The Impact of Regional and Sectoral Productivity Changes on the U.S. Economy

Transcription

1 The Impact of Regioal ad Sectoral Productivity Chages o the U.S. Ecoomy Lorezo Caliedo Yale Uiversity Ferado Parro Federal Reserve Board Esteba Rossi-Hasberg Priceto Uiversity Pierre-Daiel Sarte FRB Richmod September 16, 2013 Abstract We study the impact of regioal ad sectoral productivity chages o the U.S. ecoomy. To that ed, we cosider a eviromet that captures the effects of iterregioal ad itersectoral trade i propagatig disaggregated productivity chages at the level of a sector i a give U.S. state to the rest of the ecoomy. The quatitative model we develop features pairwise iterregioal trade across all 50 U.S. states, 26 traded ad o-traded idustries, labor as a mobile factor, ad structures ad lad as a immobile factor. We allow for sectoral likages i the form of a itermediate iput structure that matches the U.S. iput-output matrix. Usig data o trade flows by idustry betwee states, as well as other regioal ad idustry data, we calibrate the model ad carry out a variety of couterfactual experimets that allow us to gauge the impact of regioal ad sectoral productivity chages. We fid that such chages ca have dramatically differet effects depedig o the sectors ad regios affected. I extreme cases, icreases i productivity ca have egative effects o real GDP (although welfare effects remai positive). 1. INTRODUCTION Fluctuatios i aggregate ecoomic activity result from a wide variety of disaggregated pheomea. These pheomea ca reflect uderlyig chages that are sectoral i ature, such as process or product iovatios, We thak Joatho Leczar for excellet research assistace. Caliedo: lorezo.caliedo@yale.edu, Parro: ferado..parro@frb.gov, Rossi-Hasberg: erossi@priceto.edu, ad Sarte: pierre.sarte@rich.frb.org. The views expressed i this paper are those of the authors ad do ot ecessarily reflect those of the Federal Reserve Bak of Richmod, the Federal Reserve Board, or the Federal Reserve System. 1

2 or regioal i ature, such as atural disasters or chages i local regulatios. I other cases, fudametal productivity chages are actually specific to a sector ad a locatio: a large corporate bakruptcy or bailout. The heterogeeity of these potetial chages i productivity at the sectoral ad regioal levels implies that the particular sectoral ad regioal compositio of a ecoomy is essetial i determiig their aggregate impact. That is, regioal trade, the presece of local factors such as lad ad structures, regioal migratio, as well as iput-output relatioships betwee sectors, all determie the impact of a disaggregated sectoral or regioal productivity chage o aggregate outcomes. I this paper, we preset a model of the sectoral ad regioal compositio of the U.S. ecoomy ad use it to measure the elasticity of aggregate measured productivity, output, ad welfare, to disaggregated fudametal productivity chages. The maor part of research i macroecoomics has traditioally emphasized aggregate disturbaces as sources of aggregate chages. 1 Exceptios to this approach were Log ad Plosser (1983), ad Horvath (1998, 2000), who posited that because of iput-output likages, productivity disturbaces at the level of a idividual sector would be propagated throughout the ecoomy i a way that led to otable aggregate movemets. 2 More recetly, the view that idiosycratic disturbaces to idividual firms or sectors ca have sizeable effects has bee further articulated i terms of the etwork structure implied by iput-output or other likages (Acemoglu, Carvalho, Ozdaglar, Tahbaz-Salehi, 2012, ad Oberfield, 2012), ad the fact that whe the size distributio of firms or sectors is fat-tailed, idiosycratic disturbaces do ot average out eve i the absece of etwork effects (Gabaix, 2011). Foerster, Sarte, ad Watso (2011) fid empirical support for the importace of sectoral likages highlighted i these papers. To this poit, the literature studyig the aggregate implicatios of disaggregated productivity disturbaces has largely abstracted from the regioal compositio of sectoral activity. However, i the U.S., the distributio of sectoral productio across regios is far from uiform. Moreover, i previous work, Blachard ad Katz (1992) provide empirical evidece that factors related to geography, such as labor mobility across states, matter importatly for macroecoomic adustmets to disturbaces. This otio is addressed more recetly i Fogli, Hill ad Perri (2012), while Hamilto ad Owyag (2012) further establish the empirical importace of regioal characteristics for overall macroecoomic activity. What the are the mechaisms through which geographical cosideratios help determie the effects of disaggregated productivity chages? What is their quatitative importace? These are the issues that we take up i this paper. The fact that differet regios of the U.S. differ sigificatly i what they produce has two importat implicatios. First, to the degree that ecoomic activity ivolves a complex etwork of iteractios betwee sectors, these iteractios take place over potetially large distaces by way of regioal trade, but tradig across distaces is costly. Secod, sice sectoral productio has to take place physically i some locatio, it is the iflueced by a wide rage of chagig circumstaces i that locatio, from chages i policies 1 This emphasis, for example, permeates the large Real Busiess Cycles literature that followed the semial work of Kydlad ad Prescott (1982). 2 See also Jovaovic (1987) who shows that strategic iteractios amog firms or sectors ca lead micro disturbaces to resemble aggregate factors. 2

3 affectig the local regulatory eviromet or busiess taxes to atural disasters. Added to these regioal cosideratios is that some factors of productio are fixed locally ad uevely distributed across space, such as lad ad structures, while others are highly mobile, such as labor. 3 For example, i the three moths followig hurricae Katria, estimates from the Curret Populatio Survey idicate that the total populatio of Louisiaa fell by more tha 6 percet, ad is oly gettig back to its pre-katria tred six years later. To study how these differet aspects of ecoomic geography ifluece the effects of disaggregated productivity disturbaces, we develop a quatitative model of the U.S. ecoomy broke dow by regios ad sectors. I each sector ad regio, there are two factors of productio, labor ad a composite factor comprisig lad ad structures. As emphasized by Blachard ad Katz (1992), labor is allowed to move across both regios ad sectors. Lad ad structures ca be used by ay sector but are fixed locally. Sectors are itercoected by way of iput-output likages but, i cotrast to Log ad Plosser (1983) ad its esuig literature, shippig materials to sectors located i other regios is costly i a way that varies with distace. Usig ewly released data o pairwise trade flows across states by idustry, as well as other regioal ad idustry data, we calibrate the model ad explore the regioal, sectoral, ad aggregate effects of disaggregated productivity chages. Specifically, for a give productivity chage located withi a particular sector ad regio, the model delivers the effects of this chage o all sectors ad regios i the ecoomy. We fid that disaggregated productivity chages ca have dramatically differet implicatios depedig o the regios ad sectors affected. These effects arise i part by way of edogeous chages i the patter of regioal trade through a selectio effect that determies what types of goods are produced i which regios. They also arise by way of labor migratio towards regios that become more productive. Whe such migratio takes place, the iflow of workers strais local fixed factors i those regios ad, therefore, mitigates the direct effects of ay productivity icreases. I extreme cases, regioal productivity icreases ca eve have egative effects o aggregate GDP (although welfare effects are always positive). I Florida, for example, a 10 percet icrease i regioal fudametal productivity leads to a 0.3% fall i aggregate real GDP. I cotrast, i New York state, which is of comparable employmet size relative to aggregate employmet (6.1% versus 6.2%, respectively), a similar productivity chage icreases aggregate real GDP by 0.64%. Thus, the effects of disaggregated productivity chages deped i complex ways o the details of which sectors ad regios are affected, ad how these are liked through iput-output ad trade relatioships to other sectors ad regios. Ultimately, regioal trade likages, ad the fact that materials produced i oe regio are potetially used as iputs far away, are essetial i propagatig productivity chages spatially ad across sectors. Because U.S. ecoomic activity is ot distributed uiformly across regios, a full treatmet of the effects of disaggregated disturbaces caot be carried out without a explicit modelig of regioal tradig patters i differet idustries. I that cotext, distace ad other trade barriers play a key role i determiig allocatios. Thus, we fid that elimiatig U.S. regioal tradig costs associated with distace would result 3 See Kea ad Walker (2011) for a recet detailed empirical study of migratio across U.S. states. 3

4 i aggregate TFP gais of approximately 50 percet, ad i aggregate GDP gais o the order of 125 percet. These figures are evidetly sigificat, ad may be iterpreted as upper bouds o the extet to which advaces i shippig ad other trasportatio techology ca evetually cotribute to productivity ad value added. More importatly, they also represet a foudatio for the role of ecoomic geography i the study of the macroecoomic implicatios of disaggregated disturbaces. Our paper builds o the semial work of Eato ad Kortum (2002), ad the growig iteratioal trade literature that exteds their model to multiple sectors. 4 I particular, we are iflueced by recet cotributios that highlight the importace of itermediate goods ad sectoral likages i shapig the trade ad welfare effects from opeess (Caliedo ad Parro, 2011), the welfare effects arisig from reduced differeces i fudametal productivity across sectors ad coutries (Caselli, Kore, Lisicky, ad Tereyro, 2012, Costiot, Doaldso, ad Komuer, 2012, Levcheko ad Zhag, 2012), ad the spillover effects from productivity growth i Chia (Hsieh ad Ossa, 2011, di Giovai, Levcheko, ad Zhag, 2013). Relative to this literature, we develop a model that captures the iterrelatios across sectors ad regios withi the U.S. ecoomy. The geographic ature of our problem, amely the presece of labor mobility, local fixed factors, ad heterogeeous productivities, itroduce a differet set of mechaisms through which chages i fudametal productivity affect productio across sectors ad regios compared to a ope ecoomy model. From a more regioal perspective, two related papers, Reddig (2012), ad Arkolakis ad Alle (2013), study the implicatios of labor mobility for the welfare gais of trade, but abstract from studyig the role of sectoral likages or from presetig a quatitative assessmet of the effects of disaggregated fudametal productivity chages o U.S. aggregate measures of TFP, GDP, or welfare. The rest of the paper is orgaized as follows. The ext subsectio describes the compositio of U.S. ecoomic activity. We make use of maps ad figures to show how ecoomic activity varies across U.S. states ad sectors. Sectio 2 presets the quatitative model. Sectio 3 describes i detail how to compute ad aggregate measures of TFP, GDP, ad welfare across differet states ad sectors, ad shows how these measures relate to fudametal productivity chages. Sectio 4 describes the data, shows how to carry out couterfactuals, ad how to calibrate the model to 50 U.S. states ad 26 sectors. Sectio 5 quatifies the effects of differet disaggregated fudametal productivity chages. I particular, we measure the elasticity of aggregate productivity ad output to sectoral, regioal, as well as sector ad regio specific productivity chages. Sectio 6 decomposes the trade costs of movig goods across U.S. states ito a geographic distace compoet ad other regioal trade barriers. We the evaluate the importace of geographic distace for aggregate measures of TFP, GDP, ad welfare. Sectio 7 cocludes. 4 For istace, Arkolakis, Costiot, Rodriguez-Clare (2012), Burstei, Cravio, ad Vogel (2013), Burstei ad Vogel (2012), Caliedo ad Parro (2010), Chor (2010), Doaldso (2012), Dekle, Eato ad Kortum (2008), Eato, Kortum, Neima, ad Romalis (2011), Fieler (2011), Kerr (2009), Ossa (2012), Parro (2012), Ramodo ad Rodriguez-Clare (2012), ad Shikher (2011). Eato ad Kortum (2012) ad Costiot ad Rodriguez-Clare (2013) preset surveys of recet quatitative extesios of the Ricardia model of trade. 4

5 1.1 The Compositio of U.S. Ecoomic Activity Throughout the paper, we break dow the U.S. ecoomy ito 50 U.S. states ad 26 sectors pertaiig to the year 2007, our bechmark year. We motivate ad describe i detail this particular breakdow i Sectio 4. As show i Figure 1a, shares of GDP vary greatly across states. I part, these differeces stem from differeces i geographic size. However, as Figure 1a makes clear, differeces i geographic size are ot large eough to explai observed regioal differeces i GDP. New York state s share of GDP, for example, is slightly larger tha Texas eve though its geographic area is several times smaller. The remaiig differeces caot be explaied by ay mobile factor such as labor, equipmet, or other material iputs, sice those ust follow other local characteristics. I fact, as illustrated i Figure 1b, the distributio of employmet across states, although ot idetical to that of GDP, matches it fairly closely. Why the do some regios produce so much more tha others ad attract may more workers? The basic approach i this paper argues that three local characteristics, amely total factor productivity, local factors, ad access to products i other states, are essetial to the aswer. Specifically, we postulate that chages to total factor productivity (TFP) that are sectoral ad regioal i ature, or specific to a idividual sector withi a regio, are fudametal to uderstadig local ad sectoral output chages. Furthermore, these chages have aggregate effects that are determied by their geographic ad sectoral distributio. Oe iitial idicatio that differet regios ideed experiece differet circumstaces is preseted i Figure 1c, which plots average aualized percetage chages i regioal GDP across states for the period (Sectio 4 describes i detail the disaggregated data ad calculatios that uderlie aggregate regioal chages i GDP). The figure shows that aualized GDP growth rates vary across states i dramatic ways; from 7.1 percet i Nevada, to percet i Michiga. Of course, some of these chages reflect chages i employmet levels. Nevada s employmet relative to aggregate U.S. employmet grew by 3.1 percet durig this period while that of Michiga declied by percet. Figure 1d idicates that employmet levels also vary substatially over time, although somewhat less tha GDP. The latter observatio supports the view that labor is a mobile factor, drive by chages i fudametals, such as productivity. While our discussio thus far has uderscored overall ecoomic activity across states, oe may also cosider particular sectors. Doig so immediately reveals that the sectoral distributio of ecoomic activity also varies greatly across space. A extreme example is give by the Petroleum ad Coal idustry i Figure 2a. This idustry is particularly cocetrated i oly 3 states, amely Califoria, Louisiaa, ad Texas. I cotrast, Figure 2b presets GDP shares i the Wood ad Paper idustry, the most uiformly dispersed idustry i our sample. The geographic cocetratio of idustries may, of course, be explaied i terms of differeces i local productivity or access to essetial materials. I this paper, these sources of variatio are reflected i idividual idustry shares across states. For ow, we simply make the poit that variatios i local coditios are large, ad that they are far from uiform across idustries. Figure 3 shows the Herfidahl idex of GDP cocetratio across states for each idustry i our study. 5

6 Fig. 1. Distributio of ecoomic activity i the U.S. a: Share of GDP by regio (%, 2007) b: Share of Employmet by regio (%, 2007) NI NI EE 115 ET 1115 EN AI IE SE OI TT EO LR EI TT IN EO EE VL AV 2447 NA LZ LL SA LK LR NA LL SA II LL 117 ES LL II LL 117 ES LL c: Chage i GDP (%, 2002 to 2007) d: Chage i Employmet (%, 2002 to 2007) NI NH WA 8.6 ME 4.2 MT ND ID WY IN 2.2 CO 5.3 CA OH 4.5 MO AR NM OI AV LZ 477 GA SC LK LR EE EE VL EO IN WV AT 1777 AO TT AY EL AI NC EI SE EN AK DE ET IE MD AZ 2.7 PA VA 3.1 CT 2.8 IA UT 2.5 EE WI MI MA 1.5 MN SD EE OI AO 2444 AT AY AI SE AV EN EL EE LK ET IE 1177 VL LZ 1155 EE IN 718 AO 1179 AT AY EE EI 119 EL NA LL SA MS HI 2.8 LA FL II LL ES LL Fig. 2. Sectoral cocetratio across regios (shares, 2007) a: Petroleum ad Coal b: Wood ad Paper NI NI ET EN EI SE OI AO 222 IN LR EN OI AO IN SA LK LR NA LL SA LL AV LZ II NA ES LL II LL EE 2666 EE VL EO AV LL TT AT AY AI 2666 EE EI SE LK EL 169 ET IE LZ 2222 EE VL EO TT AT AY EE AI IE EL EE 222 ES LL

7 Differeces i the spatial distributio of ecoomic activity for differet sectors imply that sectoral disturbaces of similar magitudes will affect regios very differetly ad, therefore, that their aggregate impact will vary as well. Hece, to assess the implicatios of techological chages i differet sectors, oe eeds to be cogizat of how these chages are filtered through the regioal ecoomy. Studyig this process ad its quatitative implicatios is the mai purpose of this paper. F. 3. Regioal cocetratio of ecoomic activity across sectors (Herfidahl Idex, 2007) Regioal cocetratio of ecoomic acticty across sectors (Herfidahl Idex, 2007) Petroleum ad Coal Computer ad Electroic Textile, Apparel, Leather Arts ad Recreatio Chemical Iformatio Services Trasportatio Equipmet Real Estate Fiace ad Isurace Miscellaeous Educatio Other Services Machiery Costructio Wholesale ad Retail Trade Accom. ad Food Services Primary ad Fabricated Metal Furiture Food, Beverage, Tobacco Pritig Trasport Services Health Care Nometallic Mieral Electrical Equipmet Plastics ad Rubber Wood ad Paper A importat chael through which the geographic distributio of ecoomic activity, ad its breakdow across sectors, affects the impact of chages i total factor productivity relates to iterregioal trade. Trade implies that disturbaces to a particular locatio will affect prices i other locatios ad thus cosumptio ad, through iput-output likages, productio i other locatios. This chael has bee studied widely with respect to trade across coutries but much less with respect to trade across regios withi a coutry. That is, we kow little about the propagatio of local productivity chages across regios withi a coutry through the chael of iterregioal trade, whe we take ito accout that people move across states. This is perhaps surprisig give that trade is cosiderably more importat withi tha across coutries. Table 1 presets U.S. imports ad exports as a percetage of GDP i Overall, trade across regios amouts to about two thirds of the ecoomy ad it is more tha twice as large as iteratioal trade. This evidece uderscores the eed to icorporate regioal trade i the aalysis of the effects of productivity chages, as we do here. While iterregioal trade ad iput-output likages have the potetial to amplify ad propagate techological chages, they do ot geerate them. Furthermore, if all disturbaces were oly aggregate i ature, regioal ad sectoral chaels would play o role i explaiig aggregate chages. Figure 4a shows that aualized chages i sectoral measured TFP vary dramatically across sectors, from 14 percet per year i the Computer ad Electroics idustry to a declie i measured productivity of 7

8 Table 1. : Importace of Regioal Trade U.S. trade as a share of GDP (%, 2007) Exports Imports Total Iteratioal trade Iterregioal trade Source: World D evelopm et idicators ad C F S more tha 2 percet i Costructio. We describe i detail the data ad assumptios eeded to arrive at disaggregated measures of productivity by sector ad regio i Sectio 4. I that sectio, we uderscore the distictio betwee fudametal productivity ad the calculatio of measured productivity that icludes the effect of trade ad sectoral likages. I fact, the structure of the model drivig our aalysis helps precisely i uderstadig how chages i fudametal productivity affect measured productivity. 5 F. 4. Sectoral measured TFP of the U.S. ecoomy from 2002 to a: Chage Chage i sectoral i sectoral TFP ( ,%) TFP (%) b: Sectoral Sectoral cotributio cotributio to the to chage the i chage aggregate i aggregate TFP ( ,%) TFP (%) Computer ad Electroic Trasportatio Equipmet Food, Beverage, Tobacco Iformatio Services Trasport Services Machiery Miscellaeous Textile, Apparel, Leather Electrical Equipmet Plastics ad Rubber Wood ad Paper Pritig Furiture Real Estate Chemical Wholesale ad Retail Trade Fiace ad Isurace Arts ad Recreatio Accom. ad Food Services Health Care Other Services Educatio Costructio Nometallic Mieral Primary ad Fabricated Metal Petroleum ad Coal -30 Computer ad Electroic Trasportatio Equipmet Food, Beverage, Tobacco Real Estate Trasport Services Iformatio Services Wholesale ad Retail Trade Fiace ad Isurace Machiery Chemical Wood ad Paper Plastics ad Rubber Miscellaeous Textile, Apparel, Leather Electrical Equipmet Pritig Furiture Nometallic Mieral Health Care Educatio Arts ad Recreatio Accom. ad Food Services Petroleum ad Coal Costructio Other Services Primary ad Fabricated Metal Figure 4b presets the cotributio of sectoral chages i measured TFP to aggregate TFP chages. The distictio betwee Figures 4a ad 4b reflects the importace or weight of differet sectors i aggregate 5 Regioal measures of TFP at the state level are ot directly available from a statistical agecy. As explaied i Sectio 4, our calculatios of disaggregated TFP chages rely o other iformatio directly observable by regio ad sector, such as value added or gross output calculated from trade flows, as well as o uobserved iformatio iferred usig equilibrium relatioships cosistet with the model preseted i Sectio 2. Importatly, our measures of disaggregated TFP chages sum up to the aggregate TFP chage for the same period directly available from the OECD productivity database. 8

9 productivity. Oce more, the heterogeeity across sectors is surprisig. Moreover, this heterogeeity implies that chages i a particular sector will have very distict effects o aggregate productivity, eve coditioal o the size of the chages. Variatios i TFP across sectors have received cosiderable attetio i the macroecoomics literature (see Foerster et al., 2011, Gabaix, 2011, ad Acemoglu, et al., 2012, amog others). I cotrast, this literature has paid virtually o attetio to the regioal compositio of TFP chages. Figures 5a ad b shows that this lack of attetio is potetially misguided. Chages i measured TFP vary widely across regios. Furthermore, the cotributio of regioal chages i measured TFP to variatios i aggregate TFP is also very large. The differece betwee Figures 5a ad 5b reflects the weight of differet states i aggregate productivity. The chage i TFP over the period was 1.4 percet per year i Nevada but 1.1 percet i Michiga. These differeces i TFP experieces aturally cotributed to differeces i employmet ad GDP chages i those states. More geerally, variatios across states result i part from sectoral productivity chages as well as chages i the distributio of sectors across space which, as we have argued, is far from uiform. However, eve if all the variatio i Figures 5a ad b were ultimately traced back to sectoral chages, their ueve regioal compositio would ifluece their impact o trade ad, ultimately, aggregate TFP. F. 5. Regioal measured TFP of the U.S. ecoomy from 2002 to 2007 a: Chage i TFP by regios (%) b: Regioal cotributio to the chage i aggregate TFP (%) 834 LK ET 3337 IE 239 AY 3382 TT 832 AO 337 LZ II SE NI 3384 EE 3329 EL EN 3376 EI AI AT 3383 OI EE IN 833 EE VL 3397 EO 838 AV LR 838 NA LL SA ES LL LL LK ET 4111 IE 4111 AY 4141 TT 4111 AO 111 LZ II SE EN 711 EI 311 AI 717 OI IN 719 EO LR LL ES LL NI 4135 EE 4141 EL AT EE 4111 EE VL AV 4143 NA 311 SA 4194 LL 417 Oe of the key ecoomic determiats of icome across regios is the stock of lad ad structures. To our kowledge, there is o direct measure of this variable. However, as we explai i detail i Sectio 4, we ca use the equilibrium coditios from our model to ifer the regioal distributio of icome from lad ad structures across U.S. states. Figure 6 shows that per capita icome from lad ad structures i 2007 U.S. dollars varies cosiderably across states. The rage varies from a low of 4,200 ad 8,300 dollars per capita for the case of Hawaii ad Florida respectively, to a high of 70,100 dollars i Illiois. 6 We will argue 6 We obtai these figures after calibratig the model to the year 2007 ad calculatig a couterfactual sceario without trade imbalaces across states. Sectio 4 ad Appedix A.1 provide a detailed descriptio of these calculatios. 9

10 that this regioal dispersio of lad ad structures across regios i the U.S. is cetral to uderstadig the aggregate effects of disaggregated fudametal productivity chages. F. 6. Per capita regioal icome from lad ad structures (10,000 of 2007 U.S. dollars) LK IE 3889 TT 886 LZ 883 ET 882 II 3892 AY AO SE EN EO 886 LR 2 AI IN 881 LL LL 2 ES 889 EI 881 OI VL 885 LL 3883 SA 888 NI EE 889 EE 888 AV 889 NA 285 EL AT 285 EE THE MODEL Our goal is to produce a quatitative model of the U.S. ecoomy disaggregated across regios ad sectors. For this purpose, we develop a static two factor model with N regios ad J sectors. We deote a particular regio by {1,..., N} (or i), ad a particular sector by {1,..., J} (or k). The ecoomy has two factors, labor ad a composite factor comprisig lad ad structures. Labor ca freely move across regios ad sectors. Lad ad structures, H, are a fixed edowmet of each regio but ca be used by ay sector. We deote total populatio size by L, ad the populatio i each regio by L. A give sector may be either tradable, i which case goods from that sector may be traded at a cost across regios, or o-tradable. Throughout the paper, we abstract from iteratioal trade ad other iteratioal ecoomic iteractios. 2.1 Cosumers Agets i each locatio {1,..., N} order cosumptio baskets accordig to Cobb-Douglas prefereces, with shares, α, over their cosumptio of fial domestic goods, c, bought at prices, P, i all sectors {1,..., J}. Prefereces are homothetic of degree oe, so J =1 α = 1. Agets supply oe uit of labor ielastically. The icome of a aget residig i regio is I = r H L + w, where w is the wage, r is the retal rate of structures ad lad, ad r H /L is the per capita icome 10

11 from retig lad ad structures to firms i regio. 7 Thus, total icome i regio is L I = r H + w L. (1) The problem of a aget i regio is the give by v max {c } J =1 J =1 ( c ) α, subect to J It follows that total demad of fial good i regio is Agets move freely across regios. regio is where P = J =1 betwee livig i ay regio so that =1 P c = I. L c = α L I P. (2) From the household problem, the value of locatig i a particular v = r H /L + w P, ( P /α ) α is the ideal price idex i regio. I equilibrium, households are idifferet for all {1,..., N}, for some U determied i equilibrium. v = r H /L + w P = U (3) 2.2 Techology Sectoral fial goods are used for cosumptio ad as material iputs ito the productio of itermediate goods i all idustries. I each sector, fial goods are produced usig a cotiuum of varieties of itermediate goods i that sector. We refer to the itermediate goods used i the productio of fial goods as itermediates, ad to the fial goods used as iputs i the productio of itermediate goods as materials Itermediate Goods Represetative firms, i each regio ad sector, produce a cotiuum of varieties of itermediate goods that differ i their idiosycratic productivity level, z. 8 I each regio ad sector, this productivity level is a radom draw from a Fréchet distributio with shape parameter θ. Note that θ varies oly across sectors. We assume that all draws are idepedet across goods, sectors, ad regios. The productivity of all firms producig varieties i a regio-sector pair (, ) is also determied by a determiistic productivity level, T, 7 I order to abstract from the complicatios associated with the wealth effects, ad implied heterogeeity across agets withi each regio, that arise from productivity disturbaces, we assume that lad ad structures i each regio are owed by local govermets, who the ret them to firms ad distribute the proceeds to local residets. 8 I a parallel extesio of Eato ad Kortum (2002), i each sector withi a regio, each variety that is used by firms i productio withi that sector ad regio is associated with a idiosycratic productivity level. Sice techology is costatreturs-to-scale (CRS), the umber of firms producig ay give variety is idetermiate ad irrelevat for the equilibrium allocatio. Hece, throughout the aalysis, we work with firms, or represetative firms, that produce differet varieties of a sectoral good withi a regio. 11

12 specific to that regio ad sector. We refer to T as fudametal productivity. The productio fuctio for a variety associated with idiosycratic productivity z i (, ) is give by q (z ) = z [ T h (z ) β l (z ) (1 β )] γ J k=1 M k (z ) γk, (4) where h ( ) ad l ( ) deote the demad for structures ad labor respectively, M k ( ) is the demad for fial material iputs by firms i sector from sector k (variables represetig fial goods are deoted with capital letters), γ k 0 is the share of sector goods spet o materials from sector k, ad γ 0 is the share of value added i gross output. We assume that the productio fuctio has costat returs to scale, J k=1 γk = 1 γ. Observe that T scales value added ad ot gross output. This feature esures that the ecoomy does ot exhibit icreasig returs as a result of productivity chages. The uit cost of producig varieties with draw z i (, ) is give by mi w l(z ) + r h (z) + J P {h (z),l (z ),M k (z)} J k M k (z), k=1 k=1 subect to z [ T h (z ) β l (z ) (1 β )] γ J k=1 M k (z ) γk = 1, where P k is the price of fial goods i idustry k i regio. Let x deote the cost of the iput budle eeded to produce itermediate good varieties i (, ). The where B = x = B [ r β w 1 β ] γ J ( ) P k γ k, (5) k=1 [ ] γ [ (1 β ) (β 1) (β ) β J ] ( ) γ k γ k. k=1 The uit cost of a itermediate good with idiosycratic draw z i regio-sector pair (, ) is the give by z x ( ) γ. (6) T Firms located i regio ad operatig i sector will be motivated to produce the variety whose productivity draw is z as log as its price matches or exceeds x /z T ( ) γ. Assumig a competitive market for itermediate goods, firms that produce a give variety i (, ) will price it accordig to its correspodig uit cost, give by Equatio (6). Let p (z ) represet the equilibrium price of a variety for which the vector of idiosycratic productivity draws i all N regios is give by z = (z 1, z 2,...z N ). The determiatio of this price i equilibrium is discussed i detail below. Sice the productio fuctio is Cobb-Douglas, profit maximizatio implies that 12

13 iput demads, h (z ), l (z ), ad M k (z ) for all k, satisfy 9 h (z )r p (z )q (z ) l (z )w p (z )q (z ) P k M k (z ) p (z )q (z ) = γ β, (7) = γ (1 β ), (8) = γ k. (9) Fial Goods Fial goods i regio ad sector are produced by combiig itermediate goods i sector. Deote the quatity of fial goods i (, ) by Q, ad deote by q (z ) the quatity demaded of a itermediate good of a give variety such that, for that variety, the particular vector of productivity draws received by the differet regios is z = (z 1, z 2,...z N ). The productio of fial goods is give by [ Q = q (z ) 1 1/η φ ( z ) ] η 1) dz /(η, (10) { where φ (z ) = exp N ( ) =1 z θ } deotes the oit desity fuctio for the vector z, with margial { desities give by φ (z) = exp ( ) z θ }, ad the itegral is over R N +. For o-tradeable sectors, the oly relevat desity is φ ( z ) sice fial good producers use oly locally produced goods. Producers of composite sectoral goods the solve max PQ { q (z p (z ) q (z )φ ( z ) dz. )} R N + where p (z ) deotes the price of itermediate goods. The, the demad fuctio is give by where P is a price idex for sector i regio, ( p q (z ) = (z ) η ) Q, P [ P = p (z ) 1 η φ ( z ) ] 1/(1 η ) dz. There is free etry i the productio of fial goods with competitio implyig zero profits. 2.3 Prices ad Market Clearig Fial goods are o-tradable. Itermediate goods i tradable sectors are costly to trade. Oe uit of ay itermediate good i sector shipped from regio i to regio requires producig κ i 1 uits i i, with κ = 1 ad, for itermediate goods i o-tradable sectors, κ i =. Thus, the price paid for a particular 9 Factor demads are evidetly also a fuctio of the price, although we do ot ackowledge this fact explicity to ease otatio. 13

14 variety whose vector of productivity draws is z, p (z ), is give by the miimum of the uit costs across locatios, adusted by the trasport costs κ i. That is, p ( z ) = mi i z i κ i x i ( T i ) γ i. (11) Give our assumptios goverig the distributio of idiosycratic productivities, z i, we follow Eato ad Kortum (2002) to solve for the distributio of prices. Havig solved for the distributio of prices, whe sector is tradeable, the price of fial good i regio is give by P = Γ ( [ ξ ) 1 η N [ ] θ ( ) θ x i=1 i κ i T γ ] 1/θ i i, (12) where Γ ( ξ ) is a Gamma fuctio evaluated at ξ = 1 + ( 1 η ) /θ. Whe deotes a o-tradeable sector, the price idex is istead give by P = Γ ( ξ ) 1 η x ( ) T γ. (13) Regioal labor market clearig requires that J =1 L = J =1 0 l (z)φ (z) dz = L, for all {1,..., N}, (14) where L deote the umber of workers i (, ), ad atioal labor market clearig is give by N =1 L = L. I a regioal equilibrium, lad ad structures must satisfy J =1 H = J =1 where H deotes lad ad structure use i (, ). that 0 h (z)φ (z) dz = H, for all {1,..., N}, (15) Profit maximizatio by itermediate goods producers, together with these equilibrium coditios, implies r H = β 1 β w L, for all {1,..., N}. The, defiig ω (r /β ) β (w /(1 β )) (1 β ), free mobility gives us [ ] 1 ω β L = H, P U which, combied with the labor market clearig coditio, yields a expressio for labor iput i regio, L = N H [ i=1 H i ] 1/β ω P U [ ω i P iu ] 1/βi L. (16) 14

15 It remais to describe market clearig i fial ad itermediate goods markets. Regioal market clearig i fial goods is give by L c + J M k k=1 = L c + J for all {1,..., J} ad {1,..., N}. where M k k at. k=1 0 M k (z)φ k (z) dz = Q, (17) represets the use of itermediates of sector i sector Let X deote total expeditures o fial good i regio (or total reveue). The, regioal market clearig i fial goods implies that where π i X = J N k=1 γk i=1 πk ix k i + α I L, (18) deotes the share of regio s total expeditures o sector s itermediate goods purchased from regio i. As i Eato ad Kortum (2002), ad Alvarez ad Lucas (2007), it is coveiet to defie a equilibrium i the itermediate goods market, ad their associated trade flows, i terms of trade ad expediture shares, rather tha the flows ad expeditures related to idividual varieties. Defie X i = Pr z i κ i x i ( T i ) γ i mi m i κ mx m ( ) γ zm Tm m X, ad recall that, because of zero profits i fial goods sectors, total expeditures o itermediate goods i a give sector exhaust total reveue from fial goods i that sector. The, give properties of the Fréchet distributio, equilibrium i the itermediate goods market implies that π i = X i X = Γ ( ξ ) 1 η ( T i κ i x i ) γ i I o-tradable sectors, κ i = for all {1,..., N} so that π = 1. P θ. (19) I equilibrium, i ay regio, total expeditures o itermediates purchased from other regios must equal total reveue from itermediates sold to other regios, formally, J =1 N i=1 π i X = J =1 N i=1 π i X i. (20) This last coditio guaratees that trade is balaced withi each regio, although ot ecessarily i ay give sector. That is, withi a regio, a give sector may be a et exporter or importer of itermediate goods as log as the associated surplus or deficit is balaced by aother sector. I practice, regios might experiece periods i which overall imbalaces are positive or egative. I the ext sectio, we explai how to take these imbalaces ito accout i quatitative exercises. Give factor supplies, L ad {H } N =1, a competitive equilibrium for this ecoomy is a utility level U, a set of factor prices i each regio, {r, w } N =1, a set of labor allocatios, structure allocatios, fial good 15

16 expeditures, cosumptio of fial goods per perso, ad fial goods prices, { } L, H, X, c, P N,J pairwise sectoral material use i every regio, { M k { } N,N,J π i } N,J,J, ad pairwise regioal itermediate expe- =1,=1,k=1 diture shares i every sector, =1,i=1,=1 =1,=1,, such that the optimizatio coditios for cosumers ad itermediate ad fial goods producers hold, all markets clear - equatios (14), (15), (18), (19) hold -, trade is balaced - (20) holds-, ad utility is equalized across regios, - (16) holds. 3. AGGREGATION AND CHANGES IN MEASURED TFP, GDP, AND WELFARE Give the model we have ust laid out, this sectio describes how to arrive at measures of total factor productivity, GDP, ad welfare, that are disaggregated across both regios ad sectors. These calculatios of measures at the level of sector i a regio, usig available idustry ad regioal trade data for the U.S., uderlie Figures 1 through 7 ad their discussio i Sectio 1.1, as well as all calculatios i the rest of the paper. 3.1 Measured TFP Measured sectoral total factor productivity i a regio-sector pair (, ) is commoly calculated as l A = l w L + r H + J k=1 P k M k P (1 β ) γ l L β γ l H J k=1 γk l M k. (21) The first term is gross output reveue over price a measure of gross productio i (, ) which we deote by Y /P, ad which is equal to Q i the case of o-tradables, while the last three terms deote the log of the aggregate iput budle. 10 This last equatio assumes that we use gross output ad fial good prices to calculate regio-sector TFP. Observe that (7), (8), ad (9) imply that Y = w L + r H + J k=1 P k M k = w L γ (1 β ). (22) Therefore, we may calculate chages i measured TFP, Â, followig a chage i fudametal productivity, T, usig the ratio of the chage i the cost of the iput budle to the chage i the price of fial goods. 11 That is, l  = l x ˆP ( ˆT ) γ = l ) 1/θ, (23) (ˆπ 10 Oe ca prove that total gross output i (, ) uses this aggregate iput budle. To do so, aggregate Equatios (7), (8) ad (9). Usig these equatios, it is straightforward to derive that factor usage for a itermediate is ust the reveue share of that itermediate i gross reveue, Y. Substitutig i Equatio (4), ad usig the fact that prices of produced itermediates are equal to uit costs, leads to Y P = x [ (H ) β ( ) ] P L (1 β ) γ J ( k=1 M k ) γ k, where A = x /P measures regio ad sector specific TFP. 11 The hat otatio deotes A /A, where A is the ew level of total factor productivity. 16

17 where the secod equality follows from (19). Equatio (23) is cetral to uderstadig the sources of chages i measured productivity i a idividual sector withi a regio followig a chage i fudametal productivity, ˆT. Cosider first a ecoomy with ifiite tradig costs κ i = for all, so that trade is o-operative ad π = 1 i every regio. Furthermore, let us abstract from material iput use so that the share of value added i gross output is equal to oe, γ = 1. I such a ecoomy, which we abbreviate with the letters NRNS for o regioal trade ad o sectoral likages, Equatio (23) implies that chages i measured productivity  are idetical to chages i fudametal productivity, ˆT. Ay fudametal productivity chage at the level of a sector withi a regio traslates ito a idetical chage i measured productivity i that sector ad regio, ad has otherwise o effect o ay other sectors or regios. This exact relatioship betwee fudametal ad measured productivity, l  = l ˆT, o loger holds oce either trade or sectoral likages are operative. Cosider first addig sectoral likages, so that γ < 1, but still abstractig from trade (labeled NRS for o regioal trade but with sectoral likages ). that case, Equatio (23) idicates that the effect of a chage, ˆT, improves measured productivity less tha proportioally. The reaso is that the chage affects the productivity of value added i that regio ad sector but ot the productivity of sectors ad regios i which materials are produced. Therefore, i the presece of iput output likages, the effect of a fudametal productivity chage ˆT o measured productivity i (, ) falls with 1 γ = J k=1 γk. This last result follows from our assumptio that productivity chages scale value added ad ot gross output (as i Acemoglu et al. 2012). I our view, this feature of the techology is the oly reasoable way of specifyig productio i the presece of sectoral likages. Whe productivity istead affects all of gross output, sectors that ust process materials, without addig ay value by way of labor or capital, see a icrease i the umber of goods at o cost. That alterative modellig implies that aggregate fudametal productivity chages have abormally large effects o measured productivity while, with our techological assumptio, aggregate fudametal chages i the absece of trade have proportioal effects o measured productivity. This distictio matters greatly i quatitative exercises. Evidetly, with trade still shut dow, a regio ad sector specific chage i a NRS ecoomy has o effect o the measured productivity of ay other regio or sector. With trade, productivity chages are propagated across sectors ad regios. The mai effect of regioal trade o productivity arises by way of a selectio effect. Thus, let κ i I be fiite for tradable sectors, ad cosider first the regio-sector (, ) that experieces a chage or icrease i fudametal productivity, ˆT. Equatio (23) implies that the effect of trade is ultimately summarized through the chage i the regio s share of its ow itermediate goods, ˆπ. Sice a icrease i fudametal productivity i (, ) raises its regio ad sector comparative advatage, it geerally also leads to a icrease i π so that ˆπ > 1. Similarly, it reduces π k ii, for i ad all k, sice other regios ad sectors ow buy more sector- itermediates from regio. Hece, sice θ > 0, trade reduces the effect of a fudametal productivity 17

18 icrease to (, ) o measured productivity i that regio-sector while, at the same time, raisig measured productivity i other regios ad sectors. Ituitively, the selectio effect uderlyig the chage i expediture shares works as follows. As everyoe purchases more goods from the regio-sector pair (, ) that experieced a fudametal productivity icrease, that regio-sector pair ow produces a greater variety of itermediate goods. However, the ew varieties of itermediate goods, sice they were ot beig iitially produced, are associated with idiosycratic productivities that are relatively worse tha those of varieties produced before the chage. This egative selectio effect i (, ) partially offsets the positive cosequeces of the fudametal productivity chage, relative to a ecoomy with o trade, i that regio-sector pair. I other regio-sector pairs, (i, ) for i, the opposite effect takes place. As the latter regios do ot directly experiece the fudametal productivity chage, their ow trade share of itermediates decreases. As a result, the varieties of itermediate goods that cotiue beig produced i those regios have relatively higher idiosycratic productivities, thereby yieldig higher measured productivity i those locatios. All of these trade-related effects are preset whether material iputs are cosidered (case RS) or are abset from the aalysis (case RNS) Computig Aggregate, Regioal, ad Sectoral Measured TFP. Sice measured TFP at the level of a sector i a regio is calculated based o gross output i Equatio (21), we use gross output reveue shares to aggregate these TFP measures ito regioal, sectoral, or atioal measures. Chages i regioal ad sectoral measured TFP are the simply weighted averages of chages i measured TFP i each regio-sector pair (, ), where the weights are the correspodig (, ) gross output reveue shares. Thus, sice gross output reveue, Y, is give by Equatio (22), regioal chages i measured TFP are give by  = J =1 Y J =1 Y  = J J =1 =1 while sectoral chages i measured TFP ca be expressed as  = N =1 Y N =1 Y  = Similarly, chages i aggregate TFP are the give by 3.2 GDP  = J N J =1 =1 =1 Y N =1 Y  = N N =1 =1 J N =1 =1 w L γ (1 β ) w L γ (1 β ) w L γ (1 β ) w L γ (1 β ) w L γ (1 β ) J N =1 =1 Â, (24) Â. (25) w L γ (1 β ) Â. (26) Real GDP is calculated by takig the differece betwee real gross output ad expeditures o materials. Give equatios (7), (8), ad (9), as well as factor market equilibrium coditios, chages i real GDP may 18

19 be writte as l ĜDP = l ŵ ˆL ˆP This expressio simplifies further sice, from (19), = l ŵ + l ˆL l ˆP ˆP = (ˆπ ) 1 θ ( ˆT ) γ i ˆx, so that GDP chages i a regio-sector pair (, ), resultig from chages i fudametal TFP, ˆT, are give by. where the secod lie uses Equatio (23). ( ) γ ˆT l ĜDP ) (ŵ = l ) 1/θ + l (ˆπ ˆL + l x x ) (ŵ = l  + l ˆL + l, (27) Equatio (27) represets a decompositio of the effects of a chage i fudametal productivity o GDP. The first term reflects the effect of the chage o measured productivity discussed i Sectio 3.1. This effect is such that measured TFP ad output move proportioally. I other words, the selectio effect associated with itermediates ad iput-output likages acts idetically o measured TFP ad real GDP. I additio to these effects, GDP is also iflueced by two other forces captured by the secod ad third terms i Equatio (27). The secod term i Equatio (27) describes the effect of labor migratio across regios ad sectors o GDP. A positive productivity chage that attracts populatio to a give regio-sector pair (, ) will icrease GDP proportioally to the amout of immigratio, l ˆL. The reaso is that all factors i (, ) chage i the same proportios ad the productio fuctio of itermediates i Equatio (4) is costat-returs-to-scale. The effect of migratio will be positive whe the chage i fudametal TFP is positive. The third term i Equatio (27) correspods to the chage i factor prices associated with the chage i fudametal TFP. Cosider first a case without materials (RNS). I that case, l ( ŵ / x ) = β l (ŵ / r ) = β l 1/ˆL. Sice lad ad structures are fixed, ad therefore do ot respod to chages i T, while labor is mobile across locatios, a positive productivity chage that attracts people to the regio will icrease lad ad structure prices more tha wages. This mechaism leads to a reductio i real GDP, relative to the proportioal icrease associated with the first two terms. The presece of decreasig returs resultig from a regioally fixed factor implies that shiftig populatio to a locatio strais local resources, such as local ifrastructure, i a way that offsets the positive GDP respose stemmig from the iflow of workers. I regios that do ot experiece the productivity icrease, the opposite is true so that the secod ad third terms i (27) will be egative ad positive respectively. These forces are also preset whe we cosider 19

20 material iputs although, i that case, the relevat ratio is that of chages i wages to chages i the cost of the iput budle, x. The iput budle icludes the retal rate, but it also icludes the price of all materials. A overall assessmet of the effects of fudametal productivity chages the requires a quatitative evaluatio. Observe that whe cosiderig the aggregate ecoomy-wide effects of a positive ˆT, the ed result for GDP may be larger or smaller tha the origial chage. The overall impact of the last two terms i Equatio (27) will deped o whether the direct effect of migratio domiates the strai o local resources i the regio experiecig the chage,, as well as the itesiveess with which this fixed factor is used i the regios workers leave behid, i. Thus, the size ad sig of these effects deped o the overall distributio of H ad populatio L i the ecoomy ad, therefore, o whether the productivity chage icreases the dispersio of the wage-cost budle ratio, ŵ / x, across regios. If a productivity chage leads to migratio towards regios that lack abudat lad ad structures, the aggregatio of the last two terms i Equatio (27) may be egative or very small. I cotrast, if a chage moves people ito regios with a abudace of local fixed factors, the impact of these last two terms will be positive. Evidetly, whatever the case, oe must still add the direct effect of the fudametal productivity chage o measured productivity. These differet mechaisms uderscore the importace of geography, ad that of the sectoral compositio of techology chages, i order to assess the magitude of such chages. I extreme cases, these mechaisms may eve lead to egative aggregate GDP effects of productivity icreases. However, as the equilibrium allocatio is Pareto effi ciet, positive techological chages always lead to welfare gais, as discussed below. Fially, it is worth otig that i the case of aggregate productivity chages, the distributio of populatio across locatios is uchaged sice people do ot seek to move whe all locatios are similarly affected. Therefore, measured productivity ad GDP uambiguously icrease proportioally i that case Computig Aggregate, Regioal, ad Sectoral real GDP. Give that real GDP is a value added measure, we use value added shares i costat prices for aggregatio purposes. Deote sectoral ad regioal value added (, ) shares i a give bechmark year by ad υ = ξ = w L + r H ( ), J =1 w L + r H w L + r H ( ) N =1 w H + r H respectively. The, the chage i regioal real GDP arisig from a chage i fudametals is give by ĜDP = Similarly, the chage i sectoral real GDP may be expressed as ĜDP = J υ ĜDP. (28) =1 N ξ ĜDP. (29) =1 20