Internationa1 l Trade

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1 Iteratoa1 l Trade Class otes o 2/13/213 1 Eato ad Kortum (22) 1.1 Basc Assumptos N coutres, 1; :::; N Cotuum of goods u 2 [; 1] Prefereces are CES wth elastcty of substtuto : Z U Oe factor of producto (labor) 1 ( 1) q (u) ( 1) du There may also be termedate goods (more o that later) c ut cost of the commo put used producto of all goods Wthout termedate goods, c s equal to wage w coutry Costat returs to scale: Z (u) deotes productvty of (ay) rm producg u coutry Z (u) s draw depedetly (across goods ad coutres) from a Fréchet dstrbuto: wth > Pr(Z z) F (z) e Tz ; 1 (mportat restrcto, see below) Sce goods are symmetrc except for productvty, we ca forget about dex u ad keep track of goods through Z (Z 1 ; :::; Z N ). Trade s subject to ceberg costs d 1 d uts eed to be shpped from so that 1 ut makes t to All markets are perfectly compettve, 1 The otes are based o lecture sldes wth cluso of mportat sghts emphaszed durg the class. 1

2 6 1.2 Four Key Results The Prce Dstrbuto Let P (Z) c d Z be the ut cost at whch coutry ca serve a good Z to coutry ad let G (p) Pr(P (Z) p). The: G (p) Pr (Z c d p) 1 F (c d p) Let P (Z) mfp 1 (Z); :::; P N (Z)g ad let G (p) Pr(P (Z) p) be the prce dstrbuto coutry. The: G (p) 1 exp[ p ] where N X T(c d ) 1 To show ths, ote that (suppressg otato Z from here owards) Pr(P p) 1 Pr(P p) 1 [1 G (p)] Usg the G (p) 1 F (c d p) 1 [1 G (p)] 1 F (c d p) 1 e T(cd) p 1 e p The Allocato of Purchases Cosder a partcular good. Coutry buys the good from coutry f arg mfp 1 ; :::; p N g. The probablty of ths evet s smply coutry s cotrbuto to coutry s prce parameter, T (c d ) To show ths, ote that Pr P m P s If P p, the the probablty that coutry s the least cost suppler to coutry s equal to the probablty that P s p for all s 6 s 2

3 6 The prevous probablty s equal to s Pr(Ps p) s [1 G s (p)] e 6 6 p where X T (c d ) s Now we tegrate over ths for all possble p s tmes the desty dg (p) to obta Z 1 e p T T (c d ) p 1 e (c d ) p dp! Z T (c d ) 1 e p p 1 dp Z 1 dg (p)dp The Codtoal Prce Dstrbuto The prce of a good that coutry actually buys from ay coutry also has the dstrbuto G (p). To show ths, ote that f coutry buys a good from coutry t meas that s the least cost suppler. If the prce at whch coutry sells ths good coutry s q, the the probablty that s the least cost suppler s s Pr(P q) [1 G (q)] e q 6 s 6 s The jot probablty that coutry has a ut cost q of delverg the good to coutry ad s the the least cost suppler of that good coutry s the e q dg (q) Itegratg ths probablty e q dg (q) over all prces q p ad usg G (q) 1 e T(cd) p the Z p e q dg (q) Z p e q T (c d ) q 1 e T(cd) p dq T Z (c d ) p e q q 1 dq G (p) 3

4 Gve that probablty that for ay partcular good coutry s the least cost suppler, the codtoal dstrbuto of the prce charged by for the goods that actually sells s 1 Z p e q dg (q) G (p) I Eato ad Kortum (22): 1. All the adjustmet s at the extesve marg: coutres that are more dstat, have hgher costs, or lower T s, smply sell a smaller rage of goods, but the average prce charged s the same. 2. The share of spedg by coutry o goods from coutry s the same as the probablty calculated above. We wll establsh a smlar property models of moopolstc competto wth Pareto dstrbutos of rm-level productvty The Prce Idex The exact prce dex for a CES utlty wth elastcty of substtuto < 1 +, de ed as s gve by Z 1 1(1 ) p p (u) 1 du ; p 1 where 1(1 ) ; R where s the Gamma fucto, :e: (a) 1 x a 1 e x dx To show ths, ote that. Z p 1 1 Z 1 p 1 dg (p) Z 1 p (u) 1 du p 1 p 1 e p dp: 4

5 1 1 (1 ) De g x p, the dx p, p (x ), ad Z 1 p 1 (x ) (1 ) e x dx Z 1 (1 ) x (1 ) e x dx (1 ) Ths mples p wth > or 1 < fucto to be well de ed for gamma 1.3 Equlbrum Let X be total spedg coutry o goods from coutry P Let X X be coutry s total spedg We kow that X X, so T (c d ) X X (*) Suppose that there are o termedate goods so that c w. I equlbrum, total come coutry must be equal to total spedg o goods from coutry so w L X X Trade balace further requres X w L so that w L X T(w d ) P j T j(w j d j ) wl Ths provdes system of N 1 depedet equatos (Walras Law) that ca be solved for wages (w 1 ; :::; w N ) up to a choce of umerare. Ths s lke a exchage ecoomy, where coutres trade ther ow labor. Everythg s as f coutres were exchagg labor Fréchet dstrbutos mply that labor demads are so-elastc Armgto model leads to smlar eq. codtos uder assumpto that each coutry s exogeously specalzed a d eretated good 5

6 I the Armgto model, the labor demad elastcty smply cocdes wth elastcty of substtuto Uder frctoless trade (d 1 for all ; ) prevous system mples P 1+ T w P w L L j T jw j ad hece w w j T L T j L j 1(1+) 1.4 The Gravty Equato Lettg Y P X be coutry s total sales, the Y X T (c d ) X T c where X d X Solvg Tc from Y T c ad pluggg to (*) we get X Yd X Usg p 1 we ca the get X X Y d (p ) Ths s the Gravty Equato, wth blateral resstace d ad multlateral resstace terms p (ward) ad (outward) A Prmer o Trade Costs From (*) we also get that coutry s share coutry s expedtures ormalzed by ts ow share s X X S d p d X X p 6

7 Trade ad Geography 1 Normalzed mport share: (xl / x) / (xll / xl) Dstace ( mles) betwee coutres ad Image by MIT OpeCourseWare. Ths shows the mportace of trade costs ad comparatve advatage determg trade volumes. Note that f there are o trade barrers (.e, frctoless trade), the S X Lettg B X X X the B (S S ) d d Uder symmetrc trade costs (.e., d d ) the B 1 used as a measure of trade costs. d ca be We ca also see how B vares wth physcal dstace betwee ad : 2 How to Estmate the Trade Elastcty? As we wll see the trade elastcty s the key structural parameter for welfare ad couterfactual aalyss EK model Caot estmate drectly from B d emprcal couterpart of d the model because dstace s ot a Negatve relatoshp Fgure 1 could come from strog e ect of dstace o d or from mld CA (hgh ) Cosder aga the equato p d S p 7

8 Ecoometrc Socety. All rghts reserved. Ths cotet s excluded from our Creatve Commos lcese. For more formato, see If we had data o d, we could ru a regresso of l S o l d wth mporter ad exporter dummes to recover But how do we get d? EK use prce data to measure p d p : They use retal prces 19 OECD coutres for 5 maufactured products from the UNICP 199 bechmark study. They terpret these data as a sample of the prces p (j) of dvdual goods the model. They ote that for goods that mports from we should have p (j)p (j) d, whereas goods that does t mport from ca have p (j)p (j) d. Sce every coutry the sample does mport maufactured goods from every other, the max j fp (j)p (j)g should be equal to d. To deal wth measuremet error, they actually use the secod hghest p (j)p (j) as a measure of d. Let r (j) l p (j) l p (j). They calculate l(p p ) as the mea across j of r (j). The they measure l(p d p ) by max 2 j D P fr (j)g j r (j)5 p Gve S d p they estmate from l(s ) D. Method of momets: 8:28. OLS wth zero tercept: 8:3. 8

9 2.1 Alteratve Strateges Smoovska ad Waugh (211) argue that EK s procedure su ers from upward bas: Sce EK are oly cosderg 5 goods, maxmum prce gap may stll be strctly lower tha trade cost If we uderestmate trade costs, we overestmate trade elastcty Smulato based method of momets leads to a closer to 4. A alteratve approach s to use tar s (Caledo ad Parro, 211). If d t where t s oe plus the ad-valorem tar (they actually do ths for each 2 dgt dustry) ad s assumed to be symmetrc, the X X j X j d d j d j X j X j X d j d j d t t j t j tj t j t They ca the ru a OLS regresso ad recover. spec cato leads to a estmate of 8:22 Ther preferred 2.2 Gas from Trade Cosder aga the case where c w From (*), we kow that We also kow that T w X X 1 p, so! w p 1 T 1 1 : Uder autarky we have! A 1 1 T, hece the gas from trade are gve by A 1 GT!! Trade elastcty ad share of expedture o domestc goods are su cet statstcs to compute GT A typcal value for (maufacturg) s :7. Wth 5 ths mples GT :7 15 1: 74 or 7:4% gas. Belgum has :2, so ts gas are GT :2 15 1: 38 or 38%. 9

10 Oe ca also use the prevous approach to measure the welfare gas assocated wth ay foreg shock, ot just movg to autarky:!! ( ) 1 For more geeral couterfactual scearos, however, oe eeds to kow both ad Addg a Iput-Output Loop Image that termedate goods are used to produce a composte good wth a CES producto fucto wth elastcty > 1. Ths composte good ca be ether cosumed or used to produce termedate goods (put-output loop). Each termedate good s produced from labor ad the composte good wth a Cobb-Douglas techology wth labor share. We ca the wrte 1 c w p. The aalyss above mples T c p ad hece c 1 T 1 1 p Usg c w p 1 ths mples w p 1 1 T 1 1 p so w p 1 T 1 1 The gas from trade are ow!! A 1 Stadard value for s 12 (Alvarez ad Lucas, 27). For :7 ad 5 ths mples GT :7 25 1: 15 or 15% gas Addg No-Tradables Assume ow that the composte good caot be cosumed drectly. Istead, t ca ether be used to produce termedates (as above) or to produce a cosumpto good (together wth labor). 1

11 The producto fucto for the cosumpto good s Cobb-Douglas wth labor share. Ths cosumpto good s assumed to be o-tradable. The prce dex computed above s ow p g, but we care about! w p f, where p f w pg 1 Ths mples that w 1 g! (w p g ) wp 1 Thus, the gas from trade are ow!! A where 1 Alvarez ad Lucas argue that :75 (share of labor servces). Thus, for :7, 5 ad :5, ths mples GT :7 11 1: 36 or 3:6% gas 3 Comparatve statcs (Dekle, Eato ad Kortum, 28) Go back to the smple EK model above (, 1). We have X T (w d ) p X p XN T (w d ) X X w L As we have already establshed, ths leads to a system of o-lear equatos to solve for wages, 1 w L X T (w d ) P w k T L : k (w k d k ) Cosder a shock to labor edowmets, trade costs, or productvty. Oe could compute the orgal equlbrum, the ew equlbrum ad compute the chages edogeous varables. 11

12 But there s a smpler way that uses oly formato for observables the tal equlbrum, trade shares ad GDP; the trade elastcty, ; ad the exogeous shocks. Frst solve for chages wages by solvg ^ w^l Y X T^ w^d ^ P T^ ^ k k k w^kd k ad the get chages trade shares from ^ T w^d ^ ^ P : ^ k kt k w^kd ^ k w^ L ^ Y From here, oe ca compute welfare chages by usg the formula above, 1 amely!^ (^ ). To show ths, ote that trade shares are T (w d ) T (w d ) P ad k T k (w k d k ) P. k T k (w k d k ) Lettg x^ x x, the we have ^ P T (w d T^ w^d ^ ) k k k k P k Pj T j (w j d j ) T^ w^d ^ ^Tk w^kd ^ k Tk (w k d k ) P T j (w j d j ) j P k ^T w^d ^ k T^ k w^ kd ^ k : O the other had, for equlbrum we have w L X X w L ^ w L Lettg Y w L ad usg the result above for ^ we get X T^ w^d ^ w^l ^ Y P w^l ^ Y T^ w^ kd ^ k k k k 12

13 Ths forms a system of N equatos N ukows, w^, from whch we ca get w^ as a fucto of shocks ad tal observables (establshg some umerare). Here ad Y are data ad we kow d ^ ^, ^ ; T L, as well as. To compute the mplcatos for welfare of a foreg shock, smply mpose that L ^ ^ T 1, solve the system above to get w^ ad get the mpled ^ through T^ w^d ^ ^ P : ^ ^ k w^ ad use the formula to get k kt!^ ^ 1 kd k Of course, f t s ot the case that L ^ ^ T 1, the oe ca stll use ths approach, sce t s easy to show that autarky oe has w p 1 1, hece geeral T 4 Extesos of EK!^ T ^ 1 ^ 1 Bertrad Competto: Berard, Eato, Jese, ad Kortum (23) Bertrad competto ) varable markups at the rm-level Measured productvty vares across rms ) oe ca use rm-level data to calbrate model Multple Sectors: Costot, Doaldso, ad Komujer (212) T k fudametal productvty coutry ad sector k Oe ca use EK s machery to study patter of trade, ot just volumes No-homothetc prefereces: Feler (211) Rch ad poor coutres have d eret expedture shares Combed wth d ereces k across sectors k, oe ca expla patter of North-North, North-South, ad South-South trade 13

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