Online Appendix: Reciprocity with Many Goods

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1 T D T A : O A Kyle Bagwell Stanford Unversty and NBER Robert W. Stager Dartmouth College and NBER March 2016 Abstract Ths onlne Appendx extends to a many-good settng the man features of recprocty emphaszed n secton 3.1 of our Handboo chapter. Onlne Appendx: Recprocty wth Many Goods In ths Onlne Appendx we dscuss the extenson of the man features of recprocty emphaszed n secton 3.1 of our Handboo chapter to a settng wth more than two goods. We have already descrbed n the text how Bagwell and Stager (2015) show that these features are preserved n a partal equlbrum settng where the non-numerare sector s a monopolstcally compettve ndustry wth many varetes. Bagwell and Stager (2001a) demonstrate how these features extend to a 3-good partal equlbrum settng where each of the two non-numerare goods s a compettve homogeneous-good ndustry. And Bagwell and Stager (1999, note 16) descrbe how the terms-oftrade fxng property of recprocty extends to an N-good verson of the two-country compettve general equlbrum trade model featured above. 1 Here we wor wthn an N-good general equlbrum trade model and derve the three ey propertes of recprocty emphaszed n secton 3.1: () f a common terms of exchange of maret access s to be appled for both countres, then t must be one for one, the same terms of exchange emboded n GATT s recprocty prncple; () begnnng from ther Nash tarffs, both countres can gan from recprocal lberalzaton provded they do not go too far; and () begnnng from ther eff cent poltcally optmal tarffs, nether country could gan from renegotaton subject to recprocty. To accommodate N goods, we choose good 1 as the numerare and now let p w0 denote the world prce of good relatve to the world prce of the numerare good 1 under an ntal set of trade polces, and we let p w0 denote the (1 N) vector of ths set of ntal world prces (that s, p w0 s composed of the set of N 1 ntal relatve world prces, plus the frst element of p w0 whch s equal to 1). We then defne E 0 as the (N 1) vector of home country export volumes, where the j th element of E 0 equals 0 f the home country does not export good j under the ntal set of trade polces and equals the home country export volume of good j otherwse; and smlarly we defne 1 Bagwell and Stager, 2002, Appendx B.3 consders extensons of the propertes of recprocty n a many-good many-country settng when MFN s also mposed. 1

2 M 0 as the (N 1) vector of home country mport volumes, where the j th element of M 0 equals 0 f the home country does not mport good j under the ntal set of trade polces and equals the home country mport volume of good j otherwse. The analogous vectors for the foregn country are denoted by E 0 and M 0. And fnally, we denote wth a superscrpt 1 these vectors under an alternatve set of trade polces. Wth ths new mnmal notaton for the N-good model now defned, let us as n secton 3.1 begn wth the home country and consder frst a general verson of recprocty for the N-good settng defned as any change n tarffs that leads to a change n home-country export and mport volumes satsfyng p w0 [E 1 E 0 ] = γp w0 [M 1 M 0 ] (1) where as before γ s a parameter specfyng the terms of exchange of maret access. observe that maret clearng mples We next [E 1 E 0 ] = [M 1 M 0 ] and [E 1 E 0 ] = [M 1 M 0 ], whence (1) then mples p w0 [E 1 E 0 ] = γp w0 [E 1 E 0 ] and therefore p w0 [M 1 M 0 ] = γp w0 [E 1 E 0 ] or p w0 [E 1 E 0 ] = 1 γ pw0 [M 1 M 0 ], (2) whch descrbes the foregn-country terms of exchange of maret access that must accompany (1) accordng to the maret clearng requrements. From (1) and (2), t follows that for a common terms-of-exchange across countres we must have γ = 1, and thus a one-for-one exchange of mport volumes for export volumes. Hence, n ths N-good settng t remans the case that the addng-up constrant mposed by maret clearng maes t nevtable that, f governments wsh to adopt a common terms of exchange for all countres, they must adopt the one-for-one terms of exchange that characterzes GATT s recprocty prncple. We now turn to the remanng tass of ths Appendx, and show that, begnnng from ther Nash tarffs, both countres can gan from recprocal lberalzaton provded they do not go too far, and that begnnng from ther eff cent poltcally optmal tarffs, nether country could gan from renegotaton subject to recprocty. As before, for these purposes we adopt the perfectly compettve verson of the two-country general equlbrum trade model descrbed n Secton 2, extended here to the N-good case. As n our dscusson just above, there are N 1 relatve prces, and we suppose wthout loss of generalty that each country mposes trade taxes on the same N 1 goods, goods 2 through N. Let τ > 0 denote a home-country mport tarff or export tax, wth τ < 0 a home-country mport or export subsdy. Smlarly, let τ > 0 denote a foregn-country mport tarff or export tax, wth τ < 0 a foregn-country mport or export subsdy. Then, for = 2,..., N, we have world prces gven by p w (τ 2,..., τ N, τ N ), home local prces gven by p (τ, p w (τ 2,..., τ N, τ N )), and foregn local prces gven by p (τ, pw (τ 2,..., τ N, τ N )). We assume away the Lerner paradox for all, so that pw dτ < 0 for a home mport good, pw dτ > 0 for a home export good, pw dτ < 0 for a foregn mport good, and pw dτ good. And we also assume away the Metzler paradox for any ; hence, dp dτ good, dp < 0 for a Home export good, dp > 0 for a foregn export > 0 for a Home mport dτ dτ > 0 for a Foregn mport good, and dp dτ < 0 for a Foregn export good. We also assume that drect tarff effects domnate ndrect tarff effects n the followng sense. 2

3 Frst, we assume that, sgn( p w ) = sgn( pw ); sgn( τ dτ p w τ ) = sgn( pw dτ ) for = 2,..., N, (3) so that the mpact of an ncrease n a country s good- trade tax on the world prce of good domnates the ndrect mpact on that world prce of an ncrease n all of that country s other (good j ) tarffs combned. Second, we assume that dp sgn( ) = sgn( dp ) for = 2,..., N, (4) dτ dτ dp sgn( dτ ) = sgn( dp dτ ) for = 2,..., N. (5) so that the mpact of an ncrease n a country s good- trade tax on ts local prce of good domnates the ndrect mpact on that local prce of an ncrease n all of that country s other (good j ) tarffs combned. These assumptons are stronger than necessary, but serve to mae the basc argument transparent. In effect, (3) wll be used n combnaton wth (6) below and the absence of the Lerner paradox to ensure that costs are shfted abroad when a country ncreases ts trade taxes, whle (4) and (5) wll be used together wth the absence of the Metzler paradox to ensure that the recprocal trade tax changes characterzed below nvolve reductons (no ncreases) n every trade tax. Fnally, we depct home and foregn government objectves respectvely by the functons W (p 2,..., p N, p w 2,..., p w N) and W (p 2,..., p N, p w 2,..., p w N). We mpose the followng structure on these objectve functons: 2 < 0 for a home mport; > 0 for a home export; and (6) W p w < 0 for a foregn mport; W p w > 0 for a foregn export. We begn by characterzng Nash trade taxes. The Nash frst-order condtons are: W p dp dτ + W p dp dτ + W p w p w dτ = 0 for = 2,..., N, (7) p w dτ = 0 for = 2,..., N. (8) 2 Ths approach to wrtng government objectves and mposng mnmal structure on those objectves n a multgood settng s analogous to that taen n Appendx B of Bagwell and Stager (2002). 3

4 Summng each of the Nash condtons (7) and (8) over all N 1 tarffs yelds: dp W p [ ] + dτ Wp [ N dp dτ ] + [ Wp w[ N p w dτ ] = 0, (9) p w dτ ] = 0. By the assumed absence of the Lerner paradox and condtons (3) and (6), t follows that p w [ ] > 0 and dτ Wp w[ N p w dτ ] > 0. (10) In words, and referrng to (9), we then have by (10) that an ncrease n all of a country s trade taxes mples a postve nternatonal cost shftng component through the nduced world prce movements. And more specfcally, usng (10), the sum of the Nash condtons as dsplayed n (9) mples that at Nash trade taxes we must have: dp W p [ ] < 0 and dτ Wp [ N dp dτ ] < 0. (11) We next express our formal defnton of recprocty for the N-good envronment n terms of the notaton we have ntroduced just above. From an ntal set of tarffs, (τ 0 0 N, τ 0 0 N ), suppose that a tarff negotaton results n a change to the new par of tarffs, (τ 1 1 N, τ 1 1 N ). Denotng the ntal world and home local prces as p w0 p w (τ 0 0 N, τ 0 0 N ) and p0 p (τ 0, pw0 ) for = 2,..., N, and the new world and home local prces as p w1 p w (τ 1 1 N, τ 1 1 N ) and p 1 p (τ 1, pw1 ) for = 2,..., N, and wth p 0 1 = pw0 1 = p w1 1 = p for the numerare good 1, and fnally lettng M and E represent the set of home mport goods and home export goods, respectvely, we say that the tarff changes conform to the prncple of recprocty provded that M p w0 [M (p 1, p w1 ) M (p 0, p w0 )] = p w0 [E (p 1, p w1 ) E (p 0, p w0 )]. (12) E As n secton 3.1, usng the balanced trade condton that must hold both at ntal and new tarffs, t s straghtforward to show that recprocty mples [ p w1 M p w0 ]M (p 1, p w1 ) = [ p w1 p w0 ]E (p 1, p w1 ). (13) E That s, n the N-good case, tarff changes that conform to recprocty mply ether that world prces are left unchanged as a result of the tarff changes, or f world prces are altered, that they are altered n a way that leaves net trade tax revenue unchanged. Now consder, begnnng from Nash trade taxes, a small change n every home and foregn trade tax that (a) nduces a change n p equal to N dp dτ evaluated at Nash polces for = 2,..., N, thereby replcatng the local prce changes for the home country nduced by a small unlateral reducton n all of ts trade taxes begnnng from Nash, (b) nduces a change n p equal 4

5 to N dp dτ evaluated at Nash polces for = 2,..., N, thereby replcatng the local prce changes for the foregn country nduced by a small unlateral reducton n all of ts trade taxes begnnng from Nash, and (c) satsfes recprocty as defned n (12). Achevng (a), (b) and (c) s feasble, because each local prce change n each country can be targeted wth the assocated trade tax on that good, and the overall relatve magntudes of the home and foregn tarff changes can be adjusted to acheve recprocty (just as n the 2-good case). And wth the absence of the Metzler paradox and under assumptons (4) and (5) that drect tarff mpacts outwegh ndrect effects, each trade tax wll be reduced under ths maneuver. And fnally, accordng to (13), by conformng to recprocty these tarff changes ether eep all world prces fxed, or alter world prces n a way that s welfare neutral for each country. In ether case, wth recprocty ensurng that any world prce movements are mmateral for each country s welfare, the change n home and foregn welfare from these recprocal trade tax reductons begnnng from Nash s then gven by focusng only on the mpact of the local prce movements: dp W p [ ] > 0 and dτ Wp [ N dp dτ ] > 0, (14) where the nequaltes follow from (11). Gven the absence of the Metzler paradox and wth our assumptons (4) and (5) that drect tarff mpacts outwegh ndrect effects, (14) mples that both countres must gan. Hence, we have establshed that, begnnng from ther Nash tarffs, both countres can gan from at least a small amount of recprocal lberalzaton. The remanng tas of ths Appendx s to show that, begnnng from ther eff cent poltcally optmal tarffs, nether country could gan from renegotaton subject to recprocty. In ths N-good settng, the poltcal optmum s defned by the tarffs that conform to each government s poltcally optmal reacton curve and therefore satsfy W p dp dτ = 0 for = 2,..., N; and W p dp dτ = 0 for = 2,..., N. It s drect to show that the poltcal optmum s eff cent n the N-good settng. But argung as above t s also now mmedate that begnnng from the poltcal optmum, a small ncrease n any of the tarffs of one country can be met wth recprocal changes n the tarffs of ts tradng partner whch together nduce changes n the frst country s local prces whch are dentcal to those descrbed by the frst-order condton defnng ts poltcally optmal reacton curve tarffs. Wth ths, the frst-order condtons that defne the frst country s poltcally optmal reacton curve tarffs ensure that t could not gan from the local prce movements mpled by ts tarff ncreases, whle recprocty neutralzes the welfare mplcatons for the frst country of any world prce movements that are mpled by t tarff ncreases as well. Therefore, n the N-good settng, begnnng from the poltcal optmum nether country can gan from renegotaton subject to recprocty. 5