Technical Appendix: Globalization, Returns to Accumulation and the World Distribution of Output

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1 Technical Appendix: Globalizaion, Reurns o Accumulaion and he World Disribuion of Oupu Paul Beaudry and Fabrice Collard Universiy of Briish Columbia Universiy of Toulouse and nber cnrs gremaq and idei March Proofs of Proposiion In his secion we will absrac from any reference o he index of he economy, excep when sricly necessary, in order o save on noaion. Proposiion 1 In he absence of inernaional rade, he relaionship beween oupu per worker and capial per worker is given by wih ΥΓ Θ ϕ 1 Θ1 ϕ 2 ϕ ϕ 1 ϕ 1 ϕ [ϕ+1 ϕ] ϕ+1 ϕ y i = ΥΓ k ϕ+1 ϕ i 1 1 ϕ 1 ϕ 1 ϕ ϕ Γ 1 ϕ 1 ϕ [1 ϕ+1 ϕ 1 ] ϕ1 +1 ϕ1 Proof of proposiion 1: producer has o solve is In he absence of rade X 1 = X 2 = 0 so ha he problem he final good max Z ϕ 1 Z Z1 ϕ 2 P 1 Z 1 P 2 Z 2 1,Z 2 This yields he sandard inpu demand funcions ϕy = P 1 Z 1 and 1 ϕy = P 2 Z 2. Free enry on he final good marke implies ha 1 = P ϕ 1 P 1 ϕ 2 Φ he relaive price p = P 2 /P 1, we have wih Φ = ϕ ϕ 1 ϕ 1 ϕ. Then, defining P 1 = Φp ϕ 1 and P 2 = Φp ϕ 2 In he firs inermediae good secor, he represenaive producer solves max K 1,K 1 P 1 Θ 1 K 1Γ L 1 1 W 1 K 1 q K 1 The auhors would like o hank Daron Acemoglu, Ricardo Caballero, Marial Dupaigne, Parick François, David Green, Ashok Kowal, Marco Maffezoli, Fabrizzio Ziliboi for very helpful discussions. 1

2 which leads o he sandard inpu demand funcions, P 1 Z 1 = q K 1 and 1 P 1 Z 1 = W 1 L 1. Likewise, in secor 2, we have P 2 Z 2 = q K 2 and 1 P 2 Z 2 = W 2 L 2. Finally, surplus maximizaion by he rade unions subjec o labor demands leads o he wage seing rule W 2 = θ θ W 1 = W 1. In equilibrium, we have K 1 + K 2 = K and L 1 + L 2 = L such ha solving he sysem composed of demand funcions for inermediae good, capial and labor in each secor and making use of he wage seing rule, we easily ge L 1 = K 1 = Therefore, we can compue final oupu as where ΥΓ = Θ ϕ 1 Θ1 ϕ 2 ϕ ϕ + 1 ϕ K 1 ϕ K 2 = ϕ + 1 ϕ K ϕ1 1 ϕ 1 ϕ1 + 1 ϕ 1 L L 2 = ϕ1 + 1 ϕ 1 L Y = ΥΓ K ϕ+1 ϕ L 1 ϕ 1 ϕ 3 ϕ ϕ 1 ϕ 1 ϕ 1 ϕ1 ϕ [ϕ + 1 ϕ] ϕ+1 ϕ [ 1 ϕ 1 1 ϕ + 1 ϕ ϕ ] ϕ1 +1 ϕ1 Γ 1 ϕ 1 ϕ such ha oupu per worker y = Y /L expresses, in erms of capial per worker k = K /L, as y = ΥΓ k ϕ+1 ϕ 4 Hence he dynamics of he economy in inensive form may be summarized by where Υ = ΥΓ /Γ 1 ϕ 1 ϕ as seady sae. 1 + γ1 + n k +1 Γ +1 = sυ k = which admis k sυ 1 + γ1 + n + δ 1 Γ ϕ+1 ϕ + 1 δ k Γ 1 1 ϕ 1 ϕ = νυ 1 1 ϕ 1 ϕ 2

3 Proposiion 2 In he auarky equilibrium, he privae and social reurns o capial are equalized and are independen of he size he labor marke disorsion, x. Proof of proposiion 2: In he auarkic economy, privae r a and social z a reurns o capial are he same. Indeed, he renal rae of capial, in erms of good 1, is given by q = P 1 Z 1 K 1 = P 2 Z 2 K 2 In equilibrium, we have P 1 Z 1 = ϕy and P 2 Z 2 = 1 ϕy, herefore q = ϕ Y K 1 = 1 ϕ Y K 2 = ϕ + 1 ϕ Y K Hence, he reurns o capial, in erms of he final good, are given by r a = z a = q = ϕ + 1 ϕ Y K = ϕ + 1 ϕυγ k ϕ+1 ϕ 1 3

4 Proposiion 3 In he absence of inernaional rade, he seady sae disribuion of log oupu per worker relaive o a reference economy y 0, is given by µŷŷ = 1 ϕ 1 ϕ ϕ + 1 ϕ where µ ν denoes he disribuion of ν logν logν 0. 1 ϕ 1 ϕ µ ν ŷ ϕ + 1 ϕ Proof of Proposiion 3: a seady growh pah is given by In he closed economy, he aggregae producion funcion in economy i along y i = ΥΓ ϕ+1 ϕ 1 1 ϕ 1 ϕ ν 1 ϕ 1 ϕ i where ν s/1 + γ1 + n 1 δ. Le us consider he log difference beween oupu per worker in economy i and in he big economy, ŷ = logy i logy 0, where 0 denoes he big economy. Le us define ν i = logν i logν 0, we hen have ŷ i = g ν i = ϕ + 1 ϕ 1 ϕ 1 ϕ ν i Making use of he change of variable formula, and denoing by µ ν he disribuion of ν, we have µŷŷ 1 ϕ 1 ϕ 1 ϕ 1 ϕ = µ ν ŷ ϕ + 1 ϕ ϕ + 1 ϕ 4

5 Lemma 1 For given inernaional prices, here exiss wo levels of capial per worker denoed by kp, Γ and kp, Γ, such ha when endowed wih a capial per worker below kp, Γ, a small open economy specializes in he producion of good 1, while i specializes in he producion of good 2 when is capial per worker lies above kp, Γ, wih kp, Γ = Γ p Θ 2 kp, Γ = Γ p Θ 2 Θ 1 1 Θ x x 1 1 Proof of lemma 1: In he small open economy, each firm producing he final good akes he price of goods as given, such ha final oupu is given by Y = P 1 Z 1 + P 2 Z 2 The inermediae goods producers problem may be rewrien as subjec o max K 1,K 2,L 1,L 2,Z 1,Z 2 P 1 Z 1 + P 2 Z 2 q K W 1 L 1 W 2 L 2 Z 1 Θ 1 K 1Γ L 1 1 Z 2 Θ 2 K 2 Γ L 2 1 K 1 + K 2 K L 1 + L 2 L L 1 0, L 2 0, K 1 0, K 2 0 W 2 = 1 + xw 1 Since echnology is sricly increasing in inpus, he firs four consrains ough o bind, such ha he problem simplifies o max P 1 Θ 1 K K 1,K,L 1,L 1Γ L 1 1 +P 2 Θ 2 K K 1 Γ L L 1 1 q K W 1 L 1 W 1 L L 1 subjec o L 1 0, L L 1, K 1 0, K K 1 o which we associae he lagrange mulipliers λ 0 L, λ1 L, λ0 K, λ1 K. This leads o he following se of opimaliy condiions 1 K K 1 P 2 Θ 2 Γ 1 = q 5 L L 1 K K 1 1 P 2 Θ 2 Γ 1 = 1 + xw 1 6 L L 1 1 K1 P 1 Θ 1 Γ 1 K K 1 P 2 Θ 2 L 1 L L 1 1 P 1 Θ 1 K1 L 1 Γ 1 1 P 2 Θ 2 K K 1 L L 1 1 Γ 1 = λ 1 K λ 0 K 7 Γ 1 xw 1 = λ 1 L λ 0 L 8 An inerior soluion, for which K 1, K 2, L 1, L 2 > 0 which corresponds o a specializaion phase implies ha K 1, K 2, L 1 and L 2 saisfy using 6 8 K 2 L 2 = K K 1 L L 1 = x K L 1 5

6 Le us firs sudy he condiions under which an economy chooses o specialize in he producion of ype 1 inermediae good. In his case, K 1 = K and L 1 = L, which implies ha λ 0 K = λ0 L = 0 and λ1 K 0 and λ 1 L 0. Therefore, equaions 6 8, evaluaed along 9, saisfy which riggers ha 1 K1 P 1 Θ 1 Γ 1 L 1 1 P 1 Θ 1 K1 L 1 K 1 L 1 where p = P 2 /P 1. = K L x K 1 P 2 Θ 2 1 L 1 Γ x1 1 + x P 2Θ 2 1 kp, Γ Γ 1 Θ 2 p Θ 1 1 Γ x 1 K 1 Γ 1 0 L 1 1 Le us now sudy he condiions under which an economy chooses o specialize in he producion of ype 2 inermediae good. In his case, K 2 = K and L 2 = L, which implies ha λ 1 K = λ1 L = 0 and λ0 K 0 and λ 0 L 0. Therefore, equaions 6 8, evaluaed along 9, saisfy 1 P 1 Θ x 1 1 P 1 Θ x which riggers ha K K 1 L L 1 = K L 1 K K 1 Γ 1 K K 1 P 2 Θ 2 L L 1 L L 1 K K 1 L L 1 kp, Γ Γ Γ 1 1 Θ 2 p Θ x P 2Θ 2 1 Γ 1 0 K K 1 L L x 1 Γ

7 Proposiion 4 Under free rade a counry s level of oupu per worker is given by y i = where Φ = ϕ ϕ 1 ϕ 1 ϕ and Ap = Φp ϕ 1 Bp = Φp ϕ 1 Φp ϕ 1 A 1 k i Γ1 if k i kp, Γ Ap k i + Bp Γ if kp, Γ k i kp, Γ Φp ϕ A 2 k i Γ1 if k i kp, Γ p Θ 2 1 p Θ 2 1 Θ 1 Θ Proof of proposiion 4: efficien uni of labor. We have o sudy hree cases, depending on he level of he capial per k kp, Γ : In his case, he economy fully specializes in he producion of ype 1 inermediae good, we herefore have y = P 1 z 1 = P 1 Θ 1 k Γ 1, where y = Y /L and z 1 = Z 1 /L. Since P 1 is given by 2, we finally have y = Φp ϕ 1 Θ 1 k Γ 1. k kp, Γ : In his case, he economy fully specializes in he producion of ype 2 inermediae good, we herefore have y = P 2 z 2 = P 2 Θ 2 k Γ 1, where z 2 = Z 2 /L. Since P 2 is given by 2, we finally have y = Φp ϕ Θ 2 k Γ 1. kp, Γ k kp, Γ : In his case, he economy lies in he specializaion process, and we have y = P 1 z 1 + P 2 z 2 We herefore have o solve he allocaion of capial and labor problem. This implies solving he se of equaions which implies ha 1 K1 P 1 Θ 1 Γ 1 K K 1 = P 2 Θ 2 L 1 L L 1 1 P 1 Θ 1 K1 L 1 K K 1 L L 1 = K 1 L 1 = Γ p Θ 2 Θ 1 Γ 1 = x P 2Θ x 1 Γ 1 10 K K 1 L L 1 Γ x K L = kp, Γ 13 Le us hen denoe σ L = L 1 /L and σ K = K 1 /K. Solving 12 and 13, we ge σ L = k 1 1 kp, Γ 14 σ K = σ L kp, Γ k 15 7

8 We herefore easily ge P 1 z 1 = P 1 Θ 1 kp Γ 1 σ L = P 1 Θ 1 kp, Γ Γ 1 Likewise, sraighforward calculaion gives We herefore easily ge 1 σ L = 1 σ K = k 1 1 kp, Γ x x P 2 z 2 = P 2 Θ 2 kp, Γ Γ 1 1 σ L 1 Γ x = P 2 Θ 2 kp, Γ 1 P 1 Θ 1 Γ 1 kp, Γ 1 k 16 kp, Γ k 1 σ L Then, afer simple alhough edious algebra and making use of 2, we ge k kp, Γ 1 y = Bp Γ + Ap k 18 where Bp = Φp ϕ p Θ 2 Θ Ap = Φp ϕ p Θ Θ

9 Proposiion 5 If x > 0, hen under free rade he social reurns o capial are higher han he privae reurn for all counries ha do no fully specialize. Moreover, he difference beween he privae and social reurn o capial is increasing in x. Proof of proposiion 5: of lemma 1, equaion 5 which rewries r f = P 2 Θ 2 k 1 Under free rade, he privae reurns o capial, r f, are given by see proof r f 1 K K 1 = q = P 2 Θ 2 Γ 1 L L σk Γ 1 1 σ L x = P 2 Θ 2 kp, Γ 1 Plugging he definiion of kp, Γ and ha of P 2 in he laer equaion, we ge r f = Φp ϕ 1 p Θ Θ x Γ 1 Furher from he opimal allocaion of Z 1 and Z 2 in he big economy auarkic world, we have p = 1 ϕ Z 1 ϕ Z 2 Using he value of z 1 and z 2, he relaive price, p, evaluaed a he seady growh pah of he big economy indexed by 0 is given by p = Θ 1 θ1 1 ϕ + 1 ϕ Θ ϕ + 1 ϕ 1 k0 Γ 19 Plugging his expression in he definiion of kp, Γ, we can express he privae reurn o capial a he seady sae of he big economy as 1 1 ϕ ϕ ϕ ϕ1 ϕ1 1 ϕ 1 ϕ 1 ϕ 1 r f = Θ ϕ 1 Θ1 ϕ 2 1 ϕ 1 ϕ... ϕ + 1 ϕ ϕ+1 ϕ ϕ1 + 1 ϕ 1 ϕ + 1 ϕk 0 ϕ+1 ϕ 1 Γ 1 ϕ 1 ϕ or r f = ϕ + 1 ϕυγ k 0 ϕ+1 ϕ 1 = r a We now consider he social reurn o capial, which is now obained by deriving he aggregae producion funcion when he economy produces boh goods. Hence, we have z f = Ap. Using he definiion of A see proposiion 4 and he expression for p, he social reurn o capial in he seady sae of he big economy is given by = z f ϕ + 1 ϕbγ k ϕ+1 ϕ 1 = z a I is hen sraighforward o verify ha as long as, 0, 1 and x > 0 he muliplier erm is greaer han 1, such ha z f z a. 9

10 Proposiion 6 Under free rade, regardless of he value of x, all economies possess a unique non rivial seady sae. Proof of Proposiion 6: Given he form of he producion funcion, he model admis 1, 3 or an infiniy of non rivial seady sae he rivial seady sae being 0. Le us firs consider he case where we have an infiniy of seady sae. This occurs when he ray k/ν overlies he linear par of he producion funcion, which can only happen if and only if ha he linear par reduced o Apk. This siuaion would hen corresponds o Bp = 0, which is impossible if >. A second possibiliy of muliple equilibria is o have 3 seady saes. Bu he form of he funcion riggers ha a firs seady sae lies in he area where he economy is fully specialized in he producion of ype 1 inermediae goods, one in he zone of full specializaion in ype 2 inermediae goods, and one in he specializaion area. This can only be he case if he linear par of he producion funcion crosses he ray k/ν from below, which implies a negaive value for Bp which is impossible. Therefore, here exiss a unique equilibrium. 10

11 Lemma 2 The relaionship beween he seady sae level of relaive oupu-per-worker in economy i, ŷ i, and he relaive propensiy for capial deepening, ν i is given by 1 ϕ+1 ϕ log 1 1 ŷ i = logρ log 1 1 ρ exp ν i where ν = log ϕ+1 ϕ log 1 ϕ+1 ϕ 1 1 and ν = log ϕ+1 ϕ + 1 ν i ν + 1 ν i ν and ρ = if ν i < ν if ν ν i ν if ν i > ν ϕ1 +1 ϕ Proof of lemma 2: We firs sar by characerizing he wo boundary values ν and ν. Le us compue νp, Γ = kp, Γ /yp, Γ. A his paricular value, we have y i = P 1 z 1i, such ha νp, Γ = p1 ϕ kp, Γ 1 Φ Θ 1 Γ 1 Plugging ino he previous equaion he definiion of kp, Γ and he definiion of p along he seady growh pah of he big economy Equaion 19, we have k 0 νp, Γ = ϕ + 1 ϕ ΥΓ k = ϕ+1 ϕ 0 ϕ + 1 ϕ Hence, we have ν logνp, γ logν 0 = log ϕ+1 ϕ k 0 y 0 = ϕ + 1 ϕ ν 0 Le us now compue νp, Γ = kp, Γ /yp, Γ. A his paricular value, we have y i = P 2 z 2i, such ha νp, Γ = p ϕ Φ kp, Γ 1 Θ 2 Γ 1 Plugging ino he previous equaion he definiion of kp, Γ and he definiion of p a he seady sae of he big economy, we have k 0, νp, Γ = ϕ + 1 ϕ ΥΓ k = ϕ+1 ϕ 0, ϕ + 1 ϕ Hence, we have ν logνp, Γ logν 0 = log We hen have o consider 3 cases: ϕ+1 ϕ k 0, y 0, = ϕ + 1 ϕ ν 0 ν i < ν: In his case y = P 1 Θ 1 k Γ 1. Evaluaing his along a seady growh pah, we have y i = Φp ϕ 1 Θ ν 1 i Γ. Plugging he expression of he relaive price along he seady growh pah of he big economy Equaion 19 in he oupu per worker in economy i, and remembering i he definiion of ΥΓ, ii he definiion of oupu per worker in he big economy, we have ϕ + 1 ϕ y i = 1 aking log on boh sides, we have ŷ i = ϕ + 1 ϕ 1 log + log ϕ 1 1 νi ν 0 1 y0 1 ϕθ ϕθ θ1 + 1 ν i 11

12 ν ν i ν: In his case y i = Ap k i +Bp Γ. Evaluaing his along a seady growh pah, where k i = ν i y i, we have y = BpΓ 1 Ap ν i. Around he seady growh pah of he big economy, we have ϕ + 1 ϕ 1 1 y 0 Ap = 1 1 and Bp Γ = k 0 ϕ1 + 1 ϕ y 0 Remembering ha ν 0 = k0 y 0, we hen ge y i = ha ŷ = logρ log 1 1 ρ exp ν i. ρy ρ ν ν 0 where ρ = ϕ1 +1 ϕ such ν i > ν: In his case y = P 2 Θ 2 k Γ 1. Evaluaing his along a seady growh pah, we have y i = Φp ϕ Θ ν 1 i Γ. Then, using previous resuls for ν i < ν, we hen have, around he seady growh pah of he big economy, ϕ + 1 ϕ y i = aking log on boh sides, we have ŷ i = 1 log 1 ϕ + 1 ϕ ϕ1 + 1 ϕ 1 1 νi ν 0 1 y0 ϕ1 + 1 ϕ 1 + log ν i 12

13 Proposiion 7 The seady sae disribuion of oupu per worker, relaive o he reference economy, ŷ, is given by µŷŷ = 1 µ ν 1 ŷ ŷ log 1 ρ expŷ 1 1 ρ expŷ µ ν log 1 1 µ ν ŷ ŷ log ϕ+1 ϕ 1 1 ρ expŷ ρ ϕ+1 ϕ where µ ν is he disribuion of ν = logν logν 0, ρ ŷ = log Proof of proposiion 7: ϕ1 + 1 ϕ 1 1 and ŷ = log if ŷ < ŷ if ŷ ŷ ŷ if ŷ > ŷ 1 ϕ+1 ϕ1 1 1 ϕ1 + 1 ϕ 1 1 We sar by compuing he hreshold values, yp and yp, for he disribuion. These values are simply obained by plugging he values for kp and kp in he relevan producion funcions. We herefore have yp, Γ = P 1 Θ 1 kp, Γ Γ 1 using he definiion of kp, Γ, kp, Γ, P 1 and P 2, we ge and yp, Γ = ΦΓ p ϕ 1+ yp, Γ = ΦΓ p ϕ 1+ Θ Θ 1 Θ 2 1 Θ 2 and yp, Γ = P 2 Θ 2 kp, Γ Γ x x, 1 1 We now deermine he shape of he disribuion along a seady growh pah, ha is when k i = ν i y i. In his case, he relaive price is given by expression 19. Hence,denoing by y 0 he oupu level along a seady growh pah in he big economy, we can reexpress he hresholds as yp, Γ = 1 ϕ1 + 1 ϕ y 0, and yp, Γ = 1 1 ϕ1 + 1 ϕ y 1 0, We now sudy he disribuion of ŷ = logy logy 0,. The direc applicaion of he change of variable formula on he relaionship repored in lemma 2 yields 1 µ ν 1 ŷ ŷ log µŷŷ = 1 ρ expŷ 1 1 ρ expŷ µ ν log 1 1 µ ν ŷ ŷ log ϕ+1 ϕ 1 1 ρ expŷ ρ ϕ+1 ϕ where µ ν is he disribuion of ν = logν logν 0, ρ yp ϕ1 + 1 ϕ 1 ŷ log = log 1 y 0 if ŷ < ŷ if ŷ ŷ ŷ if ŷ > ŷ 1 ϕ+1 ϕ1 1 1 yp and ŷ log = log y b, ϕ1 + 1 ϕ

14 Proposiion 8 If x > 0, he firs order effec of free rade is o increase he sensiiviy of ŷ i = logy i logy 0 wih respec o ν i = logν i logν 0. However, if x=0, free rade has no firs order effec on he mapping from ν i o ŷ i. Proof of proposiion 8 Le us firs consider he auarkic case. In his siuaion, he aggregae producion funcion in economy i in he seady sae is given by y i = ΥΓ ϕ 1 ϕ 1 1 ϕ 1 ϕ ν 1 ϕ 1 ϕ i. Le us consider he log difference beween oupu per worker in economy i and in he big economy, ŷ i = logy i logy 0. Le us define ν i = logν i logν 0, we hen have ŷ i = ϕ+1 ϕ 1 ϕ 1 ϕ ν i, which is independen from he disorion induced by he exisence of rade union. Le us now consider he case of an open economy, and use he relaionship esablished in lemma 2. We compue he sensiiviy of he dispersion in he level of oupu o he dispersion in he long run propensiy o accumulae capial. Three cases should be considered ν i < ν: The sensiiviy is given by ŷi ν i = 1 ν i > ν: The sensiiviy is given by ŷi ν i = 1 ν < ν i < ν: The sensiiviy is given by ŷi ν i = ρ exp νi 1 ρ exp ν i exp ν 1 ρ exp ν 2 is unaffeced by he rade union markup. is unaffeced by he rade union markup. where ρ ϕ+1 ϕ Noe ha 2 ŷ ν ρ = > 0 such ha he sensiiviy of ŷ o ν is an increasing funcion of ρ. Then, noe ha ρ x = 1 1 ϕ + 1 ϕ 2 > x Hence, in an open economy, he larger he rade union disorsion, he greaer he sensiiviy of ŷ i o ν i. 14

15 Corollary 1 If he disribuion of ν i is concenraed around zero, hen he firs order effec of free rade on he cross counry disribuion of oupu per worker is nil when x = 0. In conras, i leads o an increase in dispersion when x > 0. Proof of Corollary 1: Le us recall ha wihin a close economy, he dispersion of log oupu per worker is deermined by ŷ i = ϕ+1 ϕ 1 ϕ+1 ϕ ν i while, when we open rade, i changes o ŷ = logρ log 1 1 ρ exp ν i which can be approximaed, around he seady sae of he big economy as ρ 1 ϕ+1 ϕ1 ŷ i 1 ρ ν i where ρ. Noe ha absen of rade union x=0, ρ = ϕ + 1 ϕ, 1 1 such ha we regain he dispersion in he close economy. Conversely, we saw in he proof of Proposiion 8 ha ρ/ x > 0, such ha ŷ x>0 > ŷ ν ν x=0 15

16 Lemma 3 Consider wo observable random variables, Y 1 and Y 2, which are boh ransformaions of a random variables ν, where he firs ransformaion is linear and resuls in he variable Y 1 = ν, > 0, while he second ransformaion is non linear coninuous and differeniable and resuls in he variable Y 2 = gν, g > 0. If he disribuion of Y 1 is uni modal, hen a necessary condiion for he disribuion of Y 2 o be bi modal is ha g no be a convex funcion. Proof of lemma 3: Since Y 1 is unimodal and is a linear ransformaion of ν, i has o be he case ha ν is also unimodal. From he change of variable formula, we know ha he disribuion of ν is given by µ y2 y 2 = µνg 1 y 2 g g 1 y. Since 2 g 0, his reduces o µ y2 y 2 = µνg 1 y 2 g g 1 y 2. A necessary condiion for he exisence of a leas wo modes in µ y2 is ha here exiss y 20, such ha µ y2 is decreasing a he lef of y 20 and increasing above i. Therefore, i has o be he case ha for any δ > 0, ε > 0, wih δ < ε, g g 1 y 20 δ g g 1 y 20 and g g 1 y 20 + δ g g 1 y 20 δ. Hence, i has o be he case ha g g 1 y 20 g g 1 y 20 δ δ 0 and g g 1 y 20+δ g g 1 y 20 δ δ 0. 16

17 Proposiion 9 If x=0, he opening of rade in inermediae goods canno generae he emergence of a bi modaliy in he cross counry disribuion of oupu per worker. Conversely, if x > 0 hen he opening up of rade may cause he disribuion o exhibi bi modaliy. Proof of Proposiion 9: g, we ge wih ρ ρ exp ν 1 ρ exp ν ϕ+1 ϕ1 1 1 Lemma 2 esablished he funcional g ha relaes ν i o ŷ i. Differeniaing g ν i = 1 ρ exp ν i 1 ρ exp ν i 1 if ν i < ν if ν ν i ν if ν i > ν. Firs, noe ha, whaever x 0, for ν ν ν, we have g ν = When x = 0, ρ = ϕ + 1 ϕ, implying ha g ν = /1 and g ν = /1. Hence, g 0 over he whole suppor of ν. From lemma 3, we now ha his rules ou bi modaliy. When x > 0, we have g ν = and g ν = Bu, since 0, 1, 1 1 > 1, such ha g is increasing for ν, ν. Conversely, as soon as x > 0, 1 1 < 1, which implies ha lim ν ν g ν > lim ν ν g ν Therefore, g is decreasing over some range of values for ν, which creaes he possibiliy of bi modaliy. 17

18 2 Allowing for Inernaional Capial Flows In his secion, we discuss he implicaions of relaxing he assumpion of no capial mobiliy in he model. More precisely, we documen he exen o which our main resuls are robusness o allowing for rade on financial capial markes. Le us firs consider he case of a perfecly fricionless inernaional financial markes. In his case, he reurns o capial are equalized across counries and he locaion of capial is independen of he counries propensiy o save. Therefore, in he absence of rade in inermediae goods and in he absence of differences in Ω i he level of oupu per worker is idenical across counries. In conras, when rade in inermediae goods is allowed, he cross counry disribuion of oupu per worker is indeerminae since here are wo mechanisms for equalizing he reurns o capial across counry: hrough rade or hrough inernaional capial flow. Hence, in he exreme case of perfec inernaional capial markes, he model has no clear predicions on how he opening up of rade will affec he cross-counry disribuion of oupu. This is a raher unsaisfacory resul. In order o have a beer sense of how our resuls can be exended in he presence of inernaional capial flows, i is useful o consider he limiing behavior of a model wih imperfecions in inernaional capial marke. To his end, le us consider he case where domesic firms face a risk premium on borrowing in he inernaional marke which is proporional o he counry s deb o GDP raio, and le us examine he oucome when his risk premium ends o zero. More precisely, le us assume ha he cos of capial in counry i, q i, is equal o he cos of capial in he large reference economy he US plus a risk premium which is proporional o he counry s deb o GDP raio as given by ki a i q i = q 0 + ρ where ρ is he gradien of he risk premium, a i is he wealh per worker in counry i and hence k i a i is he amoun of inernaional deb per worker in counry i. Through he accumulaion equaion, he wealh per worker along a seady growh pah is given by a i = ν i y i and herefore he deerminaion of he domesic cos of capial can alernaively be wrien as ki q i = q 0 + ρ ν i y i Given his equaion for he deerminaion of cos of capial, a counry level of capial per worker and oupu per worker is deermined by equaing he inernaional cos of capial o he domesic reurn on capial. In he absence of inernaional rade in inermediaes, he limiing oucome as ρ goes o zero will have all counries producing he same amoun of oupu per worker since his is he only way he reurns o capial can be equalized across counries. Hence, in his case and assuming no differences in Ω i he cross counry disribuion of oupu per worker is concenraed a a single poin. If we now open up rade in inermediaes, he deerminaion of oupu per worker in counry i depends on ν i. In paricular, if ν ν i ν hen he deerminaion of oupu per worker remains he same as in he absence inernaional capial flows since he reurns o capial are equalized recall ha wihin he incomplee specializaion area, he social reurns are consan. As a maer of fac, in he presence of inernaional capial flows, i is easy o verify ha in he limi as ρ goes o zero, he deerminaion of ŷ i y i 18

19 is given by log ŷ i = log log 1 ϕ+1 ϕ 1 1 ϕ1 +1 ϕ ϕ+1 ϕ ϕ+1 ϕ1 log 1 exp ν 1 1 i if ν < ν if ν ν ν if ν > ν As can be seen from he mapping beween ŷ i and ν i, in he limiing case where ρ ends o zero, he opening up of rade in inermediaes causes an increase in dispersion in oupu per worker. This is because, for counries wih ν i [ ν, ν], oupu per worker is no longer equalized bu insead becomes an increasing funcion of ν i. Furhermore, in addiion o his increase in dispersion for he counries wih ν i [ ν, ν], he counries wih eiher ν i < ν or ν i > ν will bunch a wo poins in he disribuion of y i. Indeed, le us consider he case of a counry wih a low propensiy o capial accumulaion ν i < ν. In he limiing case where ρ ends o zero, he only way is reurn on capial can equalize he world reurn on capial is o accumulae up o he poin is capial oupu raio reaches ν, such ha is ν i = ν. This phenomenon likely gives rise o bi-modaliy. Therefore, he main resuls presened in he paper survive he inroducion of inernaional capial flows as long as he inernaional capial marke is no perfecly fricionless. 1 3 Allowing for Endogenous World Prices In he main body of he paper, we have assumed ha under free rade he world price for inermediae goods correspond o he auarky prices of hese goods in he reference economy. Our defense for his assumpion is ha he reference economy used in our empirical work is he US economy and since he US economy is exremely large economically his may consiue a good approximaion. However, i is clearly an approximaion. Therefore in his secion we discuss how our resuls mus be modified and rephrased when his assumpion is relaxed. I is raher easy o derive he mapping beween ν i and ŷ i for he case where world prices under free rade do no correspond o he reference economy s auarky prices. To do so, le us denoe by ν he value of ν such ha a counry wih ν i = ν does no rade in equilibrium since he world prices of inermediaes are equal o is auarky prices. Then, he mapping beween ν i and ŷ i becomes log 1 ŷ i = log log 1 ϕ+1 ϕ 1 + ϕ1 +1 ϕ ϕ+1 ϕ ν i ν ν log + 1 ν i ν ν 1 ϕ+1 ϕ1 1 exp ν 1 1 i ν if ν i ν < ν if ν ν i ν ν if ν i ν > ν The previous equaion makes i clear ha he presence ν 0 simply causes a ranslaion of our original mapping beween ν i and ŷ i. However, he problem wih his mapping is ha we do no know he value of 1 In he presence of inernaional capial flows, here are wo disinc mechanisms which cause an increase in he dispersion of y i. There is an increase due o increased dispersion of capial oupu raios and, if x > 0, here is increased dispersion due o an increase in he reurn o capial. The model of Venura [1997] is an alernaive way of emphasizing he firs mechanism, while he model presened in he main body of his paper emphasizes he second mechanism. As he empirical secion of he paper has shown, he daa are more supporive of he second mechanism. 19

20 ν. Noneheless, we can sill make a condiional saemen regarding how he opening of rade will affec he disribuion of ŷ i. In paricular, in his more general case, Corollary 1 should simply be rephrased as follows If he disribuion of ν i is concenraed around ν, hen he firs order effec of free rade on he cross counry disribuion of ŷ i is nil when x = 0. In conras, i leads o an increase in dispersion when x > 0. This exended version of Corollary 1 clarifies ha our main resuls hinge on he noion ha ν be no oo differen from he mean of ν i. 2 In oher words, he key condiion for our resuls on he effec of free rade o hold is ha he average capial oupu raio across counries mus no be subsanially differen he capial oupu raio across he world. 3 2 In fac, he resul can be srenghened slighly by noing ha wha is key for our resuls is ha ν no be subsanially greaer han he mean of ν i. 3 Based on our calculaion using he World Penn ables, his condiion appears saisfied. 20

21 4 Fixed Effecs Table 1: Fixed Effecs OLS IV1 IV1 IV2 IV ŷ ν Ĥ F excl Q ν [0.000] [0.000] [0.000] [0.002] [0.001] Noe: Sandard errors in parenhesis, p values in brackes. The se of insrumens corresponds o he IV1, IV2 and IV3 ses discussed in he body ex. Table 2: Fixed Effecs and Openness OP<medOP OP medop OP<med OP OP med OP ŷ ν F excl Q ν [ 0.136] [ 0.000] [ 0.008] [ 0.000] Noe: Sandard errors in parenhesis, p values in brackes. The se of insrumens is composed of he average c/y over he sub sample and he average growh rae of populaion over he 15 firs periods of he sub sample. 21