NBER WORKING PAPER SERIES THE MACROECONOMIC IMPACT OF MICROECONOMIC SHOCKS: BEYOND HULTEN'S THEOREM David Rezza Baqaee Emmanuel Farhi Working Paper 23145 http://www.nber.org/papers/w23145 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 February 2017 We provide a nonlinear characterization of the macroeconomic impact of microeconomic productivity shocks in terms of reduced-form non-parametric elasticities for efficient economies. We also show how microeconomic parameters are mapped to these reduced-form general equilibrium elasticities. In this sense, we extend the foundational theorem of Hulten (1978) beyond the first order to capture nonlinearities. Key features ignored by first-order approximations that play a crucial role are: structural microeconomic elasticities of substitution, network linkages, structural microeconomic returns to scale, and the extent of factor reallocation. In a business-cycle calibration with sectoral shocks, nonlinearities magnify negative shocks and attenuate positive shocks, resulting in an aggregate output distribution that is asymmetric (negative skewness), fat-tailed (excess kurtosis), and a has negative mean, even when shocks are symmetric and thin-tailed. Average output losses due to short-run sectoral shocks are an order of magnitude larger than the welfare cost of business cycles calculated by Lucas (1987). Nonlinearities can also cause shocks to critical sectors to have disproportionate macroeconomic effects, almost tripling the estimated impact of the 1970s oil shocks on world aggregate output. Finally, in a long-run growth context, nonlinearities, which underpin Baumol’s cost disease via the increase over time in the sales shares of low-growth bottleneck sectors, account for a 20 percentage point reduction in aggregate TFP growth over the period 1948-2014 in the US. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications. © 2017 by David Rezza Baqaee and Emmanuel Farhi. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source. The Macroeconomic Impact of Microeconomic Shocks: Beyond Hulten's Theorem David Rezza Baqaee and Emmanuel Farhi NBER Working Paper No. 23145 February 2017 JEL No. E01,E1,E23,E32,L16 ABSTRACT We provide a nonlinear characterization of the macroeconomic impact of microeconomic productivity shocks in terms of reduced-form non-parametric elasticities for efficient economies. We also show how microeconomic parameters are mapped to these reduced-form general equilibrium elasticities. In this sense, we extend the foundational theorem of Hulten (1978) beyond the first order to capture nonlinearities. Key features ignored by first-order approximations that play a crucial role are: structural microeconomic elasticities of substitution, network linkages, structural microeconomic returns to scale, and the extent of factor reallocation. In a business-cycle calibration with sectoral shocks, nonlinearities magnify negative shocks and attenuate positive shocks, resulting in an aggregate output distribution that is asymmetric (negative skewness), fat-tailed (excess kurtosis), and a has negative mean, even when shocks are symmetric and thin-tailed. Average output losses due to short-run sectoral shocks are an order of magnitude larger than the welfare cost of business cycles calculated by Lucas (1987). Nonlinearities can also cause shocks to critical sectors to have disproportionate macroeconomic effects, almost tripling the estimated impact of the 1970s oil shocks on world aggregate output. Finally, in a long-run growth context, nonlinearities, which underpin Baumol’s cost disease via the increase over time in the sales shares of low-growth bottleneck sectors, account for a 20 percentage point reduction in aggregate TFP growth over the period 1948-2014 in the US. David Rezza Baqaee UCLA 315 Portola Plaza Los Angeles [email protected] Emmanuel Farhi Harvard University Department of Economics Littauer Center Cambridge, MA 02138 and NBER [email protected] 1 Introduction ffi The foundational theorem of Hulten (1978) states that for e cient economies and under minimal assumptions, the impact on aggregate TFP of a microeconomic TFP shock is equal totheshockedproducer’ssalesasashareofGDP: (cid:88) = λ , dlogTFP dlogA i i i λ wheredlogA isashocktoproduceriand isitssalesshareorDomarweight. i i Hulten’s theorem is a cornerstone of productivity and growth accounting: it shows how to construct aggregate TFP growth from microeconomic TFP growth, and provides structurally-interpretabledecompositionsofchangesofnationalorsectoralaggregatesinto the changes of their disaggregated component industries or firms. It also provides the benchmark answers for counterfactual questions in structural models with disaggregated production. The surprising generality of the result has led economists to de-emphasize the role of microeconomic and network production structures in macroeconomic models. After all, if sales summarize the macroeconomic impact of microeconomic shocks and we can directlyobservesales,thenweneednotconcernourselveswiththedetailsoftheunderlying disaggregated system that gave rise to these sales. Since it seems to imply that the very object of its study is irrelevant for macroeconomics, Hulten’s theorem has been something ofabugbearfortheburgeoningliteratureonproductionnetworks. Aretheseconclusionswarranted? Evenatapurelyintuitivelevel,therearereasonstobe skeptical. Take for example shocks to Walmart and to electricity production. Both Walmart and electricity production have a similar sales share of roughly 4% of U.S. GDP. It seems natural to expect that a large negative shock to electricity production would be much more damaging than a similar shock to Walmart. Indeed, this intuition will be validated by our formalresults. YetitgoesagainstthelogicofHulten’stheoremwhichimpliesthat,because the two sectors have the same Domar weight, the two shocks should have the same impact onaggregateoutput. In this paper, we challenge the view that the macroeconomic importance of a microe- conomic sector is summarized by its sales share and, more broadly, the notion that the microeconomic details of the production structure are irrelevant for macroeconomics. The key is to recognize that Hulten’s theorem only provides a first-order approximation. Non- linearities can significantly degrade the quality of the first-order approximation for large enoughshocks. Tocapturethesenonlinearities,weprovideageneralsecond-orderapprox- imation by characterizing the derivatives of Domar weights with respect to shocks. The 2 second-order terms are shaped by the microeconomic details of the disaggregated produc- tion structure: network linkages, microeconomic elasticities of substitution in production, microeconomicreturnstoscale,andthedegreetowhichfactorscanbereallocated. ffi Our results are general in that they apply to any e cient general equilibrium econ- omy. TheysuggestthatCobb-Douglasmodels,commonlyusedintheproduction-network, growth, and multi-sector macroeconomics literatures, are very special: the Domar weights, and more generally the whole input-output matrix, are constant and can be taken to be exogenous,thefirst-orderapproximationisexact,themodelislog-linear,andasaresult,the microeconomic details of the production structure are irrelevant.1 These knife-edge prop- ertiesdisappearas soonasone deviatesfromCobb-Douglas: theDomarweights andmore generallythewholeinput-outputmatrixrespondendogenouslytoshocks,andtheresulting nonlinearitiesareshapedbythemicroeconomicdetailsoftheproductionstructure. We also show that nonlinearities in production matter quantitatively for a number of ff macroeconomic phenomena operating at di erent frequencies, ranging from the role of sectoralshocksinbusinesscyclestotheimpactofoilshocksandtheimportanceofBaumol’s costdiseaseforlong-rungrowth: 1. Usingacalibratedstructuralmulti-industrymodelwithrealisticcomplementaritiesin production,wefindthatnonlinearitiesamplifytheimpactofnegativesectoralshocks andmitigatetheimpactofpositivesectoralshocks.2,3 Largenegativeshockstocrucial ff industries, like “oil and gas”, have a significantly larger negative e ect on aggregate outputthannegativeshockstolargerbutlesscrucialindustriessuchas“retailtrade”. Nonlinearities also have a significant impact on the distribution of aggregate output: theyloweritsmeanandgeneratenegativeskewnessandexcesskurtosiseventhough the underlying shocks are symmetric and thin tailed. Nonlinearities in production . . generate significant welfare costs of sectoral fluctuations, ranging from 02% to 13% dependingonthecalibration. Theseareanorderofmagnitudelargerthanthewelfare 1Amixtureofanalyticaltractability,aswellasbalanced-growthconsiderations,havemadeCobb-Douglas the canonical production function for networks (Long and Plosser, 1983), multisector RBC models (Gomme and Rupert, 2007), and growth theory (Aghion and Howitt, 2008). Recent work by Grossman et al. (2016) showshowbalancedgrowthcanoccurwithoutCobb-Douglas. 2Theempiricalliteratureonproductionnetworks, likeAtalay(2017), Boehmetal.(2017), andBarrotand Sauvagnat(2016)allfindthatstructuralelasticitiesofsubstitutioninproductionaresignificantlybelowone, andsometimesveryclosetozero,acrossintermediateinputs,andbetweenintermediateinputsandlaborat business cycle frequencies. Furthermore, a voluminous literature on structural transformation, building on Baumol (1967), has found evidence in favor of non-unitary elasticities of substitution in consumption and productionacrosssectorsoverthelong-run. 3While complementarities prevail at the sectoral level, substitutabilities dominate across firms within sectors. Thisimpliesthatwhilenonlinearitiestendtoamplifynegativesectoral-levelshocksandtoattenuate positivesectoral-levelshocks,theytendtoattenuatenegativefirm-levelshocksandtoamplifypositivefirm- level shocks. Nonlinearities therefore introduce an important qualitative difference between sectoral- and firm-levelshockswhichisabsentfromthelinearizedperspective. 3 costs of business cycles arising from nonlinearities in utility (risk aversion) identified byLucas(1987). 2. We derive and use a simple nonparametric formula, taking into account the observed changeintheDomarweightforcrudeoil,toanalyzetheimpactoftheenergycrisisof the1970suptothesecondorder. Wefindthatnonlinearitiesalmosttripledtheimpact . . oftheoilshocksfrom023%to061%ofworldaggregateoutput. 3. We show that the nonlinearities are also important for long-run growth in the pres- ence of realistic complementarities across sectors. They cause the Domar weights of bottleneck sectors with relatively low productivity growth to grow over time and ff thereby reduce aggregate growth, an e ect identified as Baumol’s cost disease (Bau- mol,1967). WecalculatethatnonlinearitieshavereducedthegrowthofaggregateTFP by20percentagepointsovertheperiod1948-2014intheUS.4 Theoutlineofthepaperisasfollows. InSection2,wederiveageneralformuladescribing ffi thesecond-orderimpactonaggregateoutputofshocksintermsofnon-parametricsu cient statistics: reduced-form general-equilibrium elasticities of substitution and input-output multipliers.5 Weexplaintheimplicationsofthisformulafortheimpactofcorrelatedshocks andfortheaverageperformanceoftheeconomy. InSection3,weusetwospecialillustrative examples to provide some intuition for the roles of the general-equilibrium elasticities of substitutionandoftheinput-outputmultipliersandfortheirdependenceonmicroeconomic primitives. InSection4,wefullycharacterizesecond-ordertermsintermsofmicroeconomic primitives for general nested-CES economies with arbitrary microeconomic elasticities of substitutionandnetworklinkages. InSection5,wefurthergeneralizetheresultstoarbitrary (potentially non-CES) production functions. In Section 6, we provide some illustrations of thequantitativeimplicationsofourresults. Relatedliterature. Gabaix(2011)usesHulten’stheoremtoarguethattheexistenceofvery large, or in his language granular firms, can be a possible source of aggregate volatility. If 4Theliteratureonstructuraltransformationemphasizestwokeyforces: non-unitaryelasticitiesofsubstitu- tionandnon-homotheticities. Bothforcescausesalessharestochangeinresponsetoexogenousshocks. Since Hulten’stheoremimpliesthatsalessharesareequaltoderivativesoftheaggregateoutputfunction,anything thatcausesthederivativetochangeisanon-linearity. Bycharacterizingthesecond-ordertermsinageneral way,ourresultsencompassbothnon-unitaryelasticitiesandnon-homotheticities. Infact,non-homotheticities canalwaysbeturnedintonon-unitaryelasticitiesofsubstitutionbyaddingmorefixed-factorstoaneconomy. 5Studyingthesecond-ordertermsisthefirststepingrapplingwiththenonlinearitiesinherentinmultisector modelswithproductionnetworks. Inthissense,ourworkillustratesthemacroeconomicimportanceoflocal andstronglynonlinearinteractionsemphasizedinreducedformbyScheinkmanandWoodford(1994). Other related work on nonlinear propagation of shocks in economic networks includes Durlauf (1993), Jovanovic (1987),Ballesteretal.(2006)Acemogluetal.(2015),Elliottetal.(2014),andespeciallyAcemogluetal.(2016). 4 there exist very large firms, then shocks to those firms will not cancel out with shocks to much smaller firms, resulting in aggregate fluctuations. Acemoglu et al. (2012), working with a Cobb-Douglas model in the spirit of Long and Plosser (1983), observed that in an economywithinput-outputlinkages,theequilibriumsizesoffirmsdependontheshapeof the input-output matrix. Central suppliers will be weighted more highly than peripheral firms,andtherefore,shockstothosecentralplayerswillnotcanceloutwithshockstosmall firms.6 Carvalho and Gabaix (2013) show how Hulten’s theorem can be operationalized to decomposethesectoralsourcesofaggregatevolatility.7 Relatedly, Acemoglu et al. (2017) deploy Hulten’s theorem to study other moments of the distribution of aggregate output. They argue that if the Domar weights are fat- tailed and if the underlying idiosyncratic shocks are fat-tailed, then aggregate output can ff exhibitnon-normalbehavior. Stateddi erently,theyshowthataggregateoutputcaninherit tail risk from idiosyncratic tail risk if the distribution of the Domar weights is fat-tailed. Our paper makes a related but distinct point. We find that, for the empirically relevant rangeofparameters,theresponseofaggregateoutputtoshocksissignificantlyasymmetric. Therefore, the nonlinearity inherent in the production structure can turn even symmetric thin-tailed sectoral shocks into rare disasters endogenously. This means that the economy could plausibly experience aggregate tail risk without either fat-tailed shocks or fat-tailed Domarweights. In a recent survey article Gabaix (2016), invoking Hulten’s theorem, writes “networks are a particular case of granularity rather than an alternative to it.” This has meant that ffi researchers studying the role of networks have either moved away from e cient models, or that they have retreated from studying aggregate output and turned their attention to themicroeconomicimplicationsofnetworks,namelythecovarianceoffluctuationsbetween differentindustriesandfirms.8,9However,ourpapershowsthatexceptinveryspecialcases, ff modelswiththesamesalesdistributionsbutdi erentnetworkstructuresonlyhavethesame 6ArelatedversionofthisargumentwasalsoadvancedbyHorvath(1998),whoexploredthisissuequanti- tativelywithamoregeneralmodelinHorvath(2000). Separately,Carvalho(2010)alsoexploreshowthelaw oflargenumbersmayfailundercertainconditionsontheinput-outputmatrix. 7ResultsrelatedtoHulten’stheoremarealsousedininternationaltrade,e.g. BursteinandCravino(2015), toinfertheglobalgainsfrominternationaltrade. 8Some recent papers have investigated aggregate volatility in production networks with inefficient equi- libria(whereHulten’stheoremdoesnothold). SomeexamplesincludeBigioandLa’O(2016),Baqaee(2018), Altinoglu(2016),Grassi(2017),andBaqaeeandFarhi(2018c). SeealsoJones(2011),Jones(2013),Bartelmeand Gorodnichenko(2015),andLiu(2017). 9Otherpapersinvestigatetheimportanceofidiosyncraticshockspropagatingthroughnetworkstogenerate cross-sectionalcovariances,butrefrainfromanalyzingaggregateoutput. SomeexamplesincludeFoersteretal. (2011),Atalay(2017),DiGiovannietal.(2014),andStella(2015),andBaqaeeandFarhi(2018b). Atalay(2017) is particularly relevant in this context, since he finds that structural elasticities of substitution in production play a powerful role in generating covariance in sectoral output. Our paper complements this analysis by focusinginsteadonthewaycomplementaritiesaffectaggregateoutput. 5 aggregate-output implications up to a fragile first-order of approximation. Their common ff salesdistributionproducethesamelinearization,buttheirdi erentnetworkstructureslead ff to di erent nonlinearities. Hence, in the context of aggregate fluctuations, networks are neither a particular case of granularity nor an alternative to it. It is simply that the sales ffi distributionisasu cientstatisticforthenetworkatthefirstorderbutnotathigherorders. 2 General Framework In this section, we set up a non-parametric general equilibrium model to demonstrate both Hulten’s theorem as well as our second-order approximation. Final demand is represented asthemaximizerofaconstant-returnsaggregatoroffinaldemandforindividualgoods = D ,..., , Y max (c c ) 1 N {c1,...,cN} subjecttothebudgetconstraint (cid:88)N (cid:88)F (cid:88)N = + π, pc w l i i f f i i=1 f=1 i=1 π wherec istherepresentativehousehold’sconsumptionofgoodi,p isthepriceand isthe i i i profit of producer i, w is the wage of factor f which is in fixed supply l . The two sides of f f the budget constraint coincide wtih nominal GDP, using respectively the final expenditure andincomeapproaches. Eachgoodiisproducedbycompetitivefirmsusingtheproductionfunction = ,..., , ,..., , y AF(l l x x ) i i i i1 iF i1 iN where A is Hicks-neutral technology, x are intermediate inputs of good j used in the i ij productionofgoodi,andl islabortype f usedbyi. Theprofitsearnedbytheproducerof if goodiare (cid:88)F (cid:88)N π = − − . p y w l p x i i i f if j ij f=1 j=1 ≤ ≤ ≤ ≤ Themarketclearingconditionsforgoods1 i N andfactors1 f Fare (cid:88)N (cid:88)N = + = . y x c and l l i ji i f if j=1 i=1 6 Competitive equilibrium is defined in the usual way, where all agents take prices as given, andmarketsforeverygoodandeverytypeoflaborclears. We interpret Y as a cardinal measure of (real) aggregate output and note that it is the correct measure of the household’s “standard of living” in this model. We implicitly rely on the existence of complete financial markets and homotheticity of preferences to ensure the existence of a representative consumer. Although the assumption of a representative consumer is not strictly necessary for the results in this section, it is a standard assumption in this literature since it allows us to unambiguously define and measure changes in real aggregateoutputwithoutcontendingwiththeissueoftheappropriatepriceindex. We assume that the production function F of each good i has constant returns to scale, i which implies that equilibrium profits are zero. This assumption is less restrictive than it may appear because decreasing returns to scale can be captured by adding fixed factors to which the corresponding profits accrue.10 A similar observation applies to the assumption thatshocksareHicks-neutral: wecanrepresentaproductivityshockaugmentingaspecific inputbyaddinganewproducerthatproducesthisinputandhittingthisnewproducerwith a Hicks-neutral shock.11 Note also that although we refer to each producer as producing one good, our framework actually allows for joint production by multi-product producers: (cid:48) for example, to capture a producer i producing goods i and i using intermediate inputs (cid:48) andfactors,werepresentgoodi asaninputenteringnegativelyintheproductionandcost functions for good i.12 Finally, note that goods could represent different varieties of goods ff ff from the same industry, goods from di erent industries, or even goods in di erent time periods,regions,orstatesofnature.13 ,..., Define Y(A A ) to be the equilibrium aggregate output as a function of the exoge- 1 N nous technology levels. Throughout the paper, and without loss of generality, we derive ff resultsregardingthee ectsofshocksinthevicinityofthesteadystate,whichwenormalize 10Ourformulascanalsoinprinciplebeappliedwithincreasing-returnstoscaleunderthejointassumption ofmarginal-costpricingandimpossibilityofshuttingdownproduction,bysimplyaddingproducer-specific fixedfactorswithnegativemarginalproductsandnegativepayments(thesefactorsare“bads”thatcannotbe freelydisposedof). 11Shocks to the composition of demand can be captured in the same way via a set of consumer-specific productivity shocks. For example, if the final demand aggregator is CES with an elasticity strictly greater thanone,anincreaseinconsumerdemandforicanbemodeledasapositiveconsumer-specificproductivity shock to i and a set of negative consumer-specific productivity shocks to all other final goods such that the consumption-share-weightedsumof theshocksis equaltozero. Thesignof theshocksmust bereversedif the elasticity of substitution is strictly lower than one, and the Cobb-Douglas case can be treated as a limit. TheseconstructionsgeneralizebeyondtheCEScase. Hulten’stheoremimpliesthatshockstothecomposition ofdemandhavenofirst-ordereffectonaggregateoutput,butingeneral,theyhavenonzerosecond-order(and moregenerallynonlinear)effects. 12Tosatisfactorilycapturesuchfeatures,oneprobablyneedstogobeyondthenested-CEScaseofSection4 anduseinsteadthenon-parametricgeneralizationtoarbitraryeconomiesprovidedinSection5. 13Ifweapplythemodeltodifferentperiodsoftimeandstatesofnature,thenYcorrespondstoanintertem- poralaggregateconsumptionindexreflectingintertemporalwelfare. 7 ,..., = ,..., tobe(A A ) (1 1). Alltherelevantderivativesareevaluatedatthatpoint. 1 N Theorem1(Hulten1978). Thefirst-ordermacroeconomicimpactofmicroeconomicshocksisgiven by:14 dlogY = λ, (1) i dlogA i (cid:80) whereλ = p y/( N p c )thesalesofproduceriasafractionofGDPorDomarweight. i i i j=1 j j Hulten’s theorem can be seen as a consequence of the first welfare theorem: since this ffi ,..., economy is e cient, Y(A A ) is also the social planning optimum and prices are the 1 N ff multipliers on the resource constraints for the di erent goods. Applying the envelope theoremtothesocialplanningproblemdeliverstheresult. Hulten’s theorem has the powerful implication that, to a first-order, the underlying mi- croeconomicdetailsofthestructuralmodelarecompletelyirrelevantaslongasweobserve theequilibriumsalesdistribution: theshapeoftheproductionnetwork,themicroeconomic elasticities of substitution in production, the degree of returns to scale, and the extent to whichinputsandfactorscanbereallocated,areallirrelevant. ff We now provide a characterization of the second-order e ects in terms of reduced- form elasticities. We need to introduce two objects: GE elasticities of substitution, and the input-output multiplier. Later on, we show how these reduced-form elasticities arise from structuralprimitivesusingastructuralmodel. We start by introducing the GE elasticities of substitution. Recall that for any homoge- ,..., neous of degree one function f(A A ), the Morishima (1967) elasticity of substitution 1 N is / 1 = dlog(MRSji) = dlog(fj fi) , ρ / / dlog(A A ) dlog(A A ) ji i j i j whereMRS istheratioofpartialderivativeswithrespecttoA andA ,and f = d f/dA.15 ij i j i i Whenthehomotheticfunction f correspondstoaCESutilityfunctionandA toquantities, i ρ is the associated elasticity of substitution parameter. However, we do not impose this ij interpretation, and instead treat this object as a reduced-form measure of the curvature of isoquants. By analogy, we define a pseudo elasticity of substitution for non-homothetic functionsinasimilarfashion. 14InthespecialcasewhereA isafactor-augmentingshock,therelevantλ correspondstoaproducer’sbill i i forthisfactorasashareofGDP.Thisisbecauseifwerelabelthelaborinputofproduceriasanewproducer, wecanrepresentafactor-augmentingshocktoi’slaborasaHicks-neutralshocktothisnewproducer. 15This is a generalization of the two-variable elasticity of substitution introduced by Hicks (1932) and analyzedindetailbyBlackorbyandRussell(1989). 8 Definition1. Foranysmoothfunction f : RN → R,thepseudoelasticityofsubstitutionis / 1 ≡ dlog(MRSji) = dlog(fj fi). ρ dlogA dlogA ji i i The pseudo elasticity of substitution is a generalization of the Moroshima elasticity of substitutioninthesensethatwhenever f ishomogenousofdegreeone,thepseudoelasticity isthesameastheMoroshimaelasticityofsubstitution. When applied to the equilibrium aggregate output function of a general equilibrium economy,wecallthepseudoelasticityofsubstitutionthegeneralequilibriumpseudoelasticity ρ of substitution or GE elasticity of substitution for short. The GE elasticity of substitution is ji interesting because it measures changes in the relative sales shares of j and i when there is anexogenousshocktoi. Thisfollowsfromthefactthat λ/λ / / dlog( i j) = dlog[(YiAi) (YjAj)] = + dlog(Yi Yj) = − 1 , 1 1 ρ dlogA dlogA dlogA i i i ji wherethefirstequalityappliesHulten’stheorem. Adecreaseintheproductivityoficauses λ/λ ρ ∈ , to increase when (0 1), and to decrease otherwise. We say that a j is a GE- i j ji ρ ∈ , complement for i if (0 1), and a GE-substitute otherwise. When f is a CES aggregator, ji thiscoincideswiththestandarddefinitionofgrosscomplementsandsubstitutes. Asusual, when f is Cobb-Douglas, i and j are neither substitutes nor complements. In general, GE-substitutabilityisnotreflexive. An important special case is when the shock dlogA hits the stock of a factor. In that f (cid:80) case,Hulten’stheoremimpliesthatdlogY/dlogA = Λ ,whereΛ = w l /( N p c )isthe f f f f f j=1 j j (cid:80) shareoffactor f inGDP.Since F w l = GDP,Euler’stheoremimpliesthattheaggregate f=1 f f output is homogenous of degree one in the supplies of the factors. This implies that the generalequilibriumpseudoelasticityofsubstitutionbetweentwofactorscanbeinterpreted asagenuineelasticityofsubstitutionbetweenthesefactorsingeneralequilibrium.16 Next,weintroducetheinput-outputmultiplier. Definition2. Theinput-outputmultiplieris (cid:88)N dlogY (cid:88)N ξ ≡ = λ. i dlogA i=1 i i=1 16Thedifferencebetweenanelasticityofsubstitutionandapseudoelasticityofsubstitutionisthattheformer istheelasticityoftheratioofmarginalratesofsubstitutionwithrespecttotheratiooftwoarguments,whereas thelatteristheelasticityofmarginalratesofsubstitutionwithrespecttoanargument. Thetwodefinitionsare equivalentwheneverthefunctiontheyareappliedtoishomogeneousofdegreeone. 9