The Gains from Input Trade in Firm-Based Models of Importing by Joaquin Blaum, Claire Lelarge and Michael Peters

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1 The Gans from Input Trade n Frm-Based Models of Importng by Joaqun Blaum, Clare Lelarge and Mchael Peters Onlne Appendx Not for Publcaton Ths Appendx contans the followng addtonal results and materal:. Detals about the dentfcaton of the Input-Output Matrx Ξ, 2. Estmates of the producton functon coeffcents, 3. A descrpton of the bootstrap procedure, 4. An extenson of the welfare equaton 4 to a mult-sector envronment, 5. Detals about the algorthm used to calbrate the model of Secton 4. A. Identfcaton of the Input-Output and Demand Structure We use the French nput-output tables from the OECD to dscplne the demand parameters [α s and the matrx of nput-output lnkages Ξ. To determne Ξ, we focus on the ntermedate supply from each ndustry j to each ndustry s. We abstract from any taxes and subsdes. As Ξ can be dentfed from expendture shares by sourcng sector, see 25, we set ζ s j = Intermedate supply from ndustry j to ndustry s Intermedate consumpton at fnal prces from ndustry s. That s, ζj s measures the mportance of sector j n the producton process of sector s. By constructon, ths ensures that S j= ζj s = for all s. We arrange the nput-output matrx so that the columns contan the dstrbuton of expendture for the dfferent sectors: ζ ζ 2 ζ S ζ2 Ξ = ζ S To determne [α s, we also use the nput-output tables as they contan nformaton on the composton of fnal demand. Snce there s no trade n fnal goods n the theory, we exclude any exports and mports n fnal goods n the data. More specfcally, the nput-output tables report fnal consumpton expendture by households on sector j, denoted by HHF C j. Followng 25, we hence set α s = ζ S S. HHF C s Sj= HHF C j.

2 Drect Data Aggregaton ISIC α Value Intermedate γ Λ Coarse α γ Added Purchases Classfcaton α V A X X X V A 2 α 2 V A 2 X 2 X 2 X 2 V A Non-Manufacturng α S from 8 γ S from 82 α V A X 6 Estmate Read off... from from Manufacturng α γ 37 α 37 V A 37 X 37 mcro-data mcro-data α 37 γ 37 4 α 4 V A 4 X 4 X 4 X 4 V A α 99 V A 99 X 99 X 99 X 99 V A 99 Table 4: Measurement of α s and γ s Non-Manufacturng α S from 8 γ S from 82 The OECD nput-output tables report ther data at the 2-dgt level of ISIC Rev. 3, whch gves 37 manufacturng ndustres. To deal wth the non-manufacturng ndustres, we group them nto a resdual sector whch we denote by S. To ncorporate ths sector n the analyss, we set α S = α j, 8 where M s the set of manufacturng sectors. Because n our theory ths sector s not engaged n foregn sourcng 76, we set j M Λ S =. The nput-output structure of sector S can be recovered from the nput-output tables. In partcular, we set ζ S j NM n= Intermedate supply from ndustry j to ndustry n NM n= Intermedate consumpton at fnal prces to ndustry n, where N M s the set of non-manufacturng sectors. To measure the materals coeffcent n the producton of sector S, we employ the Input-Output Matrx for the non-manufacturng sectors. As we observe value added and ntermedary spendng for each sector, we set γ S = NM n= X n NM n= X n V A n, 82 where X n denotes total ntermedary spendng by sector n. Table 6 summarzes how [α s and [γ s are computed. The nput-output matrx Ξ used n our emprcal analyss s contaned n Table Note that ths sector may nevertheless beneft from nput trade f t sources output from the manufacturng ndustres. 2

3 Sector S S Notes: The table contans the French nput-output matrx used n our emprcal work. We report numbers n percentage terms. Sectors are classfed at the 2-dgt-level accordng to ISIC Rev. 3. The non-manufacturng sector S s constructed as explaned n the man text and Table 6. Table 5: Input-Output Lnkages: Ξ 3

4 Industry ISIC φ k φ l γ Mnng *** ***.7.333***.43 Food, tobacco, beverages ***.4.77***.3.725***.6 Textles and leather 7-9.8***.3.293***.9.626***.2 Wood and wood products 2.3***.4.285***.6.62***.6 Paper, prntng, publshng ***.7.362***..54***. Chemcals 24.24***.8.24***..67***.4 Rubber and plastcs products 25.24***.5.289***.7.587***. Non-metallc mneral products 26.78***..294***.2.529***.5 Basc metals 27.24***..22***.5.674***.2 Metal products ex machnery and equpment 28.8***.2.42***.8.479***.9 Machnery and equpment 29.7***.3.33***.5.66***.8 Offce and computng machnery 3.37***.2.5***.32.83***.4 Electrcal machnery 3.96***.8.36***..598***.4 Rado and communcaton 32.55***.6.322*** ***.52 Medcal and optcal nstruments 33.7***.4.435*** ***.29 Motor vehcles, tralers 34.6***.9.35***.6.759***.4 Transport equpment 35.52***.9.499***.3.349***.44 Manufacturng, recyclng ***.3.283***.9.633***.2 Notes: The table contans the producton functon parameters based on observed factor shares. See Secton 3. n the man text for detals. Table 6: Producton Functon Coeffcent Estmates, by 2-dgt Sector: Factor Shares A.2 Estmatng the Parameters of the Producton Functon We report the results of estmatng the producton functon parameters usng our dfferent approaches. In Table 6, we report the results of the factor share approach. Note that ths method mposes the assumpton of constant returns, so that φ ks φ ls γ s =. Table 7 reports the results based on proxy methods akn to Levnsohn and Petrn 22 and Wooldrdge 29. We do not mpose constant returns to scale for these approaches. We assume that labor s a dynamc nput, whch seems plausble gven the strngent hrng and frng regulatons of the French economy. Note that we do not nclude frms domestc share n materal spendng n the producton functon as we estmate ε n the second stage. Fnally, Table 8 contans the results of the ntegrated GMM approach, where we treat the domestc expendture share as a dstnct nput and estmate the parameter vector φ ks, φ ls, γ s, ε s n one step. A.3 Bootstrap Procedure We sample frms from the emprcal dstrbuton wth replacement to construct 2 replcates of our mcro-data. For each of these samples, we re-calculate s and re-estmate ε and [γ s followng [ the factor shares approach explaned n Secton 3. and then re-calculate [Λ s and. Fgure 6 depcts the bootstrap dstrbutons of these varables. For the three sector-level varables, we report the dstrbuton of the sectoral averages, e.g. the upper rght panel dsplays the dstrbuton Ss= γ s. Whle the varaton n γ and s Agg D s relatvely modest, there s a qute a bt of of S s Agg Ds 4

5 Industry ISIC φ k φ l γ Mnng ***..626*** **.39 Food, tobacco, beverages ***..274***.9.538***.6 Textles and leather ***.26.53***.37.48***.96 Wood and wood products 2.38***.23.44***.24.52***.58 Paper, prntng, publshng ***.22.77***.33.6***.99 Chemcals ***.336 Rubber and plastcs products 25.48***.3.536***.5.357***.5 Non-metallc mneral products 26.22*** *** ***.9 Basc metals ***.87.48*.263 Metal products ex machnery and equpment ***.3.655***.6.23***.36 Machnery and equpment 29.86***.8.563*** ***.66 Offce and computng machnery 3.7**.8.574***..4.2 Electrcal machnery 3.44***.3.448*** ***.23 Rado and communcaton 32.23** ***.4.568***.27 Medcal and optcal nstruments 33.23***.2.5***.2.42***.73 Motor vehcles, tralers ** **.355 Transport equpment 35.94*** *** ***.256 Manufacturng, recyclng ***.8.472***.6.43***.55 Notes: The table contans the producton functon parameters based on the GMM procedure by Levnsohn and Petrn 22 and Wooldrdge 29. See Secton 3. n the man text for detals. Table 7: Producton Functon Coeffcent Estmates, by 2-dgt Sector: GMM Industry ISIC φ k φ l γ ε Mnng ***..67*** * Food, tobacco, beverages ***..278***.9.52*** ***.66 Textles and leather *** *** *** ***.743 Wood and wood products 2.72*** ***.25.44*** ***.324 Paper, prntng, publshng *** *** *** ***.383 Chemcals ***.84.64*** Rubber and plastcs products 25.63*** *** ** **.3 Non-metallc mneral products 26.22*** *** ***.2.869***.358 Basc metals *** Metal products ex machnery and equpment ***.4.667***.6.9*** ***.796 Machnery and equpment 29.83***.8.553***.24.38*** ***.279 Offce and computng machnery *** Electrcal machnery 3.38***.28.46*** ***. 2.86***.863 Rado and communcaton 32.32*.69.64*** * Medcal and optcal nstruments ***.2.498***.2.372*** ***.295 Motor vehcles, tralers ***.2.76** * 2.88 Transport equpment 35.7**.8.664*** *** Manufacturng, recyclng ***.9.468***.7.36*** ***.83 Notes: The table contans the producton functon parameters based on the GMM procedure by Levnsohn and Petrn 22 and Wooldrdge 29. See Secton 3. n the man text for detals. Table 8: Producton Functon Coeffcent Estmates, by 2-dgt Sector: GMM wth s D as nput 5

6 .5 Bootstrap Dstrbuton of 8 Bootstrap Dstrbuton of Bootstrap Dstrbuton of 6 Bootstrap Dstrbuton of agg. dom. share Notes: The upper left panel contans the bootstrap dstrbuton of ε. The remanng panels depct the bootstrap dstrbutons of S S γs, S s= S s= Λs and S S s= sagg. The pont estmates used n the man analyss are reported as Ds vertcal lnes. Fgure 6: Bootstrap Dstrbuton of Structural Parameters and Drect Prce Reductons uncertanty regardng ε. Ths s consstent wth the non-neglgble standard errors reported n Table 2. We conclude that t s the varaton n ε whch nduces most of the varaton n Λ and therefore n the consumer prce gans from nput trade reported n Tables 5-6 and shown n Fgure 5. A.4 General equlbrum and Welfare n the Model of Secton 4. Consder the setup of Secton 4.. We now consder the aggregate allocatons n ths economy. An equlbrum has the usual defnton: [ Defnton. An equlbrum s a set of prces w, [p, labor demands for producton [ and fxed costs l, l F, dfferentated product quanttes, consumpton levels and foregn demands y, c, y ROW, domestc and nternatonal nput demands by local frms [y v, [z c and sourcng strateges [n such that:. Frms maxmze profts gven by 36-37, 2. Consumers maxmze utlty gven by 2 and 3 subject to ther budget constrant p c d = wl π d, 83 6

7 3. Trade s balanced 39, 4. Labor and good markets clear L = l l F d y = c y ROW ν y v dv. We fst characterze the general equlbrum n a mult-sector verson of the economy of Secton 4.. In partcular, we consder the mult-sector structure of Secton 2.2. We derve a generalzaton of 4. We do not mpose any assumptons on how frms determne ther extensve margn. That s, we allow for an arbtrary mappng l Σ whch gves the labor resources that frm needs to spend n order to attan the sourcng strategy Σ. We assume that trade s balanced and that the value of exports n sector s s gven by αs ROW mported nputs. IM, where IM denotes the value of total spendng on Proposton 4. Let W,I and S denote welfare, consumer ncome and total spendng, respectvely. Then, the change n welfare relatve to nput autarky s gven by where I and I Aut are gven by [ and [S j and S s = α s L and S Aut j j= solve Nj l Σ d W W Aut = I P Aut IAut P, Ns I = L S s / s l Σ d, 84 s= s= I Aut = L Ss Aut / s, 85 s= ζj s α s γ j j= S j j= [ α ROW s ζ j s Nj γj S j s D ω d, 86 s = α s L S Aut j= ζ j sγ j /α s Sj Aut, 87 Furthermore, G = P Aut P s gven n Proposton 2. Proof. As labor s the only factor of producton, consumer welfare s gven by real ncome W = I/P, 7

8 consumer ncome s gven by I = L s= Ns Note that L represents total labor ncome and π denotes frm s profts. To derve π, recall that frms n sector s have a mark-up of s / s so that varable profts gross of any extensve margn resource loss are gven by Total revenue for frm s gven by π d. π V = p u y = p y / s. 88 p y = p P s s Ss, 89 where P s s the consumer prce ndex for sector s and S s denotes total spendng for sector s goods. Hence, so that π = p y / s l Σ = s p Ss l s P, Σ s I = L S s s s= s= Ns l Σ d. 9 Hence, gven [S s and [l Σ, total ncome I s fully determned. Now consder [S s s. Note that S s = S C s S X s S ROW s, 9 where Ss C, Ss X and Ss ROW denote total spendng by consumers, ntermedary producers and the rest of the world, respectvely. For our economy, we have that Sc C = α s I and Ss ROW = αs ROW Im as consumers spend a fracton α s of ther ncome I on sector s products and balanced trade requres that total spendng by the rest of the world s equal to the value of mports Im, a fracton α ROW s of whch s spent on sector s products. To derve S X s, let total domestc ntermedary purchases n sector j be gven by X j. Then Ss X = ζsx j j. 92 j= Lettng m be total materal spendng by frm and s be total spendng by frm, we know that X j = Nj s D m d = = γ j S j Nj Nj s D p P j s D γ j s d = Nj s D γ j p y d j d, 93 where we used that frms n sector j spend a fracton γ j of ther total nput spendng s on materals and that total spendng s accounts for a fracton / of revenue. Hence, 92 and 93 mply 8

9 that S X s = j= Smlarly, total mport spendng s equal to ζsγ j Nj s p j S j s D d. 94 = P s Im = = Nj Im j = s D m d j= j= Nj s p γ j S j s D, d. 95 j= j = P s Hence 94 and 95 mply that S s = α si αs ROW Nj j p γ j S j s D d P j = α si j= Usng 9, we get that S s = α s L Now note that Hence, S s = α s L j= j= j= ζ j sγ j S j Nj va Ns l Σ d j= [ α ROW s ζ j s ζj s α s γ j j= va d = p y Ns Nj l Σ d j= Nj j p γj S j s D d. P j S j j= [ α ROW s p y d = p /P s s S s Ns ζj s α s γ j j= ζsγ j Nj j p j S j s D d P j ζ j s p /P s s S s d = Nj S j j= [ α ROW s Nj j p γj S j s D d. P j s p. ζ j s P s Nj γj S j s D ω d, 96 where ω = va. Gven Ns va d LNET = L S j= l Σ d, 96 are S equatons n S unknowns S s, whch we can easly solve. Now consder the case of autarky. There we have l Σ = and s D =. Hence, 96 yelds s = α s L S Aut ζj s α s γ j S Aut j. j= j In the case of a sngle sector.e. S = t has to be the case that α S = α ROW S = ζ S S =. Hence, S Aut = L γ S Aut = γ L. 9

10 Substtutng ths n 9 yelds Smlarly, we get from 96 that j= I Aut = L [ αs ROW ζs j γ j γ S = L. γ S j Nj s D ω d = so that S = I = N L l Σ d 97 γ γ N L l Σ d. 98 γ Ths mples drectly 4. Ths concludes the proof of Proposton 4. A.5 Calbratng the Model of Secton 4 We adopt a soluton algorthm that allows us to bypass the computaton of the general equlbrum varables wthn the calbraton. Intutvely, we work wth a normalzed verson of fxed costs, where these are scaled by an approprate transformaton of the general equlbrum varables. Because the equlbrum varables depend on frms mport behavor only through the domestc shares, whch are tself a calbraton target, we can compute them after the calbraton. That s, we can frst ensure that the moments of the jont dstrbuton of value added and domestc shares are matched 77, and then back out the underlyng general equlbrum varables requred to compute welfare. We also show that the parameter z s not requred for the calbraton. We frst start wth three aggregate varables, whch are determned n equlbrum. In the snglesector verson of the model, characterzed n Secton A.4 n the Onlne Appendx, we have that aggregate spendng S and the prce level whch s also equal to the prce of domestc varetes s gven by S = P = N L l Σ d γ 99 γ γ γ γ Υ, γ γ q D where N Υ = s D, = ϕ γ/ε d. 77 For ths step, t s mportant that the dsperson and correlaton moments are n logs. See below.

11 We start by notng that the frm s optmalty condtons from the proft maxmzaton problem, contaned n Secton 7.6, can be expressed n terms of s D nstead of n. To see ths, note that 8 and 35 mply n ηε sd β ε = z ε qd β Substtutng 2 nto the frm s frst order condton 73, we obtan s D p D ε. 2 γ η ε ε η sd s D ηε β = β ε η f, 3 ϕ where where f f zq D /η ηγ Θ P /η Γ, 4 Θ = γ γ, 5 γ γ q D S Γ =. 6 P γ Smlarly, 2 and the mport status condton 79 mply that the frm s an mporter as long as where [ s γ ε D ϕ sd s D ηε γ η β β ε ε η f fi >, 7 f I Γ Θ f I. 8 3 and 7 show that we can solve for frms optmal domestc share and mport status wth knowledge of ϕ, f and f I only. Thus, we can work wth the jont dstrbuton of ϕ, f to match the moments of the jont dstrbuton of domestc shares and value added. We can then back out the exogenous component of fxed costs f I and f from f I and f usng the equlbrum varables S and P and 6. To solve for S, we requre the aggregate resource loss of fxed costs see 99. To do so, note that sd η ε l Σ = l s D = f s D P { = ΓΘ ηγ f sd /η q D z /η β s D β η ε β β ε ε η fi ε ε η fi }. Hence, N l Σ d = ΓΘ { ηγ N f sd s D η ε β β ε ε η d N f I [s D d }. 9

12 The key s now to argue that Γ s known gven the calbraton. If so, we can calculate N l Σ d from 9 gven the calbrated f and f I and parameters, as where N ηγ f Recall that 6 and 99 mply that Γ = = Solvng for Γ yelds N sd s D S P γ = P γ P γ l Σ d = Γ Θ, η ε β β ε ε η d N N L l Σ d γ γ L P γ Γ = P γ P γ γ γ γ ΓΘ. f I [s D d. Θ L. As L s a normalzaton see below, shows that Γ s fully determned as P can be evaluated from the calbrated data on domestc shares see and. Hence, N l Σ d = ΓΘ = P γ P γ γ Θ γ Θ L. Ths mples that L N L l Σ d = P γ 2 γ Θ, so that L s ndeed a normalzaton. Fnally we only have to show that 2 does not depend on q D, even though Θ does see 5. However, t can easly be shown that ΘP γ = Υ. 3 Hence, the qualty of domestc varetes q D and the foregn prce level z can be normalzed for the calbraton. The fve models we consder ft n ths framework as follows:. The aggregate model assumes that β = β and f = f I =. Hence, N l Σ d = and s D = s D can be solved from 2 usng that n = as all frms are mporters and mport from every country. The level of β s chosen to match the aggregate domestc share. The dsperson n productvty ϕ s chosen to match the dsperson n value added. 2. The homogenous bas model assumes that β = β and f = < f I. Hence, condtonal on 2

13 mportng, we have that s D = s D, whch can be solved from 2 usng that n =. The requred level fi n 7 s chosen to match the share of mporters. Gven a dstrbuton of productvty [ ϕ we can then calculate from, P from and and hence Γ from. Ths s suffcent to calculate welfare usng 2 and P Aut /P. 3. The heterogeneous bas model assumes that β vares across frms and f = < f I. As for β β. the case wth fxed costs, t s useful to consder a scaled verson of the home-bas β = ε ε In partcular, 2 shows that s D only depends on β = β p agan, we have n = as there are no fxed costs per country. Hence, we draw ϕ, β from a jont log-normal dstrbuton. Usng 2, ths generates a jont dstrbuton of ϕ, s D. We can then calbrate f I from 7 to match the share of mporters. Lke for the case of the homogenous bas model, we can then use, P and to compute all equlbrum objects. 4. For the heterogenous fxed cost model, we draw ϕ. f from a jont log-normal dstrbuton. Usng the 3, ths mples a jont dstrbuton of ϕ, s D. We can then calbrate f I from 7 to match the share of mporters. As above, we can then use, P and to compute all equlbrum objects. 5. The homogenous fxed cost model, s a specal case of the heterogenous fxed cost model where f = f. Hence, the procedure s exactly the same gven a margnal dstrbuton for ϕ. 3