Online Appendix: Measured Aggregate Gains from International Trade

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1 Ole Appedx: Measured Aggregate Gas from Iteratoal Trade Arel Burste UCLA ad NBER Javer Cravo Uversty of Mchga March 3, 2014 I ths ole appedx we derve addtoal results dscussed the paper. I the frst secto, we clude two extesos of the model wth perfect competto of Secto 3. Frst, we corporate termedate puts to the producto fucto. Secod, we cosder the case whch oe coutry operates the tradg techologes of all coutres. For each of these two extesos, we derve real GDP from the producto sde ad from the expedture sde, ad show the equvalece betwee real GDP ad real cosumpto at the world level. I dervg our secod exteso, we cosder the case (dscussed Footote 16) where the tradg techology s operated by termedares rather tha by the producers themselves. I the secod secto we show that our assumpto equato (19) of the paper that aggregate profts represet a costat share of total reveues (P t = k Y t, where Y t = W p t t q t dm t ), s satsfed the remag two specal cases of our model descrbed Secto 4. I dervg these results, we show that uder the assumptos of Collorares 2 ad 3, respose to chages varable trade costs the mass of eterg frms s costat across steady states. 1 Geeral Producto Techologes wth Perfect Competto 1.1 Allowg for termedate puts I ths secto we exted our model Secto 3 to allow for termedate puts. We allow for a geeral specfcato whch each good z ca be used both for cosumpto 1

2 ad as a termedate put. We deote by q c (z) ad qm (z) the quattes of good z from coutry used coutry for cosumpto ad as termedate puts. Market clearg mples q (z)=q c (z) + qm (z) Real GDP, producto sde We start by dervg expresso (8) the paper. The proft maxmzato problem s gve by: P t = max {y (z), x (z)} W p t (z) y (z) /t t W t x (z) dz d t p t (z) q m t (z) dz subject to y (z), q m t (z), x (z) 2 Y (z) for all z. (1) Curret-dollar GDP s gve by: GDP t = p t (z) y t (z) /t t dz W p t (z) q m t (z) dz. Real GDP from the producto sde s calculated usg the double deflato method, whch cosst deflatg output ad termedate puts usg dfferet prce dexes. I partcular, we calculate real GDP as: RGDP t = W p t (z) y t (z) /t t dz PPI t p t (z) q m t (z) dz. MPI t Where MPI t deotes the put prce dex coutry. To a frst order approxmato, the chage RGDP s gve by: dlogrgdp t = p t0 (z) y t0 (z) /t t0 (dlog (p W GDP t (z) y t (z) /t t ) dlogppi t ) dz t0 p t0 (z) q m t 0 (z) dlogmpi t dz. (2) p t0 (z) qm t0 (z) GDP t0 The chage the producer prce dex s gve by equato (7) the paper. The chage the put prce dex s gve by: dlogmpi t lt m (z) dlogp t (z) dz. 2

3 where lt m p t0 (z)q m t (z) 0 (z) = 0 W p t 0 (z)q m t (z). Substtutg dlogppi t ad dlogmpi t expresso 0 (2), ad followg the steps used Proposto 1 the paper, we ca wrte the chage aggregate productvty as: dloga t = dlogrgdp t W t0 X t0 GDP t0 dlogx t = l t0 l t0 (z)(dlogp t (z) dlogt t dlog p t (z)) dz (3) W + d t0 1 p t0 (z) qm t0 (z) dlogq m t GDP 0 (z) dz. t0 Note that the absece of tarffs o termedate puts, equato (3) cocdes wth expresso (8) the paper Real GDP, expedture sde We ow derve the equvalece betwee real GDP from the producto ad the expedture sde. To a frst order approxmato, the chage real GDP from the expedture sde s gve by, GDPt E 0 dlogrgdpt E = d t0 p t0 (z) q c t 0 (z)(dlog (d t p t (z) q c t (z)) W + p t0 (z) q t0 (z)(dlog (p t (z) y t (z) /dlogt t ) 6= 6= dlogcpi t) dz dlogepi t ) dz p t0 (z) q t0 (z)(dlog (p t (z) q t (z)) dlogipi t ) dz. (4) The term the frst le s the chage omal expedtures, deflated by the CPI. The secod ad thrd les deote exports deflated by the EPI ad mports deflated by the IPI respectvely. We ca wrte the cosumer prce dex as: E t0 d log CPI t = d t0 p t0 (z) q c t 0 (z)(dlogd t (z) + dlogp t (z)) dz (5) W The log chage the export prce dex (EPI) ca be wrtte as, p t0 (z) q t0 (z) dlogepi t = p t0 (z) q t0 (z) dlog p t (z) dz, (6) 6= 6= 3

4 ad the log chage the mport prce dex (IPI) ca be wrtte as: p t0 (z) q t0 (z) dlogipi t = p t0 (z) q t0 (z) dlogp t (z) dz. (7) 6= 6= Substtutg (5), (6) ad (7) to equato (4), ad usg q (z)=q c (z) + qm (z) we obta: GDPt E 0 dlogrgdpt E = dt0 1 p t0 (z) q c t 0 (z) dlogq c t (z) 6= W + p t0 (z) q t0 (z) dlogy t (z) p t0 (z) q m t 0 (z) dlogq m t (z) dz W + p t0 (z) q t0 (z)(dlogp t (z) dlogt t dlog p t (z)) dz. 6= Ths ca be wrtte as: dlogrgdp E t = 1 GDP E t 0 dlogrgdp t + 1 GDP E t 0 dt0 1 p t0 (z) q c t 0 (z) dlogq c t (z). 6= W World-level equvalece betwee real cosumpto ad Real GDP From the proof of Proposto 4 the paper, the equvalece betwee world real GDP ad world real cosumpto holds f the world mport prce dex equals the world export prce dex. Ths follows drectly from equatos (6) ad (7) for the case whch each coutry operates ts ow tradg techology. The case whch oe coutry operates the tradg techologes of all the other coutres s preseted below. 1.2 Tradg techologes cocetrated oe coutry We ow cosder the case whch there s a sector coutry s that operates the tradg techologes of every coutry. The secto also shows how to exted our results whe the tradg techology s operated by termedares stead of the good producers themselves. 4

5 1.2.1 Real GDP, producto sde Curret-dollar GDP coutry s s gve by: GDP s t = + W s 6= p s t (z) /t s ty s t (z) dz W p t (z) y t (z) /t t dz 6= W p t (z) /t t y t (z) dz. (8) The term the frst le deotes value added of goods producers from coutry s. The two terms the secod le capture value added the sector that operates the tradg techology. Ths sector purchases the producto from each source coutry that s boud to foreg coutres 6= at prces p t (z) /t t, ad resells a fracto 1/t t of ths producto each destato coutry at prces p t (z). Computg real GDP from the producto sde mples usg the double deflato method to deflate value added the sector operatg the tradg techology. That s: RGDP s t =( W p s t (z) /t s ty s t (z) dz s RGDP s t 1 PPI g + 6= W p t (z) y t (z) /t t dz PPI s s t s t 6= W p t (z) /t t y t (z) dz MPI s )/GDP s t 1. s t Here, PPI g s t deotes the producer prce dex of the goods sector coutry, whle PPIs s t ad MPI s s t deote the producer ad the put prce dex of the sector operatg the tradg techology. Takg a frst order approxmato aroud tme t = t 0 ad usg the fact that all deflators are ormalzed to equal 1 at date t 0 we obta: dlogrgdp s t = l s t 0 l s t 0 (z) dlogp s t (z) dlogt s t + dlogy s t (z) dlogppi g s t dz W s + 6= W p t0 (z) y t0 (z) /t t0 dz dlogmpi s GDP s t dlogppi s s t. (9) s t 0 Followg the steps used Proposto 1 the paper, the frst le equals W st 0 X st 0 GDP st 0 dlogx s t, sce prces PPI g s t do ot clude chages trade costs t s t. The secod le captures the dfferece betwee put prces ad fal prces the operatg the tradg techology. Notg that fal prces the PPI s s t equal p t (z), ad that put prces the MPI s t 5

6 equal p t (z) /t t, we obta: dlogmpi s s t dlogppist s = 6= W p t0 (z) q t0 (z) dlogt t dz. (10) W p t0 (z) q t0 (z) Substtutg to equato (8) the paper we ca wrte the chage aggregate productvty as: dloga s t = Real GDP, expedture sde GDP t0 GDP s t 0 l t0 dlogt t. 6= We ow show the relato Proposto 2 betwee chages real GDP from the producto ad the expedture sde. We assume that tarffs d s t oly apply to goods that mported to be cosumed coutry s (ad ot to goods that are mported by termedares coutry s to be resold other coutres). GDP from the expedture sde for coutry s s gve by: GDP E s t = d s tp s t (z) q s t (z) dz + 6= s 6= p t (z) /t t y t (z) dz. 6=,6= s W p t (z) y t (z) /t t dz. The frst terms ths expresso deotes curret-dollar expedtures coutry s, clusve of tarffs. The secod term deotes exports from coutry s, whch equal the foreg sales from each source coutry, except those boud to coutry s. The last term deotes mports of s, whch equal expedtures foreg goods every coutry, excludg expedture goods produced couty s. To a frst-order approxmato, the chage real GDP from the expedture sde s: GDP E s t 0 dlogrgdp E s t = d s t 0 p s t 0 (z) q ts0 (z)(dlog (d s tp s t (z) q s t (z)) + 6=,6= s 6= s 6= W p t0 (z) q t0 (z)(dlog (p t (z) q t (z)) p t0 (z) q t0 (z)(dlog (p t (z) q t (z)) dlogcpi s t) dz dlogepi s t) dz dlogipi s t) dz. We re-wrte the log-chage the cosumer prce dex coutry s gve by equato (11) 6

7 (13) the paper: d log CPI s t = W s d s t 0 p s t 0 (z) q s t 0 (z) E s t 0 (dlogd s t + dlogp s t (z)) dz. (12) All export prces for coutry s clude trade costs. The log chage the EPI s: dlogepi s t = 6=,6=s W p t0 (z) q t0 (z) dlogp t (z) dz. (13) 6=,6=s W p t0 (z) q t0 (z) dz I cotrast, mport prces do ot clude trade costs. The log chage the IPI s, d log IPI t = 6= s 6= W p t0 (z) q t0 (z)(dlogp t (z) dlogt t ) dz. (14) 6=s 6= W p t0 (z) q t0 (z) dz Substtutg (8), (12), (13) ad (14) expresso (11) we ca wrte: GDP E s t 0 dlogrgdp E s t = d s t 0 p s t 0 (z) q s t 0 (z) dlogq s t (z) dz + 6=,6= s 6= s 6= W p t0 (z) q t0 (z) dlogq t (z) dz p t0 (z) q t0 (z)(dlogq t (z) + dlogt t ) dz. Addg ad subtractg 6=s p s t 0 (z) q ts0 (z) dlogq s t (z) dz/gdp E s t 0 ad 6=s W p s t 0 (z) q ts0 (z) dlogy s t (z) dz/gdp E s t 0 we obta: GDP E s t 0 dlogrgdp E s t = p s t 0 (z) q ts0 (z) dlogy s t (z) dz W + ds t 0 1 p s t 0 (z) q s t 0 (z) dlogq s t (z) dz 6= s + 6= 6= p t0 (z) q t0 (z) dlogq t (z) dz W p t0 (z) q t0 (z)(dlogq t (z) + dlogt t ) dz. 7

8 Re-arragg, usg equatos (9) ad (10), we ca wrte: dlogrgdp E s t = GDP s t 0 GDP E s t 0 dlogrgdp s t + 6= s ds t 0 1 p s t 0 (z) q s t 0 (z) dlogq s t (z) dz GDP E s t 0. (15) whch cocdes wth expresso (11) the paper World level real cosumpto ad real GDP We ow show the equvalece betwee real cosumpto ad real GDP from the expedture sde at the world level. As Proposto 4 the paper, we eed to show that, to a frst approxmato, the chage the world s mport prce dex equals the world export prce dex. The chage the world mport prce dex s gve by: Imports t0 dlogipi t = Imports t0 dlogipi t + Imports s t 0 dlogipi s t 6= s = p t (z) y t (z) /t t dlogp t (z) dz 6= s,6= p t (z) y t (z) /t t (dlogp t (z) dlogt t ) dz. + 6= s 6= The chage the export prce dex s gve by: Exports t0 dlogepi t = Exports t0 dlogepi t + Exports s t 0 dlogepi s t 6= s p t0 (z) y t0 (z) /t t (dlogp t (z) = 6= s,6= + 6=,6= s dlogt t ) dz (16) p t0 (z) y t0 (z) /t t dlogp t (z) dz. (17) 8

9 Expressos (16) ad (17) cocde. From expresso equato (11) the paper ad (15), the chage world GDP s gve by: dlogrgdpwt E = 1 E wt0 Exports t0 dlogt t dz + 1 E wt0 W t0 X t0 dlogx t + 1 E wt0 1 p t0 (z) q t0 (z) dlogq t (z) dz. 6= W dt0 whch cocdes wth expresso (14) the paper. 2 Proofs ad Dervatos: Moopolstc competto, heterogeeous frms ad qualty 2.1 Dervg the share of profts total reveues We show that our assumpto equato (19) of the paper that aggregate profts represet a costat share of total reveues (P t = k Y t, where Y t = t p t q t dm t ), s satsfed the remag two specal cases of our model descrbed Secto 4. I the frst case, there are o fxed costs of sellg to dvdual coutres so that all frms sell each coutry. I the secod case, there are postve fxed costs of sellg dvdual coutres (curred ether the exportg or mportg coutry) ad productvtes are Pareto dstrbuted. We derve equato (19) for the geeral case whch a fracto 1 of these fxed costs are curred the exportg coutry ad a fracto f of these fxed costs are curred the mportg coutry. We also allow for a fracto 1 f y of the ovato costs to be curred the exportg coutry ad a fracto y to be curred the mportg coutry. The basele model the body of the paper assumes f = y = 0. We also show that, these two cases, the mass of frms s uchaged followg a trade lberalzato. Remember that the thrd specal case descrbed Secto 4, whe r! 0, t s straghtforward to show that the free etry codto mples that k = 0 steady-state. We start by re-wrtg some prelmary equatos of the model to make ths secto self-cotaed. Frst, aggregate profts coutry perod t et of fxed labor costs, ovato costs, ad etry costs are: P t = Y t W t L p t W 1 t f W f t f t t dm t 9 t W 1 y t W y t h t (z) dm t W t f E M Et,

10 Aggregate reveues Y t = Y t are proportoal to varable labor paymets: Y t = r r 1 W tl p t. (18) If h (z; a) = g 0 h g (z) a g, aggregate ovato expedtures are a costat fracto of aggregate reveues: W 1 y t W y t h (z; a ) dm t = 1 t rg Y t. (19) If the trade balace s a costat share k of GDP, ad codto (19) the paper holds, aggregate reveues ca be wrtte as: Y t = 1 k 1 k (1 k ) W tl t, (20) where k = y k /(y + rg) ad f there are o fxed fxed costs of sellg to dvdual coutres, ad k = k /(1 +(f (q+1 r)(g 1) rqg dvdual coutres ad productvtes are Pareto dstrbuted. + y rg ) 1 ) f there are fxed costs of sellg to Suppose we are o a steady-state equlbrum whch aggregate varables are costat. I steady-state, the dstrbuto of frms s gve by M (z) = M E d G (z) (we omt tme subscrpts for the remder of ths secto to smplfy otato). The aggregate freeetry codto (equato B.6. the paper) steady-state s: (r + d) W f E M E = Y W L p W 1 f W f [1 G ( z )] f W 1 y W y h (z) dm. I what follows, we solve for the costat of proportoalty P /Y t = k = k steady-state uder two specal cases of our model. We the show that, these two specal cases, the aggregate respose to a chage varable or fxed trade costs s mmedate (.e. there are o trasto dyamcs), so that k remas costat over tme. Case 1: No fxed costs Assume that there are o fxed costs of sellg dvdual coutres,.e. f = f = 0, so that there s o selecto. I ths case, the aggregate free etry codto (21) s: W f E M E d " = 1 Y r + d W L p W 1 t y W y h (z, a (z)) dm (z) #. (21) 10

11 ad usg (18) ad (19): Aggregate profts are: so k = r r+d W f E M E = P = Y W L = r g 1 r + d rg Y, W 1 d g 1 r + d rg Y. (22) y W y h (z) dm W M E f E g 1 rg. Note that f r > 0, aggregate cross-sectoal profts are postve eve though dscouted profts at etry are zero. The steady-state mass of eterg frms s gve by: M E = d r + d g 1 rg 1 k 1 k (1 k ) where we used (20) ad (22). Hece, the mass of etrats M E does ot chage respose to permaet chages varable or fxed trade costs. Therefore, there are o trasto dyamcs to the ew steady-state, ad k t = k. Fally, aggregate varable profts gross of etry costs are: P + W M E f E = g 1 rg Y. Hece, wth restrcted etry (so that there are o costs curred etry), equato (19) the paper holds wth k = g 1 rg. Case 2: Pareto dstrbuted productvtes Now assume that there are postve fxed costs of sellg dvdual coutres ad that the dstrbuto of eterg frms G s Pareto wth shape parameter q,.e. G = 1 for z 1. We also assume that the productvty cutoffs are teror, z > 1. We start by L f E, otg that aggregate fxed labor costs are proportoal to aggregate reveues. z q M E d W 1 f W f f [1 G ( z )] = q + 1 r q g 1 rg Y. (23) Usg (18), (19), ad (23), we ca wrte the aggregate free etry codto (21) as: M E W f E = d r 1 g 1 r + d rq g Y, (24) 11

12 Fally, combg (18), (23) ad (24), aggregate profts are P = r r 1 r + d qr g 1 g Y so k = r+d r r 1 qr g 1 g. The steady-state mass of eterg frms s gve by: M E = d r 1 r + d rq g 1 g 1 k 1 k (1 k ) where we used (20), ad (24). Hece, the mass of etrats M E does ot chage respose to permaet chages varable or fxed trade costs. Therefore, there are o trasto dyamcs to the ew steady-state. Fally, aggregate varable profts gross of etry costs are P + W M E f E = r 1 g 1 qr g Y. Hece, the model wth restrcted etry ( whch there are o etry costs), equato (19) holds wth k = g 1 g (r 1) / (qr). L f E, 12