Trade Adjustment and Productivity in Large Crises. Online Appendix May Appendix A: Derivation of Equations for Productivity


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1 Trade Adjustment Productvty n Large Crses Gta Gopnath Department of Economcs Harvard Unversty NBER Brent Neman Booth School of Busness Unversty of Chcago NBER Onlne Appendx May 2013 Appendx A: Dervaton of Equatons for Productvty Dervaton of Equaton 8 The dervaton s very smlar to Basu Fernald 2002 wth the dstncton that we have labor that s used n fxed costs. The producton functon for each frm s gven by equaton 3. Snce frms are prce takers n the prmary factor ntermedate nput markets set prces as a constant markup 1/ over margnal cost we have: Y p w L p p Y r K p Y P X X. To measure the growth rate of value added we use the conventon dvsa ndex formula: ln Y V A ln Y s Y X ln X 1 s Y X ln Y sy X ln X 1 s Y ln Q X where s Y X s the revenue share of ntermedates sy X P X X P Y whch s equal to the constant µ. We can then wrte: Y Y K + Y L p K L + Y X + Y A p X A 1 s Y X 1 s Y X ln Y s k ln K + ω L ps L ln L p + sy X ln X + F A A Y P ln A. Ths follows from the relaton s k sy k ω L 1 s Y p L p X L. Rearrangng we get: ln Y V A ln Y sq X 1 s Q X 1 µ ln X ln Y 1 µ [s k ln K + w L ps L ln L p ] + µ [ ] 1 µ 1 µ 1 µ 1 ln X ln Y + F A A 1 µ Y p ln A. 1
2 Fnally defne the welfare relevant frmlevel productvty usng the modfed Solow Resdual: ln PR t ln Y V A s k ln K s l ln L ln Y V A s k ln K s l ω L p ln L P s l 1 ω L p ln L F. Substtutng for ln Y V A we arrve mmedately at equaton 8. Dervaton of Expressons 912 We present the dervaton n the followng steps: Step 1: We express ln X ln Y as a functon of γ. It follows from equatons 4 5 p C / gven fxed w r that: ln X ln Y ln p ln P X µ 1 ln P X. Followng the defnton of γ we wrte: P Z [ p 1 ] 1 d rα w 1 α 1 µ ɛ [ rα w 1 α 1 µ P µ Z ɛ [ γ P ZZ PZ 1 P X X P X C P Z γ 1 µ 1 P X P Z γ ] 1 1 d 1 A 1 d r α w 1 α 1 µ P µ X ɛa ] 1 [ ] 1 1 µ γ 1 A 1 d } {{ } Q 1 γ where ɛ µ µ 1 µ 1 µ α α 1 α 1 α 1 µ. We can then wrte: P Z rα w 1 α ɛ 1 1 µ P X rα w 1 α ɛ 1 1 µ Q 1 1 µ γ Q 1 1 µ γ 1 γ 1 d ln P X 1 µ 1 ln Q γ + 1 ln γ. 13 Step 2: The frms decson for use of L F 1 s related to ts decson on Ω. The frm 2
3 maxmzes: Π Π wl F 1 P Y wl F subject to: We defne: wrte: Y g + z j dj j 1 D P G p P G 1 1 G G + j 1 P Z j p P Z P 1 Z Xj dj. P Xj P 1 Z Xj dj P Xj Π 1 p Y 1 p 1 D. The FOC for Ω s: whch gves the followng expressons: Π Ω w LF Ω 1 Π Π Ω wl P wl F LP L F ln Π ln1 + ln p 1 Ω Π Ω Π µ1 γ Ω 1 ln p + ln D µ ln P X 1 Ω wf v λω λ 1 Π µ1 γ wf v λω λ wλl F wl F λ 1 1 P Y µ1 γ wl P 1 µ1 αp Y 1 µ1 αp Y λ 1 1 P Y µ1 γ L P L F 1 µ1 α λ 1 1 µ1 γ. PM P X 1 1 Ω 1 µ1 α λ 1 1 µ1 γ As γ λ ncrease so does the share of labor that s used for producton. Ths s used to arrve at the expresson for ω L P : 1 ω lp L F L wl F wl F + wl P λ 1 1 µ1 γ λ 1 1 µ1 γ + 1 µ1 α. 3
4 Step 3: Express ln F as a functon of P X. We wrte: ln L F λ ln Ω λ 1 ln P M ln p m ln1 γ 1 [ln P M ln P X ] ln P M ln P X 1 ln1 γ ln L F λ ln Ω λ ln P X 1 1 ln1 γ ln p m s L wlp + wl F P V A Y V A 1 µ 1 µ s l 1 ω lp ln L F λ 1 1 µ1 γ + 1 µ1 α 1 µ s l 1 ω lp ln L F λ 1 1 µ1 γ ln L F 1 µ 1 µ 1 µ µ1 γ 1 µ λ 1 1 µ1 γ ln L F 1 µ [ ln P X 1 ln1 γ ln p m Step 4: Replace the expresson for ln P X from equaton 13 n the precedng equaton. Replacng the above terms n the expresson for frmlevel productvty equaton 8 aggregatng over all usng frm valueadded shares ω we arrve at an expresson for aggregate productvty. The last step s to relate changes n ln Q γ to changes n ω γ. We start wth an expresson for the valueadded weghts whch should relate market shares of each frm to technologes trade shares: ω p p j j 1 1 dj 1 We then substtute n usng our expresson for Q: [ Q γ ] γ µ A 1 d j γ j µ 1 γ µ 1 ω 1 A 1 1 Aj 1 1 dj A γ µ 1 1. ]. wrte: ln Q γ 1 ω ln ω + µ 1 ω ln γ gven ln A 0. 4
5 The fnal expressons 912 are arrved at through substtuton regroupng these terms usng the approxmaton ln1 γ γ 1 γ ln γ whch s vald for small shocks. In the smulaton secton we do not use ths approxmaton because we study large shocks. Appendx B: Numercal Algorthm The algorthm works as follows. Frms start wth an ntal assumpton about the prces of the domestc nput bundle PZ 0 the fnal good P G 0. Snce the mportng behavor of each frm determnes ts margnal cost thereby nfluences P Z P G ths assumpton s effectvely equvalent to takng as gven all other frms mportng decsons. Holdng these prce aggregates fxed each frm smultaneously chooses the optmal number of mported varetes Ω Wth ths new set of mport varety choces {Ω 1 } we must solve a fxed pont problem to fnd a consstent set of new prces {p 1 } because each frm s margnal cost s a functon of all other frms prces due to roundabout producton. In partcular we terate the system: P 1 µ V p A µ µ 1 µ 1 µ PZ 1 p 1 1 d 1 [ P 1 1 Z + 1 ] p µ 1 m Ω 1 1 for all frms untl the set of prces {p 1 } s consstent wth the domestc nput prce ndex PZ 1 wth all frms choces of mported varetes {Ω1 }. We then repeat ths algorthm wth frms takng as gven the prce ndces PZ 1 P G 1 generate a new set of prces { mport varetes {p 2 Ω 2 } prce ndces {PZ 2 P G 2 }. We contnue ths process untl p j } { } Ωj p j 1 Ω j 1 up to a very small tolerance. Appendx C: Comparatve Statcs of the Frm s Trade Response In ths appendx we evaluate how each frm s response to the terms of trade shock wll dffer based on ts preshock level of total mports. The ntent here s to derve an expresson that provdes some ntuton for the results n the text as such we do not provde a formal proof. We have shown that as long as λ s suffcently hgh the number of mported varetes s ncreasng n the frm s exogenous technology A. Gven ther relatve cost advantage frms wth hgher A have lower prces p consequently sell more have hgher Y. These are also the frms wth the lowest γ snce P M /P Z s lower the hghest M. The elastcty of the response n γ to the mport prce change s a functon of the ntal 48 Though our frms have fnte market shares they gnore the mpact of ther own prce changes on the aggregate prce ndex. Ths s not problematc because the largest frm n our benchmark calbraton has a market share of only 5 percent. 5
6 γ. Usng the defnton of γ we can show that: ln γ 1 γ ln PM ln P Z 1 1 γ 1 ln P Z + 1 ln Ω > 0 1 ln Ω µ 1 1 [ ] + 1 γ + ln D + 1. γ γ ln P Z λ µ + µ 1 For the second order condtons for an nteror soluton to Ω to hold the denomnator must satsfy λ µ + µ 1 γ 1 > 0. As long as the numerator s negatve ln P Z < 1 whch s not always the case frms ncrease the share spent on domestc nputs γ when mport prces ncrease. To see how ths elastcty vares across exstng mporters we wrte: ln γ γ 1 1 ln P Z + 1 As long as the parameters are such that ln P Z depends on ln Ω ln Ω 1 ln Ω 1 γ < 1 ln Ω ln Ω γ. < 0 the sgn of ths expresson / γ whch measures how the elastcty of the subextensve margn vares wth γ. If / γ > 0 ndcatng that the elastcty of the subextensve ln γ margn decreases wth the ntal γ then we know that / γ < 0 mplyng that larger mporters wll change ther mport share by a greater percentage followng an mport ln Ω prce shock. If on the other h / γ < 0 then the net effect depends on whether the drect effect of a lower γ on rasng the percent change n γ exceeds the ndrect effect that rases the relatve prce of the optmal mport bundle relatve to domestc nputs by less. ln Ω We can wrte / γ as: ln Ω 1 γ κ 2 1 µ [ µ λ 1 ln P ] Z + 1 µ ln D where κ λ µ + µ 1 1 γ. ln P Z ln D do not vary wth γ. As long as ln P Z < 1 λ s suffcently large the senstvty to γ depends on whether s postve or negatve. µ 1 1 6
7 Appendx D: Addtonal Emprcal Analyses In ths appendx we consder two addtonal emprcal analyses. Frst we study the crosssecton of manufacturng ndustres. Next we consder changes n the frequency wth whch dropped nput varetes are permanently dropped. Startng wth the crosssectonal analyss we focus on wthnmanufacturng varaton because trade may plausbly account for mportant varaton n productvty. Dfferences n the productvty declnes of the fnance government sectors for example would lkely have lttle to do wth trade. The Argentne Annual Manufacturng Census ncludes nformaton on valueadded the number of salared workers n roughly 20 2dgt ndustres. We combne ths wth nformaton also from the Argentne natonal statstcs on producer prces n those same 2dgt ndustres to construct growth n real valueadded per worker. Ths s a proxy for total factor productvty at the sector level. Next we use the nformaton from the Captal IQ database on each frms Prmary Sector to classfy some of them as belongng to these 2dgt ndustres some sectors do not match are excluded. Fnally we combne 1 the mpled growth n mports for each manufacturng subsector from our data wth 2 nformaton on ntal sector levels of γ from the 1997 nputoutput table wth 3 growth n total sectoral ntermedate spendng from the census to obtan a tmeseres for sectoral γ. Ths s essentally the same method used to determne changes n γ for the overall manufacturng sector n our calbraton. Fgures 14a 14b show the resultng relatonshp between changes n ths measure of productvty n the share of nput spendng on mports 1 γ for the perod Fgure 14a shows ths plot for all subsectors wth avalable data but we note that some of these data ponts can reflect as few as two frms each. Fgure 14b shows ths relatonshp when ncludng only those sectors wth at least 20 matched frms. Both fgures suggest that shfts n ntermedate spendng from mported nputs toward domestc nputs correlate wth measured productvty declnes though the relatonshp s clearly nosy senstve to rules on the treatment of outlers. Gven ths senstvty gven ths s not our preferred welfarerelevant productvty measure we consder ths evdence less compellng robust than the results n the prmary paper for that reason only nclude them n ths Appendx. Next we consder the share of dropped varetes that are permanently dropped as a way to address concerns that nventored varetes are used for producton. In the paper n part to deal wth ths concern we focus on a 2 year perod. Ths s a horzon much longer than that typcally used by forecasters to descrbe the nventory cycle. But we addtonally fnd t useful here to measure the share of all frm mport varety combnatons from the prevous perod that are permanently dropped.e. permanent subextensve margn adjustment n the current perod. A varety s defned as HTS10xCountry s consdered to be permanently dropped f the frm does not agan mport t through 2008 we cannot go back earler n the analyss due to the gaps n our data n 1999 as dscussed n the paper. In ths analyss we only consder frms that mported at least 1 dollar of some good sometme after 2006 therefore exclude all frms whch permanently exted trade. Ths s a conservatve treatment of these extensve margn adjusters whch f ncluded would ncrease the set of permanently dropped goods. The share of varetes that are permanently dropped s plotted n Fgure 15. Early n the crss the lne jumps above 0.2 ndcatng that more than oneffth of all prevous 7
8 Change n ValueAdded per Worker Percent Per Year TV Equp Comp Equp Leather Metals Auto Ol Food Chemcals Paper Plastcs Wood Tobacco Med Equp Machnes Trans Equp PublshngApparel Textles Furnture Change n 1g Percentage Ponts Per Year a All Sectors Change n ValueAdded per Worker Percent Per Year Comp Equp Auto Med Equp Machnes Trans Equp Metals Ol Food Chemcals Paper Wood Textles Change n 1g Percentage Ponts Per Year b Sectors wth at Least 20 Frms Fgure 14: Sectoral Input Sourcng Productvty mportervarety combnatons are permanently dropped a level far n excess of anythng seen before or after the crss. Our analyss does not requre that varetes are permanently dropped but ths nonetheless confrms that not only dd the number of dropped varetes spke dramatcally upward durng the begnnng quarters of the crss as shown n the paper but many of those dropped varetes are permanently dropped. Whle ths does not elmnate a possble role for nventores n smoothng the use of nput varetes t lmts the extent to whch nventores could have substtuted for these dropped varetes. 8
9 Fgure 15: Share of Varetes that are Permanently Dropped by Contnung Frms 9
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