binding or slack, as well as the nature of the shocks. In the case of a demand shock, gross output risesalongwithinflation,asdoesthequantityofimportedinputs. Forconstraintshocks,however, inflation and output comove negatively: a shock to the foreign firm constraint raises import price inflation while lowering imported input quantities, and a shock to the domestic constraint raises domesticgoodspriceinflationwhileloweringquantitiesproduced. 4 Accounting for Recent Inflation Experience We now apply the model to parse recent data. We describe the procedure we use to estimate the model in Section 4.1, with additional details in Appendix A. Then, we discuss data, calibration, and estimated parameters in Section 4.2. In Section 4.3, we review model fit. We then discuss whatthemodeltellsusaboutrecentinflationinSection4.4. 4.1 Estimation Framework Referring back to Section 2.6.1, the impact of a given structural shock in the model depends on whether constraints bind today following the shock, as well as the expected duration that constraints are expected to continue to bind into the future following that shock. To make this depen- 29

dence explicit, let us define a set of regimes (R ), which record which constraints are binding at a t given point in time: R ={1(Y(1)=Y¯ (1)),1(Y∗ (1)=Y¯∗ (1))}, where the indicator functions t t t Mt Mt switchonwhenindividualconstraintsbind. Givenasequence (cid:8)R (cid:9) for0≤ j≤J,togetherwith t+j theassumptionthat (cid:8)R (cid:9) ={0,0}for j>J,wecansolveforanequilibriumpathfor{X },using t+j t themethoddescribedinCagliariniandKulish(2013)andKulishandPagan(2017)(seeAppendix Afordetails). Buildingonthisidea,were-parameterizethemodelsolutioninaconvenientway. Specifically, let us define the duration that constraints are expected to bind from datet forward as d =[d ,d∗], t t t whereeachentryisanon-negativeintegerthatrecordsthenumberofperiodsthatthedomestic(d ) t orforeignconstraint(d∗)binds. Byconvention,d andd∗ takeonzerovalueswhenconstraintsare t t t slack today and expected to remain so in the absence of future shocks, and they are positive when they are binding today. As in Guerrieri and Iacoviello (2015), we construct policy matrices under theassumptionsthatagentsknowthestate(X )andthecurrentrealizationoftheshocks(ε ),but t−1 t that they do not anticipate that future shocks will occur. Under these assumptions, d summarizes t alltheinformationabouttheanticipatedsequenceofregimesthatisneededtosolveforequilibrium responses to a one-time shock in our model. Specifically, constraints may switch on immediately in response to shock at date t, then bind for some (non-negative) number of consecutive periods, and switch off thereafter. In the absence of future shocks, constraints do not then switch on again inperiodsaftertheyswitchoff(e.g.,followingashockε ,constraintscannotbeslackatdatet and t then binding at date t+1).26 With these observations, we re-write the model solution directly in termsofdurations: X =J(d ,θ)+Q(d ,θ)X +G(d ,θ)ε , (31) t t t t−1 t t wheredurationd impliesaspecificanticipatedsequenceofregimesovertime. Wereferthereader t toAppendixA.3.2fordetailsonthisresult. Following Kulish, Morley and Robinson (2017), Kulish and Pagan (2017), and Jones, Kulish and Rees (2022), our estimation framework exploits the fact that durations enter the policy function like parameters. As is standard, let us assume that observables (S ) are linearly related to the t unobserved state, as in S = H X +ν , where ν is an i.i.d. vector of normally distributed meat t t t t T surement errors. Given d ≡ {d } and θ, we can construct the piecewise linear solution with t t=1 time-varying coefficients, and then apply the Kalman filter to construct the Likelihood function L(θ,d|{S } T ). We put priors over structural parameters and independent priors over durations t t=1 26To be careful, this is not a general property of models with potentially binding constraints, but rather one that holds given the structural assumptions in our model about behavior and shock processes. While we lack a general proofofthisproperty,weverifyitholdsnumericallyinthemodelinpractice,andwecandemonstratethatimposing this criterion in the estimation procedure is reasonable via simulation analysis. One could capture a more complex structureofpotentialregimechangesviaintroductionofadditionalparameters(e.g.,durationsforbindingconstraints thatstartoneperiodforward),atthecostofaddedcomputationalcomplexity. 30

toconstructtheposterior,andthenestimatethemodelviaBayesianMaximumLikelihood. In implementing this approach to estimation, we are careful to account for the fact that the duration of binding constraints is an equilibrium object in the model – i.e., d depends on both t the state X and current shock ε in our model. Thus, we impose a rational expectations equit−1 t librium restriction on admissible durations, which requires that agents’ forecasts about how long constraints bind following a given shock are consistent with equilibrium model responses. To impose this restriction, we proceed as follows. For each proposed duration and parameter draw, we filter the data for smoothed shocks. We then evaluate whether the equilibrium model response to thosesmoothedshocksisconsistentwiththeproposeddurationdraw. Weretaintheproposeddraw ifthisrequirementissatisfied;otherwise,werejectitanddrawagain. InAppendixA.3.3,westudytheperformanceofthisprocedureusingsimulateddata,forwhich we know the true data generating process and the exact incidence of endogenously binding constraints. First, we confirm that our estimation procedure is able to recover unobserved durations from the observables that we use, by directly examining likelihood functions. Then, we also show that the reduced-form multipliers implied by the duration and parameter estimates align with true latent multipliers, which summarize the impacts of binding constraint on inflation, our key outcome. Lastly, as a practical matter to restrict the size of the parameter space, we impose priors that allow capacity constraints to bind only periods from 2020:Q2 forward. Put differently, we impose dogmatic priors that assign zero probably to binding constraints prior to 2020:Q2, thus focusing ontheroleofcapacityinexplainingtheunusualpost-pandemicinflationdynamics.27 4.2 Data and Parameters To populate Y, we collect standard macro variables together with particular series that serve to t identifywhetherconstraintsarebindingandshockstothem. Amongstandardmacrovariables,we includeconsumptionpriceinflationandthegrowthratesofconsumptionexpenditureforgoodsand services. We also use data on aggregate nominal GDP growth, the growth rate of (real) industrial production(whichwetreatasaproxyforoutputofthegoodssector),andlaborproductivitygrowth by sector (measured as real value added per worker).28 On the international side, we use data on 27Asarobustnesscheck,wehaveestimatedthemodelallowingconstraintstopotentiallybindstartingin2018:Q1, prior to the pandemic. We find that the mode of estimated durations before 2020:Q2 is zero, and that the mode of estimateddurationsafter2020:Q2isnotaffectedbytheinitialdatewhencapacityconstraintsareallowedtobind. 28We use data on labor productivity growth in manufacturing and total (private sector) labor productivity growth fromtheBureauofLaborStatistics. Weassumethatlaborproductivitygrowthinmanufacturingcoincideswithgoods labor productivity (growth in real value added per worker) in the model, while also matching aggregate (economywide) labor productivity growth in the model. While the definition of industrial production and goods output do not align exactly(industrial production includes manufacturing, mining, and electrical/gas utilities, while the BEAdefined goods sector excludes utilities and includes agriculture and construction), the dynamics of gross output for 31

importpriceinflationforconsumptiongoods,andweproxyinputpriceinflationinthemodelusing data on inflation for imported industrial materials (excluding fuels). We then also use data on the growth of total expenditure on imported consumption goods and imported materials inputs (again excludingfuels),whichweassociatewithimportedinputsofgoods.29 These data are all obtained from quarterly US national accounts produced by the Bureau of EconomicAnalysis,withtheexceptionoflaborproductivitydatafromtheBureauofLaborStatisticsandindustrialproductionfromtheFederalReserveBoard(G.17program). Havingconstructed growthratesforindividualvariablesfromthefirstquarterof1990throughthefirstquarterof2022, wedetrendthedatabyremovingthemeangrowthratefromeachseries. Finally,becauseourestimationsampleincludesasignificantperiodduringwhichinterestratesareatthezerolowerbound, weusedataonthe“shadowFedFundsrate”toestimateparametersinthemonetarypolicyrule.30 We present the full set of parameters for the model in Appendix A, which we obtain through a mix of estimation and calibration. We calibrate key value shares in the model – e.g., consumer expenditure, input use, export and import shares, etc. – to match US national accounts and inputoutput data (see Table A.4). We set a subset of the structural parameters to standard values from theliterature,includingpreferenceparametersandsomeelasticitiesofsubstitution(seeTableA.3). We also calibrate the level of excess capacity for domestic and foreign firms, settingY¯ (1)= 0 1.05Y (1)andY¯∗ (1)=1.10Y∗ (1). Theselevelsarechosentobesufficientlyhighthatconstraints 0 M0 M0 are slack prior to 2020:Q2.31 Further, note that the model and data allows us to estimate the level of capacity that actually prevailed during the pandemic. Alternative values for steady state capacity then re-scale the size of the capacity shocks needed to achieve this realized capacity level. Consistent with this observation, the level of calibrated steady state capacity is not an important parameter in understanding the key quantitative results. To demonstrate this robustness, we estimate steady-state capacity levels directly in Appendix A.6, using data from the pandemic period, thegoodssectorandindustrialproductionaresimilar. Weoptforindustrialproductiondatahere,becausewerequire asufficientlylongtimeseriestoestimatemodelparameters; quarterlygrossoutputdataisonlyavailableafter2005, whileindustrialproductiondataisavailablefrom1947. 29Weusedataforconsumergoods(exceptfoodandautomotive)toproxyforconsumptionimports,andweconstruct proxiesforimportedinputs(excludingfuels)byremovingthesubcategoryofpetroleumandproductsfromindustrial materialsandsuppliesusingstandardchainindexformulasandauxiliaryNIPAdataonthesub-categoriesofimports. 30DuringperiodswherethenominalFedFundsrateisatzero,wereplaceitwiththeshadowratefromWuandXia (2016): https://www.atlantafed.org/cqer/research/wu-xia-shadow-federal-funds-rate. Changes intheshadowratecapturetheconsequencesofunconventionalpolicyactionstakenbytheFederalReserve, suchas forwardguidanceorquantitativeeasingpolicies. Wehavecheckedtheresultsusinganalternativeshadowrateseries fromJones,KulishandMorley(2022)aswell,whichyieldssimilarresults. 31This amount of domestic excess capacity is consistent with historical fluctuations in capacity utilization for the US,asmeasuredbytheFederalReserve’sG.17dataseries,forwhichthemaximalvalueforcapitalutilizationabout five percent higher than the minimum. Further, cyclical fluctuations in this capacity utilization measure are almost entirely driven by changes in industrial production itself, rather than the Fed’s estimate of capacity (based on firm survey data). Thus, our calibration accommodates historically normal fluctuations in industrial production, absent shockstocapacity. 32

andshowthatourmaincounterfactualresultsgothroughwiththisalternativeparameterization. Turning to the final set of parameters, we estimate (a) the elasticities of substitution between home and foreign goods, in consumption and production separately; (b) the parameters in the extended Taylor rule governing the response of interest rates to inflation and output, as well as interestrateinertia,(c)parametersgoverningthestochasticprocessesforexogenousvariables,and (d)thevarianceofmeasurementerrors. Regarding(c),weassumethatexogenousvariablesevolve accordingtoAR1stochasticprocesses. We obtain an estimated mean value for the elasticity of substitution between home and foreign goods of about 1.5 in consumption and 0.5 for inputs, so consumer goods are substitutes while inputs are complements. These values are not far from standard values estimated using aggregate time series variation in the macroeconomic literature, though there is limited prior work that distinguishes consumption and input elasticities. We find that the policy rule displays inertia, and it responds to both inflation and output gaps with reasonable magnitudes.32 There is significant autoregressive persistence in most exogenous variables, and measurement error variances are plausible. SeeTableA.5forthefullsetofestimatedparameters. 4.3 Model Fit Applyingthequantitativemodelframeworktothedata,weconstructKalman-smoothedvaluesfor endogenousvariablesandobservables. InFigure11,weplotdataandsmoothedvaluesforseveral keyobservables–goods,services,andaggregatepriceinflationforconsumers,andimportedinput priceinflation–overthe2017-2022period,whereeachdatapointistheannualizedvalueofquarterly inflation. To compute the smoothed inflation series, we take 1000 draws from the posterior distribution for model parameters, compute Kalman-smoothed inflation for each draw, and then plotstatistics(themedian,5th,and95%percentiles)forthedistributionofsmoothedvalues. The model fits the dynamics of aggregate consumer price inflation well, accounting for essentially all of the four percentage point increase in headline inflation after 2020 (Figure 11a).33 It also accounts well for the two percentage point rise in inflation for the services sector (Figure 11b). Because goods price inflation is substantially more volatile than that for services, the model attributes more of its variation to measurement error. Nonetheless, smoothed values for goods 32Inunreportedanalysis,wehaveverifiedthatourresultsarerobusttoalternativeformulationsandparametersfor themonetarypolicyrule. 33Recallthataggregateconsumerpriceinflationistreatedasanunobservedvariable. Inthemodel,itisconstructed byaggregatingsector-levelconsumerpricegrowthusingfixed(steady-state)expenditureweights. Inthedata, however,thePCEdeflatorisachain-weightedindex,whichfeaturestime-varyingweights. Thus,partofthediscrepancy between aggregate inflation in the model and data is likely due to differing index number concepts. Specifically, the dramatic increase in the goods expenditure share, combined with high goods price inflation, likely pushed measuredinflationuprelativetoourfixed-weightindex. Goingforward,wefocusentirelyondecomposingmodel-based measuresofinflation,sowedonotbelaborthispoint. 33

price inflation also track the data well (Figure 11c). The model replicates the initial (roughly six percentage point) surge in goods price inflation in 2021, and goods price inflation then remains elevated into 2022. While the model captures its transitory (up/down) dynamics, it moderately undershootsthelevelofgoodspriceinflationin2022,meaningthatthemodelattributesthegapto measurementerror. Themodelalsomatchesinflationforimportedgoodsinputswell(Figure11d), matchingbothlevelsanddynamicsclosely. Forbrevityhere,wepresentsimilarfiguresillustratingmodelfitfortheremainingobservables in Appendix A.5. Together with the inflation figures here, we assess that the model captures the behaviorofeconomicvariableswellduringthepandemic,soitisausefullaboratoryforexploring thedrivingforcesunderlyingtheinflationsurge. 4.4 Explaining the Inflation Surge We provide three sets of results. The first two illustrate the role of constraints in explaining inflation. First, we examine the dynamics of the multipliers on the constraints. Second, we present counterfactualsinwhichweswitchofftheconstraints,comparingmodelresponsestothesameset of shocks with and without constraints. The third set of results focuses on how individual shocks andconstraintsshapeinflationoutcomes,bothindividuallyandviainteractionsbetweenthem. 4.4.1 MultipliersonConstraints To start, we can directly illustrate the impact of constraints by examining the smoothed value of multipliers on the domestic and foreign constraints. Because the multipliers themselves do not have intuitive economic units, we plot the reduced-form markup shocks implied by the value of (cid:16) (cid:17) (cid:16) (cid:17) themultipliers–givenby ε P 0 µˆ˜ (s)inEquation28and ε P 0 µˆ˜∗(s)inEquation29 φ(s)P H0 (s) t φ(s)P uF0 (s) ut – which summarize the impulse of binding constraints for domestic and import price inflation. As isevident,thevaluesofthemultipliersrisein2021,coincidentwiththeriseinheadlineinflation.34 On the import side, constraints appear to be slack in 2020, then bind sharply at the start of 2021, relax somewhat, then bind sharply again into 2022, and ease in the latter half of 2022. Domestic multipliers fluctuate in 2020 with gyrations in the US economy, but are near zero heading into 34Asatechnicalnote,recallweplacepositivemassonvaluesd >0inourpriorsonlystartingin2020:Q2,sothe t reduced-formmarkupshock(tiedtothemultiplier)inthefigureisidenticallyzerobeforethatdate. Asecondnoteis that while multipliers and hence reduced-form markup shocks are typically positive, they dip into negative territory at times in the simulations. This primarily reflects the fact that there is approximation error in the piecewise linear solutiontechniquethatweemploy. Whenconstraintsbind, thevalueofthemultipliersarecomputedasresidualsin thelog-linearizedPhillipsCurves. Assuch,whilethequantityconstraintbindsexactly,thecomputedmultipliersare approximationstotheexactequilibriummultipliers;further,wedonotimposeazerolowerboundonthem,aswould berequiredinthefullnon-linearsolutiontothemodel.Despitethis,wefindthattheestimatedmultipliersaretypically positive,consistentwiththeunderlyingtheory. 34

Figure11: ConsumerPriceInflationinModelandData (a)AggregateConsumerInflation stnioP egatnecreP 4 2 0 2- 4- 6- (b)ConsumerServicesInflation Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 stnioP egatnecreP 5 0 5- 01- Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 (c)ConsumerGoodsInflation stnioP egatnecreP 01 5 0 5- (d)InflationforImportedGoodsInputs Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 stnioP egatnecreP 03 02 01 0 01- 02- Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 Note: Inflationateachdateistheannualizedvaluefordemeanedquarterlyinflation,inpercentagepoints. Ifdemeanedquarterlyinflationisπ (s)=lnP(s)−lnP (s)wheret indexesquarters,thentheannualized t t t−1 inflationrateis4π (s). Dataisrawdata. Wetake1000drawsfromtheposteriordistributionofmodel t parameters,computetheKalman-smoothedvaluesformodelvariablesforeachdraw,andthenplotthe mediansmoothedvalueasthedashedline. Weshadetheareacoveringthe5%to95%percentilefor smoothedvalues(theintervalisimperceptiblysmallpriorto2020). 35

Figure 12: Smoothed Values for the Reduced-Form Markup Shock Implied by the Multipliers on Constraints (a)DomesticConstraint etatS ydaetS morf noitaiveD goL 60. 40. 20. 0 20.- (b)ForeignConstraint Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 etatS ydaetS morf noitaiveD goL 3. 2. 1. 0 Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 (cid:16) (cid:17) Note: Figure12aplotscompositevariable ε P 0 µˆ˜ (s)andFigure12bplotscompositevariable φ(s)PH0(s) t (cid:16) (cid:17) ε P 0 µˆ˜∗(s),whicharethereduced-formmarkupshocksindomesticandimportpricePhillips φ(s)PuF0(s) ut Curvesinducedbybindingconstraints. Wetake1000drawsfromtheposteriordistributionofmodel parameters,computetheKalman-smoothedvaluesformodelvariablesforeachdraw,andthenplotthe mediansmoothedvalueasthesolidline. Weshadetheareacoveringthe5%to95%percentilefor smoothedvalues. 2021. They rise steadily through 2021 into 2022, and then slacken (though still bind) through 2022:Q3. While there is limited external data to which we can benchmark the estimated multipliers, we note that their joint dynamics align well with fluctuations in the New York Federal Reserve’s Global Supply Chain Pressure Index over the post-2020 period, as we illustrate in Appendix A.5. For both multipliers, the high frequency dynamics also correspond to fluctuations in goods price inflationandimportedinputpriceinflationinFigure11,whichforeshadowsthequantitativeroleof the constraints in explaining inflation. Further, the large absolute size of increases in multipliers, and their volatility translate into large, abrupt shifts in the Phillips Curves. In Appendix A.5, we show that these quasi-markup shocks are substantially larger than would be consistent with a stochastic process for (exogenous) markup shocks estimated from pre-pandemic data. Thus, our modelappearstocaptureasourceofmarkupvariationthatisdistinctfromrun-of-the-millmarkup (elasticity of demand) shocks. We turn to model counterfactuals to parse the role of constraints further. 36

4.4.2 RelaxingConstraints We now provide counterfactual analysis as to how inflation would have evolved in the absence of capacityconstraints,giventhepathofrealizedshocksthatweinferhittheUSeconomyafter2020. To describe this exercise more precisely, the mechanics of each iteration are as follows. We first draw model parameters from the estimated posterior distributions, including the durations for binding constraints. Given these parameters, we apply the Kalman-filter to the data and construct smoothed model outcomes and shocks. Note that we construct smoothed shocks here assuming that constraints are potentially binding, in line with posterior duration estimates. Using these smoothed shocks, we then simulate the path of the economy under the counterfactual assumption thatconstraintsareslackthroughout,suchthatthesolutionconformstotheunconstrainedequilibrium dynamics of the model. We repeat this procedure for one thousand posterior draws, and we plotstatistics(meansandpercentiles)acrossthesesimulationsinFigures13and14. Figure13presentsresultsforconsumerpriceinflation. Thefigurespresentrawdataonannualized values of (de-meaned) quarterly inflation, along with data from counterfactual simulations in which we allow for measurement error in these observables.35 In Figure 13a, we see that realized inflationforconsumergoodsissubstantiallyhigherthancounterfactualinflationwithslackcapacityconstraintsduring2021andinto2022,withtheabsolutegappeakingnearsixpercentagepoints inearly2021. Putdifferently,giventheshocksweinferfromdata,bindingconstraintsaccountfor about half of the acceleration in goods price inflation from 2020:Q2 through 2021:Q2. Likewise, theyappeartoexplainabouthalfofthedeclineingoodspriceinflationinthelatterhalfof2022. Underthehood,theseinflationoutcomesaretiedtotheimpactofbindingconstraintsinholding back production of domestic goods and foreign goods inputs. In Figure 14a, we plot the path for smootheddomesticgoodsoutputalongwithcounterfactualoutput. Asisevident,intheabsenceof constraints, goods output would have risen significantly in 2021 relative to its pre-pandemic level, asaresultoftheothershocks(principally,demandshocks)thathittheeconomy. Thefactthatoutput did not rise in reality speaks directly to the role of constraints. Output of foreign goods inputs issimilarlyconstrainedinFigure14b. Correspondingly,smoothedinflationforbothdomesticallyproduced goods and foreign-produced inputs is substantially higher than counterfactual inflation inFigures14cand14d. Interestingly, binding constraints also play an important role in driving price inflation for services in Figure 13b. While services price inflation initially accelerates due to the underlying shocks, it is between one and two percentage points higher in 2021 as a result of binding con- 35Intheproceduredescribedabove,wedrawthevarianceofthemeasurementerrorfromtheposterior,andthenfilter thedatagiventhisdraw. Wethenaddadrawfromthemeasurementerrortothesmoothedcounterfactualendogenous variables to get counterfactual values for the observables that are comparable to data. An alternative approach to presenting the results would be to compare smoothed observables to model counterfactuals without measurement error;naturally,thisalternativeleadstosimilarconclusions. 37

Figure13: CounterfactualConsumerPriceInflationwithoutCapacityConstraints (a)GoodsInflation stnioP egatnecreP 01 5 0 5- (b)ServicesInflation Data Median Counterfactual Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 stnioP egatnecreP 4 2 0 2- 4- (c)AggregateInflation Data Median Counterfactual Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 stnioP egatnecreP 4 2 0 2- 4- Data Median Counterfactual Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 Note: Wetake1000drawsfromtheposteriordistributionofmodelparameters,computethe Kalman-smoothedvaluesformodelvariablesforeachdraw,addmeasurementerrortotheobservables,and thenplotthemediansmoothedvalueasthesolidline. Weshadetheareacoveringthe5%to95%percentile forsmoothedvalues. straints. In the background, this reflects both the fact services use goods as inputs, so there is a direct inflation spillover from binding constraints in the goods sector via input-output linkages. Further, binding constraints serve to tighten the labor market as well, as the price increases they generatetriggersubstitutionfromgoodsinputstowardlaborinproduction. AddinguptheseresultsinFigure13c,headlineconsumerpriceinflationisbetweenoneandtwo percentage points higher than counterfactual inflation during 2021-2022. And binding constraints account for about one third of the acceleration in headline goods price inflation from 2020:Q2 through 2021:Q2. Note further that the effect of constraints is substantially diminished late in 2022,asactualandcounterfactualinflationconvergeagain. Finally,werevisitthediscussionsurroundingprofitsperunitinthiscounterfactualexercise. In Figure 5, we presented an index of nominal profits per unit of gross output for manufacturing and the aggregate economy. In Figure 15 we present analogous results from the model for goods and services.36 Similartothedata,thesmootheddatafromourmodelyieldsasharpincreaseinprofits for the goods sector during the 2021-2022 period, even though this is not a targeted data moment. In contrast, the counterfactual economy with slack constraints yields no such goods profit surge. Moreover,profitsperunitareessentiallyflatthroughthepandemicperiod(outsidethe2020spike), forboththeeconomieswithandwithoutcapacityconstraints. Weconcludethatthemodelprovides a plausible explanation for the run-up in profits for goods producers that occurred alongside the 36In the model, the log change in nominal profits per unit of output from a given base period (t =0) is given by: (cid:2) Ξˆ (s)−yˆ(s) (cid:3) − (cid:2) Ξˆ (s)−yˆ (s) (cid:3) =[pˆ −pˆ ]+ε[rp(s)−rp (s)]−(ε−1)[rmc(s)−rmc (s)],where pˆ −pˆ = t t 0 0 Ct C0 (cid:98)t (cid:98)0 (cid:100)t (cid:100)0 Ct C0 ∑t s=0 π Cs . We add trend inflation to these log changes, to make it comparable to the date in Figure 5, and then we convert the log change to levels to plot the index. While we discuss manufacturing and aggregate profits in Figure 5duedataavailabilityinthenationalaccounts,thegoodssectorinourcalibrationincludesmanufacturingandother non-manufacturinggoodssectors. Further,againfordatareasons,wefocusedoncorporateprofitsinFigure5,while wehavenodistinctionbetweencorporateandnon-corporateentitiesinthemodel.Thisimpliesthatoneoughttofocus onqualitativecomparisonsbetweenthefigures,ratherthanamoreprecisequantitativecomparison. 38

Figure14: CounterfactualQuantitiesandInflationwithoutCapacityConstraints (a)DomesticGoodsOutput(Y(1)) t etatS ydaetS morf noitaiveD goL 50. 0 50.- 1.- 51.- (b)ImportedGoodsInputsY∗ (1) Mt Median Smoothed Value 5th-95th Percentiles Median Counterfactual Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 etatS ydaetS morf noitaiveD goL 1. 0 1.- 2.- 3.- Median Smoothed Value 5th-95th Percentiles Median Counterfactual Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 (c)DomesticGoodsPriceInflation(π(1)) t stnioP egatnecreP 01 5 0 5- 01- (d)ImportedGoodsInputInflation(π (1)) Mt Median Smoothed Value 5th-95th Percentiles Median Counterfactual Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 stnioP egatnecreP 03 02 01 0 01- 02- Median Smoothed Value 5th-95th Percentiles Median Counterfactual Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 Note: Wetake1000drawsfromtheposteriordistributionofmodelparameters,computethe Kalman-smoothedvaluesformodelvariablesforeachdraw,andthenplotthemediansmoothedvalueas thesolidline. Weshadetheareacoveringthe5%to95%percentileforsmoothedvalues. Counterfactual assumesthatconstraintsareslackinallperiods. 39

Figure15: CounterfactualProfitsperUnitwithoutCapacityConstraints (a)Goods )1=1Q7102( xednI 4.1 3.1 2.1 1.1 1 (b)Services Median Smoothed Value 5th-95th Percentiles Median Counterfactual Value 5th-95th Percentiles 2017q1 2018q1 2019q1 2020q1 2021q1 2022q1 2023q1 )1=1Q7102( xednI 4.1 3.1 2.1 1.1 1 Median Smoothed Value 5th-95th Percentiles Median Counterfactual Value 5th-95th Percentiles 2017q1 2018q1 2019q1 2020q1 2021q1 2022q1 2023q1 Note: Wetake1000drawsfromtheposteriordistributionofmodelparameters,computethe Kalman-smoothedvaluesformodelvariablesforeachdraw,andthenplotthemediansmoothedvalueas thesolidline. Weshadetheareacoveringthe5%to95%percentileforsmoothedvalues. inflationtakeoff,wherebothareexplainedinlargemeasurebybindingcapacityconstraints. 4.4.3 DecomposingtheRoleofIndividualShocksandConstraints We now examine the role of individual shocks in explaining inflation outcomes. To construct the counterfactualseries,wetakeadrawfromtheposteriordistributionsforstructuralparametersand durations. Using this draw to parameterize the state equation (Equation 31), we Kalman filter the data to obtain smoothed shocks. We then feed a subset of these shocks into the structural model (summarized by Equation 27) to compute counterfactual model outcomes. In each case, we solve forthesimulatedequilibriumpathusingDynare’sOccBinprocedure. Bydoingso,weensurethat whetherconstraintsbindatparticularpointsintimeinresponsetoshocksisendogenous. Werepeat thisprocedure1000timesandcomputethemedianacrossthesimulatedcounterfactualseries,and thesemediansareplottedinFigure16. In Figure 16a, we plot the path of aggregate consumer price inflation following four types of shocks, each fed individually into the model: demand shocks (including both the discount rate and goods-biased preference shocks), monetary policy shocks, capacity shocks, and cost shocks (including domestic productivity and foreign cost shocks). The final line is the value for inflation when all shocks are fed simultaneously into the model. At the outset, temporary negative demand shocks yield a decline then rebound of inflation in 2020. Into 2021, however, no single shock appearstoplayaparticularlyimportantroleinexplainingthepathofinflationonitsown. Theunderlyingreasonisthatnosingleshockiscapableofcausingcapacityconstraintstobind,somodel outcomesconformcloselytothoseobservedinthepriorcounterfactualsinwhichweexogenously 40

Figure16: CounterfactualConsumerPriceInflationforIndividualShocks (a)IndividualShocks stnioP egatnecreP 5 0 5- (b)CapacityShocksplusIndividualShocks All Shocks Demand Shocks Monetary Policy Shocks Capacity Shocks Cost Shocks 2020q1 2020q3 2021q1 2021q3 2022q1 2022q3 2023q1 stnioP egatnecreP 5 0 5- All Shocks Demand + Capacity Shocks Monetary Policy + Capacity Shocks Cost + Capacity Shocks 2020q1 2020q3 2021q1 2021q3 2022q1 2022q3 2023q1 Note: Eachseriesrepresentsthesimulatedpathofconsumerpriceinflation(quarterlyvalue,annualized)for theindicatedsubsetofsmoothedshocksduring2020-2022. Seetextfordefinitionofthecounterfactuals. relaxedtheconstraints. In Figure 16b, we plot a second set of counterfactuals, in which individual shocks are fed into the model in combination with shocks to capacity. In contrast to the prior figure, monetary policy standsouthere. Whilemonetarypolicyshocksplayessentiallynorolein2020,expansionarymonetarypolicyshocksin2021incombinationwithprevailingnegativecapacityshocksledtoasurge in inflation of about 4 percentage points in 2021. Put differently, while negative capacity shocks are insufficient on their own to trigger binding constraints, negative capacity shocks set the stage for demand-side shocks – especially expansionary monetary policy – to trigger the constraints. In turn,asmonetaryshocksdissipatein2021(i.e.,astheFederalReserveraisesinterestratestobring them back in line with the extended Taylor rule), then inflation falls rapidly in 2022 as constraints are relaxed again. In contrast, the role of the demand shock is much more muted in 2021 possibly reflectingthesmallerexpansionaryeffectsoffiscalvs. monetarypolicyinthatyear,butcontinued to stoke inflation in 2022.37 Thus, we conclude that the dynamics of monetary policy during this periodinteractedwithshockstocapacity,arethedrivingforcebehindtherapidriseandsubsequent fallininflationduringthepost-pandemicperiod. As a final set of counterfactuals, recall that there are both domestic and import constraints. In counterfactual results above, we have relaxed these in tandem, but it is natural to wonder what the relativecontributionofeachconstraintistotheresults. Thus,wenowplotcounterfactualsinwhich werelaxoneconstraintatatime,andthenintandem,followingthesameapproachtogeneratethe 37Bothinvarianceandhistoricaldecompositionsweobserveapredominantroleofshockstothediscountratein drivingoutputconsistentwitharealisticfiscalmultiplier. However, in2021thehistoricaldecompositionsforgoods andoveralloutputrevealastrongerroleformonetarythandiscountrateshocks. 41

Figure 17: Counterfactual Consumer Price Inflation With Relaxed Domestic versus Import Constraints stnioP egatnecreP 4 2 0 2- 4- 6- All Shocks All Shocks + Slack Import Goods Constraint All Shocks + Slack Domestic Goods Constraint All Shocks + Both Slack Goods Constraints 2020q1 2020q3 2021q1 2021q3 2022q1 2022q3 Note: Eachseriesrepresentsthesimulatedpathofconsumerpriceinflation(quarterlyvalue,annualized) forallshocksandtheindicatedsetofconstraintsduring2020-2022. Seetextfordefinitionofthe counterfactuals. counterfactuals as in the prior figure. As in Figure 13c, relaxing both constraints evidently lowers the amount of realized inflation resulting from the shocks. The domestic constraint plays a more important role in explaining the joint gap, accounting for roughly three-quarters of the overall impact of constraints. This is to be expected, in that imports account for a minority of overall spending on inputs, which limits the quantitative role of import constraints relative to domestic inputconstraints. Nonetheless,bothconstraintsplayindependentroles. 5 Extensions Inthissection,wepresenttwosetsofadditionalresults,whichprobetherobustnessofourfindings. First, we examine whether our results change when we account more carefully for energy shocks. Second,weenrichthelabormarketstructureofthemodel,toaccountforlabormarketstressduring thepandemic. 5.1 Accounting for Energy Shocks During 2021-2022, global energy prices escalated, as strong demand for energy combined with supply disruptions (e.g., following from the Ukraine war) to drive energy prices up. Further, since the middle of 2022, energy prices have receded rapidly as inflation has cooled. A natural question arises then about whether the dynamics of inflation that we attribute to occasionally binding constraintsmightinsteadbedrivenbytheseenergypricefluctuations. 42

Toframethisdiscussion,notethatourmodelabstractsfromthepeculiarfeaturesofenergymarkets–i.e.,wedonotattempttomodelenergyprices,production,anddemandexplicitly. Therefore, we think it reasonable to estimate our model using data that also excludes energy prices. In part, wehavealreadydonethisinpriorsections,inthatwehavestrippedoutpetroleumandfuelswhen we constructed the price index for imported materials. Here we also remove energy prices from thedomesticpriceindexesusedinestimation–constructingPCEinflationforgoodsandservices, excluding energy. Specifically, we remove prices for “gasoline and other energy goods” (which includesmotorvehiclefuelsandlubricants,fueloil,andotherfuels)fromthegoodsPCEpriceindex, and then we remove prices for electricity and gas utilities from the services PCE price index. Wethenre-estimatethemodelusingthemodifieddomesticpriceindexes. InFigure18a,weplottheadjustedPCEinflationseriesforgoodspricesandoverallconsumption.38 Goods price inflation is virtually indistinguishable with/without energy through 2021:Q3, during the initial inflation takeoff. Thereafter, energy prices push inflation up during early 2022, and then rapidly bring goods price inflation down thereafter. Nonetheless, the basic inverted Ushapeforgoodspriceinflationappearsinbothseries,withnon-energygoodspriceinflationfalling from 8 percent to near zero during the course of 2022. Overall PCE price inflation then reflects thesedeviationsingoodspriceinflation. In Figure 18b, we investigate the role of these differences for our conclusions about the role of constraints in explaining inflation dynamics. The simulations here follow the same scheme as in Section 4.4.3: we compare simulated inflation when all shocks are fed through the model to counterfactual inflation when one or both constraints are relaxed. As in the prior counterfactuals, bindingconstraintscontinue toplaya largequantitativerolein drivinginflation. Onedifferenceis that the foreign constraint plays a somewhat larger role in explain inflation here than in the prior simulation. To understand this, recall that we removed energy prices from import prices in all the analysis above, but we left them in domestic prices. Now, we also remove energy from domestic prices, so this increases measured import price inflation relative to domestic price inflation, and thus the import constraint appears more important. Overall, however, we read these results as confirming that constraints play an important role in explaining inflation, above and beyond any separateimpactsofenergypriceshocks. 5.2 Enriching the Labor Market Motivated by pervasive discussion of labor markets during the pandemic period and recovery, we enrich the labor market of the model in three ways. First, we allow for adjustment frictions for nominalwages,inadditiontopriceadjustmentfrictions. Second,weintroduceshockstothedisu- 38Servicesinflationlooksverysimilarwithandwithoutenergyprices,soweomititforclarityinthefigure. 43

Figure18: AccountingforEnergyShocks (a)PCEInflationwithandwithoutEnergy stnioP egatnecreP 01 8 6 4 2 0 2- 4- 6- (b)CapacityShocksplusIndividualShocks PCE PCE, excluding energy PCE Goods PCE Goods, excluding energy 2020q1 2020q3 2021q1 2021q3 2022q1 2022q3 stnioP egatnecreP 4 3 2 1 0 1- 2- 3- 4- 5- All Shocks All Shocks + Slack Import Goods Constraint All Shocks + Slack Domestic Goods Constraint All Shocks + Both Slack Goods Constraints 2020q1 2020q3 2021q1 2021q3 2022q1 2022q3 Note: InFigure18b,eachseriesrepresentsthesimulatedpathofconsumerpriceinflation(quarterlyvalue, annualized)forallshocksandtheindicatedsetofconstraintsduring2020-2022. Seetextfordefinitionof thecounterfactuals. tility of labor supply, which stand in for various pandemic-related supply shocks (e.g., responses to disease risk, the great resignation, etc.). Third, we incorporate an occasionally-binding constraintonlaborsupply,inadditiontothegoodsmarketcapacityconstraintsconsideredpreviously. Unlike normal times, labor supply constraints plausibly loomed large during the COVID period, where stay-at-home orders, school closures, and other abnormal policies constrained households’ abilitytosupplylabortothemarket. For brevity, we consign the details about this extended model to Appendix B, and we instead focusononekeyresulthere. ThemodelyieldsawagePhillipsCurvethattakestheform: (cid:18) (cid:19) (cid:18) (cid:19) ε −1 ε P π = L [mrs −rw ]+ L 0 µˆ˜ +βE (π ), (32) Wt (cid:100)t (cid:99)t Lt t Wt+1 φ φ W W W 0 where π is nominal wage inflation, mrs is the log of the marginal rate of substitution between Wt t labor supply and consumption in preferences, rw is the log real wage, and µ˜ ≡1+(µ /C −ρ) t Lt Lt t isafunctionofthemultiplieronthelaborconstraint(µ ).39 Lt TwoimportantresultsfollowfrominspectionofEquation32. Thefirst(standard)resultisthat labor (disutility) supply shocks enter the wage Phillips curve via the marginal rate of substitution (mrs ),whereincreaseddisutilityofsupplyinglaborraisesmrs andthuswageinflation. Elsewhere (cid:100)t (cid:100)t inthemodel,increasesinthedisutilityoflaborsupplyalsonaturallylowertheequilibriumquantity oflaboremployedaswell. Thesecond(non-standard)resultisthatbindinglaborconstraintsappear 39Forcompleteness,theparameterε controlssteady-statewagemarkups(thedegreeofmarketpowerexercisedby L workers)andtheparameterφ controlstheflexibilityofwages. SeeAppendixBforthedetailsunderlyingderivation W ofEquation32,andhowitfitsintotheremainderofthemodel. 44

Figure19: ModelFitwithLaborMarketExtensions (a)HoursWorked etatS ydaetS morf noitaiveD goL 0 50.- 1.- 51.- (b)RealWageGrowth Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 stnioP egatnecreP 02 01 0 01- (c) Reduced-Form Wage Markup Shock Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 etatS ydaetS morf noitaiveD goL 80. 60. 40. 20. 0 Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 Note: Wetake1000drawsfromtheposteriordistributionofmodelparameters,computethe Kalman-smoothedvaluesformodelvariablesforeachdraw,andthenplotthemediansmoothedvalueas thesolidline. Weshadetheareacoveringthe5%to95%percentileforsmoothedvalues. InFigure19c,we (cid:16) (cid:17) plotthereducedformlabormarkupshockterm εL P 0 µˆ˜ . φW W 0 Lt asreduced-form“markupshocks”inthewagePhillipsCurve. Asaresult,bindinglaborconstraints driveupwageinflation,conditionalontheotherlabormarketfundamentals. Withtheseresultsinhand,weturntoquantitativeanalysis. Wecalibrateseveralnewparameters (e.g.,ε andφ )basedonexternalreferences. Wethenre-estimatetheextendedmodelalongwith L W stochasticprocessesforlabordisutilityandlaborconstraintshocksusingtwonewobservabledata series: aggregate hours worked and real wage growth, which are constructed using data from the USBureauofLaborStatistics. DetailsonthesestepsareprovidedinAppendixB. Turning to results, we illustrate model fit and smoothed multipliers on the labor constraint in Figure 19. In Figure 19a, there is an obvious dramatic collapse in hours in early 2020:Q2, a rapid partialreboundinQ3,andthenaslowrecoverythereafterthrough2021. Themodelmatchesthese dynamics well, in large part through shocks to labor supply. In addition, Figure 19b illustrates that there were sharp gyrations in real wage growth during the early pandemic. However, real wage growth from 2020:Q4 forward was similar to the pre-pandemic period. Turning to Figure 19c, the model clearly favors a binding labor constraint in 2020:Q2, in order to explain the spike and subsequent collapse in real wage growth. Labor constraints then play a less important role in 2021-2022. The median simulation has a slack or nearly slack labor constraint in most periods, thoughlaborconstraintsdoappeartobindin2022foranon-trivialshareofthesimulations. To evaluate how incorporating labor supply shocks and constraints affect our prior results, we present two sets of counterfactuals.40 First, in Figure 20a, we illustrate how relaxing the goods 40Likepriorcounterfactuals,wedrawformtheposteriortoparameterizethemodelandfiltersmoothedshocksfrom data, and we then simulate responses to subsets of the smoothed shocks under various assumptions about whether constraints bind. Repeating this procedure 1000 times, we report median outcomes in the figures. As a technical matter,weallowgoodsconstraintstobindendogenouslyinallthesesimulations. Thelaborconstraintisathirdcon- 45

Figure20: CounterfactualswithLaborMarketExtensions (a)RelaxingGoodsCapacityorLaborConstraints stnioP egatnecreP 4 2 0 2- 4- (b)MonetaryPolicyShocksandCapacityShocks All Shocks All Shocks + Slack Goods Constraints All Shocks + Slack Labor Constraint All Shocks + Slack Goods & Labor Constraints 2020q1 2020q3 2021q1 2021q3 2022q1 2022q3 stnioP egatnecreP 4 3 2 1 0 1- All Shocks Monetary Policy Shocks Monetary Policy Shocks + Goods Capacity Shocks 2020q1 2020q3 2021q1 2021q3 2022q1 2022q3 Note: Eachseriesrepresentsthesimulatedpathofconsumerpriceinflation(quarterlyvalue,annualized)for the indicated subset of smoothed shocks and constraints during 2020-2022. See text for definition of the counterfactuals. and labor constraints separately and in combination affects inflation. When the labor constraint is assumed to be slack, inflation falls substantially at the outset of the pandemic, which is counterfactual; thus, binding labor constraints help explain the absence of disinflation in 2020. However, theirimpactdissipatesrapidly,suchthatinflationisessentiallysimilaracrossversionsofthemodel withandwithoutlaborconstraintsin2021-2022. Incontrast,assuminggoodsconstraintsareslack has little impact on inflation in 2020, but then inflation would have been significantly lower in 2021-2022 with slack goods constraints (this echoes Figure 13c). Further, we point out that the quantitativeimpactofremovingthegoodsmarketconstraintsisessentiallythesameinthismodel withlaborsupply(disutility)shocksasinthebaselinewithoutthem. Second, in Figure 20b, we investigate again how monetary policy interacts with constraint shocks in this version of the model. In these simulations, the first simulation shuts off all shocks exceptforthemonetarypolicyshocks,andthesecondconsidersthejointimpactofmonetarypolicyshocksandcapacityshocksforbothdomesticandimportedgoods. Asinthepriorsimulations, monetary policy alone has a moderate effect on inflation, while monetary policy combined with capacity shocks lead to a rapid increase in inflation in 2021, sustained high inflation through 2021 into2022,andthenacollapseininflationfrom2022:Q3forward. straint, whichcomplicatessimulation, astheDynareimplementationofOccBinonlyadmitstwopotentiallybinding constraints. Therefore,weimposethelaborconstraintbyassumingthattherearereduced-formwagemarkupshocks, which are tied to the smoothed values of the multiplier on the labor constraint. We then solve for whether the two goodsconstraintsarebindingendogenously. 46

6 Concluding Remarks We have developed a quantitative framework to study inflation that places potentially binding capacity constraints at center stage. We show that binding constraints alter the Phillips Curve relationshipbetweeninflationandrealmarginalcosts,becausefirmstaketheseconstraintsintoaccount whensettingprices. Specifically,whenconstraintsbind,firmssetpricestoequatedemandtotheir constrainedcapacity,ratherthantargetingtheiroptimalunconstrainedmarkupovermarginalcosts. This implies that binding constraints introduce a term that looks like a markup shock in both domestic and import price Phillips Curves. Applying the quantitative framework to interpret recent US data, we find that binding constraints are quantitatively important drivers of inflation, explaining half of the rise in US inflation during 2021-2022. We also find that negative capacity shocks tightened constraints during the pandemic period, which set the stage for modestly-sized demand shocks to have outsized impacts on inflation. In particular, monetary policy shocks loom large in drivinginflationin2021. Goingforward,therearevariousextensionsofthisframeworkthatwouldbeusefultoconsider. While the model includes demand-side (discount rate) shocks that capture important aspects of fiscal policy, it would be useful to extend the model to provide a more careful treatment of fiscal shocks, especially to study the tax policy instruments that supported consumption in the early stagesofthepandemic. Further,wehaveincludedcapacityasanexogenous,stochasticvariablein our framework. We also see high returns to extending the model to include endogenous capacity investment. Lastly, while we have focused on applying the framework to analyze US data in this paper, it would clearly be interesting to parse data for other countries (e.g., the UK and euro area) that experienced similar high inflation episodes. Because energy prices likely played a larger role in these related contexts, we also see value in extending the model to treat energy supply and use morecarefully. Moregenerally,theframeworkwehavedevelopedcanbedeployedtostudyoptimalpolicy,and by extension potential policy mistakes during the pandemic recovery. In our framework, binding constraintsimplythatdemandshocksworkthroughboththeISandPhillipsCurves,appearinglike amarkupshock. Thiswouldappeartocomplicatepolicydesign,relativetocanonicalframeworks in which shocks to the IS and Phillips Curves are unrelated. Further, when reduced-form markups mayreflecteithertheinfluenceofexogenousmarkupshocks,ortheimpactofbindingconstraints, optimal policy will depend on the central bank’s ability to discriminate between them. Given the importance of monetary policy shocks in our quantitative analysis, a critical analysis of policy is warranted. 47

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A Quantitative Model In this appendix, we discuss the quantitative version of the general model described in Section 2. We start by presenting the log-linear approximation of the model equilibrium conditions and the stochastic processes for exogenous variables. We then proceed to present the full details of the calibration and model estimation procedure. We provide supplemental results on parameter estimates,modelfit,androbustnesschecksinlatersections. A.1 Log-Linearization of the Model Equilibrium Conditions In Table 1, we wrote out the full nonlinear model equilibrium. In practice, we solve a piecewise linear approximation to the model, as in Guerrieri and Iacoviello (2015). This entails loglinearizingthemodelequilibriumconditionsforboththeunconstrainedandconstrainedequilibria aroundthesteadystate. We normalize Home prices relative to the domestic price level, and we denote “real” prices withtheletterr attachedtotheprice. Further,lowercasevariableswithhatsdenotelogdeviations from steady state. For example, the log deviation in the real wage from steady state is given by rw = wˆ −pˆ , while the real price of home output in sector s is rp (s) = pˆ (s)−pˆ , and so (cid:99)t t t (cid:98)Ht Ht t on.41 Foreign currency prices (denoted by stars) are normalized relative to the foreign price level; for example, foreign real marginal costs are rmc ∗ (s)=mc ∗ −pˆ∗. We also define deviations in the (cid:100)t (cid:99)t t value of constraints from steady state: yˆ¯ (1)=lnY¯ (1)−lnY¯ (1) and yˆ¯∗(1)=lnY¯∗(1)−lnY¯∗(1). t t 0 t t 0 Finally,toreducethenumberofpotentialforeignshocks,weassumethatforeignexportdemandis givenbyX∗(s)=ϖ(s) (cid:16) P t ∗ (cid:17)−σ(s) C∗,wherewetreat P t ∗ andϖ(s)asconstants,soxˆ∗(s)=cˆ∗. t P∗(s) t P∗(s) t t t t We present the log-linear equilibrium conditions in Tables A.1 and A.2. Table A.1 contains equilibrium conditions that hold in both unconstrained and constrained equilibria. Table A.2 collectsequilibriumconditionsthatdifferacrossequilibria,dependingonwhichconstraintsareslack orbinding. A.2 Stochastic Processes We collect log deviations in exogenous domestic and foreign variables – including Θˆ , ζ ˆ (1), cˆ∗, t t t and (cid:8) zˆ (s),rmc ∗(cid:9) – into vector Fˆ, and we assume that Fˆ is a first-order vector autoregressive t (cid:100)t s t t process,asinFˆ =ΛFˆ +ε ,whereΛisadiagonalmatrixthatcontainsautoregressivecoefficients t t−1 t for each series (denoted λ for variable x) and ε is a vector of shocks.42 We assume the vector of x t 41For completeness, rp(s) = pˆ (s)−pˆ , rp (s) = pˆ (s)−pˆ , rmc(s) = mc(s)−pˆ , rp (s) = pˆ (s)−pˆ , (cid:98)t t t (cid:98)Ft Ft t (cid:100)t (cid:99)t t (cid:98)Mt Mt t rp (s(cid:48),s)=pˆ (s(cid:48),s)−pˆ . (cid:98)Mt Mt t 42Note that we have imposed the restriction that foreign real marginal costs are the same for goods and services: rmc∗(s)=rmc∗. Weestimatethestochasticprocessforthisvariableusingdataforgoodsimports,roughlyspeaking. (cid:100)t (cid:100)t 53

TableA.1: CommonEquilibriumConditionsacrossUnconstrainedandConstrainedEquilibria LaborSupply −ρcˆ +rw =ψlˆ t (cid:99)t t ˆ ˆ Consumption cˆ t (s)=ζ t (s)−ϑr (cid:98) p t (s)+cˆ t with∑ s ζ 0 (s)ζ t (s)=0 cˆ (s)=−ε(s)(rp (s)−rp (s))+cˆ(s) Allocation Ht (cid:98)Ht (cid:98)t t cˆ (s)=−ε(s)(rp (s)−rp (s))+cˆ(s) Ft (cid:98)Ft (cid:98)t t EulerEquation 0=E Θˆ −Θˆ −ρ(E cˆ −cˆ)+i −E π t t+1 t t t+1 t t t t+1 (cid:20) (cid:21) (cid:16) (cid:17)1−ϑ (cid:104) (cid:105) ConsumerPrices 0=∑ s ζ 0 (s) P 0 P ( 0 s) r (cid:98) p t (s)+ 1− 1 ϑ ζ ˆ t (s) (cid:16) (cid:17)1−ε(s) (cid:16) (cid:17)1−ε(s) rp (s)=γ(s) PH0(s) rp (s)+(1−γ(s)) PCF0 (s) rp (s) (cid:98)t P 0(s) (cid:98)Ht P 0(s) (cid:98)Ft LaborDemand rw +lˆ(s)=rmc (s)+yˆ(s) (cid:99)t t (cid:100)t t rp (s)+mˆ (s)=rmc (s)+yˆ(s) (cid:98)Mt t (cid:100)t t mˆ (s(cid:48),s)=−κ(rp (s(cid:48),s)−rp (s))+mˆ (s) InputDemand t (cid:98)Mt (cid:98)Mt t mˆ (s(cid:48),s)=−η(s(cid:48))(rp (s(cid:48))−rp (s(cid:48),s))+mˆ (s(cid:48),s) Ht (cid:98)Ht (cid:98)Mt t mˆ (s(cid:48),s)=−η(s(cid:48))(rp (s(cid:48))−rp (s(cid:48),s))+mˆ (s(cid:48),s) Ft (cid:98)FMt (cid:98)Mt t MarginalCost rmc (s)=−zˆ(s)+(1−α(s))rw (s)+α(s)rp (s) (cid:100)t t (cid:99)t (cid:98)Mt InputPrices r (cid:98) p Mt (s)=∑ s(cid:48) (cid:16) α α (s ( (cid:48) s , ) s) (cid:17)(cid:16) P P 0 M ( 0 s(cid:48) ( , s s ) ) (cid:17)1−κ r (cid:98) p Mt (s(cid:48),s) rp (s(cid:48),s)= (cid:98)Mt ξ(s(cid:48),s) (cid:16) PH0(s(cid:48)) (cid:17)1−η(s(cid:48)) rp (s(cid:48))+(1−ξ(s(cid:48),s)) (cid:16) PMFt(s(cid:48)) (cid:17)1−η(s(cid:48)) rp (s(cid:48)) P 0(s(cid:48),s) (cid:98)Ht P 0(s(cid:48),s) (cid:98)FMt ConsumptionImport π (s)= ε−1(cid:0) rmc∗(s)+qˆ −rp (s) (cid:1) +βE π (s) Ft φ(s) (cid:100)t t (cid:98)Ft t Ft+1 Pricing DomesticPricingfor π (2)= ε−1(rmc (2)−rp (2))+βE π (2) Ht φ(2) (cid:100)t (cid:98)Ht t Ht+1 Services InputImportPricing π (2)= ε−1(cid:0) rmc∗(2)+qˆ −rp (2) (cid:1) +βE π (2) MFt φ(2) (cid:100)t t (cid:98)FMt t FMt+1 forServices (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) yˆ t (s)= C Y H 0 0 ( ( s s ) ) cˆ Ht (s)+∑ s(cid:48) MH Y 0 0 ( ( s s ) ,s(cid:48)) mˆ Ht (s,s(cid:48))+ X Y 0 0 ( ( s s ) ) xˆ t (s) xˆ(s)=−σ(s)(rp (s)−qˆ )+cˆ∗ t (cid:98)Ht t t MarketClearing yˆ C ∗ t (s)=cˆ Ft (s) (cid:16) (cid:17) yˆ∗ Mt (s)=∑ s(cid:48) M Y F ∗ 0( ( s s ,s ) (cid:48)) mˆ Ft (s,s(cid:48)) M0 Θˆ −ρ(cˆ −cˆ∗)+qˆ =0 t t t t (cid:16) (cid:17) ∑ s L 0 L (s) lˆ t (s)=lˆ t 0 i =ρi +ω(1−ρ)πˆ¯ +(1−ρ)ρ yˆ +Ψˆ MonetaryPolicyRule t i t−1 (cid:16) i (cid:17) t i y t t withyˆ t =∑ s P 0(s Y )Y 0(s) yˆ t (s) 0 π (s)=rp (s)−rp (s)+π Ht (cid:98)Ht (cid:98)Ht−1 t AuxiliaryInflation π (s)=rp (s)−rp (s)+π Ft (cid:98)Ft (cid:98)Ft−1 t Definitions π (s)=rp (s)−rp (s)+π FMt (cid:98)FMt (cid:98)FMt−1 t (cid:16) (cid:17)1−ϑ π¯ t =π t +∑ s ζ 0 (s) P 0 P (s) (r (cid:98) p t (s)−r (cid:98) p t−1 (s)) 0 54

TableA.2: EquilibriumConditionswithBindingConstraintsforGoods PanelA:OnlyDomesticConstraintBinds (cid:16) (cid:17) (cid:16) (cid:17) DomesticPricing π (1)= ε−1 (rmc (1)−rp (1))+ ε P 0 µˆ˜ (1)+βE π (1) Ht (cid:16) φ(1) (cid:17) (cid:100)t (cid:98)Ht φ(1)PH0(1) t t Ht+1 InputImportPricing π (1)= ε−1 (cid:0) rmc∗(1)+qˆ −rp (1) (cid:1) +βE π (1) MFt φ(1) (cid:100)t t (cid:98)MFt t FMt+1 DomesticConstraint yˆ(1)=yˆ¯(1)+ln(Y¯ (1)/Y (1)) t t 0 0 PanelB:OnlyForeignConstraintBinds (cid:16) (cid:17) DomesticPricing π (1)= ε−1 (rmc (1)−rp (1))+βE π (1) Ht φ(1) (cid:100)t (cid:98)Ht t Ht+1 (cid:16) (cid:17) (cid:16) (cid:17) InputImportPricing π (1)= ε−1 (cid:0) rmc∗(1)+qˆ −rp (1) (cid:1) + ε P 0 µˆ˜∗(1)+βE π (1) MFt φ(1) (cid:100)t t (cid:98)MFt φ(1)PMF0(1) t t MFt+1 ImportConstraint yˆ∗(1)=yˆ¯∗(1)+ln(Y¯∗(1)/Y∗(1)) t t 0 0 PanelC:BothConstraintsBind (cid:16) (cid:17) (cid:16) (cid:17) DomesticPricing π (1)= ε−1 (rmc (1)−rp (1))+ ε P 0 µˆ˜ (1)+βE π (1) Ht (cid:16) φ(1) (cid:17) (cid:100)t (cid:98)Ht φ(1)P (cid:16)H0(1) t (cid:17) t Ht+1 InputImportPricing π (1)= ε−1 (cid:0) rmc∗(1)+qˆ −rp (1) (cid:1) + ε P 0 µˆ˜∗(1)+βE π (1) MFt φ(1) (cid:100)t t (cid:98)MFt φ(1)PMF0(1) t t MFt+1 DomesticConstraint yˆ(1)=yˆ¯(1)+ln(Y¯ (1)/Y (1)) t t 0 0 ImportConstraint yˆ∗(1)=yˆ¯∗(1)+ln(Y¯∗(1)/Y∗(1)) t t 0 0 shocks has a multivariate normal distribution, with var(ε )=Σ having diagonal elements σ2 for t x eachvariablex andzerosoffdiagonal,andcov(ε ,ε )=0atallleadsandlags(s(cid:54)=0). t t+s Turning to constraints, we assume that the constraint for imports of consumption goods is not binding in all periods. In the first order approximate model, a sufficient condition to guarantee the constraint is never binding is to takeY¯∗(s) to infinity.43 Similarly, we assume that constraints are Ct not binding for services, for which taking Y¯ (2) and Y¯∗ (2) to infinity would be sufficient. This t Mt leaves Y¯ (1) and Y¯∗ (1) as the remaining constraints. We specify a stochastic process for them t Mt here, consistent with how we treat them as exogenous in the model.44 We assume they follow an Becausetheservicessectorisrelativelyclosed,thiscross-sectorrestrictionhaslittlesubstantiveimport. 43In the first order approximation, no decisions depend on the distance between the an endogenous variable and its constrained value. Thus, takingY¯∗(s) to infinity ensures the constraint is always slack, without further indirect Ct consequencesfortheapproximateequilibrium. 44Reliabledataoncapacityathighfrequenciesisgenerallynotavailable,sowecannotincludecapacityamongthe observablevariables. Existingdataoncapacity,suchastheseriescompiledbytheFederalReserveBoardtoproduce itsG.17series,arenotwellsuitedtoourexercise. Oneproblemconcernsdatafrequency. TheFederalReserverelies on underlying survey data collected at an annual frequency, so this data sheds little direct light on the dynamics of capacityathigherfrequencies(monthlyorquarterly). Asecondproblemconcernshowcapacitysurveyquestionsare posedtofirms.Specifically,thesurveyinstrumentasksfirmstoreporthowmuchtheycouldproduceiftheyhadaccess toallthelaborandmaterialstheyneedtoproduce. Thissurveyquestionfailstocapturekeyaspectsofproductionthat effectivelylimittruecapacity. Forexample,firmsmakepredeterminedchoicesaboutessentiallabor,materialinputs, andotheraspectsoftheproductionprocessthatlimittheirabilitytoproducetoday, butthiswouldbenotbepicked up by the survey. Related to these concerns, we note two features of the actual G.17 capacity data. First, capacity utilizationiswellbelow1inhistoricaldata(typicallynear0.75inrecentdata)–takenliterally, capacityconstraints areneverevenclosetobinding,whichseemsimplausible.Further,measuresofcapacityutilizationhavetrendeddown over time, as if firms are carrying more excess capacity now than in the past. This is prima facie inconsistent with auxiliary evidence of decreased slack in other dimensions of the production process (e.g., the rising prevalence of “leanproduction”methods,suchasjust-in-timeinventorymanagement). 55

TableA.3: Calibration Parameter Value Reference/Target ψ 2 Laborsupplyelasticityof0.5 ρ 2 Intertemporalelasticityofsubstitutionof0.5 β .995 Annualrisk-freerealrateof2% ϑ 0.5 Elasticityofsubstitutionacrosssectorsinconsumption ε 4 Elasticityofsubstitutionbetweenvarieties κ 0.3 Elasticityofsubstitutionforinputsacrosssectors σ(s) 1.5 Exportdemandelasticity ToyieldfirstorderequivalencetoCalvopricing, φ 35.468 withaveragepricedurationof4quarters[SimsandWolff(2017)]. autoregressiveprocess: yˆ¯ (1)=ρ yˆ¯ (1)+ε (1) (A.1) t y¯ t−1 y¯t yˆ¯ t ∗(1)=ρ y¯∗ yˆ¯ t ∗ −1 (1)+ε y¯∗t (1), (A.2) whereγ ∈(0,1)andε y¯t (1)andε y¯∗t (1)denotecapacityshocks. Weassumethecapacityshocksare independent,meanzeronormalrandomvariables,withvariancesvar(ε y¯t (1))=σ y¯ 2andvar(ε y¯∗t (1))= σ y¯ 2 ∗ ,andcov(ε y¯t (1),ε y¯,t+s (1))=cov(ε y¯∗t (1),ε y¯∗,t+s (1))=0atallleadsandlags(s(cid:54)=0). A.3 Quantitative Implementation We set parameters for quantitative analysis through a mix of calibration and estimation. We describecalibratedparametersfirst,andthenweprovidedetailsregardingtheestimationprocedure. A.3.1 Calibration We set values for a subset of the structural parameters based on standard values in the literature, which we collect in Table A.3. We use input-output data compiled by the US Bureau of Economic Analysis to pin down values for steady-state expenditure shares. We report these shares, whichreflectmeanvaluesoverthe1997-2018period,inTableA.4,alongwiththeircorresponding definitionsinthemodel. 56

TableA.4: SteadyStateShares ModelandData Description   (cid:16) (cid:17)1−ϑ  ζ 0 (1) P 0 P ( 0 1) = (cid:20) 0.26 (cid:21) Sectorsharesinconsumption  ζ 0 (2) (cid:16) P 0 P (2) (cid:17)1−ϑ 0.74 expenditure 0   (cid:16) (cid:17)1−ε γ(1) PH0(1) (cid:20) 0.80 (cid:21)  P 0(1) = Homesharesinconsumption  γ(2) (cid:16) PH0(2) (cid:17)1−ε 0.995 expenditurebysector P 0(2) (cid:20) (cid:21) (cid:20) (cid:21) α(1) 0.6 = Inputexpenditureshareof α(2) 0.4 grossoutput   (cid:16) (cid:17)(cid:16) (cid:17)1−κ (cid:16) (cid:17)(cid:16) (cid:17)1−κ α(1,1) P 0(1,1) α(1,2) P 0(1,2) (cid:20) (cid:21) 0.70 0.20  α(1) PM0(1) α(2) PM0(2) = Sectorsharesininput (cid:16) α(2,1) (cid:17)(cid:16) P 0(2,1) (cid:17)1−κ (cid:16) α(2,2) (cid:17)(cid:16) P 0(2,2) (cid:17)1−κ 0.30 0.80 expenditure α(1) PM0(1) α(2) PM0(2)   (cid:16) (cid:17)1−η (cid:16) (cid:17)1−η ξ(1,1) PH0(1) ξ(1,2) PH0(1) (cid:20) 0.77 0.84 (cid:21)  P 0(1,1) P 0(1,2) = Homesharesininput  ξ(2,1) (cid:16) PH0(2) (cid:17)1−η ξ(2,2) (cid:16) PH0(2) (cid:17)1−η 0.99 0.98 expenditure P 0(2,1) P 0(2,2) (cid:34)CH0(1) MH0(1,1) MH0(1,2) X 0(1)(cid:35) (cid:20) 0.41 0.32 0.16 0.11 (cid:21) Y 0(1) Y 0(1) Y 0(1) Y 0(1) = Domesticoutputallocation CH0(2) MH0(2,1) MH0(2,2) X 0(2) 0.61 0.07 0.29 0.03 Y 0(2) Y 0(2) Y 0(2) Y 0(2) (cid:34)M Y F M ∗ 0 0 ( ( 1 1 ,1 ) ) M Y F M ∗ 0 0 ( ( 1 1 ,2 ) )(cid:35) = (cid:20) 0.76 0.24 (cid:21) Foreignoutputallocationfor MF0(2,1) MF0(2,2) 0.08 0.92 Y∗ (2) Y∗ (2) inputs M0 M0 (cid:34) PH0(1)Y 0(1) (cid:35) (cid:20) 0.29 (cid:21) PH0(1)Y 0(1)+PH0(2)Y 0(2) = Sectorsharesingrossoutput PH0(2)Y 0(2) 0.71 PH0(1)Y 0(1)+PH0(2)Y 0(2) 57

A.3.2 EstimationProcedure Asdescribedinthemaintext,webuildonpapersbyKulish,MorleyandRobinson(2017),Kulish and Pagan (2017), and Jones, Kulish and Rees (2022) that estimate models with occasionally bindingconstraintsbytreatingthedurationofthosebindingconstraintsasanestimableparameter. Toexplainthemethodinmoredetail,wefirstdiscusshowtosolvethemodelforgivendurationsfor thebindingconstraints. Wethendescribetheestimationprocedureingreaterdetail,includinghow weconstrainadmissiblevaluesfordurationstobeconsistentwithmodelequilibriumconstraints. Solving the Model for Given Durations As in Guerrieri and Iacoviello (2015), we construct a piecewise linear approximation to the model. This consists of taking linear approximations of the model equilibrium for four regimes: the unconstrained regime, a second regime in which only domestic constraints bind, a third regime in which foreign constraints bind, and a fourth regime in which both constraints bind. Further, the linear approximations for all these regimes are taken around the non-stochastic (unconstrained) steady state of the model. The solution procedure then combinestheselocalapproximationstosolveforthepolicyfunction. Thelinearapproximationtotheunconstrainedsystemcanbewrittenas: AX =C+BX +DE X +Fε , t t−1 t t+1 t where x is an n×1 vector of model variables, ε is an l×1 vector of structural shocks, and A, C, t t B, D, and F are conformable matrices determined by the structural equations. If agents expect the economytoremainunconstrainedfromdatet forward,thenstandardrationalexpectationssolution procedures imply that the reduced-form solution is given by: X =J+QX +Gε , where J, Q, t t−1 t andGdescribethepolicyfunctionandmodeldynamics. There are three regimes in which one or both constraints bind, and let us index these by r ∈ {1,2,3}. Thenwecanexpressthelinearapproximationtothemodelequilibriumineachcaseas: A¯ X =C¯ +B¯ X +D¯ E X +F¯ ε , r t r r t−1 r t t+1 r t where A¯ , C¯ , B¯ , D¯ , and F¯ are conformable matrices that correspond to the structural equations r r r r r foreach. We summarize the expected evolution of regimes from a given datet forward by the durations that the individual constraints are expected to bind, as in d =[d ,d∗]. To fix ideas, suppose that t t t d = 1, which means that the domestic constraint binds today, and then is expected to be slack t in the future. Further, suppose that d∗ = 0, so the foreign constraint is slack today and in the t future. This implies that the constrained system governs model responses in periodt and then the 58

unconstrainedsystemappliesthereafter. Workingbackwardsfromtheunconstrainedsolution,then E X =J+QX , so then A¯ X =C¯ +B¯ X +D¯ (J+QX )+F¯ ε , where r =1 is the system t t+1 t 1 t 1 1 t−1 1 t 1 t that applies when the domestic constraint binds and the foreign constraint is slack. Solving this linearequationyieldsthereducedformsolutionforX . t Generalizingthisidea,thesystemwillevolveaccordingto: A X =C +B X +D E X +F ε , (A.3) t t t t t−1 t t t+1 t t where A , C , B , D , and F are the structural matrices that apply at date t. Then the piecewise t t t t t linearsolutionisgivenby: X =J +Q X +G ε , (A.4) t t t t−1 t t where J , Q , and G are determined via the following backward recursion, which is initialized as t t t startingfromtheunconstrainedsolution: Q =[A −D Q ] −1 B t t t t+1 t J =[A −D Q ] −1(C +D J ) (A.5) t t t t+1 t t t+1 G =[A −D Q ] −1 F . t t t t+1 t At this point, it is useful to note that this recursive solution coincides with the recursion employed by the OccBin toolkit [Guerrieri and Iacoviello (2015)] to obtain policy functions for a givenguessaboutthesequenceofregimes. TheOccbintoolkitthenproceedstoverifywhetherthe guess about the sequence of regimes is consistent with model equilibrium, given the current value of the shocks. Put differently, it solves for endogenous constraint durations given ε . While we do t not discuss this second step here, we do solve for endogenous durations (using Occbin) when we analyze counterfactual responses to shocks in the model. We also take the dependence of d on ε t t intoaccountintheestimationprocedure,withdetailsbelow. While Equations A.4 and A.5 present the model solution for a given anticipated sequence of regimes, it is important to note that the anticipated sequence changes as durations evolve over time. Thedurationd impliesaparticularsequenceofregimesanticipatedatdatest+1,t+2,etc. t Given this sequence and the maintained assumption that agents do not anticipate future shocks, one then uses the recursion above to solve for the associated policy matrices: J(d ,θ),Q(d ,θ), t t and G(d ,θ), where the notation captures the dependence of these matrices on d . At datet+1, a t t new value for durations (d ) will be realized, and one then solves the recursion anew to obtain t+1 J(d ,θ),Q(d ,θ), and G(d ,θ). And so on. The state (transition) equation of the model t+1 t+1 t+1 59

thenfeaturestime-varyingcoefficients: X =J(d ,θ)+Q(d ,θ)X +G(d ,θ)ε . (A.6) t t t t−1 t t When d = 0 , the unconstrained solution applies, so J(d ,θ) = J(θ), Q(d ,θ) = Q(θ), and t t t G(d ,θ)=G(θ)aretimeinvariant. t Joint Estimation of Durations and Structural Parameters We assume that a vector of observables(S )arelinkedtounderlyingmodelstatesviathemeasurementequation: S =H X +ν , t t t t t whereν isani.i.d. vectorofnormallydistributedmeasurementerrorsandH isaconformable(pot t tentiallytime-varying)matrixlinkingstatestoobservables. Usingthisstatespacerepresentationof the model, we can apply the Kalman filter to construct the Likelihood function L(θ,d|{S } T ), t t=1 T whered={d} isthesequenceofdurations. t=1 Weputpriorsoverstructuralparametersandindependentpriorsoverdurationstoconstructthe posterior, and then estimate the model via Bayesian Maximum Likelihood. We construct draws (cid:16) (cid:17) T from the joint posterior distribution p θ,d|{S } using a Metropolis-Hastings algorithm with t t=1 twoblocks–oneforthestructuralparameters,whicharecontinuous,andasecondforthediscrete duration parameters – as in Kulish, Morley and Robinson (2017). We use a uniform proposal density for the durations, between 0 (unconstrained) and a sufficiently large maximum duration. WediscussthepriorsinSectionA.3.4below. In evaluating proposed parameter and durations draws, we recognize that it is desirable for posterior estimates of constraint durations to be consistent with agents’ forecasts about how long constraints will endogenously bind given shocks. To this end, we constrain admissible draws to enforce this constraint, in an approximate sense. For a given proposed joint parameter (θi) and duration draw (di), we construct the piecewise linear solution for the model and use the Kalman filter to obtain smoothed structural shocks {ε˜i}T and equilibrium variables {X˜i}T given the t t=1 t t=1 data. At each sample period τ ∈ [1,...,T], we then use the piecewise linear solution to project model outcomes forward given the state and current shock – (cid:0) X˜i ,ε˜i(cid:1) , assuming that there are τ−1 τ no anticipated future shocks.45 We then check for violations of the output capacity constraints. If projected home or foreign output violates the constraints, then we reject the proposed parameter draw as inconsistent with model equilibrium. Otherwise, we accept the parameter draw, evaluate the likelihood, and proceed through the estimation algorithm. Under this procedure, we accept about 25% of the proposed parameter/duration draws, so the estimation proceeds at reasonable computationalpace. 45Recallthatintheabsenceoffutureshocks,agentsanticipatethatthemodelwillreturntotheunconstrainedstate overtime,wherethedurationofbindingconstraintsticksdowntowardzeroineachpassingperiod. Weprojectmodel outcomesforwardusingthisexpectedpathfordurations. 60

In this procedure, note that we reject the proposed draw when it implies that constraints will be violated in expectation. In turn, we accept draws for which constraints are satisfied. Strictly speaking, we do not explicitly check whether the duration d is equal to the endogenous equilib- τ rium duration consistent with (cid:0) X˜i ,ε˜i(cid:1) in the model. Nonetheless, our approach to estimation τ−1 τ provides a good approximation to model outcomes with endogenously binding constraints. To demonstratethisingreaterdetail,weturntosimulationevidence. A.3.3 ValidatingtheEstimationProcedure We provide results for two exercises to evaluate the accuracy of our estimation procedure. First, usingsimulateddata,wedemonstratethatourestimationprocedureiscapableofidentifyinglatent durations. Moreover, we show that it accurately recovers the corresponding multipliers on the constraints. Second, using results from the full estimation of the model with real world data, we comparesmoothedinflationtosimulatedmodelresults. Estimation using Simulated Data The first step is to generate simulated data from the model, forgivenparameters.46 Specifically,wedrawasetofi.i.d. shocksforallvariablesover70quarters, andthenimposeasequenceoflarge,expansionarymonetarypolicyshocksforquarters61to69of the simulation (the size of the monetary policy shocks is chosen to be three standard deviations). Theselargemonetaryshocksaresettotriggerthecapacityconstraints. Sincewecanidentifywhen theconstraintsbindinthesimulateddata,wethusknowthetruesequenceofdurations. We plot several simulated data series in Figure A.1 to illustrate this set up, under both the maintained assumption that constraints are potentially binding and the counterfactual assumption that constraints are slack in all periods. The top two panels contain simulated inflation and the policy interest rate, while the implied durations for domestic and foreign constraints are recorded inthebottomtwopanels. Theexpansionarypolicyshocksevidentlycausethepolicyratetobelow in periods 62 through 70. Further, when capacity constraints bind, inflation more than doubles at itspeakrelativetoasimulationwithoutcapacityconstraints. Treatingthesimulatedseriesasobservabledata,weillustratethatourempiricalmodeliscapable of identifying the true durations by directly examining model likelihood functions. Setting all parameters in the state and observation equations (other than durations) to their true values used to generate the simulated data, we compute the likelihood of the model for different values of the 46In contrast to the main quantitative model, we assume there is zero measurement error, so observable variables are equal to corresponding objects in the simulated data. Further, we set steady-state capacity levels so that there is 4%excesscapacityforbothhomeandforeigngoodsfirms,whichislowerthanthemainmodel. Thistightercapacity implies that we can trigger binding constraints with demand shocks alone (i.e., without negative capacity shocks). Remaining parameters are set to the mode of our baseline estimates, and (as elsewhere) we use Dynare’s OccBin toolboxtosimulatethemodel. 61

FigureA.1: Simulation (a)Inflation stnioP egatnecreP 3 2 1 0 1- (b)SimulationInterestRate Simulation Simulation with Slack Constraints 50 55 60 65 70 stnioP egatnecreP 6 4 2 0 Simulation Simulation with Slack Constraints 50 55 60 65 70 (c)DomesticDurations sdoireP fo rebmuN 3 2 1 0 (d)ForeignDurations 505152535455565758596061626364656667686970 sdoireP fo rebmuN 5 4 3 2 1 0 505152535455565758596061626364656667686970 Note: Inflation is reported at a quarterly rate in percentage points. The interest rate is also reported in percentagepoints. 62

domestic and foreign durations, at given points in time. For example, setting the duration of the foreign constraint to its true value in a given period, we then trace out the likelihood over alternative values of the duration of the domestic constraint. And vice versa. We present the results from period 60, before the constraints become binding, through period 70, when the domestic capacity constraintstopsbindingandtheforeigncapacityconstraintbindsforonemorequarter. FigureA.2plotstheinverseofthelikelihoodvalueacrossdurationsofthedomesticconstraint, where each panel corresponds to a period and the vertical line identifies the true duration. Figure A.3plotsthecorrespondingresultsfortheforeignconstraint. Asbothfiguresillustrate,theinverse likelihood is minimized at the true values in every quarter, which confirms that the likelihood procedureweimplementisabletodiscriminatebetweendurationsofdifferentlength. Importantly, for periods when the constraint does not bind, the likelihood is maximized at a duration value of zero. Turning to estimation of the multipliers, we conduct a full estimation of the model using the simulated data, in which we estimate both the structural parameters and durations, as in the main analysis.47 The posterior distributions for the structural parameters are generally well behaved, andtheirpeakslieclosetothetruevalues. Herewefocusontheestimated(smoothed)multipliers on the capacity constraints, as these play a key role in the framework. In Figure A.4, we plot the true paths for the multipliers in the simulation, along with smoothed multipliers recovered via estimation. As is evident, the smoothed values of the multipliers match the exact simulation values closely, meaning the procedure does a good job at pinning down the reduced-form impact ofconstraintsoninflation. Smoothedvs. SimulatedInflation Drawingonresultspresentedbelowandinthemaintext,we brieflycomparesmoothedinflationoutcomesobtainedviaourestimationprocedurewithoutcomes from the full structural model with endogenously binding constraints. This comparison serves to check that the empirical model with estimated durations replicates the outcomes of the structural model with endogenously binding constraints. Specifically, suppose we feed the structural shocks {ε˜i}T obtainedfromourestimationprocedurethroughthemodel,wherestructuralparametersare t t=1 settotheirmodalvaluesandweusetheOccBinproceduretosolvefortheendogenousdurationof bindingconstraintsineachperiodfollowingtherealizationofshocks. Wethenplotthissimulated inflationseriestothesmoothedinflationseriesfromourestimationinFigureA.5. Asisevident,the two series track each other closely, so we conclude that our approach to capturing endogenously bindingconstraintsintheestimationroutineperformswell. 47Inthisestimation, weallowconstraintstopotentiallybindfortwoquartersbeforethefirstperiodinwhichthey actuallybindinthesimulateddata. Further,weusethesamepriorshereasinthebaselineestimation. 63

FigureA.2: LikelihoodOverDomesticDurations (a)Period60 doohilekiL goL esrevnI 0005- 0205- 0405- 0605- 0805- (b)Period61 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 0894- 0005- 0205- 0405- 0605- 0805- (c)Period62 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 0094- 0594- 0005- 0505- 0015- 0 1 2 3 4 5 6 7 8 Duration (d)Period63 doohilekiL goL esrevnI 0005- 0205- 0405- 0605- 0805- (e)Period64 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 0584- 0094- 0594- 0005- 0505- 0015- (f)Period65 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 0064- 0074- 0084- 0094- 0005- 0015- 0 1 2 3 4 5 6 7 8 Duration (g)Period66 doohilekiL goL esrevnI 0074- 0084- 0094- 0005- 0015- (h)Period67 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 0084- 0094- 0005- 0015- (i)Period68 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 0044- 0064- 0084- 0005- 0025- 0 1 2 3 4 5 6 7 8 Duration (j)Period69 doohilekiL goL esrevnI 0084- 0094- 0005- 0015- (k)Period70 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 0005- 0205- 0405- 0605- 0805- 0 1 2 3 4 5 6 7 8 Duration Note: Theverticaldashedlinemarksthetruedurationoftheconstraintinthesimulationforeachperiod. In somefigures,thedotdenotesavalueoftheinverselikelihoodthatissubstantiallyhigherthantheother valuesplottedinthefigure;thedotislocatedatthemaximalvaluedepictedinthefigureforvisual reference. 64

FigureA.3: LikelihoodOverForeignDurations (a)Period60 doohilekiL goL esrevnI 0605- 5605- 0705- (b)Period61 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 0505- 5505- 0605- 5605- 0705- (c)Period62 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 5505- 0605- 5605- 0705- 0 1 2 3 4 5 6 7 8 Duration (d)Period63 doohilekiL goL esrevnI 9605- 5.9605- 0705- 5.0705- 1705- 5.1705- (e)Period64 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 9605- 5.9605- 0705- 5.0705- 1705- 5.1705- (f)Period65 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 5505- 0605- 5605- 0705- 0 1 2 3 4 5 6 7 8 Duration (g)Period66 doohilekiL goL esrevnI 0205- 0305- 0405- 0505- 0605- 0705- (h)Period67 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 0005- 0205- 0405- 0605- 0805- (i)Period68 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 0505- 5505- 0605- 5605- 0705- 0 1 2 3 4 5 6 7 8 Duration (j)Period69 doohilekiL goL esrevnI 5505- 0605- 5605- 0705- (k)Period70 0 1 2 3 4 5 6 7 8 Duration doohilekiL goL esrevnI 5505- 0605- 5605- 0705- 0 1 2 3 4 5 6 7 8 Duration Note: Theverticaldashedlinemarksthetruedurationoftheconstraintinthesimulationforeachperiod. In somefigures,thedotdenotesavalueoftheinverselikelihoodthatissubstantiallyhigherthantheother valuesplottedinthefigure;thedotislocatedatthemaximalvaluedepictedinthefigureforvisual reference. 65

FigureA.4: MultipliersCapacityConstraints: Simulationvs. Estimation (a)MultiplierontheDomesticConstraint etatS ydaetS morf noitaiveD goL 1 8. 6. 4. 2. 0 (b)MultiplierontheForeignConstraint Simulation Multiplier Median Smoothed Value 5th-95th Percentiles 50 55 60 65 70 etatS ydaetS morf noitaiveD goL 4 2 0 2- Simulation Multiplier Median Smoothed Value 5th-95th Percentiles 50 55 60 65 70 FigureA.5: ComparisonBetweenSmoothedInflationandOccBinSimulatedInflation stnioP egatnecreP 5 0 5- Smoothed PCE Inflation Simulated PCE Inflation using OccBin 2020q1 2020q3 2021q1 2021q3 2022q1 2022q3 2023q1 Note: SmoothedPCEInflationisKalman-smoothedconsumerpriceinflation,wherethefilterisparameterizedusingthemodalvaluesofstructuralparametersanddurationsfromtheempiricalestimation. Simulated PCEInflationusingOccBinisobtainedbysimulatingmodelresponsestosmoothedshocks(seethetextfor furtherdescription). 66

A.3.4 Priors The full set of priors for structural parameters is included in Table A.5. We use standard priors on autoregressivepersistenceofexogenousvariables,parametersinthemonetarypolicyrule,elasticities, and the standard deviations of most structural shocks. We set priors on the persistences of the exogenous capacity shocks that are wider than the priors on the other exogenous variables, as well as wide (uniform) priors on the standard deviations of the capacity shocks, since these are nonstandardparameters. Wesetuniformpriorsonmeasurementerrorsassociatedwiththreeinflationseries–consumer price inflation for goods, imported input price inflation, and imported consumption goods price inflation. Further,lookingforward,wewillreportbelowthattheposteriorestimatesarepushedtowardtheboundaryoftheallowedparameterspacefortheseparameters. Thelogicforconstraining the measurement error parameters in this way is twofold. First, because our focus is on inflation outcomes and the role of constraints in driving them, we want to lean heavily on the realized data here. Second,weestimatethemodelusingbothpre-2020andpost-2020data. Asisevidentinraw data series, the post-2020 COVID period features extreme variability in outcomes relative to the pre-2020 data. One way for the model to make sense of this is to assign very high measurement errors to the data. This is unpalatable from our perspective, as we wish to parse the actual data for this period. Thus, we effectively constrain the model to treat the post-2020 inflation data as an accurate representation of latent unobserved model variables. We envision experimenting with alternatives to this approach (e.g., allowing for different measurement error or shock processes beforeandafter2020),asthinkingabouthowtomodeltheCOVIDperiodevolves. As noted in the main text, we allow constraints to potentially bind only starting in the second quarter of 2020. That is, we put zero mass on positive durations at all dates at/before 2020:Q1, whichcanbethoughtofasadogmaticpriorthatconstraintswerenotsubstantivelyimportantprior to the pandemic. Thereafter in each period, we place equal mass on durations of 0 to 4 quarters, summing to 60% total (12% on each discrete duration). We place 30% mass on durations of 5, 6, 7, and 8 quarters, again equally spread (7.5% each). The remaining 10% mass is spread equally overdurations9through12,andweplacezeromassondurationslongerthan12quarters. A.4 Estimation Results In Table A.5, we provide the mode, mean, and 5th-95th percentiles for the posterior distributions of the structural parameters. As noted in the text, we find that domestic and foreign goods inputs are complements on the production side, while domestic and foreign goods are substitutes in consumption. The Taylor rule coefficient on inflation is near 1.5, which is standard. Interest rates also depend positively on deviations of output from steady state, and the policy rule features 67

FigureA.6: PosteriorDistributionsforConstraintDurations (a)DomesticConstraintDurations 01 5 0 (b)ForeignConstraintDurations 5th-95th Percentiles Mode Mean 2020q2 2020q4 2021q2 2021q4 2022q2 2022q4 01 8 6 4 2 0 5th-95th Percentiles Mode Mean 2020q2 2020q4 2021q2 2021q4 2022q2 2022q4 Note: Ateachdate,thereisaposteriordistributionforconstraintdurations. Eachfigurepresentsthemean, mode,andinterquartilerangeforthisposteriordistribution. a significant degree of inertia. The stochastic processes for shocks generally feature persistence, with auto-regressive coefficients generally between 0.7 and 0.9. Building on the discussion of measurement error above, we note that posterior estimates for measurement errors on consumer goodspriceinflationandimportpriceinflationarepushedtowardtheboundaryoftheirpriordistributions, reflecting tension in the model between fitting data before and during the COVID period. For all the other parameters, posterior distributions are generally well behaved, with single peaks wellinsidetheallowableparameterspaceandreasonablytightdistributions. Turning to duration estimates, we plot statistics for the posterior distributions of domestic and foreign constraint durations in Figure A.6. Due to skewness in the distributions, modal values for the duration (our preferred approach to summarizing the posterior distribution) are below the meanvalueinmostperiods. Thetimepathforthedurationestimatesmimicsthepathofestimated multipliersontheconstraints,asreportedinFigure12. A.5 Model Fit In the main text, we presented results on model fit for core inflation series in Figure 11. To evaluate model fit more broadly, we present data and smoothed values for the remaining observable variablesinFigureA.7.48 Forlegibilityinthefigures,wefocusonthe2017-2022period–thekey period leading up to and through our analysis. The model fits most series well, even capturing the whiplashdynamicsofthedatain2020. ThemodelstrugglestoreplicatedataonUSlaborproduc- 48Whilewetreattheinterestrateasanobservablevariable,weassumeitismeasuredwithouterror,soitisomitted here. 68

TableA.5: PriorandPosteriorDistributionsforStructuralParameters PanelA:ElasticityandTaylorRuleParameters Prior Posterior Parameter Dist Mean SD Mode Mean 5% 95% ConsumptionArmingtonElasticity:ι G 1.5 0.25 1.500 1.469 1.124 1.825 InputArmingtonElasticity:η G 0.5 0.15 0.549 0.563 0.362 0.796 TaylorRuleInflation:ω N 1.5 0.12 1.553 1.545 1.354 1.745 TaylorRuleInertia:αi B 0.75 0.1 0.877 0.872 0.850 0.892 TaylorRuleOutput:αy G 0.12 0.05 0.249 0.246 0.173 0.331 PanelB:StochasticProcesses Prior Posterior Parameter Dist Mean SD Mode Mean 5% 95% PreferenceforGoods:σ IG 1 2 0.314 0.403 0.192 0.642 ζ DiscountRate:σΘ IG 1 2 3.373 3.487 3.120 3.913 ForeignCosts:σrmc∗ IG 1 2 2.194 2.297 1.955 2.717 GoodsProductivity:σz(1) IG 1 2 0.192 0.196 0.128 0.273 ServicesProductivity:σz(2) IG 1 2 0.205 0.206 0.139 0.282 ForeignConstraint:σy¯∗ U 1 0.58 0.079 0.086 0.041 0.143 DomesticConstraint:σy¯ U 1 0.58 0.019 0.025 0.012 0.050 MonetaryPolicyShock:σi IG 1 2 0.153 0.154 0.134 0.176 PreferenceforGoods:ρ B 0.5 0.15 0.825 0.715 0.424 0.906 ζ DiscountRate:ρΘ B 0.5 0.15 0.727 0.727 0.670 0.780 ForeignCosts:ρrmc∗ B 0.5 0.15 0.922 0.917 0.875 0.952 GoodsProductivity:ρz(1) B 0.5 0.10 0.542 0.538 0.371 0.697 ServicesProductivity:ρz(2) B 0.5 0.15 0.911 0.813 0.434 0.946 ForeignConstraint:ρy¯∗ B 0.5 0.20 0.715 0.683 0.454 0.882 DomesticConstraint:ρy¯ B 0.5 0.20 0.914 0.863 0.698 0.957 PanelC:MeasurementError Prior Posterior Parameter Dist Mean SD Mode Mean 5% 95% GoodsPCE:σme IG 1 2 1.014 1.032 0.879 1.194 pceg ServicesPCE:σme IG 1 2 0.664 0.663 0.558 0.773 pces GoodsPCEInflation:σme U 0.25 .14 0.499 0.497 0.492 0.500 π(1) ServicesPCEInflation:σme IG 1 2 0.155 0.163 0.124 0.213 π(2) Imp.InputGoodsExpenditure:σme IG 1 2 3.205 3.253 2.929 3.629 inp Imp.ConsumptionGoodsExpenditure:σme IG 1 2 2.868 2.948 2.659 3.287 finp Imp.InputGoodsInflation:σme U 0.75 .43 1.498 1.487 1.463 1.499 inpp Imp.ConsumptionGoodsInflation:σme U 0.075 0.043 0.145 0.112 0.033 0.148 fimp GoodsProductivity:σme IG 1 2 1.151 1.163 1.012 1.324 prod1 ServicesProductivity:σme IG 1 2 1.053 1.059 0.930 1.203 prod2 IndustrialProduction:σme IG 1 2 0.930 0.961 0.836 1.100 ip AggregateNominalGDP:σme IG 1 2 0.476 0.474 0.397 0.556 nva Note: Gdenotesthegammadistribution,IGdenotestheinversegammadistribution,Udenotestheuniform distribution,Bdenotesthebetadistribution,andNdenotesthenormaldistribution. 69

tivity, particularly in 2020 for services. Through the lens of the model, this implies that the data contains substantial measurement error during the pandemic period, which seems plausible to us. More broadly, a more sensitive treatment of the impact of lockdowns on the services sector would likely be needed to match data in the middle quarters of 2020. Nonetheless, referring back to the main text, the model is able to capture the dynamics of services inflation well overall, particularly in2021-2022wheninflationescalates. Turning to “non-targeted data,” we now compare smoothed values for multipliers attached to the constraints to an external measure of supply chain disruptions. Specifically, we use the Global SupplyChainPressureIndex(GSCPI),developedbytheNewYorkFederalReserve[Benignoetal. (2022)], which combines data on transportation costs (sea and air freight rates) with elements of PurchasingManagers’Indexsurveyspertainingtosupplychainmanagementfrommajorindustrial countries (China, the Eurozone, Japan, United States, etc.). To be clear, this data is not tightly relatedtothetheoreticalconstructthatwerecoverfromthedata;italsoisnotscaledinwaythatis directly comparable to our estimates.49 Further, it is a proxy for global conditions, which doesn’t distinguishbetweenUS-basedandforeignsupplychainconstraints,sowecompareittoaweighted meanofthemedianmultipliersonthedomesticandforeignconstraints. Withallthesecaveats,we plottheGSCPIandtheweightedmeanmultiplierinFigureA.8. Asisevident,boththecomposite multiplierandtheGSCPIindexriseandfallintandem.50 Lastly,inthetext,wenotedthatfluctuationsinthereduced-formmarkupshocksinthePhillips Curves implied by binding constraints do not behave like standard markup shocks estimated from historical data. To illustrate this, we introduce an exogenous markup shock into the domestic and foreign price Phillips Curves of the baseline model, and we assume the markup shocks follow an AR1stochasticprocess. Wethenre-estimatethemodelincludingexogenousmarkupshocksusing only data from 1990:Q1-2019:Q4, under the assumption that constraints are slack throughout this period. Wethenfilterthedatatorecoversmoothedvaluesforthemarkupshocks. InFigureA.9,we plotthemediansmoothedvaluesfortheexogenousmarkupshocksovertheperiod1990-2020. We thenalsoplotthemediansmoothedvaluesforthereduced-formmarkupshocksimpliedbybinding constraintsduringthe2020:Q2-2022:Q4period,obtainedfromtheestimationabove. Asisevident, constraints induce markups shocks that are substantially larger than those that are consistent with 49TherawGSCPIindexisreportedasdeviationsfromitsmeanvalue,inunitsofthestandarddeviationof the series. The NY Fed does not report either the mean or standard deviation, only the summary index, so wecannotcomputelogchangesintheunderlyingindexovertime. Further,economicallyspeaking,thereis noobviousrelationshipbetweenunitsattachedtothemultipliers–whichsummarizeimpactsofconstraints on inflation – and units on the GSCPI. Because the GSCPI is reported at the monthly frequency, we take simplemeansacrossthreemonthintervalstoformquarterlyvalues. 50The largest deviation occurs in early 2020, when the GSCPI index escalates rapidly then falls back. While our estimated multipliers here to not reflect this, we note that the multipliers we recover from the extended model that incorporateslaborsupplyconstraintsandshocksdoesabetterjoboffittingtheseearly-pandemicdynamics. 70

FigureA.7: DataandSmoothedModelObservables (a)GoodsCons. Expenditure stnioP egatnecreP 04 02 0 02- 04- (b)ServicesCons. Expenditure Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 stnioP egatnecreP 04 02 0 02- 04- 06- (c)ImportCons. GoodsExpenditure Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 stnioP egatnecreP 001 05 0 05- 001- Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 (d)ImportGoodsInputExpenditure stnioP egatnecreP 05 0 05- 001- (e)NominalGDP Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 stnioP egatnecreP 04 02 0 02- 04- (f)ImportConsumerGoodsInflation Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 stnioP egatnecreP 6 4 2 0 2- Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 (g)IndustrialProduction stnioP egatnecreP 04 02 0 02- 04- 06- (h)GoodsProductivity Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 stnioP egatnecreP 04 02 0 02- 04- (i)AggregateProductivity Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 stnioP egatnecreP 02 01 0 01- 02- Data Median Smoothed Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 Note: Alldataandsimulatedseriesareannualizedvaluesforde-meanedquarterlygrowthratesin percentagepoints. Dataisrawdata. Wetake1000drawsfromtheposteriordistributionofmodel parameters,computetheKalman-smoothedvaluesformodelvariablesforeachdraw,andthenplotthe mediansmoothedvalueasthedashedline. Weshadetheareacoveringthethe5%to95%percentilefor smoothedvalues. 71

FigureA.8: ComparingtheNYFedGSCPItotheWeightedMeanofMultipliersonDomesticand ForeignConstraints 01.0 80.0 60.0 40.0 20.0 00.0 GSCPI (level) Composite Multiplier Term (log deviation from s.s.) 2017 2018 2019 2020 2021 2022 2023 Note: To make the scale of the GSCPI index comparable to the multiplier, we plot the raw level of (cid:16) (cid:17) the GSCPI index divided by 50. The Composite Multiplier is computed as 0.75 ε P 0 µˆ˜ (s)+ φ(s)PH0(s) t (cid:16) (cid:17) 0.25 ε P 0 µˆ˜∗(s). The weight on the domestic term is 0.75 and the weight on the foreign term is φ(s)PuF0(s) ut 0.25,whichroughlycorrespondtosharesoftotalspendingallocatedtodomesticandforeigngoods. historical data; further, the reduced-form markup shocks are also less persistent than the historical exogenousmarkupprocess. A.6 Estimated Capacity Levels In the preceding (main) model, we calibrated the levels of domestic and foreign goods capacity in steady state. However, we could instead estimate those levels, with an important caveat. The caveat is that we allow constraints to bind only after 2020 in the estimation. The “steady-state” capacity level is the level to which capacity reverts in the long run, in the absence of shocks. We areabletoestimatethislevelconditionalonthedatainperiodsinwhichconstraintsarepotentially binding. Thus, if we estimate capacity levels, we are attempting to infer the capacity level only using post-2020 data. Naturally, since constraints were binding for much of this period, plausibly due to negative shocks that pushed realized capacity down, using only this data will tend to lead ustoestimatearelativelylowlevelforsteady-statecapacity. Andinfact,thisisthatwefindwhen we treat capacity levels as parameters to be estimated: steady state goods capacity is roughly 1% abovethesteadystatelevelofgoodsoutput,whichislowerthanthecalibratedvaluewehaveused previously. Nonetheless, this difference in the level of steady-state capacity has little import for ourquantitativeassessment,aswenotedinthemaintext. To demonstrate this, we provide supplemental figures illustrating results from a version of 72

FigureA.9: ExogenousandReduced-Form(BindingConstraint)MarkupShocks (a)DomesticMarkupShocks etatS ydaetS morf noitaiveD goL 40. 20. 0 20.- (b)ForeignMarkupShocks Reduced-form Markup Induced by Domestic Constraint Exogenous Domestic Markup Shock 1990 1994 1998 2002 2006 2010 2014 2018 2022 etatS ydaetS morf noitaiveD goL 2. 51. 1. 50. 0 50.- Note: Reduced-form Markup Induced by Foreign Constraint Exogenous Foreign Markup Shock 1990 1994 1998 2002 2006 2010 2014 2018 2022 Thesolidlinesdepictreduced-formmarkupshocksinducedbybindingconstraints;thesearethesamedata asinFigures12aand12b. Thedashedlinesareexogenousmarkupshocksobtainedbyestimatingthe modelwithexogenousmarkupshocksandslackconstraintsusingonlydatafor1990:Q1-2019:Q4. Wetake 1000drawsfromtheposteriordistributionofmodelparameters,computetheKalman-smoothedvaluesfor modelvariablesforeachdraw,andthenplotthemediansmoothedvalues. Figure A.10: Counterfactual Consumer Price Inflation in Model with Estimated Steady-State CapacityLevels (a) Aggregate Consumer Price Inflation in Model and Data stnioP egatnecreP 4 2 0 2- 4- (b) Counterfactual Consumer Price Inflation for IndividualShocks Data Median Counterfacual Value 5th-95th Percentiles 2017 2018 2019 2020 2021 2022 2023 stnioP egatnecreP 5 0 5- All Shocks Demand Shocks Monetary Policy Shocks Capacity Shocks Cost Shocks 2020q1 2020q3 2021q1 2021q3 2022q1 2022q3 Note: InFigureA.10a,wetake1000drawsfromtheposteriordistributionofmodelparameters (re-estimatedforthisapplicationincludingestimatedcapacitylevels),computetheKalman-smoothed valuesformodelvariablesforeachdraw,addmeasurementerrortotheobservables,andthenplotthe mediansmoothedvalueasthesolidline. Weshadetheareacoveringthe5%to95%percentilefor smoothedvalues. InFigureA.10b,eachseriesrepresentsthesimulatedpathofconsumerpriceinflation (quarterlyvalue,annualized)fortheindicatedsubsetofsmoothedshocksduring2020-2022. Seetextfor definitionofthecounterfactuals. 73

the model in which capacity is estimated in Figure A.10.51 In Figure A.10a, we replicate the counterfactual in which we relax both the domestic and import goods constraints, as in Figure 13c. In Figure A.10b, we replicate the simulated impact of individual shocks on inflation, as presented in Figure 16. To interpret this figure, we note that these counterfactuals are comparable to those in which we feed individual shocks into the model together with capacity shocks. The reason is that capacity shocks essentially lower the average capacity level to near the estimated steady-state capacity level recovered using only post-2020 data. The results are both qualitatively and quantitatively similar to prior results, which further demonstrates that the core counterfactual resultsarelargelyrobusttothelevelofsteady-statecapacity. 51Wepresentonlytwokeyresultshereforbrevitysake;afullsetofresultsforthismodelisavailableonrequest. 74

B Labor Market Extension This appendix provides details regarding how we extend the model to address the labor market, andthenadditionaldetailsaboutquantitativeimplementationofthisextendedmodel. B.1 Model Extension Referring back the text, we now add sticky wages, potentially binding labor market constraints, and shocks to the disutility of labor to the baseline model. Conveniently, all three extensions can beformalizedbyre-writingtheconsumer-sideofthemodelasfollows. We now assume there is a unit continuum of consumers, indexed by j ∈ (0,1). Consumers are identical, with one exception: each is the monopolistic supplier of its own differentiated labor services to the market. Further, the amount of labor that each consumer is able to supply in a given period is bound above by L , which is exogenous and time varying. Different tiated labor services supplied by consumers are costlessly aggregated into a composite bundle by competitive intermediaries and sold to firms. The labor aggregation technology is given by L = (cid:16) (cid:82)1L (j)(εL −1)/εLdj (cid:17)εL /(εL −1) ,whereε >1istheelasticityofsubstitutionbetweendifferent 0 t L tiatedlaborservicesandthepriceindexforthelaborcompositeisW = (cid:16) (cid:82)1W(j)1−εLdj (cid:17)1/(1−εL ) . t 0 t Finally, each consumer faces pays Rotemberg-type adjustment costs to modify the nominal wage atwhichitsupplieslabor,asinBornandPfeifer(2020). Consumer j chooses its consumption, wage, and asset holdings to maximize utility, subject to itsbudgetconstraint,thedemandcurveforitslabor,andthelaborsupplyconstraint: (cid:34) (cid:35) ∞ (C (j))1−ρ L (j)1+ψ max E 0 ∑β tΘ t t −Λ t t (B.1) {Ct(j),Wt(j),B t+1 (j)} t ∞ =0 t=0 1−ρ 1+ψ (cid:18) (cid:19)2 φ W(j) W t s.t. PC (j)+E [S B (j)]≤B (j)+W(j)L (j)− −1 WL , (B.2) t t t t,t+1 t+1 t t t t t 2 W (j) t−1 (cid:18) W(j) (cid:19)−εL t L (j)= L , and L (j)≤L , (B.3) t t t t W t where φ is a parameter governing wage adjustment costs and Λ governs the time-varying disu- W t tility of labor supply. In a symmetric equilibrium, the first order condition for the optimal wage 75

is:   MRS +(µ /C −ρ) t Lt t 1−ε L1− (cid:16) (cid:17) −φ W (Π Wt −1)Π Wt Wt Pt (cid:34) (cid:35) Θ (cid:18) C (cid:19)−ρ 1 L +E β t+1 t+1 φ (Π −1)Π2 t+1 =0, (B.4) t Θ C Π W Wt+1 Wt+1 L t t t+1 t where µ is the multiplier on the labor constraint, Π ≡ Wt , and MRS = ΛtL t ψ is the marginal Lt Wt W t−1 t C t −ρ rate of substitution between consumption and labor supply in preferences. Further, the comple- (cid:0) (cid:1) mentaryslacknessconditionapplies: L −L µ =0,with µ ≥0. t t Lt Lt Taking a log linear approximation for this equation, we arrive at the wage Phillips Curve presentedinthemaintext: (cid:18) (cid:19) (cid:18) (cid:19) ε −1 ε P π = L [mrs −rw ]+ L 0 µˆ˜ +βE (π ), (B.5) Wt (cid:100)t (cid:99)t Lt t Wt+1 φ φ W W W 0 ˆ where π ≡wˆ −wˆ =rw −rw +π is nominal wage inflation, rw ≡wˆ−pˆ , mrs =λ + Wt t t−1 (cid:99)t (cid:99)t−1 t (cid:99)t t (cid:100)t t ψlˆ−ρcˆ withλ ˆ ≡lnΛ −lnΛ ,andµˆ˜ ≡lnµ˜ −lnµ˜ whereµ˜ ≡1+(µ /C −ρ)isafunction t t t t 0 Lt Lt L0 Lt Lt t ofthemultiplieronthelaborconstraint. To define equilibrium in this model, we modify the equilibrium conditions from Tables A.1 and A.2 as follows. First, we drop the “labor supply” condition from the baseline model, as labor supply is no longer determined by equating the marginal rate of substitution to the real wage. Second, we add the equilibrium conditions specified in Table A.6, where Panel A corresponds to anequilibriumwhenlaborconstraintsareslackatdatet,andPanelBcorrespondstothecasewhen they are binding. The new endogenous variables in the equilibrium system are: {π ,mrs } when Wt (cid:100)t thelaborconstraintareslack(whenµˆ˜ =0),and (cid:8) π ,mrs ,µˆ˜ (cid:9) whenthelaborconstraintbinds. Lt Wt (cid:100)t Lt Combinedwiththegoodsconstraints,thisdefineseightmodelregimeswithdifferentcombinations ofbindingandslackconstraints. B.2 Quantitative Details Starting with calibrated parameters, we set ε =21, following Christiano, Eichenbaum and Evans L (2005). We then choose φ so that the slope of the wage Phillips Curve is equivalent to a Calvo W model with wage adjustment parameter 0.4, when ε =21. This Calvo wage adjustment target is L taken from Fitzgerald et al. (forthcoming), who estimate it based on state-level data. The implied slope of the wage Phillips Curve is then about 0.02, which is relatively flat. We calibrate the level ofthelaborconstraint(L¯ )tobe1%higherthansteadystatelaborsupply. Becausetheactuallevel 0 76

TableA.6: EquilibriumConditionswithBindingConstraintsforLabor PanelA:LaborConstraintisSlack (cid:16) (cid:17) WageSetting π = εL −1 [mrs −rw ]+βE (π ) Wt φW (cid:100)t (cid:99)t t Wt+1 MarginalRateofSubstitution mrs =λ ˆ +ψlˆ −ρcˆ (cid:100)t t t t AuxiliaryInflationDefinition π =rw −rw +π Wt (cid:99)t (cid:99)t−1 t PanelB:LaborConstraintBinds (cid:16) (cid:17) (cid:16) (cid:17) WageSetting π = εL −1 [mrs −rw ]+ εL P 0 µˆ˜ +βE (π ) Wt φW (cid:100)t (cid:99)t φW W 0 Lt t Wt+1 MarginalRateofSubstitution mrs =λ ˆ +ψlˆ −ρcˆ (cid:100)t t t t AuxiliaryInflationDefinition π =rw −rw +π Wt (cid:99)t (cid:99)t−1 t LaborMarketConstraint lˆ =l ˆ¯ +ln(L¯ /L ) t t 0 0 of the constraint at a given point in time is a realization of a stochastic process, results are not sensitivetothisvalue. Weassumethedisutilityoflaborevolvesaccordingtoλ ˆ =ρ λ ˆ +ε ,wherevar(ε )=σ2 t λ t−1 λt λt λ and cov(ε ,ε ) = 0 for s (cid:54)= 0, and we estimate ρ and σ . Further, we assume that the λt λt+s λ λ labor constraint is subject to shocks, such that lnL¯ −lnL¯ ≡ l ˆ¯ = ε with var(ε ) = σ2 and t 0 t l¯t l¯t l¯ (cid:16) (cid:17) cov ε ,ε = 0 for s (cid:54)= 0, and we estimate σ .52 We assume observables (aggregate hours l¯t l¯,t+s l¯ worked and real wage growth) are measured with error and estimate the variance of the measurement errors. We also re-estimate all the same structural parameters and stochastic processes using thisversionofthemodel. Weconstructdataonaggregatehoursworkedandrealwagegrowthfromrawdataprovidedby the US Bureau of Labor Statistics.53 To construct real wage growth, we use hourly compensation dataforthenon-farmbusinesssectortoproxyfornominalwagegrowth(FREDseriesid: COMP- NFB), taking log growth rates of that quarterly index. We then deflate this nominal wage growth usingtheaggregatePCEpriceindex,usedinpriorsections. Tobuildanaggregatehoursseries,we combine several series. We use average weekly hours of production and nonsupervisory works in the private sector (FRED series id: AWHNONAG) to proxy hours per worker. We then compute the ratio of employment (FRED series id: CE16OV) to population (FRED series id: CNP16OV), were we smooth population estimates by taking means within two-year moving windows in order to eliminate jumps due to data revisions. We then multiply average weekly hours by the employmenttopopulationratio,takelogsofthatindex,andcomputedeviationsfromthesamplemeanof the index over the 1992:Q2 to 2019:Q4 (the pre-COVID sample). This provides an index of the 52Generalizing this description, we could alternatively think about the constraint as evolving according to l ˆ¯ = t (cid:16) (cid:17) ρ l¯ l ˆ¯ t−1 +ε l¯t ,whereρ l¯ ∈(0,1),var(ε l¯t )=σ l¯ 2,andcov ε l¯t ,ε l¯,t+s =0fors(cid:54)=0. Ourmodelimplementationcalibrates ρ l¯ =0,becauselowpersistenceintheconstrainthelpsthemodelexplainthedramaticswingsinthedatain2020. 53WeretrievethesedatafromtheFREDdatabase,maintainedbytheFederalReserveBankofSt. Louis: https: //fred.stlouisfed.org/. So,weprovideFREDseriesidentifieshere. 77

leveloflaborovertime. Resultsforthenewmodelparametersforthisextendedmodel,andre-estimationoftheremaining parameters, are included in Table A.7. Parameters estimated previously are little changed, in general. Wefindareasonabledegreeofpersistenceinlaborshocks,withthemodalautocorrelation parameternear0.7. B.3 Additional Results In Figure A.11, we present supplemental counterfactual results for the labor market extension. FigureA.11a containsthe modelresponseto constraintshocks inisolation. The remainingfigures collectmodelresponsestolaborsupply,demand,andcost(productivityandforeigncost)shocksin isolation,aswellascombinedwithsimultaneousconstraintshocks. TheseareanalogoustoFigure 20b in the main text. We note that demand and productivity shocks both depress inflation below zero in isolation and combined with constraint shocks during most of 2020-2021. As a result, labor and monetary policy shocks together account for more than 100% of the overall increase in inflation. However, because shocks interact with one another, we note that it is not correct to simplyaddtheireffectstogethertoassesstheirrelativeimportance. 78

TableA.7: PriorandPosteriorDistributionsforStructuralParameters,LaborMarketExtension PanelA:ElasticityandTaylorRuleParameters Prior Posterior Parameter Dist Mean SD Mode Mean 5% 95% ConsumptionArmingtonElasticity:ι G 1.5 0.25 1.244 1.237 0.940 1.578 InputArmingtonElasticity:η G 0.5 0.15 0.597 0.632 0.405 0.888 TaylorRuleInflation:ω N 1.5 0.12 1.379 1.400 1.214 1.587 TaylorRuleInertia:αi B 0.75 0.1 0.865 0.861 0.836 0.883 TaylorRuleOutput:αy G 0.12 0.05 0.158 0.153 0.106 0.202 PanelB:StochasticProcesses Prior Posterior Parameter Dist Mean SD Mode Mean 5% 95% PreferenceforGoods:σ IG 1 2 0.554 0.552 0.332 0.741 ζ DiscountRate:σΘ IG 1 2 4.166 4.233 3.782 4.724 ForeignCosts:σrmc∗ IG 1 2 2.096 2.129 1.745 2.528 GoodsProductivity:σz(1) IG 1 2 0.143 0.153 0.105 0.209 ServicesProductivity:σz(2) IG 1 2 0.330 0.327 0.266 0.392 ForeignConstraint:σy¯∗ U 1 0.58 0.042 0.056 0.025 0.111 DomesticConstraint:σy¯ U 1 0.58 0.068 0.088 0.049 0.153 MonetaryPolicyShock:σi IG 1 2 0.149 0.151 0.134 0.170 LaborPreference:σ IG 1 2 0.123 0.133 0.093 0.184 λ LaborConstraint:σ l¯ U 1 0.58 0.109 0.546 0.079 1.659 PreferenceforGoods:ρ B 0.5 0.15 0.833 0.789 0.578 0.918 ζ DiscountRate:ρΘ B 0.5 0.15 0.827 0.822 0.791 0.849 ForeignCosts:ρrmc∗ B 0.5 0.15 0.937 0.930 0.891 0.964 GoodsProductivity:ρz(1) B 0.5 0.10 0.556 0.551 0.381 0.707 ServicesProductivity:ρz(2) B 0.5 0.15 0.880 0.868 0.791 0.924 ForeignConstraint:ρy¯∗ B 0.5 0.20 0.903 0.854 0.669 0.962 DomesticConstraint:ρy¯ B 0.5 0.20 0.435 0.441 0.233 0.641 LaborPreference:ρ B 0.5 0.20 0.712 0.709 0.621 0.788 λ PanelC:MeasurementError Prior Posterior Parameter Dist Mean SD Mode Mean 5% 95% GoodsPCE:σme IG 1 2 1.034 1.042 0.887 1.202 pceg ServicesPCE:σme IG 1 2 0.612 0.605 0.530 0.684 pces GoodsPCEInflation:σme IG 1 2 0.779 0.800 0.713 0.898 π(1) ServicesPCEInflation:σme IG 1 2 0.100 0.102 0.079 0.129 π(2) Imp.InputGoodsExpenditure:σme IG 1 2 3.300 3.298 2.971 3.658 inp Imp.ConsumptionGoodsExpenditure:σme IG 1 2 2.864 2.846 2.542 3.167 finp Imp.InputGoodsInflation:σme IG 1 2 2.022 2.043 1.844 2.270 inpp Imp.ConsumptionGoodsInflation:σme IG 1 2 0.210 0.207 0.138 0.281 fimp GoodsProductivity:σme IG 1 2 0.957 0.976 0.865 1.101 prod1 ServicesProductivity:σme IG 1 2 0.468 0.466 0.393 0.546 prod2 IndustrialProduction:σme IG 1 2 0.990 1.010 0.886 1.147 ip AggregateNominalGDP:σme IG 1 2 0.212 0.210 0.150 0.273 nva RealWages:σme IG 1 2 0.982 0.974 0.872 1.088 drw Hours:σme IG 1 2 0.033 0.033 0.027 0.039 l Note: Gdenotesthegammadistribution,IGdenotestheinversegammadistribution,Udenotestheuniform distribution,Bdenotesthebetadistribution,andNdenotesthenormaldistribution. 79

FigureA.11: SupplementalCounterfactualswithLaborMarketExtensions (a)ConstraintShocks stnioP egatnecreP 4 3 2 1 0 1- (b)LaborSupplyShocks All Shocks Capacity Shocks 2020q1 2020q3 2021q1 2021q3 2022q1 2022q3 stnioP egatnecreP 4 3 2 1 0 1- All Shocks Labor Supply Shocks Labor Supply Shocks + Capacity Shocks 2020q1 2020q3 2021q1 2021q3 2022q1 2022q3 (c)DemandShocks stnioP egatnecreP 4 3 2 1 0 1- 2- (d)CostShocks All Shocks Demand Shocks Demand Shocks + Capacity Shocks 2020q1 2020q3 2021q1 2021q3 2022q1 2022q3 stnioP egatnecreP 4 3 2 1 0 1- 2- All Shocks Cost Shocks Cost Shocks + Capacity Shocks 2020q1 2020q3 2021q1 2021q3 2022q1 2022q3 Note: Eachseriesrepresentsthesimulatedpathofconsumerpriceinflation(quarterlyvalue,annualized)for the indicated subset of smoothed shocks and constraints during 2020-2022. See text for definition of the counterfactuals. 80