Supplemetary Appedix for Icome Differeces ad Iput-Output Structure September, 06 Appedix B: Extesios of the bechmark model 0. Log-Normally distributed IO coefficiets Cosider a more geeral versio of the model, where the elemets γ ji s of the iput-output matrix Γ are idepedet radom draws from a log-normal distributio ad are thus allowed to vary across coutries ad sectors. As we explai i more detail later, a log-normal distributio is a appropriate choice due to i equatio 9 establishig that sectoral multipliers ca be approximated by the sum of IO coefficiets i the correspodig row of the IO matrix shifted ad multiplied by /, ii the fact that sectoral multipliers are log-normally distributed, ad iii the sum of idepedet log-normal radom variables is approximately log-normal accordig to the Feto-Wilkiso method Feto, 960. Whe o-zero IO coefficiets are ot all equal to γ, the term js.t. γ ji 0 µ iγ ji log γ ji i equatio 7 is o loger equal to µ iγ log γ as i 0. Istead, we ca express it usig the approximatio of µ i i 9 ad extedig the fuctio γ ji log γ ji by cotiuity to γ ji = 0 for which i the limit it takes the value of 0: µ i γ ji log γ ji = γ is γ ji log γ ji = j= j= s= = γ is γ ji log γ ji γ is γ ii log γ ii = j i s= s= = γ is γ ji log γ ji γ is γ ii log γ ii j i s= s i γii log γ ii. To employ this i our estimatio, we eed to calculate the expectatio of this expressio. Give the
assumptio that all IO coefficiets are distributed idepedetly, we obtai that E µ i γ ji log γ ji = E [γ is ] E [γ ji log γ ji ] j= j i s= E [γ is ] E [γ ii log γ ii ] E [ γ ] ii log γ ii. s i The it remais to calculate the expectatios E [γ ij ], E [γ ji log γ ji ] ad E [ γ ii log γ ii]. First, let us deote by µ γ, σ γ the mea ad variace of the Normal distributio of logγ ij. E [γ ij ] ca be expressed i terms of these parameters usig the relatioship betwee the Normal ad log-normal distributios: E [γ ij ] = e µγ σ γ. The expressios for E [γ ji log γ ji ] ad E [ γ ii log γ ii] are less straightforward. They are established by the followig claim. Claim If x log-normal with parameters of the correspodig Normal distributio µ γ, σ γ, the E [x log x] = e µγ σ γ µγ σγ [ ad E x log x ] = e µγσ γ µγ σγ. Proof. E [x log x] = 0 x log x x e πσ γ log x µγ σ γ dx Let log x = y, so that d dx x. The E [x log x] = E [e y y] = Similarly, = πσγ µγ σγ σ4 γ σ = e γ πσγ E [ x log x ] = E [ e y y ] = = πσγ e y y e y µγ σγ d πσγ ye y µ γ yµγ σγ y σγ d 4µγ σγ 4σ4 γ σ = e γ πσγ ye [ y µγ σ σγ γ ] e y y e y µγ σγ d πσγ ye y µ γ yµγ 4σγ y σγ d ye [ y µγ σ σγ γ ] πσγ πσγ d e µγ σ γ πσγ πσγ = e µγσ γ ye y µγ σγ y d ye [ y µγ σ σγ µγ σγ. γ ] µγ σγ µ γ e σγ d ye y µγ σγ y d ye [ y µγ σ σγ µγ σγ. γ ] µγ σγ µ γ e σγ d
Collectig the terms, we obtai: E µ i γ ji log γ ji = E [γ is ] E [γ ji log γ ji ] j= j i s= E [γ is ] E [γ ii log γ ii ] E [ γ ] ii log γ ii = E [γ is ] E [γ ji log γ ji ] s i j i s= E [γ ii log γ ii ] E [γ ii log γ ii ] E [γ is ] E [ γii ] log γ ii = e µγ σ γ e µγ σ e µγσ γ γ s i µγ σ γ e µ γ σ γ µγ σγ e µ γ σ γ e µ γ σ γ µγ σγ µγ σ γ = [ e σ γ µγ e σ γ µγ ] µγ σ γ e σ γ µγ µ γ σ γ = = e σ γµ γ [ e σ γµ γ ] µγ σ γ e σ γµ γ µγ σ γ. Now, it remais to relate the distributio of γ ji s to the distributio of sectoral multipliers µ j, [ ] so as to express E j= µ iγ ji log γ ji i terms of earlier estimated parameters m µ, σµ. This relatioship is provided by equatio 9 accordig to which µ j γ ji. From this equatio it follows that Eµ = µ sum ad varµ = σ sum, where µ sum, σ sum are the mea ad the variace of the distributio of the sum γ ji. Now, while Eµ, varµ ca be expressed i terms of m µ, σ µ by meas of the relatioship betwee the Normal ad log-normal distributios, µ sum, σ sum ca be expressed i terms of µ γ, σ γ by meas of the Feto-Wilkiso method. This the provides us with the sought-after relatioship betwee parameters µ γ, σ γ ad m µ, σ µ. The Feto-Wilkiso method implies that the distributio of the sum γ ji of the idepedet log-normally distributed radom variables is approximately log-normal with e σ γ σsum = log, µ sum = log e µγ σ γ σsum = log e µ γ e σ γ σγ log. 3 Note that it is this method, i the first place, that justifies our assumptio that IO coefficiets γ ji s are log-normally distributed. Ideed, as the distributio of sectoral multipliers µ j has bee show to be log-normal, ad µ j γ ji, the sum γ ji must be distributed log-normally. By Feto-Wilkiso method, this is cosistet with γ ji s beig log-normal. Usig 3, equatios Eµ = µ sum, varµ = σ sum, ad the expressios for Eµ, varµ Eµ = e mµ/σ µ, varµ = e mµm Λσ µ [e σ µ ] 3
i footote, we derive: e σ γ = e σ sum = e varµ = e e mµm Λ σµ [e σµ ], e µγ = eµsum e σ sum = eeµ e varµ = = eemµ/σ µ e e mµm Λ σµ [e σµ ]. This is the relatioship betwee µ γ, σ γ ad m µ, σ µ. Let us deote the expressio for e σ γ the expressio for e µγ E j= by z. The usig this i, we obtai: by x ad [ µ i γ ji log γ ji = e σ γ µγ ] e µγ σ γ µγ σγ e σ γ µγ µ γ σγ = = x z[ x z]log x log z x z log z log x. [ ] Now we ca substitute this for E j= µ iγ ji log γ ji i the expressio for the expected aggregate icome, ad we arrive at E e mµm Λ/σµ σ Λ σ µ,λ γ E log γ log α logk e mµ/σ µ = e mµm Λ/σ µσ Λ σ µ,λ γ j= µ i γ ji log γ ji log Λ US i = x z[ x z ]log x log z x z log z log x log γ log α logk e mµ/σ µ log Λ US i. 4 This is the expressio for the expected aggregate icome i terms of parameter estimates used i the bechmark model aalogue of equatio 3. icome differeces i the settig with asymmetric IO likages. We brig it to estimatio ad predict cross-coutry 0. Cross-coutry differeces i fial demad structure Cosider ow the ecoomy that is idetical to our bechmark ecoomy i all but demad shares for fial goods. Namely, let us geeralize the productio fuctio for the aggregate fial good to accommodate arbitrary, coutry-sector-specific demad shares: Y = y β... yβ, where β i 0 for all i ad β i =. As before, suppose that this aggregate fial good is fully allocated to households cosumptio, that is, Y = C. 4
Usig the geeric expressio for aggregate output 7 of Propositio ad adoptig this expressio to the case of our ecoomy here, we obtai the followig formula for y: µ i λ i µ i γ ji log γ ji µ i γ i log γ i j s.t. γ ji 0 β i logβ i α log K. I this formula the vector of sectoral multipliers is defied differetly tha before, to accout for the arbitrary demad shares. The ew vector of multipliers is µ = {µ i } i = [I Γ] β. Its iterpretatio, however, is idetical to the oe before: each sectoral multiplier µ i reveals how a chage i productivity or distortio of sector i affects the overall value added i the ecoomy. Give this expressio for y, we ow derive the approximate represetatio of the aggregate output to be used i our empirical aalysis. For this purpose, we employ the same set of simplifyig assumptios as before, which results i: µ i Λ rel i γ µ i γ log γ log γ β i logβ i α logk µ i logλ US i. 5 Followig the same procedure as earlier, we use this expressio to fid the predicted value of y. First, we estimate the distributio of µ i, Λ rel i i every coutry. We fid that eve though the defiitio of sectoral multipliers is ow differet from the oe i our bechmark model, the distributio of the pair µ i, Λ rel i is still log-normal. The, usig the estimates of the parameters of this distributio, m ad Σ, together with the equatios see footote 3, we fid the predicted aggregate output Ey as a fuctio of these parameters: 3 E e mµm Λ/σµ σ Λ σ µ,λ γγ log γ log γ β i logβ i α logk e mµ/σ µ log Λ US i. 6 The resultig expressio for Ey is similar to 3 i our bechmark model. 0.3 Imported itermediates Aother extesio of the bechmark model allows for trade betwee coutries. The traded goods are used as iputs i productio of the competitive sectors, so that both domestic ad imported itermediate goods are employed i sectors productio techology. The the output of sector i is determied by the I fact, differetly from the bechmark model, the distributio is exactly log-normal ad ot trucated log-normal as it was before. 3 As before, we also assume for simplicity that all other variables o the right-had side of?? are o-radom. 5
followig productio fuctio: q i = Λ i k α i li α γi σ i d γ i i dγ i i... d γ i i f σ i i f σ i i... f σ i i, 7 where d ji is the quatity of the domestic good j used by sector i, ad f ji is the quatity of the imported itermediate good j used by sector i. The imported itermediate goods are assumed to be differet, so that domestic ad imported goods are ot perfect substitutes. Also, with a slight abuse of otatio, we assume that there are differet itermediate goods that ca be imported. 4 The expoets γ ji, σ ji [0, represet the respective shares of domestic ad imported good j i the techology of firms i sector i, ad γ i = j= γ ji, σ i = j= σ ji 0, are the total shares of domestic ad imported itermediate goods, respectively. As i our bechmark ecoomy, each domestically produced good ca be used for fial cosumptio, y i, or as a itermediate good, ad all fial cosumptio goods are aggregated ito a sigle fial good through a Cobb-Douglas productio fuctio, Y = y... y. Now, i case of a ope ecoomy cosidered here, the aggregate fial good is used ot oly for households cosumptio but also for export to the rest of the world; that is, Y = C X. The exports pay for the imported itermediate goods ad are defied by the balaced trade coditio: X = p j f ji, 8 j= where p j is the exogeous world price of the imported itermediate goods. Note that the balaced trade coditio is reasoable to impose if we cosider our static model as describig the steady state of the model. Aggregate output y is determied by equatio 7 of Propositio, adopted to our framework here: µ µ i λ i µ i γ ji log γ ji i γ i σ i j s.t. γ ji 0 µ i σ ji log p j µ i γ i σ i log γ i σ i log log j= σ i µ i α log K, j s.t.σ ji 0 µ i σ ji log σ ji where vector { µ i } i = [I Γ] is a vector of multipliers correspodig to Γ ad Γ = { γ ji } ji = { σ i γ ji } ji is a iput-output matrix adjusted for shares of imported itermediate goods. 5 I the empirical aalysis we use a approximate represetatio of aggregate output, where a rage 4 This is cosistet with the specificatio of iput-output tables i our data. 5 Observe that I Γ exists because the maximal eigevalue of Γ is bouded above by. The latter is implied by the Frobeius theory of o-egative matrices, that says that the maximal eigevalue of Γ is bouded above by the largest colum sum of Γ, which i our case is smaller tha as soo as σ i γ i < : j= σi γji}ji = σi γ i <. 6
of simplifyig assumptios is imposed. First, to be able to compare the results with the results of the bechmark model, we employ the same assumptios o i-degree ad elemets of matrix Γ. Secod, i the ew framework with imported itermediates we also impose some coditios o imports. We assume that the total share of imported itermediate goods used by ay sector of a coutry is sufficietly small ad idetical across sectors, that is, σ i = σ for ay sector i. 6 We also regard ay o-zero elemets of the vector of import shares of sector i as the same, equal to σ i such that j s.t.σ ji 0 σ i = σ. The we obtai the followig approximatio for the aggregate output y: σ γ µ i σ i j s.t.σ ji 0 µ i Λ rel i log p j log µ i γ log γ γ σ γ σ γ µ i σ log σ i log γ σ σ γ σ α log K µ i logλ US i. Now, usig equatios see footote 3 for the parameters of the bivariate log-normal distributio of µ i, Λ rel i, we ca derive the predicted aggregate output Ey: E σ γ emµm Λ/σµ σ Λ σ µ,λ σ log σ i σ i σ γ j=,j s.t. σ ji 0 log p j logλ US i e mµ/σ µ γγ log γ σ γ log σ γ log γ σ σ γ σ α logk γ σ γ. We brig this expressio to data ad evaluate predicted output i all coutries of our data sample. We ote, however, that the vector of world prices of the imported itermediates {p j } j= is ot provided i the data. The to make the compariso of aggregate icome i differet coutries possible, we assume that for ay sector i, the value of σ i j=,j s.t. σ ji 0 log p j is the same across coutries, so that this term cacels out whe the differece i coutries predicted output is cosidered. For this purpose we assume that i all coutries, the vector of shares of the imported itermediate goods used by sector i is the same ad that all coutries face the same vector of prices of the imported itermediate goods {p j } j=. 0.4 Skilled labor Cosider the ecoomy of our bechmark model where we itroduce the distictio betwee skilled ad uskilled labor. This distictio implies that the techology of each sector i : i every coutry 6 This allows approximatig log σi µi with σ µi = σ γ σ, where the equality follows from µi µ i j= σj. The latter, i tur, is a result of the approximatio of { µi}i by the first elemets of the coverget power series k=0 Γ k ad the aalogous approximatio for {µ i} see sectio 3.3. 7
ca be described by the followig Cobb-Douglas fuctio: q i = Λ i ki α u δ i si α δ γi σ i d γ i i dγ i i... d γ i i, 9 where s i ad u i deote the amouts of skilled ad uskilled labor used by sector i, γ i = j= γ ji is the share of itermediate goods i the total iput use of sector i ad α, δ, α δ 0, are the respective shares of capital, uskilled ad skilled labor i the remaider of the iputs. The total supply of skilled ad uskilled labor i the ecoomy is fixed at the exogeous levels of S ad U, respectively. I this case, the logarithm of the value added per capita, log Y/U S, is give by the expressio 7 of Propositio, adopted to our framework here. I fact, it is oly slightly differet from the expressio for y i our bechmark model cf. Propositio, where δ = 0 ad the total supply of labor is ormalized to. With skilled ad uskilled labor, the aggregate output per capita is give by: µ i λ i j s.t. γ ji 0 µ i γ ji log γ ji µ i γ i log γ i log α log K δ log U α δ log S logu S. The the approximate represetatio of y is also similar to the correspodig represetatio of y i the bechmark model cf. 0: µ i Λ rel i µ i γ log γ log γ log α logk δ log U α δ log S logu S γ where the same assumptios ad otatio as before apply. µ i logλ US i, 0 We ow employ this represetatio of y to fid the predicted value of aggregate output Ey. Note that sice the ew framework, with skilled ad uskilled labor, does ot modify the defiitio of the sectoral multipliers, the distributio of the pair µ i, Λ rel i i every coutry remais the same. It is a bivariate log-normal distributio with parameters m ad Σ that have bee estimated for our bechmark model. Usig these parameters, together with the equatios i see footote 3, we derive the expressio for the predicted aggregate output Ey i terms of the estimated parameters: E e mµm Λ/σµ σ Λ σ µ,λ γγ log γ log γ log α logk δ log U α δ log S logu S e mµ/σ µ log Λ US i. This equatio for the predicted aggregate output is aalogous to the equatio 3 that we employed i our estimatio of the bechmark model. 8
Appedix D: Additioal Figures ad Tables Figure A-: Distributio of sectoral i-degrees left ad out-degrees right GTAP sample Table A-: Coutries: WIOD Sample coutries AUS IDN AUT IND BEL IRL BGR ITA BRA LTU CAN LVA CHN MEX CYP MLT CZE NLD DEU POL DNK PRT ESP ROM EST RUS FIN SVK FRA SVN GBR SWE GRC TUR HUN USA 9
Table A-: Coutries: GTAP Sample coutries ALB LTU ARG LUX AUS LVA AUT MDG BEL MEX BGD MLT BGR MOZ BRA MWI BWA MYS CAN NLD CHE NZL CHL PER CHN PHL COL POL CYP PRT CZE ROM DEU RUS DNK SGP ESP SVK EST SVN FIN SWE FRA THA GBR TUN GRC TUR HKG TWN HRV TZA HUN UGA IDN URY IND USA IRL VEN ITA VNM JPN ZAF KOR ZMB LKA ZWE 0
Table A-3: Sector List WIOD sectors GTAP sectors Agriculture Agriculture Miig Coal 3 Food 3 Oil 4 Textiles 4 Gas 5 Leather 5 Miig 6 Wood 6 Food 7 Paper 7 Textiles 8 Refiig 8 Apparel 9 Chemicals 9 Leather 0 Plastics 0 Wood Mierals Paper Metal products Refiig 3 Machiery 3 Chemicals 4 Elec. equip. 4 Mierals 5 Trasport equip. 5 Iro 6 Maufacturig ec 6 Oth. metals 7 Electricity 7 Metal products 8 Costructio 8 Cars 9 Car retail. 9 Trasport equip. 0 Wholesale trade 0 Electric equip. Retail trade Oth. Machiery Restaurats Mauf. ec 3 Ilad trasp. 3 Electricity 4 Water trasp. 4 Gas Distr. 5 Air trasp. 5 Water Distr. 6 Trasp. ec. 6 Costructio 7 Telecomm. 7 Trade 8 Fi. serv. 8 Ilad trasp. 9 Real est. 9 Water trasp. 30 Busiess serv. 30 Air trasp. 3 Pub. admi. 3 Telecomm. 3 Educatio 3 Fiacial serv. 33 Health 33 Isurace 34 Social serv. 34 Busiess serv. 35 Household empl. 35 Recreatio 36 Educatio, Health 37 Dwelligs