ADDENDUM TO: WHAT GOODS DO COUNTRIES TRADE? A QUANTITATIVE EXPLORATION OF RICARDO S IDEAS

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1 ADDENDUM TO: WHAT GOODS DO COUNTRIES TRADE? A QUANTITATIVE EXPLORATION OF RICARDO S IDEAS ARNAUD COSTINOT, DAVE DONALDSON, AND IVANA KOMUNJER A Ths addendum provdes the proofs of Lemma 1, Theorem 1, Lemma 2, and Theorem 2 and the dervaton of Equaton 16 n Secton 412 of our man paper Lemma 1 Suppose that Assumptons A1-A4 hold Then for any mporter, j, any par of exporters, and, and any par of goods, and, A-1 ln x j x j = ln x j x j ln d j d j d j d j Proof of Lemma 1 By Assumpton A4, we now that blateral trade flows satsfy x j = p j ω ] 1 σ j ω Ω j ω Ω p j ω1 σ j α j w j L j Snce Ω j ω Ω c j ω = mn 1 I c j ω}, ths can be rearranged as x j = ω Ω p j ω1i c jω = mn 1 I c j ω}] 1 σ j ω Ω p j ω1 σ j α j w j L j, where the functon 1I } s the standard ndcator functon By Assumpton A1, ω s ndependent and dentcally dstrbuted d across varetes so the same holds for c jω In addton, ω s d across countres so 1I c jω = mn 1 I c j ω} s d across varetes as well Ths mples that p j ω 1 σ j and p j ω 1 σ j 1I c j ω = mn 1 I c ] j ω} are d across varetes Moreover, snce σ j < 1 + we have E p j ω 1 σ j < so we can use the strong law of large numbers for d random varables eg Theorem 221 n Bllngsley, 1995 and the contnuous mappng theorem eg Theorem 1810 n Davdson, Date: Aprl 18,

2 2 COSTINOT, DONALDSON, AND KOMUNJER 1994 to show that x j = E p j ω 1 σ j 1I c j ω = mn 1 I c jω}] ] α E p j w j L j j ω1 σ j Consder H jc 1j,, c Ij E p j ω 1 σ j 1I c j ω = mn 1 I c j ω}] Assumptons A1, A3 and straghtforward computatons yeld + 1 σ A-2 Hjc 1j,, c j Ij = Γ c j =1 c j ] +1 σ j /, where Γ s the Gamma functon, Γt + 0 v t 1 exp vdv for any t > 0 Note that E ] p j ω 1 σ j = so that by usng Equaton A-2 we get A-3 and hence E I Hjc 1j,, c Ij, =1 ] + 1 σ p j ω 1 σ j j = Γ A-4 x j = 1 I ], 1 σ j / =1 c j c j =1 c j α j w j L j Wth ceberg trade costs, Assumpton A2, we have c j = d jw / Combnng the prevous expresson wth Equaton A-4 gves the result of Lemma 1 Theorem 1 Suppose that Assumptons A1-A4 hold Then for any mporter, j, any par of exporters, and, and any par of goods, and, x j x A-5 ln j = ln d j d ln j, x j x j d j d j where x j x j/ π Proof of Theorem 1 We mae use of the followng Lemma Lemma 3 Suppose that Assumpton A2 holds Then, for all countres and goods, A-6 Ω = ω c ω = mn 1 I c ω }

3 ADDENDUM TO: WHAT GOODS DO COUNTRIES TRADE? 3 Proof of Lemma 3 We proceed by contradcton Fx an exporter j, and suppose there exsts a varety ω 0 of good and a country l j such that: c jl ω 0 = mn 1 I c l ω 0; c jjω 0 mn 1 I c j ω 0 Then, there must be an exporter j such that d jl w j / j ω 0 d l w / ω 0 ; d j w / ω 0 < d jj w j / j ω 0 Snce d jj = 1, multplyng the two nequaltes above gves d j d jl < d l, whch contradcts Assumpton A2 Ths completes the proof of Lemma 3 Proof of Theorem 1 contnued By defnton, we now that c ω = d w / ω Usng Lemma 3 then yelds A-7 E ω ω Ω where we have let ] = G c 1,, c I µ d w, G c 1,, c I E c ω 1 1I c ω = mn 1 I c ω }], µ Pr c ω = mn 1 I c ω } The expressons for G c 1,, c I and µ can easly be computed from the expresson for H c 1,, c I n proof of Lemma 1 when the result n Equaton A-2 s evaluated at σ = 2 and σ = 1, respectvely By Equaton A-2, we formally have 1 G c 1,, c c I = Γ =1 c ], 1/ µ = c =1 c Hence, 1 A ] 1/ = Γ =1 c ] 1/ d w = c Γ I =1 c

4 4 COSTINOT, DONALDSON, AND KOMUNJER Now, recall that we have defned π x / =1 x ] Usng the expresson for x j obtaned n A-4 t then follows that A-9 π = µ = c =1 c Combnng the two prevous equatons, we obtan 1 π A-10 = 1/ Γ Now, from Equaton A-4, we now that for every and j, x j = d jw / =1 d j w / α j w j L j, so combnng wth A-10 and usng x j = x j/π gves x j = Γ 1 ] d jw / =1 d j w / α j w j L j Analogously to Lemma 1, the result of Theorem 1 then follows Lemma 2 Suppose that Assumptons A1-A5 hold Adjustments n absolute productvty, Z } 0, can be computed as the soluton of the system of equatons A-11 K π j j=1 =1 =1 π j /Z α j γ j = γ, for all 0 /Z Proof of Lemma 2 Throughout ths proof, we use labor n country 0 as our numerare n the ntal and counterfactual trade equlbrum: w 0 = w 0 = 1 By defnton, we now that Z s chosen for any 0 such that the value of the relatve wage w /w 0 n the counterfactual equlbrum s the same as n the ntal equlbrum w /w 0 Thus Assumpton A5 mples A-12 I j=1 =1 K π j α j w j L j = w L, where π j s the share of exports from country n country j and ndustry n the counterfactual equlbrum Usng Equaton A-4, one can easly chec that A-13 π j x j =1 x j w d j/ = =1 w d, j /

5 ADDENDUM TO: WHAT GOODS DO COUNTRIES TRADE? 5 and smlarly that A-14 π j = =1 w d j/ ] w d j / ] Combnng Equatons A-13 and A-14 and usng the fact that the relatve wages reman unchanged n the counterfactual equlbrum, we get after rearrangements π A-15 π j j / ] = / ] = π j /Z =1 π j /Z, =1 π j where the second equalty uses Z 0 Equatons A-12 and A-15 mply I K j=1 =1 π j /Z α j γ j =1 π j /Z = γ, where γ w L / j=1 w jl j s the share of country n world ncome Theorem 2 Suppose that Assumptons A1-A5 hold If we remove country 0 s Rcardan comparatve advantage, then: 1 Counterfactual changes n blateral trade flows, x j, satsfy A-16 x j = =1 π j /Z /Z, for all, j, 2 Counterfactual changes n country 0 s welfare, W 0 w 0 / p 0, satsfy A-17 Ŵ 0 = K =1 I ] α 0 / =1 π 0 0 Z Proof of Theorem 2 Smlar to prevously and throughout ths proof, we use labor n country 0 as our numerare n the ntal and counterfactual trade equlbrum: w 0 = w 0 = 1 1 Counterfactual changes n blateral trade flows, x j Snce the relatve wages are unchanged n the counterfactual equlbrum, we must have ˆx j = x j /x j = π j /π j Combnng ths observaton wth Equaton A-15, we obtan ˆx j = /Z =1 π j /Z

6 6 COSTINOT, DONALDSON, AND KOMUNJER 2 Counterfactual changes n country 0 s welfare, W 0 w 0 / p 0 By defnton, we now that p 0 = p 0 /p 0 = ω Ω p 0 ω ] 1 σ 0 ω Ω p 0 ω 1 σ 0 1/1 σ 0 By nvong the strong law of large numbers for d random varables and the contnuous mappng theorem as we dd n Theorem 1, then usng Equaton A-3, we can rearrange the prevous expresson as A-18 p w d 0 = 0 / ] ] 1/ w d 0 / =1 =1 Combnng Equatons A-13 and A-18 and usng the fact that the relatve wages reman unchanged n the counterfactual equlbrum, we get after rearrangements p 0 = π ] 1/ 0 =1 0 Z By defnton of p 0 K =1 p 0 α 0, we therefore have p 0 = K π 0 =1 0 Z =1 ] α 0 /, whch mmedately mples Equaton A-17 We conclude ths onlne appendx by showng that for any par of goods, and, and any par of countres, and, Assumptons A1-A3 mply = E p ω ] Ω E p ω ] Ω E p ω ] Ω E p ω ], Ω as stated n Equaton 16 of Secton 412 n our man paper E p ω Ω ] can be readly computed from the expresson for Hc 1,, c I n the proof of Lemma 1 when the result n Equaton A-2 s evaluated at σ = 0 and σ = 1, respectvely Specfcally, we have E p ω ] Ω = Γ =1 c ] 1/

7 ADDENDUM TO: WHAT GOODS DO COUNTRIES TRADE? 7 By Equaton A-8, we also now that 1 1 = Γ =1 c ] w 1/, where we have used the fact that d = 1 Combnng the two prevous expressons for any par of goods, and, we obtan from whch the desred result follows = E p ω ] Ω E p ω ], Ω R B, P 1995: Probablty and Measure John Wley & Sons, Inc D, J 1994: Stochastc Lmt Theory Oxford Unversty Press, Oxford

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